Modular Forms and Related Topics in Number Theory: Kozhikode, India, December 10–14, 2018 9811587183, 9789811587184

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Springer Proceedings in Mathematics & Statistics

B. Ramakrishnan Bernhard Heim Brundaban Sahu   Editors

Modular Forms and Related Topics in Number Theory Kozhikode, India, December 10–14, 2018

Springer Proceedings in Mathematics & Statistics Volume 340

Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

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B. Ramakrishnan Bernhard Heim Brundaban Sahu •

•

Editors

Modular Forms and Related Topics in Number Theory Kozhikode, India, December 10–14, 2018

123

Editors B. Ramakrishnan Statistics and Applied Mathematics CUTN Thiruvarur, India

Bernhard Heim Faculty of Mathematics, Computer Science, and Natural Sciences RWTH Aachen University Aachen, Germany

Brundaban Sahu School of Mathematical Sciences NISER Bhubaneswar, Odisha, India

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-15-8718-4 ISBN 978-981-15-8719-1 (eBook) https://doi.org/10.1007/978-981-15-8719-1 Mathematics Subject Classification: 11-XX, 11Fxx, 11F37, 11F41 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This volume includes selected papers presented at the Conference on Number Theory, held during 10–14 December 2018 at the Kerala School of Mathematics (KSoM), Kozhikode, India. More than 30 talks were given at the conference by mathematicians from India and abroad. These proceedings are being published in honour of Prof. M. Manickam. Professor Manickam did his Ph.D. from Ramakrishna Mission Vivekananda College, Chennai under the guidance of Prof. T. C. Vasudevan in 1990. After finishing his Ph.D., he joined the same college as a faculty member and taught for about 20 years. In the year 2011, he moved to KSoM as Director-in-Charge. He has written about 30 research articles in the area of modular forms of integral and half-integral weights, Jacobi forms and Siegel modular forms. He has mentored many research students and teaching faculty with regard to their research works. He has organised many outreach programmes and workshops for the benefit of researchers working in the field of number theory during his time at KSoM. We invited articles from all the speakers of the conference and his well-wishers in the community of mathematics. With an overwhelming response, we received 15 contributions for this volume. We convey our heartfelt gratitude to all authors who contributed to the volume. All articles were refereed by experts in the respective field and we take this opportunity to thank them all for their timely help and support. We are sure that the articles in this volume are beneficial to the mathematics community, especially for researchers working in number theory. There are 15 chapters in this volume, out of which 11 works deal with different aspects of modular forms; two chapters are in combinatorial number theory; and one each in algebraic number theory and general mathematics. Below we mention a brief description of articles that appear in this volume: Chapter “On Vanishing of Hecke Operators” gives a brief survey of known results on Hecke operators acting on the space of cusp forms of integral weight for full modular group. A method of Ronyai and a smart application of Chevalley– Warning and Alon's Combinatorial Nullstellensatz are presented in chapter “On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems”. Chapter “p-adic Asai L-functions Attached to Bianchi Cusp Forms” deals with establishing v

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Preface

a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and constructing distributions interpolating these L-values. Chapter “Arithmetic Properties of Vector-Valued Siegel Modular Forms” describes how one can get arithmetic properties of vector-valued Siegel modular forms (more precisely: integrality properties of their Fourier coefficients) by combining the doubling method with certain holomorphic differential operators studied by T. Ibukiyama. Some omega results for Fourier coefficients of half-integral weight and Siegel modular forms are discussed in chapter “Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms”. In chapter “On Hecke Theory for Hermitian Modular Forms”, the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary quadratic number field is outlined. Sign changes of coefficients of the powers of the Dedekind eta-function, in particular, the Ramanujan tau function is studied in chapter “Sign Changes of the Ramanujan s-Function”. Chapter “The Central Limit Theorem in Algebra and Number Theory” elucidates the impact of the central limit theorem in number theory and algebra, in particular, probabilistic number theory, Goncharov’s theorem, normal number of prime factors of Fourier coefficients of modular forms and probabilistic connections to the Riemann hypothesis. In chapter “Rankin–Cohen Brackets and Identities Among Eigenforms II”, a characterisation of eigenforms which are Rankin–Cohen brackets of two quasimodular eigenforms is studied. Chapter “Rankin–Cohen Type Operators for Hilbert–Jacobi Forms” deals with construction of Rankin–Cohen type differential operators on the space of Hilbert–Jacobi forms. In chapter “Determining Modular Forms of Half-Integral Weight by Central Values of Convolution L-Functions”, it is shown that a Hecke eigenform of half-integral weight is uniquely determined by the central values of a family of convolution (Rankin–Selberg) L-functions. By constructing explicit bases for the spaces of modular forms of weight 2, level 48 with different characters, formulas for the number of representations of a positive integer n by certain quaternary quadratic forms are obtained in chapter “On the Number of Representations of a Natural Number by Certain Quaternary Quadratic Forms” along with some discussion on universal quadratic forms. A combinatorial number on the symmetric group is defined and a connection between this number and the Stirling number of first kind is established in chapter “Identities from Partition of the Symmetric Group Sn ”. In chapter “A Certain Kernel Function for L-Values of Half-Integral Weight Hecke Eigenforms”, a non-cusp form of half-integral weight in the Kohnen plus space is derived whose Petersson scalar product with a Hecke eigenform gives the special values of the L-function associated to the Hecke eigenform. Finally, in chapter “On Admissible Set of Primes in Real Quadratic Fields”, two simple families of real quadratic fields are considered and a construction of admissible set of primes in these fields is demonstrated. This proceeding will serve as an important resource for undergraduate/graduate students as well as researchers interested in broad aspects of number theory, modular forms and combinatorics.

Preface

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We take this opportunity to thank the Kerala State Council for Science, Technology and Environment (KSCSTE) and the National Board for Higher Mathematics (NBHM) for providing financial support for organising the conference. It is our pleasure to thank Mr. Shamim Ahmad for encouraging us to submit the proposal to Springer. Finally, we thank Springer Nature for accepting our proceedings to be published under the series ‘Springer Proceedings in Mathematics & Statistics’. Aachen, Germany Thiruvarur, India Bhubaneswar, India April 2020

Bernhard Heim B. Ramakrishnan Brundaban Sahu

Conference on Number Theory

December 10–14, 2018 Kerala School of Mathematics, Kozhikode Organisers: T. Jagathesan (Vivekananda College), B. Ramakrishnan (HRI, Allahabad), Sandeep E. M (KSoM)

List of Speakers S. D. Adhikari (RKMVERI, Kolkata) R. Balasubramanian (NCM, Mumbai) S. Boecherer (Mannheim, Germany) S. Das (IISc, Bengaluru) B. Heim (GUTech, Oman) J. Meher (NISER, Bhubaneswar) V. Kumar Murty (University of Toronto, Canada) V. Patankar (JNU, Delhi) G. Prakash (HRI, Allahabad) C. S. Rajan (TIFR, Mumbai) B. Ramakrishnan (HRI, Allahabad) V. P. Ramesh (CUTN, Thiruvarur) B. Sahu (NISER, Bhubaneswar) J. Sengupta (TIFR, Mumbai) K. D. Shankhadhar (IISER, Bhopal) R. Thanadurai (HRI, Allahabad) A. Vatwani (IIT, Gandhinagar) U. K. Anandavardhanan (IIT, Bombay) S. Bhattacharya (IISER, Kolkata) K. Chakraborty (HRI, Allahabad) S. Ganguly (ISI, Kolkata) K. Kumarasamy (Vivekananda College, Chennai) ix

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Conference on Number Theory

A. Mukhopadhyay (IMSc, Chennai) M. Ram Murty (Queen’s University, Kingston, Canada) P. Philippon (UPMC, France) D. Prasad (TIFR, Mumbai) R. Raghunathan (IIT, Bombay) D. S. Ramana (HRI, Allahabad) P. Rath (CMI, Chennai) A. Sankaranarayanan (TIFR, Mumbai) K. Senthilkumar (NISER, Bhubaneswar) M. K. Tamilselvi (Alpha Engineering College, Chennai) M. Waldschmidt (University Paris 6, France)

List of Participants R. Acharya (HRI, Allahabad) P. Akhilesh (CMI, Chennai) A. Bharadwaj (CMI, Chennai) S. Jyothsnaa (IMSc, Chennai) Arvind Kumar (TIFR, Mumbai) Moni Kumari (TIFR, Mumbai) Nabin Meher (NISER, Bhubaneswar) Siddhi Pathak (Queen’s University, Kingston, Canada) Anup Kumar Singh (HRI, Allahabad) Sreejith M. M. (KSoM, Kozhikode) S. Srivastav (TIFR, Mumbai) Lalit Vaishya (HRI, Allahabad) Adersh V. K. (TKM, Kollam) K. Babu (IMSc, Chennai) A. K. Jha (IISc, Bengaluru) Arpita Kar (Queen’s University, Kingston, Canada) Balesh Kumar (IMSc, Chennai) K. Mahatab (NTNU, Norway) Manish Kumar Pandey (HRI, Allahabad) Biplab Paul (IMSc, Chennai) Snehbala Sinha (IISc, Bengaluru) Srimathi (University of Bergen, Norway) M. Subramani (HRI, Allahabad)

Contents

On Vanishing of Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. K. Adersh On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. D. Adhikari, Bidisha Roy, and Subha Sarkar

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p-adic Asai L-functions Attached to Bianchi Cusp Forms . . . . . . . . . . . . Baskar Balasubramanyam, Eknath Ghate, and Ravitheja Vangala

19

Arithmetic Properties of Vector-Valued Siegel Modular Forms . . . . . . . Siegfried Böcherer

47

Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soumya Das

59

On Hecke Theory for Hermitian Modular Forms . . . . . . . . . . . . . . . . . Adrian Hauffe-Waschbüsch and Aloys Krieg

73

Sign Changes of the Ramanujan s-Function . . . . . . . . . . . . . . . . . . . . . . Bernhard Heim and Markus Neuhauser

89

The Central Limit Theorem in Algebra and Number Theory . . . . . . . . 101 Arpita Kar and M. Ram Murty Rankin–Cohen Brackets and Identities Among Eigenforms II . . . . . . . . 125 Arvind Kumar and Jaban Meher Rankin–Cohen Type Operators for Hilbert–Jacobi Forms . . . . . . . . . . . 143 Moni Kumari and Brundaban Sahu Determining Modular Forms of Half-Integral Weight by Central Values of Convolution L-Functions . . . . . . . . . . . . . . . . . . . . 157 Manish Kumar Pandey and B. Ramakrishnan

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Contents

On the Number of Representations of a Natural Number by Certain Quaternary Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . 173 B. Ramakrishnan, Brundaban Sahu, and Anup Kumar Singh Identities from Partition of the Symmetric Group Sn . . . . . . . . . . . . . . . 199 V. P. Ramesh, M. Makeshwari, M. Prithvi, and R. Thatchaayini A Certain Kernel Function for L-Values of Half-Integral Weight Hecke Eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 M. M. Sreejith On Admissible Set of Primes in Real Quadratic Fields . . . . . . . . . . . . . 227 Kotyada Srinivas and Muthukrishnan Subramani

On Vanishing of Hecke Operators V. K. Adersh

Dedicated to Murugesan Manickam on the occasion of his 60th birthday

Abstract In this survey note, we present some known results of Hecke operators acting on the space of cusp forms of weight k on the full modular group S L 2 (Z). Keywords Hecke operators · Eigenforms · τ function · Poincaré series 2010 Mathematics Subject Classification 11F11 · 11F30

1 Statement of Results For k ≥ 2 even, let Mk denote the linear space of modular forms of weight k on the full modular group S L 2 (Z) and let Sk be the subspace of Mk consisting of all cusp forms. Let f ∈ Mk and n ≥ 1 be an integer. The Hecke operator Tn is defined by [1, p. 120]   d−1  nz + bd , d −k f (Tn f )(z) = n k−1 d2 d|n b=0 where z is in the complex upper half-plane H. Then, Tn is a linear operator on Mk and it preserves the subspace Sk of cusp forms. V. K. Adersh (B) Kerala School of Mathematics, Kozhikode, Kerala , India Department of Mathematics, TKM College of Arts and Scienc,, Kollam, Kerala , India e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_1

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V. K. Adersh

The following properties of Hecke operators are well known: Theorem 1 Let m, n ≥ 1 be two positive integers. Then on Mk the following relation is satisfied: 

Tm Tn =

d k−1 T mn d2

d|(m,n)

or Tmn =



μ(d)d k−1 T md T dn .

d|(m,n)

Suppose that the dimension of Sk is l. Let B = { f 1 , . . . , fl } be an orthogonal basis with respect to the Petersson inner product of normalized Hecke eigenforms in Sk . Then for each j with 1 ≤ j ≤ l, one has Tn ( f j )=a f j (n) f j for all n ≥ 1, where the Fourier expansion of f j is given by f j (n) =

∞ 

a f j (n)e2πinz ,

n=1

where z ∈ H. Let  = SL2 (Z). For k ≥ 12 even and m ≥ 1, the mth Poincaré series on Sk is defined by  Pk;m = jγ (z)−k e2πimγ z , γ ∈∞ \



   1n ab | n ∈ Z} and for γ = , z ∈ H, jγ (z) = cz + d and 01 cd . It is well known that Pk;m ∈ Sk and γ z = az+b cz+d

where ∞ = {±

 f, Pk;m  =

(k − 1) a f (m) (4π m)k−1

(1)

 2πinz ∈ Sk , where  f, Pk;m  is the Petersson inner product for all f = ∞ n=1 a f (n)e of Pk;m and f . [4, p. 53] > Poincaré Conjecture Suppose that the dimension of Sk is at least 1. Then the Poincaré series Pk;m = 0 for all m ≥ 1.

An equivalent statement of the above conjecture is given in the following theorem. We give a proof of the theorem in Sect. 3.

On Vanishing of Hecke Operators

3

Theorem 2 Pk;n ≡ 0 if and only if Tn ≡ 0 on Sk . For k = 12, the space of cusp forms Sk has dimension one and is spanned by the form   τ (n)e2πinz = e2πi z (1 − e2πinz )24 (z) = n≥1

n≥1

, where τ (n) is the Ramanujan τ -function. The following conjecture is a special case of Poincaré conjecture due to D.H. Lehmer [5, 6]. > Lehmer Conjecture τ (m) = 0 for every m ≥ 1.

In connection with Poincaré conjecture/Lehmer conjecture, we prove the following theorem. k

k

Theorem 3 Suppose that the eigenvalues of T2 and T3 are not divisible by 2 2 and 3 2 , respectively. Then Tn annihilates the space Sk for some n if and only if T p annihilates Sk for a prime p dividing n. Corollary 1 τ ( p) = 0 if and only if τ ( pr ) = 0 for every r .

2 Proof of Theorem 3 Using Theorem 1, for prime powers we have the following polynomial expression  for T pr in T p of degree r over the ring of integers. Theorem 4 For p prime and r ≥ 1, ⎧ r r r k−r T p − αr,1 p k−1 T pr −2 + αr,2 p 2k−2 T pr −4 − · · · + (−1) 2 αr, r2 p 2 ⎪ ⎪ ⎪ ⎨ if r is even T pr = (r −1)k−(r −1) r −1 r k−1 r −2 2 ⎪ T − α T p + αr,2 p 2k−2 T pr −4 − · · · + (−1) 2 αr, r −1 p Tp r,1 p ⎪ p 2 ⎪ ⎩ if r is odd, (2) where αi, j ’s are positive integers satisfying αr,1 = r − 1, αr +1, j = αr, j + αr −1, j−1 for all j ≥ 2, r ≥ 2 and αr, j = 0 for j >

r . 2

(3)

In particular, T pr is an r th degree monic polynomial in T p with integral coefficients. Proof Letting m = p and n = pr in Theorem 1, we get

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V. K. Adersh

T p T pr = T pr +1 + p k−1 T pr −1 . That is, for r ≥ 2,

T pr = T p T pr −1 − p k−1 T pr −2 .

For r = 2, 3, we have T p2 = T p2 − p k−1 T p3 = T p3 − 2 p k−1 T p . By induction, if r is an even integer, then T pr +1 = T p T pr − p k−1 T pr −1  r r k−r = T p T pr − αr,1 p k−1 T pr −2 + αr,2 p 2k−2 T pr −4 − · · · + (−1) 2 αr, r2 p 2

− p k−1 T pr −1 − αr −1,1 p k−1 T pr −3 + αr −1,2 p 2k−2 T pr −5 − · · · (r −2)k−(r −2) r −2 2 +(−1) 2 αr −1, r −2 p T p 2 = T pr +1 − αr +1,1 p k−1 T pr −1 + αr +1,2 p 2k−2 T pr −3 − · · · + (−1) 2 αr +1, r2 p r

r k−r 2

Tp ,

where αr +1, j , j ≥ 1 are positive integers given by the relation (3). Similarly, we establish the relation in (2) when r is an odd integer.

Proof of Theorem 3 Suppose Tn annihilates Sk for some n ≥ 1. That is, Tn ( f ) = 0 for all f ∈ Sk . Let p be a prime with pr |n, pr +1  |n and T p = 0 on Sk . Then, on Sk we have Tn = 0 ⇒ T pr T pnr = 0 ⇒ T pr is singular or T pnr ≡ 0. Assume that the kernel of T pr is V ⊂ Sk which has non-zero forms. If f ∈ V , then for any Hecke operator Tm T pr (Tm ( f )) = T pr (Tm ( f )) = Tm (T pr ( f )) = 0. This proves that f ∈ V means Tm ( f ) ∈ V for all m ≥ 1. Hence, the Hilbert space V is invariant under the Hecke algebra H . So V has an orthonormal basis consisting of normalized Hecke eigenforms. Let  f ∈ V be one such normalized Hecke eigenform with the Fourier expansion f = m≥1 a f (m)q m , where a f (m) is an eigenvalue for Tm so that a f (m)’s are real algebraic integers with a f (1) = 1. Assume that r is an even integer. The case r of an odd integer follows similarly. As T pr f = 0, from the polynomial relation (2),

On Vanishing of Hecke Operators

5

a f ( p)r − αr,1 p k−1 a f ( p)r −2 + αr,2 p 2k−2 a f ( p)r −4 − · · · + (−1) 2 αr, r2 p r

r k−r 2

= 0.

This gives T p f = 0 and p k−1 |a f ( p)r . Let s be the largest integer with a f ( p) = λ f ( p) p s , for some algebraic integer λ f ( p).

(4)

Then a f ( p)r = (λ f ( p))r pr s = αr,1 p k−1 (λ f ( p))r −2 p s(r −2) − αr,2 p 2(k−1) (λ f ( p))r −4 p s(r −4) + · · · − (−1) 2 αr, r2 pr ( r

k−1 2 )

(λ f ( p))r = αr,1 (λ f ( p))r −2 p k−1−2s − αr,2 (λ f ( p))r −4 p 2(k−1)−4s + · · · − (−1) 2 αr, r2 pr ( r

k−1 2 )−r s

.

If k − 1 − 2s > 0, then p s+1 |a f ( p) which is not possible by the choice of s. ≤ s. As k is an even integer, s > k−1 . Thus, s is at least k2 . So, by the Therefore, k−1 2 2 Deligne bound [2], k

|λ f ( p)| p 2 ≤ |a f ( p)| ≤ 2 p

k−1 2

⇒

|λ f ( p)| ≤ 2 p − 2 . 1

(5)

Note that Eq. (5) is true for all the conjugates of λ f ( p) in K (see [3] Theorem 6.5.4). As λ f ( p) is a non-zero algebraic integer, product of λ f ( p) with its conjugates is a non-zero rational integer. But p = 2, 3 and Eq. (5) shows that this product has modulus smaller than 1. This is a contradiction. 

3 Proof of Theorem 2 Suppose, Pk;n ≡ 0 for some n. Then for all m ≥ 1 we have Tn (Pk;m ) = m 1−k Tn (Tm (Pk;1 )) = m 1−k Tm (Tn (Pk;1 )) = m 1−k n k−1 Tm (Pk;n ) = 0. As {Pk;m : m ≥ 1} spans Sk , we get Tn ≡ 0 on Sk . Conversely, suppose that Tn ≡ 0 on Sk for some n. Then n k−1 Pk;n = Tn (Pk;1 ) = 0, since Pk;1 ∈ Sk and Tn ≡ 0 on Sk . Thus Pk;n ≡ 0.



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V. K. Adersh

4 Further Remarks The first l = dim Sk Poincaré series, Pk;1 , Pk;2 , . . . , Pk;l , form a basis of Sk and hence none of them vanish. Since Pk;n = n 1−k Tn (Pk;1 ), the Hecke algebra H for Sk is generated by T1 , T2 , . . . , Tl . Hence, Tn (Sk ) = (0) implies n > l. We conclude this article by mentioning the following related open questions. ? Question Does Tn (Sk ) = Sk for some n ≥ 1?

? Question Does there exist l operators Tn 1 , Tn 2 , . . . , Tnl with n i > l for all 1 ≤ i ≤ l, which form a basis for the Hecke algebra H ? Or equivalently, does Pk;n 1 , Pk;n 2 , . . . , Pk;nl form a basis for Sk ?

Acknowledgements The author thanks his supervisor M. Manickam for the guidance on writing this article. The author would like to thank the referee for carefully reading the article and for giving valuable comments.

References 1. Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory, 2nd edn. Springer, New York (1990) 2. Deligne, P.: La conjecture de Weil, I. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974) 3. Diamond, F., Shurman, J.: A First Course in Modular Forms. Springer, New York (2005) 4. Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics. Am. Math. Soc. 17, 259 (1997) 5. Lehmer, D.H.: The vanishing of Ramanujan’s function τ (n). Duke Math. J. 14, 429–433 (1947) 6. Rankin, R.A.: The Vanishing of Poincaré Series. Proceedings of the Edinburgh Mathematical Society 23, 151–161 (1980)

On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems S. D. Adhikari, Bidisha Roy, and Subha Sarkar

Abstract In this article, we discuss a polynomial method due to Rónyai which was used for making some progress toward the Kemnitz’s conjecture. This method of Rónyai and some modifications of it have been successfully used by several authors since then. We shall try to explain the method by employing it in obtaining a bound for a new zero-sum constant. Keywords Polynomial method · Zero-sum

1 Introduction For a finite abelian group G (written additively) with exp(G) = n, the arithmetical invariant smn (G) is defined to be the least integer k such that any sequence S with length k of elements in G has a zero-sum subsequence of length mn. Here, by a sequence over G we mean a finite sequence of terms from G which is unordered and repetition of terms is allowed. We view sequences over G as elements of the free abelian monoid F (G) and use multiplicative notation, so our notation is consistent with [19, 22, 23]. A sequence S = g1 · . . . · gl ∈ F (G) is called a zero-sum sequence if g1 + · · · + gl = 0, where 0 is the identity element of the group. For integers m < n, we shall use the notation [m, n] to denote the set {m, m + 1, . . . , n} and for a finite set A, we denote its size by |A|, which is the number of

S. D. Adhikari (B) (Formerly at Harish-Chandra Research Institute) Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur 711202, India e-mail: [email protected] B. Roy · S. Sarkar Harish-Chandra Research Institute, HBNI, Jhunsi, Prayagraj, India e-mail: [email protected] S. Sarkar e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_2

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S. D. Adhikari et al.

elements of A. For Z/nZ, we shall often write Zn . For the direct sum of d copies of Zn , we shall write Zdn . When m = 1, it is the Erd˝os–Ginzburg–Ziv constant and is denoted by s(G). To know in detail about this and similar combinatorial constants, one may look into the expository article of Gao and Geroldinger [19] and Sect. 4.2 in the survey of Geroldinger [21]. If G is a finite abelian group with exp(G) = n, then for a non-empty subset A of [1, n − 1], one defines s A (G) to be the least integer k such that any sequence S with length k of elements in G has an A-weighted zero-sum subsequence of length exp(G) = n, that is, for any sequence x1 · . . . · xk with xi ∈ G, there exists a subset I ⊂ [1, k] with |I | = n and, for each i ∈ I , some element ai ∈ A such that 

ai xi = 0.

i∈I

Taking A = {1}, one recovers the classical Erd˝os–Ginzburg–Ziv constant s(G). The above weighted version and some other invariants with weights were initiated by Adhikari et al. [6], Adhikari and Chen [5], and Adhikari et al. [3]. For developments regarding bounds on the constant s A (G) in the case of abelian groups G with higher rank and related references, we refer to the paper of Adhikari et al. [7] and the articles [2, 4]. For a non-empty subset A of [1, n − 1], one defines smn,A (G) to be the least integer k such that any sequence S with length k of elements in G has an A-weighted zero-sum subsequence of length mn. In the next section, we shall see results on the constant s(Zdn ). While discussing the two-dimensional case, we shall describe the origin of Rónyai’s method. In Sect. 3, we shall take up the corresponding generalization with weights and shall take a particular weighted zero-sum problem in Theorem 5 to exhibit Rónyai’s method.

2 The Constant s(Znd ) We start with a theorem of Erd˝os et al. [17] (henceforth, referred to as the EGZ theorem), a prototype of zero-sum theorems. Theorem 1 (EGZ theorem) For any positive integer n, any sequence a1 · a2 · . . . · a2n−1 of 2n − 1 integers has a subsequence of n elements whose sum is 0 modulo n. With the notation introduced in the previous section, the EGZ theorem says that s(Zn )  2n − 1. Since the sequence 0n−1 · 1n−1

On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems

9

does not have any subsequence of length n whose sum is zero, one has the equality s(Zn ) = 2n − 1. For various proofs of the EGZ theorem, apart from the original paper [17], one may look into the references [1, 13, 27]. The study of the constant s(Zdn ) for positive integers d was first considered by Harborth [24]; geometrically, this is the smallest positive integer such that given a sequence of s(Zdn ) number of not necessarily distinct elements of Zd , there exists a subsequence xi1 · xi2 · . . . · xin of length n such that its centroid (xi1 + xi2 + · · · + xin )/n also belongs to Zd . Harborth [24] observed that 1 + 2d (n − 1)  s(Zdn )  1 + n d (n − 1).

(1)

Since the number of elements of Zdn having coordinates 0 or 1 is 2d , considering a sequence where each of these elements is repeated (n − 1) times, one obtains the lower bound; the upper bound is obtained by observing that in any sequence of 1 + n d (n − 1) elements of Zdn there will be at least one vector appearing at least n times. The EGZ theorem says that for d = 1, the lower bound in Eq. (1) is the value of s(Zn ). For the case d = 2 also, the lower bound in (1) was expected to give the right magnitude of s(Z2n ) and this expectation, which is known as Kemnitz’s conjecture in the literature, has been established by Reiher [28]. After a brief discussion on s(Zdn ), for general d, we shall sketch the history of the progress toward Kemnitz’s conjecture culminating in Reiher’s result. This will also give us the opportunity to describe the origin of Rónyai’s method. The general upper bound of Alon and Dubiner [14] says that there is an absolute constant c > 0 so that s(Zdn )  (cd log2 d)d n, for all n,

(2)

which shows that the growth of s(Zdn ) is linear in n. However, this result of Alon and Dubiner is far from the expected one; in the same paper [14], it has been conjectured that there is an absolute constant c such that s(Zdn )  cd n, for all n and d.

(3)

Regarding the lower bound, it has been proved by Elsholtz [15] that for an odd integer n  3, d s(Zdn )  (1.125)[ 3 ] (n − 1)2d + 1. Hence, in particular, for d  3 and an odd integer n  3,

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1 + 2d (n − 1) < s(Zdn ). Improvements in the above lower bounds have been obtained in [16]. In dimension two, regarding the Kemnitz’s conjecture, Kemnitz [25] himself had established this conjecture when n is of the form 2e 3 f 5g 7h . For general n, Alon and Dubiner [13] had established that s(Z2n )  6n − 5 (one may also look into [1]). Rónyai [29] proved that for a prime p, one has s(Z2p )  4 p − 2, and as mentioned in Rónyai [29], it is not difficult to observe that this together with an observation 41 n. made by Harborth [24] implies that s(Z2n )  10 Generalizing the above result of Rónyai, Gao [18] proved that for an odd prime p and a positive integer r , we have s(Z2pr )  4 pr − 2. A sketch of this proof of Gao and that of Reiher’s proof [28] of Kemnitz’s conjecture can be found in the expository article [11]. In the next section, in a particular weighted zero-sum problem, we exhibit Rónyai’s [29] method.

3 Weighted Zero-Sum Problems and Rónyai’s Method The polynomial method of Rónyai [29] was successfully used by Kubertin [26] in the study of skn (Zdn ) for k > 1, which had been earlier taken up by Adhikari and Rath [10] and Gao and Thangadurai [20]. A result of Kubertin [26] says that for positive integers r, d with r  d, a prime p with p > min(2r, 2d) and q a power of p, one has   3 3 2 3 d d + d − + r q − d. (4) srq (Zq )  8 2 8 For zero-sum problems with weights, when A = {±1}, for any positive integer n it was proved by Adhikari et al. [6] that s{±1} (Zn ) = n + log2 n. When n is odd, and A = {±1}, it was observed by Adhikari et al. [3] that s{±1} (Z2n ) = 2n − 1. Later, for any finite abelian group G of rank r and even exponent it was proved by Adhikari et al. [7] that there exists a constant kr , dependent only on r , such that s{±1} (G)  exp(G) + log2 |G| + kr log2 log2 |G|.

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The problem for the rank 3 case was considered in [8] and by a suitable modification of the polynomial method of Rónyai [29] it was shown that for A = {±1}, and an odd prime p, one has (9 p − 3) . s3 p,A (Z3p )  2 With further modification of the method of Rónyai [29], in [9] the following result was proved. Theorem 2 Let p be an odd prime and let k  3 be a divisor of p − 1. Let θ be an element of order k of the multiplicative group Z∗p and A be the subgroup of Z∗p generated by θ . Then, we have s3 p,A (Z2k p )  5 p − 2. Later in [12], the following corresponding result for odd rank has been proved. Theorem 3 Let p be an odd prime and let k  2 be an even integer which divides p − 1. Let θ be an element of order k of the multiplicative group (Z/ pZ)∗ and A be the subgroup of (Z/ pZ)∗ generated by θ . Then, we have s3 p,A (Zk+1 p )  4p +

p−1 − 1. k

Remark 1 We can deduce a result related to 5 p sum from Theorem 3. However, we shall then proceed to give a proof of that result without using Theorem 3, to demonstrate a modification of the method of Rónyai [29]. The following lemma was proved in [9]. Lemma 1 Let p be an odd prime and let k be a divisor of p − 1. Let θ be an element of order k of Z∗p and D = {0, θ, θ 2 , . . . , θ k }. For a positive integer m, let us consider the vector space   C = functions f : D m → Z p over the field Z p . Then the monomials C over Z p .

 1im

xiri , ri ∈ [0, k] constitute a basis of

Next we state the well-known theorem of Chevalley–Warning (see, for instance, [27]). Theorem 4 Let p be a prime number and F a finite field of characteristic p. For i = 1, 2, . . . , m, let f i ∈ F[x1 , x2 , . . . , xn ] be a non-zero polynomial of degree di in n-variables over the field F. Let N denote the number of n-tuples (x1 , x2 , . . . , xn ) of elements of F such that f i (x1 , x2 , . . . , xn ) = 0,

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for all i = 1, 2, . . . , m. If d1 + d2 + · · · + dm < n, then N ≡0

(mod p).

In particular, if N  1, then there is a non-zero simultaneous solution over F. Lemma 2 Let p be an odd prime and let k  2 be an even integer which divides p − 1. Let A = {θ, θ 2 , . . . , θ k = 1} be the subgroup of Z∗p generated by θ which is of t p−1 − 1. order k. Let S = i=1 wi ∈ F (Zk+1 p ) be a sequence of length t = 2 p + k Then S has an A-weighted zero-sum subsequence of length either p or 2 p. Proof For all integers i = 1, 2, . . . , t, we let wi = (ai1 , ai2 , . . . , ai(k+1) ) ∈ Zk+1 p . We shall consider the following system of equations over Z p : t 

ai1 xi

p−1 k

= 0,

i=1

t 

ai2 xi

i=1

p−1 k

= 0, . . . ,

t 

ai(k+1) xi

p−1 k

= 0 and

i=1

t 

p−1

xi

= 0.

i=1

Note that the sum of the degrees of the polynomials is (k + 1) p−1 + ( p − 1) = k p−1 − 2 < 2 p + − 1 = t, the number of variables. 2 p + p−1 k k Since the above system has the trivial zero solution, by Theorem 4, there exists a non-zero solution (y1 , y2 , . . . , yt ) ∈ Ztp of the above system. If we write I = {i : yi = 0 (mod p)}, then from the first (k + 1) equations, we get  ( p−1)/k yi (ai1 , ai2 , . . . , a(k+1)i ) = (0, 0, . . . , 0) i∈I

and from the last equation, we get |I | ≡ 0 (mod p). Since yi = 0 (mod p) for all ( p−1)/k ∈ A. Since t < 3 p, we get either |I | = p or |I | = 2 p. i ∈ I , we see that yi Hence, we conclude that the sequence S has an A-weighted zero-sum subsequence of length either p or 2 p.  Corollary 1 Let p be an odd prime and let k  2 be an even integer which divides p − 1. Let A = {θ, θ 2 , . . . , θ k = 1} be the subgroup of Z∗p generated by θ which is of order k. t p−1 − 1. Then Let S = i=1 wi ∈ F (Zk+1 p ) be a sequence of length t = 3 p + k S has an A-weighted zero-sum subsequence of length 2 p. Proof Since the given sequence S is of length t = 3 p + p−1 − 1 over Zk+1 p , it has k an A-weighted zero-sum subsequence T of length either p or 2 p by Lemma 2. If T is of length 2 p, then we are done. Otherwise, consider the deleted sequence ST −1 which is of length 3p +

p−1 p−1 − 1 − p = 2p + − 1, k k

On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems

13

and hence, by Lemma 2, we get ST −1 has an A-weighted zero-sum subsequence T1 of length either p or 2 p. If |T1 | = 2 p, then we are done. If |T1 | = p, then T T1 is of length 2 p and it is the required subsequence.  The following is now immediate. t wi ∈ F (Zk+1 Corollary 2 With p, k, and A as in Corollary 1, if S = i=1 p ) is a p−1 − 1, then S has an A-weighted zero-sum subsequence of length t = 5 p + k sequence of length 4 p. Corollary 3 Let p be an odd prime and let k  2 be an even integer which divides p − 1. Let A = {θ, θ 2 , . . . , θ k = 1} be the subgroup of Z∗p generated by θ which is of t p−1 − 1. wi ∈ F (Zk+1 order k. Let S = i=1 p ) be a sequence of length t = 6 p + k If S has an A-weighted zero-sum subsequence of length p or 3 p, then it has an A-weighted zero-sum subsequence of length 5 p. Proof If S has an A-weighted zero-sum subsequence T of length p, consider the deleted sequence ST −1 which is of length 5 p + p−1 − 1. Therefore, by Corollary 2, k we get an A-weighted zero-sum subsequence T1 of ST −1 of length 4 p. Hence, T T1 is the required zero-sum subsequence. If S has an A-weighted zero-sum subsequence T of length 3 p, then by the same argument, by Corollary 1, we get an A-weighted zero-sum subsequence of length 5 p.  Since by Theorem 3, the sequence S in Corollary 3 has a zero-sum subsequence of length 3 p, we get the following. Theorem 5 Let p be an odd prime and let k  2 be an even integer which divides p − 1. Let θ be an element of order k of the multiplicative group Z∗p and A be the subgroup of Z∗p generated by θ . Then, we have s5 p,A (Zk+1 p )  6p +

p−1 − 1. k

Remark 2 As mentioned earlier, we shall give a proof of the above theorem without using Theorem 3. We shall closely follow the method in [12]. By repeatedly taking out A-weighted zero-sum subsequences of length 2 p by using Corollary 1, one gets bounds for sr p,A (Zk+1 p ) for odd integers r  7. Corresponding results for even ranks can similarly be obtained by using Theorem 2. Using Corollary 1, by repeatedly taking out A-weighted zero-sum subsequences of length 2 p, one obtains bounds for sr p,A (Zk+1 p ) for even integers r  6. One can also deal with subgroup of Z∗p of odd order in Theorem 5 and the corresponding theorem for even ranks. As remarked in [12], these bounds are not always tight.

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Proof of Theorem 5. For an odd prime p and an even integer k  2 such that k divides p − 1, θ is an element of order k of the multiplicative group Z∗p and A is the subgroup of Z∗p generated by θ . We have to show that s5 p,A (Zk+1 p )  6p +

p−1 − 1. k

Note that D as defined in Lemma 1 is A ∪ {0}. m p−1 wi ∈ F (Zk+1 Let S = i=1 p ) be a sequence of length m = 6 p + k − 1. For all i = 1, 2, . . . , m, we let wi = (ai1 , ai2 , . . . , ai(k+1) ) ∈ Zk+1 p . We shall prove that S has an A-weighted zero-sum subsequence of length 5 p. If possible, suppose that the assertion is false. That is, S has no A-weighted zero-sum subsequence of length 5 p. Therefore, by Corollary 3, S cannot  have any A-weighted zero-sum subsequence of length p or 3 p. Thus, if T = j=1 wi j is a subsequence of S of length  = 3 p or p, then for any (z 1 , . . . , z  ) ∈ A , we have z 1 wi1 + · · · + z  wi ≡ (0, 0, . . . , 0)

(mod p).

(5)

In order to get a contradiction, we need to invoke Lemma 1. For this purpose, we shall introduce some polynomials as follows: Let σ (x1 , x2 , . . . , xm ) =





I ⊂[1,m], |I |= p

i∈I

xik ,

be the p-th elementary symmetric polynomial of the variables x1k , x2k , . . . , xmk . We also consider the following polynomials: P1 (x1 , x2 , . . . , xm ) = ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ p−1 p−1 p−1 m m m    ⎝ ai1 xi − 1⎠ ⎝ ai2 xi − 1⎠ . . . ⎝ ai(k+1) xi − 1⎠ , i=1

i=1

i=1

⎛ ⎞ p−1 m  P2 (x1 , x2 , . . . , xm ) = ⎝ xik − 1⎠ , i=1

P3 (x1 , x2 , . . . , xm ) = (σ (x1 , x2 , . . . , xm ) − 2)(σ (x1 , x2 , . . . , xm ) − 4)(σ (x1 , x2 , . . . , xm ) − 6) and P(x1 , x2 , . . . , xm ) = P1 (x1 , x2 , . . . , xm )P2 (x1 , x2 , . . . , xm )P3 (x1 , x2 , . . . , xm ).

On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems

15

First, we note that deg(P)  (k + 1)( p − 1) + k( p − 1) + 3kp = 5kp + p − 1 − 2k.

(6)

Claim. P(α1 , . . . , αm ) = 0 for all (α1 , . . . , αm ) ∈ D m \{(0, 0, . . . , 0)} and P(0, 0, . . . , 0) = −48. Let α = (α1 , . . . , αm ) ∈ D m \{(0, 0, . . . , 0)} be an arbitrary element. If the number of non-zero entries of α is not a multiple of p and if we take I = {1  i  m : αi = 0}, then ⎞ ⎛ ⎞ ⎛ p−1 p−1 m   ⎝ αik − 1⎠ = ⎝ αik − 1⎠ = 0 i=1

i∈I

by Fermat’s Little Theorem, and hence we get P2 (α1 , . . . , αm ) = 0. If the number of non-zero entries of α is either p, 3 p, or 5 p, then by (5), we get P1 (α1 , . . . , αm ) = 0.   If the number of non-zero entries of α is 2 p, then σ (α) = 2pp = 2 ∈ Z p , if the   number of non-zero entries of α is 4 p, then σ (α) = 4pp = 4 ∈ Z p and if the number   of non-zero entries of α is 6 p, σ (α) = 6pp = 6 ∈ Z p . Therefore, if the number of non-zero entries of α is either 2 p, 4 p, or 6 p, then P3 (α1 , . . . , αm ) = 0. Hence, the polynomial P(x1 , x2 , . . . , xm ) vanishes at all the points of D m , except at (0, 0, . . . , 0) and P(0, 0, . . . , 0) = −48, as (k + 1) is odd. This proves the claim. Consider the function P : D m → Z p in C given by the polynomial P(α1 , . . . , αm ). Now, let R = −48(1 − x1k )(1 − x2k ) . . . (1 − xmk ) ∈ Z p [x1 , . . . , xm ]. Then R(α1 , . . . , αm ) = 0 for all α = (α1 , . . . , αm ) ∈ D m \{(0, 0, . . . , 0)} and R(0, . . . , 0) = −48 . Therefore, P(x1 , . . . , xm ) and R(x1 , . . . , xm ) are equal as elements in C. ByLemma 1, we know that C has a special basis consisting of monomials of the form 1im xiri , ri ∈ [0, k]. Now, we write P as a linear combination of these basis elements by replacing each xitk+r for some integers t  1 and r ∈ [1, k] by xir and let Q be the polynomial obtained in this way. Also, in this process, the degree of the polynomial Q is not increased. Hence, by (6), we get deg Q  5kp + p − 1 − 2k. Clearly, as elements in C, P and Q are the same. Hence, Q and R are the same as elements in C. However, deg R = mk = 6kp + p − 1 − k > 5kp + p − 1 − 2k  deg Q. This will lead to a non-trivial relation among the basis elements consisting of the  monomials 1im xiri , which is impossible. Acknowledgements We thank Prof. R. Thangadurai for some useful discussions and for going through the final manuscript.

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References 1. Adhikari, S.D.: Aspects of Combinatorics and Combinatorial Number Theory. Narosa, New Delhi (2002) 2. Adhikari, S.D.: An extremal problem in combinatorial number theory. In: Berndt, B.C., Prasad, D (eds.) Proceedings of an International Conference in Celebration of the 125th Anniversary of Ramanujan’s Birth; University of Delhi, 17-22 December 2012, (Ramanujan Mathematical Society Lecture Notes Series, Number 20, 2013), pp. 1–11 3. Adhikari, S.D., Balasubramanian, R., Pappalardi F., Rath, P.: Some zero-sum constants with weights. Proc. Indian Acad. Sci. (Math. Sci.) 118, 183–188 (2008) 4. Adhikari, S.D., Balasubramanian, R., Rath, P.: Some combinatorial group invariants and their generalizations with weights. In: Granville, A., Nathanson, M.B., Solymosi, J. (eds.) Additive combinatorics, (CRM Proceedings and Lecture Notes, vol. 43, Am. Math. Soc., Providence, RI, 2007), pp. 327–335 5. Adhikari, S.D., Chen, Y.G.: Davenport constant with weights and some related questions - II. J. Comb. Theory Ser. A 115, 178–184 (2008) 6. Adhikari, S.D., Chen, Y.G., Friedlander, J.B., Konyagin, S.V., Pappalardi, F.: Contributions to zero-sum problems. Discrete Math. 306, 1–10 (2006) 7. Adhikari, S.D., Grynkiewicz, D.J., Sun, Z.-W.: On weighted zero-sum sequences. Adv. Appl. Math. 48, 506–527 (2012) 8. Adhikari, S.D., Mazumdar, E.: Modifications of some methods in the study of zero-sum constants. Integers 14, paper A 25 (2014) 9. Adhikari, S.D., Mazumdar, E.: The polynomial method in the study of zero-sum theorems. Int. J. Number Theory 11(5), 1451–1461 (2015) 10. Adhikari, S.D., Rath, P.: Remarks on some zero-sum problems. Expo. Math. 21(2), 185–191 (2003) 11. Adhikari, S.D., Rath, P.: Zero-sum problems incombinatorial number theory. The Riemann zeta function and relatedthemes: papers in honour of Professor K. Ramachandra, 1–14, Ramanujan Math. Soc. Lect. Notes Ser., 2, Ramanujan Math. Soc., Mysore (2006) 12. Adhikari, S.D., Roy, B., Sarkar, S.: Weighted Zero-Sums for Some Finite Abelian Groups of Higher Ranks. Combinatorial and Additive Number Theory III, Springer Proceedings in Mathematics & Statistics, vol. 297 (2019) 13. Alon, N., Dubiner, M.: Zero-sum sets of prescribed size, Combinatorics, Paul Erd˝os is eighty (Volume 1). Keszthely (Hungary) 33–50 (1993) 14. Alon, N., Dubiner, M.: A lattice point problem and additive number theory. Combinatorica 15, 301–309 (1995) 15. Elsholtz, C.: Lower bounds for multidimensional zero sums. Combinatorica 24(3), 351–358 (2004) 16. Edel, Y., Elsholtz, C., Geroldinger, A., Kubertin, S., Rackham, L.: Zero-sum problems in finite abelian groups and affine caps. Quart. J. Math. 58, 159–186 (2007) 17. Erd˝os, P., Ginzburg, A., Ziv, A.: Theorem in the additive number theory. Bull. Res. Counc. Isr. 10F, 41–43 (1961) 18. Gao, W.D.: A note on a zero-sum problem. J. Comb. Theory Ser. A 95(2), 387–389 (2001) 19. Gao, W.D., Geroldinger, A.: Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24, 337–369 (2006) 20. Gao, W.D., Thangadurai, R.: On zero-sum sequences of prescribed length. Aequationes Math. 72(3), 201–212 (2006) 21. Geroldinger, A.: Additive group theory and non-unique factorizations. In: Geroldinger, A., Ruzsa, I. (eds.) Combinatorial number theory and additive group theory, Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser Verlag, Basel (2009) 22. Geroldinger, A., Halter-Koch, F.: Non-unique factorizations: Algebraic, combinatorial and analytic theory, Pure and Applied Mathematics (Boca Raton), vol. 278. Chapman & Hall/CRC, Boca Raton, FL (2006)

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23. Grynkiewicz, D.J.: Structural Additive Theory. Developments in Mathematics, vol. 30. Springer, Cham (2013) 24. Harborth, H.: Ein Extremalproblem für Gitterpunkte. J. Reine Angew. Math. 262(263), 356– 360 (1973) 25. Kemnitz, A.: On a lattice point problem. Ars Combin. 16b, 151–160 (1983) 26. Kubertin, S.: Zero-sums of length kq in Zqd . Acta Arith. 116(2), 145–152 (2005) 27. Nathanson, M.B.: Inverse Problems and the Geometry of Sumsets. Additive Number Theory. Springer, Berlin (1996) 28. Reiher, C.: On Kemnitz’s conjecture concerning lattice points in the plane. Ramanujan J. 13, 333–337 (2007) 29. Rónyai, L.: On a conjecture of Kemnitz. Combinatorica 20(4), 569–573 (2000)

p-adic Asai L-functions Attached to Bianchi Cusp Forms Baskar Balasubramanyam, Eknath Ghate, and Ravitheja Vangala

Abstract We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure. Keywords Asai L-functions · p-adic L-functions · Measures and distributions · Kummer congruences · Bianchi cusp forms 2010 Mathematics Subject Classification 11F67 · 11F75 · 11F85

1 Preliminaries √ Let F be an imaginary quadratic field with ring of integers O F . Write F = Q( −D) with D > 0 and −D the discriminant of F. Let Sk (n) denote the space of Bianchi cusp forms of weight k = (k, k), k ≥ 2, and level n and central character with trivial finite part and infinity type (2 − k, 2 − k). Let f ∈ Sk (n) be a normalized eigenform and let c(m, f) be the Fourier coefficients of f, for any integral ideal m ⊂ O F . The eigenform f corresponds to a tuple ( f 1 , . . . , f h ) of classical Bianchi cusp forms, where h is the class number of F. We take f = f 1 ∈ Sk (0 (n)) and only focus on this since the Asai L-function depends only on f 1 . B. Balasubramanyam (B) Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pune 411008, India e-mail: [email protected] E. Ghate (B) · R. Vangala School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] R. Vangala e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_3

19

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Let c(r ), for r ≥ 1, denote c((r ), f ). Define the Asai L-function of f by the formula ∞  c(r ) G(s, f ) = L N (2s − 2k + 2, 11) , rs r =1 where N is the positive generator of the ideal n ∩ Z and L N (s, 11) is the L-function attached to the trivial character modulo N . The special values of this function are investigated in [4]. A generalization to cusp forms defined over CM fields can be found in [5]. Let p ∈ Z be an odd prime integer that is relatively prime to N and that is also unramified in F. Let χ : (Z/ p j Z)× → C× be a Dirichlet character with conductor dividing p j . Define the twisted Asai L-function of f by the formula G(s, χ , f ) = L N (2s − 2k + 2, χ 2 )

∞  c(r )χ (r ) r =1

rs

.

This has an Euler product expansion G(s, χ , f ) =



G l (s, χ , f ),

l

where the local L-functions at all but finitely many primes l are described as follows. Let l = p be an integer prime, not dividing N . For any l|l, let α1 (l) and α2 (l) denote the reciprocal roots of the Hecke polynomial of f at l: 1 − c(l, f )X + Nm(l)k−1 X 2 . Then ⎧ if l = l¯l, ⎨ i, j (1 − χ(l)αi (l)α j (¯l)l −s ) 1 = (1 − χ(l)α1 (l)l −s )(1 − χ 2 (l)l −2s+2k−2 )(1 − χ(l)α2 (l)l −s ) if l = l is inert, G l (s, χ, f ) ⎩ (1 − χ(l)α12 (l)l −s )(1 − χ(l)l −s+k−1 )(1 − χ(l)α22 (l)l −s ) if l = l2 is ramified.

We want to find “periods” and prove that the special values of the twisted Asai L-functions are algebraic after dividing by these periods. We also want to p-adically interpolate the special values of G(s, χ , f ) as χ varies over characters of p-power conductor.

2 Complex Valued Distributions Following Panchishkin, we now construct a complex valued distribution that is related to the twisted Asai L-function. This section basically follows Coates–Perrin-Riou [1] and Courtieu–Panchishkin [2, Sect. 1.6]. The function G(s, f ) has an Euler product formula

p-adic Asai L-functions Attached to Bianchi Cusp Forms

G(s, f ) =



G l (s, f ) =

21 ∞  d(r ) r =1

l

rs

,

and hence satisfies the hypothesis in the above references. We now assume that our fixed prime p splits as pp¯ in F. A similar argument will also work for p inert. Then the local Euler factor at p is of the form G p (s, f ) = F( p −s )−1 where F(X ) = (1 − α1 (p)α1 (p¯ )X )(1 − α1 (p)α2 (p¯ )X )(1 − α2 (p)α1 (p¯ )X )(1 − α2 (p)α2 (p¯ )X ).

In what follows, we shall assume that f is totally ordinary at p. Hence, we may assume, by possibly switching the subscripts i = 1, 2, that the inverse root κ := α1 (p)α1 (p¯ ) of the polynomial F(X ) is a p-adic unit. Also define a polynomial H (X ) as H (X ) = (1 − α1 (p)α2 (p¯ )X )(1 − α2 (p)α1 (p¯ )X )(1 − α2 (p)α2 (p¯ )X ). Let B0 = 1 and define B1 , B2 , and B3 such that H (X ) = 1 + B1 X + B2 X 2 + B3 X 3 . Let χ : (Z/ p j Z)× → C× be a character with conductor Cχ = p jχ . We want to define a complex valued distribution that interpolates the values of the twisted Lfunction: ∞  d(r )χ (r ) G(s, χ , f ) = . rs r =1 Define functions Ps : Q → C by the formula Ps (b) =

∞  d(r )e2πir b r =1

rs

which converges absolutely for (s) sufficiently large. Define a distribution μ˜ on Z×p by the formula μ˜ s (a + p j Z p ) =

3 p j (s−1)  Bi Ps (api / p j ) p −is . κ j i=0

We need to check that this satisfies the distribution relations. We will do this by showing that  χ (a)μ˜ s (a + p j Z p ) (1) a

mod p j

is independent of j as long as j ≥ jχ . For any character χ and integer M, define the generalized Gauss sum

22

B. Balasubramanyam et al.



G M, p j = a

j

χ (a)e2πia M/ p .

mod p j

It can be verified that  G M, p j =

¯ p j− jχ ) if p j− jχ |M p j− jχ G(χ )χ(M/ 0 otherwise,

where G(χ ) = G 1, p jχ is the Gauss sum of χ . From the definition, we can write the quantity in Eq. (1) as p j (s−1) κj = =

p

 a

mod p j

j (s−1)



κj p

χ (a)

js− jχ

κj

3 

Bi p −is

∞ 

i=0

d(r )e2πiap r/ p r −s i

j

r =1

Bi p −is d(r )r −s G pi r, p j

i,r

G(χ )



Bi p −is d(r )r −s χ¯

i,r



pi r p j− jχ



11Z

pi r p j− jχ

.

(2)

Here 11Z is the characteristic function of integers. It appears since the only terms that contribute to the sum are those with p j− jχ | pi r (this follows from the above formula for G pi r, p j ). Now write each r = r1r2 , where r1 is the p-power part and r2 is the away from p-part of r , respectively. We know that

i

pi r p r1 = χ ¯ (r , and ) χ ¯ 2 p j− jχ p j− jχ i

i

pr p r1 = 11Z . 11Z p j− jχ p j− jχ

χ¯

Using these in Eq. (2), we get the expression  i

i

  p js− jχ p r1 p r1 −s −s −is 11Z . G(χ ) χ(r ¯ 2 )d(r2 )r2 Bi p d(r1 )r1 χ¯ j− j j χ κ p p j− jχ r2 i,r1 (3) Note that here we have also used the fact that d(r ) = d(r1 )d(r2 ). We also know that  d(r1 )r1−s = F( p −s )−1 , r1

where the sum is taking over all powers of p. Moreover, we also have (

 i

Bi p −is )F( p −s )−1 = H ( p −s )F( p −s )−1 = (1 − κ p −s )−1 .

p-adic Asai L-functions Attached to Bianchi Cusp Forms



Hence

Bi p −is d(r1 )r1−s =



23

κ ord p r3 r3−s ,

r3

i,r1

where the r3 varies over all powers of p. We also have the relation 

κ ord p r3 =

Bi d(r1 ).

r3 = pi r1

Hence, setting r3 = pi r1 , we see that the only terms that contribute to the sum in Eq. (3) are those p-powers r3 of the form p j− jχ r4 for some p-power r4 . Also note that as r2 varies over all positive integers prime to p, we get 

χ(r ¯ 2 )d(r2 )r2−s = G(s, χ¯ , f ).

r2

We remark that if χ is the trivial character of Z×p , then G(s, χ¯ , f ) is just the pdeprived Asai L-function G p (s, f )−1 G(s, f ), where G p (s, f ) is the local Euler factor at p, since the function on Z induced by the trivial character χ is taken to vanish on pZ. In any case, we can rewrite Eq. (3) as  p js− jχ G(χ )G(s, χ¯ , f ) κ ord p r3 r3−s χ¯ j κ r 3

=



r3 p j− jχ



11Z

r3



p j− jχ

 p js− jχ G(χ )G(s, χ¯ , f ) κ j− jχ κ ord p r4 p −s( j− jχ )r4−s χ(r ¯ 4) j κ r 4

= =

p

jχ (s−1)

κ jχ p

jχ (s−1)

κ jχ

G(χ )G(s, χ¯ , f )



κ ord p r4 r4−s χ¯ (r4 )

r4

G(χ )G(s, χ¯ , f ),

since χ(r ¯ 4 ) = 0, unless r4 = 1, since by convention all Dirichlet characters of Z×p , including the trivial character, are thought of as functions on Z by requiring that they vanish on pZ. This simultaneously checks the distribution relations and establishes the relationship

p jχ (s−1) G(χ )G(s, χ¯ , f ) (4) χ d μ˜ s = κ jχ between these measures and twisted Asai L-values. We remark here that these calculations hold only for s ∈ C where G(s, f ) is absolutely convergent. And this is known for all s such that (s) > k + 1, in view of the Hecke bound c(l, f ) = O(Nm(l)k/2 ), for all but finitely many primes l of F.

24

B. Balasubramanyam et al.

In order to construct a measure, we need to show that this is a bounded distribution (after possibly dividing by some periods). We now modify the distribution μ˜ s to construct the distribution μs (a + p j Z p ) = μ˜ s (a + p j Z p ) + μ˜ s (−a + p j Z p ). The distribution relations for μs follows from those of μ˜ s . Moreover

χ dμs =

  2 χ d μ˜ s if χ is even, 0 if χ is odd.

 In the next section, we shall prove that the values of the distribution χ dμs in (4), for specific values of s, are rational, after dividing by some periods. In Sect. 6, we conjecture that these values are even p-adically bounded. We then conjecture that these values satisfy the so-called abstract Kummer congruences, and hence come from a measure.

3 Rationality Result for Twisted Asai L-values Let n = k − 2 and set n = (n, n). For any O F -algebra A, let L(n, A) denote the set of polynomials in 4 variables (X, Y, X , Y ) with coefficients in A, which are homogeneous of degree n in (X, Y ) and homogeneous of degree n in (X , Y ). We define an action of SL2 (A) on this set by γ · P(X, Y, X , Y ) = P(γ ι (X, Y )t , γ¯ ι (X , Y )t ) = P(d X − bY, −cX + aY, d¯ X − b¯ Y , −cX ¯ + aY ¯ ),   ab for γ = ∈ SL2 (A) and where γ ι = det(γ )γ −1 is the adjoint matrix of γ . Let cd 0 (n) and 1 (n) be the usual congruence subgroups of SL2 (O F ) with respect to the ideal n. Let H = {(z, t) | z ∈ C and t ∈ R with t > 0} be the hyperbolic upper half-space in R3 . There is an action of SL2 (O F ) on H which is induced by identifying H with SL2 (C)/SU2 (C) = GL2 (C)/[SU2 (C) · C× ]. The last identification is given by a transitive action of SL2 (C) on H defined via g → g · , for g ∈ SL2 (C) and = (0, 1) ∈ H. We will let L(n, A) denote the system of local coefficients associated to L(n, A). So L(n, A) is the sheaf of locally constant sections of the projection 0 (n)\(H × L(n, A)) −→ 0 (n)\H.

p-adic Asai L-functions Attached to Bianchi Cusp Forms

25

Analogous to the Eichler–Shimura isomorphism for classical cusp forms, there are isomorphisms q (0 (n)\H, L(n, C)), δq : Sk (0 (n)) −→ Hcusp for q = 1, 2. Here the cohomology on the right is cuspidal cohomology with local coefficients. We take δ = δ1 since we are interested in 1-forms. Let us now describe the image of δ( f ) under this map.  Let γ ∈ SL2 (C) and z −t z = (z, t) ∈ H. After identifying z with the matrix , recall that the action of t z¯ γ on z is by γ · z = [ρ(a)z + ρ(b)][ρ(c)z + ρ(d)]−1 , 

 α0 . Define the automorphy factor j (γ , z) = ρ(c)z + ρ(d) ∈ 0 α¯ GL2 (C). Let L(2n + 2, C) denote the space of homogeneous polynomials of degree 2n + 2 in two variables (S, T ) and coefficients in C. We will consider L(2n + 2, C) with a left action of SL2 (C). Recall that f is a function H → L(2n + 2, C) that satisfies the transformation property f (γ z, (S, T )t ) = f (z, t j (γ , z)(S, T )t ), where ρ(α) =

for γ ∈ 0 (n). There is a related “cusp form” F : SL2 (C) → L(2n + 2, C) on SL2 (C) which is defined by the formula f (z, (S, T )t ) = F(g, t j (g, )(S, T )t ), where g ∈ SL2 (C) is chosen such that g · = z. By Clebsch–Gordan, there is an SU2 (C)-equivariant homomorphism : L(2n + 2, C) → L(n, C) ⊗ L(2, C). Then δ( f ) can explicitly be described as [4, (13)] δ( f )(g) = g · ( ◦ F(g)), ∀g ∈ SL2 (C).

(5)

Note that here the action of g on L(n, C) is as described above. But the action of SL2 (C) on L(2, C) is identified with the natural action of SL2 (C) on 1 (H) = Cdz ⊕ Cdt ⊕ Cd z¯ (see [4, (6)]). The identification of L(2, C) with 1 (H) is given by sending A2 → dz, AB → −dt and B 2 → −d z¯ . With this identification, we view δ( f ) as a L(n, C)-valued differential form. It is also invariant under the action of SU2 (C) [4, (14)], so it descends to a vector-valued 1-form on H. Moreover, if γ ∈ 0 (n), then δ( f )(γ z) = γ · (δ( f )(z)). 1 (0 (n)\H, L(n, C)). So δ( f ) descends to an element of Hcusp

26

B. Balasubramanyam et al.

We make this formula more explicit. Let U and V be auxiliary variables and define the following homogeneous polynomial of degree 2n + 2 by



2n + 2 2n+2−α α 2n+2−α U V . Q= (−1) α α=0,...,2n+2 Define ψ(X, Y, X , Y , A, B)=[ψ0 (X, Y, X , Y , A, B), . . . , ψ2n+2 (X, Y, X , Y , A, B)]t by the formula (X V − Y U )n (XU + Y V )n (AV − BU )2 = Q · ψ, where each ψi is a polynomial that is homogeneous of degree n in (X, Y ), homogeneous of degree n in (X , Y ), and homogeneous of degree 2 in (A, B). Let z ∈ H, then since SL2 (C) acts transitively on H, there is a g ∈ SL2 (C) such that z = g · where = (0, 1) ∈ H. Let F α : SL2 (C) → C, for α = 0, . . . , 2n + 2, be the components of the function F : SL2 (C) → L(2n + 2, C). Then ( ◦ F)(g) = [F 0 (g), . . . , F 2n+2 (g)] · ψ(X, Y, X , Y , A, B) and δ( f )(z) = g · ( ◦ F(g)) = [F 0 (g), . . . , F 2n+2 (g)] · ψ(g ι (X, Y )t , g¯ ι (X , Y )t , t j (g −1 , )−1 (A, B)t ),

where A2 , AB, B 2 are replaced by dz, −dt, −d z¯ . Now, let β ∈ F ⊂ C and let Tβ denote the translation map H → H given by sending z = (z, t) → (z + β, t). When we view H as a quotient space of SL2 (C), this map is induced by sending the coset gSU2 (C) → γβ gSU2 (C), where 

 1β . 01

γβ := β

β

β

Let 0 (n) := γβ−1 0 (n)γβ ⊂ SL2 (F). Notice that if 0 (n)g  SU2 (C) = 0 (n)g SU2 (C), then 0 (n)γβ g  SU2 (C) = 0 (n)γβ gSU2 (C). Thus, the translation map Tβ induces a well-defined map β

Tβ

0 (n)\H −→ 0 (n)\H, which we again denote by Tβ . We now recall some basic facts about functoriality of cohomology with local coefficients. For i = 1, 2, let X i be topological spaces with universal covers X˜ i and fundamental groups i (after fixing some base points). Let Mi be local coefficient systems on X i , i.e., each Mi is an abelian group with an action of the fundamental group i . Let φ : X 1 → X 2 be a map between the spaces, it induces a map φ∗ : 1 → 2 on the fundamental groups. A map between the coefficient systems φ˜ : M2 → M1

p-adic Asai L-functions Attached to Bianchi Cusp Forms

27

is said to be compatible with φ if it satisfies ˜ ∗ (γ1 )m 2 ), ∀m 2 ∈ M2 and γ1 ∈ 1 . ˜ 2 ) = φ(φ γ1 φ(m In other words, φ˜ must be a map between representations when M2 is viewed as a ˜ there exists an representation of 1 via the map φ∗ . For any compatible pair (φ, φ), induced map φ ∗ : H q (X 2 , M2 ) → H q (X 1 , M1 ) at the level of cohomology. This map is constructed as follows. Let S∗ ( X˜ i ) denote the singular complex of the universal covers. There is a natural action on the right by i via deck transformations. Given a singular q-simplex σ : q → X˜ i and g ∈ i , we convert this right action into a left action by setting g · σ = σ · g −1 . The cohomology groups with local coefficients are given by the homology of the complex HomZi (S∗ ( X˜ i ), Mi ). The map φ ∗ is induced by the following map on the complexes (which we again denote by φ ∗ ): φ ∗ : HomZ2 (Sq ( X˜ 2 ), M2 ) → HomZ1 (Sq ( X˜ 1 ), M1 ). Given a cochain C ∈ HomZ2 (Sq ( X˜ 2 ), M2 ) and τ : q → X˜ 1 ∈ Sq ( X˜ 1 ), define ˜ φ ∗ (C)(τ ) = φ(C(φ ◦ τ )),

(6)

where we continue to denote by φ : X˜ 1 → X˜ 2 the unique lift of φ : X 1 → X 2 to the universal covers. This construction is independent of the base points chosen in the beginning. β Let us now apply this to our situation with X 1 = 0 (n)\H and X 2 = 0 (n)\H. β In this case, 1 = 0 (n) and 2 = 0 (n) and they act on M1 = M2 = L(n, C) since they are subgroups of SL2 (F). We take the map φ to be the translation map Tβ β which induces, at the level of fundamental groups, the map (Tβ )∗ : 0 (n) → 0 (n) −1 which sends γβ γ γβ → γ . It is an easy check that the map T˜β : M2 → M1 sending P → γβ−1 P, for P ∈ L(n, C), is compatible with Tβ . By the discussion above this induces a map at the level of cohomology β

Tβ∗ : H q (0 (n)\H, L(n, C)) → H q (0 (n)\H, L(n, C)). When q = 1, what is the image of the element δ( f ) ∈ H 1 (0 (n)\H, L(n, C))? After translating the above map Tβ∗ in terms of vector-valued differential forms, we get that (6)

(5)

Tβ∗ (δ( f ))(z) = γβ−1 δ( f )(γβ z) = γβ−1 γβ g · ( ◦ F(γβ g)) = g · ( ◦ F(γβ g)) = [F 0 (γβ g), . . . , F 2n+2 (γβ g)] · ψ(g ι (X, Y )t , g¯ ι (X , Y )t , t j (g −1 , )−1 (A, B)t ).

28

B. Balasubramanyam et al.

Here z ∈ H and we take g ∈ SL2 (C) such that gz = , and A2 , AB, B 2 are to be replaced by dz, −dt, −d z¯ . Following [4, Sect. 5.2], we now want to compute the restriction Tβ∗ (δ( f ))|H where H = {x + it | x, t ∈ R and t > 0} is the usual upper half-plane which is embedded into the hyperbolic 3-space H as 

 x −t x + it → . t x As in loc. cit., we make the following two simplifications. Firstly, since we wish to compute this differential form on H, we set dz =d z¯ in our  computations. Secondly,  1 −x  ∗ we only need to calculate the differential form · Tβ (δ( f ))|H , so we set 0 1 x = 0 in ψ and only calculate the modified differential form which we denote by  Tβ∗ (δ( f ))|H . Note that the components ψα of ψ, for α = 0, . . . , 2n + 2, are given by A2 cα − 2 ABcα−1 + B 2 cα−2 , ψα (X, Y, X , Y , A, B) = (−1)α 2n+2 α

where cα (X, Y, X , Y ) =

n 

(−1)k

j,k=0



n n n− j j X n−k Y k X Y . j k

n=α+ j−k



 t x . Then g · = (x, t) ∈ H ⊂ H and For x, t ∈ R and t > 0, let g = 01     1 −x 10 g ι = g¯ ι = √1t . Moreover, j (g −1 , )−1 = j (g, ) = √1t . Let f α , for 0 t 01 α = 0, . . . , 2n + 2, be the components of f : H → L(2n + 2, C). The precise relationship between f α and F α is given by √1 t

f α (z) = √

1 t

2n+2

F α (g).

Note that if z = (z, t), then Tβ (z) = (z + β, t) does not affect the t coordinate. Hence f α (Tβ z) = √

1 t

2n+2

Using this and the pullback formula, we get

F α (γβ g).

p-adic Asai L-functions Attached to Bianchi Cusp Forms

 f ))|H = Tβ∗ (δ(

2n+2 

√

t

2n+2

f α (Tβ z)ψα



α=0

=

2n+2 

29

√ √ 1 1 1 1 √ X, t Y, √ X , t Y , √ A, √ B t t t t



f α (Tβ z)ψα (X, tY, X , tY , A, B),

α=0

where we replace (A2 , AB, B 2 ) by (d x, −dt, −d x). Thus we have now constructed β β β 1  an element Tβ∗ (δ( f ))|H ∈ Hcusp (0 (N )\H, L(n, C)) where 0 (N ) := 0 (n) ∩ β 0 (N ) = 0 (n) ∩ SL2 (Z), since in the latter matrix group, the lower left entries are divisible by N . As in [4, see below Lemma 2], we have a decomposition of this cohomology group as ∼

β

1 (0 (N )\H, L(n, C)) −→ Hcusp

n 

β

1 Hcusp (0 (N )\H, L(2n − 2m, C)).

(7)

m=0

 We will call the projection of Tβ∗ (δ( f ))|H into the m-th component by Tβ∗ (δ 2n−2m ( f )), slightly abusing notation since the subscript 2n − 2m should technically be outside the parentheses. For each m, define α

g (z) =

⎧ ⎨

f α (z)+(−1)n+1−α+m f 2n+2−α (z)

if α = 0, 1, . . . , n,

⎩

f n+1 (z)

if α = n + 1.

(

2n+2 n+1

(2n+2 α )

)

(8)

Then, we have Tβ∗ (δ 2n−2m ( f ))(x, t) =

2n−2m 

(Al d x + 2Bl dt)t 2n−m−l X l Y 2n−2m−l ,

(9)

l=0

where Al =

n+1  (−1)α g α (Tβ (x, t))a(m, l, α), α=0

Bl =

n+1  (−1)α g α (Tβ (x, t))b(m, l, α), α=0

with a(m, l, α) and b(m, l, α) the integers defined at the end of [4, Sect. 5]. For any n ≥ 0 and any Z[1/n!]-algebra A, there is an SL2 (Z)-equivariant pairing [4, Lemma 4]  ,  : L(n, A) ⊗ L(n, A) → A, which induces by Poincare duality a pairing

30

B. Balasubramanyam et al. β

β

β

 ,  : Hc1 (0 (N )\H, L(n, A)) ⊗ H 1 (0 (N )\H, L(n, A)) → Hc2 (0 (N )\H, A) → A, β

where the last map Hc2 (0 (N )\H, A) → A is given by integrating a compactly β β supported 2-form on a fundamental domain [0 (N )\H] of 0 (N )\H. We will use this pairing when A = C, A = E is a p-adic number field with p > n, and with A = O E , its ring of integers. When A = C, the pairing can be extended to β

β

β

1 ( (N )\H, L(n, C)) ⊗ H 1 ( (N )\H, L(n, C)) → H 2 ( (N )\H, C) → C ∪ {∞}.  ,  : Hcusp cusp 0 0 0

β

β

For each m, there is an Eisenstein differential form E 2n−2m+2 for 0 (N ) given by 

β

E 2n−2m+2 (s, z) = γ

γ −1 · γ ∗ (ω y s ),

(10)

β β ∈ 0 (N )∞ \0 (N )

where ω = (X − zY )2n−2m dz. One may check that 

β

E 2n−2m+2 (s, z) =





β β γ = a b ∈ 0 (N )∞ \0 (N ) cd

β

1 · y s ω. (11) (cz + d)2n−2m+2 |cz + d|2s

β

We view E 2n−2m+2 as an element of H 1 (0 (N )\H, L(n, C)). We now wish to evaluate β Tβ∗ (δ2n−2m ( f )), E 2n−2m+2 , following [4, Sect. 6.3]. We have β

Tβ∗ (δ2n−2m ( f )), E 2n−2m+2  =



=

β

β [0 (N )\H]

β [0 (N )\H]

Tβ∗ (δ2n−2m ( f ))(x, t), E 2n−2m+2 (x, t) β  Tβ∗ (δ 2n−2m ( f )), E 2n−2m+2 ,

where the that we have twisted the differential forms by the action of the   indicates  1 −x matrix . Using a standard unwinding argument, the last integral becomes 0 1

0

∞

1 0

Tβ∗ (δ ωt s , 2n−2m ( f )), 

where  ω = (X − itY )2n−2m dz. Using the expression (9) for Tβ∗ (δ 2n−2m ( f ))(x, t) and the definition of the pairing, we have

p-adic Asai L-functions Attached to Bianchi Cusp Forms

∞

0

1

0

Tβ∗ (δ ωt s  = 2n−2m ( f )), 

∞

0

31

1 2n−2m  0

i l+1 Al t 2n−m+s d xdt

l=0



∞

−2 0

1 2n−2m 

0

i l Bl t 2n−m+s d xdt.

l=0

We denote the first integral by I1 and the second integral by I2 . We now compute I1 using the definition of Al as I1 =

2n−2m 

i

l+1

n+1 

α

(−1) a(m, l, α) 0

α=0

l=0

∞

1

g α (Tβ (x, t))t 2n−m+s d xdt.

0

Using the Fourier expansion for the α-th component of f , see [4, (7)] with a1 = 1, we get

2n + 2 f α (Tβ (x, t)) = t α



⎡ ⎣

⎤

ξ n+1−α c(ξ d) K α−n−1 (4π t|ξ |)e F (ξ(x + β))⎦ , i|ξ | ×



ξ ∈F

where e F (w) = e2πiTr F/Q (w) . Using (8) and plugging this into the expression for I1 , we get I1 =

2n−2m 

i l+1

1

(−1)α a(m, l, α)

ξ i|ξ |

∞

0

α=0

l=0



n 

n+1−α



c(ξ d)t 2n−m+1+s

ξ ∈F ×

K α−n−1 (4πt|ξ |) + (−1)



n+m+1−α

e F (ξ(x + β))d x +

0

2n−2m 

ξ i|ξ |

i l+1 (−1)n+1 a(m, l, n + 1) 0

l=0

1

K 0 (4πt|ξ |)dt

α−n−1

∞

K n+1−α (4πt|ξ |) dt



c(ξ d)t 2n+1−m+s

ξ ∈F ×

e F (ξ(x + β))d x.

0

The only terms c(ξ d) that survive are when ξ = 1 this case 0 e F (ξ x)d x = 1. So we have I1 =

√r , −D

for some 0 = r ∈ Z, and in

√ −r n+1−α e F (rβ/ −D)c(r ) |r | α=0 r =0 l=0 



∞ 4πt|r | 4πt|r | t 2n+1−m+s K α−n−1 + (−1)n+m+1−α K n+1−α dt √ √ D D 0

∞ 2n−2m   √ 4πt|r | dt. i l+1 (−1)n+1 a(m, l, n + 1) e F (rβ/ −D)c(r ) t 2n+1−m+s K 0 + √ D 0 r =0 l=0 2n−2m 

i l+1

n 

(−1)α a(m, l, α)



32

B. Balasubramanyam et al.

The Bessel functions have the property [4, Lemma 7]

∞

K ν (at)t

μ−1

dt = 2

μ−2 −μ

a



0

μ+ν 2





μ−ν 2

.

This implies that the two Bessel functions in the sum above will cancel each other unless α ≡ n + 1 + m mod (2). Setting s  = 2n + 2 − m + s, we have I1 =

√ s  2n−2m  (−1)n+1 D i l+1 2n+2−m+s 2(2π ) l=0

n 

(−1)m a(m, l, α)

√ e F (rβ/ −D)c(r )

0 =r ∈Z

α=0

α≡n+1+m (2)









−r n+1−α 1 3n + 3 − m − α + s n+1−m+α+s    |r | 2 2 |r |s  √ s 2n−2m n+1   √ D 1 (−1) i l+1 a(m, l, n + 1) e F (rβ/ −D)c(r ) +  4(2π )2n+2−m+s |r |s



0 =r ∈Z

l=0

2n + 2 − m + s 2

2

.

√ √ We will take β = b 2−D for some rational number b. Then the term e F (rβ/ −D) = e2πir b . Now we break the sum over r into a sum over positive integers and a sum over  n+1−α equals (−1)m when r is positive and is 1 negative integers. The term −r |r | when r is negative. The second sum over r does not have such a term, so we assume that m is even in order to be able to put these terms together into a single term. The  terms c(r ) and |r |s are obviously independent of the sign of r . So finally, we have

√ s ∞ 2n−2m   (−1)n+1 D   2πir b −2πir b c(r ) I1 = e + e i l+1  2n+2−m+s s 2(2π ) r r =1 l=0



n+1  3n + 3 − m − α + s n+1−m+α+s  , a(m, l, α) 2 2 α=0 α≡n+1+m (2)

(12) where there is an extra factor of 21 in the α = n + 1 term, which we will adjust for. By a similar computation, I2 will also have an expression in terms of b(m, l, α). Putting together these two expressions, we get that Tβ∗ (δ2n−2m ( f )),

β E 2n−2m+2 (s)

√ =

D

s

(2π)2n+2−m+s

∞   r =1

e2πir b + e−2πir b

 c(r )   · G ∞ (s, f ), rs

where we collect all the combinations of Gamma factors appearing in both I1 and I2 and denote it by G ∞ (s, f ).

p-adic Asai L-functions Attached to Bianchi Cusp Forms

33

Now let χ : (Z/ p j Z)× → C× be a primitive character (so j = jχ ) and recall that G(s, χ¯ , f ) = L N (2s − 2k + 2, χ¯ 2 )

∞  c(r )χ(r ¯ )

rs

r =1

.

Substituting the formula χ(r ¯ )=

1 G(χ )

 a

χ (a)e2πira/ p

j

mod p j

in the above equation, we get G(s  , χ¯ , f ) = L N (2n − 2m + 2 + 2s, χ¯ 2 )

1 G(χ )

 a

mod p j

χ (a)

∞  c(r ) r =1

r s

j

e2πira/ p .

Now assume that χ is an even character, i.e., χ (−1) = 1. Then grouping together the terms coming from a and −a, we get G(s  , χ¯ , f ) = L N (2n − 2m + 2 + 2s, χ¯ 2 )

∞  c(r ) 2πira/ p j 1  j χ(a) (e + e−2πira/ p ), s G(χ) r r =1 a∈R

where R is half of the representatives modulo p j such that if a ∈ R, then −a ∈ / R. We now write G(s  , χ , f ) in terms of the inner product considered earlier G(χ )G(s  , χ, ¯ f) =

 (2π )2n+2−m+s β 2 χ (a)Tβ∗ (δ2n−2m ( f )), E 2n−2m+2 (s),  L N (2n − 2m + 2 + 2s, χ¯ ) √ s G ∞ (s, f ) D a∈R

(13)

√ with β = a −D/2 p j . Let G ∞ (s, f ) = G ∞ (s, f )(s + 2n − 2m + 2). Dividing both sides of the above equation by the period G(χ¯ 2 )(2π )2n−2m+2 , we obtain G(χ )G(s  , χ, ¯ f) (2π )2n+2−m+s L N (2n − 2m + 2 + 2s, χ¯ 2 ) = · (s + 2n − 2m + 2) √ s · G(χ¯ 2 )(2π )2n−2m+2 (2π )2n−2m+2 G(χ¯ 2 ) G ∞ (s, f ) D  β χ (a)Tβ∗ (δ2n−2m ( f )), E 2n−2m+2 (s). · a∈R

We evaluate this expression at s = 0. Note that G ∞ (0, f ) = 0, by [4, Sect. 6.4] using some special arguments, and by Lanphier-Skogman and Ochiai [10] as a consequence of their proof of [4, Conjecture 1]. The special value L N (2n − 2m + 2, χ¯ 2 ) becomes rational after dividing by the period G(χ¯ 2 )(2π )2n−2m+2 . We denote this ratio by L ◦ (2n − 2m + 2, χ¯ 2 ). We get

34

B. Balasubramanyam et al.

 G(χ)G(2n − m + 2, χ¯ , f ) β = L ◦ (2n − 2m + 2, χ¯ 2 ) χ(a)Tβ∗ (δ2n−2m ( f )), E 2n−2m+2 (0). G(χ¯ 2 )∞ a∈R

(14) Here ∞ is defined as ∞ =

(2π )4n−3m+4 (2n − 2m + 2) . √ 2n−m+2 G ∞ (0, f ) D

We now conclude rationality properties of the special values G(2n − m + 2, χ¯ , f ) from Eq. (14). Choose a period ( f ) such that after dividing by this period, the differential form δ ◦ ( f ) :=

δ( f ) 1 ∈ Hcusp (0 (n)\H, L(n, E)) ( f )

takes rational values. Here E is a sufficiently large p-adic field, containing the field of rationality of the form f , which we also view as a subfield of C after fixing an isomorphism between C and Q p . Then Tβ∗ δ( f )|H = ( f ) · Tβ∗ δ ◦ ( f )|H , √ noting that if −D ∈ E, which we assume, then the image Tβ∗ δ ◦ ( f )|H of δ ◦ ( f ) under the map β

β

1 1 1 Tβ∗ |H : Hcusp (0 (n)\H, L(n, E)) → Hcusp (0 (n)\H, L(n, E)) → Hcusp (0 (N )\H, L(n, E)),

is also rational. Since Clebsch–Gordan preserves rationality, for 0 ≤ m ≤ n, we obtain that ◦ ( f )), Tβ∗ (δ2n−2m ( f )) = ( f ) · Tβ∗ (δ2n−2m β

◦ 1 where Tβ∗ (δ2n−2m ( f )) ∈ Hcusp (0 (N )\H, L(2n − 2m, E)) is also rational. ◦ ( f )) is cohomologous to a compactly supThe rational cuspidal class Tβ∗ (δ2n−2m β ported rational class which has the same value when paired with E 2n−2m+2 (0) (see β the proof of [4, Theorem 1]). Since the differential form E 2n−2m+2 (0) coming from the Eisenstein series is E-rational, at least when m = n (see Proposition 1 in Sect. 4), and the pairing between compactly supported rational classes and such classes preserves E-rationality, the following theorem follows from (14), if E contains the field of rationality of χ , which we again assume.

Theorem 1 (Rationality result for twisted Asai L-values) Let E be a sufficiently large p-adic number field with p  2N D. Let 0 ≤ m < n be even and χ be even. Then G(χ )G(2n − m + 2, χ¯ , f ) ∈ E. G(χ¯ 2 )( f )∞

p-adic Asai L-functions Attached to Bianchi Cusp Forms

35

This result matches with [4, Theorem 1] when χ is trivial. In that theorem, it was assumed that the finite part of the central character of f is non-trivial primarily to deal with the rationality of the Eisenstein series when m = n. In this paper, we have assumed (for simplicity) that the finite part of the central character of f is trivial. We could still probably include the case m = n in the theorem above, by using the β β rationality of the Eisenstein series E 2 (0, z) − pE 2 (0, pz) instead (see [4, Remark 2]).

4 Rationality of Eisenstein Cohomology Classes We start by recalling the following result that goes back to Harder [7], [8]. See also [9, Sect. 10]. Lemma 1 Eisenstein cohomology classes corresponding to Eisenstein series whose constant terms at every cusp are rational are rational cohomology classes. Proof We use notation in this proof that is independent of the rest of the paper. Let  ⊂ SL2 (Z) be a congruence subgroup and L(n, C) denote the sheaf of locally constant sections of π : \(H × L(n, C)) → \H. Consider the restriction map to boundary cohomology given by R : H 1 (\H, L(n, C)) → H∂1 (\H, L(n, C)) :=



H 1 (ξ \H, L(n, C)),

ξ

where ξ varies through the cusps of . We know that 1 1 (\H, L(n, C)) ⊕ HEis (\H, L(n, C)), H 1 (\H, L(n, C)) = Hcusp 1 1 and HEis are the cuspidal and Eisenstein part of cohomology, respecwhere Hcusp 1 tively. The restriction of R to HEis is an isomorphism 1 R : HEis (\H, L(n, C)) →



H 1 (ξ \H, L(n, C))

ξ

ω → cξ (0, ω), where cξ (0, ω) is the differential form corresponding to the “constant term” in the Fourier expansion at the cusp ξ of the differential form ω corresponding to the underlying Eisenstein series. Clearly R preserves the rational structures on both sides. The following fact is due to Harder. Fact: There exists a section M : ⊕ξ H 1 (ξ \H, L(n, C)) → H 1 (\H, L(n, C))

36

B. Balasubramanyam et al.

of R preserving rational structures on both sides. 1 Now let ω ∈ HEis (\H, L(n, C)) be such that R(ω) is rational. Then M(R(ω)) is rational and R(M(R(ω))) = R(ω). Since R is an isomorphism we have M(R(ω)) = ω. Hence, ω is rational. Thus, the Eisenstein class ω is rational if and only if the constant term in the Fourier expansion at every cusp is rational. This proves the lemma.

Proposition 1 If m = n, then the Eisenstein differential form β

β

E 2n−2m+2 (0) ∈ H 1 (0 (N )\H, L(2n − 2m, E)) is rational, for a sufficiently large p-adic number field E. √

β

Proof Recall that β = a 2 p−D if j ≥ 1 (and β = 0 if j = 0). We claim that 0 (N ) = j −1 γβ 0 (N)γβ ∩ SL2 (Z) is independent of a. We do this by showing that β

0 (N ) =

 a b  cd

∈ SL2 (Z) : a ≡ d

mod p j , c ≡ 0

 mod N p 2 j .

(15)

Indeed, if j = 0, (15) holds trivially, since in this case γβ = 1, so both sides of (15) are equal to 0 (N ). So assume that j ≥ 1. Since p is odd and we are considering representatives a ∈ R = (Z/ p j Z)× /{±1}, by replacing a by p j − a if necessary,

√ −1 ab we may assume that all a ∈ R are even, so that p j −D β ∈ Z. Let γ = ∈ cd

1β 0 (N) and γβ = . Then 01 γβ−1 γ γβ



a − cβ b − dβ + (a − cβ)β = . c d + cβ

(16)

β

Assume that the matrix in (16) is in SL2 (Z), so is in 0 (N ). Then a − cβ, b − dβ + (a − cβ)β, c, d + cβ ∈ Z. Note that c ∈ Z ⇔ c ∈ N Z. Since a − cβ and d + cβ ∈ Z, we have b − dβ + (a − cβ)β ∈ Z ⇔ (b + cβ 2 ) + (a − cβ − d − cβ)β ∈ Z ⇔ (b) + cβ 2 ∈ Z and (b) = i(a − cβ − d − cβ)β ⇔ p 2 j |c and a − cβ ≡ d + cβ

mod p j ,

√ −1 β since p  2D and both (b), D (b) ∈ 21 Z. Therefore 0 (N ) is contained in the right hand side of (15). On the other hand, if γ is any matrix on the right-hand side of (15), then by replacing β by −β in (16), one checks that γβ γ γβ−1 ∈ 0 (N). It follows that equality holds in (15). From (15), we have

p-adic Asai L-functions Attached to Bianchi Cusp Forms

SL2 (Z)∞ =

37



 ±1 n β : n ∈ Z ⊂ 0 (N ). 0 ±1



ab β Thus = SL2 (Z)∞ . Also, note that the coset in 0 (N )∞ cd β β \0 (N ) contains all the matrices of 0 (N ) whose bottom row equals ±(c, d). Hence, β β 0 (N )∞ \0 (N ) is in bijection with the set β 0 (N )∞

β 0 (N )∞

 := {(c, d) ∈ Z2  {(0, 0)} : (c, d) = 1, c ≡ 0

mod N p2 j , d ≡ ±1 mod p j }/{±1}.

For each integer k ≥ 3 and (u, v) ∈ (Z/N p 2 j )2 , consider the Eisenstein series 

E k(u,v) (z) :=

(c,d)≡(u,v) mod N p2 j (c,d)=1

1 . (cz + d)k

This Eisenstein series differs from the Eisenstein series in [3, (4.4)] by a factor of N p2 j = 21 or 1. By (11), we have β

E 2n−2m+2 (0, z) = =



1 ·ω 2n−2m+2 (cz + d) (c,d)∈ 1 2



(u,v) E 2n−2m+2 (z) · ω,

(17)

(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡±1 mod p j

noting that 2n − 2m + 2 ≥ 4, since m = n. By [3, (4.6)], for k ≥ 3, we have 

E k(u,v) (z) =

−1

ζ+l (k, μ)G lk

(u,v)

(z),

l∈(Z/N p2 j Z)×

where ζ+l (k, μ) :=

∞  m=1 m≡l mod N p2 j

μ(m) , mk

μ(·) is the Möbius function, and G (u,v) (z) := k

 (c,d)∈Z2 (c,d)≡(u,v) mod N p2 j

1 . (cz + d)−k

(18)

38

B. Balasubramanyam et al.

 β n We will obtain the q-expansion of the Eisenstein series E 2n−2m+2 (0, z) = ∞ n=0 an q , using (17), (18), and the q-expansion of the Eisenstein series above using facts from [3]. Let k ≥ 2 an integer and ϕ be the Euler totient function. For v ∈ (Z/N p 2 j )× , we have ζ v (k) :=

=

 d≡v mod ∞ 

d −k

N p2 j ∞ 

d −k + (−1)k

d=1 d≡v mod N p 2 j

d −k

d=1 d≡−v mod N p 2 j

⎛

⎞   1 ⎝ = ψ(v)−1 L(k, ψ) + (−1)k ψ(−v)−1 L(k, ψ)⎠ ϕ(N p 2 j ) 2 j 2 j ψ mod N p ψ mod N p ⎛ ⎞  1 k −1 ⎝ = (1 + (−1) ψ(−1))ψ(v) L(k, ψ)⎠ , ϕ(N p 2 j ) 2j ψ mod N p

(19) where the penultimate step follows from [3, Page 122]. If k is even, then (1 + (−1)k ψ(−1)) is equal to 2 (resp. 0) if ψ is even (resp. odd). A similar expression for ζ+l (k, μ) in terms of Dirichlet L-functions can also be derived. For l ∈ (Z/N p 2 j Z)× , by the orthogonality relations, we have ζ+l (k, μ) = =

∞ 

1 ϕ(N p 2 j ) m=1 1 ϕ(N p 2 j )

1 = ϕ(N p 2 j )

 ψ mod



ψ(l)−1 ψ(m) μ(m)m −k

N p2 j

ψ(l)−1

ψ mod N p2 j



∞ 

ψ(m) μ(m)m −k

m=1

ψ(l)−1 L(k, ψ)−1 ,

(20)

ψ mod N p2 j

where the last step follows from by multiplying the corresponding L-functions. Therefore ζ+l (k, μ) + ζ+−l (k, μ) = =

1 ϕ(N p 2 j ) 2 ϕ(N p 2 j )

By (17) and (18), we have



(ψ(l)−1 + ψ(−l)−1 )L(k, ψ)−1

ψ mod N p2 j



ψ even ψ mod N p2 j

ψ(l)−1 L(k, ψ)−1 .

(21)

p-adic Asai L-functions Attached to Bianchi Cusp Forms β

E 2n−2m+2 (0, z) =

1 2





(u,v)∈(Z/N p 2 j Z)2 u≡0 mod N p 2 j v≡±1 mod p j

l∈(Z/N p 2 j Z)×

39 l −1 (u,v)

ζ+l (2n − 2m + 2, μ)G 2n−2m+2 (z) · ω.

(22) For simplicity, let k = 2n − 2m + 2. By the description of the set , we have (c, d) ∈  which implies that N p 2 j | c and (c, d) = 1, so the congruence class v of d mod N p 2 j has order N p 2 j . Therefore, for m < n, by [3, Theorem 4.2.3], we have 



,v ) G (u (z) = ζ v (k) + k 

∞  (−2πi)k 2j (u  ,v  ) σk−1 (l)e2πilz/N p , 2 j k (k − 1)!(N p ) l=1

(23)

for tuples (u  , v  ) occurring in (22), where 





(u ,v ) σk−1 (l) =

 

sgn(l  )l k−1 e2πiv l /N p . 2j

l  |l l/l  ≡0 mod N p2 j

Constant term: β By (22) and (23), the constant term a0 in the q-expansion of E k (0, z) equals a0 =

1 2





(u,v)∈(Z/N p2 j Z)2

l∈(Z/N p2 j Z)×

u≡0 mod N p2 j v≡±1 mod p j

ζ+l (k, μ)ζ l





(u,v)∈(Z/N p2 j Z)2

l∈(Z/N p2 j Z)×

(∗)

=

1 ϕ(N p 2 j )2

−1 v

(k)

 ψ mod

1 ϕ(N p 2 j )2

(∗∗)

=

=

1 ϕ(N p 2 j )

1 2ϕ(N p 2 j )





(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡±1 mod p j

ψ,ψ1 mod N p2 j ψ1 even





(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡±1 mod p j

ψ1 mod N p2 j ψ1 even





(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡±1 mod p j

ψ1 mod N p2 j

ψ1 (v)−1

L(k, ψ1 ) L(k, ψ)

ψ1 (l −1 v)−1 L(k, ψ1 )

ψ1 mod ψ1 even

N p2 j

N p2 j

u≡0 mod N p2 j v≡±1 mod p j

=



ψ(l)−1 L(k, ψ)−1



ψ1 ψ −1 (l)

l∈(Z/N p2 j Z)×

ψ1 (v)−1

(ψ1 (v)−1 + ψ1 (−v)−1 )

(∗∗∗)

= 1,

where (∗) follows from (19) and (20), and (∗∗) and (∗ ∗ ∗) follow from the orthogonality relations.

40

B. Balasubramanyam et al.

Higher Fourier coefficients: 

(0,v ) (l) = 0 if N p 2 j  l. So assume that l is a multiple of N p 2 j . Say l = Clearly σk−1 2 j  N p l . Then    2j (0,v  ) (l) = sgn(l  )l k−1 e2πiv l /N p , σk−1 l  |l 

which is clearly E-rational if E contains a sufficiently large cyclotomic number field depending on j.  From (22) and (23), we see that the coefficient al  of q l in the Fourier expansion β of E k (0, z) equals al  =

1 2





(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡±1 mod p j

n∈(Z/N p2 j Z)×

ζ+n (k, μ)

−1 (−2πi)k σ n (0,v) (N p 2 j l  ). (k − 1)!(N p 2 j )k k−1

If j = 0, one checks that the formula for al  above reduces to a well-known expression (see [12, Theorem 7.1.3 and (7.1.30)]), and in particular al  ∈ Q is rational. So assume that j > 0. Then al  =

1 2





(u,v)∈(Z/N p2 j Z)2

n∈(Z/N p2 j Z)×

(ζ+n (k, μ) + ζ+−n (k, μ))

(−2πi)k n −1 (0,v) σ (N p 2 j l  ) (k − 1)!(N p 2 j )k k−1

u≡0 mod N p2 j v≡1 mod p j

(21)

=

(24)

=

1 ϕ(N p 2 j )

−2k ϕ(N p 2 j )





(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡1 mod p j

(u,v)∈(Z/N p2 j Z)2 u≡0 mod N p2 j v≡1 mod p j

ψ(n)−1 L(k, ψ)−1

n∈(Z/N p2 j Z)× ψ mod N p2 j ψ even









n∈(Z/N p2 j Z)× ψ mod N p2 j ψ even

ψ(n)−1



Cψ N p2 j

(−2πi)k n −1 (0,v) · σk−1 (N p 2 j l  ) (k − 1)!(N p 2 j )k

k

1 n −1 (0,v) · σk−1 (N p 2 j l  ), G(ψ)Bk,ψ¯

where in the last step we have used the following special value result for the Dirichlet L-function: L(k, ψ) = −

(−2πi)k G(ψ)Bk,ψ¯ 2k!Cψk

if ψ is even and k > 0 is even,

(24)

where Cψ denotes the conductor of ψ. Thus, al  is again E-rational for a sufficiently large p-adic number field E containing an appropriate cyclotomic number field. β Summarizing, above show that E 2n−2m+2 (0, z) has an E-rational ∞ the computations n q-expansion n=0 an q (at the cusp ∞) if E contains a sufficiently large cyclotomic number field (which depends on j). (u,v)γ (u,v) |γ = E 2n−2m+2 , for all γ ∈ SL2 (Z), the By [3, Proposition 4.2.1], since E 2n−2m+2 β Eisenstein series E 2n−2m+2 (0, z) has an E-rational q-expansion at each cusp ξ of β 0 (N ). The proposition now follows from Lemma 1.

p-adic Asai L-functions Attached to Bianchi Cusp Forms

41

5 Towards Integrality Note that the map Tβ∗ |H can also be described as the pullback of a differential form via the map β Sβ : 0 (N )\H → 0 (n)\H given by sending

 x + it → γβ

 x −t . t x

1 (0 (n)\H, L(n, O E ))[ f ], We now choose δ ◦ ( f ) such that it generates H¯ cusp 1 ¯ which is a rank one O E -submodule of Hcusp (0 (n)\H, L(n, O E )), where O E is the valuation ring of E and H¯ 1 denotes the image of the integral cohomology in the rational cohomology under the natural map. We correspondingly refine the period ( f ) so that ( f ) ∈ C× /O× E. √ a −D Since β = 2 p j , we have γβ−1 · P ∈ L(n, p12n j O E ), for P ∈ L(n, O E ). Thus, the map Sβ does not preserve cohomology with integral coefficients, but instead induces a map

1 β 1 1 (0 (n)\H, L(n, O E )) → H¯ cusp (0 (N )\H, L(n, 2n j O E )), Sβ∗ : H¯ cusp p

(25)

on cohomology. Lemma 2 Assume p > n. Then under the Clebsch–Gordan decomposition (7), we have 1 ( (n)\H, L(n, O ))  −→ Sβ∗ ( H¯ cusp 0 E

n  m=0

Proof Let ∇ =

∂2 ∂ X ∂Y

−

∂2 ∂ X ∂Y

1 ( β (N )\H, L(2n − 2m, H¯ cusp 0

1 O E )). p j (2n−m)

. By [4, Lemma 2], the projection to the m-th com-

ponent in (7) is induced by P(X, Y, X , Y ) → m!1 2 ∇ m P(X, Y, X , Y ) | X=X . Clearly Y =Y the projection continues to be defined with O E coefficients if p > n. As remarked in (25), Sβ∗ does not preserve integrality. However, since ∇(γβ−1 · X n−k Y k X

n−l l

Y )= l

∂ n−k ∂ n−k n−l n−l X (Y − β X )k X (Y + β X )l−1 − k (Y − β X )k−1 X (Y + β X)l , X ∂X ∂X

we see that if P ∈ L(n, O E ), the total power of p j in the denominator goes down by one after applying ∇ to γβ−1 · P. Iterating this, we see ∇ m (γβ−1 · P) ∈ L(2n − 1 O E ), for m = 0, . . . , n, proving the lemma. 2m, p j (2n−m)

42

B. Balasubramanyam et al.

We now assume that the prime p is greater than n, so that we may apply the lemma above. Let β ◦ 1 ( f )) ∈ H¯ cusp (0 (N )\H, L(2n − 2m, Sβ∗ (δ2n−2m

1 p j (2n−m)

O E ))

(26)

be the image of δ ◦ ( f ) under the map (25) followed by the projection to the m-th component in the Clebsch–Gordan decomposition in Lemma 2. Again note the slight abuse of notation, since the subscript 2n − 2m should be outside the brackets. By Proposition 1, we know that β

E 2n−2m+2 (0) ∈

1 ¯1 β H (0 (N )\H, L(2n − 2m, O E )), pc j

(27)

for some integer c j ≥ 0, depending on j. Let S denote the finite set of excluded primes above (i.e., p | 2N D and p ≤ n), which we extend to include the primes p < 2n + 4. We remark that if p ∈ / S, then p > 2n which ensures that the duality pairing  ,  is a well-defined pairing on cohomology with integral coefficients L(2n − 2m, O E ). For the refined period ( f ) defined above, we get the following partial integrality result. Proposition 1 Suppose p is not in the finite set of primes S, and that E is a sufficiently large p-adic number field as above. Let 0 ≤ m < n be even and χ be an even character of conductor p jχ . Then G(χ )G(2n − m + 2, χ¯ , f ) 1 ∈ jχ (4n−3m+3)+c j O E . χ ( f )G(χ¯ 2 )∞ p

(28)

Proof Indeed, this follows from the fact that by (14) the special value in the statement of the proposition has a cohomological description in terms of integrals of the form

β [0 (N )\H]

β

◦ Sβ∗ δ2n−2m ( f ) ∧ E 2n−m+2 .

These are integrals of cohomology classes with specifiable denominator over an integral cycle, and hence belong to O E with specifiable denominator. The size of the denominator can be computed from (26) and (27), taking j = jχ , and the fact that the 1 Dirichlet L-value in (14) satisfies L ◦ (2n − 2m + 2, χ¯ 2 ) ∈ p jχ (2n−2m+3) O E (which in turn follows easily from a special value result like (24), noting that the corresponding twisted Bernoulli numbers lie in p1jχ O E by a standard formula for these numbers involving the usual Bernoulli numbers up to B2n+2 , and by the well-known result of von Staudt–Clausen which says that p does not divide the denominators of these Bernoulli numbers since p − 1 > 2n + 2, by the definition of the set S).

p-adic Asai L-functions Attached to Bianchi Cusp Forms

43

6 Constructing Bounded Distributions Finally, we now define our p-adic distribution by the formula μ◦2n−m+2 =

1 · μ2n−m+2 . ( f )∞

These distributions are certainly defined whenever 2n − m + 2 ≥ k + 2 which is the same as m ≤ n − 2, but may possibly be defined for all 0 ≤ m ≤ n, by analytic continuation. We wish to show that μ◦2n−m+2 is a bounded distribution and hence a measure. To this end we recall the notion of abstract Kummer congruences. Theorem 2 (Abstract Kummer congruences). Let Y = Z×p , let O p be the ring of integers of C p and let { f i } be a collection of continuous functions in C(Y, O p ) such that the C p -linear span of { f i } is dense in C(Y, C p ). Let {ai } be a system of elements with ai ∈ O p . Then the existence of an O p -valued measure μ on Y with the property

f i dμ = ai Y

is equivalent to the following congruences: for an arbitrary choice of elements bi ∈ C p almost all zero, and for n ≥ 0, we have 

bi f i (y) ∈ p n O p , for all y ∈ Y =⇒



i

bi ai ∈ p n O p .

i

We apply this theorem with f i the collection of Dirichlet characters χ of (Z/ p j Z)× , for all j ≥ 1, thought of as functions of Y = Z×p , and with aχ ∈ O p the values of μ(χ ), for a given C p -valued distribution μ on Y . To prove that μ is an O p -valued measure on Y , it suffices to prove Kummer congruences of the more specialized form  χ

χ −1 (a)χ (y) ∈ p j−1 O p , for all y ∈ Y =⇒



χ −1 (a)μ(χ ) ∈ p j−1 O p ,

χ

(29) a fixed j ≥ 1, and where the first where χ varies over all characters mod p j , for congruence in (29) follows from the identity χ χ −1 (a)χ = φ( p j )11a+ p j Z p , for 11a+ p j Z p the characteristic function of the coset a + p j Z p ⊂ Z×p . Indeed, then the second congruence in (29) shows that μ is O p -valued on 11a+ p j Z p , whence on all O p -valued step functions on Z p , whence on all O p -valued continuous functions on Y. Claim. The Kummer congruences (29) hold for μ = μ◦2n−m+2 , for m ≤ n − 2 even.

44

B. Balasubramanyam et al.

 In order to prove this claim we must show that the second sum χ χ −1 (a)μ◦2n−m+2 (χ ) in (29) should firstly a) be integral and secondly b) be in p j−1 O p . Now (28) shows that for any even character χ and m even:

χ dμ◦2n−m+2 =

2 p jχ (2n−m+1) G(χ¯ 2 ) G(χ)G(2n − m + 2, χ¯ , f ) 1 ∈ j (2n−2m+2)+c O E , χ jχ ( f )G(χ¯ 2 )∞ κ jχ p

(30) at least if κ is a unit, which we have assumed. For odd characters χ , the integral 1 Op, above vanishes. Thus, (30) shows that the second sum above is in p j (2n−2m+2)+c j with c j = maxχ c jχ . This is still quite far from the integrality claimed in part a). Assuming that part a) holds, one must then further prove the congruence in part b). In any case, assuming the claim, we have that μ◦2n−m+2 is a measure, for 0 ≤ m ≤ n − 2 even. Let x p : Z×p → O p be the usual embedding. We now wish to glue the measures ◦ μ2n−m+2 , for 0 ≤ m even, into one measure μ◦ satisfying (see [1, Lemma 4.4], noting q(V ) there equals 1)

◦

Z×p

χ dμ = (−1)

m/2 Z×p

x mp χ dμ◦2n−m+2 .

(31)

To do this, we again appeal to the abstract Kummer congruences in the theorem above. For the f i , we consider a slightly larger class of functions than the Dirichlet characters χ above, namely, those of the form x −m p · χ , for 0 ≤ m ≤ n, with m even. We set am,χ = (−1)m/2 μ◦2n−m+2 (χ ) ∈ O p , which should be equal to μ◦ (x −m p χ ), by (31) above. We now assume that Claim. The am,χ satisfy the abstract Kummer congruences: 

j−1 bm,χ (x −m O p , for all y ∈ Y =⇒ p χ )(y) ∈ p

m,χ



bm,χ am,χ ∈ p j−1 O p .

m,χ

It would then follow from Theorem 2 that there is a measure μ◦ such that (31) holds. Note that the Kummer congruences in the latter claim actually imply the ones in the former claim for μ◦2n−m+2 , by choosing the bm  ,χ = χ −1 (a) if m  = m, and bm  ,χ = 0 if m  = m. We expect that the proof of these Kummer congruences should be similar to the Kummer congruences proved by Panchishkin in his construction of the p-adic Rankin product L-function attached to two cusp forms f and g, described in detail in [14] (see also [2], and [6] where a sign similar to the one occurring in (31) is corrected). Since μ◦ and μ◦2n+2 agree on a dense set of functions, namely, all χ , the measure ◦ μ is just the measure μ◦2n+2 . We now define the p-adic Asai L-function as the Mellin transform of the measure μ◦ = μ◦2n+2 :

L p (χ ) =

Z×p

χ (a) dμ◦2n+2 ,

for all χ : Z×p → C×p .

p-adic Asai L-functions Attached to Bianchi Cusp Forms

45

Acknowledgements The first author was supported by SERB grants EMR/2016/000840 and MTR/2017/000114. The second and third authors thank T.N. Venkataramana for useful conversations. A version of this paper has existed since about 2016. Recently, Loeffler and Williams [11] have announced a construction of the p-adic Asai L-function attached to a Bianchi cusp form using a method involving Euler systems, and Namikawa [13] has announced a construction over CM fields. We wish to thank the organizers for giving us the opportunity to submit this article to this volume in honor of Prof. M. Manickam.

References 1. Coates, J., Perrin-Riou, B.: On p-adic L-functions attached to motives over Q. In: Algebraic number theory, Advanced Studies in Pure Mathematics, vol. 17, pp. 23–54. Academic Press, Cambridge (1989) 2. Courtieu, M., Panchishkin, A.: Non-Archimedean L-functions and arithmetical Siegel modular forms. Lecture notes in mathematics, vol. 1471, 2nd edn., Springer, Berlin (2004) 3. Diamond, F., Shurman, J.: A first course in modular forms. Graduate texts in mathematics, vol. 228, Springer, New York (2005) 4. Ghate, E.: Critical values of the twisted tensor L-function in the imaginary quadratic case. Duke Math. J. 96(3), 595–638 (1999) 5. Ghate, E., Critical values of twisted tensor L-functions over CM-fields. Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1999), Proceedings of Symposia in Pure Mathematics, vol. 66, Part 1, pp. 87–109. American Mathematical Society, Providence, RI (1996) 6. Ghate, E., Vangala, R.: p-adic Rankin product L-functions. Ramanujan Mathematical Society, Lecture Notes Series, 26, 71–107 (2020) 7. Harder, G.: Period integrals of Eisenstein cohomology classes and special values of some L-functions. Progr. Math. 26, 103–142 (1982) 8. Harder, G.: Eisenstein cohomology of arithmetic groups. The case GL2 . Invent. Math. 89(1), 37–118 (1987) 9. Hida, H.: On the critical values of L-functions of GL2 and GL2 × GL2 . Duke Math. J. 74(2), 431–529 (1994) 10. Lanphier, D., Skogman, H.: Values of twisted tensor L-functions of automorphic forms over imaginary quadratic fields. With an appendix by Hiroyuki Ochiai. Canad. J. Math. 66(5), 1078– 1109 (2014) 11. Loeffler, D., Williams, C.: p-adic Asai L-functions of Bianchi modular forms. Algebra Number Theory 14(7), 1669–1710 (2020) 12. Miyake, T.: Modular forms. Springer, Berlin (1989) 13. Namikawa, K.: A construction of p-adic Asai L-functions for GL2 over CM fields, p. 39 (2019). https://arxiv.org/abs/1912.07251 14. Panchishkin, A.: Non-Archimedean Rankin L-functions and their functional equations. Izv. Akad. Nauk SSSR Ser. Math. 52(2), 336–354 (1988)

Arithmetic Properties of Vector-Valued Siegel Modular Forms Siegfried Böcherer

Abstract Many properties of Siegel modular forms of degree n can be extracted from Siegel Eisenstein series of degree 2n. In this paper, we describe how one can get arithmetic properties of vector-valued modular forms (more precisely: integrality properties of their Fourier coefficients) by combining the doubling method (as originally described by Garrett) with certain holomorphic differential operators studied by Ibukiyama. Keywords Pullback formulas · Integrality properties of Siegel modular forms 2010 Mathematics Subject Classification 11F46 · 11F30 · 11F60

1 Introduction There are many ways to see that modular forms for S L(2, Z) have an integral structure (i.e., one can, for a given weight k, find a basis consisting of forms with integral Fourier coefficients). We mention just three such methods here: (I) All modular forms can be obtained from monomials in the Eisenstein series E 4 and E 6 . (II) One can get all modular forms as linear combinations of theta series attached to positive even unimodular lattices (provided that the weight is divisible by 4). The first method would not generalize to arbitrary degree and the second one does not work for weights congruent 2 mod 4, unless one allows harmonic polynomials as coefficients; this creates trouble if one wants to include the noncusp forms. The third method, which I want to advertise in this article, is to use modular forms constructed from Cohen’s Eisenstein series, defined by S. Böcherer (B) Universität Mannheim Institut für Mathematik, Kunzenhof 4B, 79117 Freiburg, Germany e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_4

47

48

S. Böcherer

Ck−1 (z) =

∞  n=0

⎛ ⎝



⎞ H (k − 1, 4n − t 2 )⎠ e2πinz ;

t 2 4n

the H (k − 1, d) are rational numbers with bounded denominators (essentially values of L(s, χd ) at negative integers); Zagier [31] showed that this Ck−1 generates the space of modular forms over the Hecke algebra, i.e., (III) the set {Ck−1 | T (n) | n ∈ N} generates the full space of modular forms of weight k. Indeed, these Ck−1 | T (n) can be obtained from Fourier–Jacobi coefficients of index n for Siegel–Eisenstein series of degree 2, see [2, 11, 12, 31]. We will see that this construction generalizes (in this way) to arbitrary degree and general vector-valued modular forms.

2 Generalities Let Hn be the Siegel upper half-space of degree n with the usual action of the group Sp(n, R), given by (M, Z ) −→ M < Z >:= (AZ + B)(C Z + D)−1 . For a polynomial representation ρ : G L(n, C) −→ Aut (V ) on a finite-dimensional vector space V = Vρ , we define an action of Sp(n, R) on V -valued functions on Hn by ( f, M) −→ ( f |ρ M)(Z ) = ρ(C Z + D)−1 f (M < Z >). We choose the smallest nonnegative integer k such that ρ = det k ⊗ρ0 where ρ0 is still polynomial and we call this k = k(ρ) the weight of ρ; if ρ itself is scalar-valued, we often write k instead of det k . For a congruence subgroup  of  n := Sp(n, Z), we define Mρn () as the space of Siegel modular forms for ρ w.r.t., i.e., the set of all holomorphic functions F : Hn −→ V satisfying F |ρ M = F for all M ∈ ; in the case n = 1, the usual condition in cusps must be added. The subspace of cusp forms will be denoted by Sρn (). For purposes of congruences, it is convenient to realize V = Vρ as Cm in such a way that ρ(G L(n, Z)) ⊂ G L(m, Z). (1) This is always possible1 ) and we will consider this realization throughout. Indeed it is shown in [17] how one can use bideterminants to realize Vρ and how one can even obtain an explicit linearly independent subset among them with the requested property, see, in particular, the “Basis Theorem” (4.5a) in loc.cit.

1I

thank Y.Hironaka and G.Nebe, who both indicated that the requested property may be found in [17].

Arithmetic Properties of Vector-Valued Siegel Modular Forms

49

It should, however, be mentioned that most of our considerations also work with the weaker assumption that the polynomials defining ρ have rational coefficients (i.e., “ρ is defined over Q”). The Fourier expansion of F is then of type F(Z ) =



a F (T )q T ,

T

where the Fourier coefficients a F (T ) are in Cm and T runs over the set n of all symmetric half-integral matrices of size n, which are positive-semidefinite. We write q T for e2πitrace(T Z ) . It makes sense then to define integral modular forms by integrality of all components of all Fourier coefficients:  a F (T )q T ∈ Mρn | ∀T : a(T ) ∈ Zm }. Mρn (Z) := {F = T

Remark The condition (1) assures that the integrality of a F (T ) depends only on the G L(n, Z)-equivalence class of T . It is natural to ask, for which ρ does Mρn (Z) ⊗ C = Mρn

(2)

hold. For scalar-valued ρ, it is a well-known statement; we can also ask similar questions for any congruence subgroup. The aim of our work is to show that (2) always holds true. We will only consider the case of level one forms of weight k > 2n here, but we emphasize that our method allows us to include small weights; also one can extend everything to congruence subgroups (substituting Z by the ring of integers in a cyclotomic field if necessary). Remark The basic method is to use properties of Siegel–Eisenstein series of degree 2n, in particular, to use the known rationality and integrality properties of their Fourier coefficients together with the pullback formula. This is not really new: In fact, Garrett [15] employed this method to prove algebraicity properties in the scalar-valued case. Our point is that this method can be used for integrality as well, including the vectorvalued case. We emphasize that we deliberately avoid any use of theta series here (the solution of the basis problem using theta series would provide another proof of (2), see, e.g., [10] for groups 0 (N ) with N squarefree or [[24], Kua] in the context of Hilbert–Siegel modular forms), but we want to use methods applicable to arbitrary ). Finally, we mention the work of Takei [28], who proved algebraicity properties for vector-valued Siegel modular forms using the pullback formula for the special case of symmetric tensor representations (following [6]).

50

S. Böcherer

Remark In the case of scalar-valued modular forms, integrality questions have been considered by Shimura [25] and others using different methods.

3 Construction of Integral Modular Forms of Degree n from Eisenstein Series of Degree 2n We start from an (even) weight k > 2n and consider an automorphy factor ρ = ρ0 ⊗ det k : G L(n, C) −→ G L(V ). Following Ibukiyama [18], there exists a V ⊗ V -valued holomorphic differential operator Dρ on H2n , which is a polynomial in the derivatives ∂z∂i j , evaluated on Hn × Hn → H2n with equivariance property Dρ (F |k ι(M, M )) = Dρ (F) |ρz M |w ρ M ,

valid for all C ∞ -functions F on H2n and all M, M ∈ Sp(n, R). Here ι denotes the diagonal embedding of Sp(n) × Sp(n) into Sp(2n), defined by ⎛ ⎞ a0 b0     ⎜0α 0β⎟ ab αβ ⎟ ι( ( ) := ⎜ ⎝c 0 d0⎠ cd γ δ 0γ 0δ and the upper indices z and w indicate that M (M , respectively) have to be applied with respect to the variable z (w, respectively), embedded in H2n as (z, w) →   z 0 . 0w We shall apply Dρ to Siegel–Eisenstein series of degree 2n, defined by E k2n (Z ) :=

 ⎛ ⎝

⎞

∗ ∗ ⎠ 2n 2n ∈ ∞ \ C D

det(C Z + D)−k =



bk2n (T )q T ,

T

2n where ∞ consists of all M ∈  2n with C− block equal to zero (“Siegel parabolic”). The arithmetic nature of the Fourier coefficients bk2n (T ) is well explored, beginning with Siegel [27] and culminating in Katsurada’s explicit formula [20]. In particular, the Fourier coefficients are rational with bounded denominators; this property can be obtained without knowing the Fourier expansion completely explicitly, see [3]. For results concerning congruence subgroups, we refer to Kitaoka [21] and Shimura [26]. To describe the Fourier expansions of Dρ E k2n , we introduce a V ⊗ V -valued polynomial P defined on symmetric matrices T of size 2n by

Dρ (eT )(z, w) = P(T )e2πitrace(T z+Sw) ,

Arithmetic Properties of Vector-Valued Siegel Modular Forms

51



 T ∗ . ∗ S We want to normalize the differential operators Dρ in such a way that the polynomial P has rational coefficients in all its components; this can always be done: Ibukiyama’s differential operators can be chosen to have rational coefficients and then we have to divide by an appropriate power of 2πi. To handle the Fourier expansion of Dρ E k2n , it is convenient to use the standard basis (1  i  m). ai := (0, . . . , 0, 1, 0, . . . 0)t ∈ V = Cm

where eT denotes the function Z → e2πitrace(T ·Z ) on H2n and T =

We may then write the polynomial P as a linear combination P(T ) =



Pi, j (T ) · ai ⊗ a j

i, j

of scalar-valued polynomials Pi j . We consider the Fourier expansion of Dρ E k2n as a function of w:  T, j (z) ⊗ a j e2πitrace(T w) , Dρ E 2n (z, w) = T ∈n

j

where T, j is an element of Mρn . The Fourier expansion of any T, j can be written as T, j (z) =

  S∈n

i

R

 bk2n (

   S R S R )Pi, j ( )ai e2πitrace(Sz) . Rt S Rt T

The summation over R goes over all matrices in 21 Z(n,n) , but due to the condition on positive-semidefiniteness, it is a finite sum. Clearly, the properties of the bk2n (T ) imply T, j ∈ Mρn (Z) , where the prime indicates that bounded denominators (i.e., bounded independent of T ) may occur. Remark We note that the T, j are not cuspidal in general. This is due to the property k(ρ) = k, which we imposed from the beginning on our differential operator. If k(ρ) would be larger than k, we would only get cusp forms (and the pullback formulas considered later on would then be considerably simpler!).

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S. Böcherer

4 Linearized Pullback Formula Garrett [14] started to consider pullbacks of Eisenstein series. Some versions after applying differential operators appear in [4] for the scalar-valued case, and for vectorvalued cases in [9] and [7]. We need a more precise version than in [9] here, because we want to include the noncuspidal vector-valued contributions. We roughly follow Kozima’s exposition [23]:

4.1 On Vector-Valued Klingen–Eisenstein Series Throughout the rest of the paper, we stick to the case of an irreducible polynomial representation ρ : G L(n, C) −→ G L(Vρ ). As is well known, one can characterize such ρ (up to equivalence) by the n-tuple (k1 , k2 , . . . kn ) of its highest weight (with k k2  . . .  kn  0). We freely write (k1 , . . . , kn ) instead of ρ and we use the symbol ∼ to indicate equivalence of representations. As for basic properties of vector-valued modular forms (in particular, Petersson inner products) we refer to old notes of Godement [16]. As is shown by Weissauer [29], the r -fold iteration of the -operator (mapping vector-valued modular forms on Hn to vector-valued forms on Hn−r ) is zero, unless kn = · · · = kn−r +1 .

(3)

Conversely, if this condition is satisfied, we may identify a given cusp form f : Hn−r −→ Vρ with ρ of highest weight (k1 , . . . , kn−r ) with a Vρ -valued function f ∗ on Hn satisfying f ∗ |ρ γ = f ∗ for all γ in the Klingen parabolic Cn,r (Z). The reason for all this is that the invariant space V N inside Vρ under the algebraic group  N := {

 1n−r ∗ } ⊂ G L(n) 0 1r

satisfies VρN ∼ (k1 , . . . , kn−r ) ⊗ (kn−r +1 , . . . , kn ) as a G L(n − r ) × G L(r )-representation, see [29] and [30, chap.7] for details. We may now define the Klingen–Eisenstein series E ρ ,ρ ( f ) :=

 γ ∈Cn,r \n

f ∗ |ρ γ .

Arithmetic Properties of Vector-Valued Siegel Modular Forms

53

More explicit versions of this construction appear in [1] for degree 2 and in [6] for general symmetric tensor representations. These series converge for k(ρ) > n + r + 1 and we put Mρn ,ρ := {E ρ ,ρ ( f ) | f ∈ Sρr }. In the same way as in [22], we get a direct sum decomposition Mρn = ⊕ρ Mρn ρ ,

(4)

where sum goes over all ρ (or rather all r ) satisfying the condition (3), provided that k(ρ) > 2n. It will be useful later to extend the standard Petersson inner product in a rather formal algebraic way from cusp forms to the full space, by viewing (4) as a orthogonal sum and to use on each Mρn ,ρ the Petersson inner product on Sρr . We use this extended inner product, denoted by at some point.

4.2 Pullback Formula The pullback formula of Garrett now becomes, after applying Dρ with ρ irreducible,    z 0 2n Dρ E k ( )= 0w  ρ

f i,ρ

c(ρ ) E ρ ,ρ ( f i,ρ (z) ⊗ E ρ ,ρ (r ( f i,ρ ))(w). < f i,ρ , f i,ρ >

Here c(ρ ) is some nonzero constant, the summation over fi,ρ runs over an orthogonal basis of Sρr with real Fourier coefficients and the map r : Sρr −→ Sρr is defined by f −→



f | T (D)det (D)−k

D

where D runs over all r -rowed integral elementary divisor  and T (D) denotes  matrices D 0 r · r . the Hecke operator defined by the double coset  · 0 D −1 In the formula above, we avoided to write ( f ), defined by ( f )(z) := f (−¯z ) by tacitly choosing (here and in the sequel) a basis of Sρr with real Fourier coefficients. Note that (by [5]) for a Hecke eigenform f ∈ Sρr we get that r ( f ) = L(k − r, f ) · f,

54

S. Böcherer

where L(s, f ) = ξ(s) · L(s, f ). Here L(s, f ) is the Langlands standard L-function attached to f (with Euler factors of degree 2n + 1) and ξ(s) is an explicitly known quotient of shifted Riemannn zeta functions. In any case, all these L-functions are Euler products, convergent for s = k − r and hence different from zero. The pullback formula then implies for all T ∈ n> : T, j (z) =

 ρ

c(ρ )

 L(k − n, f t,ρ ) ( j) a (T ) · E ρ ,ρ ( f t,ρ )(z). < f t,ρ , f t,ρ > ft,ρ f

(5)

t,ρ

Here f t,ρ runs over an orthogonal basis of (real) Hecke eigenforms in Sρr , and denotes the Petersson inner product of Sρr . Moreover, we have decomposed the Fourier coefficients a f (T ) of E ρ ,ρ into their components: a f (T ) =



( j)

a f (T ) · a j .

j

To give a linear version of (5), we consider the linear map  : Mρn −→ Mρn , defined for a Hecke eigenform f in Sρr by E ρ ,ρ ( f ) −→ c(ρ ) · L(k − r, f ) · E ρ ,ρ ( f ). By the nonvanishing of L(k − r, f ) and in view of the Klingen decomposition (4), this defines an automorphism of Mρn for k > 2n. Then (5) implies ( j)

< f, T, j > ≈ c f (T ),

(6)

where T c f (T ) q T denotes the Fourier expansion of ( f ) and ≈ means “equality up to a nonzero constant.” Therefore, if a form f ∈ Mρn is orthogonal to all the T, j , it must be in kernel of the map . From (6), one can see that {T, j | T ∈ n , 1  j  m} generates the full space Sρn as a vector space and we have in this way confirmed (2), at least for the case of large weight.

Theorem For any irreducible polynomial representation ρ realized over Q and satisfying k(ρ) > 2n, we have Mρn (Z) ⊗ C = Mρn ;

Arithmetic Properties of Vector-Valued Siegel Modular Forms

55

more precisely, Mρn is generated by the set {T, j | T ∈ n , 1  j  m}. Comment: In the above, we only covered a special situation: We should include levels, low weights, and half-integral weights. The method above, with some efforts, allows to get similar conclusions in those more general cases. It is, however, important to remark that while an analog of (2) remains true for small weights, the stronger statement that vector-valued forms arise from scalar-valued ones of higher degree by applying Dρ may no longer be true in general !

Remark One can reformulate the results above using Jacobi forms instead of Siegel modular forms of degree 2n: Every vector-valued Siegel modular form ( of large weight, level one) arises as linear combination of D J () where  runs over scalarvalued Jacobi forms on Hn × C(n,n) of varying index and D J is an obvious Jacobi version of Ibukiymama’s differential operator Dρ . Supplement: The property Mρn (Z) ⊗ C = Mρn holds true for arbitrary weights (not only for k(ρ) > 2n).

On a (formal) Fourier series f = T a(T )q T , we define the action of Aut (C)

in the usual way by f σ := T σ (a(T ))q T , where σ acts on the components of a(T ) ∈ Cm simultaneously. We know by the theorem above that this defines an action of Aut (C) on Mρn if k(ρ) is large enough. For f ∈ Mρn with k(ρ) small, we consider, for l large enough the equality ( f · Eln )σ = f σ · Eln . The left side is a modular form, but f σ on the right side makes sense only as a Fourier series at the moment. However, fσ =

( f · Eln )σ Eln

is a meromorphic modular form, independent of l. The set {Eln | l >> 0} has no common zero (see, e.g., [13, Anhang III]) and therefore f σ is actually a holomorphic modular form. By multiplication with Eln we can identify Mρn ⊗detk where k is possibly small, with 0 a subspace W of Mρn ⊗detk+l ; this subspace is Aut (C) invariant; taking into account o n that Mρ⊗det k+l has a basis with rational Fourier coefficients (if l is large enough), we easily see (by linear algebra) that W (and then also Mρn ⊗detk ) has a basis with rational 0 Fourier coefficients. Finally, let { f j } be a basis of Mρn ⊗detk with rational Fourier o

t coefficients. After multiplication with a theta series θ Sn (Z ) := X ∈Zm,n eπitr (X S X Z )

56

S. Böcherer

for an even unimodular positive definite quadratic form for m large, these f j · θ Sn become modular forms with rational Fourier coefficients of bounded denominator. Then the f j have the same property; here, we use that theta series have integral Fourier coefficients and the constant term is equal to 1.

5 An Application to Congruences for Vector-Valued Modular Forms Here we report on our main motivation, which comes from p-adic vector-valued modular forms. In [8], we made some efforts to include vector-valued modular forms. In loc.cit. we left open, whether a vector-valued modular form for a congruence subgroup of type 0n ( p m )) with m  2 gives rise to a p-adic modular form. (Note that the case m = 1 is different and here the standard method also works for the vector-valued case, see [8]). The reason was that in the scalar-valued case the operator f −→ f p | U ( p) plays a central role to decrease the level from 0n ( p m ) to 0n ( p m−1 ) for m  2. In the vector-valued case, a natural substitute for taking a p-th power is to consider a symmetric p-power, which, however, changes the representation ρ to Sym p (ρ) with a much larger representation space. An attempt to proceed along these lines was described in [11] with an annoyingly complicated definition of p-adic vector-valued modular forms. For a geometric approach to vector-valued Siegel modular forms, see [19]. We show now how this trouble can be avoided using the methods from above (taking for granted that appropriate generalizations hold for congruence subgroups and noncuspidal forms):

Theorem For any F ∈ Mρn (0 ( p m )) with m  2 and all N  1 there exists k > k and Fˆ ∈ Mρn (0 ( p m−1 )) with ρ = ρ0 ⊗ det k such that the congruence F ≡ Fˆ

mod p N

holds. All vector-valued Siegel modular forms for 0n ( p m ) are p-adic modular forms with the same representation space V = Vρ . Proof After multiplication with a modular form congruent to 1 mod p N we may assume that the weight of F is large. By the procedure of the previous session, we may assume that there is a scalar-valued G ∈ Ml2n (0 ( p m )) such that F can be expressed by finitely many T, j arising from Dρ G(z, w). We may change G modulo

Arithmetic Properties of Vector-Valued Siegel Modular Forms

57

p N to a (again scalar-valued) Gˆ of level 0 ( p m−1 ) as in [8, Sect. 3]. Then we use ˆ Dρ G(z, w).

The main point is to do the change of level before using the differential operators. Possibly N has to be chosen larger than N because the differential operator Dρ may have powers of p in its denominator and the expression of F as a linear combination of the T, j may be not p-integral.

References 1. Arakawa,T.: Vector-valued Siegel’s modular forms of degree two and the associated Andrianov L-functions. Manuscripta Math. 44, 155–185 (1983) 2. Böcherer,S.: Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen. Math. Z. 183, 21–46 (1983) 3. Böcherer,S.: Über die Fourierentwicklung der Siegelschen Eisensteinreihen. Manuscripta Math. 45, 273–288 (1984) 4. Böcherer,S.: Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen II. Math. Z. 189, 81–110 (1985) 5. Böcherer, S.: Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe. Abh. Math. Sem. Univ. Hamburg 56, 35–47 (1986) 6. Böcherer,S.: Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen II. Math. Z. 189, 81–110 (1985) 7. Böcherer,S.: Pullbacks of Eisenstein series and special values of standard L-functions of a Siegel modular form. Unpublished notes for the Hakuba conference (2012) 8. Böcherer, S., Nagaoka,S.: On p-adic properties of Siegel modular forms. In: B.Heim et al (editors): Automorphic Forms. Springer Proceedings in Mathematics and Statistics, vol. 192. Springer (2014) 9. Böcherer, S., Schulze-Pillot, R.: Siegel modular forms and theta series attached to quaternion algebras II. Nagoya Math. J. 147, 71–106 (1997) 10. Böcherer,S., Katsurada,H., Schulze-Pillot,R.: On the basis problem for Siegel modular forms with levels. In: Edixhoven, B., Moonen, B., Geer, G.v.d. (eds.) Modular Forms on Schiermonnikoog. Birkhäuser (2009) 11. Böcherer,S.: On the notion of vector-valued p-adic modular forms. Private notes (2015) 12. Eichler,M., Zagier,D.: The Theory of Jacobi Forms. Birkhäuser (1981) 13. Freitag, E.: Siegelsche Modulfunktionen. Springer (1983) 14. Garrett, P.B.: Pullbacks of Eisenstein series; applications. In: Automorphic Forms in Several Variables. Birkhäuser (1984) 15. Garrett, P.B.: On the arithmetic of Siegel-Hilbert cusp forms: Petersson inner products and Fourier coefficients. Invent. Math. 107, 453–481 (1992) 16. Godement, R.: Exposes 4-9, Seminaire Cartan 10 (1957/58) 17. Green, J.A.: Polynomial Representations of G L n . Lecture Notes in Mathematics, vol. 830 18. Ibukiyama,T.: On differential operators on automorphic forms and invariant polynomials. Comment. Math. Univ. St. Pauli 48, 103–118 (1999) 19. Ichikawa, T.: Vector-valued p-adic Siegel modular forms. J. Reine Angew. Math. 690, 35–49 (2014) 20. Katsurada, H.: An explicit formula for Siegel series. Am. J. Math. 121, 415–452 (1999) 21. Kitaoka,Y.: Fourier coefficients of certain Eisenstein series. Proc. Jpn .Acad. Ser. A 65(7), 253–255 (1989) 22. Klingen, H.: Introductory Lectures on Siegel Modular Forms. Cambridge University Press

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23. Kozima, N.: Garrett’s pullback formula for vector-valued Siegel modular forms. J. Num. Theory 128, 235–250 (2008) 24. Kuang, J.: On the linear representability of Hilbert-Siegel modular forms. Am. J. Math. 116, 921–994 (1994) 25. Shimura,G.: On Fourier coefficients of modular forms of several variables. Nachr. Akad.d. Wiss.Göttingen. 17, 261–268 (1975) 26. Shimura, G.: On Eisenstein series. Duke Math. J. 50, 417–476 (1983) 27. Siegel, C.L.: Über die Fourierschen Koeffizienten der Eisensteinschen Reihen. Gesammelte Abhandlungen III, No.79. Springer (1966) 28. Takei, Y.: On algebraicity of vector-valued Siegel modular forms. Kodai Math. J. 15, 445–457 (1992) 29. Weissauer, R.: Vektorwertige Siegelsche Modulformen kleinen Gewichts. J. Reine Angew. Mathematik 343, 184–202 (1983) 30. Weissauer, R.: Stabile Modulformen und Eisensteinreihen. Lecture Notes in Mathematics, vol. 1219 (1986) 31. Zagier,D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular Forms in one Variab3le VI. Lecture Notes in Mathematics, vol. 627, pp. 105–169. Springer (1977)

Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms Soumya Das

Abstract We prove an -result for the Fourier coefficients of a half-integral weight cusp form of arbitrary level, nebentypus and weights. In particular, this implies that the analogue of the Ramanujan-Petersson conjecture for such forms is essentially the best possible. As applications, we show similar -results for Fourier coefficients of Siegel cusp forms of any degree and on Hecke congruence subgroups. Keywords Omega results · Fourier coefficients · Half-integral weights · Siegel modular forms

1 Introduction The role of the Ramanujan-Petersson conjecture (proved by P. Deligne [9]) cannot be over emphasised in Number theory. It says that for a Hecke normalised newform g ∈ Sk (N , χ ) (the space of cusp forms of weight k, level N and nebentypus χ ) with Fourier coefficients ag (n), (1) ag (n) ≤ σ0 (n)n (k−1)/2 . In several applications or for its intrinsic interest, one needs to know how sharp is the bound (1). This obviously leads to lower bounds for the sizes of ag (n). We mention here [14], where such sharp bounds play crucial role. Moreover, there are strong conjectures due to Serre [23] which asks for bounds of the form |ag ( p)  p (k−3)/2− , for every prime p for g ∈ Sk , where k ≥ 4 and  > 0. This does not follow from the Sato-Tate conjectures. This is a very hard question and we do not puruse this here, but we would be interested in the related problem of obtaining -type results for S. Das (B) Indian Institute of Science, Bengaluru 560012, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_5

59

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ag (n), i.e., we look for sharp lower bounds along certain subsequences of integers. Roughly speaking, these results give a measure of sharpness of bounds like (1). The answer to the question of sharpness of the Ramanujan-Petersson bound is that (1) is the best possible (see Remark 3.2) in terms of the exponent (k − 1)/2, so the question boils down to understanding the behaviour of the function ag (m)/m (k−1)/2 . One knows the following -type results about this function. In [21] Rankin proved, essentially exploiting the prime number theorem for a Hecke eigenform g ∈ Sk that it is not bounded: ag (m) (2) lim sup (k−1)/2 = ∞. m→∞ m Even a stronger result is known due to Ram Murty (cf. [17], using the holomorphy of suitable symmetric power L-functions): for some α > 0,   ag (m) α log m ) . =  exp( m (k−1)/2 log log m

(3)

Here and in the rest of the paper, for arithmetical functions f (n), g(n) with g(n) > 0 for all n ≥ 1, we use the notation f (n) = (g(n))

if and only if

lim sup n→∞

| f (n)| > 0. g(n)

(4)

In more simpler terms, this just means that | f (n)|/g(n) is bounded away from zero along a subsequence of the set of natural numbers N. The situation is not so clear for the spaces of half-integral weight modular forms. The analogue of (1) for f ∈ Sk+1/2 (4N , χ ), namely, a f (n)  f, n k/2−1/4+ ,

(5)

is open, and recently in the work of Gun-Kohnen [12] -results of the type (2), (3) +,new (4N ), where N is odd and square-free, + were obtained for newforms f in Sk+1/2 denotes Kohnen’s plus space, and new denotes the space spanned by the newforms therein. More precisely one of their main results state   a f (n) =  n k/2−1/4 exp(β(log n)δ ) ,

(6)

for some constant β > 0 and 0 < δ < 1/2. For this, the authors used the (Hecke)+,new new (4N ) to S2k (2N ) via the Shimura maps. They also extend isomorphism from Sk+1/2 the -results to arbitrary forms in the newspace by arguments from linear algebra. The aim of this paper is to improve upon all the results in [12] by generalising them to the full space Sk+1/2 (4N , χ ) with arbitrary nebentypus and level thereby obtaining sharper lower bounds. This would be achieved by using the ‘linear’ properties of the Shimura correspondence rather than the ‘multiplicative’ one used in [12]. We prove the following.

Omega Results for Fourier Coefficients of Half-Integral …

61

Let the orthogonal complement (with respect to the Petersson inner product) of the ⊥ (4N , χ ). space of unary theta series of weight 3/2 in S3/2 (4N , χ ) be denoted by S3/2 Theorem 1.1 Let k ≥ 1 and f ∈ Sk+1/2 (4N , χ ) be non-zero. Moreover, if k = 3/2 ⊥ (4N , χ ). Then assume that f belongs to S3/2  a f (n) =  n

k/2−1/4

 α log n ) , exp( log log 2n

(7)

for some constant α > 0. There should be a (suitably modified) version of this theorem for the weight 1/2 as well, using the explicit set of basis consisting of unary theta series as shown by Serre-Stark. We have not checked this. Trodding to the space of Siegel cusp forms, one can ask for questions similar g to the above. Let Sk (N , χ ) denote the space of Siegel cusp forms of degree g and g weight k > 0 on the group 0 (N ) with nebentypus χ (see Sect. 2 more details). g When N = 1, we drop it from the notation, and simply write Sk . The analogue of Ramanujan-Petersson bound states that for any ε > 0 and S > 0, the Fourier coefficients a F (S) of F (see Sect. 2) as above satisfy a(F, S)  F,ε det(S)k/2−(g+1)/4+ε .

(8)

This was proposed by Resnikoff-Saldaña [22] and is still open, even though there are some results breaking the Hecke bound (see [6]). One knows that the  cannot be dropped, see remark 3.2. Moreover, it was shown in [7] that (8) is true for the radial Fourier coefficients a F (m S0 ) (S0 fixed with disc.(S0 ) fundamental, F ∈ Sk2 not in the space of Saito-Kurokawa lifts) and that a lower bound of the form a F ( pS0 )  F det( pS0 )k/2−3/4 holds for infinitely many primes p. Since Theorem 1.1 is valid for the entire space Sk+1/2 (4N , χ ), it is possible to extend this result to the space of Siegel cusp forms (of any degree g ≥ 1) by using a suitable Fourier-Jacobi expansion. We prove -results for a F (T ) (T ∈ + g ) with a robust lower bound. g

Theorem 1.2 Let F ∈ Sk (N , χ ). Assume that k − (g − 1)/2 is at least 5/2 if g is even and at least 1 if g is odd. Then as det(T ) → ∞,   g+1 c log det(T ) k ) . a F (T ) =  det(T ) 2 − 4 exp( log log det(T )

(9)

+ We mention here that in case F is the Saito-Kurokawa lift of some h ∈ Sk−1/2 (4) (the Kohnen’s plus space of level 4), then one may obtain Theorem 1.2 immediately from the relation (cf. [10, eqn. (6), § 6], after applying Möbius inversion to the formula)

ah (det(2T )) =

 a|c(T )

μ(a)a k−1 a F (T /a),

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and using  1.1. Here c(T ) denotes the content of T (i.e., c(T ) = gcd.(n, r, m)  Theorem n r/2 if T = r/2 m ) and μ(·) the Möbius function. We can actually prove a stronger version of Theorem 1.2 when the degree g is odd. Namely, in this case, we can assure that the set in which T (as in Theorem 1.2) varies is contained in a certain subset of Sg of + g defined as Sg := {T ∈ + g |dT is odd and square-free},

(10)

where in accordance with [4, cf. eqn(1.1)], we have put dT to be the ‘absolute discriminant’ defined as (11) dT := 21 det(2T ). Note that dT ∈ N since g is odd. In particular, this implies that (−1)(n−1)/2 dT is a fundamental discriminant. We call these T accordingly as ‘odd, square-free’ or ‘fundamental’. It is well known (cf. [4]) that these Fourier coefficients play crucial roles in representation theory of automorphic forms. Among many such, let us note here that certain averages of these kind of Fourier coefficients are related to central values of (twists of) spinor L-functions when g = 2 (cf. [2, 11]). Thus, our result below may be useful in the analytic theory of such central L-values. g

Theorem 1.3 Let g be odd and F ∈ Sk be non-zero. Assume that k − (g − 1)/2 is at least 1. Then   g+1 k c log dT 2− 4 exp( ) (T ∈ Sg ). (12) a F (T ) =  dT log log dT Proof of Theorem 1.3 makes crucial use of the existence of ‘primitive’ theta components in the theta expansion of certain Jacobi forms from [4], see Sect. 4.2. It is, of course, desirable to have Theorem 1.3 for even g; however, even when g = 2, one invariably requires, via Waldsuprger’s formula, very sharp lower bounds for the central values L( f ⊗ χd , 1/2) ( f is a newform of integral weight and χd the quadratic character attached to the fundamental discriminant d). For individual f as above, such results are perhaps not known. Finally, let us mention that some version of Theorems 1.2 and 1.3 should hold for vector-valued Siegel modular forms as well.

2 Notation and Preliminaries For f ∈ Sk (N , χ ), we write its Fourier expansion as f (τ ) =

∞  n=1

a  ( f, n)n

k−1 2

e(nτ ),

Omega Results for Fourier Coefficients of Half-Integral …

63

so that by Deligne [9], we have the estimate for any ε > 0: |a  ( f, n)| ε, f n ε .

(13)

For a positive integer n with (n, N ) = 1, the Hecke operator Tn on Sk (N , χ ) is defined by d−1     k −1 2 (14) χ (a) f | a0 db . Tn f = n ad=n a>0

b=0

For k > 0 let Sk+1/2 (4N , χ) be the space of holomorphic functions f : H → C such that for all τ ∈ H, M = ac db ∈ 0 (4N ) and Dirichlet character χ mod 4N , f |k+1/2 M := jc,d (τ )−(2k+1) f ((aτ + b)(cτ + d)−1 )) = χ (d) f (τ ), where jc,d (τ ) = θ (Mτ )/θ (τ ) and θ (τ ) = n∈Z e(n 2 τ ). Here and henceforth, e(z) := exp(2πi z). Let T = Tk+1/2 (4N ) be the algebra generated by the Hecke operators T (n 2 ) (for (n, 4N ) = 1)) and U ( p 2 ) (for p | 4N ) acting on Sk+1/2 (4N , χ ). Recall that T (m) = 0 unless m is a square (cf. [24]). We would recall the Fourier expansion of T ( p 2 ) later. Let us note that when (n, 4N ) = 1), χ (n)T (n 2 ) is Hermitian and moreover T is commutative. In the sequel, we would encounter Siegel modular forms, which we review briefly. Let Hg := {Z ∈ Mg (C) | Z = Z t , (Z ) > 0} be the Siegel’s upper half-space of degree g. The symplectic group Sp(g, R) acts on Hg by Z → γ Z  = (AZ + B)(C Z + D)−1 ; and on functions F on Hg by ( f |k γ )(Z ) := det(C Z + D)−k F(γ Z ). A Siegel modular form of degree g holomorphic function F on Hg satisfying F |k γ = F for all γ ∈ Sp(g, Z) with the standard additional condition in degree 1. We g denote by Mk the vector space of all such functions and by Skn the subspace of cusp forms. An element F ∈ Mkn has a Fourier expansion with Fourier coefficients a F (S) given by  F(Z) = a F (S)e(SZ) (a F (S) ∈ C). S∈n

Here g denotes the set of all positive, half-integral g × g symmetric matrices, i.e., g = {T = T  ∈ Mg×g (Z)|Tii ∈ Z, Ti j ∈ 21 Z; T ≥ 0}.

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Further, e(SZ) := exp(2πi trace(SZ)). If F is cuspidal, then this summation is supported on T > 0 which we denote by + g. Throughout this paper, we use a decomposition for Z ∈ Hn into blocks as follows: 

τ z Z= t z Z



(z ∈ C(1,g−1) , Z ∈ Hg−1 ).

(15)

g

Clearly, every F ∈ Sk has a Fourier-Jacobi expansion with respect to the decomposition above:  φT (τ, z)e(T Z ). F(Z) = T ∈g−1

The φT are then ‘Jacobi forms’ of weight k and index T on the Jacobi group SL(2, Z)  Z1,g−1 . We refer the reader to [26] for more details. We only record here the theta expansion of such a φT which would be needed later. Namely, one has (cf. [26])  h μ (τ ) · T [μ](τ, z) φT (τ, z) = μ

with summation over μ ∈ Zg−1 /2T · Zg−1 and T [μ](τ, z) =



˜ +2zT (R+μ) ˜ e2πi(T [R+μ]τ .

R∈Zg−1

Here we use μ˜ := (2T )−1 · μ. We note here that the h μ are then modular forms of on some congruence subgroup; the Fourier expansion of h μ is of weight k − g−1 2 shape  −1 aμ (n − T −1 [μ/2])e2πi(n−T [μ/2])·τ (16) h μ (τ ) = n

and its Fourier coefficients are given by aμ (n − T −1 [μ/2]) = a F (



n μ/2 μ/2t T

 ).

(17)

3 Proof of Theorem 1.1 Let T be the Hecke algebra generated by the Hecke operators T (n 2 ) with (n, 4N ) = 1. Then it is well known that Sk+1/2 (4N , χ ) has a basis consisting of simultaneous eigenforms under T . Moreover, from standard linear algebra any subspace S of Sk+1/2 (4N , χ ) which is invariant under T can be diagonalised, since T is a commuting family of Hermitian operators when multiplied by certain non-zero scalars.

Omega Results for Fourier Coefficients of Half-Integral …

65

Therefore, letting T f := CT (n 2 ) f, n ≥ 1 (i.e., the Hecke module generated by f ), we see that T acts on T f and by the last line of the above paragraph, T f has a basis consisting of eigenforms for T . Fix one such non-zero eigenform among the basis elements (note that T f = 0 if f = 0); call it f 1 . Note that even when k = 1, ⊥ (4N , χ ). f 1 ∈ Sk+1/2 Then by definition, we have f1 =

r 

ci T (n i2 ) f,

(18)

i=1

for some c j ∈ C. Using the fact that the full Hecke algebra T (restricted to Sk+1/2 (4N , χ )) is generated as a module over Z[ζφ(N ) ] by T ( j 2 ), j = 1, 2, . . . , μ with μ depending only on k, N (see [20]), we can rewrite (18) as f1 =

μ 

a j T ( j 2 ) f.

(19)

j=1

If we assume that Theorem 1.1 is not true for f = 0, then this would imply that the Fourier coefficients of f satisfy the bound a f (n)  f n k/2−1/4 exp(

 log n ), log log 2n

(n ≥ 1)

(20)

for all  > 0. Then from the description of the Fourier coefficients of T p2 f ( p  4N ), namely,   n p k−1 a f (n) + χ 2 ( p) p k−1 a f (n/ p 2 ) aT ( p2 ) f (n) = a f ( p n) + ψ( p) p 2

(21)

it follows that the same bound (possibly with different implied constants) hold for a f1 (n) for all n ≥ 1. We note here that writing each j into product of prime powers and using (21) repeatedly, its enough to assume that each j in (19) is a prime. Now let us choose a square-free t (possibly t = 1, but see Remark 3.4) such that t-th Shimura lift g := Sh t ( f 1 ) of weight 2k is non-zero. This is possible to do, as otherwise f 1 would be zero (cf. [20, Lemma 3.2]) and because of our assumption on f (and hence on f 1 ) when the weight is 3/2 (so that the Shimura lift is still a cusp form). We refer the reader to [24] for more details on Shimura maps. We only note here that g ∈ S2k (2N , χ 2 ) and that the maps Sh t are linear. From the above and looking at the definition of the Shimura map, we have the following expression for the Fourier coefficients ag (n): ag (n) =

 d|n

ψt (d)d

k−1

n2 a f1 (t 2 ), d

  −1 k t ) ( ) , ψt (m) = χ (m)( m m

(22)

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and its clear that the following bound holds (keeping in mind that t depends only on f ): ag (n) 

 d|n

n 2 log n d k−1 ( )k−1/2 exp( ) d log log 4n

 n k−1/2 σ−1/2 (n) exp(

2 log n ), log log 4n

(23)

(t depends only on f and not on n). The Shimura maps Sh t are also Hecke equivariant (see, e.g., [20]): for any prime p  4N ,  ≥ 2, h ∈ Sk+1/2 (4N , χ ) and square-free t ≥ 1,   Sh t (h), Sh t (T ( p 2 )h) = T p Sh t (h), Sh t (T ( p 2 )h) = Tp − χ ( p 2 ) p 2k−2 T−2 p (24) where Tn is the n-th Hecke operator acting on S2k (2N , χ 2 ). Also since the Hecke algebras in question are commutative and the Hecke operators are multiplicative, it follows from (24) that g is an eigenform under all Tn ((n, 2N ) = 1). We would be done if we can prove that the bound (23) cannot hold for n large enough. From newform theory, there exists a divisor M of 2N such that conductor (χ 2 ) | M and a newform G ∈ S2k (M, χ 2 ) such that for all n with (n, 2N ) = 1, aG (n) = ag (n).

(25)

For 2 > B > 1 fixed and x ≥ 2, put S B (x) := { p ≤ x|λG ( p) > B}. We now appeal to the Sato-Tate theorem (see [16], [8, Remark 3.2 (ii)]) to get for x large enough (26) #S B (x) ≥ C · π(x), for a certain C > 0 (depending only on G and B) computed by using the Sato-Tate measure.

Now arguing in the usual manner, i.e., taking n x = p∈S B (x) p and x large enough ag (n x ) = aG (n x )  n (k−1)/2 B C·π(x) x A log n x  n (k−1)/2 exp( ), x log log n x

(27)

since log n x  x. Here A > 0 is an absolute constant. Comparing (23) and (27) and letting x → ∞, we clearly arrive at a contradiction, since

Omega Results for Fourier Coefficients of Half-Integral …

σ−1/2 (n x )  exp(

67

√ A log n x ) log log n x

for some A > 0. This formula is classical, but it seems not easy to find a reference, we quote here [18, Lemma 3.1] for a proof. This completes the proof of Theorem 1.1. Remark 3.1 It is rather clear from the above proof that the ± results with same quality as in Theorem 1.1, with obvious modifications, hold for f ∈ Sk (N ) with real Fourier coefficients. We leave this to the reader.  Remark 3.2 We remark here that the exponent k/2 − 1/4 in (20) is optimal, as can be seen by comparing the pole of the Rankin-Selberg L-function (e.g., see [3, Remark 5.3]) attached to the modular form and the enhanced region of convergence obtained if the -result is violated. This applies to bound in (1) and Siegel modular forms as well.  Remark 3.3 It is perhaps possible to reduce the question about -result for an f ∈ Sk+1/2 (4N , χ ) in the case χ quadratic and N odd, square-free to the setting of [12] by using the injective map ℘χ defined in [25, Def. 2] and using [25, Prop. 6]. However, we then get level 8N instead of 4N that is considered in [12].  Remark 3.4 We used the fact that for f ∈ Sk+1/2 (4N , χ ), the t-th Shimura lift is non-zero for some t either square-free or t = 1. If f is not in the space spanned by unary theta series (so that k + 1/2 ∈ / {1/2, 3/2}) one can show that there exists t > 1 and square-free such that Sh t ( f ) = 0. Otherwise, it follows from the definition of Sh t that the Fourier expansion of f is supported only on squares, which implies that #{n ≤ x|a f (n) = 0}  x 1/2 . However, by a theorem of Ono [19], the above inequality shows that the form is actually ‘super-lacunary’, and must be in the space spanned by theta series of weights 1/2, 3/2. This is a contradiction, as k ≥ 2. 

4 Application to Siegel Modular Forms -type results are not so well known in the context of Siegel cusp forms of degree ≥ 2, let us mention here [8, 13], where such results were obtained for Hecke eigenvalues for Saito-Kurokawa lifts. In Sect. 4.1, we show how one can use the results of the previous section to obtain -results for Fourier coefficients of Siegel cusp forms by considering suitable Fourier-Jacobi coefficients. We also show in Sect. 4.2 that one can obtain better results by considering only ‘fundamental’ Fourier coefficients (see below for the definition) when the degree is odd. The statements of the results proved in this section can be found as stated in the Introduction.

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4.1 Proof of Theorem 1.2 The idea is to use a suitable Fourier-Jacobi expansion of F, and reduce the problem to elliptic modular forms by passing through some (non-zero) Fourier-Jacobi coefficient. We look at the Fourier-Jacobi expansion F(Z ) =



φT (τ, z)e(T · τ  ).

(28)

T ∈+ g−1

Let us choose T ∈ g−1 such that φT = 0. Among the theta components of φT , let us choose one, say, h := h μ ∈ Sk−(g−1)/2 ( (4D N )) (D = det(2T )) which is non-zero (cf. [4, Sect. 4]). Let the Fourier expansion of h be given by (cf. (17), (16)) h(τ ) =



 −1  μ/2 )q −T [μ/2] . t μ /2 T

 aF (

>T −1 [μ/2]

(29)

Put H := h(Dτ ) ∈ Sk−(g−1)/2 ( 1 (16D 2 N 2 )).

(30)

For convenience, let us also set M = 16D 2 N 2 and k  = k − (g − 1)/2. Then the argument presented for the proof of Theorem 1.1 works with slight modification. Namely, we note that the Hecke module TH is left invariant by the operators T (m 2 ), (m, M) = 1 and the diamond operators d, ((d, M) = 1). Then as before, we get hold of some 0 = H1 ∈ Sk  (M, χ ) (for some χ mod M since we have included the diamond operators) which is an eigenfunction of T (m 2 ), (m, M) = 1. From now on, following the proof verbatim the same as that of Theorem 1.1 if k  ∈ 21 Z (note that since k  ≥ 5/2, the hypotheses of Theorem 1.1 are satisfied) and [17, Theorem 2] when k  ∈ Z (where the result obviously generalises to any level) shows that there exist infinitely many u ≥ 1 such that for some c > 0, 

a H (u)  u (k −1)/2 exp(

c log u ). log log 2u

(31)

We let T run through the matrices (recall D = det(2T )) 

 u/D + T −1 [μ/2] μ/2 ; (u ≥ 1, u ≡ −D · T −1 [μ/2] mod D). T μt /2

(32)

From the well-known Jacobi identity  det(

  μ/2 ) = ( − T −1 [μ/2]) det(T ), μt /2 T

(33)

Omega Results for Fourier Coefficients of Half-Integral …

69

we see that the determinant of a generic matrix in (32) is up to a constant, u. Now referring to (29), (30) and (31), we therefore finish the proof (possibly with some constant different than that in (31)) by noting that (k  − 1)/2 = k/2 − (g + 1)/4.  Remark 4.1 If we consider non-cuspidal Siegel modular forms of degree 2, then a result similar in spirit to that of Theorem 1.3 is contained in the proof of [5, Prop. 7.7].

4.2 Proof of Theorem 1.3 The proof follows the same lines as that of Theorem 1.2, and the room for improvement comes from the fact that one can choose, in the proof of Theorem 1.2, the index T of the Jacobi form φT (= 0) in Sg−1 (cf. [4, Proposition 3.8]). However, there are certain subtleties over the proof of Theorem 1.2, and we prefer to give all the details. From [4], let us recall that this leads one, via the theta decomposition of φT and considering the theta components, to an elliptic cusp form (of weight k  = k − (g − 1)/2 ≥ 1) h = h μ ∈ Sk  ( (d))

(d = det(2T ), μ ∈ (2T )−1 Zg−1 /Zg−1 ),

(34)

note that d ∈ N since g is odd) of integral weight which is referred to a ‘primitive ’ theta component in [4]. Primitive theta components have an important property that their Fourier coefficients only survive away from the level; this means for our h that we can rewrite its Fourier expansion in the form (cf. (29), [4, Sect. 3.4.1]) h(τ ) =



ah (α)q α/d ,

(35)

α≥1, (α,d)=1

where we have put (cf. (16), (17)) ah (α) := ah ( − T −1 [μ]) = a F (



  μ/2 ) μt /2 T

(36)

keeping in mind the relation (by primitivity of T )  − T −1 [μ/2] =:

α d

((α, d) = 1).

(37)

We would henceforth work with this h, or rather with H ∈ Sk  ( 1 (D)) (D = d 2 ) defined by f (τ ) = h(dτ ) (38) and try to reduce the situation to that of newforms, where we can use the Sato-Tate machinery, see (43).

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The following setup can also be found in [4, Appendix], we mention some of the details here to maintain continuity. Since F D1 = ∪χ F D,χ where χ runs over Dirichlet characters mod D such that the conductor dχ satisfies dχ |D, we can write f (τ ) =

 χ

cχ, j,δ f j (δτ ),

(39)

δ

j

where χ runs over Dirichlet characters mod D, f j runs over ∪χ F D,χ for all χ that m χ | M, δ runs over the divisors of D (such that δ · (level of f j )|D), and cχ, j,δ are scalars. By classical (new)old-form theory it follows that f cannot entirely lie in the old space (see [15, Theorem 4.6.8]), and hence must have a non-zero new component. In terms of (39), this means that there exists an χ  , j  such that cχ  , j  ,1 = 0. For ease of notation, we can assume without loss that j  = 1 and omit the χ  from display. We would now move on to the setup of another argument given in [1] which involves sieving the entire old classes corresponding to all f j with j = 1 by means of Hecke operators T p with p  D (for 1 (D)) and using multiplicity one. We do not repeat this formalism again here, but refer the reader, along with the outline above, to [1, p. 332, proof of Thm. 6]. For example, to remove the old class corresponding to b2 , we apply the operator T p − b2 ( p)I d. to f , where p  D is chosen so that b1 ( p) = b2 ( p). One can clearly repeat this process enough times, as claimed above, to remove all classes except that corresponding to j = 1. After removing the old classes as mentioned above, we arrive at an F ∈ Sk  (D, χ  ) such that ∞   F(τ ) = A(n)q n := c1,δ f 1 (δτ ). (40) δ|N

n=1



Let f j (τ ) = n b j (n)q n be its Fourier expansion. By the above, we get finitely many algebraic numbers βt and positive square-free (including 1) rational numbers γt such that for every n A(n) =



c1,δ b1 (n/δ) =

δ|N



βt a f (γt n), (t  #F D1 ).

(41)

t

Clearly, there are finitely many primes required for the above procedure, let us call them {qm }m . At each step, we choose primes bigger than D. For the m-th step, the smallest odd prime qm > qm−1 such that b1 (qm ) = bm (qm ) for m = 1 would work, cf. [1, proof of Thm. 6].

In (41), let us focus on those n ≥ 1 which are co-prime to D · m qm where {qm }m were the finitely many primes used to sieve out the old classes above. Then (recalling j  = 1 by convention) for these n, c1,1 b1 (n) =

 t

βt a f (γt n).

(42)

Omega Results for Fourier Coefficients of Half-Integral …

71

Note that c1,1 = 0. Thus, taking n to be any square-free integer with (n, D ·

q m m ) = 1, we see that c1,1 b1 (n) =



βt a f (κt n), (κt ∈ N, κ(t) square-free).

(43)

t

Suppose now the assertion of Theorem 1.3 is false. As in the proof of Theorem 1.1, this implies that for all  > 0, one has the bound (after possibly tweaking the constants involved) k  log dT − g+1 ) (dT ∈ Sg )and (44) a F (T )  dT2 4 exp( log log dT via the Fourier expansion (29) and (35), this implies that for all odd, square-free α with (α, D) = 1 g+1  log α k ah (α)  α 2 − 4 exp( ), (45) log log α where we have used the crucial relation  (37) and that given α ∈ N with (α, d) = 1,   μ/2 ∈ Sg such that dT = α. there exists unique T = μt /2 T From (45) and (38), it follows that the same bound as in (45) holds for f as well. Now referring

to (43), we see that b1 (n)

also satisfies the bound in (45). However, taking nx = ℘∈S B (x) ℘ with (℘, D · m qm ) = 1 as in the proof of Theorem 1.1 gives us the bound (27) for b1 (nx ) which is incompatible with that in (45), since  > 0 can be chosen arbitrary small. This finishes the proof. Acknowledgements The author thanks the referee for useful comments. He also thanks IISc. Bangalore, DST (India) and UGC centre for advanced studies for financial support. During the preparation of this work the author was supported by a MATRICS grant MTR/2017/000496 from DST-SERB, India, and during the final revision of the paper by a Humboldt fellowship from Alexander von Humboldt Foundation.

References 1. Anamby, P., Das, S.: Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients. Publ. Math. 63(1), 307–341 (2019) 2. Böcherer, S.: Bemerkungen über die Dirichletreihen von Koecher und Maaß, Math. Gottingensis 68 3. Böcherer, S., Das, S.: Characterization of Siegel cusp forms by the growth of their Fourier coefficients. Math. Ann. 359(1–2), 169–188 (2014) 4. Böcherer, S., Das, S.: On fundamental Fourier coefficients of Siegel modular forms. Preprint https://arxiv.org/abs/1810.00762 5. Böcherer, S., Das, S.: Petersson norms of not necessarily cuspidal Jacobi modular forms and applications. Adv. Math. 336, 335–376 (2018) 6. Böcherer, S., Kohnen, W.: Estimates for Fourier coefficients of Siegel cusp forms. Math. Ann. 297(3), 499–517 (1993)

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7. Das, S., Kohnen, W.: Some remarks on the Resnikoff-Saldaña conjecture. The legacy of Srinivasa Ramanujan, Ramanujan Mathematical Society Lecture Notes Series, vol. 20, pp. 153–161. Ramanujan Mathematical Society, Mysore (2013) 8. Das, S., Sengupta, J.: An Omega-result for Saito-Kurokawa lifts. Proc. Am. Math. Soc. 142(3), 761–764 (2014) 9. Deligne, P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. No. 43, 273–307 (1974) 10. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser, Boston-Basel-Stuttgart (1985) 11. Furusawa, M., Morimoto, K.: On special Bessel periods and the Gross-Prasad conjecture for S O(2n + 1) × S O(2). Math. Ann. 368(1–2), 561–586 (2017) 12. Gun, S., Kohnen, W.: On the Ramanujan-Petersson conjecture for modular forms of halfintegral weight. Forum Math. 31(3), 703–711 (2019) 13. Gun, S., Paul, B., Sengupta, J.: On Hecke eigenvalues of Siegel modular forms in the Maass space. Forum Math. 30(3), 775–783 (2018) 14. Kohnen, W.: A short note on Fourier coefficients of half-integral weight modular forms. Int. J. Number Theory 6(6), 1255–1259 (2010) 15. Miyake, T.: Modular forms. Translated from the 1976 Japanese original by Yoshitaka Maeda. Reprint of the first 1989 English edition. Springer Monographs in Mathematics. Springer, Berlin (2006). x+335 pp 16. Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R.: A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011) 17. Ram Murty, M.: Oscillations of Fourier coefficients of modular forms. Math. Ann. 262(4), 431–446 (1983) 18. Norton, K.K.: Upper bounds for sums of powers of divisor function. J. Num. Theory 40, 60–85 (1992) 19. Ono, K.: Gordon’s - conjecture on the lacunarity of modular forms. C. R. Math. Acad. Sci. Soc. R. Can. 20(4), 103–107 (1998) 20. Purkait, S.: Hecke operators in half-integral weight. J. Théor. Nombres Bordeaux 26(1), 233– 251 (2014) 21. Rankin, R.A.: An - result for the coefficients of cusp forms. Math. Ann. 203, 239–250 (1973) 22. Resnikoff, H.L., Saldaña, R.L.: Some properties of Fourier coefficients of Eisenstein series of degree 2. J. Reine angew Math. 265, 90–109 (1974) 23. Serre, J.P.: Divisibilité de certaines fonctions arithmétiques. L’ Ens. Math. 22, 227–260 (1976) 24. Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97, 440–481 (1973) 25. Ueda, S.: Yamana: On newforms for Kohnen plus spaces. Math. Z. 264(1), 1–13 (2010) 26. Ziegler, C.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59, 191–224 (1989)

On Hecke Theory for Hermitian Modular Forms Adrian Hauffe-Waschbüsch and Aloys Krieg

Abstract In this paper, we outline the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary-quadratic number field. The Hecke algebra turns out to be commutative. Its inert part has a structure analogous to the case of the Siegel modular group and coincides with the tensor product of its p-components for inert primes p. This leads to a characterization of the associated Siegel-Eisenstein series. The proof also involves Hecke theory for particular congruence subgroups. Keywords Hecke algebra · Hermitian modular group · Cusp forms · Siegel-Eisenstein series 2010 Mathematics Subject Classification 11F55 · 11F60

1 Introduction The Hermitian modular group associated with an imaginary-quadratic number field K was introduced by Braun [1, 2] as an analog of the Siegel modular group. The case of class number >1 leads to number theoretical complications. If one wants to consider the Hecke theory as, for instance, by Freitag [8], there are only a few concrete results (cf. [5, 11]). Most authors consider the situation over local fields (cf. [16]). In this paper, we show that each double coset contains a matrix in block diagonal form. Hence, the Hecke algebra is commutative. Moreover, we characterize Dedicated to Murugesan Manickam on the occasion of his 60th birthday. A. Hauffe-Waschbüsch · A. Krieg (B) Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany e-mail: [email protected] A. Hauffe-Waschbüsch e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_6

73

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a particular subalgebra of the Hecke algebra, which is related to inert primes. As a consequence, we obtain a characterization of the Siegel-Eisenstein series, which was available up to now only in the case of class number 1 (cf. [13]). Many of our results are similar to the investigations by Manickam [14] on Jacobi forms.

2 The Hecke Algebra for the Hermitian Modular Group Throughout the paper let √ K = Q( −m) ⊂ C, m ∈ N squarefree be an imaginary-quadratic number field. Its discriminant and ring of integers are dK =

 −m −4m

and O K = Z + ZωK =

 √ Z + Z(1 + −m)/2 if m ≡ 3 (mod4), √ if m ≡ 1, 2 (mod4). Z + Z −m

Denote its class number by h K and the associated primitive real Dirichlet character mod |dK | by χK . Define the set of integral unitary similitudes of factor q ∈ N by 

tr

Δn (q) := {M ∈ O 2n×2n ; J [M] := M J M = q J }, J = K

Moreover, let Δn,q :=

∞ 

Δn (q  ), Δn =

=0





⎛

1

0

⎞

0 −I ⎜ ⎟ , I = ⎝ ... ⎠ . I 0 0 1

Δn (q).

q∈N

Γn := Δn (1) ⊆ U (n, n; C) := {M ∈ C2n×2n ; J [M] = J } denotes the Hermitian modular group of degree n. Given q ∈ N let Γn [q] = {M ∈ Γn ; M ≡ I (modq)} stand for the principal congruence subgroup of level q. We will always assume a block decomposition 

 A B M= ∈ Δn , A, B, C, D ∈ O n×n K . C D Lemma 2.1 Given M ∈ Δn (q) then

On Hecke Theory for Hermitian Modular Forms

75

(Γn \Γn MΓn ) < ∞. Proof Use M −1 Γn M ∩ Γn ⊇ Γn [q], hence

2 (Γn \Γn MΓn ) = Γn : Γn ∩ M −1 Γn M]  [Γn : Γn [q]  q 8n < ∞.  Hence (Γn , Δn ) fulfills the Hecke condition (cf. [8, 12]). Let ∂k (G) ⊆ O K stand for the ideal generated by all k × k subdeterminants of an integral matrix G, which is invariant under multiplication with unimodular matrices. Then [1], Theorem 1, resp. [2], Lemma 1, implies Lemma 2.2 If M ∈ Δn there exist L ∗ , L  ∈ Γn such that 

A∗ 0   A M L = C

L∗ M =

 B∗ , D∗  0 , D

  A , C

OK

det A∗ = ∂n

OK

det A = ∂n (A, B).

The next step is a block diagonal decomposition in double cosets. Lemma 2.3 Given M ∈ Δn there exist L 1 , L 2 ∈ Γn such that  L1 M L2 =

A∗ 0 0 A∗ H

 for some H = H

tr

∈ O n×n K .

Proof Choose A∗ such that | det A∗ | is minimal among all the matrices 

 A B ∈ Γn MΓn with det A = 0. C D

Let M ∗ =

 A∗

B∗ C ∗ D∗



∈ Γn MΓn . Then

M ∗ Γn =

 A 0 Γn and ∂n (A∗ , B ∗ ) = (det A )O K C  D



follow from Lemma 2.2. However, the minimality of | det A∗ | shows (det A∗ )O K = (det A )O K . The same holds for the first block column. Hence A∗−1 B ∗ and C ∗ A∗−1 are integral and Hermitian. Therefore, we get a matrix

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A. Hauffe-Waschbüsch and A. Krieg



As

 A∗

D 0 D



A∗ 0 0 D

 ∈ Γn MΓn .

∈ Γn MΓn we conclude ∂n (A∗ , D  ) = (det A∗ )O K

hence

tr

and H = H . A∗−1 D  = H ∈ O n×n K



A simple consequence is given as follows. Corollary 2.4 Given M ∈ Δn then Γn MΓn = Γn M tr Γn .   Proof We assume A0 D0 ∈ Γn MΓn due to Lemma 2.3. By means of [6], Theorem 2.2, there are U, V ∈ G L n (O K ) such that U AV = Atr . Hence    tr    U 0 V 0 A 0 A 0 = , tr −1 tr −1 0 D∗ 0 D 0 U 0 V tr

J [M] = q J then implies A D = AD ∗ = q I , hence D ∗ = D tr .



As M → M tr is an involution, which keeps the double cosets invariant, we conclude from [8] or [12] the following. Theorem 2.5 (Γn , Δn ) is a Hecke pair. The Hecke algebra H (Γn , Δn ) is commutative. Our next aim is to describe particular products in this Hecke algebra. Therefore, we need Lemma 2.6 Let q, r ∈ N be coprime and dK = −3, −4. Then Γn [q] · Γn [r ] = Γn . Proof As the principal congruence subgroups are normal, we may restrict to generators of Γn . We use the generators from [4], Theorem 2.1, for which the claim follows by a simple calculation of the form 

for H = H

tr

I H 0 I



I H 0 I

∈ O n×n and some  ∈ N, q | . K

 ∈ Γn [r ] 

On Hecke Theory for Hermitian Modular Forms

77

An application is described in Corollary 2.7 Given M ∈ Δn (q), r ∈ N, gcd(q, r ) = 1 then Γn MΓn = Γn MΓn [r ]. Proof Clearly M −1 Γn M ∩ Γn ⊇ Γn [q] holds. Now apply Lemma 2.6.



We consider a particular case. Let M ∈ Δn (q), gcd(q, r ) = 1 and M ≡ I (mod r ) as well as •  Γn L j , L j ≡ I (mod r ) Γn MΓn = 1 j

due to Corollary 2.7. Then we immediately obtain Γn [r ]MΓn [r ] =

• 

Γn [r ]L j

(1)

1 j

as well as Γn MΓn =

• 

Γn R L j R −1 for R ∈ Δn (r ).

(2)

1 j

An immediate consequence is Corollary 2.8 Given M ∈ Δn (q), L ∈ Δn (r ) with coprime q, r ∈ N, then Γn MΓn · Γn LΓn = Γn M LΓn . Proof We choose decompositions Γn MΓn =

• 

Γn M K i , K i ∈ Γn [r ], Γn LΓn =

i

• 

Γn L R j

j

due to Corollary 2.7. Clearly the right cosets Γn M K i L R j are mutually disjoint and contained in Γn M LΓn . Thus, the claim follows.



In the case of h K = 1, the Hecke algebra coincides with the tensor product of its primary components Hn, p = H (Γn , Δn, p ), p prime. In this situation, the structure is described in [11]. If h K > 1 this result is no longer √ true (cf. [5], 3.3.6), e.g., for K = Q( −5)

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Γ2 diag (1, 1 +

 √ √ −5, 6, 1 + −5)Γ2 ∈ / Hn, p . p

Many authors define the Hecke algebra as the tensor product of its p-components (cf. [16]). But the tensor product is a proper subalgebra of H (Γn , Δn ) in general. The example shows that it is much more difficult to look at the decomposition of double cosets. Lemma 2.9 Let M ∈ Δn (q), q = r1r2 ∈ N, where r1 is a product of split or ramified primes and r2 a product of inert primes. Then there exist M j ∈ Δn (r j ), j = 1, 2, such that Γn MΓn = Γn M1 Γn · Γn M2 Γn .   Proof We may assume M = A0 D0 due to Lemma 2.3 and consider the determinantal divisors. Let ∂k (A) = Ik · O K ak , k = 1, . . . , n, where ak ∈ N divides r2n and Ik is not divisible by p O K for any inert prime p. In view of [6], Theorem 2.1, there exist A j ∈ O n×n K , j = 1, 2, ∂k (A1 ) = Ik , ∂k (A2 ) = O K ak , k = 1, . . . , n. tr −1

Define D j = r j A j

. Then, we have  Mj =

Aj 0 0 Dj

 ∈ Δn (r j ), j = 1, 2 ,

Γn M 1 Γn · Γn M 2 Γn = Γn M 1 M 2 Γ2 by means of Corollary 2.8. As Ik and O K ak are coprime, we conclude ∂k (A1 A2 ) = ∂k (A1 ) · ∂k (A2 ) = ∂k (A), k = 1, . . . , n , from [6], Theorem 4.2, or [5], Satz 2.6.8. Then Γn M1 M2 Γ2 = Γn MΓn follows from [6], Theorem 2.2.



3 The Inert Part of the Hecke Algebra Lemma 2.9 shows that it is interesting to have a closer look at the inert part defined by

On Hecke Theory for Hermitian Modular Forms

Δinert = n

79



Δn (q)

q∈N p|q⇒ p inert

and call Hninert = H (Γn , Δinert n ) the inert part of the Hecke algebra. Given M ∈ Δn (q), where q is only divided by inert primes, we conclude that ∂k (M) = O Kr , where r | q n . Thus, we can apply Theorem 2.5 as well as [6], Theorem 2.2, in order to obtain the elementary divisor theorem similar to the case of the Siegel modular group (cf. [8, 12]). the double coset Γn MΓn contains a unique Theorem 3.1 Given M ∈ Δn (q) ⊆ Δinert n representative diag (a1 , . . . , an , d1 , . . . , dn ), a j , d j ∈ N, a j d j = q , a1 |a2 | . . . , |an |dn |dn−1 | . . . |d1 . In this case, the elementary divisor theorem holds. Next we have a look at right coset representatives. the right coset Γn M possesses a repreCorollary 3.2 Given M ∈ Δn (q) ⊆ Δinert n sentative of the form   tr A B , AD = q I, 0 D where D is an upper triangular matrix with diagonal entries d j ∈ N, d j | q, j = 1, . . . , n. Now we use Corollary 2.8 in order to get  H (Γn , Δn, p ). Corollary 3.3 Hninert = p inert

In this case, one can directly adopt the proofs, which are given for the Siegel modular group in [8] or [12]. Next we consider generators. Corollary 3.4 Let p be an inert prime. Then H (Γn , Δn, p ) is generated by the double cosets 

 I 0 Γn , 0 pI   Tn, j ( p 2 ) = Γn diag 1, . . . , 1, p, . . . , p, p2 , . . . , p2 , p, . . . , p Γn j = 0, . . . , n − 1,             Tn ( p) = Γn

j

n− j

j

n− j

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A. Hauffe-Waschbüsch and A. Krieg

which are algebraically independent. Given M ∈ Δn (q) ⊆ Δinert we choose a representative n ∗



M =

       B1 ∗ D1 d A B A1 0 , B= , D= , A = tr a α ∗ ∗ 0 δ 0 D

in Γn M and define for k ∈ Z, n  2 

−k

φk (Γn M) = δ Γn−1 M1 , M1 =

A1 B1 0 D1

 ∈ Δn−1 (q).

This map can be extended to a homomorphism of Hecke algebras (cf. [8, 10–12]). The main result is given as follows. Corollary 3.5 If p is an inert prime and n  2 one has φk (Tn ( p)) = ( p 2n−1−k + 1)Tn−1 ( p). Note that we also need the Hecke algebra for Γn [r ], i.e.,  Tnr ( p) = Γn [r ]

 I 0 Γn [r ]. 0 pI

If p ≡ 1(mod r ) we have the same result as above due to (1).

4 Hermitian Modular Forms Let Hn := {Z ∈ Cn×n ;

1 (Z 2i

tr

− Z ) > 0}

denote the Hermitian half-space of degree n, where > resp.  0 stands for positive  definite resp. positive semi-definite. Given f : Hn → C, M = CA DB ∈ Δn we define for k ∈ Z   f | M : Hn → C, Z → det(C Z + D)−k f (AZ + B)(C Z + D)−1 . k

The vector space M (Γn , k) of Hermitian modular forms consists of all holomorphic functions f : Hn → C satisfying f | M = f for all M ∈ Γn k

On Hecke Theory for Hermitian Modular Forms

81

with the usual additional condition of boundedness for n = 1, where we deal with classical elliptic modular forms for S L 2 (Z). Each f ∈ M (Γn , k) possesses a Fourier expansion of the form 

f (Z ) =

α f (T ) e2πi trace (T Z ) ,

T ∈ n , T 0 tr

where T = (ti j ) ∈ n means T = T , 1 t j j ∈ Z, ti j ∈ √ O K for i = j. dK The subspace of cusp forms C (Γn , k) is characterized by α f (T ) = 0 ⇒ T > 0. Moreover, we define the Siegel φ -operator by  f | φ : Hn−1 → C, Z 1 → lim f y→∞

Z1 0 0 iy





=

T1 ∈ n−1 ,T1 0

 αf

 T1 0 2πi trace (T1 Z 1 ) . e 0 0

If h K = 1 then f is a cusp form if and only if f | φ ≡ 0. This is more complicated for h K > 1 (cf. [3], Lemma 1). Therefore, let RU =

  tr U 0 ∈ U (n, n; K) for U ∈ G L n (K). 0 U −1

Theorem 4.1 Let n  2 and let I j = u j , 1, u j ∈ K, j = 1, . . . , h, h = h K , be a set of representatives of the ideal classes in K. Then f ∈ M (Γn , k) is a cusp form if and only if ⎛

f | RU(n)j k

1 ⎜ .. ⎜. | φ ≡ 0, U j = ⎜ ⎜. ⎝ .. 0

⎞ 0 .. ⎟ .⎟ ⎟ , j = 1, . . . , h. .. ⎟ . 0⎠ ··· uj 1 0 ··· .. .

Proof Let T0 ∈ n , T0  0, det T0 = 0. Then there exists 0 = g ∈ O nK with T0 g = 0. Next we determine U ∈ G L n (O K ) and 1  j  n such that

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⎛ ⎞ 0 ⎜ .. ⎟ ⎜.⎟ tr −1 ⎜ ⎟ U g = ⎜ 0 ⎟ · λ, 0 = λ ∈ O K , ⎜ ⎟ ⎝u j ⎠ 1 hence

  tr tr tr tr ∗0 g = U U j en · λ, T0 U U j = , ∗ ∈ K(n−1)×(n−1) . 00

In view of f | RU j = f | RU | RU j = f | RU j U k

k

= (det U )

k



k

k

α f (T ) e

tr

tr

2πi trace (U j U T U U j ·Z )

T ∈ , T 0

the application of φ yields α f (T0 ) = 0. Hence, f is a cusp form.



Now we have a closer look at the choice of u j in Theorem 4.1. Lemma 4.2 Let dK = −4, −8 and p be an odd prime, p | dK . Then representatives of the ideal classes I j = u j , 1, u j ∈ K, may be chosen such that we find an N ∈ N with the properties p  N and N u j ∈ O K , j = 1, . . . , h K . Proof According to [7], p. 211, u j may be chosen in the form uj =

√ β j + dK , β 2j − dK = 4α j γ j , α j , γ j ∈ N, β j ∈ Z. 2α j

As α j ∈ N let N j ∈ N be minimal such that N j u j ∈ O K , we may assume p | α j as we are done otherwise. Then p | β j follows. As p 2  dK we obtain p 2  (β 2j − dK ), p 2  α j . Thus, we may choose u ∗j = and Nj =

2α j √ , u ∗j , 1K∗ = u j , 1K∗ β j + dK

β 2j − dK p

∈ N satisfies N j u ∗j ∈ O K , p  N .

On Hecke Theory for Hermitian Modular Forms

83

Then N = N1 · . . . · Nh K is a solution.



Next we need a purely number theoretical assertion on the existence of such primes. Lemma 4.3 Let dK = −4, −8 and suppose that there is an odd prime divisor of dK , which does not divide N ∈ N. Then there exist infinitely many inert primes satisfying p ≡ 1(mod N ). Proof At first, assume m ≡3 (mod4). Let  = gcd(N , m). Then m =  because of  a m  N . We find a ∈ N with m/ = −1. Dirichlet’s prime number theorem asserts the existence of infinitely many primes p satisfying p ≡ 1 (mod 4N ), p ≡ a (mod m/), since the modules are coprime. Quadratic reciprocity yields  χK ( p) =

−m p



 =

−1 p



  p p = −1. m/  

The other cases are dealt with in a similar way.

5 Hecke Operators Given f ∈ M (Γn , k) we define the Hecke operator Γn MΓn , M ∈ Δn , acting on f by  f | L ∈ M (Γn , k). (3) f | Γn MΓn = k

L:Γn \Γn MΓn

k

This definition is linearly extended on H (Γn , Δn ). Moreover, we apply the analogous definition for subgroups of Γn . Lemma 5.1 Hecke operators map cusp forms on cusp forms.   Proof We may choose L = A0 DB in (3) due to Lemma 2.2. f | k

A B 0 D

(Z ) =



(det D)−k α f (T ) e2πi trace (T B D

−1

+T [A]Z /q)

(4)

T ∈ n , T >0

if M ∈ Δn (q). Hence, only positive definite matrices appear in the Fourier expansion.  Next we consider the eigenvalues of Hecke operators.

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Lemma 5.2 Let p be an inert prime and let f ∈ Mk (Γn , k) with α f (0) = 0 as well as f | Tn ( p) = λ f for some λ ∈ C. Then k

λ=

n 

( p 2 j−1−k + 1).

j=1

Proof Use Corollary 3.5 as well as f | Tn ( p) | φ = f | φ | φk (Tn ( p)) k

k

= ( p 2n−1−k + 1) f | φ | Tn−1 ( p) k

as well as f | φ n = α f (0). 

After n steps, the result follows. Next we consider the other extreme case of cusp forms.

Lemma 5.3 Let f ∈ M (Γn [q], k) be a cusp form. Assume that p is an inert prime, p ≡ 1(modq), M ∈ Δn ( p), and f | Γn [q]MΓn [q] = λ f . Then k

|λ|  p −kn/2

n 

( p 2 j−1 + 1).

j=1

Proof There exists Z 0 ∈ Hn such that the function Hn → R, Z → (det Y )k/2 | f (Z )| attains its maximum at Z 0 due to [3]. Then the result follows in the same way as in [8], Hilfssatz IV.4.8, because of n      ( p 2 j−1 + 1)  Γn [q]\Γn [q]MΓn [q] =  Γn \Tn ( p) = j=1

due to (2) as well as the case k = 0 in Lemma 5.2. Next we need an assertion on iterative φ-operators. Lemma 5.4 Let f ∈ M (Γn , k), R j ∈ U ( j, j; K), j = 1, . . . , n. Then f | Rn | φ | Rn−1 | φ . . . | R1 | φ = c. lim f (i y I ) = c · α f (0) k

k

k

y→∞



On Hecke Theory for Hermitian Modular Forms

85

for some c = 0. Proof As f | Rn | φ | Rn−1 = f | Rn (Rn−1 × I ) | φ (cf. [9]), we get k

k

k

f | Rn | φ . . . | R1 | φ = f | R | φ n , k

k

k

where R = Rn · (Rn−1 × I ) · . . . · (R1 × I ) ∈ U (n, n; K). 

Now use Lemma 2.2 and (4).

We give an application to the characterization of cusp forms. Therefore, we use the special matrices RU (n) from Theorem 4.1. 

Lemma 5.5 Let f ∈ M (Γn , k), 1  j  n. Then f | R | φ j ≡ 0 for all R ∈ U (n, n; K) k

holds if and only if this is true for     R = RU (n) RVn · RU (n−1) RVn−1 × I · . . . · RU (n− j+1) RVn− j+1 × I ∈ U (n, n; K) in

i n−1

i n− j+1

(5) for all V ∈ G L  (O K ) and i  ∈ {1, . . . , h K },  = n, . . . , n − j + 1. Proof Apply the same arguments as in the proof of Theorem 4.1 and Lemma 5.4.  Remark If f ∈ M (Γn , k) is symmetric, i.e., f (Z tr ) = f (Z ), and M ∈ Δn satisfies det M ∈ R+ , we observe f | Γn MΓn (Z tr ) = f | Γn MΓn (Z ). k

k

We conclude Γn MΓn = Γn MΓn for M ∈ Δinert from Theorem 3.1. Thus, these Hecke n operators map the subspace of symmetric Hermitian modular forms on itself.

6 The Siegel-Eisenstein Series According to [1], we may define the Siegel-Eisenstein series E k(n) (Z ) =

 M:Γn,0 \Γn

for even k > 2n, dK = −3, −4, where

1 | M(Z ), Z ∈ Hn , k

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 Γn,0 = We have

  A B ∈ Γn . 0 D

E k(n) | φ = E k(n−1) , E k(0) := 1.

We can take the same proof as in [8], IV.4.7, in order to get Lemma 6.1 Let k > 2n be even, dK = −3, −4, M ∈ Δn . Then there exists a λ ∈ C such that E k(n) | Γn MΓn = λE k(n) . k

We obtain our final result and recall the definition of N from Lemma 4.2. Theorem 6.2 Let k > 2n be even, dK = −3, −4. Let p be an inert prime p ≡ 1 mod N 2n−2 and f ∈ Mk (Γn , k) satisfying α f (0) = 1 and f | Tn ( p) = λ f k

for some λ ∈ C. Then

f = E k(n) .

Proof The case n = 1 is clear from the classical theory as E k(1) coincides with the normalized elliptic Eisenstein series. Let n  2. Since the constant term of the Fourier expansion is non-zero, we can apply Lemma 5.2. If f = E k(n) , there exists a minimal j, 1  j  n such that ( f − E k(n) ) | R | φ j ≡ 0 for all R ∈ U (n, n; K). k

This means that the non-zero Fourier coefficients have rank > n − j. Now apply Lemma 5.5 and assume  f := ( f − E k(n) ) | R | φ j−1 ≡ 0 k

for an R ∈ U (n, n; K) of the form (5) quoted there. Thus  f ∈ M (Γn− j+1 [N 2 j−2 ], k) is a cusp form. We conclude from Lemma 5.2

On Hecke Theory for Hermitian Modular Forms

87

N 2 j−2   f | Tn− j+1 ( p) = λ f , k



n− j−1

λ=

( p 2−1−k + 1) > 1.

=1

But  f is a cusp form. Therefore, we can apply Lemma 5.3 in order to get 

n− j+1

|λ|  p −k(n− j+1)/2 =p



n− j+1

( p 2−1 + 1) < p −k(n− j+1)/2

=1 (n− j+1)(n− j+2−k/2)

p 2

=1

1

in view of k > 2n. This contradicts λ > 1 and yields the claim.



Remark (a) The cases dK = −3, −4 are excluded because of the additional units. As h K = 1 in these cases, the results are contained in [13], where the proof is only valid for class number 1. Due to our proof here the results in [15] are also valid for arbitrary K. Moreover, these considerations fill the gap in [13] such that the results of Sect. 8 there are true for arbitrary h K . (b) If dK = −3, −4 one has to impose the condition that k is divisible by the number of units in O K . Alternatively, for arbitrary even k, one has to restrict the summation to Γn ∩ S L 2n (O K ) or to insert the factor (det M)−k/2 in the definition of E k(n) .

References 1. Braun, H.: Hermitian modular functions. Ann. Math. 50, 827–855 (1949) 2. Braun, H.: Hermitian modular functions III. Ann. Math. 53, 143–160 (1950) 3. Braun, H.: Darstellung hermitischer Modulformen durch Poincarésche Reihen. Abh. Math. Semin. Univ. Hamb. 22, 9–37 (1958) 4. Dern, T.: Multiplikatorsysteme und Charaktere Hermitescher Modulgruppen. Monatsh. Math. 126, 109–116 (1998) 5. Ensenbach, M.: Hecke-Algebren zu unimodularen und unitären Matrixgruppen. Ph.D. thesis, RWTH Aachen. http://publications.rwth-aachen.de/record/50390/files/Ensenbach_Marc.pdf (2008) 6. Ensenbach, M.: Determinantal divisors of products of matrices over Dedekind domains. Linear Algebra Appl. 432, 2739–2744 (2010) 7. Forster, O.: Algorithmische Zahlentheorie, 2nd edn. Springer, Berlin (2015) 8. Freitag, E.: Siegelsche Modulfunktionen. Grundlehren der mathematischen Wissenschaften, vol. 254. Springer, Berlin (1983) 9. Krieg, A.: Modular Forms on Half-Spaces of Quaternions. Lecture Notes in Mathematics, vol. 1143. Springer, Berlin (1985) 10. Krieg, A.: Das Vertauschungsgesetz zwischen Hecke-Operatoren und dem Siegelschen φOperator. Arch. Math. 46, 323–329 (1986) 11. Krieg, A.: The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions. Proc. Indian Acad. Sci. Math. Sci. 97, 201–229 (1987)

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12. Krieg, A.: Hecke algebras. Mem. Am. Math. Soc. 435 (1990) 13. Krieg, A.: The Maaß spaces on the Hermitian half-space of degree 2. Math. Ann. 289, 663–681 (1991) 14. Manickam, M.: On Hecke theorie for Jacobi forms. In: Venkataramana, T.N. (ed.) Cohomology of arithmetic groups. L-functions and automorphic forms, pp. 89–93. Narosa Publishing House, New Delhi (2001) 15. Nagaoka, S., Nakamura, Y.: On the restriction of the Hermitian Eisenstein series and its applications. Proc. Am. Soc. 139, 1291–1298 (2011) 16. Raum, M.: Hecke algebras related to the unimodular and modular groups over quadratic field extensions and quaternion algebras. Proc. Am. Math. Soc. 139, 1301–1331 (2011)

Sign Changes of the Ramanujan τ -Function Bernhard Heim and Markus Neuhauser

Dedicated to Murugesam Manickam on the occasion of his 60th birthday

Abstract The signs and vanishing of Fourier coefficients of modular forms are important properties of modular forms and are closely related. The focus of this paper is on the coefficients of the powers of the Dedekind η-function, in particular the discriminant function  and the Ramanujan τ -function. To all nth coefficients we attach polynomials Pn (x). The root distribution of the Pn (x) dictates the sign of all coefficients. In particular it determines the first non-sign change of τ (n). We further show the influence of this property on the neighbours η23 and η25 of  = η24 , which leads to a number of conjectures. Keywords Dedekind η-function · Fourier coefficients · Polynomials · Ramanujan τ -function

B. Heim (B) · M. Neuhauser German University of Technology in Oman (GUtech), PO Box 1816, Athaibah PC 130, Muscat, Oman e-mail: [email protected] M. Neuhauser e-mail: [email protected] Sultanate of Oman and Faculty of Mathematics, Computer Science, and Natural Sciences, RWTH Aachen University, 52056 Aachen, Germany © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_7

89

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1 Introduction Ramanujan’s fundamental paper [24] entitled “On certain arithmetical functions” has been a source of inspiration over the last 100 years. Ramanujan introduced the integer-valued arithmetic function τ (n) (see Eq. (92) in his paper). In honour of Ramanujan, it is now called the Ramanujan τ -function [18, 26]. The present paper is devoted to the non-vanishing and sign changes of Fourier coefficients of the powers of the Dedekind η-function, and in particular that of the Ramanujan τ -function. Ramanujan recorded the first 30 values of τ (n). τ (n)

n

τ (n)

1 1 11 534612 21 2 −24 12 −370944 22 3 252 13 −577738 23 4 −1472 14 401856 24 5 4830 15 1217160 25 6 −6048 16 987136 26 7 −16744 17 −6905934 27 8 84480 18 2727432 28 9 −113643 19 10661420 29 10 −115920 20 −7109760 30

−4219488 −12830688 18643272 21288960 −25499225 13865712 −73279080 24647168 128406630 −29211840

n

n

τ (n)

The discriminant function (ω), the unique normalized cusp form of weight 12 on the upper half space H with respect to SL(2, Z), has the following definition (ω) := q

∞ ∞   24  1 − qn = τ (n) q n , n=1

(q := e2πiω ) .

n=1

Ramanujan observed several surprising properties. The most prominent are 1. Multiplicative law: τ (n m) = τ (n)τ (m) for gcd(n, m) = 1 and τ ( pl+1 ) = τ ( pl )τ ( p) − p 11 τ ( pl−1 ), for p prime, l ∈ N. 2. Growth rate: |τ ( p)| ≤ 2 p 11/2 for p prime. 3. Congruences: τ ( p) ≡ (1 + p 11 ) (mod 691) for p prime. Ramanujan proved the congruence property 3 and Mordell [17] proved the multiplicative law 1. Finally, Deligne [3] proved the growth condition 2, the celebrated Ramanujan–Petersson conjecture for modular forms in 1974, and was awarded the Fields medal. The first thirty values of τ (n) are non-zero. The signs of the values change frequently. Indeed, Lehmer’s long-standing conjecture predicts that τ (n) never vanishes [10, 14, 15, 18, 23]. It was recently [5] described as one of the fundamental open problems in studying the arithmetic properties of Fourier coefficients of modular forms.

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Ramanujan’s table shows that the signs of the coefficients strictly alternate from τ (1) until τ (6), after which the first non-sign change τ (6)τ (7) > 0 occurs. In this paper we present a new theory, which explains this pattern in a broader context. In the past, starting with the work by Murty [19] and Knopp, Kohnen, and Pribitkin [11] many extraordinary results on the sign changes of modular forms have been obtained. Most of the results are based on the cuspidality of the involved modular form, a variant of a theorem by Landau and properties of L-functions. This implies, in particular, that the values of the Ramanujan τ -function are infinitely many times positive and negative. Also, most results [4, 5, 13, 14, 16] assume that the involved modular forms are of half-integral or integral weight, and are Hecke eigenforms. They study the eigenvalues of cusp forms. The approach taken by Ghosh and Sarnak [4], in a broader sense, matches with our work. The sign changes of the Fourier coefficients are linked with the zeros of the modular form on some geodesic segments. We connect the sign changes with the root distribution of polynomials. Our method works for η-products, but not for modular forms in general. The polynomials and their root distributions dictate the sign changes of all Fourier coefficients for all powers of the Dedekind η-function. In this paper, this will be applied to the Ramanujan τ -function.

2 Root Distribution In this section, we recall a theorem by Kostant [12] on the non-vanishing and sign changes of the first coefficients of Euler products associated to affine Lie algebras. This does not cover the properties of τ (7). The first non-sign change τ (6) τ (7) > 0 is determined by the root distribution of certain polynomials Pn (x) associated with the powers of the Dedekind η-function.

2.1 Kostant’s Theorem We start with Kostant’s Theorem ([12], Theorem 0.11). The theorem implies that (−1)n+1 τ (n) > 0 for n = 1, 2, 3, 4, 5, 6, with no implication given for τ (7). Note also that the multiplicative law 1 does not give any information on the value of τ (7). Let H be the upper complex half space. Let q := e2πiω , where ω ∈ H. Then the Dedekind η-function is defined by 1

η(ω) = q 24

∞  n=1

(1 − q n ) .

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The Dedekind η-function is a modular form of weight 1/2 with a multiplier system [22]. Let r ∈ N. We put ∞ 

an (r )q n =

n=0

∞ 

(1 − q n )r .

(1)

n=1

Theorem 1 (Kostant  2004) Letnmm 2∈−1N. Then the coefficients of the q-expansion of the infinite product ∞ satisfy n=1 (1 − q ) (−1)n an (m 2 − 1) = dim Cn for m ≥ max {4, n}, where Cn is a certain vector space. The vector space Cn is non-trival if and only if n ≤ m. We also refer to the work by Han [6], where the Nekrasov–Okounkov hook length formula [20] was used to extend the result by Kostant.

2.2 The Polynomials Pn (x) We recall the definition of the family of polynomials Pn (x) of degree n satisfying Pn (−r ) = an (r ) for all n, r ∈ N0 (see [2, 9, 21, 25]).  Definition 1 Let σ (n) := d|n d. Let P0 (x) := 1. For n ∈ N we put n x Pn (x) := σ (k)Pn−k (x) . n k=1

(2)

For example, P1 (x) = x, P2 (x) = x2 (x + 3), and P3 (x) = 3!x (x + 1)(x + 8). All the coefficients of Pn (x) are non-negative. Hence all real roots except 0 are negative. It is known that P10 (x) also has non-real roots [10]. Further, it is conjectured [1] that all roots are simple. We have checked this for n ≤ 500. In Fig. 1, we have used PARI/GP to check this plotted the roots of the first ten polynomials. This root distribution explains the non-vanishing and signs of the first coefficients of Ramanujan’s τ function. We highlighted the values x = −25, x = −24 and x = −23. Note that Pn (−24) = an (24) = τ (n + 1). We explain how to deduce the signs of τ (n) from Fig. 1. Let x0 = −24. Let n 0 be the smallest integer n such that Pn (x) has a root left of x0 . Then (−1)n Pn (x) > (−1)n Pn (x0 ) = (−1)n τ (n + 1) > 0 for 0 ≤ n < n 0 = 6, x ≤ x0 . This shows the strict sign changes. Since the parity of the number of roots appearing left from x0 for n 0 is odd, we obtain that an 0 −1 (24) an 0 (24) > 0. This implies τ (6) τ (7) > 0.

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93

Fig. 1 Real roots of Pn (x)

Let n ∈ N. Let

α1(n) ≤ α2(n) ≤ · · · ≤ · · · < αd(n) n

be the ordered list of the real roots of Pn (x). Note, we also recorded the multiplicities. It is believed by strong numerical evidence (n ≤ 500) and conjectured [1], that = 0. Let r ∈ N. We define multiplicities do not occur. Here 1 ≤ dn ≤ n and αd(n) n  (n) rn := 1 ≤ j ≤ dn | α (n) j ≤ −r, for α j root of Pn (x) .

(3)

Theorem 2 Let n, r ∈ N. Then (−1)n+rn an (r ) ≥ 0 .

(4)

Let −r be smaller than the smallest real root of Pn (x). Then we have Corollary 1 Let r, n ∈ N and rn = 0. Then an (r ) < 0 if n is even and an (r ) > 0 if n is odd.

(5)

Recently [8], Kostant’s and Han’s result was improved. Theorem 3 (Heim, Neuhauser 2020) Let κ = 15. For all x ∈ C and n ∈ N with |x| > κ(n − 1) we obtain Pn (x) = 0. Corollary 2 Let x < 0 real be given. Then Pn (x) Pn+1 (x) < 0 for all x < −κ n.

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3 Modular Forms with Real Weight We recall a result of Knopp, Kohnen, and Pribitkin [11]. Let  be a discrete subgroup of SL(2, R) with a fundamental region of finite hyperbolic volume. Let f (ω) =



a(n) e2πi(n+κ)ω/λ

n+κ>0

be a non-trivial cusp form, with a multiplier system, with respect to . Here λ is a positive real number and 0 ≤ κ < 1. Then the sequence {a(n)} has infinitely many changes of sign in the strong sense. This means we have an infinite number of coefficients such that a(n) > 0 and an infinite number of coefficients such that a(n) < 0. Let x < 0 be a real number. Let Fx (ω) := q − 24 x

∞ ∞    −x x 1 − qn = q − 24 Pn (x) q n n=1

(ω ∈ H) .

(6)

n=0

It is well known that Fx is a cusp form of positive real weight (−x)/2. The result of [11] implies that the coefficients {Pn (x)} have infinitely many changes of sign in the strong sense. In their paper [13] “On modular signs,” Kowalski, Lau, Soundararajan, and Wu considered the question to what extent the signs of the coefficients of classical modular forms uniquely determine the modular form. They focused on primitive forms of even weight k for 0 (N ), where N ∈ N, of trivial Nebentypus. In this paper, we are mainly interested in the η-products provided by the set := {Fx | x < 0}. The results of [13] could only be applied to F−24 . Nevertheless, the question of recovering a form by its signs is interesting, and seems to be very challenging. We follow their relaxed definition of signs of a sequence of real numbers. We view 0 as being of both signs simultaneously. This helps to avoid the difficult task of studying the vanishing of the coefficients and, as they state, increases the possibility of having equal signs. Let sign(F) be the sequence of signs of F. Question  n −x Is it possible to recognize ∞ n=1 (1 − q ) , x < 0, by the sign of the coefficients. Let F, G ∈ , so does sign(F) = sign(G) imply F = G?

The question appears to be too general, since F−1 and F−3 , have, for example, the same sign sequence. Let {n k }∞ k=0 be the sequence of coefficients of F−1 and F−3 , with Pn k (−1) Pn k (−3) = 0 .

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95

The sequence is given by 0, 1, 15, 210, 2926, 40755, . . .. The generating function is given by the coefficients of the t expansion of t . 1 − 15 t + 15 t 2 − t 3

(7)

In the OEIS database this is sequence A076139. We have Theorem 4 Let n be any integer such that Pn (−1) Pn (−3) = 0. Then there exists a suitable k such that n = n k and Pn k (−1) Pn k (−3) > 0. Moreover Pn k (−1) > 0 for k ≡ 0, 3 (mod 4) and Pn k (−1) < 0 otherwise. Sketch of Proof of Theorem 4. Euler and Jacobi proved that for r = 1 and r = 3: ∞     1 − qn = (−1)m q (3m+1)m/2 , n=1

m∈Z

∞ ∞   3  1 − qn = (−1)m (2m + 1) q (m+1)m/2 . n=1

m=0

That is an (1) = (−1)m = 0 if and only if there is an m ∈ Z such that n = (3m + 1) m/2. Further we have an (3) = (−1)m (2m + 1) = 0 if and only if there is an m ≥ 0 such that n = (m + 1) m/2 and otherwise they vanish. Theorem 4 can be proven using the following lemmata, whose proof we leave to the reader. To determine the signs we have to determine the respective m’s to the n’s. The first lemma yields a criterion on the n’s when the coefficients are non-zero. Lemma 1 We have an (1) = 0 if and only if 24n + 1 is a square. Similarly an (3) = 0 if and only if 8n + 1 is a square. In the second lemma we recursively define sequences exhausting all the squares from the previous lemma, which can be proven by induction. Lemma 2 Let  sk+1= 2sk + tk and tk+1 = 3sk + 2tk for k ≥ 0. Then  s0 = 1 = t0 and for all k ≥ 0 sk2 − 1 /8 = tk2 − 1 /24 =: n k ∈ Z. Example 1 Starting with s0 = 1 = t0 we obtain the sequences s1 = 3, t1 = 5, s2 = 11, t2 = 19, s3 = 41, t3 = 71, s4 = 153, …, and from this n 0 = 0, n 1 = 1, n 2 = 15, n 3 = 210, n 4 = 2926, …. The last lemma, which can again be proven by induction, almost completes the proof for Theorem 4. Lemma 3 For all k ≥ 0

  m 1,k 1. for r = 1 and m 1,k = (−1)k tk − 1 /6 holds an k (1) = (−1) ,   m 3,k 2m 3,k + 1 , and 2. for r = 3 let m 3,k = (sk − 1) /2 then an k (3) = (−1) 3. m 1,k ≡ m 3,k (mod 2). The last step in the proof of Theorem 4 is to determine the signs of the coefficients. This can be done using induction and by employing Lemma 3.

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4 Patterns of Sign Distribution Let F = Fx ∈ and n ∈ N. We denote by signn (F) the sequence of the signs of P0 (x) up to Pn (x). Then for all x < 0 we obtain

sign1 (F) = (+, −) and sign2 (F) = From P3 (x) =

x 6

(+, −, −) (+, −, +)

for − 3 < x < 0 , for x < −3 .

(x 2 + 9x + 8) we deduce

⎧ (+, −, −, −) ⎪ ⎪ ⎪ ⎨(+, −, −, +) sign3 (F) = ⎪ (+, −, +, +) ⎪ ⎪ ⎩ (+, −, +, −)

for − 1 < x < 0 , for − 3 < x < −1 , for − 8 < x < −3 , for x < −8 .

(8)

This gives a first indication of how the root distribution dictates the sign distribution. in Tabel 1 we used the symbol ∞ to indicate that all compared coefficients have the same sign. It is interesting to note that F−1 and F−2 , and F−2 and F−3 may have the same signs. We have checked this for the first 106 coefficients. Remark 1 For r = 3 and s = 4 the signs of the first coefficients up to n = 170 are equal: sign170 (−3) = sign170 (−4), but P171 (−3)P171 (−4) < 0. Table 1 Smallest n such that Pn (−r ) Pn (−s) < 0 r\s 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

∞ ∗ ∞ 2 2 2 2 2 2 2 2 2 2 2 2

∗ ∞ ∗ 2 2 2 2 2 2 2 2 2 2 2 2

∞ ∗ ∞ 171 15 6 6 6 3 3 3 3 3 3 3

2 2 171 ∞ 13 6 5 5 3 3 3 3 3 3 3

2 2 15 13 ∞ 6 5 5 3 3 3 3 3 3 3

2 2 6 6 6 ∞ 28 16 3 3 3 3 3 3 3

2 2 6 5 5 28 ∞ 16 3 3 3 3 3 3 3

2 2 6 5 5 16 16 ∞ 12 9 6 6 6 6 4

9

10

11

12

13

14

15

2 2 3 3 3 3 3 12 ∞ 9 6 6 6 6 4

2 2 3 3 3 3 3 9 9 ∞ 8 8 8 8 4

2 2 3 3 3 3 3 6 6 8 ∞ 19 12 10 4

2 2 3 3 3 3 3 6 6 8 19 ∞ 12 10 4

2 2 3 3 3 3 3 6 6 8 12 12 ∞ 10 4

2 2 3 3 3 3 3 6 6 8 10 10 10 ∞ 7

2 2 3 3 3 3 3 4 4 4 4 4 4 7 ∞

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Table 2 Exactly one real root between −25 and −23 for n ≤ 100 n n 13 15 17 27 29 30 36 38 41 42 44 45 47 57 59 62 63

−23.150927 −24.864805 −23.074977 −23.440509 −23.615150 −23.178879 −23.081453 −24.447298 −24.200070 −23.962299 −24.771644 −24.395443 −24.238858 −23.206444 −24.305099 −24.354004 −24.574897

64 65 68 74 76 77 79 80 81 82 85 86 87 89 91 97 99

−23.619361 −23.374262 −23.398381 −23.309977 −24.558757 −23.877250 −24.227541 −24.050739 −23.657095 −23.031029 −23.990449 −23.196264 −24.986967 −24.277215 −23.433527 −23.167067 −23.203932

5 Neighbours of  In the following we identify  with F−24 . We show that  can be distinguished by the signs from the neighbours F−25 and F−23 . The root distribution of the polynomials Pn (x) displayed in Fig. 1 for 1 ≤ n ≤ 10 shows that sign10 (F−25 ) = sign10 () = sign10 (F−23 ) = (+, −, +, −, +, −, −, +, −, −) . Tables 2 and 3 record all the real roots between −25 and −23 for the first 100 polynomials Pn (x). The first time a root appears is for n = 13. This root is between −24 and −23. This implies that sign13 () = sign13 (F−23 ) and sign13 (F−25 ) = sign13 (). The case n = 15 leads to sign15 (F−25 ) = sign15 (). We also note that sign13 (F−25 ) = sign13 (F−23 ). Table 3 shows that for n = 90 two real roots exist between −25 and −24. This implies that the signs of P90 (−25), P90 (−24), and P90 (−23) are equal.

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Table 3 Exactly two real roots between −25 and −23 for n ≤ 100 n −24.860664 −24.412396 −24.785486 −24.578305

61 69 90 92

−23.556942 −23.211492 −24.403536 −23.374025

6 Open Questions Let x < 0. The signs of the coefficients of Fx are dictated by the root distribution of {Pn (x)}n . Since Fx is a modular form of real weight, Fx has an infinite number of coefficients which are positive and an infinite number of coefficients which are negative [11]. Challenge Let x < 0. Show that {Pn (x)}n has an infinite number of values which are positive and an infinite numbers of values which are negative, using only the n definition: P0 (x) = 1 and Pn (x) := nx k=1 σ (k) Pn−k (x). In other words, find a modular-free proof. Although the coefficients of F−1 and F−3 have equal signs, based on extensive calculations, we believe that the signs of the coefficients of F ∈ determine the F already for x < −3. Conjecture 1 Let x, y ∈ Z and x, y < −3. Then sign(Fx ) = sign(Fy ) implies x = y. It would be interesting to find out if this can be proven by combinatorial or analytic methods. Remark 2 We could also speculate that the conjecture is true for x, y ∈ R and x, y ≤ −3. This would be the strong version of the conjecture. We have extended the data in Fig. 1 up to n = 500 and observed that the smallest roots have some kind of interlacing property. This leads to Conjecture 2 Let x < 0. Let n 0 be the smallest number such that Pn 0 −1 (x) Pn 0 (x) ≥ 0. Then we have Pn 0 (x) Pn 0 +1 (x) ≤ 0. This means that the first non-sign change is followed by a sign change. We end with a quotation of Hardy regarding the Ramanujan τ -function [7, 18]: We may seem to be straying into one of the backwaters of mathematics, but the genesis of τ (n) as a coefficient in so fundamental a function compels us to treat it with respect.

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Acknowledgements The authors would like to thank Robert Tröger for many useful discussions and providing us with Fig. 1. The authors thank the anonymous referee for useful comments.

References 1. Amdeberhan, T.: Theorems, problems and conjectures. arXiv:1207.4045v6 [math.RT]. Accessed 13 July 2015 2. D’Arcais, F.: Développement en série. Intermédiaire Math. 20, 233–234 (1913) 3. Deligne, P.: La conjecture de Weil. I. Publ. Math. IHES 43, 273–307 (1974) 4. Ghosh, A., Sarnak, P.: Real zeros of holomorphic Hecke cusp forms. J. Eur. Math. Soc. (JEMS) 14(2), 465–487 (2012) 5. Gun, S., Kumar, B., Paul, B.: The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newforms. J. Number Theory 200, 161–184 (2019) 6. Han, G.: The Nekrasov-Okounkov hook length formula: refinement, elementary proof and applications. Ann. Inst. Fourier (Grenoble) 60(1), 1–29 (2010) 7. Hardy, G.H.: Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, 3rd edn. Chelsea, New York (1978) 8. Heim, B., Neuhauser, M.: The Dedekind eta function and D’Arcais type polynomials. Res. Math. Sci. 7 Paper No. 3, 8pp. (2020). https://doi.org/10.1007/s40687-019-0201-5 9. Heim, B., Neuhauser, M., Rupp, F.: Imaginary powers of the Dedekind eta function. Experimental Mathematics 29(3), 317-325 (2020). https://doi.org/10.1080/10586458.2018.1468288 10. Heim, B., Neuhauser, M., Weisse, A.: Records on the vanishing of Fourier coefficients of powers of the Dedekind eta function. Res. Number Theory 4(3), 12 (2018), Article 32 11. Knopp, M., Kohnen, W., Pribitkin, W.: On the signs of Fourier coefficients of cusp forms. Ramanujan J. 7, 269–277 (2003) 12. Kostant, B.: Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra. Invent. Math. 158, 181–226 (2004) 13. Kowalski, E., Lau, Y.-K., Soundararajan, K., Wu, J.: On modular signs. Math. Proc. Camb. Phil. Soc. 149, 389–411 (2010) 14. Kumari, M., Murty, M.R.: Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms. Int. J. Number Theory 14(8), 2291–2301 (2018) 15. Lehmer, D.H.: The vanishing of Ramanujan’s τ (n). Duke Math. J. 14, 429–433 (1947) 16. Matomäki, K., Radziwill, M.: Sign changes of Hecke eigenvalues. Geom. Funct. Anal. 25, 1937–1955 (2015) 17. Mordell, L.J.: On Mr. Ramanujan’s empirical expansions of modular functions. Proc. Camb. Philos. Soc. 19, 117–124 (1920) 18. Murty, M.R., Murty, V.K.: The Mathematical Legacy of Srinivasa Ramanujan. Springer, New Delhi (2012) 19. Murty, M.R.: Oscillations of Fourier coefficients of modular forms. Math. Ann. 262, 431–446 (1983) 20. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. In: Etingof, P., Retakh, V.S., Singer, I.M. (eds.) The Unity of Mathematics. Progress in Mathematics, vol. 244, pp. 525–596. Birkhäuser, Boston (2006) 21. Newman, M.: An identity for the coefficients of certain modular forms. J. Lond. Math. Soc. 30, 488–493 (1955) 22. Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and qseries. Conference Board of Mathematical Sciences, vol. 102. American Mathematical Society Providence (2004) 23. Ono, K.: Lehmer’s conjecture on Ramanujan’s tau-function. J. Indian Math. Soc. (Special Centenary Issue), 149–163 (2008)

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24. Ramanujan, S.: On certain arithmetical functions. Trans. Cambridge Philos. Soc. 22, 159–184 (1916). In: Hardy, G.H., Seshu Aiyar, P.V., Wilson, B.M. (eds.) Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publishing, American Mathematical Society, Providence (2000) 25. Serre, J.-P.: Sur la lacunarité des puissances de η. Glasgow Math. J. 27, 203–221 (1985) 26. Williams, K.: Historical remarks on Ramanujan’s tau function. Amer. Math. Monthly 122(1), 30–35 (2015)

The Central Limit Theorem in Algebra and Number Theory Arpita Kar and M. Ram Murty

Abstract As its name suggests, the central limit theorem occupies a central position in all of mathematics and perhaps all of science. From its humble origins in combinatorics, it has evolved into a powerful tool through which we can understand the mysteries of nature. In this paper, we survey how it has led to the development of new insights in algebra and number theory. Keywords Central limit theorem · Erd˝os-Kac theorem, Goncharov’s theorem, Riemann Hypothesis 2010 Mathematics Subject Classification 11M06 · 20C15

1 Introduction The central limit theorem is one of the remarkable theorems of twentieth-century mathematics, and there are even good reasons to say it is the most remarkable discovery of our time. With its humble origins emanating from a simple combinatorial problem, it has evolved over decades into a profound principle of probability theory. It gave birth to statistical methods and now influences disciplines outside of mathematics such as biology, medical science, economics and artificial intelligence. This paper will not expound these connections and ramifications outside the mathematical field, but rather, our goal is to elucidate its impact on number theory and algebra, which surprisingly, is not so well-known even among pure mathematicians. Research of the M. R. Murty partially supported by an NSERC Discovery grant. A. Kar · M. R. Murty (B) Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada e-mail: [email protected] A. Kar e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_8

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Apart from its intrinsic beauty, the central limit theorem offers us a metaphor for approaching and understanding a notorious unsolved problem in number theory, namely the Riemann hypothesis. This celebrated problem and its manifold incarnations as the Generalized Riemann hypothesis, both in the context of algebraic number fields and the wider context of automorphic L-functions, occupies a pivotal place in the landscape of pure mathematics. Like the Himalayan peaks, these hypotheses brood ominously over all of mathematics and invite us to scale their celestial summits. The central limit theorem, and probability theory in general, offers us a method to approach these summits. To motivate our discussion, let us begin with a study of the Liouville function λ(n) defined as follows. If n = p1a1 p2a2 . . . pkak is the unique factorization of a natural number n into distinct prime powers, then λ(n) = (−1)(n)

where

(n) = a1 + a2 + · · · ak .

In other words, λ(n) is +1 if n has an even number of prime factors and −1 if n has an odd number of prime factors (counted with multiplicity). The Riemann hypothesis is then equivalent to the assertion that for every  > 0, 

λ(n) = O(N 2 + ). 1

(1)

nN

This was first noted by Pólya [28] in 1919. This suggests the following “thought experiment” where we would treat the λ(n) as “random variables” taking the values ±1. Indeed, consider the set § N := {σ = (a1 , a2 , . . . , a N ) :

ai = ±1}.

Evidently, |§ N | = 2 N . For each σ ∈ § N , we define s(σ ) = a1 + a2 + · · · + a N . Since each of ±1 is taken with probability 1/2, we would expect s(σ ) to be zero “on average.” In fact,  1  s(σ ) = (s(σ ) + s(−σ )) = 0. 2 σ ∈§ σ ∈§ N

N

Writing σ = (a1 (σ ), . . . , a N (σ )), we also see that

The Central Limit Theorem in Algebra and Number Theory





s(σ )2 =

σ ∈§ N



103

ai (σ )a j (σ )

σ ∈§ N 1i, jN

=



⎛ ⎝N +

σ ∈§ N

= N 2N +



⎞ ai (σ )a j (σ )⎠

i= j



ai (σ )a j (σ ).

σ ∈§ N i= j

For each σ ∈ § N and i = j, define σˇ to be the same as σ except ai (σˇ ) = −ai (σ ), a j (σˇ ) = a j (σ ). It is then transparent that interchanging sums in the last summation and pairing σ with σˇ , the sum vanishes and we obtain 

s(σ )2 = N 2 N .

σ ∈§ N

In other words, s(σ ) has mean zero and variance N . Inspired by the Chebycheff inequality, we deduce P(σ : |s(σ )| > N 1/2+ ) 

1 , N 2

which goes to zero as N tends to infinity. In other words, we can expect that for any random sequence a1 , a2 , . . . of ±1’s that a1 + a2 + · · · + a N = O(N 2 + ) 1

with probability 1. This is essentially Chebycheff’s inequality (1867). In fact, more can be shown: Theorem 1.1 (de Moivre (1738), Laplace (1812))   β 1 s(σ ) 2 e−t /2 dt P σ :α √ β → √ 2π α N as N → ∞. This beautiful theorem originates from the combinatorial problem of counting the number of heads or tails one can expect in N random flips of a fair coin. What is intriguing about the theorem is its suggestive connection to the Riemann hypothesis where we can view the values of λ(n) as a random collection of ±1’s. Viewed in this way, Eq. (1) seems plausible. We will return later to the theme of the Riemann hypothesis and its connection to probability theory.

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2 Review of Some Probabilistic Concepts Some remarks pertaining to “probabilistic thought” are worth repeating here. Let X be a discrete random variable assuming the values x1 , x2 , . . .. The probability of the event X = x j will be denoted as P(X = x j ). The function f (x j ) := P(X = x j ) is called the probability distribution of the random variable X . Clearly, f (x j )  0 and



f (x j ) = 1.

(2)

j

It is possible to go in the “reverse” direction. Suppose we are given a set of points x1 , x2 , . . . and a function f defined on these points satisfying (2), then it is customary to speak of a random variable X assuming values x1 , x2 , . . . with probabilities f (x1 ), f (x2 ), . . .. Thus, given f satisfying (2), we say “let X be a random variable with distribution f .” The notion of “independent random variables” is undoubtedly familiar to the reader. However, it may help to elucidate our understanding if we recall this notion from the “probabilistic mind set.” Indeed, we say X 1 , X 2 , . . . X n are independent if P(X = α1 , X 2 = α2 , . . . X n = αn ) = P(X 1 = α1 )P(X 2 = α2 ) · · · P(X n = αn ). Thus, if X k depends only on the outcome of the kth trial and not on the previous outcomes, then the variables X 1 , X 2 , . . . X n are mutually independent. We refer the reader to p. 205 of [17] for further clarification of these concepts. We have discussed these ideas in the discrete random variable case with a view to our applications below. They undoubtedly apply in an analogous fashion to the non-discrete case.

3 The Evolution of the Central Limit Theorem The de Moivre–Laplace theorem evolved for almost a century into the modern central limit theorem. Beginning with the work of Chebycheff (1887), and then his two pupils Markov (1898) and Lyapunov (1901), Lindeberg (1922), and finally Levy (1935) and Feller (1935), the central limit theorem morphed into its modern form. Feller [16] writes that “For more than one hundred years a great many mathematicians have been working on the problem discovering many special cases to which the theorem applies and gradually establishing, and relaxing step for step, sufficient conditions under which the theorem holds. To the less critical mind, the law appeared as a universal law or, occasionally, as a law of nature.”

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Theorem 3.1 (The central limit theorem) Suppose for each n, X n1 , X n2 , . . . , X nrn are independent random variables on some probability space ϒn with measure Pn . Put Sn = X n1 + X n2 + · · · + X nrn . Assume the expectation E(X nk ) = 0 and set σn2 =

rn 

2 σnk

k=1 2 2 where σnk = E(X nk ). Then

  β 1 Sn (ω) 2 β → √ e−t /2 dt, P ω:α σn 2π α and max krn

2 σnk →0 σn2

as n → ∞ if and only if rn  1 2 X nk d Pn = 0 2 n→∞ σ k=1 n |X nk |>σn lim

(3)

for every  > 0. Remark 1 The sufficiency was shown by Lindeberg (1922) and the necessity by Feller (1935). The condition (3) is often referred to as the Lindeberg condition. According to the historical article [31], Alan Turing also discovered this independently in 1934 while still an undergraduate at the age of 22. The version of the central limit theorem given in our theorem can be found on p. 408 of [4]). Here is an elegant application of the central limit theorem (which was discovered by Ramanujan independently and without considerations of probability). Ramanujan ([29], p. 323) showed n  1 nk lim e−n = . n→∞ k! 2 k=0 We can deduce this via the central limit theorem as follows. If X 1 , X 2 , . . . is a sequence of independent random variables each having the Poisson distribution with parameter 1, then each has mean 1 and variance 1. By the central limit theorem, the

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sum

X1 + X2 + · · · + Xn − n √ n

approaches the normal distribution as n → ∞. However, the sum of two Poisson random variables with parameter x and y is again Poisson with parameter x + y. Thus, X1 + X2 + · · · + Xn is again Poisson with parameter n with mean n and variance n. We immediately deduce that  P

 0 n  1 nk 1 X1 + X2 + · · · + Xn − n 2 −n →√ e−t /2 dt = . 0 =e √ k! 2 n 2π −∞ k=0

4 The Evolution of Probabilistic Number Theory In 1917, Hardy and Ramanujan [20] proved the following theorem. Theorem 4.1 Let ω(n) denote the number of distinct prime factors of n. For any fixed  > 0, the number of n  x such that |ω(n) − log log n| > (log log n)1/2+ is at most

x . (log log x)2

Remark 2 In other words, almost all numbers have log log n prime factors (in the sense of natural density). Their proof was technically complicated. Precisely, we say ω(n) has normal order log log n. In general, an arithmetic function f (n) has normal order g(n) (where g(n) is a continuous monotone function) if given any  > 0, the number of n  x such that | f (n) − g(n)| > g(n) is o(x) as x → ∞. Now, ω(n) and (n) are examples of additive functions. In other words, (mn) = (m) + (n) whenever m and n are coprime. Thus, one can write ω(n) =

 p|n

ω( p),

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and it is suggestive to view ω(n) as a sum of “independent random variables” when looked at this way. However, this viewpoint was long in coming and had a roundabout emergence taking several decades. In 1934, Paul Turán [30] gave an elementary proof of the Hardy–Ramanujan theorem which was extremely simple. He did this by showing that 

(ω(n) − log log n)2 = O(x log log x).

nx

In retrospect, one can see that what Turán did was a number theoretic analogue of the celebrated Chebycheff inequality in probability theory. His two-page paper gave rise to a spectacular cascade of events and ultimately led to the development of probabilistic number theory. In a letter to Elliott (p. 18 of [15]) written in 1976, Turán recalls “When writing Hardy first in 1934 on my proof of Hardy–Ramanujan’s theorem I did not know what Tchebycheff’s inequality was and a fortiori of the central limit theorem. Erd˝os, to my best knowledge, was at that time not aware too. It was Mark Kac who wrote to me a few years later that he discovered when reading my proof in JLMS that this is basically probability and so was his interest turned to this subject.” Apparently, Kac asked if Turán could prove similar estimates for the higher moments:  (ω(n) − log log n)k . nx

Apparently, Kac hinted in his letter that if one could derive similar estimates, then there is a normal distribution law for ω(n). Turán continues that though he realized he could estimate the higher moments, he “found absolutely no interest to do it actually.” His reasons, he confesses, for not doing so were that he saw no applications of such results! It was Mark Kac who picked up the sequence of ideas leading to the possibility of applying the central limit theorem to additive number theoretic functions. He recalls (see page 24 of [15]) that “If I remember it correctly I first stated (as a conjecture) the theorem on the normal distribution of the number of prime divisors during a lecture in Princeton in March 1939. Fortunately for me and possibly for Mathematics, Erd˝os was in the audience, and he immediately perked up. Before the lecture was over he had completed the proof, which I could not have done not having been versed in the number theoretic methods, especially those related to the sieve.” He continues, “With Erd˝os’s contribution it became clear that we have had a beginning of a nice chapter of Number Theory, bringing upon it to bear the concepts and methods of Probability Theory.” So in 1940, Erd˝os and Kac [14] proved that   β 1 ω(n) − log log n 2 e−t /2 dt. β → √ P nN :α √ log log n 2π α

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Although the metaphor of the central limit theorem suggests this result, it is not a corollary of it. In the 1950s Kubilius [22] generalized this result to arbitrary additive functions.

5 Goncharov’s Theorem Around 1942, unaware of the developments in number theory along the lines of the Erd˝os–Kac theorem, Goncharov considered the following problem related to the symmetric group on n letters, denoted n . For each σ ∈ n , it is elementary that it can be written uniquely (unique up to ordering) as a product of disjoint cycles and one can define ω(σ ) to be the number of cycles in its unique factorization, viewing it as a group-theoretic analogue of ω(n) of Hardy and Ramanujan. For suggestive reasons, we write ω(σ ) and ω(n), it being clear from the discussion the context we are in. One can directly apply Theorem 3.1 to show that ω(σ ), when appropriately normalized, is normally distributed (no pun intended). The reader will recall from basic algebra that every permutation σ ∈ n can be written as a product of disjoint cycles in a canonical way as follows. We begin with 1 which is mapped to σ (1) which in turn is mapped to σ 2 (1) and so forth. In other words, the canonical decomposition of σ as a product of disjoint cycles is written as an ordered sequence of orbits, beginning with the orbit of 1 under σ , then the orbit of the smallest number not in the orbit of 1 and so on. To illustrate, let us consider the permutation σ of 8 given by  σ =

12345678 35416872



This has the canonical disjoint cycle decomposition σ = (134)(2568)(7),

(4)

so that ω(σ ) = 3. We define random variables X k (1  k  n) on n by setting X k = 1 if a cycle is completed at the kth step and otherwise we set X k = 0. For example, with σ as in (4), we have X 3 (σ ) = X 7 (σ ) = X 8 (σ ) = 1, X 1 (σ ) = X 2 (σ ) = X 4 (σ ) = X 5 (σ ) = X 6 (σ ) = 0. Clearly, X 1 (σ ) = 1 if and only if σ (1) = 1 and the number of such permutations is 1 (n − 1)! so that P(X 1 = 1) = . n 1 since there are n − More generally, we see that P(X k = 1) is given by (n−(k−1)) (k − 1) choices at the kth step to complete a cycle. Indeed, the permutations σ ∈ n

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109

for which X k (σ ) = 1 have a disjoint cycle decomposition where at the kth position, we also have a “close bracket.” Since the outcome at the kth step does not depend on the earlier outcomes, the X k ’s are independent. We will now write X nk to denote X k for obvious reasons. This example and formulation are discussed on p. 242 of [17] and p. 78 of [4]. Clearly Sn (σ ) = X n1 (σ ) + X n2 (σ ) + · · · + X nn (σ ) is the number of cycles in the cycle representation of σ . The mean m nk of X nk is the probability that X nk equals 1,

1 1 2 and variance is σnk = m nk (1 − m nk ). If L n = nk=1 , then Sn which is n−k+1 k has mean n n   1 mk = = Ln n − k+1 k=1 k=1 and variance

n 

m nk (1 − m nk ) = L n + O(1).

k=1

Applying Chebychev’s inequality and the fact that L n = log n + O(1), one can prove an analogue of Theorem 4.1 (as outlined in the next section) concluding that most permutations on n letters have about log n cycles in their unique disjoint cycle decomposition. We can now apply the central limit theorem to deduce Goncharov’s theorem because the Lindeberg condition is vacuously satisfied. Indeed, the X nk ’s are bounded random variables and σn → ∞ so that for n large, the sequence in Lindeberg’s limit condition eventually becomes the zero sequence. Thus, unlike the Erd˝os–Kac theorem, Goncharov’s theorem can be deduced via the central limit theorem. Goncharov’s paper is long and complicated. The underlying conceptual fabric is missing. Harper [21] is less sympathetic. He writes, “Goncharov …by brute force tortuously manipulates the characteristic functions of the distributions until they approach exp(−x 2 /c), c a positive constant.”

6 The Connection to Stirling Numbers of the First Kind One can also approach the problem in the previous section from a different combinatorial perspective. Following Turán, we can consider  σ ∈n

ω(σ ),

and

 σ ∈n

ω2 (σ ),

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although this is not the way Goncharov thought about the problem. However, interpreted this way, the question becomes an elementary problem in basic combinatorics. Denoting s(n, k) to be the signed Stirling number of the first kind, and recalling that |s(n, k)| is the number of permutations in n with exactly k disjoint cycles in its factorization (see pg. 26 of [10]), we immediately see that 

ω(σ ) =

σ ∈n

n 

k|s(n, k)|,



and

ω2 (σ ) =

σ ∈n

k=1

n 

k 2 s(n, k)|.

k=1

Now it is elementary that x(x + 1) · · · (x + n − 1) =

n 

|s(n, k)|x k .

(5)

k=1

That is, the unsigned Stirling numbers |s(n, k)| are defined algebraically as the coefficients of the rising factorial. Differentiating both sides of this polynomial identity and setting x = 1, we deduce 

ω(σ ) = n!

σ ∈n

n  1 j j=1

= n!Hn , n 1 where Hn denotes the nth harmonic number. Since 1 d x = log n, we get (by an x application of the method of proof for the integral test) log n +

1  Hn  log n + 1. n

More precisely, Hn = log n + γ +

1 1 (n) + + , 2 2n 12n 120n 4

where γ denotes Euler’s constant and 0 < (n) < 1 for all n. So, we get  σ ∈n

  1 . ω(σ ) = n! log n + γ + O n 

In other words, the average number of disjoint cycles of a random permutation of n is log n. Similarly, in order to calculate the second moment of ω(σ ), we differentiate (5) twice and set x = 1 to get,

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111

⎞ n n  n   1 1 ⎠ |s(n, k)|k(k − 1) = n! ⎝ − 2 i j i k=0 i=1 j=1 i=1 ⎛ ⎞ n n n n  n     1 1 ⎠. =⇒ − |s(n, k)|k 2 = k|s(n, k)| + n! ⎝ 2 i j i k=0 k=0 i=1 j=1 i=1 ⎛

n 

Now, n 2 n n   1 1 = ij i i=1 j=1 i=1   2 1 = log n + γ + O n   log n 2 . = (log n) + 2γ log n + O n Also,

n

1 = O (1). Putting these together, we get that the second moment is i2     log n . ω(σ )2 = n! (2γ + 1) log n + γ + (log n)2 + O n σ ∈ i=1

n

Then the variance   1  log n 2 . (ω(σ ) − log n) = log n + γ + O n! σ ∈ n

(6)

n

From here, one can now derive an analogue of Turán’s theorem for symmetric groups. To show ω(σ ) has normal order log n, one needs to show that given any  > 0, the number of σ ∈ n such that |ω(σ ) − log n| >  log n is o(n!). Using Chebychev’s inequality, this is  O

n!  2 log n



which is o(n!) as n → ∞ for any  > 0. More precisely, it is worth noting here that given a fixed  A > 0,the approach of n! Turán applied to this problem gives a nice estimate of O for the excepA log n

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tional set, that is, the number of permutations for which ω(σ ) deviates from its mean value log n by more than A log n. In order to understand the distribution of ω(σ ), one can now consider the higher moments  (ω(σ ) − log n)k σ ∈n

for k  2. Using the binomial theorem, one can show that this is k     k (log n)k− j ω(σ ) j j σ ∈ j=0

=

k   j=0

n

 n  k (log n)k− j m j |s(n, m)|. j m=0

Denoted by M j the inner sum M j :=

n 

m j |s(n, m)|.

(7)

m=0

Estimation of this sum for a general j seems quite complicated and hence we do not pursue this method in this paper any further. One can, however, apply   β 1 ω(σ ) − log n 2 e−t /2 dt, β → √ P σ :α √ log n 2π α as n tends to infinity, to deduce the behaviour of (7). Goncharov in [18] takes another approach to prove the above result. Though the Erd˝os–Kac theorem cannot be deduced from the central limit theorem, it is possible to derive a modified version of the central limit theorem to do so. We refer the reader to the papers of Billingsley [5, 6] for further amplification of this discussion. The polynomial identity (5) can be given a probabilistic interpretation. Indeed, if we write  n n    t +n−k |s(n, k)|t k = , (8) k! n−k+1 k=1 k=1 we can interpret each factor on the right as the probability generating function of X k . Since the X k ’s are independent and ω(σ ) =

 k

X k (σ ),

The Central Limit Theorem in Algebra and Number Theory

we have

E(t ω ) =



113

E(t X k ).

k

In other words, we can deduce (8) as a consequence of the independence of the X k ’s from purely probabilistic considerations. There is a well-known duality principle in combinatorics that relates certain theorems about the symmetric group n to theorems about partitions of sets of n elements. This is not a rigid principle but only a metaphor. As such, the analogue of Goncharov’s theorem has been worked out by Harper, and we refer the reader to [21] for further details.

7 Normal Number of Prime Factors of Fourier Coefficients of Modular Forms The generalization of the Erd˝os–Kac theorem to study the normal number of prime factors of Fourier coefficients of modular forms was initiated in 1984 by the second author and Kumar Murty in [25, 26]. We now describe their work and very briefly indicate future directions. To expedite our exposition, we only discuss the case of the Ramanujan τ -function and refer the reader to [25, 26] for the general case of modular forms. Recall that the Ramanujan τ -function is defined via the infinite product: q

∞  n=1

(1 − q n )24 =

∞ 

τ (n)q n ,

q = e2πi z

n=1

with (z) > 0. Ramanujan conjectured that τ (n) is a multiplicative function of n and that |τ ( p)|  2 p 11/2 . As is well-known, the Ramanujan conjecture was proved by Deligne [12] as a culmination of his work on the Weil conjectures. In [25], the second author and K. Murty show subject to a generalized quasiRiemann hypothesis (more precisely, there exists a 1/2 < δ < 1 such that all Artin L-functions have no zeros in (s) > δ) that 

(ω(τ ( p)) − log log p)2 = O(π(x) log log x),

px τ ( p)=0

where the summation is over primes p. In other words, the normal order of ω(τ ( p)) is log log p. In [26], they extend this work and establish the analogue of the Erd˝os– Kac theorem, again subject to the same generalized quasi-Riemann hypothesis. More generally, they studied ω(τ (n)) and showed that

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ω(τ (n)) − 21 (log log n)2 P n:α β √ (log log n)3/2 / 3

1 →√ 2π



β

e−t

2

/2

dt

α

In current work in progress, the authors hope to extend these studies to shifts of primes. We expect that, for example, ω(τ ( p + a)) with a = 0 has normal order (log log p)2 /2. We also expect an analogue of the Erd˝os–Kac theorem to hold for these shifts.

8 Probabilistic Connections to the Riemann Hypothesis The Riemann zeta function ζ (s), originally defined as a Dirichlet series ζ (s) =

∞  1 , ns n=1

for (s) > 1 can be analytically continued to the entire complex plane except for s = 1 where it has a simple pole. The celebrated Riemann hypothesis is the statement that the real part of all the non-trivial zeroes of ζ (s) is 21 . This fugitive Riemann hypothesis has been both a source of inspiration and frustration for many generations of mathematicians. It is said that Hilbert and Pólya were the first to suggest that if we could interpret the non-trivial zeroes of ζ (s) as related to eigenvalues of some Hermitian operator, the Riemann hypothesis would follow. But the hypothesized Hermitian operator has not been found yet. Probability theory may offer us a window into interpreting the zeroes of ζ (s) and give them new meaning. Indeed, in a paper of Biane, Pitman and Yor [3], we find the following exposition. Let ∞  2 e−n πt θ (t) = n=−∞

be the classical Jacobi theta function. It is well known that it satisfies the modular transformation √ tθ (t) = θ (1/t), t > 0. (9) By means of this transformation, one can show that if we define ξ(s) := then

1 s(s − 1)π −s/2 (s/2)ζ (s), 2

4ξ(s) = s(s − 1)

0

∞

(θ (t) − 1)t s/2

dt , t

(10)

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115

which is valid for (s) > 1, can be extended analytically to the entire complex plane and deduce that ξ(s) = ξ(1 − s), which is the celebrated functional equation of the Riemann zeta function. Now, if we define ∞  2 2 2 e−n π y , G(y) := θ (y ) = n=−∞

then (9) becomes G(1/y) = yG(y).

(11)

Set d H (y) = dy



d y G(y) dy



2

= 2yG (y) + y 2 G

(y), which by definition of G(y) becomes H (y) = 4y

2

∞ 

(2π n y − 3π n )e 2 4 2

2

−πn 2 y 2

n=1

and H (y) satisfies the same functional equation as G, namely y H (y) = H (y −1 ),

y > 0.

Then Riemann’s formula (10) for ξ(s) becomes

∞

2ξ(s) =

H (y)y s

0

dy . y

Since 2π 2 n 4 y 2 > 3π n 2 for y  1, we see H (y) > 0 in this region. But then, by the functional equation, we have H (y) > 0 for y > 0 also. A routine computation shows that 2ξ(0) = 2ξ(1) = 1 and so

0

∞

H (y) dy = y

0

∞

H (y)dy = 1.

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Thus, the function H (y) can be viewed as the density function of a probability distribution on (0, ∞) with mean 1. If Y is a random variable with this distribution, the functional equation for H translates as E( f (1/Y )) = E(Y f (Y )). Also, one can view 2ξ(s) as E(Y s ). In 1997, Xian-Jin Li [23] derived a remarkable criterion for the truth of the Riemann hypothesis. To state Li’s criterion, we define for each natural number n,     1 n 1− 1− , λn = ρ ρ where the sum is over the non-trivial zeros of the Riemann zeta function. Then, Li’s criterion is that the Riemann hypothesis is true if and only if λn  0, for all natural numbers n. It is not difficult to see that λn =

  d n  n−1 1 log ξ(s) s  . s=1 (n − 1)! ds n

Using Leibniz’s rule,

 n−1   kn− j n−1 , j (n − j)! j=0

λn = n where kn =

(12)

 dn  ξ(s)) (log  . s=1 ds n

If all the kn ’s were positive, then by (12), the Riemann hypothesis follows. However, this is not always the case. For example, k3 = −0.000222316, k4 = −0.0000441763 according to the table on pg. 441 of [3]. In fact, one can interpret the kn ’s as cumulants of the random variable log(1/Y ) with Y as before. Recall that given a random variable X , the moment generating function is E(et X ) and the cumulant generating function is log E(et X ). If we let L := log(1/Y ), then as

The Central Limit Theorem in Algebra and Number Theory

117

2ξ(s) = E(Y s ) = E(Y 1−s ) = E(e(s−1)L ), we see that

∞ 

log(e(s−1)L ) = log 2ξ(s) =

kn

n=1

(s − 1)n . n!

The cumulants kn ’s are thus related to the moments of L as follows μn = E(L n ) =

∞

(− log y)n

0

through the formula μn =

 n−1   n−1 j=0

j

H (y) dy y

μ j kn− j .

(13)

(14)

To see this, observe that taking the derivative with respect to t of the equation log E(et X ) =

∞  n=1

kn

tn , n!

we see (upon setting X = log(1/Y )) that (14) follows. The positivity of the first cumulant k1 is assured by Jensen’s inequality since k1 = μ1 = −E(log Y )  − log E(Y ) = 0.

(15)

Recall that for any convex function φ defined on the range of a random variable X , Jensen’s inequality states that E(φ(X ))  φ(E(X ))

(16)

so that (15) follows on applying (16) to φ(x) = − log x which is convex (as it has a positive second derivative). The positivity of the second cumulant can also be deduced through considerations of probability. Indeed, k2 = μ2 − μ21 is the variance of L, and therefore positive. However, as stated earlier, k3 , k4 are negative. Let us record here the following curious related fact. In [19], it was shown that the Riemann hypothesis is true if and only if  1 = 2 + γ − log 4π, |ρ|2 ρ

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where γ is Euler’s constant and the sum is over non-trivial zeroes of ζ (s). The cumulants kn are related to the Stieltjes constants γn which are defined as the coefficients of the Laurent expansion of ζ (s) about s = 1, ∞

ζ (s) =

 1 + γn (s − 1)n , s − 1 n=0

where γ0 = γ is Euler’s constant and one views the γn ’s as generalizations of this. In fact, it is not difficult to show that γn are given by the limits γn = lim

m→∞

m  logn k k=1

k

logn+1 m − n+1

.

We are more interested in the generalized Stieltjes constants ηn given by the Laurent ζ

expansion of (s) about s = 1. Thus, ζ ∞

−

 1 ζ

(s) = + ηn (s − 1)n . ζ s − 1 n=0

By basic algebra, it is easy to express the ηn ’s as polynomials in the constants γn with rational coefficients. For example, 1 η0 = −γ0 , η1 = −γ1 + γ02 . 2 The significance of these constants lies in an important arithmetic formula for the λn ’s derived by Bombieri and Lagarias [7]: λn = 1 − where S1 (n) = and S2 (n) =

n (γ + log 4π ) + S1 (n) + S2 (n), 2

(17)

  n    n 1 (−1) j 1 − j ζ ( j) j 2 j=2

n   n     n n η j−1 = −nγ0 + η j−1 . j j j=1 j=2

(18)

They also showed that the condition of positivity can be considerably weakened to deduce the Riemann hypothesis. In fact, they show that if for any  > 0, there is a constant c() > 0 such that for all n  1,

The Central Limit Theorem in Algebra and Number Theory

119

λn  −c()en then the Riemann hypothesis follows. Clearly, this is a substantial weakening of Li’s criterion. Coffey [11] has shown that for n  2, 1 (n(log n + γ − 1) + 1)  S1 (n)  (n(log n + γ + 1) − 1). 2 In particular, S1 (n) is non-negative for all n  2 so that apart from S2 (n), the contributions of the other terms to λn in (17) is O(n log n). Thus the truth of Riemann hypothesis hinges on S2 (n) and the growth of the generalized Stieltjes constants. Omar and Bouanani [8] extend Li’s criterion to the function field setting. Clearly, we should, therefore, focus our attention on η j ’s. As noted in [7], one can show that ⎛ ⎞ j j j+1  (log x) ⎠ (m) log m (−1) lim ⎝ − ηj = , j! x→∞ mx m j +1 

where (m) =

log p, m = p α , α  1, 0, otherwise.

This expression is unwieldly. We offer an alternate expression via Laguerre polynomials.

For ψ(x) = nx (n), we know ζ

− (s) = s ζ



∞

ψ(x) dx x s+1 1 ∞ s ψ(x) − x = +s dx s−1 x s+1 1 ∞ 1 ψ(x) − x = +1+s d x. s−1 x s+1 1

Here, we can write the integral as

∞

(ψ(x) − x) −(s−1) log x e dx x2 1 ∞  (−1) j (s − 1) j ∞ (x) log j x =(s − 1 + 1) d x, j! x2 1 j=0 (s − 1 + 1)

where (x) = ψ(x) − x. If we let

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∞

δj = 1

then ηj =

(x) log j x d x, x2

(19)

(−1) j δ j (−1) j−1 δ j−1 + . j! ( j − 1)!

(20)

It is possible to derive estimates for η j , δ j and γ j but all of these estimates have exponential growth. So we need to explore cancellations. In this context, we have the Laguerre polynomials: L n (x) =

n    n (−x) j

j

j=0

j

.

The generalized Laguerre polynomials which can be defined by the following generating function (see [1, 9, 24]): ∞ 

n L (α) n (x)t =

n=0

  1 tx exp − , (|t| < 1) (1 − t)α+1 1−t

can also be defined by the closed formula for n  0, L (α) n (x)

=

n 

 (−1)

k=0

where

r  k

k

 n + α xk n − k k!

(21)

denote the binomial coefficients given by     r r (r − 1)(r − 2) · · · (r − k + 1) r = 1 and = 0 k k(k − 1) · · · 1

for positive integer k and any complex number r . In order to understand S2 (n), we can try to get the expression of η j−1 in (18) in terms of integrals of generalized Laguerre polynomials. Using (18) and (20), we get S2 (n) = −nγ0 +

n    (−1) j−1 δ j−1 n j=2

j

( j − 1)!

= −nγ0 + T1 (n) + T2 (n) where using (19)

+

(−1) j−2 δ j−2 ( j − 2)!



The Central Limit Theorem in Algebra and Number Theory

T1 (n) =

121

n    n (−1) j−1 δ j−1

( j − 1)! ⎛ ⎞ n   (x) ⎝ n (− log x) j−1 ⎠ dx j x2 ( j − 1)! j=2 j

j=2



∞

= 1

and n    n (−1) j−2 δ j−2

T2 (n) =

( j − 2)! ⎛ ⎞ n   (x) ⎝ n (− log x) j−2 ⎠ d x. j x2 ( j − 2)! j=2 j

j=2



∞

= 1

Writing j = j − 1 and using expression for T1 (n) as

n  k

=



n n−k



for 0  k  n, we can rewrite the sum in

 n−1  j

 n j (log x) (−1) j + 1 ( j )! j =1 =

 n−1   (n − 1) + 1 j =1

(n − 1) − j

(−1) j



(log x) j ( j )!



= − n + L (1) n−1 (log x). A similar calculation for the sum in the expression for T2 (n) gives n    n (− log x) j−2 j=2

=

j

( j − 2)!



n−2   (n − 2) + 2 (− log x) j j =0

(n − 2) − j

( j )!

=L (2) n−2 (log x). Thus, we get the following expression for S2 (n) in terms of the generalized Laguerre polynomials



 (x)  (1) (2) L (log x) + L (log x) d x. n−1 n−2 x2 1 1 (22) The integral in the last term of (22) can be simplified as follows: S2 (n) = −nγ0 − n

∞

(x) dx + x2

∞

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(2) L (1) n−1 (log x) + L n−2 (log x)   n−1  n−2  k   n n (log x)k k (log x) = (−1) (−1)k + n−1−k n−2−k k! k! k=0 k=0    n−2  n n (log x)n−1  (log x)k = + (−1)k + n−1−k n−2−k (n − 1)! k! k=0  n−2  n+1 (− log x)k (log x)n−1  = + n−1−k (n − 1)! k! k=0    n−1  n−1+2 (log x)n−1  n − 1 + 2 (− log x)k (log x)n−1 − + = n−1−k (n − 1)! n − 1 − (n − 1) (n − 1)! k! k=0  n−1   n − 1 + 2 (− log x)k = n−1−k k! k=0

=L (2) n−1 (log x). Thus, we get

∞

S2 (n) = −nγ0 + 1

(x) (2) L (log x)d x − n x 2 n−1



∞ 1

(x) d x. x2

(23)

The last term in the above equation is actually O(n) by a simple application of the unconditional error term in the prime number theorem. So, to prove the Riemann hypothesis, we need to focus on the integral in the second term in view of the results of Bombieri and Lagarias.

9 Concluding Remarks The discussion of the preceding section can also be applied to study the generalized Riemann hypothesis from a probabilistic perspective. In fact, this perspective is the origin of the large sieve method. The probabilistic model can give us some idea of what we can expect. Indeed, treating for example, the Legendre symbols χ p (n) := (n/ p) with p prime, as a random variable, we can prove the following: let π˜ (x) denote the number of primes between x and 2x. Let z = z(x) be such that log z →∞ log x as x → ∞. Then, for any continuous real-valued function h, we have

The Central Limit Theorem in Algebra and Number Theory

1 lim h x→∞ z



x p2x



nz

χ p (n)

π(x) ˜

1 =√ 2π

123



∞

h(t)e−t

2

/2

dt.

−∞

In particular, if h(x) = |x|, this says that GRH holds “on average.” We refer the reader to the forthcoming paper [27]. To keep our survey succinct, we have refrained from giving an encyclopedic treatment of this topic. However, there is one result from probability theory that requires highlighting. In 1918, Hardy and Ramanujan developed their celebrated circle method to study the partition function. The reader will recall that the number of partitions of n is denoted p(n) and has the generating function ∞ 

p(n)t = n

n=0

∞ 

(1 − t n )−1 .

n=1

Thus, p(4) = 5 since 1 + 1 + 1 + 1, 1 + 1 + 1 + 2, 1 + 3, 2 + 2, 4 are the five partitions of 4. Hardy and Ramanujan proved that p(n) ∼

√ 1 √ eπ 2n/3 . 4n 3

(24)

The second author, in joint work with Dewar [13] showed how one can derive this using arithmetic. Báez-Duarte [2] showed that a local central limit theorem can be used to establish (24). These remarkable symbiotic developments will be studied in a later paper. Acknowledgements We would like to thank the referee and Brundaban Sahu for their detailed comments and helpful suggestions on a previous version of the paper.

References 1. Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists: A Comprehensive Guide. Academic, Oxford (2011) 2. Báez-Duarte, L.: Hardy-Ramanujan’s asymptotic formula for partitions and the central limit theorem. Adv. Math. 125(1), 114–120 (1997) 3. Biane, P., Pitman, J., Yor, M.: Probability laws related to the Jacobi theta and Riemann zeta functions and Brownian excursions. Bull. Amer. Math. Soc. 38, 435–465 (2001) 4. Billingsley, P.: Probability and Measure. Wiley Series in Probability and Statistics, Anniversary edn. Wiley, Hoboken (2012) 5. Billingsley, P.: On the central limit theorem for the prime divisor functions. Amer. Math. Monthly 76, 132–139 (1969) 6. Billingsley, P.: Prime numbers and Brownian motion. Amer. Math. Monthly 80, 1099–1115 (1973) 7. Bombieri, E., Lagarias, J.: Complements to Li’s criterion for the Riemann hypothesis. J. Number Theory 77(2), 274–287 (1999)

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8. Omar, S., Bouanani, S.: Li’s criterion and the Riemann hypothesis for function fields II. Finite Fields Appl. 45, 53–58 (2017) 9. Carlitz, L.: Some generating functions for Laguerre polynomials. Duke Math. J. 35, 825–827 (1968) 10. Cioab˘a, S.M., Murty, M.R.: A first course in graph theory and combinatorics. Texts and Readings in Mathematics, vol. 55. Hindustan Book Agency, New Delhi (2009) 11. Coffey, M.W.: Toward verification of the Riemann hypothesis: application of the Li criterion. Math. Phys. Anal. Geom. 8, 211–255 (2005) 12. Deligne, P.: La conjecture de Weil. I. Publications Mathématiques de l’IHÉS 43, 273–307 (1974) 13. Dewar, M., Murty, M.R.: A derivation of the Hardy-Ramanujan formula from an arithmetic formula. Proc. Amer. Math. Soc. 141(6), 1903–1911 (2013) 14. Erd˝os, P., Kac, M.: The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62, 738–742 (1940) 15. Elliott, P.D.T.A.: Probabilistic Number Theory II. Central Limit Theorems. Springer, Berlin (1980) 16. Feller, W.: The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51, 800–832 (1945) 17. Feller, W.: An Introduction to Probability Theory and Its Applications, 1st, 2nd edn. Wiley, New York (1957) 18. Goncharov, V.: On the field of combinatory analysis. Amer. Math. Soc. Transl. 2(19), 1–46 (1962) 19. Gun, S., Murty, M.R., Rath, P.: Transcendental sums related to the zeros of zeta functions. Mathematika 64(3), 875–897 (2018) 20. Hardy, G.H., Ramanujan, S.: The normal number of prime factors of a number n. Q. J. Math. 48, 76–92 (1917) 21. Harper, L.H.: Stirling behaviour is asymptotically normal. Ann. Math. Stat. 38, 410–414 (1967) 22. Kubilius, J.: Probabilistic methods in the theory of numbers. Translations of Mathematical Monographs, vol. 11. American Mathematical Society, Providence (1964) 23. Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis. J. Number Theory 65, 325–333 (1997) 24. Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, New York (1966) 25. Murty, M.R., Murty, V.K.: Prime divisors of Fourier coefficients of modular forms. Duke Math. J. 51, 57–77 (1984) 26. Murty, M.R., Murty, V.K.: An analogue of the Erd˝os-Kac theorem for Fourier coefficients of modular forms. Indian J. Pure Appl. Math. 15, 1090–1101 (1984) 27. Murty, M.R., Prabhu, N.: Central limit theorems for sums of quadratic characters, modular forms and elliptic curves. To appear in Proceedings of the American Mathematical Society 28. Pólya, G.: Verschiedene Bemerkungen zur Zahlentheorie. Jahresber. Deutsch. Math.-Verein. 28, 31–40 (1919) 29. Ramanujan, S.: Collected Papers, Chelsea (1962) 30. Turan, P.: On a theorem of Hardy and Ramanujan. J. Lond. Math. Soc. 9, 274–276 (1934) 31. Zabell, S.L.: Alan turing and the central limit theorem. Amer. Math. Monthly 102, 483–494 (1995)

Rankin–Cohen Brackets and Identities Among Eigenforms II Arvind Kumar and Jaban Meher

Abstract In this article, we completely characterize all the cases for which Rankin– Cohen brackets of two quasimodular eigenforms are again eigenforms on the full modular group S L 2 (Z). This is a continuation of our work initiated in Kumar and Meher (Pac J Math 297:381–403, 2018) for a subclass of the space of quasimodular forms. Keywords Eigenforms · Nearly holomorphic modular forms · Quasimodular forms · Rankin–Cohen brackets 2010 Mathematics Subject Classification 11F25 · 11F30 · 11F37

1 Introduction and Statement of the Main Result For any even integer k ≥ 2, let Mk and Sk denote the respective spaces of modular forms and cusp forms of weight k on the full modular group S L 2 (Z). Furthermore, the Eisenstein series of weight k is defined by E k (z) = 1 −

∞ 2k  σk−1 (n)e2πinz , Bk n=1

A. Kumar School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] J. Meher (B) School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, P. O. Jatni, Khurda, Bhubaneswar 752050, Odisha, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_9

125

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 where Bk is the kth Bernoulli number, σk−1 (n) = d|n d k−1 and z is in the complex upper-half plane H. It is well known that E k ∈ Mk for k ≥ 4 but E 2 is a quasimodular form of weight 2 on S L 2 (Z) which is not a modular form. Due to the finite-dimensional nature of the space Mk , one expects many identities among modular forms. In some cases, we get nice relations among the Fourier coefficients of the modular forms from the identities of modular forms. One such identity is the following. σ7 (n) = σ3 (n) + 120

n−1 

σ3 (m)σ3 (n − m).

(1)

m=1

The above identity which involves the divisor functions follows from the identity E 42 = E 8 . The identity E 42 = E 8 is of a special kind among modular forms on S L 2 (Z) in which the product of two eigenforms gives rise to an eigenform. Therefore, one natural question is: Is it possible to get an exhaustive list of cases in which the product of two eigenforms gives an eigenform? Duke [3] and Ghate [5] independently addressed this question for eigenforms of the full modular group, proving there are only 16 such cases. One can generalize the above phenomenon in several directions, namely, considering the Rankin–Cohen brackets or arbitrary products of eigenforms; also, there are similar problems for higher level and also for other kinds of automorphic forms such as quasimodular forms and nearly holomorphic modular forms. Since then several mathematicians have worked in these directions (see [2, 8–10, 12] etc.). The phenomenon of products of arbitrary number of eigenforms giving eigenforms on S L 2 (Z) has been studied by Emmons and Lanphier [4]. It should be noted that the 0th Rankin–Cohen bracket of two modular forms is the product of these two modular forms. The problem of Rankin–Cohen brackets of eigenforms giving eigenforms on S L 2 (Z) has been studied by Lanphier and Takloo-Bighash [10]. We know that the space of quasimodular forms is a generalization of the space of modular forms. Therefore, one expects to obtain more identities in the case of quasimodular forms which is, in fact, true. One of the reasons to study the phenomenon 1 d is in the case of quasimodular forms is the identity D = E 2 , where D = 2πi dz the differential operator and  is the Ramanujan delta function. This identity can be interpreted as an identity among quasimodular forms, in which the product of two quasimodular eigenforms on S L 2 (Z) results in an eigenform. The case of the product of two quasimodular eigenforms on S L 2 (Z) has been studied in [2, 12]. The case of arbitrary products of quasimodular eigenforms on S L 2 (Z) has been studied in [8]. To state our main result, we first define Rankin–Cohen brackets of two quasimodular forms which is defined by Martin and Royer [11]. Let f and g be two quasimodular forms of weights k and l with depths s and t on S L 2 (Z), respectively. For any integer ν ≥ 0, the νth Rankin–Cohen bracket of f and g is defined by [ f, g]ν :=

ν  α=0

(−1)α

   k−s+ν−1 l −t +ν−1 D α f D ν−α g. ν−α α

(2)

Rankin–Cohen Brackets and Identities Among Eigenforms II

127

It is well known that [ f, g]ν is a quasimodular form of weight k + l + 2ν and depth at most s + t on S L 2 (Z). Throughout the paper, k denotes the unique normalized cusp form of weight k on S L 2 (Z) for k ∈ {12, 16, 18, 20, 22, 26}. In our previous paper [9], we initiated the study of Rankin–Cohen brackets of quasimodular eigenforms. More precisely, we proved the following result [9, Theorem 1.1]. Theorem 1 Let f and g be two quasimodular eigenforms such that the depth of each of the forms f and g is strictly less than half of the weight of the form. Then there are finitely many triples ( f, g, ν) with the property that f and g are quasimodular eigenforms and [ f, g]ν is again an eigenform. All the possible cases (up to some constant multiple) are the following: • [E 4 , E 4 ]0 = E 8 , [E 4 , E 6 ]0 = E 10 , [E 4 , E 10 ]0 = [E 6 , E 8 ]0 = E 14 , [E 4 , D E 4 ]0 = 21 D E 8 ; • if k, l ∈ {4, 6, 8, 10, 14} and ν ≥ 1 with k + l + 2ν ∈ {12, 16, 18, 20, 22, 26}, then [E k , El ]ν = cν (k, l)k+l+2ν , where cν (k, l) = −

    2k ν + k − 1 2l ν + l − 1 + (−1)ν+1 ; ν ν Bl Bk

• if k ∈ {4, 6, 8, 10, 14} and ν ≥ 0 with l, k + l + 2ν ∈ {12, 16, 18, 20, 22, 26}, then [E k , l ]ν = cν (l)k+l+2ν , where cν (l) = • [E 4 , D E 4 ]1 [E 6 , D E 6 ]1 [E 4 , D E 4 ]3 [E 4 , D E 4 ]5

  ν +l −1 ; ν

= 96012 , [E 4 , D E 8 ]1 = [E 8 , D E 4 ]1 = 192016 , = −302416 [E 4 , D E 6 ]2 = −504016 , [E 6 , D E 4 ]2 = 504016 , = 480016 , [E 8 , D E 8 ]1 = 384020 , [E 6 , D E 6 ]3 = −2822420 , = 1344020 ;

• [E 4 , D 2 E 4 ]1 = 960D12 , [E 4 , D E 6 ]1 = −2016D12 , [E 6 , D E 4 ]1 = 1440D12 , [E 4 , D E 4 ]2 = 2400D12 , [E 6 , D 2 E 6 ]1 = −3024D16 , [E 6 , D E 6 ]2 = −10584D16 , [E 4 , D 2 E 4 ]3 = 4800D16 , [E 4 , D E 4 ]4 = 8400D16 , [E 8 , D 2 E 8 ]1 = 3840D20 [E 8 , D E 8 ]2 = 17280D20 , [E 6 , D 2 E 6 ]3 = −28224D20 , [E 6 , D E 6 ]4 = −63504D20 ,

[E 4 , D 2 E 4 ]5 = 13440D20 , [E 4 , D E 4 ]6 = 20160D20 .

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Therefore, in [9], we could obtain an explicit list of cases for which Rankin–Cohen brackets of quasimodular eigenforms on S L 2 (Z) are again eigenforms, but only for a subspace consisting of forms whose depths are strictly less than half of their weights. The aim of this paper is to remove this condition from Theorem 1 and get a complete list of cases for the full space of quasimodular forms on S L 2 (Z). We now state the main result of this paper. Theorem 2 Let f and g be two quasimodular eigenforms on the full modular group S L 2 (Z). Then there are finitely many triples ( f, g, ν) (up to some constant multiples of f and g) with the property that f and g are quasimodular eigenforms such that [ f, g]ν is a quasimodular eigenform. All the possible cases are the following: 1 D 2 E , [E , D 2 E ] = − 1 D 3 E , [E , D E ] = − 1 D 3 E , • [E 2 , D E 2 ]1 = − 10 4 2 2 1 4 2 2 2 4 10 10 1 2 [E 2 , E 2 ]2 = − 5 D E 4 , [E 2 , D E 2 ]3 = −2412 , [E 2 , D 2 E 2 ]3 = −24D12 , [E 2 , D E 2 ]4 = −24D12 , [E 2 , E 2 ]4 = −4812 ; 4 D2 E6, [E 2 , D E 4 ]1 = − 10 D 2 E 6 , [E 4 , E 2 ]1 = 23 D E 6 , • [E 4 , D E 2 ]1 = 21 21 3 [E 6 , E 2 ]1 = 4 D E 8 , [E 4 , D E 2 ]2 = −24012 , [E 4 , D 2 E 2 ]2 = −240D12 ,

[E 6 , D E 2 ]2 = −504D12 , [E 2 , D E 4 ]2 = 24012 , [E 2 , D 2 E 4 ]2 = 240D12 , [E 8 , E 2 ]2 = −384D12 , [E 2 , D E 6 ]2 = −504D12 , [E 6 , E 2 ]2 = −100812 , [E 4 , D E 2 ]3 = −480D12 , [E 2 , D E 4 ]3 = 240D12 , [E 10 , E 2 ]2 = −158416 ,

[E 4 , E 2 ]3 = −72012 , [E 6 , E 2 ]3 = −840D12 , [E 4 , E 2 ]4 = −600D12 , [E 6 , E 2 ]4 = −352816 , [E 2 , 12 ]1 = 16 ; • [E 2 , 12 ]0 = D12 , • and all the cases presented in Theorem 1.

[E 8 , E 2 ]3 = −336016 , [E 4 , E 2 ]5 = −158416 ;

The only essentially new tool in this paper as compared to [9] is the proof of Theorem 8. Due to the non-availability of this result, we could not prove the result for the full space of quasimodular forms in [9]. The paper is organized as follows. In Sect. 2, we state some basic results in the theory of nearly holomorphic modular forms and quasimodular forms. Section 3 contains some results on Eisenstein series. In Sect. 4, we establish some intermediate results which are crucial to the proof of our main result. In Sect. 5, we prove our main result. At the end, we give a remark on the Chazy equation in Sect. 6.

2 Preliminaries We follow the notations and definitions used in [9] and refer to it for more details. We recall the following definitions. Definition 1 A nearly holomorphic modular form f of weight k and depth ≤ p for 1 S L 2 (Z) is a polynomial in Im(z) of degree ≤ p whose coefficients are holomorphic functions on H with moderate growth such that −k

(cz + d)

 f

az + b cz + d

 = f (z) (z ∈ H),

Rankin–Cohen Brackets and Identities Among Eigenforms II

for all

129

  ab ∈ S L 2 (Z), where Im(z) is the imaginary part of z. c d

Definition 2 A holomorphic function f on H is called a quasimodular form of weight k and depth p for S L 2 (Z) if there exist holomorphic functions f 0 , f 1 , f 2 ,…, f p on H such that (cz + d)−k f



az + b cz + d

 =

p  j=0

 f j (z)

c cz + d

j (z ∈ H),

  ab for all ∈ S L 2 (Z), f p is not identically vanishing and f has no terms with c d negative exponents in its Fourier expansion. k≤ p (resp. M k≤ p ) the complex vector space of all nearly holomorWe denote by M phic modular forms (resp. quasimodular forms) of weight k and depth at most p on k and M k be the space of all nearly holomorphic modular forms and S L 2 (Z). Let M quasimodular forms of weight k on S L 2 (Z), respectively. k≤ p , defined by The Maass–Shimura operator Rk on f ∈ M 1 Rk f (z) = 2πi ≤p



k ∂ + 2i Im(z) ∂z

 f (z),

≤ p+1

k into M k+2 . maps M Shimura [14, pp. 32] introduced the notions of slowly increasing and rapidly k are such that the product f g is a rapidly k . If f, g ∈ M decreasing functions in M decreasing function, the Petersson inner product of f and g is defined by   f, g :=

SL 2 (Z)\H

f (z)g(z)y k

d xd y , y2

(3)

where z = x + i y. We have the following orthogonality result for nearly holomorphic modular forms [9, Theorem 2.12]. l . Assume that r and s are positive integers such Theorem 3 Let f ∈ Sk and g ∈ M that k + 2r = l + 2s. Then cr  f, g if r = s, r s Rk f, Rl g = 0 if r = s, where cr =

r! k(k 4r

+ 1) · · · (k + r − 1).

We also recall the following result of Shimura [14, Corollary 6.9].

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k−2 such that both g Lemma 1 Let f ∈ Sk . Then  f, Rk−2 g = 0 for every g ∈ M and Rk−2 g are slowly increasing functions. Rankin–Cohen brackets for nearly holomorphic modular forms are defined in [9]. Let F and G be two nearly holomorphic modular forms of weights k and l and with depths s and t, respectively, on S L 2 (Z). For any integer ν ≥ 0, the νth Rankin–Cohen bracket of F and G is defined by    k−s+ν−1 l −t +ν−1 α [F, G]ν := Rk F Rlν−α G. (−1) ν − α α α=0 ν 

α

(4)

It is known [9, Theorem 4.1] that [F, G]ν is a nearly holomorphic modular form of weight k + l + 2ν and depth at most s + t on S L 2 (Z).

2.1 Eigenforms The action of Hecke operators on nearly holomorphic modular forms and on quasimodular forms are the same as on modular forms. For any integer n ≥ 1, the operator k and M k . A quasimodular form is called a quasimodular Tn preserve the spaces M eigenform (or simply an eigenform) if it is an eigenvector for every Hecke operator Tn (n ≥ 1). Similarly, we define nearly holomorphic eigenforms. We have the following characterization of eigenforms. Proposition 1 ([8, Theorem 1.1]) Let f be a nearly holomorphic eigenform of weight k and depth p for the full modular group S L 2 (Z). If p < k/2 then f = p

k

Rk−2 p f p , where f p ∈ Mk−2 p is an eigenform, and if p = k/2 then f ∈ CR22

−1

E 2∗ .

Proposition 2 ([2, Proposition 3.1]) Let f be a quasimodular eigenform of weight k and depth p for S L 2 (Z). If p < k/2 then f = D p f p , where f p ∈ Mk−2 p is an k eigenform, and if p = k/2 then f ∈ CD 2 −1 E 2 . We also recall the following results on quasimodular eigenforms [8, Lemma 4.3, 4.4].  2πinz k is a non-zero eigenform. Then Lemma 2 Suppose that f = ∞ ∈M n=0 a(n)e a(1) = 0. Also we have a(0) = 0 if and only if f ∈ CE k .

2.2 The Isomorphism ∗ and M ∗ denote the algebras of all nearly holomorphic modular forms and all Let M quasimodular forms on S L 2 (Z), respectively. We have an isomorphism between the two algebras given in the following theorem [7, Proposition 2.3.4].

Rankin–Cohen Brackets and Identities Among Eigenforms II

131

Theorem 4 The function defined by f (z) =

p  f j (z)

→ f 0 (z) j Im(z) j=0

∗ to M ∗ is a Hecke equivariant algebra isomorphism. Moreover, the isomorfrom M ∗≤ p bijectively. k≤ p onto M phism maps M From the above theorem, we observe that a polynomial relation among eigenforms in ∗ and vice versa. Suppose ∗ gives rise to a corresponding polynomial relation in M M that F and G are two nearly holomorphic modular forms of weights k and l with depths s and t, respectively, on S L 2 (Z) such that f and g are their respective images of the isomorphism defined in Theorem 4. Then for any integer ν ≥ 0, the image of [F, G]ν by the isomorphism is [ f, g]ν .

2.3 L-Functions Attached to Modular Forms For f (z) :=

∞ 

a(m)e2πimz ∈ Mk , the L-function attached to f is defined by

m=0

L( f, s) =

∞  a(m) m=1

ms

.

It is well known that if f ∈ Sk , then the L-function L( f, s) is analytically continued to the whole complex plane C and it satisfies the functional equation L ∗ ( f, s) := (2π )−s (s)L( f, s) = (−1)k/2 L ∗ ( f, k − s). If f (z) =

∞  m=0

a(m)e2πimz and g(z) =

∞ 

b(m)e2πimz are two modular forms of

m=0

weights k and l on S L 2 (Z), respectively, the Rankin–Selberg L-function associated with f and g is defined by L( f × g, s) :=

∞  a(m)b(m) m=1

ms

.

We now recall the following interesting non-vanishing result of the L-function L( f, s) associated with a cusp form at the center ([10, Corollary 3.2] and [9, Remark 4.7]). Although the result is stated in terms of the completed L-function L ∗ ( f, s) in [10, Corollary 3.2], one can see that it is valid for the L-function L( f, s).

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Lemma 3 Suppose that k > 20 and k ≡ 0 (mod 4). Then there are two eigenforms f, g ∈ Sk with L ∗ ( f, k/2) = 0 and L ∗ (g, k/2) = 0. Moreover, for each integer k ≥ 12 with k ≡ 0 (mod 4), there exists a non-zero eigenform f ∈ Sk such that L ∗ ( f, k/2) = 0.

3 Eisenstein Series

   1m Let ∞ = ± |m ∈ Z . For any integer k ≥ 0, z ∈ H and s ∈ C, the non01 holomorphic normalized Eisenstein series E k (z, s) is defined by E k (z, s) =



j (γ , z)−k | j (γ , z)|−2s ,

γ ∈∞ \SL 2 (Z)

  ab . The above series for E k (z, s) is absoc d lutely convergent for Re(2s) > 2 − k. It is well known that where j (γ , z) = (cz + d) with γ =

(k + r ) E k+2r (z, −r ), (k)

Rkr E k (z) = (−4π y)−r

(5)

where y = Im(z). To get the Fourier expansion of the above Eisenstein series, we first observe that 2ζ (k + 2s)E k (z, s) =

∞ 

(mz + n)−k | mz + n |−2s .

(6)

m,n=−∞ (m,n) =(0,0)

We have the following result which gives the Fourier expansion for 2ζ (k + 2s) E k (z, s) [13, Theorem 7.2.9]. Theorem 5 For any integer k, the Eisenstein series E k (z, s) is analytically continued to a meromorphic function on the whole s-plane and has the Fourier expansion: √  2ζ (k + 2s)E k (z, s) = 2ζ (k + 2s) + 2 πi −k + 2k+1 i −k

2s+k−1



 2s+k ζ (2s + k − 1) 1−k−2s 2 y (s + k) (s) 2

 π s+k σk+2s−1 (n)n −s e2πinz ω(4π ny; k + s, s) y −s (s + k) n≥1

+2

1−k −k

i

π s −k−s  σk+2s−1 (n)n −k−s e−2πinz ω(4π ny; s, k + s), y (s) n≥1

where the function

Rankin–Cohen Brackets and Identities Among Eigenforms II

zβ ω(z; α, β) = (β)



∞

133

e−zu (u + 1)α−1 u β−1 du

0

is holomorphic on H × C × C and H = {z ∈ C | Re(z) > 0}. Furthermore, for any compact subset K of C, there exist positive constants A and B satisfying | ω(4π ny; k + s, s) |≤ A(1 + y −B ) for any positive integer n and any s ∈ K .

3.1 Eisenstein Series of Weight 2 It is well known that the Eisenstein series E 2∗ (z) := E 2 (z) −

3 π Im(z)

is a nearly holomorphic modular form of weight 2 and depth 1 on S L 2 (Z). Moreover, we have  y 1/n |2 γ , E 2∗ (z) = lim n→∞

γ ∈∞ \SL 2 (Z)

  ab where for any function f on the upper-half plane H, γ = ∈ S L 2 (Z) and any c d integer k, the slash operator |k on f is defined by f |k γ (z) = (cz + d)−k f



 az + b . cz + d

Applying induction on r , one verifies that for any s ∈ C, we have Rkr y s =

(k + s + r ) s−r 1 y . r (−4π ) (k + s)

Thus using the Weierstrass theorem for interchanging the limit and differentiation, we get R2r E 2∗ (z) = lim

n→∞

= =

 γ ∈∞ \SL 2 (Z)

(2 + r ) lim (−4π )r n→∞





R2r y 1/n |2 γ = lim

n→∞





R2r y 1/n |2+2r γ

γ ∈∞ \SL 2 (Z)

y 1/n−r |2+2r γ

γ ∈∞ \SL 2 (Z)

(2 + r ) lim E 2+2r (z, 1/n − r ). (−4π y)r n→∞

(7)

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4 Auxiliary Results We state the following two results which will be used in the proof of the main result. The proofs of Theorems 6 and 7 are along the same lines of the proofs of Theorems 5.4 and 5.5 of [9], respectively. Therefore, we do not give proofs of these results here. Theorem 6 Let k1 be an even non-negative integer and r1 , r2 ≥ 0 be integers. For any integer ν ≥ 0, let k = k1 + 2 + 2r1 + 2r2 + 2ν. Then for any normalized eigenform f ∈ Sk , we have 

f, [Rkr11 E k1 ,

R2r2 E 2∗ ]ν 

= c(k; k1 , r1 , 2, r2 ) · L

where c(k; k1 , r1 , 2, r2 ) =

∗



   k1 k1 k k ∗ −1 L , f, + f, + 2 2 2 2

  (−1)k1 /2−r1 24 2k1 k1 + r1 + r2 + 2ν . 2k−1 Bk1 ν

Theorem 7 Let r1 , r2 and ν be non-negative integers and let k = 4 + 2r1 + 2r2 + 2ν. Then for any normalized eigenform f ∈ Sk−2 we have Rk−2 f, [R2r1 E 2∗ ,

R2r2 E 2∗ ]ν 

= c (k; 2, r1 , 2, r2 ) · L 1

∗



   k k ∗ f, − 1 L f, + 1 , 2 2

where ν −242  c (k; 2, r1 , 2, r2 ) = k−1 A α tα , 2 α=0 1

   r1 + ν + 1 r2 + ν + 1 , with Aα = ν−α α

and tα = (−1)r1 −1 (r1 + α)(r1 + α + 1) + (−1)r2 +ν−1 (r2 + ν − α)(r2 + ν − α + 1). Moreover, if r2 = 0 then c1 (k; 2, 0, 2, r2 ) = 0. We also need the following result to prove our main result. In fact, we need a particular case of the following result for our purposes. To prove Theorem 8, we use the dominated convergence theorem which is not used in the proof of Theorem 6. The absolute convergence provides the justification in applying Rankin’s unfolding argument in the proof of Theorem 6. A similar argument can not be applied in the proof of the following result. Theorem 8 Let k, k1 , r1 , r2 , ν be non-negative integers such that k + 2r = k1 + 2 + ∞ ∞   a(n)e2πinz ∈ Sk and g(z) = b(n)e2πinz 2r1 + 2r2 + 2ν. Suppose that f (z) = n=1

∈ Sk1 . Then we have Rkr

f, [Rkr11 g,

R2r2 E 2∗ ]ν 

n=0

 = c(k, r ; k1 , r1 , 2, r2 ) · L

k k1 f × g, + 2 2

 ,

Rankin–Cohen Brackets and Identities Among Eigenforms II

135

where c(k, r ; k1 , r1 , 2, r2 ) = ×

(−1)r2 +ν (4π )k+2r −1 ν 

Aα

α=0



with Aα =

r r 1 +α  u=0 v=0

(r ) (r +α)

(−1)−u−v Pu,k Pv,k1

1

k1 + r1 + ν − 1 ν−α

and (r ) = Pu,k

(k + 2r − r2 − ν + α − u − v − 1),

  r2 + ν + 1 (2 + r2 + ν − α), α

(8)

  r (k + r ) . u (k + r − u)

Proof Using the definitions of Rankin–Cohen brackets and the Petersson inner product we have Rkr

f, [Rkr11 g,

R2r2 E 2∗ ]ν 

   k1 + r1 + ν − 1 r2 + ν + 1 = (−1) ν−α α α=0  dx dy × Rkr f Rkr11 +α g R2r2 +ν−α E 2∗ y k+2r . y2 SL 2 (Z)\H ν 

α

Using (7) in the above expression we obtain Rkr f, [Rkr11 g, R2r2 E 2∗ ]ν  =  ×

ν  α=0

(−1)α

   k1 + r1 + ν − 1 r2 + ν + 1 (2 + r2 + ν − α) ν−α α (−4π)r2 +ν−α

lim Rkr f Rkr11 +α g E 2+2r2 +2ν−2α (z, 1/n − r2 − ν + α) y k+2r−r2 −ν+α+1/n

S L 2 (Z)\H n→∞

dx dy . y2

(9) Now we justify the interchange of the integration and the limit in the above equation. For this we first note that |Rkr f (z) Rkr11 +α g(z) E 2+2r2 +2ν−2α (z, 1/n − r2 − ν + α) y k+2r −r2 −ν+α+1/n |  e−2π y |E 2+2r2 +2ν−2α (z, 1/n − r2 − ν + α)|. Then using Theorem 5 we observe that for large y, there exists a positive constant M such that |E 2+2r2 +2ν−2α (z, 1/n − r2 − ν + α)|  (1 + y M ), where the implied constant does not depend on n.

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Combining the last two inequalities and using the dominated convergence theorem, we can interchange the integral sign and the limit in (9). Applying the standard Rankin unfolding argument, we obtain Rkr f, [Rk1 g, R22 E 2∗ ]ν  r

r

1

= (−1)r2 +ν

 ∞ 1 r +α lim Rkr f (z) Rk1 g(z) y k+2r −r2 −ν+α−2+1/n d x d y, 1 (4π )r2 +ν−α n→∞ 0 0 α=0 ν 

Aα

where Aα is as given in (8). Substituting the Fourier expansions of f and g in the above identity Rkr f, [Rkr11 g, R2r2 E 2∗ ]ν  = (−1)r2 +ν  × lim

n→∞ 0

∞



1



ν  α=0

r r 1 +α  Aα (r ) (r +α) Pu,k Pv,k11 (−4π)−u−v r2 +ν−α (4π) u=0 v=0

(10)

a(m)b(t)m r −u t r1 +α−v e2πi x(m−t) e−2π y(m+t) y k+2r −r2 −ν+α−2−u−v+1/n d x d y,

0 m1 t1

where (s) Pw,l

  s (l + s) · = w (l + s − w)

Using the bounds of Fourier coefficients, we deduce that we can interchange the sum and the integration in the right-hand side of (10), and using the fact that the integral over the x variable will survive only when t = m, we obtain r r 1 +α  Aα (r ) (r +α) Pu,k Pv,k11 (−4π)−u−v r2 +ν−α (4π) α=0 u=0 v=0  ∞  lim a(m)b(m)m r +r1 +α−u−v e−4π ym y k+2r −r2 −ν+α−2−u−v+1/n dy.

Rkr f, [Rkr11 g, R2r2 E 2∗ ]ν  =(−1)r2 +ν

n→∞

ν 

0

m1

Now the definitions of the Gamma function and the Rankin–Selberg L-function gives Rkr f, [Rkr11 g, R2r2 E 2∗ ]ν  = (−1)r2 +ν

ν  α=0

Aα (4π)

k+2r −1

r r 1 +α  (r ) (r1 +α) (−1)−u−v Pu,k Pv,k1 u=0 v=0

× (k +2r −r2 −ν +α−u −v−1)L

This completes the proof.

 f × g,

k1 k + 2 2

 .



Remark Note that for r = 0, the constant c(k, r ; k1 , r1 , 2, r2 ) appearing in Theorem 8 is non-zero which has been proved in [9, Theorem 5.2]. More precisely, we have c(k, 0; k1 , r1 , 2, r2 ) =

(−1)r2 +ν (2 + r1 + r2 + ν)(k1 + r1 + r2 + 2ν + 1) = 0. (ν + 1) (4π )k−1

Rankin–Cohen Brackets and Identities Among Eigenforms II

137

5 Proof of Theorem 2 Let [ f, g]ν be a quasimodular eigenform, for some eigenforms f and g. Since in [9], we have already considered all the cases where the depth is strictly less than half of the weight for both the forms f and g, and therefore, without loss of generality, we assume that g has depth equal to half of the weight. Using Proposition 2, g is either E 2 or its derivative and so we need to check only the following three cases of Rankin–Cohen brackets to see if they give rise to eigenforms. (i) [Dr1 E k1 , Dr2 E 2 ]ν for r1r2 = 0 and k1 ≥ 4, (ii) [E 2 , Dr2 E 2 ]ν , (iii) [Dr1 f, E 2 ]ν for any cusp form f of weight k1 which is an eigenform, where r1 and r2 are non-negative integers. Let k be the corresponding weight of any form in the above cases, i.e., k is 2 + k1 + 2r1 + 2r2 + 2ν, 4 + 2r2 + 2ν, or 2 + k1 + 2r1 + 2ν in the above three cases, respectively. If k ≤ 26 or k = 24, we use SAGE mathematical software to find all of the cases when it gives an eigenform. All of the obtained identities are listed in Theorem 2. Therefore, hereafter we assume that either k = 24 or k ≥ 28. To complete the proof, we will show that in the remaining cases the Rankin–Cohen brackets of two eigenforms do not give eigenforms. For this purpose, we use the theory of nearly holomorphic modular forms. Applying Theorem 4, we notice that it is equivalent to consider the following cases of nearly holomorphic modular forms. (i) [Rkr11 E k1 , R2r2 E 2∗ ]ν for r1r2 = 0 and k1 ≥ 4, (ii) [E 2∗ , R2r2 E 2∗ ]ν , (iii) [Rkr11 f, E 2∗ ]ν for any cusp form f which is an eigenform. Here k = 24 or k ≥ 28. We want to prove that we would not get an eigenform in any of the cases listed above.

5.1 First Case Assume that [Rkr11 E k1 , R2r2 E 2∗ ]ν is an eigenform for r1r2 = 0 and k1 ≥ 4. By Proposition 1, we have r g, [Rkr11 E k1 , R2r2 E 2∗ ]ν = Rk−2r for some eigenform g ∈ Mk−2r . First consider the case r = 0. Since the dimension of Sk is at least 2, one deduces that there exists a non-zero normalized eigenform h ∈ Sk such that h, g = h, [Rkr11 E k1 , R2r2 E 2∗ ]ν  = 0. Therefore, by Theorem 6 we get

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    k1 k k1 k − 1 L h, + = 0. L h, + 2 2 2 2

(11)

Since L(h, s) has an Euler product in the region Re(s) > k+1 , L(h, s) does not 2 . Since k ≥ 4, the left-hand side of (11) is non-zero. This vanish for Re(s) > k+1 1 2 gives a contradiction. Thus if r = 0, [Rkr11 E k1 , R2r2 E 2∗ ]ν is not an eigenform. If r ≥ 1, by employing Lemma 1 and as done in the case when r = 0, we deduce that the Rankin–Cohen brackets do not result in eigenforms.

5.2 Second Case If ν = 0, then this reduces to the case of product of two nearly holomorphic eigenforms. This case has been done in [8] and it follows that E 2∗ (R2r2 E 2∗ ) does not result in an eigenform for any integer r2 ≥ 0. Therefore, we assume that ν ≥ 1. Let [E 2∗ , R2r2 E 2∗ ]ν be an eigenform. Then as we have done in the first case, we deduce that r g [E 2∗ , R2r2 E 2∗ ]ν = Rk−2r for some eigenform g ∈ Mk−2r . Let us first consider the case r = 0. By looking at the Fourier expansions in the above identity we deduce that g has to be a cusp form. Since g, g = 0, we have g, [E 2∗ , R2r2 E 2∗ ]ν  = 0. Thus, by Theorem 6 we obtain L ∗ (g, k/2 + 1)L ∗ (g, k/2) = 0. Since k/2 + 1 lies in the region in which L(g, s) has an Euler product, L(g, k/2 + 1) = 0 and hence L ∗ (g, k/2 + 1) = 0. Also from the functional equation of L ∗ (g, s) we see that if k ≡ 2 (mod 4) then L ∗ (g, k/2) = 0. Therefore, from the above discussion, we deduce that k ≡ 0 (mod 4). Then by Lemma 3 and Theorem 6, there exist two eigenforms f 1 , f 2 ∈ Sk such that  f 1 , g = 0 and  f 2 , g = 0. This contradicts the fact that g is an eigenform. Let r ≥ 1. Then by Lemma 1, for any eigenform f ∈ Sk , we have  f, [E 2∗ , R2r2 E 2∗ ]ν  = 0.

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As done in the case when r = 0, we deduce in this case that L ∗ ( f, k/2) = 0. By Lemma 3 this implies that k ≡ 2 (mod 4). Now if r = 1 then [E 2∗ , R2r2 E 2∗ ]ν = Rk−2 g and k − 2 ≡ 0 (mod 4). By Lemma 3 there exist two normalized eigenforms f 1 and f 2 in Sk−2 such that L

∗



k−2 f1, 2

 = 0 and L

∗



k−2 f2 , 2

 = 0.

Then by Theorem 3, we have  f 1 , g =

1 1 Rk−2 f 1 , Rk−2 g = Rk−2 f 1 , [E 2∗ , R2r2 E 2∗ ]ν  c1 c1

 f 2 , g =

1 1 Rk−2 f 2 , Rk−2 g = Rk−2 f 1 , [E 2∗ , R2r2 E 2∗ ]ν . c1 c1

and

Thus by applying Theorem 7, we deduce that there are two normalized eigenforms f 1 and f 2 in Sk−2 such that  f 1 , g = 0 and  f 2 , g = 0, which gives a contradiction. Now assume that r ≥ 2. If f ∈ Sk−2 is any eigenform, then Theorem 3 implies that r g = 0. Rk−2 f, [E 2∗ , R2r2 E 2∗ ]ν  = Rk−2 g, Rk−2r



Thus by Theorem 7, the above identity implies that L ∗ f, k−2 = 0. Since k − 2 ≡ 0 2 (mod 4) and f is an arbitrary eigenform, Lemma 3 gives a contradiction.

5.3 Third Case Assume that [Rkr11 f, E 2∗ ]ν is an eigenform, where f ∈ Sk1 is an eigenform. Then we have r g [Rkr11 f, E 2∗ ]ν = Rk−2r for some eigenform g ∈ Mk−2r . If r = 0, there exists a non-zero eigenform h ∈ Sk such that h, g = h, [Rkr11 f, E 2∗ ]ν  = 0 Applying Theorem 8, we obtain

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  k1 k = 0. L h × f, + 2 2

(12)

Since h and f are eigenforms of weight k and k1 , respectively, L(h × f, s) has an Euler product in the region Re(s) > 2k + k21 and thus L h × f, k2 + k21 = 0. This contradicts (12). Therefore, when r = 0, the Rankin–Cohen brackets do not result in eigenforms. If r ≥ 1, by employing Lemma 1 and as done in the case when r = 0, we deduce that the Rankin–Cohen brackets of eigenforms do not result in eigenforms. This proves Theorem 2.

6 Further Remarks The Chazy equation is the differential equation 2D 3 E 2 − 2E 2 D 2 E 2 + 3(D E 2 )2 = 0.

(13)

We have Ramanujan’s formula for D E 2 : D E2 =

E2 −E 4 + 2. 12 12

Using (13) and the above Ramanujan’s formula for D E 2 , one easily deduces that [E 4 , E 2 ]2 = 0.

(14)

Conversely using the Ramanujan’s formula for D E 2 and D E 4 in (14) we deduce (13). Thus the differential equations in (13) and (14) are equivalent. In (14), the 2nd Rankin–Cohen bracket of two quasimodular eigenforms is zero. We are not aware of any other such non-trivial identity. It would be interesting to find out an explicit list of all such identities. Acknowledgements We have used the open source mathematics software SAGE to do our calculations. We would like to thank the referee for giving useful suggestions which improved the presentation of the manuscript. The second author was supported by the DST-SERB grant MTR/2017/000022 for carrying out this research work.

References 1. Bruinier, J.H., van der Geer, G., Harder, G., Zagier, D.: The 1-2-3 of Modular Forms. Springer, Berlin (2008) 2. Das, S., Meher, J.: On quasimodular eigenforms. Int. J. Number Theory 11, 835–842 (2015) 3. Duke, W.: When is the product of two Hecke eigenforms an eigenform? In: Number Theory in Progress, vol. 2 (Zakopane-Ko´scielisko, 1997), de Gruyter, Berlin (1999), pp. 737–741

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4. Emmons, B.A., Lanphier, D.: Products of an arbitrary number of Hecke eigenforms. Acta Arith. 130, 311–319 (2007) 5. Ghate, E.: On monomial relations between Eisenstein series. J. Ramanujan Math. Soc. 15, 71–79 (2000) 6. Johnson, M.L.: Hecke eigenforms as products of eigenforms. J. Number Theory 133(7), 2339– 2362 (2013) 7. Kumar, A.: On some problems involving nearly holomorphic modular forms and an estimate for Fourier coefficients of Hermitian cusp forms. Ph.D. Thesis, HBNI (2017) 8. Kumar, A., Meher, J.: On arbitrary products of eigenforms. Acta Arith. 173, 283–295 (2016) 9. Kumar, A., Meher, J.: Rankin-Cohen brackets and identities among eigenforms. Pac. J. Math. 297, 381–403 (2018) 10. Lanphier, D., Takloo-Bighash, R.: On Rankin-Cohen brackets of eigenforms. J. Ramanujan Math. Soc. 19, 253–259 (2004) 11. Martin, F., Royer, E.: Rankin-Cohen brackets on quasimodular forms. J. Ramanujan Math. Soc. 24, 213–233 (2009) 12. Meher, J.: Some remarks on Rankin-Cohen brackets of eigenforms. Int. J. Number Theory 8, 2059–2068 (2012) 13. Miyake, T.: Modular Forms. Springer, Berlin (1989) 14. Shimura, G.: Modular forms: Basics and Beyond. Springer, New York (2012)

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms Moni Kumari and Brundaban Sahu

Abstract We construct Rankin–Cohen type differential operators on the space of Hilbert–Jacobi forms. This generalizes a result of Choie and Eholzer (J Number Theory, 68:160–177, 1998) in the case of Jacobi forms to Hilbert–Jacobi forms. Keywords Hilbert–Jacobi forms · Rankin–Cohen brackets 2010 Mathematics Subject Classification 11F41 · 11F50 · 11F60

1 Introduction There are many interesting connections between differential operators and modular forms, and many interesting results have been studied. In particular, Rankin [8, 9] gave a general description of the differential operators which send modular forms to modular forms. Cohen [5] constructed certain covariant bilinear operators and obtained modular forms with interesting Fourier coefficients. Zagier [11, 12] called these covariant bilinear operators as Rankin–Cohen operators and studied their algebraic properties. Rankin–Cohen type operators for Jacobi forms on H × C have been studied using heat operators in [2, 3]. Using Maass operator Böcherer [1] showed that the space of bilinear holomorphic differential operators raising the weight ν is in general of dimension 1 + [ν/2] for Jacobi forms on H × C. In [4], Choie and Eholzer explicitly give a family of bilinear holomorphic differential operators using Rankin–Cohen type operators of right dimension 1 + [ν/2] and also remark (in Sect. 8) that it would be M. Kumari (B) Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] B. Sahu National Institute of Science Education and Research, HBNI, P. O. Jatni, Khurda, Bhubaneswar 752050, Odisha, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_10

143

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interesting to understand how their construction can be generalized to higher Jacobi forms. Skogman [10] extended the theory of Jacobi forms over a totally real number field, known as Hilbert–Jacobi forms. In this paper, we study differential operators of Rankin–Cohen type on the space of Hilbert–Jacobi forms which give an answer to the question posed by Choie and Eholzer in [4]. The paper is organized as follows. In Sect. 2, we recall basic facts about Hilbert– Jacobi forms and define Rankin–Cohen type operators for the Hilbert–Jacobi forms and state the main result. We develop certain tools for our proof in Sect. 3 and give a proof of the main result in Sect. 4. We follow the same exposition as given in [4]. This article is a part in the Ph. D. thesis [7] of the first author.

2 Preliminaries and Statement of Result Let K be a totally real-number field of degree g := [K , Q] over Q with ring of its algebraic integers O K and we denote its g real embedding by σ1 , . . . , σg . We denote ith embedding of an element α ∈ K by α (i) := σi (α) for any 1  i  g. An (i) element α ∈ K is said to be totally positive, α > 0, if all its embeddings g α (i) into R are positive. g The trace and norm of α ∈ K are defined by tr(α) = i=1 α and N (α) = i=1 α (i) , respectively. The trace and norm of an element α ∈ Cg are given by the sum and by the product of its components, respectively. More generally, for c = (c1 , . . . , cg ), d = (d1 , . . . , dg ), k = (k1 , . . . , k g ) and m = (m 1 , . . . , m g ) ∈ Cg , we define the following: tr(mz) :=

g 

m i z i and (cz + d)k :=

i=1

g 

(ci z i + di )ki .

i=1



 ab : a, b, c, d ∈ O K , ad − bc = 1 . We denote cd the Hilbert–Jacobi group as  J (K ) defined by Let  K := S L 2 (O K ) =

 J (K ) := S L 2 (O K )  (O K × O K ), with the group multiplication  γ1 .γ2 :=

a1 b1 c1 d1



a2 b2 c2 d2



 , (λ1 , μ1 )

a2 b2 c2 d2



 + (λ2 , μ2 ) ,

  ai bi , (λi , μi ) for i = 1, 2. The Hilbert–Jacobi group  J (K ) ci di acts on the space Hg × Cg by 

where γi :=

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms

145



  a b , (λ, μ) ◦ (τ1 , . . . , τg , z 1 , . . . , z g ) c d

 a (g) τg + b(g) z 1 + λ(1) τ1 + μ(1) z g + λ(g) τg + μ(g) a (1) τ1 + b(1) = , . . . , (g) , ,..., , c(1) τ1 + d (1) c τg + d (g) c(1) τ1 + d (1) c(g) τg + d (g)

  a b , (λ, μ) ∈  J (K ) and (τ , . . . , τ , z , . . . , z ) ∈ Hg × Cg . 1 g 1 g cd g → For an integer x ∈ N0 , we denote − x := (x, . . . , x) ∈ N0 . For ν = (ν1 , . . . , νg ) ∈ g g g N0 , l = (l1 , . . . , l g ) ∈ N0 and z = (z 1 , . . . , z g ) ∈ C , we denote

where

|ν| =

g 

νi , ν! =

i=1

g 

νi ! and z ν =

i=1

g 

z iνi .

i=1

Also we denote ν  l if ν j  l j for all 1  j  g and e[z] for e2πi z for z ∈ C. For a holomorphic function φ : Hg × Cg → C, we define the following two slash g operators. For a fixed k ∈ N0 and m ∈ O K ,         mcz 2 ab φ|k,m M (τ, z) := (cz + d)−k e tr − φ ◦ (τ, z) , cd cτ + d  for M =

ab cd

(1)

 ∈ S L 2 (O K ) and

    φ|m (λ, μ) (τ, z) := e[tr m(λ2 τ + 2λz) ]φ ((λ, μ) ◦ (τ, z)) ,

(2)

for (λ, μ) ∈ O K × O K . Definition 1 A Hilbert–Jacobi form of weight k and index m for a totally realfield K is a holomorphic function φ : Hg × Cg → C which satisfies the following conditions: 1. φ|k,m γ = φ, for all γ ∈  K , 2. φ|m (λ, μ) = φ, for all (λ, μ) ∈ O K × O K , 3. φ has a Fourier expansion of the form, φ(τ, z) =



cφ (n, r )e[tr(nτ + r z)],

n,r ∈O ∗K 4nm−r 2 0

where O∗K = {μ ∈ K | tr(μλ) ∈ Z for all λ ∈ O K }. We note that O∗K is δ −1 K , the inverse of the different ideal of the number field K . Moreover, such a form φ is called Hilbert–Jacobi cusp form if cφ (n, r ) = 0, whenever K ,cusp K 4nm − r 2 = 0. Let Jk,m (Jk,m ) denote the space of Hilbert–Jacobi forms (Hilbert–

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Jacobi cusp forms) of weight k and index m for the field K . For more details on the theory of Hilbert–Jacobi forms we refer to [10]. Now we define the heat operators. Definition 2 For 1  j  g, let e j be jth unit vector in Rg . For a given m ∈ O K , we define the mth heat operator, L m :=

e g   ∂ ∂2 j 8πim . − 2 ∂τ ∂z j=1

(3)

 In the above definition, we denote “ ” for the composition of operators. Now we state some properties of these operators which can be proved as in the case of Jacobi forms [2]. Lemma 1 Let φ(τ, z) be a holomorphic function on the space Hg × Cg , k ∈ Zg and m ∈ O K . Then 1. for X ∈ O K × O K , (L m φ)|m X = L m (φ|m X ),

(4)

g

2. for any ν ∈ N0 and M ∈ S L 2 (O K ), we have L m ν (φ)|k+2ν,m M =

 ν  (8πimc)ν−l (α + ν − 1)! l

g l∈N0

(cτ + d)ν−l (α + l − 1)!

L l m (φ|k,m M),

(5)

lν

where α = k − 21 . We define Rankin–Cohen type differential operators on the space of Hilbert– Jacobi forms using the heat operators. Definition 3 Let φ, φ  : Hg × Cg → C be two holomorphic functions and let g g k, k  , m, m  be complex numbers. Then for any X ∈ Cg , ν ∈ N0 and l ∈ N0 with li ∈ {0, 1} for all 1  i  g, define 



,m,m [φ, φ  ]k,k X,2ν+l =







,m,m ,l (−1) j m l− j m  j [∂zj φ, ∂zl− j φ  ]k,k , X,2ν

g

j∈N0 jl

where for any two holomorphic functions f and f  on Hg × Cg 



,m,m ,l [ f, f  ]k,k := X,2ν



Ar,s, p (k, k  , l)(1 + m X )s (1 − m  X )r L m+m  (L rm ( f )L sm  ( f  )), p

g r,s, p∈N0 ,

r +s+ p=ν

with Ar,s, p (k, k  , l) =

(−(k + k  + l − 3/2 + r + s + p))r +s . r ! s! p! (k − 3/2 + r )! (k  − 3/2 + s)!

(6)

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms

Here (x)n =



0in−1 (x

147

− i).

Remark 1 In the above definition, we have the following convention. 







,m,m ,0 ,m,m [φ, φ  ]k,k = [φ, φ  ]k,k . X,2ν X,2ν

Remark 2 Note that constants Ar,s, p (k, k  , l) are different than Cr,s, p (k, k  ), which appeared in [4] for the field K = Q. Now we state the main result. Theorem 1 Let φ, φ  be Hilbert–Jacobi forms of weight and index k, m and k  , m  g g respectively. Then for any X ∈ Cg , ν ∈ N0 and l ∈ N0 with li ∈ {0, 1} for all 1  i  g,  ,m,m  (7) [φ, φ  ]k,k X,2ν+l is a Hilbert–Jacobi form of weight k + k  + 2ν + l and index m + m  . There are two known methods to prove result like Theorem 1. First one, by showing  ,m,m  that [φ, φ  ]k,k X,2ν+l satisfy all the required conditions to be a Hilbert–Jacobi form (see, [4, Sect. 4]) and second one, by using generating series (see, [6, Theorem 3.2], [4, Sect. 5]). We prove our result by using generating series. In the next section, we shall develop some tools for the proof of Theorem 1.

3 Intermediate Results K and α = k − 21 . Then the formal power series Proposition 1 Let φ(τ, z) ∈ Jk,m associated with the Jacobi form φ defined by

(τ, z; W ) := φ

 L ν (φ)(τ, z) m W ν, g ν!(α + ν − 1)!

(8)

ν∈N0

satisfies the following functional equation,   Mτ, φ

z W ; cτ + d (cτ + d)2



      mcW mcz 2 (τ, z; W ), (9) = (cz + d)k e tr e 4tr φ cτ + d cτ + d

for all M = ∗c d∗ ∈ S L 2 (O K ).

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, we have Proof From the definition of φ 

 Mτ, φ



z W ; cτ + d (cτ + d)2

  z ν (φ) Mτ, L  m Wν cτ + d = ν!(α + ν − 1)! (cτ + d)2ν g ν∈N0

=

   mcz 2 k e tr (cτ + d) (L νm φ)|k+2ν,m M(τ, z)  cτ + d ν!(α + ν − 1)!

g ν∈N0

Wν.

K Using (5) and the assumption that φ ∈ Jk,m , the right-hand side of the above equation is equal to

  (cτ + d)k e tr

mcz 2 cτ + d



= (cτ + d)k e tr 

= (cz + d)k e tr

 

    ν (8πimc)ν−l (α + ν − 1)! l 1 L (φ| M) Wν m k,m ν−l (α + l − 1)! ν!(α + ν − 1)! g l (cτ + d)

g

ν∈N0

mcz 2 cτ + d mcz 2 cτ + d



   g

ν∈N0

g

l∈N0 lν

l∈N0 lν

 1 (8πimc)ν−l l L (φ) Wν m l!(ν − l)!(α + l − 1)! (cτ + d)ν−l

    mcW (τ, z, W ). e 4tr φ cτ + z



This completes the proof.

Let  f (τ, z; W ) be a power series in W whose coefficients are holomorphic  functions on Hg × Cg , i.e.,  f (τ, z; W ) = ν∈N0g χν (τ, z)W ν . For M = ac db ∈ S L 2 (O K ), we define 

       mcW mcz 2  e −4tr f |k,m M (τ, z; W ) := (cτ + d)−k e −tr cτ + d cτ + d   z W aτ + b . , ; ×  f cτ + d cτ + d (cτ + d)2

Next, we show that for a given formal power series satisfying certain conditions, one can construct a family of Hilbert–Jacobi forms like in the case of Jacobi forms [Theorem 5.1, [4]]. (τ, z; W ) be a formal power series in W, i.e., Theorem 2 Let φ (τ, z; W ) = φ

 g ν∈N0

satisfying the functional equation

χν (τ, z)W ν ,

(10)

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms

  |k,m M (τ, z; W ) = φ (τ, z; W ), f or all M = a b ∈ S L 2 (O K ), φ cd

149

(11)

g

for some k ∈ N0 and m ∈ O K . Furthermore, assume that the coefficients χν (τ, z) are holomorphic functions on Hg × Cg with Fourier expansion of the form, 

χν (τ, z) =

c(n, r )e[tr(nτ + r z)],

(12)

n,r ∈O ∗K 4nm−r 2 0

satisfying χν |m Y = χν for all Y ∈ O K × O K .

(13)

g

Then for each ν ∈ N0 , the function ξν (τ, z) defined by ξν (τ, z) :=

 (−(k − 3/2 + ν))ν− j L mj (χν− j ), j! g

(14)

j∈N0 jν

is a Hilbert–Jacobi form of weight k + 2ν and index m. Remark 3 We call k and m that appeared in Eq. (11) is the weight and index of the , respectively. power series φ Proof We show that ξν (τ, z), defined by (14) is invariant under S L 2 (O K ) action. For 1  j  g, let e j be the jth unit vector in Rg . Define the jth differential operator ∂ ∂2 ∂2 ∂ ej  := 8πim ( j) − 2 − (k j − 1/2) − Wj , L k,m ∂τ j ∂Wj ∂z j ∂ W j2 k,m be the collection of all functions where k = (k1 , k2 , . . . , k g ) and m ∈ O K . Let M  ν  f (τ, z; W )= ν∈N0g χν (τ, z)W which satisfy the condition: 

  f |k,m M (τ, z; W ) =  f (τ, z; W ),

for all M = ac db ∈ S L 2 (O K ).

We note that the constant term χ0 (τ, z) in the power series expansion of  f (τ, z; W ) ∈ k,m satisfy the following: M    χ0 |k,m M (τ, z) = χ0 (τ, z),

for all M = ac db ∈ S L 2 (O K ).

Then using the definition of slash operator (11), one can show that ej ej  )|k+2e j ,m M, |k,m M) = ( φ L k,m (φ L k,m

(15)

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for all M ∈ S L 2 (O K ). We note that 1  j  g), denoted by  L k,m satisfy

g j=1

ej ej  (the composition of all  L k,m for L k,m

 )|k+2,m M, for all M ∈ S L 2 (O K ). |k,m M) = ( L k,m φ L k,m (φ k,m to M k+2,m which is given in terms of In other words,  L k,m is a map from M power series by  L k,m :



χλ (τ, z)W λ →

g λ∈N0

     (−1)1+ j 1j (λ + 1 − j)!(λ + α − j)!L mj (χλ+1− j )  g λ∈N0

λ!(λ + α − 1)!

g j∈N0

W λ,

j1

with α = k − 1/2. Composing the maps  L k+i,m for 1  i  ν − 1, 





L k,m L k+2,m L k+2ν−2,m k+2,m − k+2ν,m k,m − −→ M −−→ · · · −−−−−→ M M

then it maps



g

λ∈N0

χλ (τ, z)W λ to

   (−1)ν+ j g λ∈N0

g j∈N0

ν  j (λ + ν − j)!(λ + 2ν + α − j − 2)!L m (χλ+ν− j )  j λ!(λ + α + ν − 2)!

W λ.

jν

− → We note that the constant term, i.e., λ = 0 in the above series is ν! times ξν . Hence from (15), ξν is invariant under S L 2 (O K ) action. The other conditions hold easily  from given hypothesis on function χν (τ, z). In the next two lemmas, we show how the operator ∂z behaves under the group and lattice actions. g

Lemma 2 Let φ be a Hilbert–Jacobi form of weight k and index m. For j ∈ N0 with ji ∈ {0, 1} for all 1  i  g, we have  aτ + b z W , ; cτ + d cτ + d (cτ + d)2          4πimcz a j−a mcW mcz 2 (τ, z; W ). e 4tr ∂z φ = (cτ + d)k+ j e tr cτ + d cτ + d cτ + d g

 ∂z/cτ +d φ



j

(16)

a∈N0 a j

Proof This Lemma is an easy consequence of Proposition 1.



Lemma 3 Suppose f (z) is a holomorphic function on the space Hg and Y = g (λ, μ) ∈ O K × O K . Then for j ∈ N0 with ji ∈ {0, 1} for all 1  i  g, we have

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms

(∂zj f )|m Y =



(−4πimλ)a ∂zj−a ( f |m Y ).

151

(17)

g a∈N0

a j

Proof One can prove this result using the definition of the action “|m Y ”.



4 Proof of Theorem 1 − → − → First, we prove for case l = 0 and then for general case l = 0 . − → Case I: l = 0 . For a fixed X ∈ Cg , consider the series FX (τ, z; W ) defined by      τ, z; (1 − m X )W ,  τ, z; (1 + m  X )W φ FX (τ, z; W ) = φ  are defined by Eq. (8). We shall show that the function FX (τ, z; W )  and φ where φ satisfy all the necessary conditions for Theorem 2 and consequently deduce the result.  given in the Propo and φ Using the corresponding functional equation for φ sition 1, one can easily show that the function FX (τ, z; W ) also satisfy the same functional equation as (11) with weight k + k  and index m + m  .  with  and φ Now we shall look at the power series expansion of FX . Replacing φ their corresponding expressions (8) in FX , we get FX (τ, z; W )    (1 + m  X )ν L νm (φ) ν (1 − m X )ν L νm (φ  ) ν W W =  g ν! (k − 3/2 + ν)! g ν! (k − 3/2 + ν)! ν∈N0 ν∈N0   (1 + m  X )a (1 − m X )ν−a ν−a a  L m (φ)L m  (φ ) W ν =  g g a! (ν − a)! (k − 3/2 + a)! (k − 3/2 + ν − a)! ν∈N0

=



a∈N0 aν

χν,F (τ, z)W ν ,

g

ν∈N0

where χν,F (τ, z) :=

 g

a∈N0 aν

(1 + m  X )a (1 − m X )ν−a  L a (φ)L ν−a m  (φ ). a! (ν − a)! (k − 3/2 + a)! (k  − 3/2 + ν − a)! m

(18) g g g H × C for all ν ∈ N . We note that if φ has Clearly χν,F (τ, z) is holomorphic on 0  cφ (n, r )e[tr(nτ + r z)], then for any t ∈ N, the Fourier expansion φ(τ, z) = n,r ∈O ∗K 4nm−r 2 0

the function L tm (φ) has the Fourier expansion

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M. Kumari and B. Sahu



L tm (φ)(τ, z) =

cφ (n, r )(4nm − r 2 )t e[tr(nτ + r z)].

(19)

n,r ∈O ∗K 4nm−r 2 0

Replacing φ and φ  by their Fourier expansions and using the repeated action of the heat operator from (19), we have χν,F (τ, z) =

 g

a∈N0 aν

 ×  ×

(1 + m  X )a (1 − m X )ν−a a! (ν − a)! (k − 3/2 + a)! (k  − 3/2 + ν − a)!  (4nm − r 2 )a cφ (n, r )e[tr(nτ + r z)]

 n,r ∈O ∗K 4nm−r 2 0



 (4n  m  − r 2 )ν−a cφ  (n  , r  )e[tr(n  τ + r  z)]

n  ,r  ∈O ∗K 4n  m  −r 2 0



=

N ,R∈O ∗K 4N (m+m  )−R 2 0

g

a∈N0 aν

(1 + m  X )a (1 − m X )ν−a a! (ν − a)! (k − 3/2 + a)! (k  − 3/2 + ν − a)!

 (4nm − r 2 )a (4n  m  − r 2 )ν−a cφ (n, r )cφ  (n  , r  ) e[tr(N τ + Rz)].



×



n,n  ,r,r  ∈O ∗K n+n  =N , r +r  =R, 4nm−r 2 0, 4n  m  −r 2 0

One can check that 4N (m + m  ) − R 2  0 for the above choices of N and R and the last sum is a finite sum for a given N and R. From (4), it is clear that χν,F |m+m  Y = χν,F for all Y ∈ O K × O K . Hence from Theorem 2, ξν,F (τ, z) is a Hilbert–Jacobi form of weight k + k  + 2ν and index m + m  . This completes the  ,m,m  (τ, z) = ξν,F (τ, z). proof in this case because [φ, φ  ]k,k X,2ν − → g Case II: l = 0 . For a fixed X ∈ C , consider the function G X (τ, z; W ) defined by G X (τ, z; W ) =



 l− j   j  τ, z; (1 + m  X )W ∂z φ τ, z; (1 − m X )W . (−1) j m l− j m  j ∂z φ

g

j∈N0 jl

(20) functional equation as (11) with We show that the function G X satisfies the same

a b   weight k + k + l and index m + m . Let c d ∈ S L 2 (O K ). Using (20), we have

Rankin–Cohen Type Operators for Hilbert–Jacobi Forms

153



 aτ + b z W , ; cτ + d cτ + d (cτ + d)2    z (1 + m  X )W aτ + b j  , ; = (−1) j m l− j m  j ∂z/cτ +d φ cτ + d cτ + d (cτ + d)2 g

GX

j∈N0 jl

×

l− j  ∂z/cτ +d φ



aτ + b z (1 − m X )W , ; cτ + d cτ + d (cτ + d)2

 .

Using Lemma 2, the above equation becomes 

 aτ + b z W , ; cτ + d cτ + d (cτ + d)2       cz 2 cW k+k  +l   e 4tr (m + m ) = (cτ + d) e tr (m + m ) cτ + d cτ + d      4πimcz a    τ, z; (1 + m  X )W (−1) j m l− j m  j ∂zj−a φ × cτ + d g g

GX

j∈N0 jl

a∈N0 a j

   4πim  cz b   l− j−b  φ τ, z; (1 − m X )W . × ∂z cτ + d g b∈N0 bl− j

Now we split the above sum into two parts, 

 aτ + b z W , ; GX cτ + d cτ + d (cτ + d)2       cz 2 cW k+k  +l   e 4tr (m + m ) e tr (m + m ) = (cτ + d) cτ + d cτ + d       τ, z; (1 − m X )W  τ, z; (1 + m  X )W ∂zl− j φ × (−1) j m l− j m  j ∂zj φ g

j∈N0 jl

+

  g

α,β∈N0 α+β 0, then g1 = ±g2 . This question was posed by Kohnen in [5]. In 1999, Luo [6] proved the following result. Let Hk (N ) denote the orthogonal basis of normalized Hecke eigenforms of weight k on 0 (N ). Suppose that f and g are two normalized newforms of weight 2k (resp. 2k  ) on 0 (N ) (resp. 0 (N  )). If there exist a positive integer  and infinitely many primes p such that for all forms h ∈ H2 ( p), the central values of the Rankin– Selberg L-functions are equal (i.e., L(1/2, f ⊗ h) = L(1/2, f  ⊗ h)), then k = k  and N = N  . This result of Luo can be viewed as the G L(2) analog of the result of Luo and Ramakrishnan which is mentioned above. As a variant of Luo’s result, in [1], Ganguly, Hoffstein, and Sengupta considered twists by Hecke eigenforms of fixed level and varying weight. More precisely, if g ∈ H (1) and g  ∈ H (1) are such that L(1/2, f ⊗ g) = L(1/2, f ⊗ g  ), f ∈ Hk (1) for infinitely many k, then  =  and g = g  . (Here, k, ,  are all even positive integers.) There are other generalizations in the case of eigenforms of integral weight (see, for example, [9, 12]). In this paper, we generalize the work of Ganguly et al. in the case of forms of half-integral weight. We consider Hecke eigenforms of half-integral weight on 0 (4) which lies in the Kohnen plus space. The method is very similar to the one carried out by Ganguly et al. in [1]. However, we need the development of Rankin–Selberg convolutions and their approximate functional equations, etc., for which we use a result of Munshi [8], who has done the basics of this theory for the purpose of proving some simultaneous non-vanishing of twisted L-functions of integral weight. We now state the main result of this paper. Main Theorem Let g, g  be two Hecke eigenforms belonging to the Kohnen plus space of weights  + 1/2 and  + 1/2 on 0 (4), respectively. Suppose that L(1/2, f ⊗ g) = L(1/2, f ⊗ g  ),

(1)

for any Hecke eigenform f of weight k + 1/2 on 0 (4) belonging to the Kohnen plus space. Then, we have  =  and g = g  . As mentioned before, we follow the method of Ganguly et al. [1]. After recalling the basic properties of Rankin–Selberg convolution L-functions and its approximate functional equation in Sects. 3 and 4, we first prove an auxiliary result in Sect. 5 giving an asymptotic expression for the spectral average of central values of the Rankin– Selberg convolution L-functions and use it to prove our main theorem. In Sects. 6 and 7, we give estimates for the main and error terms appearing in the auxiliary theorem. Finally, in Sect. 8, we demonstrate the method of proving our main theorem. As a first reduction, we use the auxiliary theorem to show that the normalized |D|th Fourier

Determining Modular Forms of Half-Integral Weight …

159

coefficients of g and g  are equal and then use the explicit Waldspurger theorem obtained by Kohnen [5] to deduce that the special values of the twisted L-functions of the corresponding (via the Shimura–Kohnen lift) integral weight eigenforms are equal upto some constant. We then use the result of Luo and Ramakrishnan [7] to finally prove the main theorem. As mentioned above, in [1], the authors consider the twist by level 1 Hecke eigenforms and they had mentioned that it is possible to consider twists by higher level newforms also to get similar result and in [12], Zhang had carried out this task. It is likely that in the case of half-integral weight also one may carry out similar twists by newforms in the Kohnen plus space of higher (square-free) level and obtain analogous result as in our main theorem. We plan to do it in our forthcoming work. The contents of this paper are part of the first author’s thesis [10].

2 Notations and Preliminaries + Let k ≥ 2 be a natural number. We denote by Sk+1/2 (0 (4)), the Kohnen plus space of the space of modular forms f of weight k + 1/2 on 0 (4) whose Fourier coefficients a f (n) satisfy the following property that a f (n) = 0 implies that (−1)k n ≡ 0, 1 (mod 4). Let m ∈ N such that (−1)k m ≡ 0, 1 (mod 4). Then, the mth Poincaré + (0 (4)) is characterized by series in the Kohnen plus space Sk+1/2 +  f, Pk+1/2,m =

(k − 1/2) a f (m), 6(4π m)k−1/2

+ (0 (4)). The factor 6 in the denominator comes from the index for all f ∈ Sk+1/2 + (z) in of 0 (4) in S L 2 (Z). The Fourier expansion of the Poincaré series Pk+1/2,m Sk+1/2 (0 (4)) is given by

Pk+1/2,m (z) =



gk+1/2,m (n)q n ,

(2)

n≥1 (−1)k n≡0,1 (mod 4)

where ⎡ ⎤  √   √ k+1 mn π 2⎣ ⎦ gk+1/2,m (n) = Hc (m, n)Jk−1/2 δm,n + (−1)[ 2 ] π 2(n/m)(k/2−1/4) 3 c c1

and q = e2πi z , z ∈ H, the complex upper half-plane. In the above,

(3)

      1  4c −4 k+1/2 4 Hc (m, n) = (1 − (−1)k i) 1 + e4c (mδ + nδ −1 ), c 4c δ δ ∗ δ(4c)

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M. K. Pandey and B. Ramakrishnan

where δ −1 is an integer such that δδ −1 ≡ 1 (mod 4c) and Jk−1/2 (x) is the Bessel 2πi x/c function of order k − /12. Further, we have also  c  used the notation ec (x) = e for a complex x and an integer c. The symbol d denotes the generalized quadratic residue symbol as described in [3, 11]. Now, we have the following Petersson formula:  a f (m) + Pk+1/2,m (z) = ω f k−1/2 f, (4) m + f ∈Fk+1/2 (4)

+ + (4) denotes an orthogonal basis for the plus space Sk+1/2 (0 (4)) and where Fk+1/2 ( − 1/2) 1 . By comparing the nth Fourier coefficients of both the sides ωf = 6(4π )k−1/2  f, f of the above equation, we get



ωf

+ f ∈Fk+1/2 (4)

 √ a f (m)a f (n) 2 k+1 δm,n +(−1)[ 2 ] π 2(n/m)(k/2−1/4) = k−1/2 m 3 ×

 c1

Hc (m, n)Jk−1/2

(5)

 √  π mn , c

where ω f is defined after Eq. (4). For details on modular forms of half-integral weight and the plus space, we refer to [3–5].

3 Rankin–Selberg L-Functions In this section, we shall obtain a functional equation satisfied by the Rankin–Selberg L-function associated to forms of half-integral weight. Such a study for forms of halfintegral weight was first done by Munshi [8], and we refer the reader

to this work for more details. Let f i ∈ Sk+i +1/2 (0 (4)), i = 1, 2, and f i (z) = n≥1 a fi (n)e(nz)

be their Fourier expansions, where the sum  varies over all natural numbers n the same with (−1)ki n ≡ 0, 1 (mod 4). We also assume that k1 and k2 have   parity, ab i.e., k1 ≡ k2 (mod 2). Set H (z) = f 1 (z) f 2 (z). Then for any γ = ∈ 0 (4), cd we get k1 +1/2 (cz + d)k2 +1/2 f 1 (z) f 2 (z). H (γ (z)) = (cz + d)   10 = , 01

Note that the group 0 (4) has three cusps ∞, 0, 1/2 and the matrices g∞  −1  −1 0 2 1 2 , g1/2 = take the cusp ∞ to the corresponding cusps ω = g0 = 20 20

Determining Modular Forms of Half-Integral Weight …

161

∞, 0, 1/2, respectively. Corresponding to each cusp ω = ∞, 0, 1/2, there is an integral weight Eisenstein series of level 4 (and with weight k  ) given by the following. E ω (z, s; k  ) =





j (gω−1 γ , z)k Im(gω−1 γ z)s .

(6)

γ ∈ω \0 (4)

In the above, j (γ , z) = (cz + d)(cz + d)−1 and the stabilizer ω of the cusps ω is given by ω = gω ∞ gω−1 , ω = ∞, 0, 1/2. It is known that these Eisenstein series converge absolutely for Re(s) > 1 and that they have analytic continuations to the whole of C. Further, they satisfy a functional equation. In particular, we have E ∞ (z, 1 − s; k  ) = φ∞ (s)E ∞ (z, s; k  ) + φ0 (s)E 0 (z, s; k  ) + φ1/2 (s)E 1/2 (z, s; k  ), (7) where 24s−3 ζ (2s)(s + k  )π −s (8) φ∞ (s) = (1 − 22s−2 ζ (2 − 2s)(1 − s + k  )π −(1−s) ) 

and φ0 (s) = φ1/2 (s) =

1 22s−1

 − 1 φ∞ (s).

(9)

  ab Also, for any γ = ∈ 0 (4), we have c d −k 

E ω (γ z, s; k  ) = (cz + d)



(cz + d)k E ω (z, s; k  ).

k1 +k2 +1

2 So, the function H (z)E ω (z, s; k1 −k )y 2 is invariant under 0 (4) and, therefore, 2 one can consider the integral    k1 +k2 +1 d xd y k1 − k2 y 2 H (z)E ω z, s; . (10) Rω = 2 y2

0 (4)\H

Now following the standard unfolding argument, we obtain the Rankin–Selberg Lfunction as follows: ∞  a f1 (n)a f2 (n) L(s, f 1 × f 2 ) = . (11) n s+(k1 +k2 −1)/2 n=1 In the following, we define the completed Rankin–Selberg L-functions associated to f 1 and f 2 corresponding to each of the three cusps ∞, 0, 1/2.

162

M. K. Pandey and B. Ramakrishnan ∗

∞ (s, f 1 ⊗ f 2 ) = π −2s−k (s + k  )(s + k ∗ )ζ (2s) ∗

0 (s, f 1 ⊗ f 2 ) = π −2s−k (s + k  )(s + k ∗ )ζ (2s)

∞  a f1 (n)a f2 (n) , n s+k ∗ n=1 ∞  n=1 n≡0 (mod 4)

∗

∞ (s, f 1 ⊗ f 2 ) = π −2s−k (s + k  )(s + k ∗ )ζ (2s)

a f1 (n)a f2 (n) , n s+k ∗

∞  n=1 n≡(−1)k1 (mod 4)

a f1 (n)a f2 (n) . n s+k ∗

(12) In the above, we made the following substitutions, k  = (k1 − k2 )/2 and k ∗ = (k1 + k2 − 1)/2. We also assume that k1 > k2 . Note that our definition of completed Rankin–Selberg L-function differs from the definition given by Munshi [8, p. 671] by some power of 2. The functional equation we obtain below is the same as obtained by Munshi. However, he combines both the cases of k  even and odd into a single functional equation. Note that as k1 and k2 have the same parity, k  is an integer. These completed Rankin–Selberg L-functions together satisfy a functional equation, which is given below. ∞ (1 − s, f 1 ⊗ f 2 ) = ψ∞ (s)∞ (s, f 1 ⊗ f 2 ) + ψ0 (s)0 (s, f 1 ⊗ f 2 ) + ψ1/2 (s)1/2 (s, f 1 ⊗ f 2 ),

(13)

where 

1 (−1)k (1 − 22s−1 ) , ψ . (s) = ψ (s) = ψ∞ (s) = 0 1/2 2(1 − 22s−2 ) 2(1 − 22s−2 )

(14)

Note that ψ∞ (1/2) = 1. Since both f 1 and f 2 belong to the Kohnen plus space, we see that ∞ (s, f 1 ⊗ f 2 ) = 0 (s, f 1 ⊗ f 2 ) + 1/2 (s, f 1 ⊗ f 2 ). So, with this observation, we have the following.  ∞ (s, f 1 ⊗ f 2 ), if k  is even, ∞ (1 − s, f 1 ⊗ f 2 ) = (2ψ∞ (s) − 1)∞ (s, f 1 ⊗ f 2 ), if k  is odd.

(15)

4 Approximate Functional Equation In this section, we determine approximate functional equation for the completed Rankin–Selberg L-function and use it to get an expression for the central value of the Rankin–Selberg L-function. The methods used in this section are standard. For

Determining Modular Forms of Half-Integral Weight …

163

details we refer to [1, 2]. As in Sect. 3, we assume that f i ’s are modular forms in Sk+i +1/2 (0 (4)) and also we use the same notation for k  and k ∗ as given in Sect. 3. Let G(u) be a holomorphic function on an open set containing |Re(u)|  3/2 and bounded therein. We also choose the function G such that G(0) = 1, G(−u) = G(u) 2 (later we will be taking G(u) = eu ). For X > 0, we consider the integral I (X, s) =

1 2πi

 (3/2)

X u ∞ (s + u, f 1 ⊗ f 2 )

G(u) du. u

We now move the line of integration from 3/2 to −3/2, which will pick up the residue at u = 0 and so we get I (X, s) =

1 2πi

 (−3/2)

X u ∞ (s + u, f 1 ⊗ f 2 )

G(u) du + ∞ (s, f 1 ⊗ f 2 ). u

Therefore, 1 ∞ (s, f 1 ⊗ f 2 ) = I (X, s) − 2πi

 (−3/2)

X u ∞ (s + u, f 1 ⊗ f 2 )

G(u) du. u

Using the functional equation given by (15), the above expression is given by ⎧ ⎨ I (X, s) + I (X −1 , 1 − s), if k  is even ∞ (s, f 1 ⊗ f 2 ) = I (X −1 , 1 − s) ⎩ I (X, s) + , if k  is odd. 2ψ∞ (s) − 1

(16)

(Recall that k  = (k1 − k2 )/2.) We now define L(s, f 1 ⊗ f 2 ) := ζ (2s)L(s, f 1 × f 2 ),

(17)

where L(s, f 1 × f 2 ) is defined by (11). We write the Dirichlet series corresponding to ζ (2s)L(s, f 1 × f 2 ) as follows. L(s, f 1 ⊗ f 2 ) = ζ (2s)L(s, f 1 × f 2 ) =

∞ 

b f1 ⊗ f2 (n)n −s ,

(18)

n=1

where the coefficients are given by b f1 ⊗ f2 (n) =

 a f (m)a f (m) 1 2 . k∗ m 2

(19)

n=mt

(Note: k ∗ = (k1 + k2 − 1)/2.) Substituting the above notation, the integral I (X, s) becomes

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M. K. Pandey and B. Ramakrishnan

I (X, s) =

1 2πi



∗

(3/2)

X u π −2s−2u−k (s + u + k ∗ )(s + u + k  )

∞  b f1 ⊗ f 2 (n) G(u) du. n s+u u n=1

Now, interchanging the order of integration and summation, we get ∗

I (X, s) = π −2s−k (s + k ∗ )(s + k  )

∞  bf

1 ⊗ f 2 (n)

ns

n=1

where



 Vs

π 2n X

 ,

(20)



 1 Xu G(u) Vs = γ (s, u) du, 2πi (3/2) (π 2 n)u u (s + u + k ∗ )(s + u + k  ) . γ (s, u) = (s + k ∗ )(s + k  ) π 2n X

(21)

Substituting X by X −1 and s by 1 − s in (20), we get ∗ I (X −1 , 1 − s) = π −2+2s−k (1 − s + k ∗ )(1 − s + k  )

∞  b f 1 ⊗ f 2 (n) V1−s n 1−s



n=1

π 2n X −1

 .

Therefore, using (16), when k  is even, we get ∞  b f 1 ⊗ f 2 (n) V1−s n 1−s n=1   ∞  b f 1 ⊗ f 2 (n) ∗ π 2n −2s−k ∗  (s + k )(s + k ) Vs +π ns X



∗ ∞ (s, f 1 ⊗ f 2 ) = π −2+2s−k (1 − s + k ∗ )(1 − s + k  )

π 2n X −1



n=1

and when k  is odd, it follows that ∗

∞ (s, f 1 ⊗ f 2 ) = π −2s−k (s + k ∗ )(s + k  )

 2  ∞  b f1 ⊗ f2 (n) π n Vs s n X n=1

+

∗ π −2+2s−k (1 − s

+ k ∗ )(1 − s

2ψ∞ (s) − 1

 2  ∞ π n + k  )  b f1 ⊗ f2 (n) V . 1−s n 1−s X −1 n=1

Observe that at the point s = 1/2, both sides of the above expressions have the same gamma factor and the same power of π , and so after cancelation of these terms, we get L(1/2, f 1 ⊗ f 2 ) =

∞  bf n=1

1 ⊗ f 2 (n)

n 1/2



 V1/2

π 2n X



 + V1/2

π 2n X −1

 .

Determining Modular Forms of Half-Integral Weight …

165

Now, substituting X = 1 in the above, we have the following expression for the central value: ∞  b f1 ⊗ f2 (n) L(1/2, f 1 ⊗ f 2 ) = 2 V1/2 (π 2 n). (22) 1/2 n n=1

5 An Auxiliary Theorem + Let g ∈ S+1/2 (0 (4)) be Hecke eigenform with Fourier coefficients ag (n). For a fixed fundamental discriminant D with (−1)k D > 0, the following average simplifies using (22) as follows.



ω f L(1/2, f ⊗ g)

f ∈Fk+1/2 (4)

a f (|D|) =2 |D|k/2−1/4

∞ 



ωf

f ∈Fk+1/2 (4) n=1

b f ⊗g (n) a f (|D|) V1/2 (π 2 n). n 1/2 |D|k/2−1/4

(23) From now onwards, we use the following notation κ and κ ∗ (instead of k  and k ∗ ): κ = (k − )/2 and κ ∗ = (k +  − 1)/2. Now substituting for b f ⊗g (n) from Eq. (19), we get, 

ω f L(1/2, f ⊗ g)

f ∈Fk+1/2 (4)

a f (|D|) |D|k/2−1/4 =2



ωf

∞ 

⎛ ⎝

f ∈Fk+1/2 (4) n=1

=2

 a f (m)ag (m)

n=mt 2

∞  V1/2 (π 2 n)  n=1

n 1/2

n=mt 2

∗ mκ

⎠

a f (|D|) V1/2 (π 2 n) |D|k/2−1/4



ag (m) m /2−1/4

⎞

f ∈Fk+1/2 (4)

ωf

n 1/2

a f (m)a f (|D|) (m|D|)k/2−1/4

.

(24) Using the Petersson formula (Eq. (5)), the above becomes  f ∈Fk+1/2

a f (|D|) 4 ω f L(1/2, f ⊗ g) = k/2−1/4 |D| 3 (4)



 ag (|D|) M|D| (k, ) + E g,|D| (k, ) , |D|/2−1/4 (25)

where M|D| (k, ) is the main term given by M|D| (k, ) = |D|−1/2 and E g,|D| (k, ) is the error term given by

∞  V1/2 (π 2 |D|t 2 ) t t=1

(26)

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M. K. Pandey and B. Ramakrishnan

E g,|D| (k, ) =

    √ ∞  k+1 √  V1/2 (π 2 n)  ag (m) π m|D| 2 , (−1) π 2 H (|D|, m)J c k−1/2 n 1/2 c m /2−1/4 2 n=1

c1

n=mt

(27) with V1/2 (π 2 x) is given by (21). Thus, we obtained the following (auxiliary) theorem to prove our main result. + (0 (4)) and Theorem A Let g be a cusp form in the Kohnen plus space S+1/2 + Fk+1/2 (4) be an orthogonal basis for the space Sk+1/2 (0 (4)). Then, we have the following formula for the spectral average of the central value of the Rankin–Selberg (convolution) L-functions.

 f ∈Fk+1/2

a f (|D|) 4 ω f L(1/2, f ⊗ g) = k/2−1/4 |D| 3 (4)



 ag (|D|) M|D| (k, ) + E g,|D| (k, ) , |D|/2−1/4

(28) where M|D| (k, ) and E g,|D| (k, ) are given by Eqs. (26), (27) (ω f and L(1/2, f ⊗ g) are defined in Sects. 2 and 3, respectively). In the next two sections, we shall give estimates for these main and error terms in order to get our main result.

6 Estimation of the Main Term M| D| (k, ) We have M|D| (k, ) = |D|−1/2

∞  V1/2 (π 2 |D|t 2 ) t t=1

 −u ∞  1 1 (π 2 |D|t 2 ) (u + a)(u + b)G(u) du (by using (21)). = |D|−1/2 t 2πi (3/2) (a)(b)u t=1

Here, we have put a = κ ∗ + 1/2 and b = κ + 1/2, where κ, κ ∗ are defined as in Sect. 5. So, we have M|D| (k, ) =



|D|−1/2 2πi

(3/2)

−u

(π 2 |D|) (u + a)(u + b)G(u) ζ (2u + 1)du. (a)(b)u

Now by moving the line of integration to Re(u) = −1/2 and noticing that the integrand has a double pole at u = 0, with the residue at u = 0 given by 



  (a) + (b) + 2γ0 − log(π 2 |D|),   where γ0 is Euler’s constant. Therefore, we have

Determining Modular Forms of Half-Integral Weight … 

|D|

1/2

167



  M|D| (k, ) = (a) + (b) + 2γ0 − log(π 2 |D|) + I,  

where I denotes the following integral along the line (−1/2). I =

1 2πi



−u

(−1/2)

(π 2 |D|) (u + a)(u + b)G(u) ζ (2u + 1)du. (a)(b)u

For estimating the integral I , we need the following known estimate for the ratio of -functions, using the Stirling bound. See, for example, [1, Lemma 2]. For A(> 0), c real with |c| < A/2, one has (A + c + it)

|A + it|c , (A + it)

(29)

where the implied constant depends on c. Using this estimate along with the fact that |(x + i y)|  |(x)|, we get the following estimate of the integral I along the line Re(u) = −1/2. |D|1/2 I  2κ



∞ −∞

|

G(−1/2 + iv) ||ζ (2iv)|dv. (−1/2 + iv)

Further, using the fact that ζ (it) |t|1/2 , it follows that I |D|κ . Finally, combining everything, we have the following estimate for the main term: 1/2

 M|D| (k, ) = |D|

1/2

     2 (a) + (b) + 2γ0 − log(π |D|) + O(1/k).  

(30)

7 Estimation of the Error Term E g,| D| (k, ) Before we proceed to estimate the error term, we obtain some preliminary results on certain Dirichlet series in the following section.

7.1 Preliminary Results In this section, we prove the functional equation of certain Dirichlet series associated to of half-integral weight in the Kohnen plus space. Let

modular forms + 2πinz ∈ S+1/2 (4). We consider the following Dirichlet series g(z) = ∞ n=1 ag (n)e associated to g, defined by

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     ∞ ag (n)e αn β α = L g s, , /2−1/4+s β n n=1

(31)

where α, β are positive integers with (α, β) = 1. In the above, we have used the following notation: e(x) = e2πi x . Next, we prove the functional equation satisfied by the above Dirichlet series. Suppose that t ∈ R+ . Since g is invariant under the action of 0 (4), we get g(γ z) = where γ =

d

d

  ab ∈ 0 (4). Now, taking z = cd 

g

 c   −4 −−1/2

it −d + c c

 =

 c   −4 +1/2 d

d

(cz + d)+1/2 g(z), −d c

+ itc , we get γ z =

(cz + d)−−1/2 g



+

a c

i , ct

so that



i a + c ct

.

(32)

The Fourier expansions of g(z) and g(γ z) (for the above value of z) are given by 

    ∞ −d it −nd −2πtn g + = e c , ag (n)e c c c n=1    ∞  na  −2πn a i g + = e ct . ag (n)e c ct c n=1 Using (32), the Mellin transform of g becomes 



∞

g 0

−d it + c c

 t s+/2−1/4

dt = t

∞  c   −4 +1/2

 0

d

d

(it)−−1/2 g



a i + c ct

 t s+/2−1/4

dt . t

(33) Now substituting the Fourier expansion of g(z) and g(γ z) as given above, we get

(c/2π)s+/2−1/4 (s + /2 − 1/4)

∞  ag (n)e( −nd c ) n s+/2−1/4 n=1

= (c/2π)/2+3/4−s (/2 + 3/4 − s)

 c   −4 +1/2 d

d

ι−−1/2

∞  ag (n)e( na c ) n /2+3/4−s n=1

.

(34)

Determining Modular Forms of Half-Integral Weight …

169

7.2 Error Term Estimation For simplicity, we write the error term as E and it is given by   √ ∞  √  V1/2 (π 2 n)  ag (m) π m|D| [ k+1 2 ]π . (−1) 2 H (|D|, m)J c k−1/2 n 1/2 m /2−1/4 c 2 n=1 c1 n=mt

So, we write it as 

E = (−1)

k+1 2



  √ ∞ ∞ √  ag (m)  V1/2 (π 2 mt 2 )  π m|D| . π 2 H (|D|, m)J c k−1/2 t c m /2+1/4 m=1

t=1

c1

Proceeding as done in [1, Sect. 4], (by using the inverse Melling transform of the J -Bessel function), we can write E as √ k+1 (−1)[ 2 ] π 2 E= 2(2πi)2    G(u)  k2 −  × ζ (2u + 1) u  k2 + (3/2) (α)

1 4 3 4

+ −



s 2  s 2



   + u + κ  21 + u + κ ∗      21 + κ  21 + κ ∗

1 2

× 2s π −2u−s |D|−s/2 Su duds, where Su =

∞   m=1 c1

ag (m) /2+1/4+u+s/2 m

Hc (|D|, m) . c−s

Using the Weil bound for the Kloosterman sum Hc (|D|, m), the series converges absolutely, and so we can change the order of summation in Su to get     s−1       4c c −4 k+1/2 4 k Su = 1 − (−1) i 1 + c 4 a (mod 4c)∗ a a c1   ∞  ag (m) |D|a + md × e m /2+1/4+u+s/2 4c m=1       4c   −4 k+1/2  4 k = 1 − (−1) i 1 + c a a a (mod 4c)∗ c1     e |D|a 4c d 1/2 + u + s/2, , × L g c1−s 4c

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where d is an integer such that ad ≡ 1 (mod 4c), and we have denoted the sum over m by the Dirichlet series L g (s, d/4c) as defined in Sect. 7.1. Now by applying the functional equation for the Dirichlet series given by (34), we get    2 + 41 − u − 2s i +1/2 2u+s   (π/2) Su =(1 − (−1) i) 4  2 + 41 + u + 2s      −4 k− 4 −1−2u 1+ c × c a a (mod 4c)∗ c1     −a |D|a e . × L g 1/2 − u − s/2, 4c 4c k

Now, we move the line of integration in the s variable to Re(s) = α = −7. Since Re(−u − s/2 + 1/2) = 5/2, the Dirichlet series L g (1/2 − u − s/2, −a/4c) is absolutely convergent. Therefore, we can write 

 a (mod 4c)∗

−4 a

 k−    −a |D|a L g 1/2 − u − s/2, e 4c 4c =

∞  m=1



ag (m) m /2+1/4−u−s/2

a (mod 4c)∗



−4 a

k−   (|D| − m)a e 4c .

Since we are interested in the case where k and  having the same parity, the sum over a reduces to the Ramanujan sum and so we have the following estimate for Su : Su (π/2)2u+s

(/2 + 1/4 − u − s/2) . (/2 + 1/4 + u + s/2)

Thus, the estimate of the error term E simplifies as 



(π/2)2u+s ζ (2u + 1)

E (3/2) (−7)

×

G(u) (k/2 − 1/4 + s/2) (1/2 + u + κ) u (k/2 + 3/4 − s/2) (1/2 + κ)

(1/2 + u + κ ∗ ) (/2 + 1/4 − u − s/2) duds. (1/2 + κ ∗ ) (/2 + 1/4 + u + s/2)

(35) Now writing u = 3/2 + iv and s = −7 + it and integrating with respect to the t variable we get,  E

∞ −∞

e−v (2 + iv + κ ∗ )(2 + iv + κ) |dv. (9/4 + v2 )1/2 (1/2 + κ ∗ )(1/2 + κ) 2

k −7 (k 2 + v2 )1/2 |

Finally, using the bound for the ratio of -functions and the estimate |(x + i y)|  |(x)|, we have

Determining Modular Forms of Half-Integral Weight …

E k

−7



∞ −∞

171

e−v dv 1. (9/4 + v2 )1/2 2

(v2 + k 2 )7/2

8 Proof of the Main Theorem Using our auxiliary result (Theorem A), we shall prove the main result in this section. + (0 (4)) and g  ∈ S+ +1/2 (0 (4)) are Hecke By assumption, the functions g ∈ S+1/2 eigenforms such that L(1/2, f ⊗ g) = L(1/2, f ⊗ g  ), for all Hecke eigenforms + (0 (4)). Therefore, Theorem A implies that for all fundamental discrimf ∈ Sk+1/2 inants D with (−1)k D > 0, ag (|D|) ag (|D|) M|D| (k, ) + E g,|D| (k, ) = M|D| (k,  ) + E g ,|D| (k,  ). |D|/2−1/4 |D| /2−1/4 (36) Using Stirling’s formula for the derivatives of (s), it follows that for k large, M|D| (k, ) = log k + O(1). Also the error terms are bounded for large k. Using these two observations in (36), we get ag (|D|) ag (|D|) = , |D|/2−1/4 |D| /2−1/4

(37)

for all fundamental discriminants D with (−1)k D > 0. Let F and F  be the normalized Hecke eigenforms of weights 2 and 2 on S L 2 (Z), corresponding to the Hecke eigenforms g and g  (via the Shimura–Kohnen maps). Using the corresponding Waldspurger’s formula for g and g  , obtained by Kohnen [5, Corollary 1] and using (37), we see that L(F, χ D , ) = C L(F  , χ D ,  ),

(38)

for all fundamental discriminants with (−1)k D > 0 and C > 0 is a constant. (Here, L(F, χ D , ) denotes the usual   L-function associated to the modular form F twisted with the character χ D = D· .) Using Theorem B of Luo–Ramakrishnan [7], this implies that  =  and F = F  . Our main theorem now follows using the ‘multi+ (0 (4)). plicity 1’ result in S+1/2

References 1. Ganguly, S., Hoffstein, J., Sengupta, J.: Determining modular forms on S L(2, Z) by central values of convolution L-functions. Math. Ann. 345, 843–857 (2009) 2. Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)

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3. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Studies in Mathematics, vol. 97. Springer, Berlin (1993) 4. Kohnen, W.: Modular forms of half-integral weight on 0 (4). Math. Ann. 248(3), 249–266 (1980) 5. Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271, 237–268 (1985) 6. Luo, W.: Special L-values of Rankin-Selberg convolutions. Math. Ann. 314(3), 591–600 (1999) 7. Luo, W., Ramakrishnan, D.: Determination of modular forms by twists of critical L-values. Invent. Math. 130(2), 371–398 (1997) 8. Munshi, R.: A note on simultaneous nonvanishing twists. J. Number Theory 132, 666–674 (2012) 9. Pi, Q.: Determining cusp forms by central values of Rankin-Selberg L-functions. J. Number Theory 130, 2283–2292 (2010) 10. Pandey, M.K.: Explicit Shimura liftings of certain class of forms and some problems involving modular l-functions. Ph.D Thesis, Homi Bhabha National Institute (HRI, Prayagraj), August 2019 11. Shimura, G.: On modular forms of half integral weight. Ann. Math (2) 97, 440–481 (1973) 12. Zhang, Y.: Determining modular forms of general level by central values of convolution Lfunctions. Acta Arith. 150, 93–103 (2011)

On the Number of Representations of a Natural Number by Certain Quaternary Quadratic Forms B. Ramakrishnan, Brundaban Sahu, and Anup Kumar Singh

Abstract In this paper, we find a basis for the space of modular forms of weight 2 on 1 (48) and then use this basis to find formulas for the number of representations of  a positive integer are of 2 n by2 certain quaternary2 quadratic 2forms which 4 ai xi2 , i=1 bi (x2i−1 + x2i−1 x2i + x2i ) and a1 x1 + a2 x22 + b1 (x32 + the form i=1 x3 x4 + x42 ), where ai ’s belong to {1, 2, 3, 4, 6, 12} and bi ’s belong to {1, 2, 4, 8, 16}. In [1], A. Alaca et al. considered similar problem for the quaternary forms (which are diagonal) with coefficients 1, 2, 3, 6. Thus, our work extends their results with additional coefficients 4 and 12, and further in our work, we consider two more types of quaternary quadratic forms which are not diagonal. Moreover, our formulas for the diagonal quaternary quadratic forms with coefficients in {1, 2, 3, 4, 6, 12} include explicit formulas for the number of representations (of a natural number) by 8 of the Ramanujan’s universal quaternary quadratic forms [19]. We also determine some of the universal quadratic forms in the other two types of forms considered in our work. Keywords Representation numbers of quaternary quadratic forms · Modular forms of one variable 2010 Mathematics Subject Classification 11E25 · 11F11 · 11E20 B. Ramakrishnan (B) (on lien) Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019, India e-mail: [email protected]; [email protected] Department of Statistics and Applied Mathematics, Central University of Tamil Nadu, Neelakudi, Thiruvarur 610 005, India B. Sahu School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, HBNI, Via- Jatni, Khurda 752 050, Odisha, India e-mail: [email protected] A. K. Singh Indian Institute of Science Education and Research, Berhampur 760 010, Odisha, India e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_12

173

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1 Introduction In this paper, we consider the problem of finding the number of representations of a natural number by the following three types of quaternary quadratic forms given by Q1 : a1 x12 + a2 x22 + a3 x32 + a4 x42 , Q2 : b1 (x12 + x1 x2 + x22 ) + b2 (x32 + x3 x4 + x42 ), Q3 : a1 x12 + a2 x22 + b1 (x32 + x3 x4 + x42 ), where the coefficients ai ∈ {1, 2, 3, 4, 6, 12}, 1 ≤ i ≤ 4 and bi ∈ {1, 2, 4, 8, 16}, i = 1, 2. Without loss of generality, we may assume that a1 ≤ a2 ≤ a3 ≤ a4 , b1 ≤ b2 and gcd(a1 , a2 , a3 , a4 ) = 1, gcd(b1 , b2 ) = 1, gcd(a1 , a2 , b1 ) = 1. Finding explicit formula for the number of representations of n by these types of quadratic forms is a classical problem in number theory. The classical formula of Jacobi for the sum of four squares corresponds to the quadratic form Q1 with (a1 , a2 , a3 , a4 ) = (1, 1, 1, 1). There are several works in the literature which give formulas for the representation numbers corresponding to quaternary quadratic forms with coefficients. We list some of them here [1–9, 22]. In [1], A. Alaca et al. considered 35 quadratic forms of type Q1 with coefficients ai ∈ {1, 2, 3, 6}. They obtained explicit bases for the spaces of modular forms of  weight 2 on 0 (24) with character χ0 (trivial character modulo 24) or χd = d· for d = 8, 12, 24 and used these bases to determine formulas for the number of representations of a natural number by Q1 , with ai ∈ {1, 2, 3, 6}. However, out of these 35 quadratic forms of type Q1 , formulas for 18 forms appeared in the works [2, 4, 5, 9]. More precisely, denoting the quadratic forms in Q1 by the quadruple (a1 , a2 , a3 , a4 ), the forms (1, 1, 1, 1), (1, 1, 2, 2), (1, 1, 3, 3), (1, 1, 6, 6), (1, 2, 3, 6), (2, 2, 3, 3) were considered in [2], the forms (1, 1, 1, 3), (1, 1, 2, 6), (1, 2, 2, 3), (1, 3, 3, 3), (1, 3, 6, 6), (2, 3, 3, 6) were considered in [4], the forms (1, 1, 1, 2), (1, 2, 2, 2) were considered in [5, 22], and the forms (1, 1, 2, 3), (1, 2, 2, 6), (1, 3, 3, 6), (2, 3, 6, 6) were considered in [9]. There are several works which deal with some of these cases. For details we recommend the reader to look at the references appearing in the works of Williams and his co-authors mentioned here. The total number of quadratic forms Q1 , with ai ∈ {1, 2, 3, 4, 6, 12} is 126. Out of this, 35 cases come from the works of [1, 2, 4, 5, 9] (when ai = 4, 12), and so we do not consider these cases in our present work. Further, there are 36 cases which have the property that gcd(coefficients) > 1. Therefore, in our work, we consider only the remaining 55 cases of quadratic forms Q 1 . There are only 4 quadratic forms of type Q2 and there are 65 quadratic forms of type Q3 (such that the coefficients have no common factors). In (Sect. 4 Table 1), we give the list of quadratic forms Qi , i = 1, 2, 3 considered in our work (55 forms in Q1 , 4 forms in Q2 and 65 forms in Q3 ). These are listed according to the modular forms space (in which the corresponding theta series belong). Note that in place of M2 (48, χ ), we mention only the character χ which is either χ0 (trivial character modulo 48) or χd , d = 8, 12, 24.

On the Number of Representations of a Natural Number by Certain …

175

Some of our formulas were also proved in works of K. S. Williams and his coauthors [2–8], which we mention in Table 2 (Sect. 4) the table below. (These formulas were obtained using different methods.) Let N, Z, Q, and C denote the sets of natural numbers, integers, rational numbers, and complex numbers, respectively. For n ∈ N, let the number of representations of n by the quadratic forms Q1 , Q2 , and Q3 be denoted, respectively, by N1 (a1 , a2 , a3 , a4 ; n) = #{(x1 , x2 , x3 , x4 ) ∈ Z4 : a1 x12 + a2 x22 + a3 x32 + a4 x42 = n}, (1) N2 (b1 , b2 ; n) = #{(x1 , x2 , x3 , x4 ) ∈ Z4 : b1 (x12 + x1 x2 + x22 ) + b2 (x32 + x3 x4 + x42 ) = n}, (2) and N3 (a1 , a2 , b1 ; n) = #{(x1 , x2 , x3 , x4 ) ∈ Z4 : a1 x12 + a2 x22 + b1 (x32 + x3 x4 + x42 ) = n}.

(3) We observe that the generating functions corresponding to the quaternary quadratic forms considered in our work are modular forms of weight 2 on 1 (48). So, we construct explicit bases for the spaces of modular forms of weight 2 on 0 (48) with character χ (modulo 48) and use them to give formulas for N1 (a1 , a2 , a3 , a4 ; n), N2 (b1 , b2 ; n), and N3 (a1 , a2 , b1 ; n). It is to be noted that in his work [19], S. Ramanujan gave the list of 55 universal quadratic forms of type Q1 . Our work includes 8 out of these 55 forms which are given by (a1 , a2 , a3 , a4 ) = (1, 1, 1, 4), (1, 1, 2, 4), (1, 1, 2, 12), (1, 1, 3, 4), (1, 2, 3, 4), (1, 2, 4, 4), (1, 2, 4, 6), and (1, 2, 4, 12). We give explicit formulas for the number of representations of these 8 quadratic forms in Theorem 2.1. In §2.1, we give simplified expressions for some of the formulas obtained in our work and as a consequence deduce that the quadratic form x12 + x1 x2 + x22 + (x32 + x3 x4 + x42 ) is universal when  = 2 and non-universal when  = 4. Using the formulas for N2 (b1 , b2 ; n) and N3 (a1 , a2 , b1 ; n), we show the universality and non-universality of some of the forms in these two types.

2 Preliminaries and Statement of Results We use the theory of modular forms to prove our results, and so we first fix our notations and present some of the basic facts on modular forms. For positive integers k, N ≥ 1 and a Dirichlet character χ modulo N with χ (−1) = (−1)k , let Mk (N , χ ) denote the C- vector space of holomorphic modular forms of weight k for the congruence subgroup 0 (N ), with character χ . Let us denote by Sk (N , χ ), the subspace of cusp forms in Mk (N , χ ). The modular forms space is decomposed into the space of Eisenstein series (denoted by Ek (N , χ )) and the space of cusp forms Sk (N , χ ) and one has (4) Mk (N , χ ) = Ek (N , χ ) ⊕ Sk (N , χ ).

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Explicit basis for the space Ek (N , χ ) can be obtained using the following construction. For details we refer to [17, 21]. Suppose that χ and ψ are primitive Dirichlet characters with conductors N and M, respectively. For a positive integer k ≥ 2, let ⎛ ⎞   ⎝ E k,χ,ψ (z) := c0 + ψ(d)χ (n/d)d k−1 ⎠ q n , n≥1

(5)

d|n

where q = e2πi z (z ∈ C, Im(z) > 0) and

0 c0 = Bk,ψ − 2k

if N > 1, if N = 1,

with Bk,ψ denoting the generalized Bernoulli number with respect to the character ψ. Then, the Eisenstein series E k,χ,ψ (z) belongs to the space Mk (N M, χ /ψ), provided χ (−1)ψ(−1) = (−1)k and N M = 1. We give a notation to the inner sum in (5): σk−1,χ,ψ (n) :=



ψ(d)χ (n/d)d k−1 .

(6)

d|n

In this paper, we use the Eisenstein series of the above type with the following 11 pairs of characters given by (1, χ8 ), (χ8 , 1), (1, χ12 ), (χ12 , 1), (1, χ24 ), (χ24 , 1), (χ−4 , χ−4 ), (χ−4 , χ−3 ), (χ−3 , χ−4 ), (χ−3 , χ−8 ), (χ−8 , χ−3 ). For a square-free integer d ≡ 1 (mod  4), the Dirichlet character χd (modulo |d|) denotes the real quadratic character d· , whereas for a square-free integer d ≡ 2, 3 (mod 4), the Dirichlet   character χ4d (modulo 4|d|) is the real quadratic character 4d· . These characters are nothing but the Kronecker symbol. The character 1 is the trivial character given by 1(n) = 1 for all n ≥ 1. The constant term c0 corresponding to each of these 11 pairs is given in the following table. (χ , ψ) (χ8 , 1), (χ12 , 1), (χ24 , 1), (χ−4 , χ−4 ), (χ−4 , χ−3 ), (χ−3 , χ−4 ), (χ−3 , χ−8 ), (χ−8 , χ−3 ) (1, χ8 ) (1, χ12 ) (1, χ24 )

c0 0 − 21 −1 −3

Bk When χ = ψ = 1 (i.e., when N = M = 1) and k ≥ 4, we have E k,χ,ψ (z) = − 2k E k (z), where E k (z) is the normalized Eisenstein series of weight k in the space Mk (1), defined by 2k  E k (z) = 1 − σk−1 (n)q n . (7) Bk n≥1

In the above, σr (n) is the sum of the r th powers of the positive divisors of n and ∞  Bm m x = x . We also need the Bk is the kth Bernoulli number defined by x e − 1 m=0 m!

On the Number of Representations of a Natural Number by Certain …

177

Eisenstein series of weight 2, which is a quasimodular form of weight 2, depth 1 on S L 2 (Z) and is given by  E 2 (z) = 1 − 24 σ (n)q n . n≥1

(Note that σ (n) = σ1 (n).) Let η(z) = q 1/24



(1 − q n )

n≥1

denote the Dedekind eta function. Then, is a finite product of integer s an eta-quotient powers of η(z) and we denote it as i=1 ηri (di z), where di ’s are positive integers and ri ’s are non-zero integers. For more details on holomorphicity/modularity of eta-quotients, one may refer to [14]. In the case of the space of cusp forms Sk (N , χ ), we use a basis consisting of newforms of level N and oldforms generated by the newforms of lower level d, d|N , χ modulo d, d = N . However, when χ = χ12 , we construct a basis for the space of newforms, which are not Hecke eigenforms. For a basic theory of newforms, we refer to [10, 16] and for details on modular forms, we refer to [15, 17, 21]. We now state the main results of this paper. In the following statements, χ denotes a Dirichlet character modulo 48, which is either  the principal character modulo 48, denoted as χ0 or the Kronecker symbol χd = d· , where d = 8, 12, or 24. For each such χ , let χ denote the dimension of the C- vector space M2 (48, χ ). Then

χ =

14 if χ = χ0 or χ12 , 12 if χ = χ8 or χ24 .

Theorem 2.1 Let n ∈ N. For each entry (a1 , a2 , a3 , a4 ) corresponding to Q1 in Table 1, the associated theta series is a modular form of weight 2 on 0 (48) with character χ . Therefore, using the basis given in (Sect. 4 Table 3), we have N1 (a1 , a2 , a3 , a4 ; n) =

χ 

αi,χ Ai,χ (n),

(8)

i=1

where Ai,χ (n) are the Fourier coefficients of the basis elements f i,χ and the values of the constants αi,χ ’s are given (in Sect. 4, Tables 5, 6, 7, 8). Explicit formulas for N1 (a1 , a2 , a3 , a4 ; n) are given below for the 8 universal quadratic forms (obtained in Ramanujan’s work [19]) corresponding to (a1 , a2 , a3 , a4 ) = (1, 1, 1, 4), (1, 1, 2, 4), (1, 1, 2, 12), (1, 1, 3, 4), (1, 2, 3, 4), (1, 2, 4, 4), (1, 2, 4, 6), and (1, 2, 4, 12).

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N1 (1, 1, 1, 4; n) = 4σ (n) − 20σ (n/4) + 24σ (n/8) − 32σ (n/16) + 2σ2,χ−4 ,χ−4 (n), N1 (1, 1, 2, 4; n) = 4σ2,χ8 ,1 (n) − 2σ2,1,χ8 (n/2), 1 2 N1 (1, 1, 2, 12; n) = 2σ2,χ24 ,1 (n) − σ2,1,χ24 (n/2) + σ2,χ−3 ,χ−8 (n) + σ2,χ−8 ,χ−3 (n/2) 3 3 4 16 + 16τ2,24,χ24 ;1 (n/2) + τ2,24,χ24 ;2 (n) + τ2,24,χ24 ;2 (n/2), 3 3 1 N1 (1, 1, 3, 4; n) = 3σ2,χ12 ,1 (n) − 3σ2,χ12 ,1 (n/2) + 12σ2,χ12 ,1 (n/4) − σ2,1,χ12 (n) 2 1 3 3 + σ2,1,χ12 (n/2) − σ2,1,χ12 (n/4) + σ2,χ−3 ,χ−4 (n) + σ2,χ−3 ,χ−4 (n/2) 2 2 2 + 3σ2,χ−3 ,χ−4 (n/4) − σ2,χ−4 ,χ−3 (n) − σ2,χ−4 ,χ−3 (n/2) − 4σ2,χ−4 ,χ−3 (n/4) N1 (1, 2, 3, 4; n) =

N1 (1, 2, 4, 4; n) = N1 (1, 2, 4, 6; n) = N1 (1, 2, 4, 12; n) =

+ τ2,48,χ12 ;1 (n) − τ2,48,χ12 ;2 (n), 1 2 2σ2,χ24 ,1 (n) − σ2,1,χ24 (n/2) + σ2,χ−3 ,χ−8 (n) + σ2,χ−8 ,χ−3 (n/2) 3 3 2 8 − 8τ2,24,χ24 ;1 (n/2) − τ2,24,χ24 ;2 (n) − τ2,24,χ24 ;2 (n/2), 3 3 2σ2,χ8 ,1 (n) − 2σ2,1,χ8 (n/2), 3 1 −σ2,1,χ12 (n/4) + σ2,χ12 ,1 (n) + σ2,χ−4 ,χ−3 (n) − 3σ2,χ−3 ,χ−4 (n/4), 2 2 1 1 σ2,χ24 ,1 (n) − σ2,1,χ24 (n/2) + σ2,χ−8 ,χ−3 (n) + σ2,χ−3 ,χ−8 (n/2) 3 3 4 2 + 4τ2,24,χ24 ;1 (n/2) + τ2,24,χ24 ;2 (n) + τ2,24,χ24 ;2 (n/2). 3 3

Theorem 2.2 Let n ∈ N. Then, we have N2 (1, 2; n) = 6σ (n) − 12σ (n/2) + 18σ (n/3) − 36σ (n/6), N2 (1, 4; n) = 6σ (n) − 18σ (n/2) − 18σ (n/3) + 24σ (n/4) + 54σ (n/6) − 72σ (n/12), 3 9 9 27 N2 (1, 8; n) = σ (n) − σ (n/2) + σ (n/3) + 9σ (n/4) − σ (n/6) − 12σ (n/8) 2 2 2 2 9 (9) + 27σ (n/12) − 36σ (n/24) + τ2,24 (n), 2 3 9 9 27 N2 (1, 16; n) = σ (n) − σ (n/2) − σ (n/3) + 9σ (n/4) + σ (n/6) − 18σ (n/8) 2 2 2 2 9 − 27σ (n/12) + 24σ (n/16) + 54σ (n/24) + 72σ (n/48) + τ2,48 (n), 2

where τ2,24 (n) and τ2,48 (n) are the nth Fourier coefficients of 2,24 (z) and 2,48 (z), respectively, defined in Sect. 3.1. Theorem 2.3 Let n ∈ N. For each entry (a1 , a2 , b1 ) corresponding to Q3 in Table 1, the associated theta series is a modular form of weight 2 on 0 (48) with character χ . Therefore, using the basis given in (Sect. 4Table 3), we have N3 (a1 , a2 , b1 ; n) =

χ  i=1

βi,χ Ai,χ (n),

(10)

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where Ai,χ (n) are the Fourier coefficients of the basis elements f i,χ and the values of the constants βi,χ ’s are given in (Sect. 4, Tables 9, 10, 11 and 12). Note: Explicit formulas for some of the cases (a1 , a2 , b1 ) in the above theorem are given in Sects. 2.1 and 2.2.

2.1 Simplification of Some of the Formulas and Determining Universal Property In this section, we shall simplify some of the formulas given in Theorems 2.1–2.3 and discuss about the universal property of the corresponding quadratic forms. In Theorem 2.1, we consider three formulas corresponding to (1, 1, 1, 4), (1, 1, 2, 4), and (1, 2, 4, 4). We first consider the formula for N1 (1, 1, 1, 4; n), given in Theorem 2.1: N1 (1, 1, 1, 4; n) = 4σ (n) − 20σ (n/4) + 24σ (n/8) − 32σ (n/16) + 2σ2,χ−4 ,χ−4 (n), where σ2,χ−4 ,χ−4 (n) =

 d|n

 −4  −4  d  −4 

n/d

d. This twisted divisor sum vanishes when

n is even and it is equal to n σ (n), when n is odd. Therefore, when n is odd,    σ (n). When n is even, using the the representation number becomes 4 + 2 −4 n multiplicative property, it is easy to see that it takes the value λσ (m), where n = 2α m, α ≥ 0 and λ = 12, 8, or 24 according as α = 1, 2 or ≥ 3, respectively. Thus, we have ⎧   (4 + 2 −4 )σ (n) ⎪ n ⎪ ⎪ ⎨12σ (m) N1 (1, 1, 1, 4; n) = ⎪8σ (m) ⎪ ⎪ ⎩ 24σ (m)

if if if if

2  |n, n = 2m, m is odd, (11) n = 4m, m is odd, n = 2α m, m is odd and α ≥ 3.

Note that in the case when n is odd, the formula is nothing but 6σ (n) if n ≡ 1 (mod 4) and 2σ (n) if n ≡ 3 (mod 4). From this formula, it is clear that N1 (1, 1, 1, 4; n) > 0 for all n ≥ 1. This shows that the form x12 + x22 + x32 + 4x42 is universal. Next, we consider the quadratic forms corresponding to the cases (1, 1, 2, 4) and (1, 2, 4, 4). For an odd natural number n, we have     2  2 2 d = 4 d. (12) N1 (1, 1, 2, 4; n) = 4σ2,χ8 ,1 (n) = 4 n/d n d d|n d|n When n = 2α m, α ≥ 1, m odd, then the formula simplifies as follows.

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N1 (1, 1, 2, 4; n) = 4σ2,χ8 ,1 (n) − 2σ2,1,χ8 (n/2)  2  2  d −2 d = 4 n/d d d|n d|n/2      2 2 α+2 = 2 −2 d. m d d|m

(13)

Combining the above two cases, we get

N1 (1, 1, 2, 4; n) =

⎧     2 2 ⎪ ⎨4 n d|n d d

if n is odd,

⎪  2 ⎩ α+2  2  α −2 2 d|m d d if n = 2 m, α ≥ 1, m is odd. m      (14) Now, it is easy to see that for an odd positive integer n, both n2 and d|n d2 d have   2 d is positive when n the same sign (positive or negative). Therefore, n2  d  2   α+2  2  d|n − 2 is odd. Using this fact, it also follows that 2 d|m d d is positive for m all odd positive integers m, when α ≥ 1. (Note that one can give explicit values for these twisted divisor sums similar to the one given in Eq.(22).) Thus, for all natural numbers n, we have N1 (1, 1, 2, 4; n) > 0, which implies that the quaternary form x12 + x22 + 2x32 + 4x42 is universal. Using similar arguments as in the case of N1 (1, 1, 2, 4; n), the formula for N1 (1, 2, 4, 4; n) given in Theorem 2.1 simplifies as follows (with n = 2α m, α ≥ 0 and m is odd). N1 (1, 2, 4, 4; n) = 2σ2,χ8 ,1 (n) − 2σ2,1,χ8 (n/2) ⎧     2 2 ⎪ if n is odd, ⎨2 n d|n d d = ⎪  2 ⎩ α+1  2  α −2 2 d|m d d if n = 2 m, α ≥ 1, m is odd. m (15) Thus, the form x12 + 2x22 + 4x32 + 4x42 is also universal. Next, we simplify the two formulas for N2 (1, 2; n) and N2 (1, 4; n) given in Theorem 2.2. Using the multiplicative property of the divisor function, it can be seen that N2 (1, 2; n) = 6σ (n) − 12σ (n/2) + 18σ (n/3) − 36σ (n/6) = 6(3β+1 − 2)σ (m), if n = 2α 3β m,

gcd(m, 6) = 1,

(16)

which is always positive and, therefore, the form x12 + x1 x2 + x22 + 2(x32 + x3 x4 + x42 ) is a universal quadratic form.

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Now for the other case N2 (1, 4; n), Theorem 2.2 gives the following formula N2 (1, 4; n) = 6σ (n) − 18σ (n/2) − 18σ (n/3) + 24σ (n/4) + 54σ (n/6) − 72σ (n/12).

Now, write n = 2α 3β m, with gcd(m, 6) = 1, the above formula reduces to    6σ (m) σ (3β ) − 3σ (3β−1 ) σ (2α ) − 3σ (2α−1 ) + 4σ (2α−2 ) . The value of the factor (σ (2α ) − 3σ (2α−1 ) + 4σ (2α−2 )) is 1 when α = 0, and 0 if α = 1, whereas, it is equal to 2(2α−1 − 1) for α ≥ 2. On the other hand, for all β ≥ 0, we have (σ (3β ) − 3σ (3β−1 )) = 1. It turns out that the formula does not depend on β and so assuming n = 2α 3β m, m a positive integer with gcd(m, 6) = 1, the formula for N2 (1, 4; n) becomes ⎧ ⎪ if α = 0, ⎨6σ (m) N2 (1, 4; n) = 0 if α = 1, ⎪ ⎩ 12(2α−1 − 1)σ (m) if α ≥ 2.

(17)

Since N2 (1, 4; n) = 0 for all positive integers n ≡ 2 (mod 4), it follows that the corresponding quadratic form x12 + x1 x2 + x22 + 4(x32 + x3 x4 + x42 ) is not a universal form. Further, if n = 3β , β ≥ 1, then N2 (1, 4; n) = 6 and for n = 2α 3β , α ≥ 2, we have N2 (1, 4; n) = 12(2α−1 − 1). Finally, we give formulas for the quadratic forms Q 3 corresponding to the cases (1, 3, 1), (1, 3, 2), and (1, 3, 4), which involve only divisor functions σ (n). Using these formulas, we show that two of them (corresponding to (1, 3, 1) and (1, 3, 2)) are universal forms and the third one (corresponding to (1, 3, 4)) is non-universal. Using Table 3 for the basis elements and Table 9 for the linear combination coefficients, formulas for the cases (1, 3, 1), (1, 3, 2), and (1, 3, 4) are given below. N3 (1, 3, 1; n) = 8σ (n) − 12σ (n/2) − 24σ (n/3) + 16σ (n/4) + 36σ (n/6) − 48σ (n/12), N3 (1, 3, 2; n) = 2σ (n) + 6σ (n/3) − 8σ (n/4) − 24σ (n/12), N3 (1, 3, 4; n) = 2σ (n) − 6σ (n/2) − 6σ (n/3) + 16σ (n/4) + 18σ (n/6) − 48σ (n/12).

By writing n = 2α 3β m, α, β ≥ 0 and gcd(m, 6) = 1, the above formulas reduce to the following simplified expressions.

8σ (m) if α = 0, N3 (1, 3, 1; n) = (18) α 12(2 − 1)σ (m) if α ≥ 1.

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2(3β+1 − 2)σ (m) if α = 0, 6(3β+1 − 2)σ (m) if α ≥ 1.

(19)

⎧ ⎪ if α = 0, ⎨2σ (m) N3 (1, 3, 4; n) = 0 if α = 1, ⎪ ⎩ 12(2α−1 − 1)σ (m) if α ≥ 2.

(20)

N3 (1, 3, 2; n) =

The above formulas directly imply that the quadratic forms x12 + 3x22 + (x32 + x3 x4 + x42 ) are universal where  = 1, 2 and the quadratic form x12 + 3x22 + 4(x32 + x3 x4 + x42 ) is non-universal. Among the 65 cases of type Q3 considered in this work, 16 forms are universal (this is verified using the famous ‘290’ theorem [11]). They are given by the following triplets: (a1 , a2 , b1 ) ∈ {(1, 3, 1), (1, 3, 2), (1, 12, 1), (2, 6, 1), (1, 6, 1), (1, 6, 2), (2, 3, 1), (1, 1, 1), (1, 1, 2), (1, 4, 1), (1, 4, 2), (2, 2, 1), (1, 2, 1), (1, 2, 2), (1, 2, 4), (2, 4, 1)}. Among these 16 cases, the formulas for the 6 cases (1, 3, 1), (1, 3, 2), (1, 1, 1), (1, 1, 2), (1, 4, 2), (2, 2, 1) involve only the divisor functions. So, it is easy to verify the universal property from our formulas for these cases. We have just shown (using (18) and (19)) that the cases (1, 3, 1) and (1, 3, 2) are universal. In a similar way, we show that the remaining 4 cases are also universal by using explicit formulas given by Theorem 2.3. Writing n = 2α 3β N , α, β ≥ 0, gcd(N , 6) = 1, the following formulas are obtained for the cases (1, 1, 1), (1, 1, 2), (1, 4, 2), (2, 2, 1) using Theorem 2.3: 

 N )F12 (N ), 3   N −1 N )F12 (N ), N3 (1, 1, 2; n) = (2α+1 − (−1)α+β+ 2 )(3β+1 + (−1)α+β 3 ⎧

 N −1 ⎪ (1 − 21 (−1)β+ 2 )(3β+1 + (−1)β N3 )F12 (N ) ⎪ ⎪ ⎨

 N3 (1, 4, 2; n) = 3(3β+1 − (−1)β N3 )F12 (N ) ⎪

 ⎪ N −1 ⎪ α−1 ⎩ (2 − (−1)α+β+ 2 )(3β+1 + (−1)α+β N3 )F12 (N ) ⎧

 ⎨3(3β+1 − (−1)β N )F12 (N ) if 3

 N3 (2, 2, 1; n) = N −1 α+β+ N α β+1 α+β ⎩(2 + (−1) 2 )(3 − (−1) 3 )F12 (N ) if N −1 N3 (1, 1, 1; n) = (2α+2 + (−1)α+β+ 2 )(3β+1 − (−1)α+β

if α = 0, if α = 1, if α ≥ 2. α = 0, α ≥ 1.

(21) In the above, we have used the following notation defined in [6, p. 1542]: For a natural number n,

λ+1 12  p λ+1 − p   12 

 . d= F12 (n) = n/d p − 12p d|n pλ n

(22)

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It is clear from the above definition that for all natural numbers n, F12 (n) > 0. With the same assumption that n = 2α 3β N , where α, β ≥ 0 and gcd(N , 6) = 1, one can check easily the following facts (which are used to prove (21)):   12  d n/d d|n   12  d d d|n   −3   −4  d d n/d d|n   −4   −3  d d n/d d|n

= 2α 3β F12 (N ),  =

12 N



= (−1)

 F12 (N ) =

α+β α

2



N 3

−4 N





N 3

 F12 (N ), (23)

F12 (N ),

= (−1)α+β+(N −1)/2 3β F12 (N ).

Since F12 (n) is positive for all n ≥ 1, it follows from (21) that all the four quadratic forms corresponding to (1, 1, 1), (1, 1, 2), (1, 4, 2), and (2, 2, 1) are universal. As mentioned before, there are 10 quadratic forms (of type Q3 ) considered in our work, which are universal (by using the ‘290’ theorem) for which our explicit formulas involve Fourier coefficients of cusp forms. So, it will be interesting to get this property using our explicit formulas. Here, we would like to mention a very interesting and motivating survey article by J. H. Conway [13] on the 15 and 290 Theorem on the universal property of integral quadratic forms.

2.2 Remarks on Equivalence of Formulas As mentioned in the introduction, our results include 36 known formulas (19 corresponding to Q1 , 2 corresponding to Q2 , 15 corresponding to Q3 ), which are obtained using different methods. The cases (1, 2) and (1, 4) corresponding to Q2 were obtained in [6]. Formula for the case (1, 2) given in [6] is same as our formula (Theorem 2.2) and the formula for (1, 4) given in [6, Theorem 15] is equivalent to (17). However, for the remaining 34 cases, some of the earlier formulas have been expressed in a different way. We would like to remark that our formulas are equivalent to these formulas obtained earlier. Here, we indicate how these equivalence properties can be realized. The formulas deduced in the previous section (§2.1) (i.e., simplified versions of actual formulas obtained from our theorems) are exactly the same formulas obtained in the earlier works [2, 4, 5, 7, 8]. Below we mention the formulas along with reference to the earlier result: N1 (1, 1, 1, 4; n) [2, Theorem 1.7], N1 (1, 1, 2, 4; n), N1 (1, 2, 4, 4; n) [5, Theorems 5.3, 5.4], N3 (1, 1, 1; n), N3 (1, 1, 2; n) [4, Theorems 11.1, 12.1], N3 (1, 3, 1; n), N3 (1, 3, 2; n), N3 (1, 3, 4; n) [8, Theorem 1.2 (iii)], N3 (1, 4, 2; n) [7, Theorem 1.3], and N3 (2, 2, 1; n) [8, Theorem 1.4].

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We now show one more formula corresponding to Q3 for the case (a1 , a2 , b1 ) = (1, 1, 8) and deduce the formula obtained in [6, Theorem 1.4] for this case. By using Table 11 (for the character χ12 ) and the basis for the space given in Table 3, our formula for N3 (1, 1, 8; n) is obtained by comparing the nth Fourier coefficients, which is given below. 1 3 3 1 σ2,1,χ12 (n) − σ2,1,χ12 (n/2) + σ2,χ12 ,1 (n) + σ2,χ−4 ,χ−3 (n) 2 2 2 2 9 3 + σ2,χ−3 ,χ−4 (n) + σ2,χ−3 ,χ−4 (n/2) 2 2          3  12 3  12 1  −3 −4 1  12 d− d+ d+ d = 2 d 2 d 2 n/d 2 d n/d d|n d|n/2 d|n d|n       3  −4 −3 9  −4 −3 + d+ d. 2 d n/d 2 d n/d

N3 (1, 1, 8; n) =

d|n

d|n/2

Now using (22), (23) in the above, we get the following explicit formula for N3 (1, 1, 8; n): Let n = 2α 3β N , where α, β ≥ 0 and gcd(N , 6) = 1. Then, N3 (1, 1, 8; n) ⎧ ⎪ if n ≡ 3, 6, 7(mod 8), ⎨0 N β+1 β = (3 + (−1) 3 )F12 (N ) if n ≡ 1, 5(mod 8), ⎪   ⎩ α−1 α+β+ N 2−1 β+1 α+β N )F12 (N ) if n ≡ 0, 2, 4(mod 8). − (−1) )(3 + (−1) (2 3 (24) The above formula is the same as Theorem 1.4 of [6]. Note that the above formula implies that the corresponding quadratic form is non-universal.

3 Proofs In this section, we shall take χ to be one of the four characters χ0 , χ8 , χ12 , or χ24 and χ is the dimension of the space of modular forms M2 (48, χ ). The main ingredient in proving our theorems is the construction of explicit bases for the spaces M2 (48, χ ). For uniformity, we shall denote these basis elements as { f i,χ (z) : 1 ≤ i ≤ χ } and write their Fourier expansions as f i,χ (z) =



Ai,χ (n)e2πinz .

n≥0

The basis elements f i,χ (z) are explicitly given in (Sect. 4 Table 3).

(25)

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3.1 A Basis for M2 (48, χ0 ) The vector space M2 (48, χ0 ) has dimension 14 with dimC E2 (48, χ0 ) = 11 and dimC S2 (48, χ0 ) = 3. For a, b divisors of N with a|b, (b > a), we define φa,b (z) to be φa,b (z) :=

1 (bE 2 (bz) − a E 2 (az)). b−a

(26)

It is easy to see that φa,b ∈ M2 (N , χ0 ). We need the following two eta-quotients: 2,24 (z) = η(2z)η(4z)η(6z)η(12z) = 2,48 (z) =

η4 (4z)η4 (12z) = η(2z)η(6z)η(8z)η(24z)

∞ 

τ2,24 (n)q n ,

n=1 ∞ 

τ2,48 (n)q n .

(27) (28)

n=1

Using the above functions, we give a basis for the space M2 (48, χ0 ) in the following proposition. Proposition 3.1 A basis for the space of Eisenstein series E2 (48, χ0 ) is given by {φ1,b : b|48(b > 1), E 2,χ−4 ,χ−4 (z), E 2,χ−4 ,χ−4 (3z)} and a basis for the space of cusp forms S2 (48, χ0 ) is given by {2,24 (z), 2,24 (2z), 2,48 (z)}.

3.2 A Basis for M2 (48, χ8 ) The vector space M2 (48, χ8 ) has dimension 12 with dimC E2 (48, χ8 ) = 8 and dimC S2 (48, χ8 ) = 4. For the space of cusp forms, we need the following etaquotients. ∞

2,24,χ8 ;1 (z) =

 η(z)η4 (6z)η2 (8z) = τ2,24,χ8 ;1 (n)q n , η(2z)η(3z)η(12z) n=1

2,24,χ8 ;2 (z) =

η2 (z)η(8z)η4 (12z)  = τ2,24,χ8 ;2 (n)q n . η(4z)η(6z)η(24z) n=1

∞

The following proposition gives a basis of the space M2 (48, χ8 ).

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Proposition 3.2 A basis for the space of Eisenstein series E2 (48, χ8 ) is given by {E 2,1,χ8 (az), a|6, E 2,χ8 ,1 (bz), b|6} and a basis for the space of cusp forms S2 (48, χ8 ) is given by {2,24,χ8 ;1 (z), 2,24,χ8 ;1 (2z), 2,24,χ8 ;2 (z), 2,24,χ8 ;2 (2z)}.

3.3 A Basis for M2 (48, χ12 ) The vector space M2 (48, χ12 ) has dimension 14 with dimC S2 (48, χ12 ) = 2. For the space of cusp forms, we use the following eta-quotient. ∞

2,48,χ12 (z) =

η11 (2z)η(6z)η(8z)η(24z)  a2,48,χ12 (n)q n . = η4 (z)η5 (4z)η(12z) n=1

Using the above eta-quotient, we define the following two cusp forms (which are obtained by considering the character twists of the above eta-quotient). 2,48,χ12 ;1 (z) =



a2,48,χ12 (n)q n ,

2,48,χ12 ;2 (z) =

n≥1 n≡1 (mod 4)



a2,48,χ12 (n)q n .

n≥1 n≡3 (mod 4)

(29) Using these functions, we give a basis for the space M2 (48, χ12 ) in the following proposition. Proposition 3.3 A basis for the space of Eisenstein series E2 (48, χ12 ) is given by {E 2,1,χ12 (az); a|4, E 2,χ12 ,1 (bz); b|4, E 2,χ−4 ,χ−3 (t1 z); t1 |4, E 2,χ−3 ,χ−4 (t2 z); t2 |4} and a basis for the space of cusp forms S2 (48, χ12 ) is given by {2,48,χ12 ;1 (z), 2,48,χ12 ;2 (z)}.

3.4 A Basis for M2 (48, χ24 ) The vector space M2 (48, χ24 ) has dimension 12 with dimC S2 (48, χ24 ) = 4. We need the following eta-quotients.

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∞

2,24,χ24 ;1 (z) =

η(z)η(4z)η4 (6z)η2 (24z)  = τ2,24,χ24 ;1 q n , η(2z)η(3z)η2 (12z) n=1

2,48,χ24 ;2 (z) =

η2 (z)η4 (4z)η(6z)η(24z)  τ2,24,χ24 ;2 q n . = η2 (2z)η(8z)η(12z) n=1

∞

In the following, we give a basis for the space M2 (48, χ24 ). Proposition 3.4 A basis for the space of Eisenstein series E2 (48, χ24 ) is given by {E 2,1,χ24 (az); a|2, E 2,χ24 ,1 (bz); b|2, E 2,χ−3 ,χ−8 (t1 z); t1 |2, E 2,χ−8 ,χ−3 (t2 z); t2 |2} and a basis for the space of cusp forms S2 (48, χ24 ) is given by {2,24,χ24 ;1 (z), 2,24,χ24 ;1 (2z), 2,24,χ24 ;2 (z), 2,24,χ24 ;2 (2z)}.

3.5 Proofs of Theorms All the bases given in Propositions 3.1–3.4 are given in tabular form (Sect. 4, Table 3) along with identifying the elements f i,χ (z), 1 ≤ i ≤ χ We are now ready to prove the theorems. The generating functions for the two types of quadratic forms considered in this paper, viz., sum of squares and forms of type x 2 + x y + y 2 are given, respectively, by the classical theta function  2 e2πin z , (30) (z) = n∈Z

and the function F (z) =



e2πi(m

2

+mn+n 2 )z

.

(31)

m,n∈Z

The theta function (z) is a modular form of weight 1/2 on 0 (4) and F (z) is a  modular form of weight 1 on 0 (3) with character 3· (see [15], [20, Theorem 4], [12] for details). To each quadratic form (a1 , a2 , a3 , a4 ) as in the Table 1 (corresponding to the quadratic forms Q1 ), the associated theta series is given by (a1 z)(a2 z)(a3 z)(a4 z).

(32)

By using [18, Lemmas 1–3], we see that the above function is a modular form in M2 (48, χ ), where χ is one of the four characters that appear in Table 1. Now using

188

B. Ramakrishnan et al.

the bases constructed as in Table 3, one can express each of the theta products (32) as a linear combination of the respective basis elements. Since N1 (a1 , a2 , a3 , a4 ; n) is the nth Fourier coefficient of the theta product (32), by comparing the nth Fourier coefficients, we get the required formulae in Theorem 2.1. We now briefly demonstrate the case (1, 1, 1, 4). The linear combination coefficients in this case are given by (from Table 5, character χ0 ) 0, 0, 5/8, 0, −7/8, 0, 5/4, 0, 0, 2, 0, 0, 0, 0. Therefore,     5 4 1 7 8 1 E 2 (4z) − E 2 (z) − E 2 (8z) − E 2 (z) θ 3 (z)θ (4z) = 8 3 3 8 7 7   5 16 1 + E 2 (16z) − E 2 (z) + 2E 2,χ−4 ,χ−4 (z) 4 15 15 1 5 4 = − E 2 (z) + E 2 (4z) − E 2 (8z) + E 2 (16z) + 2E 2,χ−4 ,χ−4 (z). 6 6 3 Comparing the nth Fourier coefficients of both the sides, we get N1 (1, 1, 1, 4; n) = 4σ (n) − 20σ (n/4) + 24σ (n/8) − 32σ (n/16) + 2σ2,χ−4 ,χ−4 (n). Next, for the four quadratic forms given by the pairs (1, 2), (1, 4), (1, 8), (1, 16) in Table 1, the corresponding theta series is the product of the forms F (b1 z) and F (b2 z). Again by using Lemmas 1 and 3 in [18], these forms belong M2 (48, χ0 ). So, we can express these 4 forms as a linear combination of the basis elements of M2 (48, χ0 ), which we denote as follows. Let (b1 , b2 ) ∈ {(1, 2), (1, 4), (1, 8), (1, 16)}. Then N2 (b1 , b2 ; n) =

14 

ci Ai,χ0 (n),

(33)

i=1

where Ai,χ0 (n) are the Fourier coefficients of the basis elements f i,χ0 (z) (given in Table 3). The values of the constants ci for each pair (b1 , b2 ) are given in Table 4 (Sect. 4). The values of ci are non-zero only in the case of basis elements which are either φ1,b (z), b|48 and b > 1 or one of the cusp forms 2,24 (z), 2,48 (z). The Fourier expansion of the Eisenstein series φa,b (z) is given as follows. φa,b (z) = 1 +

24a  24b  σ (n/a)q n − σ (n/b)q n . b − a n≥1 b − a n≥1

By substituting the values of the constants ci in the expression along with the Fourier expansion of the above basis elements, we get the required formulas in Theorem 2.2. Finally, the theta series corresponding to each quadratic form Q3 represented by the triplets (a1 , a2 , b1 ) in Table 1 is the product (a1 z)(a2 z)F (b1 z). By using Lemmas 1–3 of [18], it can be observed that this theta product is a modular form of

On the Number of Representations of a Natural Number by Certain …

189

Table 1 (List of quadratic forms) Space

(a1 , a2 , a3 , a4 ) for Q1 (1, 1, 1, 4), (1, 1, 4, 4), (1, 1, 3, 12), (1, 1, 12, 12), (1, 2, 2, 4), (1, 2, 6, 12),

χ0

(1, 3, 3, 4), (1, 3, 4, 12), (1, 4, 4, 4), (1, 4, 6, 6), (1, 4, 12, 12), (2, 2, 3, 12), (2, 3, 4, 6), (3, 3, 4, 4), (3, 4, 4, 12) (1, 1, 2, 4), (1, 1, 6, 12), (1, 2, 4, 4), (1, 2, 3, 12), (1, 2, 12, 12),

χ8

(1, 3, 4, 6), (1, 4, 6, 12), (2, 3, 3, 4), (2, 3, 4, 12), (3, 4, 4, 6) (1, 1, 1, 12), (1, 1, 3, 4), (1, 1, 4, 12), (1, 2, 2, 12), (1, 2, 4, 6), (1, 3, 3, 12),

χ12

(1, 3, 4, 4), (1, 3, 12, 12), (1, 4, 4, 12), (1, 6, 6, 12), (1, 12, 12, 12), (2, 2, 3, 4), (2, 3, 6, 12), (3, 3, 3, 4), (3, 3, 4, 12), (3, 4, 4, 4), (3, 4, 6, 6), (3, 4, 12, 12) (1, 1, 2, 12), (1, 1, 4, 6), (1, 2, 3, 4), (1, 2, 4, 12), (1, 3, 6, 12), (1, 4, 4, 6),

χ24

(1, 6, 12, 12), (2, 3, 3, 12), (2, 3, 4, 4), (2, 3, 12, 12), (3, 3, 4, 6), (3, 4, 6, 12) (b1 , b2 ) for Q2

χ0

(1, 2), (1, 4), (1, 8), (1, 16) (a1 , a2 , b1 ) for Q3 (1, 3, 1), (1, 3, 2), (1, 3, 4), (1, 3, 8), (1, 3, 16), (1, 12, 1), (1, 12, 2), (1, 12, 4), (1, 12, 8),

χ0

(1, 12, 16), (2, 6, 1), (3, 4, 1), (3, 4, 2), (3, 4, 4), (3, 4, 8), (3, 4, 16), (4, 12, 1) (1, 6, 1), (1, 6, 2), (1, 6, 4), (1, 6, 8), (1, 6, 16), (2, 3, 1),

χ8

(2, 3, 2), (2, 3, 4), (2, 3, 8), (2, 3, 16), (2, 12, 1), (4, 6, 1) (1, 1, 1), (1, 1, 2), (1, 1, 4), (1, 1, 8), (1, 1, 16), (1, 4, 1), (1, 4, 2), (1, 4, 4),

χ12

(1, 4, 8), (1, 4, 16), (2, 2, 1), (3, 3, 1), (3, 3, 2), (3, 3, 4), (3, 3, 8), (3, 3, 16), (3, 12, 1), (3, 12, 2), (3, 12, 4), (3, 12, 8), (3, 12, 16), (4, 4, 1), (6, 6, 1), (12, 12, 1) (1, 2, 1), (1, 2, 2), (1, 2, 4), (1, 2, 8), (1, 2, 16), (2, 4, 1), (3, 6, 1),

χ24

(3, 6, 2), (3, 6, 4), (3, 6, 8), (3, 6, 16), (6, 12, 1)

weight 2 on 0 (48) with one of the characters χ0 or χd , d = 8, 12, 24 (depending on the triplets (a1 , a2 , b1 )). Formulas in Theorem 2.3 now follow from comparing the Fourier coefficients of these associated modular forms. This completes the proofs of the theorems.

4 Tables for Theorems 2.1 and 2.3 In this section, we shall give all the tables, including the tables which give explicit coefficients αi,χ and βi,χ that appear in Theorems 2.1 and 2.3.

190

B. Ramakrishnan et al.

Table 2 (List of earlier results) Type Q1

Cases

References

(1, 1, 1, 4), (1, 1, 4, 4), (1, 1, 3, 12), (1, 1, 12, 12), (1, 2, 2, 4), (1, 3, 3, 4), (1, 3, 4, 12), (1, 4, 4, 4), (1, 4, 6, 6), (1, 4, 12, 12), (2, 2, 3, 12), (3, 3, 4, 4), (3, 4, 4, 12)

[2]

(1, 1, 1, 12), (1, 1, 3, 4), (1, 1, 4, 12), (1, 2, 2, 12), (1, 3, 3, 12), (1, 3, 4, 4), (1, 3, 12, 12), (1, 4, 4, 12), (1, 6, 6, 12), (1, 12, 12, 12), (2, 2, 3, 4), (3, 3, 3, 4),

Q2 Q3

(3, 3, 4, 12), (3, 4, 4, 4), (3, 4, 6, 6), (3, 4, 12, 12)

[3]

(1, 2, 4, 6)

[4]

(1, 1, 2, 4), (1, 2, 4, 4)

[5]

(1, 2), (1, 4)

[6]

(1, 1, 1), (1, 1, 2), (1, 1, 4), (3, 3, 1), (3, 3, 2), (3, 3, 4)

[4]

(1, 1, 8), (1, 4, 2), (3, 12, 2)

[7]

(1, 3, 1), (1, 3, 2), (1, 3, 4), (1, 4, 4), (2, 2, 1), (6, 6, 1)

[8]

Table 3 (List of basis elements) f 1,χ0 (z) = φ1,2 (z), f 2,χ0 (z) = φ1,3 (z), f 3,χ0 (z) = φ1,4 (z), f 4,χ0 (z) = φ1,6 (z), f 5,χ0 (z) = φ1,8 (z), f 1,χ8 (z) = E 2,1,χ8 (z), f 2,χ8 (z) = E 2,1,χ8 (2z), f 3,χ8 (z) = E 2,1,χ8 (3z), f 4,χ8 (z) = E 2,1,χ8 (6z), f 1,χ12 (z) = E 2,1,χ12 (z), f 2,χ12 (z)) = E 2,1,χ12 (2z), f 3,χ12 (z) = E 2,1,χ12 (4z), f 4,χ12 (z) = E 2,χ12 ,1 (z), f 5,χ12 (z) = E 2,χ12 ,1 (2z), f 1,χ24 (z) = E 2,1,χ24 (z), f 2,χ24 (z) = E 2,1,χ24 (2z), f 3,χ24 (z) = E 2,χ24 ,1 (z), f 4,χ24 (z) = E 2,χ24 ,1 (2z),

f 6,χ0 (z) = φ1,12 (z), f 7,χ0 (z) = φ1,16 (z), f 8,χ0 (z) = φ1,24 (z), f 9,χ0 (z) = φ1,48 (z), f 10,χ0 (z) = E 2,χ−4 ,χ−4 (z),

f 11,χ0 (z) = E 2,χ−4 ,χ−4 (3z), f 12,χ0 (z) = 2,24 (z), f 13,χ0 (z) = 2,24 (2z), f 14,χ0 (z) = 2,48 (z).

f 5,χ8 (z) = E 2,χ8 ,1 (z), f 6,χ8 (z) = E 2,χ8 ,1 (2z), f 7,χ8 (z) = E 2,χ8 ,1 (3z), f 8,χ8 (z) = E 2,χ8 ,1 (6z), f 6,χ12 (z) = E 2,χ12 ,1 (4z), f 7,χ12 (z) = E 2,χ−4 ,χ−3 (z), f 8,χ12 (z) = E 2,χ−4 ,χ−3 (2z), f 9,χ12 (z) = E 2,χ−4 ,χ−3 (4z), f 10,χ12 (z) = E 2,χ−3 ,χ−4 (z), f 5,χ24 (z) = E 2,χ−3 ,χ−8 (z), f 6,χ24 (z) = E 2,χ−3 ,χ−8 (2z), f 7,χ24 (z) = E 2,χ−8 ,χ−3 (z), f 8,χ24 (z) = E 2,χ−8 ,χ−3 (2z),

f 9,χ8 (z) = 2,24,χ8 ;1 (z), f 10,χ8 (z) = 2,24,χ8 ;1 (2z), f 11,χ8 (z) = 2,24,χ8 ;2 (z), f 12,χ8 (z) = 2,24,χ8 ;2 (2z). f 11,χ12 (z) = E 2,χ−3 ,χ−4 (2z), f 12,χ12 (z) = E 2,χ−3 ,χ−4 (4z), f 13,χ12 (z) = 2,48,χ12 ;1 (z), f 14,χ12 (z) = 2,48,χ12 ;2 (z). f 9,χ24 (z) = 2,24,χ24 ;1 (z), f 10,χ24 (z) = 2,24,χ24 ;1 (2z), f 11,χ24 (z) = 2,24,χ24 ;2 (z), f 12,χ24 (z) = 2,24,χ24 ;2 (2z).

On the Number of Representations of a Natural Number by Certain …

191

Table 4 . b1 , b2 1, 2 1, 4 1, 8 1, 16

c1 1 4 3 8 3 32 3 32

c2 −1 2 1 2 −1 8 1 8

c3

c4 5 4 −15 8 15 32 −15 32

0 −3 4 −9 32 −9 32

c5

c6

c7

c8

c9

c10

c11

c12

c13

c14

0

0

0

0

0

0

0

0

0

0

0

11 4 −33 32 33 32

0

0

0

0

0

0

0

0

0

23 16 −69 32

0

0

0

9 2

0

0

47 16

0

0

0

0

9 2

7 16 21 32

−15 16

Table 5 For the character χ0 a1 , a2 , a3 , a4 α1,χ0

α2,χ0

α3,χ0

α4,χ0

α5,χ0

α6,χ0

α7,χ0

α8,χ0

α9,χ0

α10,χ0 α11,χ0 α12,χ0 α13,χ0 α14,χ0

1, 1, 1, 4

0

0

5 8

0

− 78

0

0

0

2

0

0

0

0

1, 1, 4, 4

1 24 1 12 1 48 1 24 1 96 1 12 1 16 1 16 1 48 1 32 1 48 1 96 1 48 1 32

0

0

0

7 − 24

0

5 4 5 4

0

0

2

0

0

0

0

1 6 1 12

5 − 16

5 − 12

55 48

− 58

− 23 16

3

0

2

1

5 − 48

0

− 58

− 23 48

47 24 47 24

1

0

7 16 7 48

1

3

1

2

1

0

0

0

7 − 24

0

0

0

0

0

0

0

0

1 − 24

0

5 96

7 − 96

0

5 4 5 16

− 23 96

0

1 2

1

1

5 − 16 5 − 16 5 − 16

5 − 12 5 − 16

7 16 7 16

55 48 55 48

− 23 16 − 23 16

-1

-3

0

-2

1

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

1 12 1 24 1 12

0

5 − 48

− 23 48

0

1

-2

0

1 2

3 2

1 2

0

1 2

0

5 − 48

0

0

-1

2

0

1 − 24

0

5 96

7 − 96

0

0

0

1 2

1

-1

0

5 − 48 5 − 32

7 48 7 24

47 24 47 24 47 24 47 48 47 24 47 24

0

5 − 32

7 48 7 24 7 48

0

5 − 32

− 58 − 58 5 4 − 58 − 58 − 58 5 16 − 58 − 58

47 48 47 24 47 24

0

1 6 1 12

-1

-3

-1

-2

1

− 21

− 23

− 21

0

1 2

1, 1, 3, 12 1, 1, 12, 12 1, 2, 2, 4 1, 2, 6, 12 1, 3, 3, 4 1, 3, 4, 12 1, 4, 4, 4 1, 4, 6, 6 1, 4, 12, 12 2, 2, 3, 12 2, 3, 4, 6 3, 3, 4, 4 3, 4, 4, 12

0

1 12 1 24

5 − 32

55 96

0 0 55 96

23 − 24

− 23 48 − 23 96 − 23 48 23 − 24

Table 6 For the character χ8 a1 , a2 , a3 , a4

α1,χ8

α2,χ8

α3,χ8

α4,χ8

α5,χ8

α6,χ8

α7,χ8

α8,χ8

α9,χ8

α10,χ8 α11,χ8 α12,χ8

1, 1, 2, 4

0

−2

0

0

4

0

0

0

0

0

0

0

1, 1, 6, 12

0

− 45

0

− 65

8 5

0

− 12 5

0

8 5

32 5

4 5

− 85

1, 2, 4, 4

0

−2

0

0

2

0

0

0

0

0

0

0

1, 2, 3, 12

0

0

− 12 5

24 5 24 5

− 16 5

− 12 5

− 45

− 16 5

0

− 65 − 65 − 12 5 − 12 5 − 65

4 5 12 5 8 5 4 5

2 5

0

24 5 12 5

0

0

4 5 2 5 8 5 4 5 4 5 2 5 4 5

0

1, 2, 12, 12

2 5 2 5

− 85 12 5 − 16 5 4 5 − 85

− 65 2 5 12 5 6 5 − 85

− 85

1, 3, 4, 6

0

1, 4, 6, 12

0

2, 3, 3, 4

0

2, 3, 4, 12

0

3, 4, 4, 6

0

− 45 − 45 2 5 2 5 − 45

0 0 0 0

0 0 0 0 0 0

− 12 5 − 65 24 5 12 5 − 65

0 0 0 0 0 0

− 16 5 − 85 4 5

− 85 24 5 4 5

− 85

1 2 1 2 1 2

0

1 2

0

1 4

0

− 41

0

1 2 1 2

− 41

0

1 4

− 21

− 21

− 21

0

0

− 21

0

0

− 41

0

1 4

0

0

− 21

− 21

1 4

0

− 41

1, 1, 1, 12

1, 1, 3, 4

1, 1, 4, 12

1, 2, 2, 12

1, 2, 4, 6

1, 3, 3, 12

1, 3, 4, 4

1, 3, 12, 12

1, 4, 4, 12

1, 6, 6, 12

1, 12, 12, 12

2, 2, 3, 4

2, 3, 6, 12

3, 3, 3, 4

3, 3, 4, 12

3, 4, 4, 4

3, 4, 6, 6

3, 4, 12, 12

0

0

0

α2,χ12

α1,χ12

a1 , a2 , a3 , a4

Table 7 For the character χ12

−1

−1

−1

−1

−1

−1

−1

1

−1

−1

1 2 3 4 1 2 1 4

1

− 21

0

− 23

0

1

0

0

− 21

0

− 23

−1

−3

−1

−1

−1

3 2 1 2 3 4 1 2 1 4 3 2 1 2

0

0

0

−3

3

α5,χ12

1

3 2 3 2 3 2

3

3

α4,χ12

−1

−1

−1

−1

−1

−1

α3,χ12

2

0

6

0

−4

0

0

2

0

6

4

12

4

0

0

0

12

−12

α6,χ12

0

1 2

− 41 1 2 1 4

1 2

1

− 21 1 2 − 41 1 2 1 4 − 21 − 21

1 2

0

− 21

0

−1

0

0

1 2

0

− 21

1

−1

1

0

− 21 1

0

−1

1

α8,χ12

− 21

−1

−1

α7,χ12

2

0

−2

0

−4

0

0

2

0

−2

4

−4

4

0

0

0

−4

4

α9,χ12

− 41

0

− 43

− 21

− 21

0

0

1 4

0

3 4

0

0

− 21

0

0

3 2 3 2 3 2

α10,χ12

− 41

0

− 43

− 21

− 21

0

0

1 4

0

3 4

0

0

− 21

0

0

3 2 3 2 3 2

α11,χ12

−1

−1

3

−1

−1

1

3

−1

−1

3

−1

3

−1

−3

3

3

3

3

α12,χ12

0

−1

0

0

−1

0

−1

1

1

1

1

1

1

0

1

2

1

3

α13,χ12

1 3 1 3

−1

2 3

1

0

−1

0

− 31

0

1 3

−1

1 3

0

1

0

−1

1

α14,χ12

192 B. Ramakrishnan et al.

α1,χ24

0

0

0

0

0

0

0

0

0

0

0

0

a1 , a2 , a3 , a4

1, 1, 2, 12

1, 1, 4, 6

1, 2, 3, 4

1, 2, 4, 12

1, 3, 6, 12

1, 4, 4, 6

1, 6, 12, 12

2, 3, 3, 12

2, 3, 4, 4

2, 3, 12, 12

3, 3, 4, 6

3, 4, 6, 12

0

2

1

2 3

− 13

− 13 − 13 − 13 − 13 − 13 − 13 − 13 − 13 − 13

1 3 2 3 1 3

1

1 3 2 3

1

0

2

− 13

0

0

0

0

0

0

0

0

0

0

2

− 13

α4,χ24

α3,χ24

α2,χ24

Table 8 For the character χ24

0

1 3

0

0

0

0

− 13 2 3 1 3

0

− 23

0

0

0

0

0

0

α6,χ24

1 3

− 13

2 3 1 3 2 3

− 23

2 3

α5,χ24

0

0

0

0

0

0

0

0

0

0

0

0

α7,χ24

− 31 1 3 1 3

1

− 31

1 3

-1

1 3

1

1

-1

1

α8,χ24

− 43 − 83 − 43

0

− 83

8 3

4

4 3

0

0

8

0

α9,χ24

16 3 − 16 3 − 43

-8

8 3 16 3

0

8 3

4

-8

0

16

α10,χ24

− 43 − 23

0

− 43

0

2 3 2 3 4 3 4 3

− 23

4 3 8 3

α11,χ24

0

0

8 3

− 38

8 3

0

− 38

0

4 3

− 38

− 38

16 3

α12,χ24

On the Number of Representations of a Natural Number by Certain … 193

0

1, 3, 2

4, 12, 1

3, 4, 16

3, 4, 8

3, 4, 4

1 16 1 64 1 64 3 16

0

3, 4, 2

3, 4, 1

2, 6, 1

1, 12, 8

1, 12, 4

1, 12, 16

0

1 16 1 64 1 64 7 48 1 8

1, 12, 2

1, 12, 1

1, 3, 16

1, 3, 8

1 8 1 32 1 32 1 8

2 3 −1 6 1 6 −1 24 1 24 1 3 −1 12 1 12 −1 48 1 48 −1 4 1 3 −1 12 1 12 −1 48 1 48 1 4

1 4

1, 3, 1

1, 3, 4

β2,χ0

β1,χ0

a 1 , a 2 , b1

Table 9 For the character χ0

−1 2 1 4 −1 2 −7 32 −7 32 −5 16 5 32 −5 16 −5 64 −5 64 −3 16 −5 16 5 32 −5 16 −5 64 −5 64 −7 16

β3,χ0

−5 16 5 64 −5 64 −15 16

0

−5 16 5 64 −5 64 35 48 −5 8

0

−5 8 5 32 −5 32 −5 8

0

−5 4

β4,χ0

7 32 7 16

0

7 32 7 24 7 16 −7 32 7 16

0

7 16 21 32 7 16 −7 32 7 16

0

0

0

β5,χ0 11 6 11 12 11 6 −77 96 77 96 55 48 55 96 55 48 −55 192 55 192 −11 16 55 48 55 96 55 48 −55 192 55 192 77 48

β6,χ0

−5 8 5 16 −5 8 5 16 −5 8 −5 8

0

−15 16 −5 8 5 16 −5 8 5 16 −5 8

0

0

0

0

β7,χ0

−23 32 −23 16

0

−23 32 23 24 −23 16 −23 32 −23 16

0

23 16 −69 32 −23 16 −23 32 −23 16

0

0

0

β8,χ0

47 24 47 48 47 24 47 48 47 24 47 24

0

47 16 47 24 47 48 47 24 47 48 47 24

0

0

0

0

β9,χ0

0

3 4 −3 4

0

-3 −1 4 −1 4

0

3 4 −3 4

0

0

−3 2 1 2

-1

0

-3

3 -1

0

3 4 3 4

−3 4 3 4

0

0

3

0

0

1 4 1 4

3 2

0

3 2

0

0

0

β12,χ0

1

3

−1 2

0

0

0

0

0

β11,χ0

1

0

0

0

0

0

β10,χ0

0

0

0

0

3

-6

0

0

0

0

3

6

0

0

0

0

0

β13,χ0

3

−3 4 3 4

0

−3 2

3

0

3 4 3 4

0

3 2

3

3 2

0

0

0

0

β14,χ0

194 B. Ramakrishnan et al.

4, 6, 1

2, 3, 16

2, 3, 8

2, 3, 4

2, 3, 2

2, 3, 1

1, 6, 16

1, 6, 8

1, 6, 4

1, 6, 2

1, 6, 1

a 1 , a 2 , b1

0

−4 5 2 5 −4 5 2 5 2 5 2 5 −4 5 2 5 −4 5 −1 5

β1,χ8

3 5 −4 5

0

0

0

0

−6 5

0

0

0

0

β2,χ8

Table 10 For the character χ8

0

−6 5 −12 5 −6 5 −12 5 3 5 −12 5 −6 5 −12 5 −6 5 6 5

β3,χ8

−18 5 −6 5

0

0

0

0

−9 5

0

0

0

0

β4,χ8 32 5 8 5 8 5 2 5 2 5 16 5 16 5 4 5 4 5 1 5 24 5

β5,χ8

−32 5

0

0

0

0

0

0

0

0

0

0

β6,χ8 −48 5 48 5 −12 5 12 5 −3 5 96 5 −24 5 24 5 −6 5 6 5 −36 5

β7,χ8

48 5

0

0

0

0

0

0

0

0

0

0

β8,χ8

−12 5 −12 5 6 5 −6 5 24 5

0

6 5 6 5

0

24 5 12 5

β9,χ8

0

6 5

0

0

0

0

18 5

0

0

0

0

β10,χ8

6 5 −6 5 6 5 −18 5

6 5 −24 5

0

0

0

0 0

−12 5 12 5

0

0

0

0

β12,χ8

0

0

−12 5 −12 5 6 5

β11,χ8

On the Number of Representations of a Natural Number by Certain … 195

0

−3 2 3 4 1 2 1 2 1 2 −1 4 1 8

-1

1 2 −1 4 −1 2 −1 2 −1 2 1 4 −1 8

1, 1, 4

−3 2 3 4 1 2 1 2 1 2 −1 4 1 8

-1

1 2 −1 4 −1 2 −1 2 −1 2 1 4 −1 8

3, 3, 4

0

0

0

4, 4, 1

6, 6, 1

12, 12, 1

3, 12, 16

3, 12, 8

3, 12, 4

3, 12, 2

3, 12, 1

3, 3, 16

3, 3, 8

0

-1

3, 3, 2

0

-1

0

0

0

-1

0

-1

3, 3, 1

0

2, 2, 1

1, 4, 16

1, 4, 8

1, 4, 4

1, 4, 2

1, 4, 1

1, 1, 16

1, 1, 8

-1

1, 1, 2

0

-1

1, 1, 1

β2,χ12

β1,χ12

a1 , a2 , b1

Table 11 For the character χ12

-1

0

-1

-1

-1

-1

-1

-1

−3 2

0

0

0

0

0

-1

-1

-1

-1

-1

−3 2

0

0

0

0

β3,χ12

-4 -3

3 2

-9

−1 2 −1 4

-1

1

-1

0

0

0

0

0

-12

−3 2 −3 4

-3

3

-3

0

0

0

0

0

β5,χ12

3

1 2 1 4 1 8 9 2

1

2

1 2 1 4

1

2

4

9

3 2 3 4 3 8

3

6

3 2 3 4

3

6

12

β4,χ12

4

0

12

1

2

4

-4

4

0

0

0

0

0

0

3

6

12

-12

12

0

0

0

0

0

β6,χ12

3 2

3

1 2 −1 4 1 8 −3 2

-1

2

−1 2 1 4

1

-2

4

-3

−1 2 1 4 −1 8

1

-2

1 2 −1 4

-1

2

-4

β7,χ12

3

4

-3

−1 2 1 4

1

1

1

0

0

0

0

0

-4

1 2 −1 4

-1

-1

-1

0

0

0

0

0

β8,χ12

4

0

-4

1

-2

4

4

4

0

0

0

0

0

0

-1

2

-4

-4

-4

0

0

0

0

0

β9,χ12

0

0

0

−1 2 −1 4 −1 2 1 2 −1 2 −1 4 −1 8

-1

1

-1

0

3 2 3 4 3 2 −3 2 3 2 3 4 3 8

3

-3

3

β10,χ12

0

1

0

−3 2 −3 4 −1 2 1 2 −1 2 −1 4 −1 8

0

0

0

-3

9 2 9 4 3 2 −3 2 3 2 3 4 3 8

0

0

0

β11,χ12

-1

0

3

-1

1

-1

1

-1

−3 2

0

0

0

0

0

3

-3

3

-3

3

9 2

0

0

0

0

β12,χ12

3

0

3

0

0

0

0

3

0

0

0

0

0

0

1

0

-3

1 2

0

0

0

1

1

0

0

0

0

0

0 0

3 2

0

0

-3

0

0

0

0

0

β14,χ12

0

0

0

3

3

0

0

0

0

β13,χ12

196 B. Ramakrishnan et al.

On the Number of Representations of a Natural Number by Certain …

197

Table 12 For the character χ24 a1 , a2 , b1 β1,χ24

1, 2, 16

−1 3 −1 3 −1 3 −1 3 1 6

2, 4, 1

0

1, 2, 1 1, 2, 2 1, 2, 4 1, 2, 8

3, 6, 16

−1 3 −1 3 −1 3 −1 3 1 6

6, 12, 1

0

3, 6, 1 3, 6, 2 3, 6, 4 3, 6, 8

β2,χ24

β3,χ24

β4,χ24

β5,χ24

β6,χ24

β7,χ24

β8,χ24

β9,χ24

β10,χ24 β11,χ24 β12,χ24

0

8

0

0

-1

0

0

0

0

4

0

0

1

0

-4

0

0

2

0

0

-1

0

0

0

0

1

0

0

1

0

2

0

−1 2 −1 3

1 2

0

8 3 −4 3 2 3 −1 3 1 6

0

1 2

3 2

0

6

−4 3 −4 3 2 3 2 3 2 3

6

-8

2

8 3

0

1

0

-16

-2

0

8 3 −4 3 2 3 −1 3 1 6

0

0

8 3 4 3 −4 3 −2 3 −4 3

0

4 3

0

0

0

0

0

−2 3

0

0

0

0

0

−1 3 1 3 −1 3 1 3 1 6

−16 3

8 3 4 3 2 3 1 3 1 6

-2

−2 3

0

2

8 3

0

4

16 3

2

0

0 0 0 0 −1 2 −1 3

2

0 0 0 0 −8 3

0 0 0

0 0 0 1 2 1 3

0 0 0 0 2

Acknowledgements We thank the referee for making some useful suggestions. We have used the open-source mathematics software SAGE (www.sagemath.org) for carrying out our calculations. The second author is partially funded by SERB grant MTR/2017/000228. Part of the work was done when the third named author was visiting NISER, Bhubaneswar and he thanks the institute for the warm hospitality.

References 1. Alaca, A., Alaca, S., Aygin, Z.S.: Theta products and eta quotients of level 24 and weight 2. Funct. Approx. Comment. Math. 57, 205–234 (2017) 2. Alaca, A., Alaca, S., Lemire, M.F., Williams, K.S.: Nineteen quaternary quadratic forms. Acta Arith. 130, 277–310 (2007) 3. Alaca, A., Alaca, S., Lemire, M.F., Williams, K.S.: Theta function identities and representations by certain quaternary quadratic forms II. Int. Math. Forum 3(12), 539–579 (2008) 4. Alaca, A., Alaca, S., Lemire, M.F., Williams, K.S.: Theta function identities and representations by certain quaternary quadratic forms. Int. J. Number Theory 4, 219–239 (2008) 5. Alaca, A., Alaca, S., Lemire, M.F., Williams, K.S.: The number of representations of a positive integer by certain quaternary quadratic forms. Int. J. Number Theory 5, 13–40 (2009) 6. Alaca, A., Alaca, S., Williams, K.S.: On the two dimensional theta functions of the Borweins. Acta Arith. 124, 177–195 (2006) 7. Alaca, A., Alaca, S., Williams, K.S.: Representation numbers of certain quaternary quadratic forms in a genus consisting of a single class. Int. J. Number Theory 12, 1529–1573 (2016) 8. Alaca, S., Pehlivan, L., Williams, K.S.: On the number of representations of a positive integer as a sum of two binary quadratic forms. Int. J. Number Theory 10, 1395–1420 (2014) 9. Alaca, A., Williams, K.S.: On the quaternary forms x 2 + y 2 + 2z 2 + 3t 2 , x 2 + 2y 2 + 2z 2 + 6t 2 , x 2 + 3y 2 + 3z 2 + 6t 2 and 2x 2 + 3y 2 + 6z 2 + 6t 2 . Int. J. Number Theory 8, 1661–1686 (2012) 10. Atkin, A.O.L., Lehner, J.: Hecke operators on 0 (m). Math. Ann. 185, 134–160 (1970) 11. Bhargava, M., Hanke, J.: Universal quadratic forms and the 290-theorem, preprint (2005) 12. Borwein, J.M., Borwein, P.B., Garvan, F.G.: Some cubic modular identities of Ramanujan. Trans. Am. Math. Soc. 343, 35–47 (1994)

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13. Conway, J.H.: Universal quadratic forms and the fifteen theorem. In: Quadratic Forms and Their Applications. Contemporary Mathematics, vol. 272, pp. 23–26. American Mathematical Society, Providence, RI (2000) 14. Dummit, D., Kisilevsky, H., McKay, J.: Multiplicative products of η -functions, in Finite Groups–Coming of Age 15. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms, 2nd edn., Graduate Texts in Mathematics, vol. 97. Springer (1993) 16. Li, W.-W.: Newforms and functional equations. Math. Ann. 212, 285–315 (1975) 17. Miyake, T.: Modular Forms. Springer, Berlin (1989) 18. Ramakrishnan, B., Sahu, B., Kumar Singh, A.: On the representations of a positive integer by certain classes of quadratic forms in eight variables. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series, Springer Proceeding Mathematic Statistics, vol. 221, pp. 641–66. Springer, Cham (2017) 19. Ramanujan, S.: On the expression of a number in the form ax 2 + by 2 + cz 2 + du 2 . Proc. Camb. Philos. Soc. 19, 11–21 (1917) 20. Schoeneberg, B.: Elliptic Modular Functions: an Introduction. Translated from the German by J. R. Smart and E. A. Schwandt. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 203. Springer, New York–Heidelberg (1974) 21. Stein, W.: Modular Forms, a Computational Approach, Graduate Studies in Mathematics 79. American Mathematical Society, Providence (2007) 22. Williams, K.S.: On the representations of a positive integer by the forms x 2 + y 2 + z 2 + 2t 2 and x 2 + 2y 2 + 2z 2 + 2t 2 . Int. J. Mod. Math. 3, 225–230 (2008)

Identities from Partition of the Symmetric Group Sn V. P. Ramesh, M. Makeshwari, M. Prithvi, and R. Thatchaayini

Abstract In this article, we define a combinatorial number, denoted by n Tk , and defined to be the number of permutations σ ∈ Sn which can be written as a product of k-number of transpositions but not m-number of transpositions for any integer m with 1  m < k. We establish a connection between this number and the Stirling number of the first kind and study some properties. Furthermore, we give a lexicographic order relation among the elements of the symmetric group on n-symbols and study, mainly, a concept, namely directional change of lexicographic permutations of Sn . The directional change of a permutation is a positive integer which indicates the hierarchy with its successive permutation in lexicographic ordering. We introduce a number, n Dk (for integers n  2), which counts the number of lexicographic permutations of Sn having the directional change value as k. We compute this number by means of recurrence relations, etc. Keywords Symmetric group · Orbits · Transpositions · Stirling number · Permutations · Lexicographic sorting · Ordering · Directional change 2010 Mathematics Subject Classification 11B73 · 20B30

V. P. Ramesh (B) · M. Makeshwari · M. Prithvi · R. Thatchaayini Department of Mathematics, Central University of Tamil Nadu, Thiruvarur 610005, Tamil Nadu, India e-mail: [email protected] M. Makeshwari e-mail: [email protected] M. Prithvi e-mail: [email protected] R. Thatchaayini e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_13

199

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1 Introduction It is well known that any permutation can be written as a product of transpositions but not a unique fashion. However, in the literature, many authors [2, 4, 5, 10, 11] studied permutations that can be written as a product of minimal number of transpositions. In this article, we shall be studying similar but a counting problem related to this problem and the connection with the Stirling number. In the study of partitions in Combinatorics, Stirling numbers play a vital role and further very close to the binomial coefficients. There are two kinds of Stirling numbers, namely, first and second kind. For notations, we  follow Knuth [7] and  denote the unsigned Stirling number of the first kind by mn and second kind by mn . Among these two, the second kind appears in numerous applications and Stirling himself had analyzed the second kind first in his book [1, 7, 14]. In this article, we deal with the unsigned Stirling number of the first kind. Let Sn be the symmetric group of all permutations on n-symbols, namely, {1, 2, 3, . . . , n} and let σ ∈ Sn be any arbitrary element. It is well known that the relation, for any i, j ∈ {1, 2, 3, . . . , n} i ∼ j ⇐⇒ there exists k ∈ Z such that σ k (i) = j is an equivalence relation and the equivalence classes are called orbits of σ . It is to be noted that a permutation σ ∈ Sn is said to be a cycle of length , if one of its orbits has  elements and rest of them have only one element. And two cycles are said to be disjoint, if the intersection of the orbits having more than  element is empty.  one The unsigned Stirling number of first kind, denoted by mn , and is defined to be the number of permutations of Sn having m number of  orbits. Note that since every permutation of Sn has at least one orbit, we see that n0 = 0 for any n  1, and since  the identity permutation alone can have n orbits, we have nn = 1 for any n  1. We know that every permutation can be written as a product of disjoint cycles and further every cycle can be written as a product of transpositions (a transposition is a cycle of length two). It is also known that writing a permutation into product of transpositions is not unique. However, we define another combinatorial constant involving transpositions as follows. Definition 1 A combinatorial number, namely, n Tk for any integers 0  k  n, is defined as follows.   n Tk = # σ ∈ Sn : k = min{σ = γ1 . . . γs : γi is a transposition} . s

It is to be noted that n Tn = 0 and also that n T0 = 1. Since every permutation can be written as a product of transpositions, it may be natural to expect that the above two numbers have a connection. In this article, we prove the following identities on this combinatorial constant.

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201

Theorem 1 For any given integer k with 0  k  n, we have  n = n Tn−k . k In other words, the number of permutations having k number of orbits is equal to the number of permutations having n − k minimal number of transpositions. Corollary 1 For any integer n  2, we have, n 

n k=0 2|k

k

=

n 

n k=1 2k

k

=

n! . 2

In other words, the number of even permutations of Sn is equal to the number of permutations having even number of orbits and hence, we have, n

k=0 2|k

n

Tk =

n 

n k=0 2|k

k

Theorem 1 says that the total number of even permutations is equal to the total number of permutations having even number of orbits. This motivated us to look  at the examples more closely. For instance, the permutation 17 62 93 84 55 26 37 18 49 ∈ S9 has the orbits {1, 3, 4, 7, 8, 9}, {2, 6} and {5}. It has of orbits while it  odd number is an even permutation. Whereas the permutation 17 62 43 14 85 26 37 58 ∈ S8 has the orbits {1, 3, 4, 7}, {2, 6}, {5, 8}. This permutation has odd number of orbits and it is an odd permutation.     Since 4 = 11, these eleven permutations in S are 1 2 3 4 , 1 2 3 4 , 1 2 3 4 ,  1 2 3 4 21 2 3 4  1 2 3 4  1 2 3 4  1 2 3 4  1 2 43 4  11 23 34 42 1 4 2 13 2 3 4 2 1 4 3 2 3 1 4 , 2 4 3 1 , 3 1 2 4 , 3 2 4 1 , 3 4 1 2 , 4 1 3 2 , 4 2 1 3 , and 4 3 2 1 . Note that these permutations in S4 are even permutations. These observations led us to connect the number of orbits and the number of transpositions of any permutation in Sn . With these observations, we have the following corollary which predict that a given permutation is odd or even based on its number of orbits. More precisely, we prove the following. Corollary 2 Let n  2 be any integer and σ be an element in Sn . Then, we have the following. (a) If 2|n, then σ has even number of orbits if and only if σ is an even permutation; (b) If 2  |n, then σ has even number of orbits if and only if σ is an odd permutation. This combinatorial number enjoys the following recurrence relation.

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Corollary 3 We have (n+1)

Tn−k+1 = n n Tn−k +n Tn−k+1 .

Now, we introduce another combinatorial constant using lexicographic ordering of the symmetric group Sn . For convenience, we denote the permutation that maps, i to ai by 

1 2 ... n − 1 n . [a1 , a2 , . . . , an ] := a1 a2 . . . an−1 an Let us define a Lexicographic order relation for permutations of Sn as follows. The order property of natural number system (law of trichotomy) is used to establish a hierarchy in Sn . Definition 2 (Lexicographic ordering of Permutations) Let [a1 , a2 , . . . , an ] and [b1 , b2 , . . . , bn ] be two different permutations in Sn . We say that [a1 , a2 , . . . , an ] predecessor of (appears before) [b1 , b2 , . . . , bn ] in the lexicographic ordering if a1 < b1 or there exist a least integer i such that ai < bi and a j = b j , for all j ∈ {1, 2, . . . , i − 1}. For further information on lexicographic permutations and order, the reader may refer to [3, 6, 8, 9, 12, 13]. When comparing a permutation with its successive permutation, the ith entry of the permutation which decides the hierarchy with its successive permutation (as defined above) is referred as the hierarchical entry of the permutation. Hereafter, we shall denote the listing of lexicographic permutations of Sn by σ1 , σ2 , . . . , σn! . It is to be noted that σi+1 is the successor of σi in this ordering. Based on the hierarchical entry of permutation in lexicographic ordering, we shall introduce a notation called directional change, denote it by d, of a given permutation and is defined as follows. Definition 3 (Directional Change (d)) Directional change of any lexicographic permutation σi is said to be m if the mth entry of σi is the hierarchical entry when compared with its successor σi+1 and is denoted by d(σi ) = m. In other words, directional change is an onto map from d : Sn \ σn! → {1, 2, 3, . . . , n − 1}. Remark 1 Let σi = [a1 , a2 , . . . , an ] and σi+1 = [b1 , b2 , . . . , bn ] be any two successive lexicographic permutations of Sn , for some i ∈ {1, 2, . . . , n! − 1}, then   d(σi ) = m if, and only if, for all j ∈ 1, 2, . . . , m − 1 , a j = b j and am < bm . While ordering the permutations of Sn , we naturally needed this “Directional Change” which motivated us to define this definition. With this introduction to the directional change of lexicographic permutations of Sn , we have the following natural questions, namely, (i) Is d a one-one function? (ii) For any given k ∈ {1, 2, . . . , n − 1}, what is the cardinality of the set of all preimages of k, i.e., |d−1 (k)|. In other words, how many permutations have the same directional change value?

Identities from Partition of the Symmetric Group Sn

203

Since n! − 1 > n − 1 for any integer n  3, one can immediately see that d is not a one-one function, when n  3. To answer the second question, we first introduce the following notation. Definition 4 For any integers n  2 and 1  k < n, we define the number n Dk which counts the number of lexicographic permutations of Sn having the directional change value as k. Note that when n = 1, the symmetric group on one symbol S1 consists of only one permutation and since there is no hierarchy in S1 , we did not define n Dk for n = 1. Furthermore, in this article we prove the following results on directional change, Theorem 2 For any integer n > 1, we have, n

Dn−1 =

n! . 2

(1)

Since we know that there are n! permutations in Sn and with the application of above theorem we get the following corollary, Corollary 4 For any integer n > 2, 1+

n−2

n

Di = n Dn−1 =

i=1

n! 2

In the following result, we establish the link between n Dk and the well-understood permutations and combinations. Theorem 3 For any integer n > 1 and for any integer k with 1  k < n, we have n

Dk = n Pk−1 ×

n−k

C1 ,

where n Pk denotes the number of ways of arranging n objects in k places and n Ck denotes the number of ways of selecting k objects from n objects. Now, we state the following lemma which is useful in the proof of Theorem 3. Lemma 1 For any integer m < n, let σi = [a1 , . . . , am , . . . , an ] be any lexicographic permutations of Sn . Then d(σi ) = m ⇐⇒ am < am+1 and am+1 > am+2 > · · · > an . Also, we establish the following recursive relation.

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Theorem 4 For any integer n > 1 and for any integer k with 1  k < n, n+1

Dk+1 = (n + 1) n Dk .

Finally, we prove the following formula. Theorem 5 For any integer n > 1 and for any integer  with 1   < n, we have 

i=1

n

Di =

n! − 1. (n − )!

2 Proof of Theorem 1  First we prove that nk  n Tn−k . Let σ ∈ Sn be arbitrary element such that σ has k number of orbits. We have to show that the least number of transpositions of σ is n − k. Let P1 , P2 , . . . , Pk be the orbits of σ . Case 1. |Pi | > 1 for every i ∈ {1, 2, . . . , k}. In this case, since each orbit Pi can be considered as a cycle σi , we have σ1 , . . . , σk disjoint cycles such that σ = σ1 σ2 . . . σk . Let the length of σi be i > 1 for every i ∈ {1, 2, . . . , k}. Note that 1 + 2 + · · · + k = n. For each cycle σi of length i , there are i − 1 transpositions. This implies that there are (1 − 1) + (2 − 1) + · · · + (k − 1) = n − k transpositions for σ . Hence, σ contributes 1 to the number n Tn−k . Since σ is arbitrary, we are done. Case 2. |Pi | = 1 for some i ∈ {1, 2, . . . , k}. In this case, let P j1 , P j2 , . . . , P jh be the orbits such that |P jr | > 1 for every r ∈ {1, 2, . . . , h}. Since for each orbit P jr there exists a cycle σ jr , we can write σ = σ j1 σ j2 . . . σ jh , where σ j1 , σ j2 , . . . , σ jh are disjoint cycles. Let the length of σ jr be  jr > 1 for every r ∈ {1, 2, . . . , h}. Hence, we get  j1 +  j2 + · · · +  jh = n − (k − h). This implies that there are ( j1 − 1) + ( j2 − 1) + · · · + ( jh − 1) = n − (k − h) − h = n − k transpositions for σ . Thus, both the cases together imply the claim. Now we prove the other way inequality. Let σ ∈ Sn be any element such that the least number of transpositions for σ is n − k. We claim that σ has only k orbits.

Identities from Partition of the Symmetric Group Sn

205

Let σ = (a1 , b1 )(a2 , b2 ) . . . (an−k , bn−k ) be the product of transpositions. If (ai , bi )’s are disjoint cycles, then there are n − 2(n − k) singleton orbits for σ . and we get (n − k) + (n − 2(n − k)) = k number of orbits, as we wanted. By renaming the indices, if necessary, we assume that there are natural numbers 1  2  · · ·  s > 1 such that σ = (a1 , b11 ) . . . (a1 , b11 ) . . . (as , bs1 ) . . . (as , bss )(c1 , b1 ) . . . (cr , br ), where ci ’s are distinct and 1 + · · · + s + r = n − k. With this notation, we have r = n − k − 1 − · · · − s number of 2-length orbits, ( j + 1)-length orbits for all j = 1, 2, . . . , s, and some singleton orbits in σ . Thus, the number of singleton orbits is n − 2r − (1 + 1) − (2 + 1) − · · · − (s + 1) = −r + k − s. Therefore, the number of orbits of σ is r + s + k − r − s = k. This proves that n  n  T . This proves the theorem.  n−k k

3 Proofs of Corollaries 1, 2 and 3 3.1 Proof of Corollary 1  By Theorem 1, we know that nk =n Tn−k for all 1  k  n. If n is even, n  n



n n! n = Tn−k = 2 k k=0 k=0 2|k

2|k

which counts the number of even permutations and n 

n k=1 2k

k

=

n

k=0 2k

n

Tn−k =

n! 2

which counts the number of odd permutations. n   n n      n n n If n is odd, = = Tn−k = number of odd permutations and k k n  k=0 2k

k=0 2|k

n

k=0 2|k

k=1 2k

Tn−k = number of even permutations, and therefore both are equal to

n! . 2



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3.2 Proof of Corollary 2 Let n  2 and σ ∈ Sn be a given permutation. Note that σ has k number of orbits if and only if σ can be written as a product of n − k least number of transpositions. When 2|n, then note that 2|k if and only if 2|n − k. Therefore (a) follows. When 2  n, then note that 2|k if and only if 2  n − k. Therefore (b) follows. 

3.3 Proof of Corollary 3 Knuth [7] proved that

   n+1 n n =n + . k k k−1 

Then, by Theorem 1, the result follows.

4 Proofs of Theorem 2 and Corollary 4 4.1 Proof of Theorem 2 We prove this by mathematical induction. When n = 2, there are only two lexicographic permutations, namely, [1, 2] and [2, 1]. Therefore, d([1, 2]) = 1, since there is a change in the first place, when compared with its successive permutation, namely, [2, 1]. Since the permutation [2, 1] is being the last permutation in lexicographic order, there is no value of d on this. Hence, we get, 2 D1 = 1 = 2!2 . . To prove the Let us assume the result for n = m − 1 which is m−1 Dm−2 = (m−1)! 2 identity for n = m, we establish the following identity. For any integer m  3, we have m

Dm−1 = (m − 1)! + (m − 2)

m−1

Dm−2 .

m−1

Dm−2 .

(2)

Claim 1. First, we prove that m

Dm−1  (m − 1)! + (m − 2)

We will prove by the method of insertion. We know that in Sm−1 , there will be Dm−2 permutations having the value of d as m − 2. Let [a1 , a2 , . . . , am−2 , am−1 ] be any permutation of Sm−1 with d([a1 , a2 , . . . , am−2 , am−1 ]) = m − 2. The idea here is to insert m and collect the permutations(σi s) of Sm such that d(σi ) = m − 1. Let us start with any of the lexicographic permutations of {1, 2, . . . , m − 1}, i.e., Sm−1 .

m−1

Identities from Partition of the Symmetric Group Sn

207

Fig. 1 Generating permutations of Sm with k = m − 1 by method of insertion from permutations of Sm−1

Fig. 2 Generating permutations of Sm with d value as m − 1 by method of insertion from permutations of Sm−1 with d value as m − 2

It is clear that there will be m places to insert m, as shown in Fig. 2. We insert m in every place and finally add up all numbers to get Claim 1. (1) Placing m after m − 1th entry as indicated in Fig. 1. In this case, we get a permutation σ of Sm with d(σ ) = m − 1. That is, if [a1 , a2 , . . . , am−1 ] is the permutation we start with, then after insertion, we get the permutation [a1 , a2 , . . . , am−1 , m]. Since m is greater than am−1 , the successor of [a1 , a2 , . . . , am−1 , m] is [a1 , a2 , . . . , m, am−1 ]. Therefore, d([a1 , a2 , . . . , am−1 , m]) is equal to m − 1. By this, we have generated (m − 1)! permutations of Sm with d value as m − 1. (2) Placing m at (m − 1)th entry. The insertion of m in (m − 1)th entry will generate a permutation with d value as m − 2. Let [a1 , a2 , . . . , am−2 , m, am−1 ] be the permutation generated with m in (m − 1)th entry. Then the successive permutation will be [a1 , a2 , . . . , am−1 , am−2 , m], since (i) am−2 < am−1 and (ii) am−1 < m and am−2 < m. Therefore, the d value of [a1 , a2 , . . . , am−2 , m, am−1 ] is m − 2. Since we count the number of σ with d value m − 1, it is clear that this will generate no permutation whose d is m − 1. (3) Placing m in any of the first m − 2 places as in Fig. 2. Since [a1 , a2 , . . . , am−2 , am−1 ] has d value as m − 2, it is clear that am−1 > am−2 . Without loss of generality, let us assume that we insert m before r th entry of the permutation where 1  r  m − 2 (as indicated by the arrows). We get, [a1 , a2 , . . . , ar −1 , m, ar , . . . , am−2 , am−1 ] ∈ Sm . Since am−1 > am−2 , the successor of [a1 , a2 , . . . , ar −1 , m, ar , . . . , am−2 , am−1 ] must be the permutation [a1 , a2 , . . . , ar −1 , m, ar , . . . , am−1 , am−2 ] whose d value is m − 1. Therefore, we have generated (m − 2) ×m−1 Dm−2 permutations of Sm whose d value is m − 1. By adding up the numbers in (1) and (3), we get Claim 1. Claim 2: m Dm−1  (m − 1)! + (m − 2) m−1 Dm−2 .

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Let σ = [a1 , a2 , . . . , am−1 , am ] be any lexicographic permutation of Sm with d(σi ) = m − 1. (1) am = m. If we remove am from σ , we get a permutation [a1 , a2 , . . . , am−1 ] of Sm−1 . This implies that σ can be generated by (1) of Claim 1. (2) am = m. Since am = m and d(σ ) = m − 1, there exists i with i ∈ {1, 2, . . . , m − 2} such that ai = m and am−1 = m. Now we remove ai = m from σ to get a permutation [a1 , a2 , . . . , ai−1 , ai+1 , . . . , am−1 , am ] = σ1 of Sm−1 with d(σ1 ) = m − 2 (since d(σ ) = m − 1). The insertion of m in between ai and ai+1 will give σ . This is nothing but (3) of Claim 1. Thus, by (1) and (2) proves that m

Dm−1  (m − 1)! + (m − 2)

m−1

Dm−2

Therefore, we have m

Dm−1 = (m − 1)! + (m − 2) m−1 Dm−2 

(m − 1)! (by induction assumption) = (m − 1)! + (m − 2) 2 

m−2 = (m − 1)! 1 + 2 m = (m − 1)! 2 m! , = 2 

which proves the theorem.

4.2 Proof of Corollary 4 For any integer n > 2, the cardinality of the symmetric group Sn is n!, and every permutation has a unique d value, except for the last permutation. Therefore, by Theorem 2, we have 1+

n−2

i=1

n

Di = n! − n Dn−1 = n! −

n! n! = . 2 2 

Identities from Partition of the Symmetric Group Sn

209

5 Proofs of Lemma 1 and Theorems 3, 4, 5 5.1 Proof of Lemma 1 This is a consequence of the lexicographic ordering of permutations. Suppose d(σi ) = m. Then we prove that am < am+1 and am+1 > am+2 > · · · > an . If possible, we suppose am > am+1 or there exists j ∈ {m + 1, m + 2, . . . , n − 2, n − 1} such that a j < a j+1 . If there exists j ∈ {m + 1, m + 2, . . . , n − 2, n − 1} such that a j < a j+1 , then by definition, d(σi ) = m since m < j. Therefore, clearly, d(σi ) = m implies that am+1 > am+2 > · · · > an . Now, suppose, am > am+1 . Since am+1 > am+2 > · · · > an is true, we get am > am+1 > . . . > an . Therefore, by the definition of lexicographic ordering, d(σi ) = m, which is a contradiction. Hence, d(σi ) = m implies that am < am+1 and am+1 > am+2 > · · · > an . The converse follows by the definition of lexicographic ordering. 

5.2 Proof of Theorem 3 Now, we are ready to prove Theorem 3. The idea here is to collect all the lexicographic permutations having the directional change value as k and prove that the cardinality of such set is n Pk−1 × n−k C1 . By Lemma 1, any permutation σ = [a1 , a2 , . . . , ak−1 , ak , ak+1 , . . . , an ] has directional change as k if and only if ak < ak+1 and ak+1 > ak+2 > · · · > an . Now we have to collect all the lexicographic permutations [a1 , a2 , . . . , an ] such that ak < ak+1 and ak+1 > ak+2 > · · · > an . From Fig. 3, it is evident that we have n Pk−1 possibilities to choose a1 , a2 , . . . , ak−1 and n−k C1 possibilities to choose ak , the rest, namely, ak+1 , ak+2 , . . . , an can be chosen in exactly one way. Therefore, we  get, n Dk = n Pk−1 × n−k C1 .

Fig. 3 Counting the lexicographic permutations having directional change as k

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5.3 Proof of Theorem 4 By Theorem 3, we have, n+1

Dk+1 = = = = =

Pk · (n+1)−(k+1) C1 (n + 1)! ((n + 1) − (k + 1))! × ((n + 1) − k)! (n − k − 1)! (n − k)! (n + 1)! × (n − k + 1)! (n − k − 1)! n! (n − k)! (n + 1) × × (n − k + 1)! (n − k − 1)! (n + 1) n Dk .

n+1



5.4 Proof of Theorem 5 Let us prove this theorem by the principle of mathematical induction on . For  = 1, it is easy to observe that n D1 = n − 1. Hence it is true for  = 1. Now let us assume that it is true for  = m and prove that it is true for  = m + 1. Therefore, by the induction hypothesis, we have, m

n

Di =

i=1

Claim:

m+1

n

Di =

i=1

n! − 1. (n − m)!

n! − 1. (n − (m + 1))!

Now, m+1

i=1

n

Di =

m

n

Di + n Dm+1 =

i=1

n! − 1 + n Dm+1 (n − m)!

 n! − 1 + n Pm · n−(m+1) C1 (n − m)!

 n! (n − m − 1)! n! −1+ × = (n − m)! (n − m)! (n − m − 2)! (n)! × (n − m − 1) n! −1+ = (n − m)! (n − m)! n! + (n! × (n − m − 1) = −1 (n − m)!

=

Identities from Partition of the Symmetric Group Sn

1 + (n − m − 1) (n − m)! n! = − 1. (n − (m + 1))!

= n!

Hence the claim.

211

 −1



References 1. Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. Dordrecht (1974). https://doi.org/10.1007/978-94-010-2196-8 2. Denes, J.: The representation of a permutation as a the product of a minimal number of transpositions, and its connection with the theory of graphs. Publ. Math. Inst. Hung. Acad. Sci. 4, 63–71 (1959) 3. Djamegni, C.T., Tchuente, M.: A cost-optimal pipeline algorithm for permutation generation in lexicographic order. Parallel Distrib. Comput. 44, 153–159 (1997) 4. Feit, W., Lyndon, R., Scott, L.L.: A remark about permutations. J. Comb. Theory (Ser. A) 18, 234–235 (1975) 5. Fialkow, L., Salas, H.: Data exchange and permutation length. Math. Mag. 65, 188–193 (1992) 6. Knuth, D.E.: Lexicographic permutations with restrictions. Discret. Appl. Math. 117–125 (1979) 7. Knuth, D.E.: Two notes on notation. Am. Math. Mon. 99(5), 403–422 (1992) 8. Knuth, D.E.: Art of Computer Programming, Vol. 1. Fundamental Algorithms, vol. 3 (1998). ISBN: 0-201-89683-4 9. Knuth, D.E.: The Art of Computer Programming, Vol. 4A. Combinatorial Algorithms, Part 1. Pearson Education (2011). ISBN: 978-81-317-6193-9 10. Lossers, O.P.: Solution to problem E3058. Am. Math. Mon. 93, 820–821 (1986) 11. Mackiw, G.: Permutations as products of transpositions. Am. Math. Mon. 105(5), 438–440 (1995) 12. Reingold, E.M., Nievergelt, J., Deo, N.: Combinatorial Algorithms: Theory and Practice. Prentice-Hall Inc., Englewood Cliffs (1997) 13. Spoletini, E.: Generation of permutation following Lehmer and Howell. Am. Math. Soc. Math. Comput. 43, 565–572 (1984) 14. Stirling, J.: Methodus Differentialis (London, 1930). English Translation, The Differential Method (1749)

A Certain Kernel Function for L-Values of Half-Integral Weight Hecke Eigenforms M. M. Sreejith

Dedicated to Murugesan Manickam on the occasion of his 60th birthday.

Abstract In this note, we derive a non-cusp form of weight k + 1/2 (k ≥ 2, even) for 0 (4) in the Kohnen plus space whose Petersson scalar product with a cuspidal Hecke eigenform f is equal to a constant times the L value L( f, k − 1/2). We also prove that for such a form f and the associated form F under the Dth Shimura– a (D)L(F,2k−1) Kohnen lift the quantity π fk−1  f, f L(D,k) is algebraic. Keywords L-values · Kernel functions · Half-integral weight · Hecke operators · Shimura–Kohnen lift 2010 Mathematics Subject Classification 11F37 · 11F11 · 11F27 · 11F30 · 11F67

1 Introduction Let D be a positive fundamental discriminant and k ≥ 2 be an even integer. Kohnen and Zagier [7] considered a modular form  D G k,4D |T r44D Pr+ of weight k + 1/2 for 0 (4) in the Kohnen plus space. They proved that its image under Dth Shimura– Kohnen lift equals a known constant times the square of an Eisenstein series of weight 2k, level 1 (see Proposition 3 of [7]). In this paper, we consider a similar M. M. Sreejith (B) Kerala School of Mathematics, Kozhikode, Kerala, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_14

213

214

M. M. Sreejith

modular form  D E k,4D,( D. ) |T r44D Pr+ of weight k + 1/2 (k ≥ 2, even) for 0 (4) in the Kohnen plus space. We first sum up its cusp parts over all positive fundamental discriminants D and get a cusp form of weight k + 1/2 for 0 (4) in the plus space, and characterize the resulting cusp form (see Theorem 1 in Sect. 3). Finally, we characterize the above modular form  D E k,4D,( D. ) |T r44D Pr+ for each positive fundamental discriminant D, and get the following algebraic information. The results of Shimura and Manin [9, 11] show that the values of L-functions associated with a normalized newform of integral weight belong to a number field. new (N ) and L(F, s) = n≥1 a F (n)n −s (Re(s) >> 1) is More precisely, if F ∈ S2k the associated L-series, and if Q F denotes the number field generated by all the eigenvalues of F over Q, then they proved the existence of a real number ω such that L(F, n) ∈ QF , πnω

where 1 ≤ n ≤ 2k − 1.

In this paper, we let 2|k and consider a normalized Hecke eigenform F of weight 2k, level 1 and the associated unique non-zero Hecke eigenform f of weight k + 1/2 for 0 (4). Both of them are cusp forms with f |S D = a f (D)F for all positive fundamental discriminants D, where S D denotes the Dth Shimura–Kohnen lift (see [6]). We prove that a f (D)L(F, 2k − 1) ∈ Q. π k−1  f, f L(D, k) To get this result, we use the fact that the Fourier coefficients of θ D E k,4D,( D. ) |T r44D Pr+ are rational numbers.

2 Notations and Preliminaries Let k ≥ 2 be an even integer. Let M2k denote the space of modular forms of weight 2k for S L 2 (Z) and S2k denote the space of cusp forms of weight 2k for S L 2 (Z). For an integer M, let 0 (M) denote the congruence subgroup of S L 2 (Z) defined by 0 (M) :=

 a b  cd

 ∈ S L 2 (Z) : c ≡ 0 (mod |M|) .

Let Mk+1/2 (4N ) denote the space of modular forms of weight k + 1/2 for 0 (4N ) and Sk+1/2 (4N ) denote the space of cusp forms of weight k + 1/2 for 0 (4N ). + (4N ) denote the Kohnen ‘plus’ space For N odd and square-free integer, let Mk+1/2 containing the forms f in Mk+1/2 (4N ) whose Fourier coefficients a f (n) are 0 unless + + (4N ) = Mk+1/2 (4N ) ∩ Sk+1/2 (4N ). We define (−1)k n ≡ 0, 1 (mod 4). We let Sk+1/2 the stroke operator in the integral and half-integral weight respectively as follows:

A Certain Kernel Function for L-Values of Half-Integral …

F|k

a b cd

(z) := (ad − bc)k/2 (cz + d)−k f

f |k+1/2

 a b  cd



, φ(z) (z) := φ(z)

−2k−1



az + b cz + d

 f

215

 for k ∈ Z,

az + b cz + d

 for all

a b cd

a b cd

∈ G L+ 2 (Q),

∈ G L+ 2 (Q),

, t 2 = 1. Using these we can define modular forms and Hecke with φ 2 (z) = t √cz+d ad−bc type operators on them. For details, we refer to [5]. Let Tm be the mth Hecke operator on M2k . These operators preserve the space of cusp forms. Suppose a non-zero form F ∈ S2k satisfies the relation F|Tm = a F (m)F for every m ≥ 1 and a F (1) = 1, then F is known as a normalized Hecke eigenform. The space S2k has an orthonormal basis of normalized eigenforms of all Hecke operators (for example, see Theorem 6.15 in [4]); let B be such a basis. Similarly for + (4), we denote the Hecke operators the space of half-integral weight forms in Sk+1/2 by Tm+2 (see [6], p. 250 for definition; the operators Tm+2 for m ≥ 1 are generated by operators T p+2 where p varies over all primes). Let B + denotes an orthogonal basis of + eigenforms for all Hecke operators Tm+2 (m ≥ 1) on Sk+1/2 (4), (refer Theorem 1 of [6] for existence of such a basis). Note that elements of B + can be chosen in such a way that their Fourier coefficients are real and algebraic numbers (refer [5], p. 216, line 12). For k ≥ 2, Proposition 1 of [6] gives the decomposition: + (4) = CHk+1/2 Mk+1/2



+ Sk+1/2 (4),

where Hk+1/2 is an Eisenstein series: i∞ 0 + 2−2k−1 (1 − (−1)k i)E k+1/2 , Hk+1/2 := E k+1/2 i∞ 0 and E k+1/2 are as defined in [6], p. 254. Hk+1/2 takes the value 1 at where E k+1/2 infinity; this series was first studied by Cohen [2]. If f ∈ Mk+1/2 (4) with Fourier  expansion f = n≥1 a f (n)e2πinz , define the standard operators U4 and W4 by

f |U4 =

n≥1

 a f (4n)e

2πinz

, and f |W4 =

2z i

−k−1/2

 f

 −1 . 4z

+ Now, define the projection map Pr+ (see [7], p. 185) from Mk+1/2 (4) into Mk+1/2 (4) by 1 k+1 −k Pr+ = (−1)[ 2 ]2 W4 U4 + . 3

216

M. M. Sreejith

+ The map Pr+ satisfies  f |Pr+ , g =  f, g|Pr+  for f, g ∈ Mk+1/2 (4) where f or g is a cusp form (see [7], p. 186, line 13). Thus, the projection map Pr+ preserves + (4), and it is hermitian on this space. It also the plus space of cusp forms Sk+1/2 preserves the space generated by the Cohen–Eisenstein series as it is the orthogonal complement of the space of cusp forms.

Let D be a positive fundamental discriminant and s ∈ C with Re(s) >> 1, define L(D, s) :=

D n≥1

n

n −s .

 2 Let (z) = n∈Z e2πin z be the standard theta function, where z ∈ H (here, H denotes the upper half plane). It is a modular form of weight 1/2 for 0 (4) (we refer to Chap. 15 of [3] or Chap. 3, Sect. 1 of [5] for the details). Let k ≥ 2 be a positive integer and D ≡ 0, 1 (mod 4) be a positive fundamental discriminant. Define  D (z) := (Dz) =



2

e2πi Dn z .

n∈Z

  It is a modular form of weight 1/2 for 0 (4D) with character D. of conductor D or 4D according as D ≡ 1(4) or D/4 ≡ 2, 3(4) (refer [10], p. 32, lines 16–18). We also define an Eisenstein series E k,4D,( D. ) by   D 1

(cDz + d)−k . E k,4D,( D. ) = 2 (c,d)=1 d 4|c

  It is a modular form of weight k, level 4D, and character D. , and has rational Fourier coefficients (see below appendix). Let  

  1 ab f k+1/2 f T r44D = cd [0 (4) : 0 (4D)] a b ∈ (4D)\ (4) 0 0 cd

denote the trace operator (adjoint to the inclusion map under Petersson scalar product), and it maps Sk+1/2 (4D) into Sk+1/2 (4). We take 

1 0 4|D1 | 1



  1 μ  D = D D and μ (mod D ) 1 2 2 01 

as a set of representatives to define the above trace operator (see p. 196 of [7]).

A Certain Kernel Function for L-Values of Half-Integral …

217

3 Statement of Results  + Let λ D,k be a constant such that  D E k,4D,( D. ) T r44D Pr+ − λ D,k Hk+1/2 ∈ Sk+1/2 (4). Also let k ≥ 2 be even. Then we have Theorem 1



  D E k,4D,( D. ) T r44D Pr+ − λ D,k Hk+1/2

0 0 and the sum in right hand side is taken over an orthogonal basis of Hecke eigenforms + of Sk+1/2 (4). + In the simplest case where k = 6, we use S13/2 (4) = Cδ and S12 = C (where δ and are as defined in p. 177 of [7]) and we have δ|S1 = (this can be seen by comparing the Fourier coefficient of δ and ). Here, S1 is the first Shimura–Kohnen + (4) into S2k (the definition of Shimura map defined by Kohnen is lift from Sk+1/2 given in p. 176 of [7]). We have,

Corollary 1  D E

0

4 Proof of the Theorem  Observe that for 

   D 

1/2



a b Dc d

a b Dc d





∈ 0 (4D), 4|c and 2|k (hence

 −4 k d

= 1) one has

    az + b −4 1/2 Dc (Dcz + d)−1/2  D d d Dcz + d   D  D (z). = d 

(z) :=

That is  D (z) =

c  −4 1/2 d

d

(Dcz + d)−1/2

n∈Z

e2πi Dn

2 az+b Dcz+d

218

M. M. Sreejith

=

c  −4 k+1/2 d

d



D d





e2πi Dn

2 az+b Dcz+d

.

n∈Z

Multiplying with 21 (cDz + d)−k 1 (cDz + d)−k 2

(Dcz + d)−1/2

D d

on both sides, we get

    D c −4 k+1/2 1 −k  D (z) = (cDz + d) 2 d d d

az+b −1/2 2πi Dn 2 Dcz+d (Dcz + d) e . n∈Z

Sum over (c, d)’s (with gcd(c, d) = 1 and 4|c)      

1

1 D D c −4 k+1/2 −k −k  D (z) = (cDz + d) (cDz + d) 2 d 2 d d d (c,d)=1 (c,d)=1 4|c

4|c

(Dcz + d)−1/2



e2πi Dn

2 az+b Dcz+d

.

n∈Z

We get    −4 k+1/2 Dc 1

2 az+b  D E k,4D,( D. ) = (Dcz + d)−k−1/2 e2πi Dn Dcz+d 2 n∈Z (c,d)=1 d d 4|c

   Dc −4 k+1/2 1

= (Dcz + d)−k−1/2 2 (c,d)=1 d d 4|c

 Dc   −4 k+1/2 2 az+b + (Dcz + d)−k−1/2 e2πi Dn Dcz+d d d n≥1 (c,d)=1 4|c

= E k+1/2,4D +



Pk+1/2,4D;n 2 D .

n≥1

In the above E k+1/2,4D =

   −4 k+1/2 Dc 1

(Dcz + d)−k−1/2 2 (c,d)=1 d d 4|c

is an Eisenstein series of weight k + 1/2 for 0 (4D). For each n ≥ 1, the Poincarè series

A Certain Kernel Function for L-Values of Half-Integral …

Pk+1/2,4D;n

219

c  −4 k+1/2 az+b = (cz + d)−k−1/2 e2πin cz+d d d (c,d)=1  4D c

is a cusp form of weight k + 1/2 for 0 (4D) and characterized by g, Pk+1/2,4D;n  =

(k − 1/2) 1 ag (n), [SL2 (Z) : 0 (4D)] (4π n)k−1/2

 where g = n≥1 ag (n)e2πinz is an arbitrary cusp form of weight k + 1/2, level 4D. Using straightforward computations we have noticed that, T r44D maps Poincarè series to Poincarè series and Eisenstein series to Eisenstein series. We have  Pk+1/2,4D;n 2 D T r44D = Pk+1/2,4;n 2 D , and

 E k+1/2,4D T r44D Pr+ = Hk+1/2 ,

where Hk+1/2 is the Cohen–Eisenstein series defined in Sect. 2. To get this, let f ∈ Sk+1/2 . Now,      E k+1/2,4D T r44D , f = E k+1/2,4D , f |ι = 0, since f |ι is cusp form (here, ι is the inclusion map, adjoint to the trace map T r44D ). This proves T r44D preserves the space of Eisenstein series, which is orthogonal complement of the space of cusp forms with respect to the Petersson scalar product. So, we get a constant λ D,k such that 



D E k,4D, D.

 4D T r Pr+ = λ D,k Hk+1/2 + P+ 4 n≥1

k+1/2,4;n 2 D



(k − 1/2) f a f (n 2 D)  f, f i 4 (4π n 2 D)k−1/2 n≥1 f ∈B+ ⎛ ⎞ (k − 1/2) ⎝ a f (n 2 D) ⎠ f = λ D,k Hk+1/2 + , i 4 (4π )k−1/2 (n 2 D)k−1/2  f, f  +

= λ D,k Hk+1/2 +

f ∈B

n≥1

(1) where i 4 = [SL2 (Z) : 0 (4)]. Now, by summing both sides over all the fundamental discriminants D > 0, we get      D E k,4D,( D. ) T r44D Pr+ − λ D,k Hk+1/2

D fund. disc. D>0

=





D fund. disc. f ∈B+ D>0



(k − 1/2) a f (n 2 D) i 4 (4π )k−1/2 n≥1 (n 2 D)k−1/2



f .  f, f 

(2)

220

M. M. Sreejith

As D varies over all the positive fundamental discriminants and n 2 (n ≥ 1) varies over all squares, then n 2 D varies over all integers m ≥ 1 such that m ≡ 0, 1 (mod 4) where k is even (see appendix for a proof). Thus, the above becomes 



   4D   D E k,4D,( ) T r4 Pr+ − λ D,k Hk+1/2 D .

D fund. disc. D>0

=

⎛

(k − 1/2) ⎜ ⎜ ⎝ i 4 (4π )k−1/2 + f ∈B

=

⎞

m≥1 m≡0,1 (mod 4)

a f (m) ⎟ ⎟ f ⎠  f, f  k−1/2 m

f (k − 1/2)

. L( f, k − 1/2) i 4 (4π )k−1/2  f, f +

(3)

f ∈B

Note that in (2), before summing over all positive fundamental discriminants, right hand side is a finite linear combination of absolutely convergent series. Each of the  a f (m) series in the finite sum becomes an absolute convergent series after m k−1/2 taking the sum over all fundamental discriminant D > 0.

m≥1 m≡0,1 (mod 4)



5 Applications We derive a certain algebraic nature involving some L-series associated with a Hecke + (4) eigenform F and Fourier coefficient of f . We now use multiplicity result for Sk+1/2 and its relation with S2k via Shimura correspondence (as in [6], pp. 176–177). Let F ∈ S2k be the normalized Hecke eigenform which corresponds to f via the identity: If D is a positive fundamental discriminant, then there exists a unique (upto a scalar + (4) such that the following holds multiple) non-zero Hecke eigenform f ∈ Sk+1/2 a f (n D) = a f (D) 2

d|n

 d

k−1

μ(d)

D d

 a F (n/d).

(4)

+ (4) states that if F ∈ S2k is a Hecke eigenform and The multiplicity result for Sk+1/2 + f ∈ Sk+1/2 (4) is a corresponding eigenform for all T p+2 (with eigenvalue λ p ) via above Eq. (4) such that F|T p = λ p f for all primes p, then the eigenspace generated by f has dimension one [6]. The corresponding basis element f via (4) can be chosen in such a way that its Fourier coefficients are real and algebraic (refer [7], p. 177, lines 4–6). Now, Eq. (4) gives

A Certain Kernel Function for L-Values of Half-Integral …

a f (n 2 D) a f (D)

= k−1/2 2 k−1/2 (n D) D n≥1 n≥1

 d|n

221

d k−1 μ(d)

d

a F (n/d)

n 2k−1

a f (D) n k−1 μ(n) = k−1/2 D n 2k−1 n≥1 =

D

D n

a F (m) m≥1

m 2k−1

a f (D) L(F, 2k − 1) . D k−1/2 L(D, k)

Substituting this in Eq. (1), we get D E



(k − 1/2) T r 4D Pr = λ + D,k Hk+1/2 + 4 k,4D, D. i 4 (4π )k−1/2

f ∈B+

a f (D) L(F, 2k − 1) f L(D, k)  f, f 

(D)k−1/2

(k − 1/2)D −k+1/2 a f (D)L(F, 2k − 1) f . = λ D,k Hk+1/2 +  f, f  i 4 (4π )k−1/2 L(D, k) + f ∈B

(5)  Claim:  D E k,4D,( D. ) T r44D Pr+ has rational Fourier coefficients. Proof of claim: Let G =  D E k,4D,( D. ) ∈ Mk+1/2 (4D). Note that since  D and E k,4D,( D. ) have rational Fourier coefficient, so has G. The projection operator Pr+ picks up the coefficients such that (−1)k n ≡ 0, 1 (mod 4), so it does not change the nature of Fourier coefficients. It is enough to prove that G|T r44D has rational Fourier coefficients. We take     1 01 μ  D = D1 D2 and μ (mod D2 ) 4|D1 | 1 0 1  as a set of representatives for the action of trace operator. Note that 

1 0 4|D1 | 1

1 μ 0 1

=



0 1/4D −1 0

  −1 −1   0 D2

0 −1 4D1 0

1 μ 0 1

.

By considering the action of the above representative matrices individually, one by one (in this order) and by proceeding along the standard arguments (see calculations in the appendix of [7]), we conclude that the Fourier coefficients of G|T r44D are rational.  Finally, we have the following + (4) is a Hecke eigenform and F ∈ S2k is the correspondProposition 1 If f ∈ Sk+1/2

ing eigenform via (4), then

a f (D)L(F,2k−1) π k−1  f, f L(D,k)

is algebraic over rationals.

+ Proof Let d  = dim(Mk+1/2 (4)), and { f 0 = Hk+1/2 , f 1 , . . . , f d  −1 } be a basis of + Hecke eigenforms for Mk+1/2 (4), whose Fourier coefficients are real and algebraic (coefficients of Hk+1/2 are rationals, refer [2]). Let jr ≡ 0, 1 (mod 4) where 0 ≤ r ≤ d  − 1. Let j0 = 0 and ej0 = Hk+1/2 so that aej ( j0 ) = 1. Let us consider the 0

222

M. M. Sreejith

+ j1th Poincarè series P j1 in Sk+1/2 (4) which is not identically zero. Let ej1 =

P j1 . a P j ( j1 ) 1

+ Then ej1 ∈ Sk+1/2 (4) with aej ( j1 ) = 1 and aej ( j0 ) = 0. We pick ej2 from the orthog1 1 onal complement of span of {ej0 , ej1 }. To select this, we find a non-zero Poincarè series indexed by j2 in this orthogonal complement and in the plus space. We consider + (4) and take its projection inside the orthogonal comthe Poincarè series P j2 ∈ Sk+1/2   plement of span of {e j0 , e j1 }. Denote this projection by P j2 (P j2 = P j2 ⊕ P j2 , where

P j2 is in the linear span of {ej1 }). Let ej2 =

P j 2 . a P  ( j2 ) j2

Then ej2 is orthogonal to both

the forms ej0 and ej1 with aej ( j2 ) = 1 and aej ( j0 ) = aej ( j1 ) = 0. By proceeding in 2 2 2 + (4), this way, we have obtained a basis {ej0 , . . . , ejr , . . . , ejd  −1 } for the space Mk+1/2 where aejr ( jr ) = 1 and aejr ( ji ) = 0 for i = 0, 1, . . . , r − 1. + (4) such that Using this basis, we get another basis {e j0 , e j1 , . . . , e jd  −1 } of Mk+1/2  ae jr ( ji ) = 1 if i = r and 0 otherwise, where 0 ≤ i, r ≤ d − 1. A direct computation gives this set of basis from the constructed basis {ej0 , . . . , e jd  −1 } (for a detailed one such proof, we refer to the proof of Lemma 4.1 in [8]). Now, consider the system of linear equations formed by writing { f 0 , . . . , f d  −1 } in terms of the new basis {e j0 , . . . , e jd  −1 }. We note that in this system of equations, + (4). Hence, the correboth { f 0 , . . . , f d  −1 } and {e j0 , . . . , e jd  −1 } are bases for Mk+1/2 sponding matrix (a fi ( jr ))d  ×d  is invertible. Now, Eq. (5) gives

D E

k,4D,



D .

 T r 4D Pr + 4

= λ D,k Hk+1/2 +

(k − 1/2)D −k+1/2 a f (D)L(F, 2k − 1) f . i 4 (4π )k−1/2 L(D, k)  f, f  + f ∈B

By comparing the consecutive j0 , j1 , . . . , jd  −1 Fourier coefficients from the above, we get a system of equations:   a fi ( jr ) d  ×d  X = Y.  Entries of Y are Fourier coefficients of  D E k,4D,( D. ) T r44D Pr+ , which are rationals. Entries of X are  λ D,k ,

(k − 1/2)D −k+1/2 a f 1 (D)L(F1 , 2k − 1) (k − 1/2)D −k+1/2 a f d  −1 (D)L(Fd  −1 , 2k − 1) ,...,  f1 , f1   f d  −1 , f d  −1  i 4 (4π )k−1/2 L(D, k) i 4 (4π )k−1/2 L(D, k)

 ,

where {F1 , . . . , Fd  −1 } are normalized cuspidal Hecke eigenforms corresponding to the eigenforms { f 1 , . . . , f d  −1 }, respectively, via Shimura correspondence as given by Eq. (4). If K f = Q(a fi ( jr ))0≤i, j≤d  −1 , the number field, then entries of X are in a (D)L(F,2k−1) K f . Thus, we get that the entries of X are algebraic numbers. Hence, π fk−1  f, f L(D,k) is algebraic. 

A Certain Kernel Function for L-Values of Half-Integral …

223

6 Appendix (A) Fourier Expansion of the Eisenstein Series In the following computation, we have assumed that D is a positive fundamental  k+1/2 = 1. discriminant. Also k ≥ 2 is even, hence −4 d E



k,4D, D .

D (cDz + d)−k d

= 1

2

= =

=

1 2 1 2 1 2

(c,d)=1 4|c



(4c,d)=1

D d



(4cDz + d)−k

D 1

(d)−k + d 2

c =0

d=1,−1



=1+

D 1



(1)−k +

D −1

D

c≥1 d∈Z

=1+





c≥1 δ|4c

d





d∈Z (4c,d)=1

(−1)−k



+

D d



(4cDz + d)−k 





c≥1

d∈Z (4c,d)=1

D d



(4cDz + d)−k

μ(δ)(4cDz + d)−k

δ|4c δ|d

μ(δ)

D (4cDz + dδ)−k dδ

d∈Z

−k

μ(δ)  D   4cD =1+ z+d k dδ δ δ c≥1 δ|4c

d∈Z

d  → Dn + d  =1+

−k      4cD 

D 1 μ(δ) D δ z+d +n δ d D Dk δk c≥1 δ|4c n∈Z d  (mod D) 

  (−2πi)k k−1 μ(δ) D =1+ n k k δ D (k − 1)! δ c≥1 δ|4c

n≥1

  (−2πi)k k−1 μ(δ) D =1+ n δ D k (k − 1)! δk c≥1 δ|4c

n≥1

=1+



d (mod D)

d (mod D)



D d D d

 2πin e  e

4cD z+d  δ +n D



d 2πin 4c z 2πin D δ e

√     (−2πi)k D k−1 D D μ(δ) 2πin 4c z δ n e k n δ D (k − 1)! δk c≥1 δ|4c

n≥1

  



μ(4c/δ) D D (−2πi)k e2πinδz =1+ n k−1 n (4c/δ)k 4c/δ D k−1/2 (k − 1)! n≥1 c≥1

=1+

δ|4c

   



μ(4c/δ) D D (−2πi)k e2πimz (m/δ)k−1 m/δ (4c/δ)k 4c/δ D k−1/2 (k − 1)! ⎛

m≥1 δ|m c≥1 δ|4c

⎞     ⎜ ⎟ k



(−2πi) μ(c/δ) D D ⎜ ⎟ e2πimz ⎟ =1+⎜ (m/4δ)k−1 ⎝ D k−1/2 (k − 1)! ⎠ m/4δ (c/δ)k c/δ ⎛

m≥1 δ|m/4 c≥1 δ|c

⎜ (−2πi)k 2−k D

2 ⎜ −⎜ ⎝ D k−1/2 (k − 1)!

⎞

 (m/2δ)k−1

m≥1 δ|m/2 c≥1 δ|c,δ odd

D m/2δ



μ(c/δ) (c/δ)k



D c/δ



⎟ ⎟ e2πimcz ⎟ ⎠

224

M. M. Sreejith ⎛ =1+⎝

⎛ =1+⎝



⎞   μ(c) D



D (−2πi)k c/δ (n/δ)k−1 e2πin4z ⎠ n/δ ck D k−1/2 (k − 1)! n≥1 c≥1 δ|n ⎛ ⎞

k −k D   μ(c) D



⎜ (−2πi) 2 ⎟ D c/δ 2 −⎜ (n/δ)k−1 e2πin2z ⎟ ⎝ D k−1/2 (k − 1)! ⎠ n/δ ck n≥1 c≥1 δ|n δ odd

⎞  

(−2πi)k D δ k−1 e2πin4z ⎠ δ D k−1/2 (k)L(D, k) n≥1 δ|n ⎛ ⎞ k −k D  



⎜ (−2πi) 2 ⎟ D 2 k−1 2πin2z ⎟ −⎜ δ e ⎝ D k−1/2 (k)L(D, k) ⎠ δ n≥1 δ|n δ odd

2 D



2 2 2πin4z ∗ =1+ σk−1,D (n)e − σk−1,D (n)e2πin2z , L(D, 1 − k) L(D, 1 − k)2k n≥1

where σk−1,D (n) =

n≥1

D d|n

d

∗ d k−1 and σk−1,D (n) =

 D d|n d odd

d

d k−1 . We have used

functional equation for Dirichlet L function (notations as given in [1], Theorem 12.11):     D k−1 (k) D −πik πik 2 2 e L(D, 1 − k) = e G(1, D)L(D, k) + (2π )k −1 D k−1 (k) √ = 2. DL(D, k), (−2πi)k or,

(−2πi)k 2 = . k−1/2 D (k)L(D, k) L(D, 1 − k)

Since L(D, 1 − k)/ζ (1 − 2k) ∈ Q and ζ (1 − 2k) ∈ Q, we have L(D, 1 − k) ∈ Q and hence the coefficients of the defined Eisenstein series are rational. (B) Claim: As D varies over all the positive fundamental discriminants and n 2 (n ≥ 1) varies over all squares, then n 2 D varies over all integers m ≥ 1 such that m ≡ 0, 1 (mod 4). Proof of claim: Let m ≥ 1 be an integer such that m ≡ 0, 1 (mod 4). If m is an odd integer, split m = Dn 2 where D ≡ 1 (mod 4) and square-free with n ≥ 1 since m ≡ 1 (mod 4). If m is an even integer then 4|m. We write m/4 = dn 2 where d is a square-free integer and n ≥ 1. If d ≡ 1 (mod 4), replace d by D so that m = D(2n)2 . If d ≡ 2, 3 (mod 4), let D = 4d so that m = Dn 2 . Since d is square-free integer, in all the cases, D is a fundamental discriminant. This proves the claim. 

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225

Acknowledgements The author thanks his supervisor M. Manickam for framing the problem and guidance. The author would like to thank IMSc, Chennai for the hospitality, and the author also thanks referee for carefully reading the article and valuable comments made which improved the style of the article. The author is supported by INSPIRE Fellowship (IF170843).

References 1. Apostol, T.M.: Introduction to Analytic Number Theory, Springer International Student Edition. Narosa Publishing House (1998) 2. Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217(3), 271–285 (1975) 3. Cohen, H., Strömberg, F.: Modular Forms - A Classical Approach. Graduate Studies in Mathematics, vol. 179. American Mathematical Society, Providence (2017) 4. Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence (1997) 5. Koblitz, N.I.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (1993) 6. Kohnen, W.: Modular forms of half-integral weight on 0 (4). Math. Ann. 248, 249–266 (1980) 7. Kohnen, W., Zagier, D.: Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64, 175–198 (1980) 8. Manickam, M., Ramakrishnan, B., Vasudevan, T.C.: Hecke operators on modular forms of half-integral weight. Arch. Math. 51, 343–352 (1988) 9. Manin, Y.: Periods of parabolic forms and p-adic Hecke series. Math. USSR Sb. 21, 371–393 (1973) 10. Serre, J.P., Stark, H.M.: Modular forms of weight 1/2. Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627, pp. 27–67. Springer, Berlin (1976) 11. Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976)

On Admissible Set of Primes in Real Quadratic Fields Kotyada Srinivas and Muthukrishnan Subramani

Abstract The concept of admissible set of primes, as propounded by Ram Murty and his collaborators, has been instrumental in establishing Euclidean algorithm in certain Number fields. In this article, we consider two simple families of real quadratic fields and demonstrate how to construct admissible set of primes in them. Keywords Euclidean algorithm · Admissible primes · Chowla fields

1 Introduction An integral domain D is called a Euclidean domain if there exist a Euclidean function on D, that is, if there exists a mapping φ : D → Z satisfying: φ(ab) ≥ φ(a), for all a, b ∈ D, b = 0, and if a and b with b = 0 are elements of D, then there exists q, r ∈ D such that a = bq + r and φ(r ) < φ(b). By abuse of language, we shall say that a number field K admits a Euclidean algorithm if a Euclidean function is defined on its ring of integers O K . The classification of algebraic number fields which admits a Euclidean algorithm is an important problem in number theory. Important contributions were made by Motzkin [6], Weinberger [11] and many others (see the Ph.D. thesis [10] of the second author and the references therein). Recently, the problem of determining the existence of Euclidean algorithm for Galois number fields have been considered by Ram Murty and his school with phenomenal success (see [5]). The novelty in their approach is tweaking Motzkin’s criteria (mentioned above) in such a way that the main problem reduces to showing K. Srinivas (B) Institute of Mathematical Sciences, HBNI CIT Campus, Taramani, Chennai 600113, India e-mail: [email protected] M. Subramani Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, Vandalur-Kelambakkam Road, Chennai 600127, India e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 B. Ramakrishnan et al. (eds.), Modular Forms and Related Topics in Number Theory, Springer Proceedings in Mathematics & Statistics 340, https://doi.org/10.1007/978-981-15-8719-1_15

227

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K. Srinivas and M. Subramani

the existence of what are called admissible set of primes in the ring of integers. We now define this concept below. Definition 1 A set of prime ideals {p1 , p2 , . . . , ps } in O K is called an admissible set of primes (henceforth we shall simply say admissible set) if, for all β = pa11 pa22 . . . pas s with ai ∈ N ∪ {0}, every element in the coprime residue class (mod β) is rep× × resented by a unit in O× K . That is, the natural canonical map O K → (O K /β) is surjective. Just for illustration, we mention a result of Harper and Ram Murty below Theorem 1 (Murty, Harper [5]) Let K /Q be abelian of degree n with O K having class number one that contains a set of admissible primes with s elements. Let r be the rank of the unit group. If r + s ≥ 3, then O K is Euclidean. More recently, this concept has been used to show the existence of Euclidean algorithm in certain number fields of lower unit rank r (see [7, 9]). Note that by Dirichlet unit theorem, r ∼ O× K = UK × Z ,

where U K is the torsion group and r is the rank of K . Since the unit group O× K is generated by r + 1 elements, so is the group (O K /pa11 pa22 . . . pas s )× . Now, by Chinese remainder theorem s (O K /piai )× , (O K /pa11 pa22 . . . pas s )× ∼ = ⊕i=1

(1)

which implies that s ≤ r + 1. In [4], Murty and Harper suggested that s = r + 1 can always be attained for every algebraic number field with unit rank r . In the particular case of real quadratic fields, it asserts that we can always find an admissible set with two elements. The aim of this article is two-fold. First, to demonstrate to the readers how to explicitly construct admissible set of primes in a number field. For this purpose, we consider two simple families of quadratic fields and do the analysis. The second point is to show that the above-mentioned assertion of Murty and Harper is true for these families. It must be remarked that a different and more non-trivial construction of a family of real quadratic fields having an admissible set with two elements was constructed in √ [7]. However,√ in this note, we consider the Chowla fields, i.e., fields of the form Q( n 2 + 1), Q( n 2 + 2) and show the existence of the admissible set. More precisely, we prove the following √ Theorem 2 Each member of the family of real quadratic fields A := {Q( n 2 + 1) : n = 6(38 + 32 173 k), k ∈ Z} admits an admissible set of primes with two elements. Theorem 3 Let p1 = 13, p2 = 11 be two odd primes. Consider the family B := √ {Q( n 2 + 2) : p1 p2 |n, n ∈ N}. Then, each element in the family B contains a set of admissible primes with two elements.

On Admissible Set of Primes in Real Quadratic Fields

229

The proof of the theorem is based on the following propositions which are inspired from the work of Harper √ [4] in which he constructed a set of admissible primes for the ring of integers Z[ 14] (see also [7]). Proposition 1 Let p ⊂ O L be an unramified prime with inertial degree one and lying above the odd rational prime p. If ρ is a primitive root modulo p and ρ p−1 ≡ 1 (mod p2 ), then ρ generates the group (O K /p2 )× . Proposition 2 Let L be an algebraic number field and O L be its ring of integers. Assume that ρ ∈ O L be a unit of infinite order. If p1 and p2 are distinct, unramified primes with inertial degree one, and lying above the odd rational primes p1 and p2 and if 1. ρ generates (O L /p21 )× ; 2. ρ has order p2 ( p2 − 1)/2 in (O L /p22 )× ; 3. p2 ≡ 3 (mod 4); and   4. gcd p1 ( p1 − 1), p2 ( p2 − 1)/2 = 1,   2 2 × then O× . L maps onto O L /p1 p2

2 Proof of the Theorem 2 From now onwards, throughout this section, we shall fix the notations: √ K = Q( d), d = n 2 + 1, n = 6(38 + 32 173 k), k ∈ Z. Our aim is to find two prime ideals p1 , p2 in O K satisfying conditions in Proposition 2, thereby producing the required√admissible set (with two primes). We first observe that ε = n + d is the fundamental unit in K . We need to select two rational primes p1 and p2 such that all the conditions in Proposition 2 are satisfied. Our choice of the primes are p1 = 17 and p2 = 3. We need only to ascertain that the conditions (1) and (2) in Proposition 2 are satisfied. Let d K denotes the discriminant of K . Since       dK d 1 = = = 1, 3 3 3 the prime 3 splits in O K . This means the primes in O K which lies above 3 are unramified with inertial degree one. A simple computation gives

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N (ε + 1) = (ε + ε¯ ) = 2n, where n ≡ 0 (mod 3), but n ≡ 0 (mod 32 ). Therefore, there exists a prime ideal p2 lying above 3 such that p2 |(ε + 1) but p22  (ε + 1). As ε ≡ −1 (mod p2 ) and the group (O K /p2 )× has order 2, the unit ε generates the group (O K /p2 )× . Again by the choice of p2 , ε2 ≡ 1 (mod p22 ). Otherwise, ε2 ≡ 1 (mod p22 ), i.e., 2 p2 | (ε2 − 1) = (ε − 1)(ε + 1). But p2  (ε − 1) implies that p22 | (ε + 1). This is a contradiction, since N (ε + 1) ≡ 0 (mod p22 ). It now follows from Lemma 1 that the unit ε is a generator in (O K /p22 )× . In other words, the unit ε has order p2 ( p2 − 1) modulo p22 . Now consider the prime p1 = 17. We see that 17 splits completely in O K since 

dK 17



 =

n2 + 1 17



 =

−1 17

 = 1.

We now show that N (ε8 − 1) ≡ 0 (mod 17). This will imply that every prime ideal p lying above 17 is unramified with inertial degree one and the unit ε is a primitive root modulo p. A simple computation gives N (ε8 − 1) = N (ε − 1)N (ε + 1)N (ε2 + 1)N (ε4 + 1) = −2n(2n)(2 + 2(n 2 + d))(2 + 2(n 4 + 6n 2 d + d 2 )) ≡ 3 · 14 · 13 · 2 ≡ 4 ≡ 0 (mod 17). Thus, N (ε8 − 1) ≡ 0 (mod 17). Now we show that ε16 ≡ 1 (mod p2 ) for some prime ideal p lying above 17. Again, by routine computation, we get obtain N (ε16 − 1) = N (ε − 1)N (ε + 1)N (ε2 + 1)N (ε4 + 1)N (ε8 + 1) = −2n(2n)(2 + 2(n 2 + d))(2 + 2(n 4 + 6n 2 d + d 2 )) (2 + 2(n 8 + 28n 6 d + 70n 4 d 2 + 28n 2 d 3 + d 4 )) ≡ 4457 · 456 · 1594 · 4269 · 4629 ≡ 309 ≡ 0

(mod 173 ).

Therefore, for some prime ideal p1 in O K lying above 17, ε16 ≡ 1 (mod p21 ). Thus, again using Proposition 1, we see that the unit ε is a generator modulo p21 . Now, put ρ = −ε. As ε is a generator in (O K /p22 )× and p2 ≡ 3 (mod 4), the unit ρ has order p2 ( p2 − 1)/2 modulo p22 . Since p1 ≡ 1 (mod 4) and the prime p1 has norm p1 , the unit ρ is also a generator for the group (O K /p21 )× . Thus, for the unit ρ and primes p1 , p2 the conditions in Proposition 2 are satisfied. Therefore, every real quadratic field K in the family A has an admissible set of primes of cardinality two.

On Admissible Set of Primes in Real Quadratic Fields

231

Remark 1 The family A contains infinitely many real quadratic fields. This can be seen from the following result. Lemma 1 (Ricci [8]) Let f (x) ∈ Z[x] be a separable quadratic polynomial. Assume that gcd { f (n) : n ∈ Z} is square free. Then the set { f (n) : n ∈ Z} contains infinitely many square free values. We note that the quadratic polynomial f (k) = (6(38 + 32 173 k))2 + 1 is a separable polynomial and f (0) = 1. Thus, f (k) satisfies all condition of above Lemma 1. Therefore, f (k) assumes infinitely many square free values, √ as k runs over all integers. Hence, there are infinitely many real quadratic fields Q( d) where d = n 2 + 1 and d square free.

3 Proof of Theorem 3 √ Let us consider quadratic fields K := Q( n 2 + 2), and fix p1 = 13, p2 = 11. We further assume that n ≡ ±1 (mod p1 p2 ). Following the line of argument of Theorem 2, we prove that there exist prime ideals above p1 and p2 satisfying all the conditions in the Proposition 2.√ From [3] (Problem 8.2.11 (d)), the fundamental unit of K is of the form n 2 + 1 + n n 2 + 2. We first observe that √the rational primes p1 and p2 splits completely in each of the quadratic fields Q( n 2 + 2). Since d K is equal to either n 2 + 2 or 4(n 2 + 2), in either case,       2 n +2 3 dK = = p1 p1 p1 and

We recall that



dK p2



 =

n2 + 2 p2



 =

3 p2

 .

   3 1, if p ≡ 1, 11 (mod 12) = p −1, if p ≡ 5, 7 (mod 12).

(2)

    Since p1 ≡ 1 (mod 12), p2 ≡ −1 (mod 12) we get dpK1 = 1, and dpK2 = 1. Now, we check the conditions of Proposition 2 one by one. Note that condition 3 and 4 are trivially satisfied. To check conditions 1 and 2 we invoke Proposition 1. Now, we shall prove condition 1 first. For this, we need to prove that ε generates the group (O K /p1 )× , and ε p1 −1 ≡ 1 (mod p21 ), for some primes ideal p1 in K lying above p1 , i.e., we need to prove that ε6 ≡ 1 (mod p1 ) and ε12 ≡ 1 (mod p21 ). By simple calculations, we obtain

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K. Srinivas and M. Subramani

N (ε3 − 1) ≡ 2

(mod p1 )

(3)

and ε3 + ε¯ 3 ≡ 0

(mod p1 ).

(4)

Since N (ε3 + 1) = N (ε3 − 1) − 2(ε3 + ε¯ 3 ), from (3) and (4), we get N (ε6 − 1) = N (ε3 − 1)N (ε3 + 1) ≡ 4

(mod p1 ).

Thus, for any prime ideal p above p1 , ε is a primitive root modulo p. Again, a similar calculation gives, N (ε12 − 1) = N (ε6 − 1)N (ε6 + 1) ≡ 1191 ≡ 0

(mod p13 ).

Therefore, there exists a prime ideal p1 above p1 such that p1 is a non-Wieferich prime with respect to the base ε. Thus, ε is a primitive root modulo (mod p21 ). Now, we prove that ε has order p2 ( p22 −1) modulo p22 . By explicit computation, we see that N (ε11 − 1) ≡ 86 ≡ 0

(mod p22 )

and N (ε5 − 1) ≡ 4 ≡ 0

(mod p22 ).

Thus, ε has order p2 ( p2 − 1) or p2 ( p22 −1) . If ε has order p2 ( p2 − 1), we replace ε by −ε. Then −ε has order p2 ( p22 −1) modulo p22 . Finally, we can see that ε satisfies all the four conditions of the Proposition 2. This establishes that {p1 , p2 } is an admissible set as claimed. This completes the proof of Theorem 3. Remark 2 In Theorem 3, we have chosen the rational primes p1 = 13 and p2 = 11 for convenience. In√general, Theorem 3 can be stated as follows: Let K n = Q( n 2 + 2) : p1 p2 |n, n ≡ ±1 mod 12 be a family of number fields where p1 and p2 are rational primes satisfying the following conditions: for each n, let 1. p1 and p2 be distinct, unramified primes in O K n with inertial degree one, and lying above odd rational primes p1 and p2 ; 2. ε generates (O K n /p21 )× ; 3. ε has order p2 ( p22 −1) in (O K n /p22 )× ; 4. p2 ≡ 3 (mod 4); and 5. gcd( p1 ( p1 − 1), p2 ( p22 −1) ) = 1,

On Admissible Set of Primes in Real Quadratic Fields

233

for some unit ε ∈ O× K n . Then, for each n, K n contains a set of admissible primes with two elements. Acknowledgements The authors wish to thank the referee for some valuable suggestions. This work is partly supported by SERB MATRICS Project No. MTR /2017/001006.

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