Jacobi-like forms, pseudodifferential operators, and quasimodular forms 9783030291228, 9783030291235

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Springer Monographs in Mathematics

YoungJu Choie Min Ho Lee

Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

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YoungJu Choie Min Ho Lee •

Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms

123

YoungJu Choie Department of Mathematics POSTECH Pohang, Korea (Republic of)

Min Ho Lee Department of Mathematics University of Northern Iowa Cedar Falls, IA, USA

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-29122-8 ISBN 978-3-030-29123-5 (eBook) https://doi.org/10.1007/978-3-030-29123-5 Mathematics Subject Classification (2010): 11F11, 11F50 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is concerned with various topics centered around connections of quasimodular forms with Jacobi-like forms and automorphic pseudodifferential operators and is intended for researchers and graduate students in number theory. In particular, it contains a detailed exposition of such connections for the first time in book form. Modular forms for a discrete subgroup Γ of SL(2, R) have been playing an important role in number theory for a long time. Quasimodular forms generalize modular forms and were introduced by Kaneko and Zagier in 1995 and have been studied in connection with various topics in number theory and other areas of mathematics in recent years. Jacobi-like forms for Γ are formal power series, whose coefficients are holomorphic functions on the Poincar´e upper half plane H, invariant under a certain right action of Γ . They are also closely linked to pseudodifferential operators, which are formal Laurent series in ∂ −1 having holomorphic functions on H as coefficients. The operation of SL(2, R) on H given by linear fractional transformations determines a natural right action of SL(2, R) on the space of pseudodifferential operators, and pseudodifferential operators that are invariant under the restriction of this action to Γ are automorphic pseudodifferential operators. There are mutual correspondences among Jacobi-like forms, automorphic pseudodifferential operators, and sequences of modular forms of certain type. On the other hand these objects are closely linked to quasimodular forms due to the fact that each coefficient function of a Jacobi-like form or an automorphic pseudodifferential operator is a quasimodular form. As a result, there is a natural projection map carrying a Jacobi-like form to one of its coefficients. On the other hand, this map has a right inverse which may be regarded as a lifting map carrying a quasimodular form to a Jacobi-like form and having the given quasimodular form as one of its coefficients. Similar projection and lifting maps can also be considered between automorphic pseudodifferential operators and quasimodular forms.

v

vi

Preface

The main goal of this book is to explore relations among quasimodular forms, Jacobi-like forms, automorphic pseudodifferential operators and sequences of modular forms and discuss various applications of such relations.

The first author is partially supported by NRF-2018R1A4A1023590 and NRF-2017R1A2B2001807.

Pohang, Korea, September 30, 2019 Cedar Falls, Iowa, USA September 30, 2019

YoungJu Choie Min Ho Lee

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Formal Power Series and Pseudodifferential Operators . . . . 7 1.1 Correspondences of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Actions of SL(2, R) on Power Series . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Jacobi-like Forms and Modular Series . . . . . . . . . . . . . . . . . . . . . 18 1.4 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2

Jacobi-like Forms and Pseudodifferential Operators . . . . . . . 2.1 Power Series Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Automorphic Pseudodifferential Operators of Mixed Weight . . 2.3 Bundles of Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . 2.4 Poincar´e Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Linear Maps and Rankin–Cohen Brackets . . . . . . . . . . . . . . . . .

31 31 36 40 43 46

3

Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Jacobi-like Forms, Modular Series, and Hecke Operators . . . . . 3.2 Hecke Operators on Pseudodifferential Operators . . . . . . . . . . . 3.3 Differential Equations and Modular Forms . . . . . . . . . . . . . . . . . 3.4 Hecke Operators and Differential Equations . . . . . . . . . . . . . . . .

53 53 60 67 72

4

Lie 4.1 4.2 4.3 4.4

Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lie Algebras of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lie Algebra Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivariant Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lie Algebras of Jacobi-like Forms . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 80 83 87

5

Heat Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Radial Heat Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Modular Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 100 102

vii

viii

Contents

6

Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Cohomology of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Jacobi-like Forms and Group Cohomology . . . . . . . . . . . . . . . . .

107 107 111 115

7

Quasimodular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Quasimodular and Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Polynomials and Formal Power Series . . . . . . . . . . . . . . . . . . . . . 7.3 Quasimodular Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 127 130 137 141

8

Quasimodular and Modular Polynomials . . . . . . . . . . . . . . . . . . 8.1 Correspondences of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modular and Quasimodular Polynomials . . . . . . . . . . . . . . . . . . 8.3 Poincar´e Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Heat Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 153 157 161

9

Liftings of Quasimodular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Liftings of Modular Forms to Quasimodular Forms . . . . . . . . . 9.2 Liftings of Quasimodular Forms to Jacobi-like Forms . . . . . . . . 9.3 Lie Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Rankin–Cohen Brackets on Quasimodular Forms . . . . . . . . . . .

165 165 168 176 181

10 Quasimodular Forms and Vector-valued Modular Forms . . 10.1 Vector-valued Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Symmetric Tensor Representations . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Scalar- and Vector-valued Modular Forms . . . . . . . . . . . . . . . . . 10.4 Quasimodular Polynomials and Vector-valued Modular Forms

185 185 189 194 202

11 Differential Operators on Modular Forms . . . . . . . . . . . . . . . . . 11.1 Jacobi-like Forms and Quasimodular Forms with Character . . 11.2 Differential Operators on Modular Forms . . . . . . . . . . . . . . . . . . 11.3 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Differential Operators Associated to Theta Functions . . . . . . .

207 207 211 215 218

12 Half-integral Weight Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Quasimodular Forms of Half-integral Weight . . . . . . . . . . . . . . . 12.2 Shimura Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Shintani Liftings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 223 227 231

13 Projective Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Projective Structures on Riemann Surfaces . . . . . . . . . . . . . . . . 13.2 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 The Degree Four Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 The Degree Five Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 238 244 246

Contents

14 Applications of Quasimodular Forms . . . . . . . . . . . . . . . . . . . . . . 14.1 Quasimodular Forms for the Full Modular Group . . . . . . . . . . . 14.2 The Work of Kaneko–Zagier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Covers of Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Modular Linear Differential Equations . . . . . . . . . . . . . . . . . . . . 14.6 Holomorphic Anomaly Equations . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.1 Gromov–Witten Invariants . . . . . . . . . . . . . . . . . . . . . . . . 14.7.2 Covers of Pillowcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.3 Square-tiled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.4 Curves on Abelian Surfaces . . . . . . . . . . . . . . . . . . . . . . . .

ix

251 251 256 263 268 276 281 284 284 286 287 289

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 List of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Introduction

Quasimodular forms, which generalize classical modular forms, were introduced by Kaneko and Zagier in [54] in connection with counting covers of an elliptic curve, and they also occur as holomorphic parts of nearly holomorphic modular forms studied by Shimura in [110]. Since then various applications of quasimodular forms have been investigated in many papers. The goal of this book is to discuss various properties of quasimodular forms in connection with Jacobi-like forms and automorphic pseudodifferential operators and describe some of their applications. The problem of counting covers of a Riemann surface with fixed ramification type can be traced back to the days of Hurwitz, who determined the number of covers of a sphere. In 1995 Dijkgraaf studied the number of connected covers of an elliptic curve with simple branching by considering an associated generating series given in terms of powers of q = e2πiz with z in the Poincar´e upper half plane. In the case of covers of genus g, he showed that this generating series is a quasimodular form of weight 6g − 6 for the full modular group SL(2, Z). A rigorous proof of this result in a more general setting was obtained by Kaneko and Zagier in [54], where in fact quasimodular forms were first introduced. Bloch and Okounkov [10] extended the idea of Dijkgraaf and Kaneko–Zagier by proving that the q bracket associated to a shifted symmetric polynomial function on the set of all partitions is a quasimodular form. The quasimodularity of the same type of function for a more general branching profile was obtained by Eskin and Okounkov [39] (see also [91], [92]). They also proved that generating functions for enumeration of branched coverings of the pillowcase orbifold are quasimodular forms of level two, which can be used to compute the volumes of the strata of the moduli space of quadratic differentials (cf. [40]). The quasimodularity for the case of covers weighted with Siegel–Veech weights was studied by Chen, M¨ oller and Zagier in [14], and they used this result to prove conjectures of Eskin and Zorich [41] about the large genus limits of Masur–Veech volumes and of Siegel–Veech constants.

1

2

Introduction

Quasimodular forms also appear in the problem of counting curves on abelian surfaces (cf. [3]), counting square-tiled surfaces in covers of a torus (see [80]), as the constant term in the Fourier expansion of the Weierstrass ℘function, or in the study of moments of periodic functions or Chazy’s equation in Painlev´e theory (see Chapter 1 of the book [11]). They also occur in string theory in modern physics. For example, the B-model topological string theory on a Calabi–Yau threefold X has a symmetry group Γ ⊂ SL(2, Z), which acts on the topological string wave function, and the associated string amplitude is a quasimodular forms for Γ (see e.g [1] and [99]). Many more papers devoted to various aspects of quasimodular forms have appeared in recent years (cf. [2], [5], [6], [13], [35], [83], [87], [88], [89], [98], [100], [102], [104], [116]). Given a discrete subgroup Γ of SL(2, R) and integers w, m with m ≥ 0, if R denotes the ring of polynomially bounded holomorphic functions on the Poincar´e upper half plane H, a quasimodular form for Γ of weight w and depth at most m is an element of R such that there exist f0 , f1 , . . . , fm ∈ R satisfying   m    az + b c c 1 = f + · · · + f f (z) + f (z) (z) 0 1 m (cz + d)w cz + d cz + d cz + d   for all z ∈ H and ac db ∈ Γ . Assuming that Γ is commensurable with SL(2, Z), the functions fk are uniquely determined by f , and therefore we can consider the associated polynomial F (z, X) =

m 

fr (z)X r

r=0

called a quasimodular polynomial. Then the condition for f to be a quasimodular form is equivalent to the condition for the polynomial F (z, X) to be invariant under a certain right action of Γ on the space of polynomials Rm [X] of degree at most m over R. In fact, there is a one-to-one correspondence between quasimodular forms and quasimodular polynomials, and it is often convenient to deal with quasimodular polynomials for the study of quasimodular forms. Jacobi-like forms for a discrete subgroup Γ of SL(2, R) are formal power series over the ring R that are invariant under a certain right action of Γ (see [31], [114]). In fact, there is an action of the group SL(2, R) on the algebra R[[X]] of formal power series over R whose restriction to Γ is the Γ -action that determines Jacobi-like forms. The Γ -invariance property of a Jacobilike form is essentially one of the two conditions that must be satisfied by a Jacobi form introduced by Eichler and Zagier in [37], and the same property determines relations among the coefficients of the given Jacobi-like form. These relations can be used to express each coefficient of a Jacobi-like form as linear combinations of derivatives of some modular forms for Γ . If the weight of a Jacobi-like form is λ, then the associated modular forms belong

Introduction

3

to a sequence of the form {fw }w≥0 , where fw is a modular form of weight 2w + λ. Such a sequence determines a modular series of weight λ, which is a formal power series belonging to R[[X]] whose coefficients are the terms of the given sequence. Modular series can also be regarded as Γ -invariant formal power series by modifying the SL(2, R)-action mentioned above, and there is a one-to-one correspondence between Jacobi-like forms and modular series of the same weight. In particular, given a modular form f , there is a Jacobi-like form f(z, X) corresponding to the modular series whose only nonzero term is the constant term given by f , and it is known as the Cohen–Kuznetsov lifting of f (cf. [31]). Pseudodifferential operators are formal Laurent series in the formal inverse ∂ −1 of the derivative operator ∂ whose coefficients are complex-valued functions, and they have been studied extensively over the years in connection with a variety of topics in pure and applied mathematics. For example, they play a critical role in the theory of nonlinear integrable partial differential equations, also known as soliton equations (see e.g. [32]). In this book we consider pseudodifferential operators whose coefficients are holomorphic functions on H. In this case the usual linear fractional action of SL(2, R) on H induces an operation of the same group on such operators. One interesting property of this operation is that it is compatible with the SL(2, R)-action on formal power series described above under an isomorphism defined in such a way that the coefficients of a pseudodifferential operator are some constant multiples of the corresponding formal power series. Pseudodifferential operators that are Γ -invariant under this operation are automorphic pseudodifferential operators for Γ , and they are in one-to-one correspondence with Jacobi-like forms for Γ due to the above-mentioned compatibility. Various topics related to mutual correspondences among Jacobi-like forms, automorphic pseudodifferential operators, and modular series were investigated by Cohen, Manin, and Zagier (see [31] and [114]). One of the goals in this book is to discuss connections of quasimodular forms with Jacobi-like forms and automorphic pseudodifferential operators. What allows us to establish such connections is the fact that each coefficient of a Jacobi-like form or an automorphic pseudodifferential operator is a quasimodular form. More specifically, there is a complex linear map from formal power series to polynomials over R that is SL(2, R)-equivariant with respect to the above-mentioned actions that determine Jacobi-like forms and quasimodular polynomials. By restricting this map to Γ -invariant elements we obtain a complex linear map from Jacobi-like forms to quasimodular polynomials. If quasimodular polynomials are identified with quasimodular forms, this map simply carries a Jacobi-like form to one of its coefficients. The same map is equivariant with respect to appropriate Hecke operator actions on Jacobi-like forms and quasimodular polynomials. Furthermore, there is a right inverse of such a map, which may be regarded as a lifting map from quasimodular forms to Jacobi-like forms or to pseudodifferential operators. The Jacobi-like form obtained by applying this lifting map to a quasimod-

4

Introduction

ular polynomial may be considered as the Cohen–Kuznetsov lifting of the given quasimodular polynomial. Using the correspondence between Jacobilike forms and automorphic pseudodifferential operators, we can also consider a map from automorphic pseudodifferential operators to quasimodular polynomials as well as its right inverse. Various applications of correspondences and liftings mentioned above can be studied. For example, using the Lie bracket for pseudodifferential operators associated to the natural noncommutative multiplication, we can determine Lie algebra structures on the spaces of Jacobi-like forms and quasimodular polynomials. As another application, we can consider the Shimura correspondence for quasimodular forms. Given an odd integer k, the usual Shimura correspondence associates to each modular form of weight λ = k − 12 a modular form of weight 2λ − 1 (see [109]). On the other hand, there is also a map carrying modular forms in the opposite direction known as the Shintani lifting (cf. [111]). By introducing the notion of quasimodular forms and Jacobi-like forms of half integral weight, we can construct a quasimodular analog of Shimura isomorphisms and Shintani liftings. There are certain differential operators related to Jacobi-like forms and quasimodular forms. We can consider differential operators on Jacobi-like forms, which may be called radial heat operators. On the other hand, quasimodular forms determine differential operators which carry modular forms to modular forms. Such differential operators can be used to obtain another interpretation of Rankin–Cohen brackets for modular forms. This book is organized as follows. In Chapter 1 we consider formal power series and pseudodifferential operators over the ring of holomorphic functions on H and introduce SL(2, R)-actions on those objects. We look at two such actions on formal power series and one on pseudodifferential operators. Given a discrete subgroup Γ of SL(2, R), the Γ -invariant objects for these actions are Jacobi-like forms, modular series, and automorphic pseudodifferential operators. We discuss mutual correspondences among these objects. In Chapter 2 we discuss the interpretation of Jacobi-like forms and automorphic pseudodifferential operators as sections of vector bundles. We also study Poincar´e series and Rankin–Cohen brackets for Jacobi-like forms and automorphic pseudodifferential operators. Hecke operators for Jacobi-like forms, modular series, automorphic pseudodifferential operators, and their connections with certain linear ordinary differential equations are discussed in Chapter 3. Chapter 4 is concerned with various Lie algebra structures, and Chapter 5 is about heat operators. Relations between the cohomology of the discrete group Γ and Jacobi-like forms are the subject of Chapter 6, and quasimodular forms and quasimodular polynomials are discussed in Chapter 7. In Chapter 8 we construct liftings of quasimodular forms to Jacobi-like forms and discuss some of their applications including Lie brackets and Rankin–Cohen brackets for quasimodular forms. The correspondence between quasimodular polynomials and modular polynomials are described in Chapter 9, which is used to construct Poincar´e series for quasimodular forms. In Chapter 10 we study

Introduction

5

connections between quasimodular forms and certain vector-valued modular forms associated to symmetric tensor representations. We discuss in Chapter 11 certain differential operators on modular forms that can be constructed using quasimodular forms. In Chapter 12 we extend some of the results in the previous chapters to the case of half-integral weights, and use these to obtain the quasimodular analogs of Shimura correspondences and Shintani liftings. Projective structures on Riemann surfaces and their connections with pseudodifferential operators are discussed in Chapter 13. Finally in Chapter 14, we present a survey of various applications of quasimodular forms including the pioneering work on covers of elliptic curves by Dijkgraaf, Kaneko and Zagier as well as some of the work of Okounkov and his collaborators.

Chapter 1

Formal Power Series and Pseudodifferential Operators

Jacobi-like forms are formal power series, whose coefficients are holomorphic functions on the Poincar´e upper half plane, satisfying a certain transformation formula under the operation of a discrete subgroup Γ of SL(2, R). Pseudodifferential operators are formal Laurent series in the formal inverse ∂ −1 of the derivative operator ∂ whose coefficients are also holomorphic functions on the Poincar´e upper half plane. There is a natural action of the discrete group Γ on the space of pseudodifferential operators, and the invariant elements under this action are automorphic pseudodifferential operators. Cohen, Manin and Zagier [31] established correspondences among Jacobi-like forms, automorphic pseudodifferential operators and certain sequences of modular forms. In this chapter we discuss such correspondences and their applications by introducing formal power series, called modular series, whose coefficients are modular forms of certain type.

1.1 Correspondences of Power Series In this section we introduce automorphisms of the space of formal power series over the ring of holomorphic functions on the upper half plane. Let H be the Poincar´e upper half plane, and let R be the ring of holomorphic functions f : H → C that are polynomially bounded, meaning that there is a constant N > 0 such that f (z) = O((1 + |z|2 )/y N )

(1.1)

for z ∈ H with y = Im(z) as y → ∞ and y → 0 (cf. [30, Section 5.1]). We denote by R[[X]] the complex algebra of formal power series in X with coefficients in R. If δ is a nonnegative integer, we set R[[X]]δ = X δ R[[X]],

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_1

(1.2)

7

8

1 Formal Power Series and Pseudodifferential Operators

so that an element Φ(z, X) ∈ R[[X]]δ can be written in the form Φ(z, X) =

∞ 

φk (z)X k+δ

(1.3)

k=0

with φk ∈ R for each k ≥ 0. Given such Φ(z, X) ∈ R[[X]]δ and λ ∈ Z with λ > −2δ, we consider two other formal power series (Λλ,δ Φ)(z, X), (Ξλ,δ Φ)(z, X) ∈ R[[X]]δ defined by (Λλ,δ Φ)(z, X) =

∞ 

r+δ φΛ , r (z)X

(1.4)

r+δ φΞ , r (z)X

(1.5)

r=0

(Ξλ,δ Φ)(z, X) =

∞  r=0

where φΛ k =

k  r=0

1 (r) φ , r!(2k + 2δ + λ − r − 1)! k−r

φΞ k = (2k + 2δ + λ − 1)

k 

(−1)r

r=0

(2k + 2δ + λ − r − 2)! (r) φk−r r!

(1.6)

(1.7)

(r)

for each k ≥ 0; here φk−r denotes the r-th derivative of φk−r . Lemma 1.1 Let u and v be positive integers with u ≤ v. Then we have u 

(−1)r

r=0

   v−r u = 0. u−1 r

Proof. Given positive integers u and v with u ≤ v and a real variable x, from the relation   du−1 v−r v − r v−r−u+1 x x = (u − 1)! u−1 dxu−1 we obtain





v−r u−1

 1 du−1 v−r  = x  (u − 1)! dxu−1 x=1

for each nonnegative integer r with r ≤ u. Thus we see that u  r=0

(−1)r

   v−r u u−1 r

 u     (−1)u du−1  u u+r v−r  = (−1) x  (u − 1)! dxu−1 r=0 r x=1

1.1 Correspondences of Power Series

9

  u     u (−1)u du−1 u−r  v−u (−x) x ·  u−1 r (u − 1)! dx x=1 r=0    (−1)u du−1  v−u = (1 − x)u  x (u − 1)! dxu−1 x=1  u−1    (−1)u  u − 1 du−1−μ v−u dμ u  = (x ) · (1 − x) = 0;  u−1−μ μ μ (u − 1)! μ=0 dx dx x=1 =



hence the lemma follows.

Proposition 1.2 The maps Λλ,δ , Ξλ,δ : R[[X]]δ → R[[X]]δ given by (1.4) and (1.5) are complex linear isomorphisms with (Ξλ,δ )−1 = Λλ,δ

(1.8)

for all λ, δ ∈ Z with λ > −2δ and δ ≥ 0. Proof. Given δ ≥ 0 and λ > −2δ, we first consider a power series Φ(z, X) and its image (Λλ,δ Φ)(z, X) under Λλ,δ as in (1.3) and (1.4), respectively. Then from (1.6) we obtain φΛ k−r =

k−r  =0

1 () φ !(2k − 2r + 2δ + λ −  − 1)! k−−r

for 0 ≤ r ≤ k. Thus, if we set ((Ξλ,δ ◦ Λλ,δ )Φ)(z, X) =

∞ 

φ r (z)X r+δ ,

r=0

from (1.6) and (1.7) we see that φ k 2k + 2δ + λ − 1 k  (−1)r (r) = (2k + 2δ + λ − r − 2)!(φΛ k−r ) r! r=0 =

k k−r   (−1)r (2k + 2δ + λ − r − 2)! (+r) φ r!!(2k − 2r + 2δ + λ −  − 1)! k−−r r=0 =0

=

k k   r=0 u=r

=

u k   u=0 r=0

(−1)r (2k + 2δ + λ − r − 2)! (u) φ r!(u − r)!(2k − r + 2δ + λ − u − 1)! k−u (−1)r (2k + 2δ + λ − r − 2)! (u) φ r!(u − r)!(2k − r + 2δ + λ − u − 1)! k−u

(1.9)

10

1 Formal Power Series and Pseudodifferential Operators

=

(2k + 2δ + λ − 2)! φk (2k + 2δ + λ − 1)!

u k  1 (u)  (−1)r u!(2k + 2δ + λ − r − 2)! φk−u + u r!(u − r)!(2k − r + 2δ + λ − u − 1)!(u − 1)! u=1 r=0  u   k  1 (u)  u 2k + 2δ + λ − r − 2 φk + φk−u = u−1 r (2k + 2δ + λ − 1) u=1 u r=0

for each k ≥ 0. Since λ > −2δ, if 1 ≤ u ≤ k, we have 2k + 2δ + λ − r − 2 ≥ u. Thus, using Lemma 1.1, we see that  u    u 2k + 2δ + λ − r − 2 r=0

u−1

r

=0

for 1 ≤ u ≤ k. Hence (1.9) can be written in the form φ k φk = , (2k + 2δ + λ − 1) (2k + 2δ + λ − 1) and therefore we obtain ((Ξλ,δ ◦ Λλ,δ )Φ)(z, X) = Φ(z, X). We now assume that (Ξλ,δ Φ)(z, X) is as in (1.5) and that ((Λλ,δ ◦ Ξλ,δ )Φ)(z, X) =

∞ 

φr (z)X r+δ .

r=0

Thus, in particular, (1.7) is valid for k ≥ 0. We shall verify that φk = φk using induction by first assuming, given a nonnegative integer n, that φk = φk =

k  =0

1 (φΞ )() !(2k + 2δ + λ −  − 1)! k−

holds for each k ≤ n. Then from (1.7) we obtain φΞ n+1 = (2n + 2δ + λ + 1)

n+1 

(−1)r

r=0

= (2n + 2δ + λ + 1)!φn+1

(2n + 2δ + λ − r)! (r) φn+1−r r!

(1.10)

1.1 Correspondences of Power Series

− (2n + 2δ + λ + 1)

11 n 

(−1)r

r=0

(2n + 2δ + λ − r − 1)! (r+1) φn−r , (r + 1)!

which can be written in the form φΞ n+1 (2n + 2δ + λ + 1)!

φn+1 =

(1.11)

 (2n + 2δ + λ − r − 1)! (r+1) 1 φn−r . (−1)r (2n + 2δ + λ)! r=0 (r + 1)! n

+

Since (1.10) holds for k ≤ n, the sum on the right hand side of (1.11) can be written as n 

(−1)r

r=0

= = =

(2n + 2δ + λ − r − 1)! (r+1) φn−r (r + 1)!

n n−r   r=0 =0 w n   w=0 r=0 n 

(−1)r (2n + 2δ + λ − r − 1)! (φΞ )(+r+1) !(r + 1)!(2n − 2r + 2δ + λ −  − 1)! n−−r (−1)r (2n + 2δ + λ − r − 1)! (φΞ )(w+1) (w − r)!(r + 1)!(2n − r + 2δ + λ − w − 1)! n−w

1  (−1)r (w + 1)! w + 1 r=0 (w − r)!(r + 1)! w=0 w

(2n + 2δ + λ − r − 1)! (φΞ )(w+1) w!(2n − r + 2δ + λ − w − 1)! n−w    n w  2n + 2δ + λ − r − 1 1  r w+1 (w+1) (φΞ = (−1) . n−w ) w r + 1 w + 1 w=0 r=0 ×

On the other hand, using Lemma 1.1, we have   2n + 2δ + λ − r − 1 (−1) w r=0      w+1  2n + 2δ + λ 2n + 2δ + λ − r r−1 w + 1 . = = (−1) w w r r=1

w 



r

w+1 r+1

Thus (1.11) can now be written in the form φn+1 =

φΞ n+1 (2n + 2δ + λ + 1)!

  n  2n + 2δ + λ 1 1 (w+1) (φΞ + n−w ) w (2n + 2δ + λ)! w=0 w + 1

12

1 Formal Power Series and Pseudodifferential Operators

=

φΞ n+1 (2n + 2δ + λ + 1)!

n+1  1 2n + 2δ + λ 1 (w) (φΞ + n+1−w ) w−1 (2n + 2δ + λ)! w=1 w

=

n+1 

1 (w) (φΞ = φn+1 . n+1−w ) w!(2n + 2δ + λ − w + 1)! w=0

Hence we see that φk = φk holds for each positive integer n by induction, and therefore it follows that ((Λλ,δ ◦ Ξλ,δ )Φ)(z, X) = Φ(z, X), 

which completes the proof of the proposition.

1.2 Actions of SL(2, R) on Power Series In this section we study two types of actions of SL(2, R) on the space of formal power series with respect to which the automorphisms Λλ,δ and Ξλ,δ considered in Section 1.1 are equivariant. The group SL(2, R) acts on the Poincar´e upper half plane H as usual by linear fractional transformations. Thus we may write γz = for all z ∈ H and γ =

a b c d

az + b cz + d

∈ SL(2, R). For the same z and γ, we set

J(γ, z) = cz + d,

K(γ, z) = cJ(γ, z)−1 =

c . cz + d

(1.12)

The map J : SL(2, R) × H → C determined by the first formula is a wellknown automorphy factor satisfying the cocycle condition J(γγ  , z) = J(γ, γ  z)J(γ  , z)

(1.13)

for γ, γ  ∈ SL(2, R) and z ∈ H. Lemma 1.3 The map K : SL(2, R) × H → C defined by the second formula in (1.12) satisfies K(γ, γ  z) = J(γ  , z)2 (K(γγ  , z) − K(γ  , z))

(1.14)

for all z ∈ H and γ, γ  ∈ SL(2, R).      Proof. Let γ = ac db and γ  = ac db  be elements of SL(2, R), so that

1.2 Actions of SL(2, R) on Power Series

γγ  =



13

 

aa + bc ab + bd . ca + dc cb + dd

Then, given z ∈ H, we have c ca + dc −     + dc )z + (cb + dd ) c z + d c(a d − b c ) =   ((ca + dc )z + (cb + dd ))(c z + d ) c , = J(γγ  , z)J(γ  , z)

K(γγ  , z) − K(γ  , z) =

(ca

where we used the relation a d − b c = 1. On the other hand, we have c(c z + d ) c(a z + b ) + d(c z + d ) cJ(γ  , z) = = J(γ  , z)2 (K(γγ  , z) − K(γ  , z)); J(γγ  , z)

K(γ, γ  z) =



hence the lemma follows.

We note that the identity (1.14) can also be obtained formally from the relation d K(γ, z) = ln J(γ, z). dz Indeed, using (1.13), we have K(γγ  , z) =

d d d ln J(γγ  , z) = ln J(γ, γ  z) + ln J(γ  , z) dz dz dz d(γ  z) d d = ln J(γ, γ  z) + ln J(γ  , z) dz d(γ  z) dz = J(γ  , z)−2 K(γ, γ  z) + K(γ  , z)

for all z ∈ H and γ, γ  ∈ SL(2, R). Since J(1, z) = 1 and K(1, z) = 0, from (1.13) and (1.14), we see that J(γ −1 , γz) = J(γ, z)−1 ,

K(γ −1 , γz) = −J(γ, z)2 K(γ, z)

(1.15)

for γ ∈ SL(2, R) and z ∈ H. Given a function f ∈ R, a formal power series Φ(z, X) ∈ R[[X]], an integer λ, and an element γ ∈ SL(2, R), we set (f |λ γ)(z) = J(γ, z)−λ f (z)

(1.16)

(Φ |Jλ γ)(z, X) = J(γ, z)−λ e−K(γ,z)X Φ(γz, J(γ, z)−2 X), (Φ

|M λ

γ)(z, X) = J(γ, z)

−λ

Φ(γz, J(γ, z)

−2

X)

(1.17) (1.18)

14

1 Formal Power Series and Pseudodifferential Operators

for z ∈ H. Using (1.13), we see easily that f |λ (γγ  ) = (f |λ γ) |λ γ  for all γ, γ  ∈ SL(2, R). Similar identities for the other operations also hold as shown in the following lemma. Lemma 1.4 If λ ∈ Z and Φ(z, X) ∈ R[[X]], we have Φ |Jλ (γγ  ) = (Φ |Jλ γ) |Jλ γ  ,

 M M  Φ |M λ (γγ ) = (Φ |λ γ) |λ γ

(1.19)

for all γ, γ  ∈ SL(2, R). Proof. Given elements λ ∈ Z, Φ(z, X) ∈ R[[X]] and γ, γ  ∈ SL(2, R), using (1.17), we obtain (Φ |Jλ γ |Jλ γ  )(z, X) = J(γ  , z)−λ e−K(γ × e−K(γ,γ





z)J(γ ,z)



,z)X

−2

X

= J(γγ  , z)−λ e−(K(γ



J(γ, γ  z)−λ

Φ(γγ  z, J(γ, γ  z)−2 J(γ  , z)−2 X)

,z)+K(γ,γ  z)J(γ  ,z)−2 )X

× Φ(γγ  z, J(γγ  , z)−2 X) = (Φ |Jλ (γγ  ))(z, X), which verifies the first relation in (1.19). The second relation follows easily by suppressing the exponential factors from the above computation.  From Lemma 1.4 it follows that the operations |Jλ and |M λ determine right actions of SL(2, R) on R[[X]]. Before we state the next proposition, we note   that the binomial symbol nk can be considered for any integer n by setting     n n(n − 1) · · · (n − k + 1) n = = 1, (1.20) k 0 k! for each positive integer k. Proposition 1.5 Let k, m ∈ Z with m > 0. Given a holomorphic function f ∈ R on H and an element γ ∈ SL(2, R), we have   m m−r  dm m−r m! k + m − 1 K(γ, z) (f | γ)(z) = (−1) f (r) (γz) (1.21) k k+2r m − r dz m r! J(γ, z) r=0 for all z ∈ H. Proof. Given m > 0 and γ ∈ SL(2, R), using the relations d J(γ, z) = J(γ, z)K(γ, z), dz

d K(γ, z) = −K(γ, z)2 , dz

(1.22)

1.2 Actions of SL(2, R) on Power Series

15

we have   d K(γ, z)m−r = −(k + 2r)J(γ, z)−k−2r−1 (J(γ, z)K(γ, z))K(γ, z)m−r dz J(γ, z)k+2r + (m − r)J(γ, z)−k−2r K(γ, z)m−r−1 (−K(γ, z)2 ) = −(k + m + r) From this and the fact that d dz



d dz (γz)

K(γ, z)m−r (r) f (γz) J(γ, z)k+2r



K(γ, z)m+1−r . J(γ, z)k+2r

= J(γ, z)−2 , for f ∈ R, we see that K(γ, z)m+1−r (r) f (γz) J(γ, z)k+2r K(γ, z)m−r (r+1) f (γz). + J(γ, z)k+2r+2

= −(k + m + r)

Thus, assuming that (1.21) is true for m > 0, we obtain dm+1 (f |k γ)(z) dz m+1   m  K(γ, z)m+1−r (r) m+1−r m! k + m − 1 (k + m + r) = (−1) f (γz) m−r r! J(γ, z)k+2r r=0   m  K(γ, z)m−r (r+1) m−r m! k + m − 1 + (−1) f (γz) m−r r! J(γ, z)k+2r+2 r=0   m  K(γ, z)m+1−r (r) m! k + m − 1 (k + m + r) = (−1)m+1−r f (γz) m−r r! J(γ, z)k+2r r=0   m+1  k + m − 1 K(γ, z)m+1−r (r) m! m+1−r (−1) f (γz). + (r − 1)! m + 1 − r J(γ, z)k+2r r=1 However, for 1 ≤ r ≤ m we have     m! k+m−1 m! k + m − 1 (k + m + r) + m−r r! (r − 1)! m + 1 − r     m! k+m−1 m! k + m − 1 (k + m) + = m−r m−r r! (r − 1)!   k+m−1 m! + (r − 1)! m + 1 − r     k+m m! k+m−1 m! (m + 1 − r) + = m−r r! m + 1 − r (r − 1)!   k+m−1 m! + (r − 1)! m + 1 − r

16

1 Formal Power Series and Pseudodifferential Operators

 

 k+m m! k+m (m + 1)! − = m+1−r r! (r − 1)! m + 1 − r     k+m−1 k+m−1 − − m+1−r m−r   (m + 1)! k+m = . m+1−r r! 

Thus we obtain   dm+1 k+m−1 K(γ, z)m+1 m+1 (k + m) (f | γ)(z) = (−1) m! f (γz) k m dz m+1 J(γ, z)k 1 f (m+1) (γz) + J(γ, z)k+2m+2   m  k+m (m + 1)! + (−1)m+1−r m+1−r r! r=1 K(γ, z)m+1−r (r) f (γz) J(γ, z)k+2r   m+1  k+m m+1−r (m + 1)! (−1) = m+1−r r! r=0 ×

×

K(γ, z)m+1−r (r) f (γz) , J(γ, z)k+2r

which is the relation (1.21) with m replaced by m + 1; hence the proposition follows by induction.  Theorem 1.6 The isomorphisms Λλ,δ , Ξλ,δ : R[[X]]δ → R[[X]]δ in Proposition 1.2 are SL(2, R)-equivariant under the operations |Jλ and |M λ given by (1.17) and (1.18), that is, ((Λλ,δ Φ) |Jλ γ)(z, X) = ((Λλ,δ (Φ |M λ γ))(z, X),

(1.23)

J ((Ξλ,δ Φ) |M λ γ)(z, X) = ((Ξλ,δ (Φ |λ γ))(z, X)

for each γ ∈ SL(2, R) and Φ(z, X) ∈ R[[X]]δ . Proof. Since (1.8) holds, it suffices to verify (1.23). Given γ ∈ SL(2, R) and Φ(z, X) ∈ R[[X]]δ as in (1.3), using (1.4) and (1.17), we have ((Λλ,δ Φ) |Jλ γ)(z, X) = J(γ, z)−λ

 ∞

 (−1)r K(γ, z)r X r r! r=0   ∞ −2−2δ +δ φΛ (γz)J(γ, z) X ×  =0

1.2 Actions of SL(2, R) on Power Series

=

∞ ∞   (−1)r

r!

r=0 =0

=

k ∞   (−1)r

r!

k=0 r=0

17

+r+δ J(γ, z)−λ−2−2δ K(γ, z)r φΛ  (γz)X

k+δ J(γ, z)−λ−2k+2r−2δ K(γ, z)r φΛ . k−r (γz)X

From this and (1.6) we obtain ∞ 

((Λλ,δ Φ) |Jλ γ)(z, X) =

ξk (z)X k+δ ,

k=0

where ξk (z) =

k  (−1)r r=0

=

r!

J(γ, z)−λ−2k+2r−2δ K(γ, z)r φΛ k−r (γz)

k k−r   (−1)r J(γ, z)−λ−2k+2r−2δ K(γ, z)r φ() k−r− (γz)

r!!(2k − 2r + 2δ + λ −  − 1)!

r=0 =0

(1.24)

.

On the other hand, from (1.18) we see that −λ (Φ |M λ γ)(z, X) = J(γ, z)

∞ 

φk (γz)J(γ, z)−2k−2δ X k+δ

k=0

=

∞ 

(φk |λ+2k+2δ γ)(z)X k+δ ;

k=0

hence we have (Λλ,δ (Φ |M λ γ))(z, X) =

∞ 

ηk (z)X k+δ ,

k=0

where ηk =

k  r=0

1 (φk−r |λ+2k−2r+2δ γ)(r) r!(λ + 2k + 2δ − r − 1)!

for k ≥ 0. However, from (1.21) we obtain (φk−r |λ+2k−2r+2δ γ)

(r)

(z) =

r  =0

(−1)

 λ + 2k − r + 2δ − 1 r− !

r− r!

×



K(γ, z)r− () φ (γz). J(γ, z)λ+2k−2r+2δ+2 k−r

18

1 Formal Power Series and Pseudodifferential Operators

Thus we have ηk (z) =

r k   r=0 =0

(−1)r− !(r − )!(2k − 2r + 2δ + λ +  − 1)! ×

=

k k   =0 r=

K(γ, z)r− () φ (γz) J(γ, z)λ+2k−2r+2δ+2 k−r

(−1)r− !(r − )!(λ + 2k − 2r + 2δ +  − 1)! ×

K(γ, z)r− () φ (γz). J(γ, z)λ+2k−2r+2δ+2 k−r

Changing the index r to r + , we see that ηk (z) =

k− k   =0 r=0

(−1)r !r!(λ + 2k − 2r + 2δ −  − 1)! ×

K(γ, z)r () φ (γz). J(γ, z)λ+2k−2r+2δ k−−r

Comparing this with (1.24), we have ξ k = ηk for each k ≥ 0, and therefore (1.23) follows.



1.3 Jacobi-like Forms and Modular Series In this section we review some basic properties of Jacobi-like forms and modular series, which are formal power series invariant under a discrete subgroup of SL(2, R) with respect to the actions considered in Section 1.2. In particular, we show that the automorphisms of the space of power series considered in Section 1.1 induce isomorphisms between the space of Jacobi-like forms and that of modular series (cf. [25]). Let Γ be a discrete subgroup of SL(2, R), and let |λ , |Jλ and |M λ with λ ∈ Z be the operations defined by (1.16), (1.17) and (1.18), respectively. Definition 1.7 (i) A holomorphic function f ∈ R is a modular form of weight λ for Γ if it satisfies f |λ γ = f (1.25) for all γ ∈ Γ . (ii) A formal power series Φ(z, X) ∈ R[[X]] is a Jacobi-like form of weight λ for Γ if it satisfies

1.3 Jacobi-like Forms and Modular Series

(Φ |Jλ γ)(z, X) = Φ(z, X)

19

(1.26)

for all z ∈ H and γ ∈ Γ . (iii) A formal power series Φ(z, X) ∈ R[[X]] is a modular series for Γ of weight λ if it satisfies (Φ |M λ γ)(z, X) = Φ(z, X)

(1.27)

for all z ∈ H and γ ∈ Γ . We denote by Mλ (Γ ) the space of modular forms of weight λ for Γ and by Jλ (Γ ) and Mλ (Γ ) the spaces of Jacobi-like forms and modular series, respectively, of weight λ for Γ . Remark 1.8 (i) We note that, because of our definition of R, a modular form f ∈ Mλ (Γ ) is polynomially bounded and therefore satisfies (1.1) in addition to the condition (1.25) (see [30, Section 5.1]). (ii) Jacobi-like forms generalize Jacobi forms in the following sense. Let ψ(z, X) ∈ R[[X]] be a Jacobi-like form belonging to Jλ (Γ ), so that it satisfies (1.26). We set X = ζ 2 , and consider the formal power series Ψ (z, ζ) ∈ R[[ζ 2 ]] in ζ 2 over R given by Ψ (z, ζ) = ψ(z, 2πiζ 2 ). Then, using (1.26), we obtain Ψ (γz, J(γ, z)−1 ζ) = ψ(γz, J(γ, z)−2 2πiζ 2 ) = J(γ, z)λ exp(2πiζ 2 K(γ, z))ψ(z, 2πiζ 2 ) = J(γ, z)λ exp(2πiζ 2 K(γ, z))Ψ (z, ζ). Thus we see that Ψ (z, ζ) satisfies one of the two transformation formulas which define Jacobi forms for Γ of weight λ and index 1 (see [37, Chapter I]). (iii) Modular series may be regarded as special types of Jacobi-like forms in some sense as follows. In addition to the weight, we may also introduce an index to a Jacobi-like form, so that a Jacobi-like form of weight λ and index μ ∈ R is defined by using the operation (Φ |Jλ,μ γ)(z, X) = J(γ, z)−λ e−μK(γ,z)X Φ(γz, J(γ, z)−2 X) for γ ∈ Γ instead of (1.26). Then Jacobi-like forms of index 1 are simply Jacobi-like forms in the sense of Definition 1.7. On the other hand, modular series for Γ of weight λ are Jacobi-like forms of weight λ and index 0. Given a nonnegative integer δ, we denote by Jλ (Γ )δ and Mλ (Γ )δ the subspaces of Jλ (Γ ) and Mλ (Γ ), respectively, defined by Jλ (Γ )δ = Jλ (Γ ) ∩ R[[X]]δ ,

Mλ (Γ )δ = Mλ (Γ ) ∩ R[[X]]δ ,

(1.28)

20

1 Formal Power Series and Pseudodifferential Operators

where R[[X]]δ is as in (1.2). Lemma 1.9 The formal power series Φ(z, X) in (1.3) is a modular series belonging to Mλ (Γ )δ if and only if φk ∈ M2k+2δ+λ (Γ ) for each k ≥ 0. Proof. Using (1.18), (1.27) and Definition 1.7(iii), the formal power series Φ(z, X) in (1.3) belongs to Mλ (Γ )δ if and only if ∞ 

φk (z)X k+δ = Φ(z, X) = (Φ |M λ γ)(z, X)

k=0

= =

∞  k=0 ∞ 

J(γ, z)−λ φk (γz)(J(γ, z)−2 X)k+δ J(γ, z)−2k−2δ−λ φk (γz)X k+δ

k=0

for all z ∈ H and γ ∈ Γ , which is equivalent to the condition that φk (z) = J(γ, z)−2k−2δ−λ φk (γz) = (φk |2k+2δ+λ γ)(z) for each k ≥ 0. Thus φk belongs to M2k+2δ+λ (Γ ), which proves the lemma.  Proposition 1.10 The automorphisms Λλ,δ , Ξλ,δ : R[[X]]δ → R[[X]]δ in Proposition 1.2 induce isomorphisms Λλ,δ : Mλ (Γ )δ → Jλ (Γ )δ ,

(1.29)

Ξλ,δ : Jλ (Γ )δ → Mλ (Γ )δ

(1.30)

for λ, δ ∈ Z with δ ≥ 0 and λ > −2δ. Proof. From Theorem 1.6 and Definition 1.7 we see that Λλ,δ (Mλ (Γ )δ ) ⊂ Jλ (Γ )δ , hence the proposition follows.

Ξλ,δ (Jλ (Γ )δ ) ⊂ Mλ (Γ )δ ; 

The next theorem is a slightly modified version of Proposition 2 in [31] by Cohen, Manin and Zagier.

1.3 Jacobi-like Forms and Modular Series

21

Theorem 1.11 Let δ and λ be integers with δ ≥ 0 and λ > −2δ, and consider the formal power series Φ(z, X) =

∞ 

φk (z)X k+δ ∈ R[[X]]δ .

(1.31)

k=0

Then the following conditions are equivalent: (i) The formal power series Φ(z, X) is a Jacobi-like form belonging to Jλ (Γ )δ . (ii) The coefficients of Φ(z, X) satisfy (φk |2k+2δ+λ γ)(z) =

k  1 K(γ, z)r φk−r (z) r! r=0

(1.32)

for all k ≥ 0 and γ ∈ Γ . (iii) Each coefficient of Φ(z, X) can be written in the form φk =

k  r=0

1 (r) f r!(2k + 2δ + λ − r − 1)! k−r

(1.33)

for k ≥ 0, where f is a modular form belonging to M2+λ (Γ ) for each  ≥ 0. (iv) The function hk given by hk = (2k + 2δ + λ − 1)

k 

(−1)r

r=0

(2k + 2δ + λ − r − 2)! (r) φk−r r!

(1.34)

is a modular form belonging to M2k+2δ+λ (Γ ) for each k ≥ 0. Proof. Given z ∈ H and γ ∈ Γ , using (1.17) and (1.26), we see that Φ(z, X) belongs to Jλ (Γ )δ if and only if ∞  k=0

φk (γz)J(γ, z)−2k−2δ−λ X k+δ =

   ∞ ∞ 1 K(γ, z)r X r φ (z)X +δ r! r=0 =0

∞ ∞   1 K(γ, z)r φ (z)X r++δ = r! r=0 =0

k ∞   1 K(γ, z)r φk−r (z)X k+δ . = r! r=0 k=0

Hence we obtain the equivalence of (i) and (ii) by comparing the coefficients of X k+δ . The equivalence of (iii) and (iv) follows from Proposition 1.2 by setting fk = hk . If Φ(z, X) belongs to Jλ (Γ )δ and if (Ξλ,δ Φ)(z, X) is as in (1.5), then (Ξλ,δ Φ)(z, X) ∈ Mλ (Γ ) by Proposition 1.10; hence from Lemma

22

1 Formal Power Series and Pseudodifferential Operators

1.9 we see that φΞ  ∈ Mλ+2+2δ (Γ ) for each  ≥ 0. Thus we obtain (iv) by setting hk = φΞ k for k ≥ 0. On the other hand, if the sequence {h }≥0 is as in (1.34), then by Lemma 1.9 the power series F (z, X) =

∞ 

hk (z)X k+δ

k=0

is a modular series belonging to Mλ (Γ )δ and Φ(z, X) = (Λλ,δ F )(z, X), which implies that Φ(z, X) ∈ Jλ (Γ )δ . Thus we obtain the equivalence of (i) and (iv), and therefore the proof of the theorem is complete.  If Φ(z, X) in (1.31) is a Jacobi-like form belonging to Jλ (Γ )δ and if the sequence {hk }k≥0 is as in (1.34), we note that Ξλ,δ (Φ(z, X)) =

∞ 

hk (z)X k+δ ,

k=0

Λλ,δ

 ∞

 hk (z)X k+δ

= Φ(z, X),

k=0

where Ξλ,δ : Jλ (Γ )δ → Mλ (Γ )δ and Λλ,δ : Mλ (Γ )δ → Jλ (Γ )δ are the isomorphisms in (1.29) and (1.30). Example 1.12 (Cohen–Kuznetsov liftings) Given a modular form f ∈ M2w+λ (Γ ) with w ≥ 0, we consider the sequence {g }∞ =0 of functions on H given by f if  = w − δ; g = 0 if  = w − δ. Then the formal power series Ff (z, X) =

∞ 

g (z)X +δ

=0

is a modular series belonging to Mλ (Γ )δ . If the corresponding Jacobi-like form is given by Λλ,δ (Ff (z, X)) =

∞  =0

from (1.6) we see that

φ (z)X +δ ∈ Jλ (Γ )δ ,

1.3 Jacobi-like Forms and Modular Series

φ =

  r=0

=

23

1 (r) g r!(2 + 2δ + λ − r − 1)! −r

f (−w+δ) ( − w + δ)!( + w + δ + λ − 1)!

assuming that p! = 0 for p < 0. Hence we obtain a Jacobi-like form Φf (z, X) = =

∞  =w−δ ∞  =0

f (−w+δ) X +δ ( − w + δ)!( + w + δ + λ − 1)!

(1.35)

f () X +w . !( + 2w + λ − 1)!

Thus the map f → Φf determines a lifting Lw λ,δ : M2w+λ (Γ ) → Jλ (Γ )δ

(1.36)

of modular forms to Jacobi-like forms known as the Cohen–Kuznetsov lifting (see [31]). Given δ ≥ 0, we consider the symbol map Sδ : R[[X]]δ → R

(1.37)

for formal power series defined by Sδ F = f0

∞ for F (z, X) = k=0 fk (z)X k+δ ∈ R[[X]]δ . Then there is a short exact sequence of the form S

δ R → 0, 0 → R[[X]]δ+1 → R[[X]]δ −→

(1.38)

where the second arrow is the natural inclusion map. If F (z, X) is a Jacobilike form belonging to Jλ (Γ )δ , then we see easily that Sδ F ∈ M2δ+λ (Γ ); hence the exact sequence (1.38) induces the short exact sequence S

δ 0 → Jλ (Γ )δ+1 → Jλ (Γ )δ −→ M2δ+λ (Γ ) → 0.

The formula (1.35) for w = δ can be written as (Lδλ,δ f )(z, X) =

∞  =0

f () (z) X +δ !( + 2δ + λ − 1)!

(1.39)

24

1 Formal Power Series and Pseudodifferential Operators

for f ∈ M2δ+λ (Γ ), so that (2δ + λ − 1)!Sδ ◦ Lδλ,δ = 1M2δ+λ (Γ ) ; hence it follows that the short exact sequence (1.39) splits.

1.4 Pseudodifferential Operators Modular series and Jacobi-like forms for a discrete group Γ ⊂ SL(2, R) correspond to certain Γ -invariant pseudodifferential operators, as was discussed in [31] by Cohen, Manin and Zagier. We discuss pseudodifferential operators and such correspondences in this section. If R is the ring of holomorphic functions on H as before, a pseudodifferential operator over R is a formal Laurent series in the formal inverse ∂ −1 of ∂ = d/dz with coefficients in R of the form u 

Ψ (z) =

hk (z)∂ k

k=−∞

with u ∈ Z and hk ∈ R for each k ≤ u. We denote by Ψ D(R) the space of all pseudodifferential operators over R. Using the relation  d(γz) −1 d d = = J(γ, z)2 ∂ d(γz) dz dz for γ ∈ SL(2, R) and z ∈ H, we see that the group SL(2, C) acts on Ψ D(R) on the right by (Ψ ◦ γ)(z) = Ψ (γz) =

u 

hk (γz)(J(γ, z)2 ∂)k

(1.40)

k=−∞

for all γ ∈ SL(2, R), where J(γ, z) is as in (1.12). If α is an integer, we denote by Ψ D(R)α the subspace of Ψ D(R) consisting of the pseudodifferential operators of the form ∞  ψk (z)∂ α−k (1.41) k=0

with ψk ∈ R for all k ≥ 0. We can consider the symbol map Sα∂ : Ψ D(R)α → R

(1.42)

for pseudodifferential operators which sends the pseudodifferential operator in (1.41) to ψ0 . Then there is a short exact sequence of the form

1.4 Pseudodifferential Operators

25 S∂

α 0 → Ψ D(R)α+1 → Ψ D(R)α −−→ R → 0,

(1.43)

where the second arrow is the natural inclusion map. Definition 1.13 Given a discrete subgroup Γ of SL(2, R), an automorphic pseudodifferential operator for Γ is a pseudodifferential operator that is invariant under the action of each element of Γ . We denote by Ψ D(R)Γ the subspace of Ψ D(R) consisting of all automorphic pseudodifferential operators for Γ , that is, Ψ D(R)Γ = {Ψ (z) ∈ Ψ D(R) | Ψ ◦ γ = Ψ for all γ ∈ Γ }, and set Ψ D(R)Γα = Ψ D(R)Γ ∩ Ψ D(R)α for α ∈ Z. Lemma 1.14 Given γ ∈ SL(2, R), we have 

J(γ, z) ∂ 2

m

   ∞  m m−1 J(γ, z)2m K(γ, z) ∂ m− = !  

(1.44)

=0

for each integer m. Proof. In order to use induction we assume that (1.44) holds for a positive integer m. Then we have 

J(γ, z) ∂ 2

m+1

   ∞  m m−1 J(γ, z)2m K(γ, z) = J(γ, z) !   =0   × (2m − )K(γ, z) + ∂ ∂ m−    ∞  m m−1 (2m − )J(γ, z)2m+2 K(γ, z)+1 ∂ m− = !   =0    ∞  m m−1 J(γ, z)2m+2 K(γ, z) ∂ m+1− , + !   2

=0

where we used the relations in (1.22). Using 2m −  = m + (m − ), the first of the previous two sums can be written as   m−1 J(γ, z)2m+2 K(γ, z) ∂ m+1− −1 =1    ∞  m−1 m J(γ, z)2m+2 K(γ, z) ∂ m+1− , + ( − 1)!(m −  + 1) −1 −1

∞ 



m ( − 1)!m −1

=1

26

1 Formal Power Series and Pseudodifferential Operators

where we shifted the index by 1. Using the identities       m m m−1 m , = ! ( − 1)!m  −1 −1 −1       m m−1 m−1 m , = ! ( − 1)!(m −  + 1) −1  −1 −1 we see that 

J(γ, z) ∂ 2

m+1

     

 ∞  m m−1 m−1 m + + = !   −1 −1 =1

× J(γ, z)2m+2 K(γ, z) ∂ m+1− + J(γ, z)2m+2 ∂ m+1    ∞  m+1 m J(γ, z)2m+2 K(γ, z) ∂ m+1− , = !   =0

which is simply (1.44) with m replaced by m+1, and therefore (1.44) holds for m ≥ 0. This can be easily extended to the case of negative m, and therefore the lemma follows.  We now introduce isomorphisms between the space of formal power series and that of pseudodifferential operators defined as follows. Given a formal power series ∞  φk (z)X k+δ ∈ R[[X]]δ (1.45) Φ(z, X) = k=0

and a pseudodifferential operator Ψ (z) =

∞ 

ψk (z)∂ −k−ε ∈ Ψ D(R)−ε

k=0

with δ, ε > 0, we set ∂ Φ)(z) = (Iξ,δ

∞ 

Ck+δ+ξ φk (z)∂ −k−δ−ξ ,

(1.46)

k=0

X Ψ )(z, X) = (Iξ,ε

∞ 

−1 Ck+ε ψk (z)X k+ε−ξ

(1.47)

k=0

for each nonnegative integer ξ ≤ ε, where Cν with ν > 0 denotes the integer Cν = (−1)ν ν!(ν − 1)!.

(1.48)

Then it can be easily seen that X ∂ ◦ Iξ,δ )Φ = Φ, (Iξ,δ+ξ

∂ X (Iξ,ε−ξ ◦ Iξ,ε )Ψ = Ψ,

(1.49)

1.4 Pseudodifferential Operators

27

and therefore the resulting maps ∂ : R[[X]]δ → Ψ D(R)−δ−ξ , Iξ,δ

X Iξ,ε : Ψ D(R)−ε → R[[X]]ε−ξ

(1.50)

X −1 ∂ are complex linear isomorphisms with (Iξ,ε ) = Iξ,ε−ξ .

Proposition 1.15 Let Φ(z, X) ∈ R[[X]]δ and Ψ (z) ∈ Ψ D(R)−ε with δ and ε being positive integers. If 0 ≤ ξ ≤ ε, then we have ∂ ∂ Φ) ◦ γ = Iξ,δ (Φ |J2ξ γ), (Iξ,δ

X X (Iξ,ε Ψ ) |J2ξ γ = Iξ,ε (Ψ ◦ γ)

(1.51)

for all γ ∈ SL(2, C). Proof. Given γ ∈ SL(2, C), z ∈ H and Φ(z, X) ∈ R[[X]]δ as in (1.45), using (1.40) and (1.46), we have ∂ ∂ Φ) ◦ γ)(z) = (Iξ,δ Φ)(γz) ((Iξ,δ ∞  Ck+δ+ξ φk (γz)(J(γ, z)2 ∂)−k−δ−ξ . = k=0

Using (1.44), we obtain ∂ Φ) ◦ γ)(z) ((Iξ,δ    ∞ ∞   −k − δ − ξ − 1 −k − δ − ξ Ck+δ+ξ φk (γz)r! = r r r=0 k=0

× J(γ, z)−2k−2δ−2ξ K(γ, z)r ∂ −k−δ−ξ−r    k ∞   r−k−δ−ξ−1 r−k−δ−ξ = Ck−r+δ+ξ φk−r (γz)r! r r r=0 k=0

× J(γ, z)−2k−2δ−2ξ+2r K(γ, z)r ∂ −k−δ−ξ . From this and the identity   −α (−1)r (α − 1 + r)! = r r!(α − 1)! for α ≥ 1, we see that ∂ ((Iξ,δ Φ)

◦ γ)(z) =

k ∞   (−1)r k=0 r=0

r!

Ck+δ+ξ J(γ, z)−2k−2δ−2ξ+2r × K(γ, z)r φk−r (γz)∂ −k−δ−ξ .

On the other hand, using (1.17), we have

(1.52)

28

1 Formal Power Series and Pseudodifferential Operators

(Φ |J2ξ γ)(z, X) = J(γ, z)−2ξ

 ∞ r=0

(−1)r K(γ, z)r X r r! ×

∞ 



φk (γz)(J(γ, z)−2 X)k+δ

k=0

=

∞ ∞   (−1)r k=0 r=0

=

r!

∞  k  (−1)r k=0 r=0

r!

J(γ, z)−2k−2δ−2ξ K(γ, z)r φk (γz)X k+r+δ J(γ, z)−2k−2δ−2ξ+2r K(γ, z)r φk−r (γz)X k+δ .

∂ to this relation and comparing it with (1.52), we see that Applying Iξ,δ ∂ ∂ Iξ,δ (Φ |J2ξ γ)(z) = ((Iξ,δ Φ) ◦ γ)(z),

which verifies the first relation in (1.51). We now apply this to Φ(z, X) = X Ψ )(z, X), so that (Iξ,δ ∂ X ∂ X Iξ,δ ((Iξ,δ Ψ ) |J2ξ γ) = Iξ,δ (Iξ,δ Ψ ) ◦ γ = Ψ ◦ γ. X to Then the second relation in (1.51) is obtained by applying the map Iξ,δ this relation. 

Corollary 1.16 Given a discrete subgroup Γ of SL(2, R), the maps in (1.50) determine the complex linear isomorphisms ∂ : J2ξ (Γ )δ → Ψ D(R)Γ−δ−ξ , Iξ,δ

with

X Iξ,ε : Ψ D(R)Γ−ε → J2ξ (Γ )ε−ξ

X −1 ∂ ) = Iξ,ε−ξ (Iξ,ε

for integers δ, ε > 0 and ξ ≥ 0 with ξ ≤ ε. Proof. This follows immediately from Proposition 1.15 and (1.50).



From Theorem 1.6 and Proposition 1.15 it follows that the diagram Λ2ξ,δ

∂ Iξ,δ

Λ2ξ,δ

∂ Iξ,δ

R[[X]]δ −−−−→ R[[X]]δ −−−−→ Ψ D(R)−δ−ξ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐(◦γ) (|J (|M  2ξ γ) 2ξ γ) R[[X]]δ −−−−→ R[[X]]δ −−−−→ Ψ D(R)−δ−ξ is commutative for each γ ∈ SL(2, R), and this induces mutual isomorphisms among the three spaces M2ξ (Γ )δ ∼ = J2ξ (Γ )δ ∼ = Ψ D(R)Γ−δ−ξ

1.4 Pseudodifferential Operators

29

for δ ≥ 0 and ξ ∈ Z.

Applying Corollary 1.16 to the case where ξ = 0 and δ = ε, we obtain the isomorphisms ∂ I0,δ : J0 (Γ )δ → Ψ D(R)Γ−δ ,

X I0,δ : Ψ D(R)Γ−δ → J0 (Γ )δ

X ∂ −1 with I0,δ = (I0,δ ) . From this and Corollary 1.16 we see that there are canonical isomorphisms X I0,δ

Ξ0,δ

Ψ D(R)Γ−δ −−→ J0 (Γ )δ −−−→ M0 (Γ )δ for each positive integer δ. On the other hand, the Cohen–Kuznetsov lifting map Lw λ,δ in (1.36) with λ = 0 induces the linear map ∂

Γ Lw 0,δ : M2w (Γ ) → Ψ D(R)−δ ,

(1.53)

where ∂ w ( ∂ Lw 0,δ f )(z) = ((I0,δ ◦ L0,δ )f )(z) ∞  C+δ f (+δ−w) ∂ −−δ = ( − w + δ)!( + w + δ − 1)! =0

∞  (−1)+δ ( + δ)!( + δ − 1)! (+δ−w) −−δ f = ∂ ( − w + δ)!( + w + δ − 1)! =0

for all f ∈ M2w (Γ ). In particular, for w = δ we obtain (∂ Lδ0,δ f )(z) =

∞  (−1)+δ ( + δ)!( + δ − 1)! =0

!( + 2δ − 1)!

f () ∂ −−δ .

(1.54)

∂ is the symbol map in (1.42), then we see that If S−δ ∂ Ψ ∈ M2δ (Γ ) S−δ

for Ψ (z) ∈ Ψ D(R)Γ−δ . Hence the exact sequence (1.43) induces the short exact sequence ∂ S−δ

0 → Ψ D(R)Γ−δ−1 → Ψ D(R)Γ−δ −−→ M2δ (Γ ) → 0. On the other hand, from (1.54) we obtain ∂ (S−δ ◦ ∂ Lδ0,δ )f =

(−1)δ δ!(δ − 1)! f, (2δ − 1)!

(1.55)

30

1 Formal Power Series and Pseudodifferential Operators

which implies that     2δ − 1 ∂ δ ∂ S−δ ◦ (−1)δ L0,δ = 1M2δ (Γ ) . δ Thus it follows that the short exact sequence (1.55) splits.

(1.56)

Chapter 2

Jacobi-like Forms and Pseudodifferential Operators

It is well known that modular forms over a discrete subgroup Γ ⊂ SL(2, R) can be regarded as sections of a line bundle over the modular curve associated to Γ . It is also known that Jacobi forms can be interpreted as sections of certain vector bundles (cf. [59], [62], [113]). In light of these facts and the presence of close connections among modular forms, Jacobi-like forms, and pseudodifferential operators discussed in Chapter 1, it would be natural to seek an interpretation of Jacobi-like forms as well as pseudodifferential operators in terms of sections of vector bundles. In this chapter, we construct vector bundles over the quotient Γ \H of the Poincar´e upper half plane by Γ whose sections can be identified with Jacobi-like forms or automorphic pseudodifferential operators (cf. [64], [65]). We also discuss automorphic pseudodifferential operators of mixed weight, Poincar´e series, Rankin–Cohen brackets, and some linear maps of Jacobi-like forms (cf. [23], [69]).

2.1 Power Series Bundles Let Γ be a discrete subgroup of SL(2, R), which acts on the Poincar´e upper half plane H as in Chapter 1. In this section we construct a vector bundle over the quotient space Γ \H whose sections can be identified with Jacobi-like forms (cf. [64]). Let C[[X]] be the set of formal power series in X with coefficients in C. Given a nonnegative integer δ, we denote by C[[X]]δ = X δ C[[X]]

(2.1)

the subspace of C[[X]] consisting of formal power series of the form ∞ 

cr X r+δ

r=0

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_2

31

32

2 Jacobi-like Forms and Pseudodifferential Operators

with cr ∈ C for each r ≥ 0. If λ, , m ∈ Z with  ≥ 0, we set Mλ,m (γ, z) =

(−1) K(γ, z) !J(γ, z)2m+λ

(2.2)

for all z ∈ H and γ ∈ SL(2, R), where J(γ, z) and K(γ, z) are as in (1.12). Then for δ ≥ 0 we see that ∞ 

∞  1 (−K(γ, z)X) !

Mλ,m (γ, z)X +δ = J(γ, z)−2m−λ

=0

= J(γ, z) Given γ ∈ SL(2, R) and f (X) = as in (2.1), we set  γ · (z, f (X)) =

γz,

=0 −2m−λ −K(γ,z)X

e

∞

r=0 cr X

∞ ∞  

r+δ

(2.3)

.

∈ C[[X]]δ with C[[X]]δ being

Mλ,r+δ (γ −1 , γz)cr X +r+δ

 (2.4)

r=0 =0

for z ∈ H. Proposition 2.1 The formula (2.4) determines an action of SL(2, R) on the space H × C[[X]]δ . Proof. Let γ, γ  ∈ SL(2, R), z ∈ H, and f (X) = Using (2.2), (2.3) and (2.4), we have 





γ · (γ · (z, f (X))) = γ · γz, 

= γ  γz,

∞ ∞  

∞

r=0 cr X

r+δ

Mλ,r+δ (γ −1 , γz)cr X +r+δ

∈ C[[X]]δ .



r=0 =0 ∞ ∞  ∞  

Mλm,+r+δ (γ 

−1

, γ  γz)

r=0 =0 m=0

×

Mλ,r+δ (γ −1 , γz)cr X m++r+δ



 ∞ ∞   = γ  γz, cr X r+δ Mλ,r+δ (γ −1 , γz)X  r=0

=0 ∞ 

×

m=0

−1 Mλm,+r+δ (γ  , γ  γz)X m



2.1 Power Series Bundles

33

 ∞ ∞   = γ  γz, cr X r+δ Mλ,r+δ (γ −1 , γz) r=0  −1

× (J(γ

=0 

, γ γz)−2 X) J(γ 

−1

, γ  γz)−2r−2δ−λ

× exp(−K(γ  

= γ  γz,

∞ 

−1

, γ  γz)X)



cr X r+δ J(γ −1 , γz)−2r−2δ−λ

r=0

× exp(−K(γ −1 , γz)J(γ  × J(γ 

−1

−1

, γ  γz)−2 X)

, γ  γz)−2r−2δ−λ exp(−K(γ 

−1

 , γ  γz)X) .

However, using (1.13) and Lemma 1.3, we obtain J((γ  γ)−1 , γ  γz) = J(γ −1 γ 

−1

K((γ  γ)−1 , γ  γz) = K(γ −1 γ 

, γ  γz) = J(γ −1 , γz)J(γ 

−1

−1

, γ  γz),

, γ  γz)

= K(γ −1 , γz)J(γ 

−1

, γ  γz)−2 + K(γ 

−1

, γ  γz).

Thus we see that  ∞  γ  · (γ · (z, f (X))) = γ  γz, J((γ  γ)−1 , γ  γz)−2r−2δ−λ r=0 

× exp(−K((γ γ)

−1



, γ γz))cr X

 r+δ

  ∞ ∞   = γ  γz, Mλ,r+δ ((γ  γ)−1 , γ  γz)cr X +r+δ r=0 =0 

= (γ γ) · (z, f (X)); hence the proposition follows.



We denote the quotient of the space H × C[[X]]δ by the discrete subgroup Γ ⊂ SL(2, R) with respect to the action shown in Proposition 2.1 by Pλδ = Γ \(H × C[[X]]δ ), and let U = Γ \H be the quotient space associated to Γ . Since the natural projection map H × C[[X]]δ → H is clearly Γ -equivariant by (2.4), it induces a surjective map π : Pλδ → U such that π −1 (x) is isomorphic to C[[X]]δ for each x ∈ U . Thus Pλδ has the structure of a complex vector bundle over the Riemann surface U whose fiber is the complex vector space C[[X]]δ of formal power series in X. We denote by Γ 0 (U, Pλδ ) the space of all holomorphic

34

2 Jacobi-like Forms and Pseudodifferential Operators

sections of Pλδ over U , that is, the set of holomorphic functions s : U → Pλδ such that π ◦ s = 1U . Lemma 2.2 If Φ(z, X) is a Jacobi-like form belonging to Jλ (Γ )δ , then we have γ · (z, Φ(z, X)) = (γz, Φ(γz, X)) (2.5) for all γ ∈ Γ . Proof. We assume that the Jacobi-like form Φ(z, X) ∈ Jλ (Γ )δ is of the form Φ(z, X) =

∞ 

fk (z)X k+δ .

k=0

Then, given γ ∈ Γ and z ∈ H, using (2.2), (2.3) and (2.4), we have  γ · (z, Φ(z, X)) =

γz, 

=

γz,

Mλ,r+δ (γ −1 , γz)fr (z)X +r+δ

r=0 =0 ∞ 

J(γ −1 , γz)−2r−2δ−λ fr (z)e−K(γ

−1

 (2.6) 

,γz)X

X r+δ

r=0

 =

∞ ∞  

γz, J(γ −1 , γz)−λ e−K(γ

−1

,γz)X

Φ(z, J(γ −1 , γz)−2 X)



for all γ ∈ Γ . However, since Φ(z, X) ∈ Jλ (Γ )δ , from (1.26) we obtain Φ(z, J(γ −1 , γz)−2 X) = Φ(γ −1 (γz), J(γ −1 , γz)−2 X) = J(γ −1 , γz)λ eK(γ

−1

,γz)X

Φ(γz, X).

Combining this with (2.6), we obtain (2.5)



Theorem 2.3 The space Γ 0 (U, Pλδ ) of holomorphic sections of the vector bundle Pλδ over the quotient space U = Γ \H is canonically isomorphic to the space Jλ (Γ )δ of Jacobi-like forms for Γ of weight λ. Proof. Let s : U → Pλδ be a holomorphic section of Pλδ over U , and denote by  : H → U the natural projection map. Then, given z ∈ H, we have s((z)) =



  ∞ ∈ Γ \H × C[[X]]δ ck,z X k+δ z, k=0

for some sequence {ck,z }∞ k=0 of complex numbers, where [(·)] denotes the Γ orbit of the element (·) of H × C[[X]]δ . We define the sequence {fks }∞ k=0 of C-valued functions on H by fks (z) = ck,z

(2.7)

2.1 Power Series Bundles

35

for all z ∈ H and k ≥ 0. Given γ ∈ Γ , since (γz) = (z), we have

 s((z)) = s((γz)) =

γz,

∞ 

 ck,γz X k+δ

.

(2.8)

k=0

On the other hand, using (2.4) and the fact that [γ · w] = [w] for each w ∈ H × C[[X]]δ , we obtain

   ∞ s((z)) = γ · z, ck,z X k+δ k=0

 =

γz,

Mλ,r+δ (γ −1 , γz)cr,z X +r+δ



r=0 =0

 =

∞ ∞  

γz,

k ∞  

Mλ,k+δ− (γ −1 , γz)ck−,z X k+δ

 .

k=0 =0

Comparing this with (2.8) and using (2.2) and (2.7), we see that fks (γz) = ck,γz =

k 

Mλ,k+δ− (γ −1 , γz)ck−,z

=0

=

k 

s Mλ,k+δ− (γ −1 , γz)fk− (z)

=0

=

k  (−1) =0

!

s K(γ −1 , γz) J(γ −1 , γz)−2k−2δ+2−λ fk− (z).

Using this and the relations (1.13) and (1.15), we obtain fks (γz) =

k  1 s K(γ, z) J(γ, z)2k+2δ+λ fk− (z) ! =0

for each k ≥ 0. From this and Theorem 1.11(ii) it follows that the formal power series ∞  fks (z)X k+δ k=0

belongs to Jλ (Γ )δ . On the other hand, suppose that Ψ (z, X) =

∞  k=0

fk (z)X k+r

36

2 Jacobi-like Forms and Pseudodifferential Operators

is a Jacobi-like form belonging to Jλ (Γ )δ . We define the map sΨ : U → Pλδ by

   ∞ z, fk (z)X k+δ sΨ ((z)) = k=0

for all z ∈ H. Using (2.5), we have      ∞ ∞  k+δ k+δ = γ · z, , fk (γz)X fk (z)X γz, k=0

k=0

and therefore we see that sΨ is well-defined. Since clearly π ◦ sΨ = 1U , it follows that sΨ is a holomorphic section of Pλδ over U ; hence the proof of the theorem is complete. 

2.2 Automorphic Pseudodifferential Operators of Mixed Weight In this section we consider another action of SL(2, R) on the space Ψ D(R) of pseudodifferential operators which extends the action given by (1.40) and was introduced by Cohen, Manin and Zagier in [31], which extends the action given by (1.40). We discuss a condition for an element of Ψ D(R) to be invariant under a discrete subgroup of SL(2, R). Throughout the rest of this section we fix integers λ and μ. Given an element Ψ of Ψ D(R) and γ ∈ SL(2, R), we denote by Ψ |λ,μ γ the element of Ψ D(R) given by (Ψ |λ,μ γ)(z) = J(γ, z)−λ (Ψ ◦ γ)(z)J(γ, z)μ = J(γ, z)

−λ

Ψ (γz)J(γ, z)

(2.9)

μ

for all z ∈ H. Lemma 2.4 The formula (2.9) determines a right action of SL(2, R) on Ψ D(R). Proof. Given elements γ1 , γ2 ∈ SL(2, R) and Ψ ∈ Ψ D(R), using (1.15) and the fact that the map Ψ → Ψ ◦ γ with γ ∈ SL(2, R) determines a right action of SL(2, R) on Ψ D(R), we have (Ψ |λ,μ (γ1 γ2 ))(z) = J(γ1 γ2 , z)−λ (Ψ ◦ (γ1 γ2 ))(z)J(γ1 γ2 , z)μ = (J(γ1 , γ2 z)J(γ2 , z))−λ ((Ψ ◦ γ1 ) ◦ γ2 )(z) × (J(γ1 , γ2 z)J(γ2 , z))μ = J(γ2 , z)−λ J(γ1 , γ2 z)−λ (Ψ ◦ γ1 )(γ2 z) × J(γ1 , γ2 z)μ J(γ2 , z)μ

2.2 Automorphic Pseudodifferential Operators of Mixed Weight

37

= J(γ2 , z)−λ ((Ψ |λ,μ γ1 ) ◦ γ2 )(z)J(γ2 , z)μ = ((Ψ |λ,μ γ1 ) |λ,μ γ2 )(z) for all z ∈ H, which proves the lemma.



If Ψ is a pseudodifferential operator belonging to Ψ D(R)w with w ∈ Z, it can be easily seen from (2.9) that Ψ |λ,μ γ belongs to Ψ D(R)w for all γ ∈ SL(2, R). Thus the action in Lemma 2.4 induces an action of SL(2, R) on Ψ D(R)w for each w ∈ Z. For nonnegative integers k,  and r with k ≥ 1, we set   (−1)r (k + )!(k +  + r − 1)! μ λ,μ (2.10) Mk,,r (γ, z) = r !k!(k − 1)! × J(γ, z)−2k−λ+μ K(γ, z)+r for all z ∈ H and γ ∈ SL(2, R). Lemma 2.5 For each positive integer k, we have J(γ, z)−λ (J(γ, z)2 ∂)−k J(γ, z)μ =

∞ ∞  

−k−−r Mλ,μ k,,r (γ, z)∂

=0 r=0

for all z ∈ H and γ ∈ SL(2, R). Proof. Given γ ∈ SL(2, R), from (1.44) we see that 2

(J(γ, z) ∂)

−k

   ∞  −k − 1 −k J(γ, z)−2k K(γ, z) ∂ −k− . = !  

(2.11)

=0

On the other hand, we have ∂

−k−

J(γ, z) = μ

=

 ∞   −k −  r=0 ∞   r=0

r −k −  r

(∂ r J(γ, z)μ )∂ −k−−r

  μ r!J(γ, z)μ K(γ, z)r ∂ −k−−r . r

From (2.11), (2.12) and the relations     −k (−1) (k +  − 1)! −k − 1 (−1) (k + )! = = , ,   !(k − 1)! !k!   −k −  (−1)r (k +  + r − 1)! = , r r!(k +  − 1)! we obtain

(2.12)

38

2 Jacobi-like Forms and Pseudodifferential Operators

J(γ, z)−λ (J(γ, z)2 ∂)−k J(γ, z)μ   ∞ ∞   (−1)r (k + )!(k +  + r − 1)! μ = r !k!(k − 1)! r=0 =0

× J(γ, z)−2k−λ+μ K(γ, z)+r ∂ −k−−r . Thus the lemma follows from this and (2.10).



We again consider a discrete subgroup Γ of SL(2, R) as well as its action on Ψ D(R) obtained by restricting the action of SL(2, R) in Lemma 2.4 to Γ . Definition 2.6 An element Ψ ∈ Ψ D(R) is an automorphic pseudodifferential operator of mixed weight (λ, μ) for Γ if Ψ |λ,μ γ = Ψ for all γ ∈ Γ . We denote by Ψ D(R)λ,μ the set of such automorphic pseudodifferential operators of mixed weight. For each w ∈ Z we set λ,μ ∩ Ψ D(R)w , Ψ D(R)λ,μ w = Ψ D(R)

(2.13)

where Ψ D(R)w is as in Section 1.4. We note that Γ Ψ D(R)0,0 w = Ψ D(R)w . ∂ is the symbol map in (1.42). Using (2.9) and the identity ∂ ◦ γ = If Sw ∂ Ψ of an element Ψ ∈ J(γ, z)2 ∂ for γ ∈ Γ , we see easily that the image Sw ∂ under S is a modular form belonging to M Ψ D(R)λ,μ λ−μ−2w (Γ ) and that w w the short exact sequence in (1.43) induces an exact sequence of the form S∂

w λ,μ −→ Mλ−μ−2w (Γ ) → 0. 0 → Ψ D(R)λ,μ w−1 → Ψ D(R)w −

The following theorem generalizes a part of Theorem 1.11. Theorem 2.7 Let u be a nonnegative integer, and let Ψ be a pseudodifferential operator belonging to Ψ D(R)−u given by Ψ (z) =

∞ 

(−1)k+u (k + u)!(k + u − 1)!φk (z)∂ −k−u

(2.14)

k=0

for all z ∈ H. Then Ψ belongs to Ψ D(R)λ,μ −u if and only if (φk |2k+2u+λ−μ γ)(z)

  s k   (−1)r (k + u − r)! μ K(γ, z)s φk−s (z) = r (k + u)!(s − r)! s=0 r=0

for all γ ∈ Γ and k ≥ 0.

(2.15)

2.2 Automorphic Pseudodifferential Operators of Mixed Weight

39

Proof. Given γ ∈ Γ and Ψ ∈ Ψ D(R)−u as in (2.14), using (2.9) and Lemma 2.5, we have (Ψ |λ,μ γ −1 )(z) =

∞ 

(−1)+u ( + u)!( + u − 1)!φ (γ −1 z)

=0

× J(γ −1 , z)−λ (J(γ −1 , z)2 ∂)−−u J(γ −1 , z)μ ∞ ∞  ∞   = (−1)+u ( + u)!( + u − 1)! =0 m=0 r=0 −1 × φ (γ −1 z)Mλ+u,μ , z)∂ −−u−m−r . ,m,r (γ

Introducing the indices k =  + m + r and s = m + r, we obtain (Ψ |λ,μ γ −1 )(z) =

k  s ∞  

(−1)k+u−s (k + u − s)!(k + u − s − 1)!

k=0 s=0 r=0 −1 × φk−s (γ −1 z)Mλ,μ , z)∂ −k−u . k+u−s,s−r,r (γ

Since Ψ |λ,μ γ −1 = Ψ , we have φk (z) =

s k   (−1)s (k + u − s)!(k + u − s − 1)!

(k + u)!(k + u − 1)!

s=0 r=0

−1 × φk−s (γ −1 z)Mλ,μ , z). k+u−s,s−r,r (γ

From this and (2.10) we see that φk (γz) =

s k   (−1)s (k + u − s)!(k + u − s − 1)! s=0 r=0

(k + u)!(k + u − 1)!

φk−s (z)

(2.16)

  μ (−1)r (k + u − r)!(k + u − 1)! × (s − r)!(k + u − s)!(k + u − s − 1)! r

× J(γ −1 , γz)−2k−2u+2s−λ+μ K(γ −1 , γz)s   s k   (−1)r+s (k + u − r)! μ = r (k + u)!(s − r)! s=0 r=0 × J(γ −1 , γz)−2k−2u+2s−λ+μ K(γ −1 , γz)s . However, using (1.15), we have J(γ −1 , γz)−2k−2u+2s−λ+μ K(γ −1 , γz)s = J(γ, z)2k+2u−2s+λ−μ (−1)s J(γ, z)2s K(γ, z)s . Hence (2.16) can be written in the form

40

2 Jacobi-like Forms and Pseudodifferential Operators

φk (γz) =

  s k   (−1)r (k + u − r)! μ s=0 r=0

(k + u)!(s − r)!

r

J(γ, z)2k+2u+λ−μ K(γ, z)s φk−s (z), 

which is equivalent to (2.15).

2.3 Bundles of Pseudodifferential Operators

In this section we interpret Γ -automorphic pseudodifferential operators of the type discussed in Section 2.2 as holomorphic sections of some vector bundle over the quotient space Γ \H (cf. [65]). Given a nonnegative integer u, let Ψ D(R)0−u be the subspace of Ψ D(R)−u consisting of pseudodifferential operators with constant coefficients, that is, the set of elements of the form ∞ 

Ak ∂ −k−u

k=0

with Ak ∈ C for each k ≥ 0. Let Γ be a discrete subgroup as before, and set    ∞ −k−u Ak ∂ γ · z, k=u

 =

γz,

∞  ∞ ∞  

(2.17) −1 Ak Mλ,μ , γz)∂ −k−u−−r k+u,,r (γ



k=0 =0 r=0 −1 for all γ ∈ Γ and z ∈ H, where Mλ,μ , γz) is as in (2.10). Using k+u,,r (γ Lemma 2.4 and Lemma 2.5, we see that (2.17) determines a left action of Γ on the space H × Ψ D(R)0−u . We denote the associated quotient space by

[V]u = Γ \H × Ψ D(R)0−u . If U = Γ \H, then from (2.17) we see that the natural projection map H × Ψ D(R)0−u → H is Γ -equivariant; hence it induces a map π : [V]u → U with π −1 (x) ∼ = Ψ D(R)0−u for each x ∈ U . Thus [V]u may be regarded as a homogeneous vector bundle over U whose fiber is the complex vector space Ψ D(R)0−u . We denote by Γ 0 (U, [V]u ) the space of holomorphic sections of [V]u over the Riemann surface U , that is, the set of holomorphic maps s : U → [V]u such that π ◦ s = 1U .

2.3 Bundles of Pseudodifferential Operators

41

Theorem 2.8 The space Γ 0 (U, [V]u ) is isomorphic to the space of sequences {φk }∞ k=u of holomorphic functions on H such that (φk |2k+2u+λ−μ γ)(z) =

s k   s=0 r=0

(2.18)

  (−1)r (k + u − r)! μ K(γ, z)s φk−s (z) (k + u)!(s − r)! r

for all z ∈ H, γ ∈ Γ and k ≥ u. Proof. We consider a holomorphic section s ∈ Γ 0 (U, [V]u ) of the bundle [V]u over U . If  : H → U = Γ \H denotes the natural projection map, then for each z ∈ H we have

   ∞ s((z)) = z, Ak,z ∂ −k−u k=0

for some Ak,z ∈ C, where [(·)] denotes the Γ -orbit of (·) in the quotient space [V]u = Γ \H × Ψ D(R)0−u . For each integer k ≥ 0, we define the function φk : H → C by (−1)k+u Ak,z (2.19) φk (z) = (k + u)!(k + u − 1)! for all z ∈ H. Given γ ∈ Γ , since (z) = (γz), we have

 s((z)) = s((γz)) =

γz,

∞ 

Ak,γz ∂

−k−u

 .

(2.20)

k=0

On the other hand, since [γ · w] = [w] for all w ∈ H × Ψ D(R)0−u , by using (2.17) we also see that

   ∞ s((z)) = γ · z, Ak,z ∂ −k−u k=0

 =

γz,

∞ ∞  ∞  

−1 A,z Mλ,μ , γz)∂ −−u−m−r +u,m,r (γ

 .

=0 m=0 r=0

Introducing the indices k =  + m + r and p = m + r, we obtain

 s((z)) =

γz,

p k  ∞  

−1 Ak−p,z Mλ,μ , γz)∂ −k−u k+u−p,p−r,r (γ

k=0 p=0 r=0

By comparing this with (2.20) we have Ak,γz =

s k   p=0 r=0

−1 Ak−p,z Mλ,μ , γz) k+u−p,p−r,r (γ

 .

42

2 Jacobi-like Forms and Pseudodifferential Operators

for all k ≥ u. Thus, using (2.19), we see that φk (γz) =

p k   (−1)p (k + u − p)!(k + u − p − 1)!

(k + u)!(k + u − 1)!

p=0 r=0

−1 × φk−p (z)Mλ,μ , γz). k+u−p,p−r,r (γ

However, from (2.10) we obtain −1 , γz) Mλ,μ k+u−p,p−r,r (γ

  μ (−1)r (k + u − r)!(k + u − 1)! = (p − r)!(k + u − p)!(k + u − p − 1)! r × J(γ −1 , γz)−2k−2u+2p−λ+μ K(γ −1 , γz)p   μ (−1)r (k + u − r)!(k + u − 1)! = (p − r)!(k + u − p)!(k + u − p − 1)! r × J(γ, z)2k+2u−2p+λ−μ (−J(γ, z)2 K(γ, z))p   μ (−1)r+p (k + u − r)!(k + u − 1)! = (p − r)!(k + u − p)!(k + u − p − 1)! r × J(γ, z)2k+2u+λ−μ K(γ, z)p ,

where we used (1.15). Hence we have φk (γz) =

  p k   (−1)r (k + u − r)! μ p=0 r=0

(k + u)!(p − r)!

r

J(γ, z)2k+2u+λ−μ K(γ, z)p φk−p (z),

which is equivalent to (2.18). We now consider a sequence satisfying (2.18), and define the map sφ : U → [V]u by 

  ∞ k −k sφ ((z)) = z, (−1) k!(k − 1)!φk (z)∂ k=u

for z ∈ H. Using computations similar to the ones used above, it can be shown that

  ∞  sφ ((γz)) = γz, (−1)k k!(k − 1)!φk (γz)∂ −k

k=u

   ∞ k −k = γ · z, (−1) k!(k − 1)!φk (z)∂ k=u

   ∞ = z, (−1)k k!(k − 1)!φk (z)∂ −k ; k=u

hence sφ is well-defined. Since π ◦ sφ = 1U , we see that sφ is a holomorphic  section of the bundle [V]u , and therefore the theorem follows.

2.4 Poincar´ e Series

43

Corollary 2.9 Let u be a nonnegative integer, and let Ψ D(R)λ,μ −u be the associated Γ -invariant subspace of Ψ D(R)−u in (2.13). Then Ψ D(R)λ,μ −u is canonically isomorphic to the space Γ 0 (U, [V]u ) of global sections of [V]u over U . Proof. This follows immediately from Theorem 2.7 and Theorem 2.8.



2.4 Poincar´ e Series Poincar´e series provide examples of modular forms. In this section we construct analogues of Poincar´e series for Jacobi-like forms and automorphic pseudodifferential operators (cf. [61]). In this section we assume that the discrete subgroup Γ of SL(2, R) is a Fuchsian group of the first kind and denote by F the subring of R consisting of holomorphic functions f : H → C that are bounded at each cusp of Γ . Given a cusp x ∈ Q ∪ {∞} of Γ , if Γx = {γ ∈ Γ | γx = x}, then there is an element σ ∈ SL(2, R) and a positive real number h such that    n∈Z . (2.21) σΓx σ −1 · {±1} = ± ( 10 nh 1 ) Given integers w ≥ 3 and u ≥ 0, we set  Pwx,u (z) = J(σγ, z)−w eu/h (σγz) = γ∈Γx \Γ



(eu/h |w σγ)(z)

(2.22)

γ∈Γx \Γ

for all z ∈ H, where eμ (·) = exp(2πiμ(·)) for μ ∈ C. Then it is well known that the series in (2.22) converges absolutely and uniformly on any compact subset of H, and the resulting function Pwx,u : H → C is a Poincar´e series for modular forms belonging to Mw (Γ ) (see e.g. [85]). If α, u ∈ Z, we set ηαx,u (z) = J(σ, z)−α eu/h (σz)

(2.23)

for all z ∈ H, where h is as in (2.21). Then, using (1.13), we have (ηαx,u |α γ)(z) = J(γ, z)−α J(σ, γz)−α eu/h (σγz) = J(σγ, z)−α eu/h (σγz). Thus the Poincar´e series (2.22) can be written in the form  x,u (ηw |w γ)(z). Pwx,u (z) = γ∈Γx \Γ

(2.24)

44

2 Jacobi-like Forms and Pseudodifferential Operators

Given ξ ∈ Z, we consider the formal power series Gx,u 2ξ,δ (z, X) =

∞ 

x,u (z)X r+δ ∈ F[[X]], η2(ξ+r+δ)

(2.25)

J ((Λ2ξ,δ Gx,u 2ξ,δ ) |2(ξ+r+δ) γ)(z, X)

(2.26)

r=0

and set



x,u (z, X) = P 2ξ,δ

γ∈Γx \Γ

for z ∈ H, where x is a cusp of Γ as above and Λ2ξ,δ is as in (1.4). x,u (z, X) can be written in the form Theorem 2.10 The series P 2ξ,δ x,u (z, X) = P 2ξ,δ

∞    r   (−1)−j (2πiu)j γ∈Γx \Γ r=0 j=0 =0

×

(2.27)

hj j!( − j)!

K(σγ, z)−j eu/h (σγz) X r+δ , + 2δ + 2ξ − 2 + j − 1)!

J(σγ, z)2ξ+2δ+2r (2r

and it is a Jacobi-like form belonging to F J2ξ (Γ )δ , which is a subspace of J2ξ (Γ )δ consisting of Jacobi-like forms with coefficients in F. Proof. Using (1.18), (2.24), (2.25), (2.26) and Theorem 1.6, we have  M x,u (z, X) = Λ2ξ,δ (Gx,u (2.28) P 2ξ,δ 2ξ,δ |2(ξ+r+δ) γ)(z, X) γ∈Γx \Γ

=



Λ2ξ,δ

γ∈Γx \Γ

= Λ2ξ,δ

 ∞

 ∞

 (η2(ξ+δ+r) |2(ξ+r+δ) γ)(z)X r+δ

r=0



x,u P2(ξ+δ+r) (z)X r+δ

.

r=0 x,u ∈ M2(ξ+δ+r) (Γ ) for each r ≥ 0, from Lemma 1.9 we see that Since P2(ξ+δ+r) the formal power series

F (z, X) =

∞ 

x,u P2(ξ+δ+r) (z)X r+δ ∈ F[[X]]δ

(2.29)

r=0

is a modular series belonging to Mm 2ξ (Γ )δ , and therefore by Proposition 1.10 its image x,u (z, X) = (Λ2ξ,δ F )(z, X) (2.30) P 2ξ,δ under Λ2ξ,δ is a Jacobi-like form belonging to J2ξ (Γ )δ ; hence it follows that x,u (z, X) ∈ F J2ξ (Γ )δ . P 2ξ,δ

2.4 Poincar´ e Series

45

On the other hand, from (1.4), (1.6), (2.22) and (2.29) we obtain (Λ2ξ,δ F )(z, X) =



Λ2ξ,δ

 ∞

γ∈Γx \Γ

=

 (eu/h |2(ξ+δ+r) σγ)(z)X r+δ

(2.31)

r=0

∞  

φ r (z)X r+δ ,

γ∈Γx \Γ r=0

where φ r =

r  =0

1 (eu/h |2(ξ+δ+r−) σγ)() . !(2r + 2δ + 2ξ −  − 1)!

(2.32)

However, from (1.21) we see that (eu/h |2(ξ+δ+r−) σγ)() (z)     −j ! 2ξ + 2δ + 2r −  − 1 = (−1) −j j! j=0

(2.33) (2.34)

K(σγ, z)−j (eu/h )(j) (σγz) J(σγ, z)2ξ+2δ+2r     (−1)−j (2πiu)j ! 2ξ + 2δ + 2r −  − 1 = −j hj j! j=0 ×

×

K(σγ, z)−j eu/h (σγz). J(σγ, z)2ξ+2δ+2r

Thus we obtain (2.27) by combining (2.31), (2.32) and (2.33).



x,u (z, X) in (2.27) can be regarded as a Poincar´e series for The series P 2ξ,δ x,u (z, X) is the Jacobi-like form Jacobi-like forms. From (2.28) we see that P 2ξ,δ corresponding to the modular series (2.29). We note that a more general type of Poincar´e series can be constructed as follows. Let F (z, X) ∈ F[[X]]δ be a formal power series that is invariant under Γx , and set  P x,F (z, X) = (F |J2ξ γ)(z, X). γ∈Γx \Γ

Then P x,F (z, X) is a Jacobi-like form belonging to

F

J2ξ (Γ )δ .

Example 2.11 Using the notation in (2.22), we consider F (z, X) = eu/h (σz)X δ , which is regarded as a formal power series belonging to F[[X]]. Then the associated Poincar´e series is a Jacobi-like form belonging to J2ξ (Γ ) given by

46

2 Jacobi-like Forms and Pseudodifferential Operators

P x,F (z, X) =



(F |J2ξ γ)(z, X)

γ∈Γx \Γ

=



J(γ, z)−2ξ eu/h (σγz)e−K(γ,z)X

γ∈Γx \Γ

=

∞   (−1)k K(γ, z)k eu/h (σγz) γ∈Γx \Γ k=0

(k!)J(γ, z)2ξ

Xk.

∂ in Corollary 1.16, we can also consider the Using the isomorphism Iξ,δ x,u (z, X) for automorphic pseudodifferential corresponding Poincar´e series ∂ P 2ξ,δ operators by setting ∂

x,u (z, X) P 2ξ,δ ∂ x,u (z, X)) (P = Iξ,δ 2ξ,δ

=

  r ∞    (−1)r+δ+ξ+−j (2πiu)j

hj j!( − j)!

γ∈Γx \Γ r=0 j=0 =0

×

(r + δ + ξ)!(r + δ + ξ − 1)!K(σγ, z)−j eu/h (σγz) −r−δ−ξ ∂ , J(σγ, z)2ξ+2δ+2r (2r + 2δ + 2ξ − 2 + j − 1)!

which belongs to Ψ D(F)Γ−δ−ξ .

2.5 Linear Maps and Rankin–Cohen Brackets In this section we discuss some applications of the correspondences among Jacobi-like forms, modular series, and pseudodifferential operators. We consider linear maps of Jacobi-like forms associated to modular forms and discuss Rankin–Cohen brackets for modular forms (cf. [15], [19], [69]). We fix an integer w and a modular form h : H → C belonging to Mw (Γ ). Given a formal power series Φ(z, X) =

∞ 

φk (z)X k+δ ∈ R[[X]]δ

k=0

with δ ≥ 0, we set Lh (Φ(z, X)) =

∞ 

h(z)φk (z)X k+δ .

k=0

Then we see easily that Lh (Mλ (Γ )δ ) ⊂ Mλ+w (Γ )δ ,

Lh (Jλ (Γ ))δ ⊂ Jλ+w (Γ )δ

2.5 Linear Maps and Rankin–Cohen Brackets

47

for each λ ∈ Z. Thus Lh induces, by restriction, the linear maps LM h : Mλ (Γ )δ → Mλ+w (Γ )δ ,

LJh : Jλ (Γ )δ → Jλ+w (Γ )δ .

(2.35)

We construct below a linear map of Jacobi-like forms compatible with LM h as well as a linear map of modular series compatible with LJh with respect to the correspondence between modular series and Jacobi-like forms discussed in Section 1.3. We define linear maps LJh : Jλ (Γ )δ → R[[X]]δ ,

LM h : Mλ (Γ )δ → R[[X]]δ

(2.36)

by setting LJh (Φ(z, X)) =

s k−s ∞  k    (−1)r (2k + 2δ − 2s + λ − 1)

j!r!(s − j)!

k=0 s=0 j=0 r=0

×

LM h (F (z, X)) =

(2.37)

(2k + 2δ − 2s + λ − r − 2)! (j) (r+s−j) k+δ h φk−r−s X , (2k + 2δ + λ + w − s − 1)!

r k−r k  ∞    (−1)r (2k + 2δ + λ + w − 1)

j!s!(r − j)!

k=0 r=0 j=0 s=0

×

(2.38)

(2k + 2δ + λ + w − r − 2)! (j) (r+s−j) k+δ h fk−r−s X (2k − 2r + 2δ + λ − s − 1)!

for Φ(z, X) =

∞ 

φk (z)X k+δ ∈ Jλ (Γ )δ ,

(2.39)

k=0

F (z, X) =

∞ 

fk (z)X k ∈ Mλ (Γ )δ .

(2.40)

k=δ

Theorem 2.12 (i) The image of the linear map LJh in (2.36) is contained in Jλ+w (Γ )δ . Furthermore, if LJh : Jλ (Γ )δ → Jλ+w (Γ )δ is the same linear map whose codomain is restricted to Jλ+w (Γ )δ , then the diagram J L

Jλ (Γ )δ −−−h−→ Jλ+w (Γ )δ ⏐ ⏐ ⏐ ⏐Ξ Ξλ,δ   λ+w,δ LM

Mλ (Γ )δ −−−h−→ Mλ+w (Γ )δ is commutative, where Ξλ,δ , Ξλ+w,δ and LJh are as in (1.5) and (2.35).

48

2 Jacobi-like Forms and Pseudodifferential Operators

(ii) The image of the linear map LM h in (2.36) is contained in Mλ+w (Γ )δ . Furthermore, if the codomain of LM is restricted to Mλ+w (Γ )δ , the diagram h LJ

Jλ (Γ )δ −−−h−→ Jλ+w (Γ )δ ⏐ ⏐ ⏐ ⏐Ξ Ξλ,δ   λ+w,δ M L

Mλ (Γ )δ −−−h−→ Mλ+w (Γ )δ is commutative.

Proof. Given a Jacobi-like form Φ(z, X) ∈ Jλ (Γ )δ as in (2.39), we set Ξλ,δ (Φ(z, X)) =

∞ 

f (z)X +δ ,

=0

which belongs to Mλ (Γ )δ by (1.30). Then from (1.5) and (1.7) we obtain fk = (2k + 2δ + λ − 1)

k 

(−1)r

r=0

(2k + 2δ + λ − r − 2)! (r) φk−r , r!

which is a modular form belonging to M2k+2δ+λ (Γ ) for each k ≥ 0 by Lemma 1.9. Thus, if we write (LM h ◦ Ξλ,δ )(Φ(z, X)) =

∞ 

f  (z)X +δ ∈ Mλ+w (Γ )δ ,

=0

we see that f k = hfk = (2k + 2δ + λ − 1) ×

k 

(−1)r

r=0

(2k + 2δ + λ − r − 2)! (r) hφk−r ∈ M2k+2δ+λ+w (Γ ) r!

for each k ≥ 0. We now set −1 M (Ξλ+w,δ ◦ LM h ◦ Ξλ,δ )(Φ(z, X)) = (Λλ+w,δ ◦ Lh ◦ Ξλ,δ )(Φ(z, X))

=

∞ 

k+δ . φ(z)X

k=0

Then it is a Jacobi-like form belonging to Jλ+w (Γ )δ by (1.29), and from (1.4) and (1.6) we obtain

2.5 Linear Maps and Rankin–Cohen Brackets

φ k =

k  s=0

=

k  s=0

1 (hfk−s )(s) s!(2k + 2δ + λ + w − s − 1)! 2k + 2δ − 2s + λ − 1 s!(2k + 2δ + λ + w − s − 1)! ×

k−s 

(−1)r

r=0

=

49

(2k + 2δ − 2s + λ − r − 2)! (r) (hφk−r−s )(s) r!

k−s  s k  

2k + 2δ − 2s + λ − 1 s!(2k + 2δ + λ + w − s − 1)! s=0 r=0 j=0   s (2k + 2δ − 2s + λ − r − 2)! (j) (r+s−j) h φk−r−s × (−1)r j r!

for each k ≥ 0. From this and (2.37) it follows that J (Λλ+w,δ ◦ LM h ◦ Ξλ,δ )(Φ(z, X)) = Lh (Φ(z, X)). In particular, we see that LJh (Φ(z, X)) ∈ Jλ+w (Γ )δ J and LM h ◦ Ξλ,δ = Ξλ+w,δ ◦ Lh , which verifies (i). Given F (z, X) ∈ Mλ (Γ )δ as in (2.40), if we write Λλ,δ (F (z, X)) =

∞ 

φk (z)X k+δ ,

k=0

then it belongs to Jλ (Γ )δ by (1.29) and from (1.6) we obtain φk =

k  s=0

1 (s) f s!(2k + 2δ + λ − s − 1)! k−s

(2.41)

for each k ≥ 0. Then it follows that LJh (Λλ,δ (F (z, X)))

=

∞ 

h(z)φk (z)X k+δ

k=0

is a Jacobi-like form belonging to Jλ+w (Γ )δ by (2.35). Applying Ξλ+w,δ to this, we have (Ξλ+w,δ ◦ LJh ◦ Λλ,δ )(F (z, X)) =

∞  k=0

fk (z)X k+δ ,

(2.42)

50

2 Jacobi-like Forms and Pseudodifferential Operators

where by (1.7) and (2.41) the coefficients are given by fk = (2k + 2δ + λ + w − 1) ×

k 

(−1)r

r=0

(2k + 2δ + λ + w − r − 2)! (hφk−r )(r) r!

= (2k + 2δ + λ + w − 1)   r k   r r (2k + 2δ + λ + w − r − 2)! (j) (r−j) h φk−r × (−1) j r! r=0 j=0 =

r k−r k    r (2k + 2δ + λ + w − r − 2)! r=0 j=0 s=0

j

r! ×

(−1)r (2k + 2δ + λ + w − 1) (j) (r+s−j) h fk−r−s s!(2k − 2r + 2δ + λ − s − 1)!

for all k ≥ 0. Thus by comparing (2.42) with (2.38) we obtain −1 )(F (z, X)) = LM (Ξλ+w,δ ◦ LJh ◦ Ξλ,δ h (F (z, X)).

In particular, we see that LM h (F (z, X)) ∈ Mλ+w (Γ )δ and Ξλ+w,δ ◦ LJλ = LM h ◦ Ξλ,δ , which proves (ii).



Example 2.13 If f is a modular form belonging to M2δ+λ (Γ ), by the Cohen– Kuznetsov lifting in Example 1.12 the series Φf (z, X) =

∞  k=0

f (k) (z) X k+δ k!(k + λ + 2δ − 1)!

is a Jacobi-like form belonging to Jλ (Γ )δ such that Ξλ,δ (Φf (z, X)) = f (z)X δ , which belongs to Mλ (Γ ) by (1.29). Thus we have δ LM h (Φf (z, X)) = h(z)f (z)X ∈ Mλ+w (Γ ),

and, as in Example 1.12, we see that LJh (Φf (z, X)) = (Λλ+w,δ ◦ LM h ◦ Ξλ,δ )(Φf (z, X)) ∞  (hf )(k) (z) = X k+δ , k!(k + λ + w + 2δ − 1)! k=0

2.5 Linear Maps and Rankin–Cohen Brackets

51

which is a Jacobi-like form belonging to Jλ (Γ )δ+w . Example 2.14 Let Ff (z, X) = f (z)X δ be a modular series belonging to Mλ (Γ ) with f ∈ M2δ+λ (Γ ). Then as in Example 1.12 we see that −1 Ξλ,δ (Ff (z, X)) =

∞  k=0

f (k) (z) X k+δ ∈ Jλ (Γ )δ ; k!(k + λ + 2δ − 1)!

hence we have −1 LJh (Ξλ,δ (Ff (z, X))) =

∞  k=0

h(z)f (k) (z) X k+δ ∈ Jλ+w (Γ )δ . k!(k + λ + 2δ − 1)!

Thus, using (1.5), we obtain −1 J LM h (Ff (z, X)) = (Ξλ+w,δ ◦ Lλ ◦ Ξλ,δ )(Ff (z, X)) =

∞ 

fk (z)X k+δ ,

k=0

where fk = (2k + 2δ + λ + w − 1)

k 

(−1)r

r=0

(2k + 2δ + λ + w − r − 2)!(hf (k−r) )(r) r!(k − r)!(k − r + λ + 2δ − 1)!

for each k ≥ 0. As another application, we now consider bilinear maps on modular forms known as Rankin–Cohen brackets (see e.g. [23], [31]). Lemma 2.15 Given Jacobi-like forms Φ1 (z, X) ∈ Jλ1 (Γ )δ1 and Φ2 (z, X) ∈ Jλ2 (Γ )δ2 , the formal power series Φ(z, X) = Φ1 (z, X)Φ2 (z, −X) is a modular series belonging to Mλ1 +λ2 (Γ ). Proof. If γ ∈ Γ and z ∈ H, we have Φ(γz, J(γ, z)−2 X) = Φ1 (γz, J(γ, z)−2 X)Φ2 (γz, −J(γ, z)−2 X) = J(γ, z)λ1 e−K(γ,z)X Φ1 (z, X) × J(γ, z)λ2 eK(γ,z)X Φ2 (z, −X) = J(γ, z)λ1 +λ2 Φ(z, X); 

hence the lemma follows. We now consider modular forms f ∈ M2δ1 +λ1 (Γ ),

g ∈ M2δ2 +λ2 (Γ )

for Γ . Given a nonnegative integer n, we set

(2.43)

52

2 Jacobi-like Forms and Pseudodifferential Operators n 

[f, g]n (z) =

=0

(−1)n−+δ2 f () (z)g (n−) (z) (2.44) !(n − )!( + 2δ1 + λ1 − 1)!(n −  + 2δ2 + λ2 − 1)!

for all z ∈ H. Theorem 2.16 For each nonnegative integer n the function [f, g]n on H given by (2.44) is a modular form for Γ of weight 2n + 2δ1 + 2δ2 + λ1 + λ2 . Proof. If f and g are as in (2.43), by (1.35) and (1.36) their liftings (Lδλ11 ,δ1 f )(z, X) and (Lδλ22 ,δ2 g)(z, X)δ1 belonging to Jλ1 (Γ )δ1 and Jλ2 (Γ )δ2 , respectively, are given by (Lδλ11 ,δ1 f )(z, X) =

∞ 

f () (z) X +δ1 , !( + 2δ1 + λ1 − 1)!

=0

(Lδλ22 ,δ2 g)(z, X)

=

∞  r=0

g (r) (z) X r+δ2 . r!(r + 2δ2 + λ2 − 1)!

Thus we have (Lδλ11 ,δ1 f )(z, X) · (Lδλ22 ,δ2 g)(z, −X) =

∞ ∞   =0 r=0

=

n ∞   n=0 =0

(−1)r+δ2 f () (z)g (r) (z) X +r+δ1 +δ2 !r!( + 2δ1 + λ1 − 1)!(r + 2δ2 + λ2 − 1)! (−1)n−+δ2 f () (z)g (n−) (z) X n+δ1 +δ2 . !(n − )!( + 2δ1 + λ1 − 1)!(n −  + 2δ2 + λ2 − 1)!

Comparing this expression with (2.44), we see that (Lδλ11 ,δ1 f )(z, X)

·

(Lδλ22 ,δ2 g)(z, −X)

=

∞ 

[f, g]n (z)X n+δ1 +δ2 ,

n=0

which is a modular series belonging to Mλ1 +λ2 (Γ )δ1 +δ2 by Lemma 2.15. Hence the theorem follows from this and Lemma 1.9.  From Theorem 2.16 we obtain the bilinear map [·, ·]n : M2δ1 +λ1 (Γ ) × M2δ2 +λ2 (Γ ) → M2n+2δ1 +2δ2 +λ1 +λ2 (Γ ), which is known as the Rankin–Cohen bracket for modular forms (cf. [29], [31], [114]).

Chapter 3

Hecke Operators

One of the effective tools in the theory of modular forms is the notion of Hecke operators (see e.g. [85]). Hecke operators can also be considered in connection with various other topics which involve actions of a discrete subgroup of a semisimple Lie group. In this chapter we introduce Hecke operators acting on the spaces of Jacobi-like forms, modular series, and pseudodifferential operators. We show that these Hecke operator actions are compatible with the mutual correspondences among those three objects studied in Chapter 1. We also discuss connections of these actions with Hecke operator actions on certain types of linear ordinary differential equations.

3.1 Jacobi-like Forms, Modular Series, and Hecke Operators In this section we review the usual Hecke operators on modular forms and introduce Hecke operators on Jacobi-like forms and modular series (cf. [79]). Let GL+ (2, R) be the group of 2 × 2 real matrices of positive determinant. Then, as in the case of SL(2, R), it acts on the Poincar´e upper half plane H by linear fractional transformations. We extend the maps J and K given by (1.12) to the maps J, K : GL+ (2, R) × H → C by setting J(α, z) = cz + d, for z ∈ H and α = condition

a b c d

K(α, z) =

c cz + d

(3.1)

∈ GL+ (2, R). Then the map J satisfies the cocycle

J(αα , z) = J(α, α z)J(α , z)

(3.2)

for all z ∈ H and α, α ∈ GL+ (2, R). On the other hand, the identity (1.14) should be modified in such a way that

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_3

53

54

3 Hecke Operators

(det γ  )K(γ, γ  z) = J(γ  , z)2 (K(γγ  , z) − K(γ  , z))

(3.3)

for all z ∈ H and γ, γ  ∈ GL+ (2, R). We now extend the right action of SL(2, R) on functions on H in (1.16) by setting (f |k α)(z) = det(α)k/2 J(α, z)−k f (αz)

(3.4)

for all z ∈ H and α ∈ GL+ (2, R). Using (3.2), it can be shown that f |k (αα ) = (f |k α) |k α

(3.5)

for all α, α ∈ GL+ (2, R). Given a discrete subgroup Γ of SL(2, R), Hecke operators acting on the space Mk (Γ ) of modular forms of weight k for Γ can be described as follows. Two subgroups Γ1 and Γ2 of GL+ (2, R) are said to be commensurable if Γ1 ∩ Γ2 has finite index in both Γ1 and Γ2 , in which case we write Γ1 ∼ Γ2 .  ⊂ GL+ (2, R) is given If Δ is a subgroup of GL+ (2, R), its commensurator Δ by  = {g ∈ GL+ (2, R) | gΔg −1 ∼ Δ}. Δ We now consider an element α ∈ Γ and assume that the associated double coset Γ αΓ has a decomposition of the form Γ αΓ =

s 

Γ αi

(3.6)

i=1

for some αi ∈ GL+ (2, R) with i = 1, . . . , s (see e.g. [85]). For each modular form f ∈ Mk (Γ ), we define the function Tk (α)f on H by Tk (α)f =

s 

(f |k αi ).

(3.7)

i=1

Using (3.5) and the decomposition (3.6), it can be shown that Tk (α)f is also an element of Mk (Γ ). Definition 3.1 The Hecke operator on Mk (Γ ) for k ∈ Z associated to the double coset Γ αΓ with α ∈ Γ is the linear map Tk (α) : Mk (Γ ) → Mk (Γ ) defined by (3.7). Given an integer λ, an element ξ ∈ GL+ (2, R), and a formal power series Φ(z, X) ∈ R[[X]], we set (Φ |Jλ ξ)(z, X) = (det ξ)λ/2 J(ξ, z)−λ e−K(ξ,z)X × Φ(ξz, (det ξ)J(ξ, z)

(3.8) −2

X)

for all z ∈ H, where J is as in (3.1). This formula determines a right action of GL+ (2, R) on R[[X]], which extends the action of SL(2, R) given by (1.17).

3.1 Jacobi-like Forms, Modular Series, and Hecke Operators

55

In particular, we have (Φ |Jλ ξ) |Jλ ξ  = Φ |Jλ (ξξ  )

(3.9)

for all ξ, ξ  ∈ GL+ (2, R) and Φ(z, X) ∈ R[[X]]. Furthermore, if Γ is a discrete subgroup of SL(2, R) ⊂ GL+ (2, R), then the formal power series Φ(z, X) is a Jacobi-like form belonging to Jλ (Γ ) if and only if (Φ |Jλ γ)(z, X) = Φ(z, X) for all γ ∈ Γ . If Φ(z, X) is a Jacobi-like form belonging to Jλ (Γ ) and if the double coset of an element α ∈ Γ is as in (3.6), we set (TλJ (α)Φ)(z, X) =

s 

(Φ |Jλ αi )(z, X)

(3.10)

i=1

for all z ∈ H. Proposition 3.2 For each α ∈ Γ the power series TλJ (α)Φ given by (3.10) is independent of the choice of the coset representatives α1 , . . . , αs , and the map Φ → TλJ (α)Φ determines a linear endomorphism TλJ (α) : Jλ (Γ ) → Jλ (Γ ) on the space Jλ (Γ ) of Jacobi-like forms of weight λ for Γ . Proof. Suppose that {β1 , . . . , βs } is another set of coset representatives with βi = γi αi and γi ∈ Γ for 1 ≤ i ≤ s. Using (3.9) and the fact that Φ is a Jacobi-like form belonging to Jλ (Γ ), we have s 

(Φ |Jλ βi )(z, X) =

i=1

=

s  i=1 s 

((Φ |Jλ γi ) |Jλ αi )(z, X)

(3.11)

(Φ |Jλ αi )(z, X),

i=1

and hence TλJ (α)Φ is independent of the choice of the coset representatives. Since the linearity of the map Φ → TλJ (α)Φ is clear, it suffices to show that TλJ (α)(Jλ (Γ )) ⊂ Jλ (Γ ). If Φ(z, X) ∈ Jλ (Γ ) is as above and if γ ∈ Γ , from (3.9), (3.10) and (3.11) we see that (TλJ (α)Φ) |Jλ γ =

s  i=1

(Φ |Jλ αi ) |Jλ γ =

s 

Φ |Jλ (αi γ).

i=1

However, the set {α1 γ, . . . , αs γ} is another complete set of coset representatives, and therefore we have

56

3 Hecke Operators s 

Φ |Jλ (αi γ) =

i=1

s 

Φ |Jλ αi .

i=1

Thus we obtain (TλJ (α)Φ) |Jλ γ = (TλJ (α)Φ); hence the proposition follows.



Definition 3.3 A linear operator of the form TλJ (α) : Jλ (Γ ) → Jλ (Γ ) for α ∈ Γ given by (3.10) is a Hecke operator on the space Jλ (Γ ) of Jacobi-like forms. If δ ≥ 0, we see that TλJ (α)(Jλ (Γ )δ ) ⊂ Jλ (Γ )δ , and therefore we also obtain the Hecke operator TλJ (α) : Jλ (Γ )δ → Jλ (Γ )δ for each δ ≥ 0.

∞ Lemma 3.4 Let Φ(z, X) = k=0 φk (z)X k+δ be a Jacobi-like form belonging to Jλ (Γ )δ , and let α be an element of Γ ⊂ GL+ (2, R) such that the associated double coset Γ αΓ has a decomposition as in (3.6), then we have TλJ (α)Φ(z, X) =

k s  ∞   (−1)

!

i=1 k=0 =0

(det αi )k+δ−+λ/2 K(αi , z)

(3.12)

× J(αi , z)−2k+2−2δ−λ φk− (αi z)X k+δ for all z ∈ H. Proof. From (3.10) we see that (TλJ (α)Φ)(z, X) =

s 

(Φ |Jλ αi )(z, X) ∈ R[[X]].

i=1

However, using (3.8), we have ((Φ |Jλ αi )(z, X) = (det αi )λ/2 J(αi , z)−λ e−K(αi ,z)X × Φ(αi z, (det αi )J(αi , z)−2 X) ∞   (−1)r = (det αi )λ/2 J(αi , z)−λ K(αi , z)r X r r! r=0 ×

∞ 

k=0

φk (αi z)(det αi )k+δ J(αi , z)−2k−2δ X k+δ



3.1 Jacobi-like Forms, Modular Series, and Hecke Operators

=

k ∞   (−1)

!

k=0 =0

57

(det αi )k+δ−+λ/2 J(αi , z)−2k+2−2δ−λ × K(αi , z) φk− (αi z)X k+δ

for 1 ≤ i ≤ s; hence the lemma follows.



We recall from Chapter 1 that there is an isomorphism Ξλ,δ : Jλ (Γ )δ → Mλ (Γ )δ

(3.13)

as in (1.30) between the spaces of Jacobi-like forms and modular series characterized as follows. The elements Φ(z, X) =

∞ 

φk (z)X k+δ ∈ Jλ (Γ )δ ,

k=0

F (z, X) =

∞ 

hk (z)X k+δ ∈ Mλ (Γ )δ

k=0

satisfy the relation Ξλ,δ Φ = F if and only if φk =

k  r=0

1 (r) h r!(2k + 2δ + λ − r − 1)! k−r

(3.14)

for all k ≥ 0 by (1.6), or equivalently h = (2 + 2δ + λ − 1)

  r=0

(−1)r

(2 + 2δ + λ − r − 2)! (r) φ−r r!

(3.15)

for all  ≥ 0 by (1.7). From Lemma 1.9 we see that {h }∞ =0 is a sequence of modular forms with h ∈ M2+2δ+λ (Γ ). We now consider the expression of the image of a Jacobi-like form under the Hecke operator in terms of the associated modular forms.

∞ Theorem 3.5 Let Φ(z, X) = k=0 φk (z)X k+δ be a Jacobi-like form belonging to Jλ (Γ )δ , and let {h }∞ =0 be the corresponding sequence of modular forms given by (3.15). If α is an element of Γ ⊂ GL+ (2, R) such that the associated double coset Γ αΓ has a decomposition as in (3.6), then we have TλJ (α)Φ(z, X) =

μ s  ∞  k   k=0 i=1 μ=0 r=0

(3.16) (−1)μ−r (det αi )k−μ+r+δ+λ/2 (μ − r)!r!(2k − 2μ + 2δ + λ + r − 1)! (r)

×

K(αi , z)μ−r hk−μ (αi z) J(αi , z)2k−2μ+2r+2δ+λ

X k+δ

58

3 Hecke Operators

for all z ∈ H. Proof. If k ≥  ≥ 0, from (3.14) we obtain φk− (αi z) =

k−  r=0

1 (r) h (αi z). r!(2k − 2 + 2δ + λ − r − 1)! k−−r

From this and (3.12) it follows that TλJ (α)Φ(z, X) =

s  k− ∞  k   k=0 i=1 =0 r=0

(−1) (det αi )k−+δ+λ/2 !r!(2k − 2 + 2δ + λ − r − 1)! (r)

×

K(αi , z) hk−−r (αi z) k+δ X . J(αi , z)2k−2+2δ+λ

Using the index μ =  + r, we have TλJ (α)Φ(z, X) =

k s  ∞  k   k=0 i=1 r=0 μ=r

(−1)μ−r (det αi )k−μ+r+δ+λ/2 (μ − r)!r!(2k − 2μ + 2δ + λ + r − 1)! (r)

×

K(αi , z)μ−r hk−μ (αi z) J(αi , z)2k−2μ+2r+2δ+λ

X k+δ .

k μ

k k Replacing r=0 μ=r by μ=0 r=0 in this equation we obtain (3.16), and therefore the proof of the theorem is complete.  Given α ∈ Γ as in Theorem 3.5, we now introduce the Hecke operator TλM (α) : Mλ (Γ )δ → Mλ (Γ )δ on the space Mλ (Γ )δ of modular series defined by TλM (α)

 ∞ =0

 h (z)X +δ

=

∞ 

(T2+2δ+λ (α)h )(z)X +δ ,

(3.17)

=0

where T2+2δ+λ (α) is the Hecke operator in (3.7) on the space M2+2δ+λ (Γ ) of modular forms. Theorem 3.6 For each α ∈ Γ the diagram T J (α)

Jλ (Γ )δ −−λ−−→ Jλ (Γ )δ ⏐ ⏐ ⏐ ⏐Ξ Ξλ,δ   λ,δ T M (α)

Mλ (Γ )δ −−λ−−→ Mλ (Γ )δ is commutative, where Ξλ,δ is as in (1.30).

(3.18)

3.1 Jacobi-like Forms, Modular Series, and Hecke Operators

59

∞  Proof. Let Φ(z, X) ∈ Jλ (Γ ) δ , {hν }ν=0 and α ∈ Γ be as in Theorem 3.5. ∞ k+δ , from (3.17) we obtain Then, since (Ξλ,δ Φ)(z, X) = k=0 hk (z)X

(TλM (α) ◦ Ξλ,δ )Φ(z, X) =

∞ 

(T2+2δ+λ (α)h )(z)X +δ .

=0

Using this and (3.14), we have −1 ◦ TλM (α) ◦ Ξλ,δ )Φ(z, X) (Ξλ,δ

=

(3.19)

k ∞   k=0

1 dμ (T2k−2μ+2δ+λ (α)hk−μ (z))X k+δ . μ μ!(2k + 2δ + λ − μ − 1)! dz μ=0

Assuming that the double coset Γ αΓ associated to α ∈ Γ has a decomposition as in (3.6) and using (1.21) and (3.7), we have dμ (T2k−2μ+2δ+λ (α)hk−μ (z)) dz μ s dμ  = μ (hk−μ |2k−2μ+2δ+λ αi )(z) dz i=1   μ s   μ! 2k − μ + 2δ + λ − 1 = μ−r r! i=1 r=0 ×

(−1)μ−r (det αi )k+δ−μ+r+λ/2 K(αi , z)μ−r (r) hk−μ (αi z). J(αi , z)2k+2δ−2μ+2r+λ

Thus the right-hand side of (3.19) can be written as μ ∞  s  k   (−1)μ−r (det αi )k+δ−μ+r+λ/2 k=0 i=1 μ=0 r=0

=

r!(2k − μ + 2δ + λ − 1)!   2k − μ + 2δ + λ − 1 × μ−r K(αi , z)μ−r (r) × h (αi z)X k+δ J(αi , z)2k−2μ+2δ+2r+λ k−μ

μ s  ∞  k   k=0 i=1 μ=0 r=0

(−1)μ−r (det αi )k+δ−μ+r+λ/2 (μ − r)!r!(2k − 2μ + 2δ + λ + r − 1)! ×

Hence we obtain

K(αi , z)μ−r (r) h (αi z)X k+δ . J(αi , z)2k−2μ+2δ+2r+λ k−μ

−1 Ξλ,δ ◦ TλM (α) ◦ Ξλ,δ = TλJ (α),

and therefore the theorem follows.



60

3 Hecke Operators

Example 3.7 Given nonnegative integers δ and w with w ≥ δ and a modular form f ∈ M2w+λ (Γ ) with w ≥ δ, we consider the formal power series Ff (z, X) =

∞ 

g (z)X +δ ,

=0



where g =

f 0

if  = w − δ; if  = w − δ.

Then by Example 1.12 the Cohen–Kuznetsov lifting of f can be written as −1 (Lw λ,δ f )(z, X) = (Λλ,δ Ff )(z, X) = (Ξλ,δ Ff )(z, X)

=

∞  k=0

f (k+δ−w) (z) X k+δ , (k − w + δ)!(k + w + δ + λ − 1)!

where Λλ,δ and Lw λ,δ are as in (1.30) and (1.36). Thus we see that the diagram Jλ (Γ )δ  ⏐ Lw λ,δ ⏐

T J (α)

−−λ−−→

Jλ (Γ )δ  ⏐ Lw ⏐ λ,δ

(3.20)

Tλ (α)

M2w+λ (Γ ) −−−−→ M2w+λ (Γ ) is commutative for each α ∈ Γ, so that the Cohen–Kuznetsov lifting Lw λ,δ is Hecke equivariant.

3.2 Hecke Operators on Pseudodifferential Operators As was discussed in Chapter 1 there are correspondences among Jacobi-like forms, modular series, and pseudodifferential operators. In this section we introduce Hecke operators on pseudodifferential operators and discuss their compatibility with those on Jacobi-like forms and on modular series under the respective correspondences. If Ψ (z) ∈ Ψ D(R)−ε with ε ∈ Z is a pseudodifferential operator of the form Ψ (z) =

∞ 

ψk (z)∂ −k−ε

(3.21)

k=0

and if β ∈ GL+ (2, R), we define the corresponding pseudodifferential operator (Ψ |∂−ε β)(z) ∈ Ψ D(R)−ε by

3.2 Hecke Operators on Pseudodifferential Operators

(Ψ

|∂−ε

   k ∞   k+ε k+ε−1 β)(z) = !  

61

(3.22)

k=0 =0

×

(det β)k+ε− K(β, z) ψk− (βz)∂ −k−ε J(β, z)2(k+ε−)

for all z ∈ H. Lemma 3.8 Given Ψ (z) ∈ Ψ D(R)−ε , we have Ψ |∂−ε (ββ  ) = (Ψ |∂−ε β) |∂−ε β 

(3.23)

for all β, β  ∈ GL+ (2, R). Proof. For Ψ (z) ∈ Ψ D(R)−ε and β, β  ∈ GL+ (2, R), using (3.22), we have ((Ψ |∂−ε β) |∂−ε β  )(z)    k ∞   k + ε k + ε − 1 (det β)k+ε−j K(β, z)j = j! j j J(β, z)2(k+ε−j) j=0 k=0

  k−j   k+ε−j−1 k+ε−j × !   =0

× =

k−j k  ∞   k=0 j=0 =0

(det β)k+ε−j− K(β, z) ψk−j− (ββ  z)∂ −k−ε J(β, β  z)2(k+ε−j−)

(k + ε)!(k + ε − 1)! j!!(k + ε − j − )!(k + ε − 1 − j − )!

(det ββ  )k+ε−j− J(ββ  , z)2(k+ε−j−)   (det β  )K(β, β  z) × K(β  , z)j ψk−j− (ββ  z)∂ −k−ε . J(β  , z)2 ×

Thus, introducing the index w = j +  and replacing j by w − , we obtain ((Ψ |∂−ε β) |∂−ε β  )(z)

  k + ε k + ε − 1 (det ββ  )k+ε−w = w! w w J(ββ  , z)2(k+ε−w) k=0 j=0   w      w  w− (det β )K(β, β z) K(β , z) ×  J(β  , z)2 ∞  k 



=0

× ψk−w (ββ  z)∂ −k−ε

62

3 Hecke Operators

  k + ε k + ε − 1 (det ββ  )k+ε−w = w! w w J(ββ  , z)2(k+ε−w) k=0 j=0 w  (det β  )K(β, β  z)  × K(β , z) + ψk−w (ββ  z)∂ −k−ε J(β  , z)2    k ∞   k+ε k+ε−1 = w! w w j=0 k ∞  



k=0

×

(det ββ  )k+ε−w K(ββ  , z)w ψk−w (ββ  z)∂ −k−ε J(ββ  , z)2(k+ε−w)

= (Ψ |∂−ε (ββ  ))(z), 

where we used (3.3).

We consider a discrete subgroup Γ of SL(2, R) as before, and let α ∈ Γ be an element of the commensurator of Γ such that the corresponding double coset has a decomposition of the form Γ αΓ =

s 

Γ αi

(3.24)

i=1

with α1 , . . . , αs ∈ GL+ (2, R). We then define the associated linear endomorphism Ψ (α) : Ψ D(R)−ε → Ψ D(R)−ε T−ε with ε ∈ Z by Ψ (α)Ψ )(z) = (T−ε

s 

(Ψ |∂−ε αi )(z)

(3.25)

i=1

for all Ψ (z) ∈ Ψ D(R)−ε . Using (3.23), it can be shown as usual that Ψ (α)(Ψ D(R)Γ−ε ) ⊂ Ψ D(R)Γ−ε ; T−ε

hence the formula (3.25) determines the endomorphism Ψ (α) : Ψ D(R)Γ−ε → Ψ D(R)Γ−ε T−ε

of automorphic pseudodifferential operators for each ε ∈ Z. Given a positive integer ε and a nonnegative integer ξ ≤ ε, let X : Ψ D(R)Γ−ε → J2ξ (Γ )ε−ξ , Iξ,ε

∂ Iξ,ε−ξ : J2ξ (Γ )ε−ξ → Ψ D(R)Γ−ε

be the linear isomorphisms in Corollary 1.16.

(3.26)

3.2 Hecke Operators on Pseudodifferential Operators

63

Lemma 3.9 If Ψ (z) ∈ Ψ D(R)Γ−ε , then we have ∂ X ((Iξ,ε Ψ ) |J2ξ β) = Ψ |∂−ε β, Iξ,ε−ξ

X X (Iξ,ε Ψ ) |J2ξ β = Iξ,ε (Ψ |∂−ε β)

(3.27)

for all β ∈ GL+ (2, R) and ξ ∈ Z. Proof. Assuming that Ψ (z) ∈ Ψ D(R)Γ−ε is given by Ψ (z) =

∞ 

ψk (z)∂ −k−ε ,

k=0

from (1.47) we obtain X Ψ )(z, X) = (Iξ,ε

∞  j=0

(−1)j+ε ψj (z)X j+ε−ξ . (j + ε)!(j + ε − 1)!

Using this and (3.8), we have X Ψ ) |J2ξ β)(z, X) ((Iξ,ε

= (det β)ξ J(ξ, z)−2ξ e−K(ξ,z)X ∞  (−1)j+ε ψj (βz) × (j + ε)!(j + ε − 1)! j=0 × (det β)j+ε−ξ J(β, z)−2j−2ε+2ξ X j+ε−ξ =

∞  ∞  (−1)j++ε (det β)j+ε K(β, z) ψj (βz) j=0 =0

!(j + ε)!(j + ε − 1)!J(β, z)2(j+ε)

X j++ε−ξ .

Thus by using the index k = j +  and replacing j by k −  we obtain X ((Iξ,ε Ψ ) |J2ξ β)(z, X)

=

k ∞   k=0 =0

(−1)k+ε (det β)k+ε− K(β, z) ψk− (βz) X k+ε−ξ . !(k + ε − )!(k + ε −  − 1)!J(β, z)2(k+ε−)

From this and (1.46) we see that ∂ X (((Iξ,ε Ψ ) |J2ξ β))(z) (Iξ,ε−ξ

=

k ∞   (k + ε)!(k + ε − 1)!(det β)k+ε− K(β, z) ψk− (βz) k=0 =0

!(k + ε − )!(k + ε −  − 1)!J(β, z)2(k+ε−)

∂ −k−ε ,

which is equal to the right-hand side of (3.21); hence we obtain the first relation in (3.27). The second relation follows from this and the fact that X ∂ is the inverse of Iξ,ε .  Iξ,ε−ξ

64

3 Hecke Operators

If α ∈ Γ satisfies (3.24), then by (3.12) the Hecke operator J (α) : J2ξ (Γ )ε−ξ → J2ξ (Γ )ε−ξ T2ξ

on J2ξ (Γ )ε−ξ is given by J T2ξ (α)Φ(z, X)

=

s  k ∞   (−1) i=1 k=0 =0

!

(det αi )k+ε− K(αi , z)

(3.28)

× J(αi , z)−2k+2−2ε φk− (αi z)X k+ε−ξ for Φ(z, X) =

∞ 

φk (z)X k+ε−ξ ∈ J2ξ (Γ )ε−ξ .

k=0

Proposition 3.10 Given integers ξ and ε with ε ≥ 1, we have ∂ J X Ψ ◦ T2ξ (α) ◦ Iξ,ε = T−ε (α), Iξ,ε−ξ

J X X Ψ T2ξ (α) ◦ Iξ,ε = Iξ,ε ◦ T−ε (α)

(3.29)

for each α ∈ Γ. Proof. Let α ∈ Γ be an element such that the corresponding double coset has X Ψ )(z, X) a decomposition as in (3.24), and let Ψ (z) ∈ Ψ D(R)Γ−ε , so that (Iξ,ε is a Jacobi-like form belonging to J2ξ (Γ )ε−ξ by Corollary 1.16. Using (3.10), (3.25) and the second relation in (3.27), we have J X ((T2ξ (α) ◦ Iξ,ε )Ψ )(z, X) =

s 

X ((Iξ,ε )Ψ ) |J2ξ αi )(z, X)

i=1

= =

s 

X Iξ,ε (Ψ |∂−ε αi )(z, X)

i=1 X Ψ Iξ,ε (T−ε (α)Ψ )(z, X),

which implies the second relation in (3.29). The first relation follows from X ∂ is the inverse of Iξ,ε−ξ by (1.49).  this and the fact that Iξ,ε Ψ (α) is compatible with the Proposition 3.10 shows that the operator T−ε J Hecke operator T2ξ (α) on Jacobi-like forms.

Definition 3.11 If α ∈ Γ is an element such that the corresponding double coset has a decomposition as in (3.24), then the linear endomorphisms Ψ (α) : Ψ D(R)Γ−ε → Ψ D(R)Γ−ε T−ε

given by (3.25) is the Hecke operator on Ψ D(R)Γ−ε associated to α (see [18] for another description of Hecke operators on pseudodifferential operators).

3.2 Hecke Operators on Pseudodifferential Operators

65

From Proposition 3.10 and (3.18) we obtain the commutative diagram Ψ (α) T−ε

Ψ D(R)Γ−ε −−−−−→ Ψ D(R)Γ−ε ⏐ ⏐ ⏐I X X ⏐ Iξ,ε   ξ,ε J T2ξ (α)

J2ξ (Γ )ε−ξ −−−−→ J2ξ (Γ )ε−ξ ⏐ ⏐ ⏐ ⏐Ξ Ξ2ξ,ε−ξ   2ξ,ε−ξ

(3.30)

M T2ξ (α)

M2ξ (Γ )ε−ξ −−−−→ M2ξ (Γ )ε−ξ for each α ∈ Γ and ε ≥ ξ. The Hecke operators on pseudodifferential operators can also be expressed as follows. Given an automorphic pseudodifferential operator of the form ∞ 

Ψ (z) =

ψ (z)∂ −−ε ∈ Ψ D(R)Γ−ε

(3.31)

=0

and a nonnegative integer k, we define the associated holomorphic function (AΨ )k : H → C by (AΨ )k (z) = (2k + 2ε − 1)

k  (−1)k+ε (2k + 2ε − r − 2)! (r) ψ (z) (3.32) r!(k + ε − r)!(k + ε − r − 1)! k−r r=0

for all z ∈ H. Lemma 3.12 Given integers ξ and ε with ε ≥ ξ, let Ξ2ξ,ε−ξ : J2ξ (Γ )ε−ξ → M2ξ (Γ )ε−ξ ,

X Iξ,ε : Ψ D(R)Γ−ε → J2ξ (Γ )ε−ξ

be isomorphisms in (3.13) and (3.26). If Ψ (z) ∈ Ψ D(R)Γ−ε , then we have ((Ξ2ξ,ε−ξ ◦

X Iξ,ε )Ψ )(z, X)

=

∞ 

(AΨ )k (z)X k+ε−ξ ,

(3.33)

k=0

where (AΨ )k is as in (3.32). In particular, the function (AΨ )k is a modular form belonging to M2k+2ε (Γ ) for each k ≥ 0. Proof. If Ψ (z) ∈ Ψ D(R)Γ−ε is given by (3.31), using (1.47) and Corollary 1.16, we see that X (Iξ,ε Ψ )(z, X) =

∞  k=0

(−1)k+ε ψk (z)X k+ε−ξ (k + ε)!(k + ε − 1)!

(3.34)

66

3 Hecke Operators

is a Jacobi-like form belonging to J2ξ (Γ )ε−ξ . Thus, given k ≥ 0, by (3.15) the function hk = (2k + 2ε − 1)

k 

(−1)r

r=0

× = (2k + 2ε − 1)

(2k + 2ε − r − 2)! r!

(−1)k−r+ε (r) ψ (k + ε − r)!(k + ε − r − 1)! k−r k  (−1)k+ε (2k + 2ε − r − 2)! (r) ψ = (AΨ )k r!(k + ε − r)!(k + ε − r − 1)! k−r r=0

is a modular form belonging to M2k+2ε (Γ ), and the corresponding formal power series ∞ ∞   hk (z)X k+ε−ξ = (AΨ )k (z)X k+ε−ξ k=0

k=0

is a modular series belonging to M2ξ (Γ )ε−ξ and satisfies (3.33).



Proposition 3.13 Given α ∈ Γ and Ψ ∈ Ψ D(R)Γ−ε , we have Ψ (α)Ψ )(z) = (T−ε

k ∞   (−1)k+ε (k + ε)!(k + ε − 1)!

r!(2k + 2ε − r − 1)!

k=0 r=0

(3.35)

× (T2k−2r+2ε (α)(AΨ )k−r (z))(r) ∂ −k−ε for each ε ≥ 0. Proof. The Jacobi-like form X Ψ )(z, X) ∈ J2ξ (Γ )ε−ξ (Iξ,ε

corresponding the automorphic pseudodifferential operator Ψ ∈ Ψ D(R)Γ−ε is given by (3.34). On the other hand, if the functions (AΨ )k are as in (3.32), the corresponding modular series is given by X Ψ ))(z, X) = (Ξ2ξ,ε−ξ (Iξ,ε

∞ 

(AΨ )k (z)X k+δ ∈ M2ξ (Γ )ε−ξ

k=0

as in (3.33). Using this and (3.19), we have J X (α) ◦ Iξ,ε )Ψ )(z, X) ((T2ξ

=

k ∞   k=0 r=0

1 (T2k−2r+2ε (α)(AΨ )k−r (z))(r) X k+ε−ξ r!(2ε + 2k − r − 1)!

for α ∈ Γ. From this, (1.48), (1.50) and (3.35) we obtain

3.3 Differential Equations and Modular Forms

67

∂ J X ((Iξ,ε−ξ ◦ T2ξ (α) ◦ Iξ,ε )Ψ )(z, X)

=

∞  k  (−1)k+ε (k + ε)!(k + ε − 1)! k=0 r=0

r!(2ε + 2k − r − 1)! × (T2k−2r+2ε (α)(AΨ )k−r (z))(r) ∂ −k−ε ,

which coincides with the right-hand side of (3.35). Thus the proposition follows from this and (3.29). 

3.3 Differential Equations and Modular Forms In this section we review connections between meromorphic modular forms and a certain class of linear ordinary differential equations following closely the work of Stiller in [112]. Given a Fuchsian group Γ ⊂ SL(2, R) of the first kind, we first introduce meromorphic modular forms as follows.

Definition 3.14 A meromorphic modular form of weight k for Γ is a meromorphic function f : H → C which is meromorphic at the cusps of Γ and satisfies f |k γ = f k (Γ ) the space of all such meromorphic modfor all γ ∈ Γ . We denote by M ular forms.

Throughout the rest of this chapter we fix a meromorphic modular form 1 (Γ ) of weight one for Γ . Then the associated compact Riemann ϕ ∈ M surface X = Γ \H∗ may be considered as an algebraic curve over C. We denote by K(X) the function field of the algebraic curve X, and choose a nonconstant element x of K(X). If the functions ϕ(z) and zϕ(z) on H are regarded as functions on X, they satisfy a second order homogeneous linear ordinary differential equation Dϕ,X f = 0 on X with Dϕ,X =

d2 d + QX (x) + PX (x) 2 dx dx

(3.36)

that has regular singular points, where PX (x) and QX (x) are elements of K(X). Given an element f ∈ K(X), we have df df dz = , dx dx dx

68

3 Hecke Operators

 d2 f d2 f d  df dz  dz = = dx2 dz dz dx dx dz 2  d2 f = dz 2  d2 f = dz 2

dz df d  dx −1  dz + dx dz dz dz dx df  dx −1 d2 x  dz 2 − dz dz dz 2 dx dx  dz 2 df d log − , dz dz dz dx

where z is the standard coordinate in C. Using this, we can pull the differential operator (3.36) back via the natural projection H∗ → X = Γ \H∗ . Then the homogeneous equation Dϕ,X f = 0 on X is equivalent to the equation Dϕ f = 0 on H with Dϕ =

d2 d + P (z) + Q(z), 2 dz dz

(3.37)

where P (z) and Q(z) are meromorphic functions on H given by P (z) = PX (x(z))

d dx dx − log , dz dz dz

Q(z) = QX (x(z))

 dx 2 dz

(cf. [112, p. 63]). Thus the functions zϕ(z) and ϕ(z) for z ∈ H are linearly independent solutions of the associated homogeneous equation Dϕ f = 0, and the regular singular points of Dϕ coincide with the cusps of Γ (see [112] for details). Given a positive integer m, let S m Dϕ be the linear ordinary differential operator of order m + 1 such that the solutions of the corresponding homogeneous equation S m Dϕ f = 0 are of the form m 

Ci (zϕ(z))m−i (ϕ(z))i =

i=0

m 

Ci z m−i ϕ(z)m

(3.38)

i=0

for some constants Ci ∈ C. In order to describe residue conditions we consider a more general ordinary differential operator of order n of the form D=

dn−1 d dn + P0 , + P + · · · + P1 n−1 dxn dxn−1 dx

where Pi ∈ K(X) for 0 ≤ i ≤ n − 1. Let S ⊂ X be the set of singular points of P0 , . . . , Pn−1 , and let X0 = X − S. We choose a base point x0 ∈ X0 and let ω1 , . . . , ωn be a basis for the space of local solutions of Df = 0 near x0 . Then the Wronskian (3.39) WD = det MD is the determinant of the n × n matrix MD = (dj−1 ωi /dxj−1 ) whose (i, j) entry is dj−1 ωi /dxj−1 for 1 ≤ i, j ≤ n. Given x ∈ X, let η = {η1 , . . . , ηn−1 } be the set of n − 1 local solutions of Df = 0 near x, and let Aη be the

3.3 Differential Equations and Modular Forms

69

(n − 1) × (n − 1) matrix whose (i, j) entry is dj−1 ηi /dxj−1 for 1 ≤ i, j ≤ n − 1. Then a function ψ ∈ K(X) is said to satisfy the residue conditions with respect to D if the differential (Aη ψ/WD )dx has zero residue at every x ∈ X0 = X − S for each set η of n − 1 local solutions of Df = 0 near x. Definition 3.15 An element ψ ∈ K(X) is said to satisfy the parabolic residue conditions with respect to D if it satisfies the residue conditions and if for each η the differential (Aη ψ/WD )dx has zero residue at every singular point x ∈ S whenever Aη is single-valued. Let ν be a positive integer, and let Pν,ϕ be the set of meromorphic functions ψ on H such that the corresponding elements ψX ∈ K(X) satisfy the parabolic residue conditions with respect to S 2ν Dϕ . Given ψ ∈ Pν,ϕ , we denote by S(ψ) a solution of the differential equation S 2ν Dϕ f = ψ, and set   d2ν+1 S(ψ) . (3.40) ρν,ϕ (ψ) = 2ν+1 dz ϕ2ν Note that ρν,ϕ (ψ) is independent of the choice of the solution S(ψ) because we have    2ν  2ν 1  d2ν+1 d2ν+1  2ν−i 2ν 2ν+i ai z ϕ ai z = 2ν+1 =0 dz 2ν+1 ϕ2ν i=0 dz i=0 for any constants ai ∈ C. Lemma 3.16 The function ρν,ϕ (ψ) on H given by (3.40) is a meromorphic modular form for Γ of weight 2ν + 2, and the associated map 2ν+2 (Γ ) ρν,ϕ : Pν,ϕ → M is a one-to-one linear map of complex vector spaces. 2ν+2 (Γ ) follows from results Proof. The fact that ρν,ϕ (ψ) is an element of M in [112, p. 32]. Since the map ρν,ϕ is clearly complex linear, it suffices to show that its kernel is zero. Suppose ρν,ϕ (ψ) = 0 for some ψ ∈ Pν,ϕ . Then by (3.40) we see that 2ν  S(ψ) = ϕ(z)2ν ai z i i=0

for some constants ai ∈ C. Since S(ψ) is a solution of the differential equation S 2ν Dϕ f = ψ, we have  ψ = S Dϕ S(ψ) = S Dϕ ϕ(z) 2ν

2ν

2ν

2ν  i=0

 ai z

i

.

70

3 Hecke Operators

2ν However, by (3.38) the functions ϕ(z)2ν i=0 ai z i are solutions of the homo  geneous equation S 2ν Dϕ f = 0, and therefore it follows that ψ = 0.

We now extend the notion of pseudodifferential operators and Jacobi-like be the ring of meromorforms to the case of meromorphic coefficients. Let R phic functions on H, and denote by Ψ D(R) the set of all pseudodifferential If α ∈ Z, we use Ψ D(R) α to denote the subspace of Ψ D(R) operators over R. consisting of the elements of the form ∞ 

ψk (z)∂ −k+α

k=0

for each k ≥ 0. Similarly, let Ψ D(R) Γ be the subspace of Ψ D(R) with ψk ∈ R consisting of Γ -invariant elements, and set Γ ∩ Ψ D(R) α. Γ = Ψ D(R) Ψ D(R) α We also denote by Jλ (Γ ) the space of Jacobi-like forms of weight λ for Γ with meromorphic coefficients, and set Jλ (Γ )δ = X δ Jλ (Γ ) for δ ≥ 0. Then we see that most of the results on pseudodifferential operators and Jacobi-like forms obtained in the previous chapters remain valid for the meromorphic cases. Given a nonnegative integer ξ, we extend the map Iξ∂ in (1.46) to the case of meromorphic coefficients, so that we obtain the map Iξ∂ : R[[X]] δ → Ψ D(R)−δ−ξ given by (Iξ∂ Φ)(z) =

∞ 

−δ−ξ Ck+δ+ξ φk (z)∂ −k−δ−ξ ∈ Ψ D(R)

(3.41)

k=0

∞ for Φ(z, X) = k=0 φ(z)X k+δ ∈ R[[X]] δ , where Ck+δ+ξ is as in (1.48). By Corollary 1.16 the pseudodifferential operator in (3.41) is Γ -invariant if and only if Φ(z, X) ∈ J2ξ (Γ )δ . On the other hand, by Theorem 1.11 the same condition is satisfied if and only if there is a meromorphic modular form 2j+2ξ (Γ ) for each j ≥ 0 such that hj ∈ M φk (z) =

k  r=0

1 (r) h (z) r!(2k + 2δ + 2ξ − r − 1)! k−r

(3.42)

3.3 Differential Equations and Modular Forms

71

for all z ∈ H and k ≥ 0. Furthermore, by Proposition 1.2 the formula (3.42) is equivalent to the relation hk (z) = (2k + 2δ + 2ξ − 1)

k  (−1)r (2k + 2δ + 2ξ − r − 2)!

r!

r=0

(r) φk−r (z)

for each k ≥ 0. ∞ Let ν=ξ−1 Pν,ϕ be the set of sequences (ψν )∞ ν=ξ−1 of meromorphic functions on H, which has the natural structure of a complex vector space. Given ∞ ∈ P , we set a sequence ψ = (ψν )∞ ν,ϕ ν=ξ−1 ν=ξ−1 Eϕδ,ξ (ψ) =

k ∞   k=0 r=0

Ck+δ+ξ (3.43) r!(2k + 2δ + 2ξ − r − 1)! (2k+2ξ−r−1)  S(ψ k+ξ−r−1 ) ∂ −k−δ−ξ , × ϕ2k+2ξ−2r−2

where ξ is a nonnegative integer with ξ > 1.

Theorem 3.17 The formula (3.43) determines a linear map ∞ 

Eϕδ,ξ :

Γ Pν,ϕ → Ψ D(R) −δ−ξ

ν=ξ−1

of complex vector spaces for δ, ξ ∈ Z with δ ≥ 0 and ξ ≥ 1.

Proof. Given a sequence ψ = (ψν )∞ ν=ξ−1 ∈ η =

 S(ψ

+ξ−1 ) 2+2ξ−2 ϕ

∞ ν=ξ

Pν,ϕ , we set

(2+2ξ−1) (3.44)

for integers  ≥ 0. Then from Lemma 3.16 and (3.40) it follows that η ∈ 2+2ξ (Γ ) for all  ≥ 0. Thus by Theorem 1.11(iii) the formal power series M k ∞   k=0 r=0

1 (r) η X k+δ r!(2k + 2δ + 2ξ − r − 1)! k−r

is a Jacobi-like form belonging to J2ξ (Γ )δ . Using this, Corollary 1.16 and (1.46), we see that the pseudodifferential operator

72

3 Hecke Operators k ∞   k=0

(r)

Ck+δ+ξ ηk−r ∂ −k−δ−ξ r!(2k + 2δ + 2ξ − r − 1)! r=0 =

k ∞   k=0 r=0

Ck+δ+ξ r!(2k + 2δ + 2ξ − r − 1)! (2k+2ξ−r−1)  S(ψ k+ξ−r−1 ) ∂ −k−δ−ξ × 2k+2ξ−2r−2 ϕ

Γ is Γ -invariant; hence it follows that Eϕδ,ξ (ψ) ∈ Ψ D(R) −δ−ξ . Since the linearity δ,ξ of Eϕ is clear, the proof of the theorem is complete. 

3.4 Hecke Operators and Differential Equations As was discussed in Section 3.3, modular forms are related to certain linear ordinary differential equations. In this section, using such relations, we introduce operators which may be regarded as Hecke operators on differential equations (cf. [63]). Let Γ ⊂ SL(2, R) be a Fuchsian group of the first kind as in Section 3.3, and let Γ be its commensurator. We consider another Fuchsian group of the first kind Γ0 whose commensurator coincides with Γ. Then for each α ∈ Γ the double coset Γ αΓ0 has a decomposition of the form Γ αΓ0 =

d 

Γ α

(3.45)

=1

k (Γ ) associated for some α1 , . . . , αd ∈ GL+ (2, R). The Hecke operator on M k (Γ ) → M k (Γ0 ) defined to the double coset Γ αΓ0 is the linear map Tk (α) : M by d  (f |k α ) (3.46) Tk (α)f = =1

k (Γ ), where f |k α is as in (3.4). In particular, if Γ0 = Γ , then for all f ∈ M k (Γ ). Tk (α) is a linear endomorphism of M 1 (Γ ) and Let Γ, Γ0 ⊂ SL(2, R) with Γ ∼ Γ0 be as above, and let ϕ ∈ M S Dϕ be as in Section 3.3. If ϕ0 : H → C is a nonzero meromorphic modular form of weight one for Γ0 , then, as in (3.37), we can consider the associated differential operator m

D ϕ0 =

d2 d + P0 (z) + Q0 (z) 2 dz dz

3.4 Hecke Operators and Differential Equations

73

such that P0 (z) and Q0 (z) are meromorphic functions on H, the functions zϕ0 (z) and ϕ0 (z) are linearly independent solutions of the associated homogeneous equation Dϕ0 f = 0, and the regular singular points of Dϕ0 coincide with the cusps of Γ0 . Thus S m Dϕ0 is the differential operator such that {z m−i ϕ0 (z)m | 0 ≤ i ≤ m} is the set of linearly independent solutions of the homogeneous equation S m Dϕ0 f = 0. We also consider the associated space Pν,ϕ0 of meromorphic functions on H satisfying parabolic residue conditions with respect to S 2ν Dϕ0 and the complex linear map 2ν+2 (Γ0 ) ρν,ϕ0 : Pν,ϕ0 → M using (3.40) with ϕ replaced by ϕ0 . Let α be an element of the commensurator Γ of Γ such that Γ αΓ0 is as in 2ν+2 (Γ ) → M 2ν+2 (Γ0 ) be the associated Hecke (3.45), and let T2ν+2 (α) : M operator in (3.46) with k = 2ν + 2. Given ψ ∈ Pν,ϕ , we set   Fν,α (z) = ϕ0 (z)2ν · · · (T2ν+2 (α))(ρν,ϕ (ψ)) dz · · · dz, (3.47)

(TνP (α)ψ)(z) = det(α)ν

d  det(α )ν+1 ψ(α z)

(3.48) J(α , z)2ν+2 2ν+2    WDϕ0 (z) 2ν+1 ϕ(α z) × ϕ0 (z) WDϕ (α z)   for all z ∈ H, where ρν,ϕ (ψ) is as in (3.40) and · · · dz · · · dz denotes the (2ν + 1)-fold indefinite integral with respect to z. =1

Theorem 3.18 The formula (3.48) determines the complex linear map TνP (α) : Pν,ϕ → Pν,ϕ0 satisfying

TνP (α)(ψ) = S 2ν Dϕ0 (Fν,α )

(3.49)

for all ψ ∈ Pν,ϕ , where S 2ν Dϕ0 (Fν,α ) is the function obtained by applying the differential operator S 2ν Dϕ0 to the function Fν,α given by (3.47). Proof. Given ψ ∈ Pν,ϕ , using [112, Theorem 3 bis. 5], we have  S(ψ) (2ν+1) ϕ2ν

= (−1)2ν+1

ϕ2ν+2 ψ ϕ2ν+2 ψ = − 2ν+1 , 2ν+1   W W Dϕ

Dϕ

74

3 Hecke Operators

D (z) is the pullback of the Wronskian WD (x) defined as in (3.39) where W ϕ ϕ via the natural projection map H → X0 = Γ \H. From this and Lemma 3.16 we see that the function ρν,ϕ (ψ) = −

ϕ2ν+2 ψ  2ν+1 W Dϕ

2ν+2 (Γ ). Thus, using on H is a meromorphic modular form belonging to M (3.46) and (3.48), we obtain T2ν+2 (α)(ρν,ϕ (ψ))(z) = − det(α)ν =−

d  det(α )ν+1 ϕ2ν+2 (α z)ψ(α z)

=1 2ν+2

 2ν+1 (α z) J(α , z)2ν+2 W Dϕ

(3.50)

ϕ0 (z) (T P (α)ψ)(z)  2ν+1 (z) ν W Dϕ0

= ρν,ϕ0 (TνP (α)(ψ))(z) for all z ∈ H. On the other hand, since Fν,α is a solution of the differential equation S 2ν Dϕ0 f = S 2ν Dϕ0 (Fν,α ), it follows from (3.40), (3.47) and (3.50) that d2ν+1  Fν,α  d2ν+1  S(S 2ν Dϕ0 (Fν,α ))  = dz 2ν+1 ϕ2ν dz 2ν+1 ϕ2ν P = T2ν+2 (α)(ρν,ϕ (ψ)) = ρν,ϕ0 (Tν (α)(ψ)).

ρν,ϕ0 (S 2ν Dϕ0 (Fν,α )) =

Since ρν,ϕ0 is injective by Lemma 3.16, we obtain (3.49), and therefore the proof of the theorem is complete.  The linear map TνP (α) in Theorem 3.18 may be regarded as the Hecke operator on Pν,ϕ associated to α, and such operators allow us to define the map ∞ ∞   Pν,ϕ → Pν,ϕ0 T P (α) : ν=1

ν=1

associated to α by setting T P (α)ψ = (TνP (α)ψν ) ∞ for each sequence ψ = (ψν )∞ ν=1 ∈ ν=1 Pν,ϕ . Given an automorphic pseudodifferential operator Φ(z) =

∞  =0

Γ φ (z)∂ −−ε ∈ Ψ D(R) −ε

3.4 Hecke Operators and Differential Equations

75

and a nonnegative integer k, by extending (3.32) to the meromorphic case we obtain the meromorphic function (AΨ )k : H → C given by k  (−1)k+ε (2k + 2ε − r − 2)! (r) ψ , (AΨ )k = (2k + 2ε − 1) r!(k + ε − r)!(k + ε − r − 1)! k−r r=0

(3.51)

2k+2ε (Γ ). The meromorphic version of (3.35) provides which belongs to M Ψ Γ associated to an element α ∈ Γ us the Hecke operator T−ε (α) on Ψ D(R) −ε given by Ψ (T−ε (α)Ψ )(z) =

k ∞   (−1)k+ε (k + ε)!(k + ε − 1)! k=0 r=0

r!(2k + 2ε − r − 1)!

(3.52)

× (Tk−r (α)(AΨ )k−r (z))(r) ∂ −k−ε with

2k+2ε−2r (Γ ), A(Φ)k−r ∈ M

Ψ (α) is compatible with for each k ≥ 0. The following theorem shows that T−ε P the Hecke operator T (α) on Pν,ϕ .

Theorem 3.19 Let ε and ξ be integers with ε > 0 and ε ≥ ξ ≥ 0. For Γ the pseudodifferential operator T Ψ (α)Φ given by (3.52) is each Φ ∈ Ψ D(R) −ε Γ -invariant, and the diagram ∞

δ,ξ Eϕ

Γ Pν,ϕ −−−−→ Ψ D(R) −ε ⏐ ⏐ ⏐ ⏐ Ψ (α) T P (α) T−ε ∞ Ξϕ0 Γ ν=1 Pν,ϕ0 −−−−→ Ψ D(R)−ε ν=1

is commutative. ∞ Proof. Let ψ = (ψν )∞ ν=ε−ξ ∈ ν=1 Pν,ϕ . Then, using (3.40) and (3.43) with δ = ε − ξ, we obtain an automorphic pseudodifferential operator Eϕδ,ξ (ψ) ∈ Γ given by Ψ D(R) −ε ∞  Ck+ε φk ∂ −k−ε , (3.53) Eϕδ,ε (ψ) = k=0

where φk =

k  r=0

1 (ρk+ε−r−1,ϕ (ψk+ε−r−1 ))(r) r!(2k + 2ε − r − 1)!

(3.54)

for all k ≥ 0. From (1.7) we see that (3.54) is equivalent to the relation

76

3 Hecke Operators

ρk+ε−1,ϕ (ψk+ε−1 ) = (2k + 2ε − 1)

k  (−1)r (2k + 2ε − r − 2)!

r!

r=0

φk−r .

On the other hand, from (3.51) and (3.53) we obtain A(Eϕδ,ξ (ψ))k = (2k + 2ε − 1) = (2k + 2ε − 1)

k  (−1)k+ε (2k + 2ε − r − 2)! Ck+ε−r φk−r r!(k + ε − r)!(k + ε − r − 1)! r=0 k  (−1)r (2k + 2ε − r − 2)!

r!

r=0

φk−r ,

2k+2ε (Γ ). Thus we have which is a meromorphic modular form belonging to M ρk+ε−1,ϕ (ψk+ε−1 ) = A(Eϕδ,ξ (ψ))k for each k ≥ 0. Hence from (3.35) we see that Ψ (α)(Eϕδ,ξ (ψ)) = T−ε

k ∞  

(−1)k+ε

k=0 r=0

(k + ε)!(k + ε − 1)! r!(2k + 2ε − r − 1)!

× (T2k+2ε−2r (α)(ρk+ε−r−1,ϕ (ψk+ε−r−1 ))(r) ∂ −k−ε . Using (3.53) and the relation T P (α)ψ = (TνP (α)(ψν )), we have Ξϕ0 (T P (α)ψ) =

k ∞   (−1)k+ε (k + ε)!(k + ε − 1)!

r!(2k + 2ε − r − 1)!

k=0 r=0

P × (ρk+ε−r−1,ϕ0 (Tk+ε−r−1 (α)(ψk+ε−r−1 )))(r) ∂ −k−ε .

However, from (3.50) we obtain P ρk+ε−r−1,ϕ0 (Tk+ε−r−1 (α)(ψk+ε−r−1 ))

= T2k+2ε−2r (α)(ρk+ε−r−1,ϕ (ψk+ε−r−1 )). Hence we see that Ψ (α)(Eϕδ,ξ (ψ)) = Eϕδ,ξ (T P (α)ψ), T−ε 0

and therefore the theorem follows.



Chapter 4

Lie Algebras

Given a pseudodifferential operator, as was discussed in Section 1.4, we can obtain the corresponding formal power series by using some constant multiples of its coefficients in such a way that the correspondence is SL(2, R)equivariant. The space of pseudodifferential operators is a noncommutative algebra over C and therefore has a natural structure of a Lie algebra. In this chapter we determine the corresponding Lie algebra structure on the space of formal power series and study some of its properties. We also discuss these results in connection with automorphic pseudodifferential operators, Jacobilike forms, and modular series for a discrete subgroup of SL(2, R) (cf. [66]).

4.1 Lie Algebras of Power Series In this section we introduce a Lie algebra structure on the space of formal power series that is compatible with the natural Lie algebra structure on the noncommutative algebra of pseudodifferential operators. We determine an explicit formula for the associated Lie bracket. Given nonnegative integers δ1 , δ2 and pseudodifferential operators Ψ (z) ∈ Ψ D(R)−δ1 ,

Φ(z) ∈ Ψ D(R)−δ2 ,

of the form Ψ (z) =

∞ 

ψk (z)∂ −k−δ1 ,

Φ(z) =

k=0

∞ 

φk (z)∂ −k−δ2 ,

k=0

we see that Ψ (z)Φ(z) =

 ∞  ∞  ∞   −k − δ1 k=0 =0 q=0

q

ψk (z)φ (z)∂ −k−−q−δ1 −δ2 . (q)

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_4

77

78

4 Lie Algebras

Changing the indices from k, , q to r, p, q with r = k +  + q and p =  + q, we have  p  r  ∞   −r − δ1 + p (q) ψr−p (z)φp−q (z)∂ −r−δ1 −δ2 . Ψ (z)Φ(z) = q r=0 p=0 q=0 From this we see that the Lie bracket [·, ·]∂ associated to the noncommutative product operation on Ψ D(R) can be written as [Ψ (z), Φ(z)]∂ = Ψ (z)Φ(z) − Φ(z)Ψ (z) =

∞ 

η(Ψ, Φ)r (z)∂ −r−δ1 −δ2 ,

(4.1)

r=0

where η(Ψ, Φ)r (z) =

 p  r   −r − δ1 + p p=0 q=0

(q)

ψr−p (z)φp−q (z)

q

 −

  −r − δ2 + p (q) φr−p (z)ψp−q (z) q

for all r ≥ 0. If r, p, q, α and β are integers with r ≥ p ≥ q and α, β > 0, we set r,p,q = Ξα,β

(r − p + α)!(r − p + q + α − 1)! q!(r + α + β)! (p − q + β)!(p − q + β − 1)! × . (r + α + β − 1)!

(4.2)

We then consider formal power series F (z, X) ∈ R[[X]]δ1 and G(z, X) ∈ R[[X]]δ2 given by F (z, X) =

∞ 

fk (z)X k+δ1 ,

G(z, X) =

k=0

∞ 

gk (z)X k+δ2 ,

(4.3)

k=0

and define the associated formal power series [F (z, X), G(z, X)]X ∈ R[[X]]δ1 +δ2 by [F (z, X), G(z, X)]

X

=

p  r  ∞  

(q)

fr−p (z)gp−q (z) Ξδr,p,q 1 +ξ1 ,δ2 +ξ2

r=0 p=0 q=0

−

(q) gr−p (z)fp−q (z) Ξδr,p,q 2 +ξ2 ,δ1 +ξ1

(4.4)

 X r+δ1 +δ2 .

4.1 Lie Algebras of Power Series

79

Theorem 4.1 The formula (4.4) determines a bilinear map [·, ·]X : R[[X]]δ1 × R[[X]]δ2 → R[[X]]δ1 +δ2 of formal power series satisfying   Iξ∂1 +ξ2 ,δ1 +δ2 [F (z, X), G(z, X)]X =

(4.5)

[Iξ∂1 ,δ1 (F (z, X)), Iξ∂2 ,δ2 (G(z, X))]∂

for F (z, X) ∈ R[[X]]δ1 , G(z, X) ∈ R[[X]]δ2 and ξ1 , ξ2 ≥ 0, where the maps ∂ are as in (1.46). I∗,∗

Proof. Let F (z, X) ∈ R[[X]]δ1 and G(z, X) ∈ R[[X]]δ2 be as in (4.3). Then by (1.46) the pseudodifferential operators Iξ∂1 ,δ1 (F (z, X)) and Iξ∂2 ,δ2 (G(z, X)) are given by

Iξ∂1 ,δ1 (F (z, X)) = Iξ∂2 ,δ2 (G(z, X)) =

∞  k=0 ∞ 

Ck+δ1 +ξ1 fk (z)∂ −k−δ1 −ξ1 , Ck+δ2 +ξ2 gk (z)∂ −k−δ2 −ξ2

k=0

for ξ1 , ξ2 ≥ 0. From these relations and (4.1) we obtain

[Iξ∂1 ,δ1 (F (z, X)), Iξ∂2 ,δ2 (G(z, X))]∂  p  r  ∞   −r − δ1 − ξ1 + p Cr−p+δ1 +ξ1 = q r=0 p=0 q=0 (q)

× Cp−q+δ2 +ξ2 fr−p (z)gp−q (z)   −r − δ2 − ξ2 + p Cr−p+δ2 +ξ2 − q  (q) × Cp−q+δ1 +ξ1 gr−p (z)fp−q (z) ∂ −r−δ1 −δ2 −ξ1 −ξ2 .

Using this and (1.47), we have

80

4 Lie Algebras

IξX1 +ξ2 ,δ1 +δ2 +ξ1 +ξ2 [Iξ∂1 ,δ1 (F (z, X)), Iξ∂2 ,δ2 (G(z, X))]∂  p  r  ∞   −r − δ1 − ξ1 + p Cr−p+δ1 +ξ1 = q r=0 p=0 q=0 (q)

× Cp−q+δ2 +ξ2 fr−p (z)gp−q (z)  −r − δ2 − ξ2 + p Cr−p+δ2 +ξ2 − q  X r+δ1 +δ2 (q) × Cp−q+δ1 +ξ1 gr−p (z)fp−q (z) , Cr+δ1 +δ2 +ξ1 +ξ2 

which can be easily seen to coincide with the right-hand side of (4.4); hence we obtain IξX1 +ξ2 ,δ1 +δ2 +ξ1 +ξ2 [Iξ∂1 ,δ1 (F (z, X)), Iξ∂2 ,δ2 (G(z, X))]∂ = [F (z, X), G(z, X)]X . Using this and (1.49), we have Iξ∂1 +ξ2 ,δ1 +δ2 [F (z, X), G(z, X)]X = (Iξ∂1 +ξ2 ,δ1 +δ2 ◦ IξX1 +ξ2 ,δ1 +δ2 +ξ1 +ξ2 )[Iξ∂1 ,δ1 (F (z, X)), Iξ∂2 ,δ2 (G(z, X))]∂ = [Iξ∂1 ,δ1 (F (z, X)), Iξ∂2 ,δ2 (G(z, X))]∂ , 

which verifies (4.5).

From Theorem 4.1 it follows that [ , ]X is a Lie bracket on R[[X]], and therefore R[[X]] together with this bracket is a complex Lie algebra.

4.2 Lie Algebra Homomorphisms In this section we determine a Lie algebra homomorphism from sl(2, C) to R[[X]] which corresponds to the natural action of SL(2, C) on Ψ D(R). Let End(Ψ D(R)) be the space of complex linear endomorphisms of Ψ D(R), and let Aut(Ψ D(R)) be the subgroup End(Ψ D(R)) consisting of the invertible elements. We consider the left action ρ : SL(2, C) → Aut(Ψ D(R)) of SL(2, C) on Ψ D(R) defined by (ρ(γ)Ψ )(z) = (Ψ ◦ γ −1 )(z) = Ψ (γ −1 z)

(4.6)

for all γ ∈ SL(2, C) and Ψ (z) ∈ Ψ D(R), where (Ψ ◦ γ −1 )(z) is as in (1.40). Then, as usual, the homomorphism ρ determines a linear map σ : sl(2, C) → End(Ψ D(R))

4.2 Lie Algebra Homomorphisms

81

defined by σ(v)Ψ (z) =

  d d   ρ(etv )Ψ (z) = Ψ (e−tv z) dt dt t=0 t=0

(4.7)

for v ∈ sl(2, C), Ψ (z) ∈ Ψ D(R) and z ∈ C, which satisfies the relation σ([v, w]) = σ(v)σ(w) − σ(w)σ(v) for all v, w ∈ sl(2, C). We shall show below that each endomorphism σ(v) ∈ End(Ψ D(R)) of Ψ D(R) can be given by left multiplication by some element of Ψ D(R). We consider the standard basis {h, e+ , e− } for the Lie algebra sl(2, C) given by       1 0 01 00 h= , e+ = , e− = , (4.8) 0 −1 00 10 which satisfy the relations [h, e+ ] = 2e+ ,

[h, e− ] = −2e− ,

[e+ , e− ] = h.

Lemma 4.2 If v ∈ sl(2, C) is an element given by v = K 1 h + K 2 e + + K3 e − with K1 , K2 , K3 ∈ C, then we have σ(v)Ψ (z) = ((−2K1 z − K2 + K3 z 2 )∂)Ψ (z)

(4.9)

for all Ψ (z) ∈ Ψ D(R). Proof. For the basis vectors of sl(2, C) in (4.8) we see that  −t      e 0 1 −t 1 0 −te+ −te− , e = , e = e−th = 0 1 −t 1 0 et for all t ∈ C. From these relations and (4.7) we obtain   d   Ψ (e−2t z) = −2e−2t z∂Ψ (e−2t z) = −2z∂Ψ (z), dt t=0 t=0   d   = −∂Ψ (z − t) = −∂Ψ (z), σ(e+ )Ψ (z) = Ψ (z − t) dt t=0 t=0    z z d  z2   = ∂Ψ σ(e− )Ψ (z) = Ψ   dt −tz + 1 t=0 (−tz + 1)2 −tz + 1 t=0 σ(h)Ψ (z) =

= z 2 ∂Ψ (z) for all z ∈ H. Thus the lemma follows from these by linear extension.



82

4 Lie Algebras

Lemma 4.2 shows in particular that each σ(v) with v ∈ sl(2, C) may be regarded as an element of Ψ D(R) itself in such a way that σ(v)Ψ (z) is the product of the pseudodifferential operators σ(v) and Ψ (z). On the other hand, the noncommutative complex algebra Ψ D(R) has the structure of a Lie algebra whose bracket operation is given by [Ψ1 , Ψ2 ]∂ = Ψ1 Ψ2 − Ψ2 Ψ1 for all Ψ1 , Ψ2 ∈ End(Ψ D(R)). Thus we see that the map σ may be regarded as the Lie algebra homomorphism σ : sl(2, C) → Ψ D(R). As a result, the adjoint representation of the Lie algebra Ψ D(R) determines the representation A : sl(2, C) → End(Ψ D(R)) of sl(2, C) on Ψ D(R) given by A(v)Ψ (z) = ad(σ(v))Ψ (z) = [σ(v), Ψ (z)]

(4.10)

for all v ∈ sl(2, C) and Ψ (z) ∈ Ψ D(R). Proposition 4.3 The values of A at the standard basis vectors of sl(2, C) in (4.8) are given by A(h)(Ψ (z)) = −2 A(e+ )(Ψ (z)) = − A(e− )(Ψ (z)) =

 z(∂ψν )(z) − (α − ν)ψν (z) ∂ α−ν

∞  

ν=0 ∞ 

(∂ψν )(z)∂ α−ν

ν=0 ∞  

z 2 (∂ψν )(z) − 2(α − ν)zψν (z)

ν=0

with ψ−1 = 0 for Ψ (z) =

(4.11) (4.12) (4.13)

 − (α − ν)(α − ν + 1)ψν−1 (z) ∂ α−ν ,

∞ ν=0

ψν (z)∂ α−ν ∈ Ψ D(R)α with α > 0.

Proof. Consider an element of the form g(z)∂ ω ∈ Ψ D(R) with g ∈ R and ω ∈ Z. Then, using (4.9) and (4.10), we have 1 − A(h)(g(z)∂ ω ) = [z∂, g(z)∂ ω ] = z∂(g(z)∂ ω ) − g(z)∂ ω (z∂) 2 = z(∂g)(z)∂ ω + zg(z)∂ ω+1 − g(z)z∂ ω+1 − ωg(z)∂ ω = (z(∂g)(z) − ωg(z))∂ ω −A(e+ )(g(z)∂ ω ) = [∂, g(z)∂ ω ] = ∂(g(z)∂ ω ) − g(z)∂ ω+1

4.3 Equivariant Splittings

83

= (∂g)(z)∂ ω + g(z)∂ ω+1 − g(z)∂ ω+1 = (∂g)(z)∂ ω A(e− )(g(z)∂ ω ) = [z 2 ∂, g(z)∂ ω ] = z 2 ∂(g(z)∂ ω ) − g(z)∂ ω (z 2 ∂) = z 2 (∂g(z))∂ ω + z 2 g(z)∂ ω+1 − g(z)(z 2 ∂ ω+1 + 2ωz∂ ω + ω(ω − 1)∂ ω−1 ) = (z 2 (∂g(z)) − 2ωzg(z))∂ ω − ω(ω − 1)g(z)∂ ω−1 . From these relations the identities (4.11), (4.12) and (4.13) can be obtained easily.  Using the notation in (1.47), we see that the map X ◦ σ : sl(2, C) → R[[X]] σX = I0,0

is a Lie algebra homomorphism, where the Lie algebra structure on R[[X]] is given by the bracket in (4.4) with δ1 = δ2 = 0 = ξ1 = ξ2 . Indeed, for ∂ −1 X ) = I0,0 , we have v, w ∈ sl(2, C), using (4.5) and the relation (I0,0 [(σX (v))(z, X), (σX (w))(z, X)]X X ∂ ∂ = I0,0 [(I0,0 ◦ σX )(v))(z), ((I0,0 ◦ σX )(w))(z)]∂ X = I0,0 [(σ(v))(z), (σ(w))(z)]∂ X = (I0,0 ◦ σ)([v, w])(z) = σX ([v.w])(z, X).

Thus it follows that the Lie algebra sl(2, C) acts on R[[X]] by the adjoint representation, so that v · F (z, X) = ad(v)F (z, X) = [(σX v)(z, X), F (z, X)]X for v ∈ sl(2, C) and F (z, X) ∈ R[[X]].

4.3 Equivariant Splittings Given α ∈ Z, there is a complex linear map from Ψ D(R)α to R which carries a pseudodifferential operator to its coefficient of ∂ α . In this section we obtain a right inverse of this map that is equivariant with respect to an action of sl(2, C). If {h, e+ , e− } is the standard basis in (4.8) for sl(2, C), using the inner product v, w = Tr(vw) for v, w ∈ sl(2, C), we see that h, h = 2,

h, e+  = 0,

h, e−  = 0,

e+ , e−  = 1.

84

4 Lie Algebras

Thus the corresponding dual basis {h∗ , (e+ )∗ , (e− )∗ } is given by h∗ = h/2,

(e+ )∗ = e− ,

(e− )∗ = e+ ,

and the associated Casimir operator C belonging to the universal enveloping algebra U (sl(2, C)) of sl(2, C) can be written as C = hh∗ + e+ (e+ )∗ + e− (e− )∗ =

1 2 h + e+ e− + e− e+ . 2

(4.14)

The action of sl(2, C) given by (4.10) induces the action of C on Ψ D(R)α , whose formula is given in the next theorem.

∞ Proposition 4.4 If Ψ (z) = ν=0 ψν (z)∂ α−ν ∈ Ψ D(R)α , then we have A(C)Ψ (z) = 2

∞ 

(α − ν)(α − ν + 1)(ψν (z) + (∂ψν−1 )(z))∂ α−ν

(4.15)

ν=0

with ψ−1 = 0. Proof. Using (4.11), (4.12) and (4.13), we have A(h2 )Ψ (z) = 4

∞  

z∂(z(∂ψν )(z) − (α − ν)ψν (z))

ν=0

 + (α − ν)(z(∂ψν )(z) + (α − ν)ψν (z)) ∂ α−ν

=4

∞  

z(∂ψν )(z) + z 2 (∂ 2 ψν )(z) − (α − ν)z(∂ψν )(z)

ν=0

=4

 − (α − ν)z(∂ψν )(z) + (α − ν)2 ψν (z) ∂ α−ν

∞  

(1 − 2α + 2ν)z(∂ψν )(z)

ν=0

A(e+ e− )Ψ (z) = −

 + z 2 (∂ 2 ψν )(z) + (α − ν)2 ψν (z) ∂ α−ν ,

∞   ∂ z 2 (∂ψν )(z) − 2(α − ν)zψν (z) ν=0

=−

 − (α − ν)(α − ν + 1)ψν−1 (z) ∂ α−ν

∞  

2z(∂ψν )(z) + z 2 (∂ 2 ψν )(z)

ν=0

− 2(α − ν)ψν (z) − 2(α − ν)z(∂ψν )(z)

 − (α − ν)(α − ν + 1)(∂ψν−1 )(z) ∂ α−ν

4.3 Equivariant Splittings

=−

85

∞  

2(1 − α + ν)z(∂ψν )(z) + z 2 (∂ 2 ψν )(z)

ν=0

 − 2(α − ν)ψν (z) − (α − ν)(α − ν + 1)(∂ψν−1 )(z) ∂ α−ν ,

A(e− e+ )Ψ (z) = −

∞  

z 2 (∂ 2 ψν )(z) − 2(α − ν)z(∂ψν )(z)

ν=0

 + (α − ν)(α − ν + 1)(∂ψν−1 )(z) ∂ α−ν

with ψ−1 = 0. From these relations and (4.14) we see that A(C)Ψ (z) =

∞  

2(1 − 2α + 2ν)z(∂ψν )(z) + 2z 2 (∂ 2 ψν )(z)

ν=0

+ 2(α − ν)2 ψν (z) − 2(1 − α + ν)z(∂ψν )(z) − z 2 (∂ 2 ψν )(z) + 2(α − ν)ψν (z) + (α − ν)(α − ν + 1)(∂ψν−1 )(z) − z 2 (∂ 2 ψν )(z) + 2(α − ν)z(∂ψν )(z)  + (α − ν)(α − ν + 1)(∂ψν−1 )(z) ∂ α−ν , 

which is equivalent to (4.15). Let Sα∂ : Ψ D(R)α → R be the symbol map given by (Sα∂ Ψ )(z) = ψ0 (z) for Ψ (z) =

∞ 

ψk (z)∂ α−k ∈ Ψ D(R)α

(4.16)

(4.17)

k=0

as in (1.42). We now assume that there is an sl(2, C)-equivariant splitting η : R → Ψ D(R)α of Sα∂ , meaning that η is a linear map satisfying Sα∂ (ηf ) = f,

η(A(v)f ) = A(v)(ηf )

(4.18)

for all f ∈ R and v ∈ sl(2, C), where A is as in (4.10) and f is regarded as an element of Ψ D(R). Proposition 4.5 If η : R → Ψ D(R)−ε satisfies (4.18) with α = −ε < 0, then we have  ∞  2ε − 2  (−1)ν (ν + ε)!(ν + ε − 1)! (ν) (ηf )(z) = ε f (4.19) ε ν!(ν + 2ε − 1)! ν=0 for all f ∈ R.

86

4 Lie Algebras

Proof. Given f ∈ R, from (4.18) we obtain η(A(C)f ) = A(C)(ηf ),

(4.20)

where C is the Casimir operator in (4.14). Thus we see that ∂ ∂ ◦ η)(A(C)f ) = S−ε (A(C)(ηf )). A(C)f = (S−ε

If ηf is given by (ηf )(z, X) =

∞ 

(4.21)

ψν (z)∂ −ε−ν

ν=0

with ψ0 = f , then from (4.15) we obtain (4.22) (A(C)(ηf ))(z) = 2ε(ε − 1)f (z)∂ α ∞  +2 (ν + ε)(ν + ε − 1)(ψν + ∂ψν−1 )(z)∂ −ε−ν . ν=1

Using this, (4.15) and (4.21), we have A(C)f = 2ε(ε − 1)f, η(A(C)f ) = 2ε(ε − 1)(ηf ). From this, (4.20) and (4.22) we see that 0 = (A(C)(ηf ))(z) − (η(A(C)f ))(z) = (A(C)(ηf ))(z) − 2ε(ε − 1)(ηf )(z) ∞    (ν + ε)(ν + ε − 1)(ψν + ∂ψν−1 )(z) − ε(ε − 1)ψν (z) ∂ −ε−ν =2 =2

ν=1 ∞  

 ν(ν + 2ε − 1)ψν (z) + (ν + ε)(ν + ε − 1)(∂ψν−1 )(z) ∂ −ε−ν ,

ν=1

which implies that (ν + ε)(ν + ε − 1))∂ψν−1 ν(ν + 2ε − 1) (ν + ε)(ν + ε − 1) · · · ε = (−1)ν ν! (ν + ε − 1)(ν + ε − 2) · · · (ε − 1) × ∂ ν ψ0 (ν + 2ε − 1)(ν + 2ε − 2) · · · (2ε)(2ε − 1) (ν + ε)!(ν + ε − 1)! (2ε − 2)! f (ν) = (−1)ν ν!(ν + 2ε − 1)! (ε − 1)!(ε − 2)!

ψν = −

4.4 Lie Algebras of Jacobi-like Forms

 = (−1)ν ε

87



2ε − 2 (ν + ε)!(ν + ε − 1)! (ν) f ε ν!(ν + 2ε − 1)!

for all ν ≥ 1. Hence the proposition follows.



4.4 Lie Algebras of Jacobi-like Forms In this section we apply some of the results obtained in the previous sections to the cases of automorphic pseudodifferential operators and Jacobi-like forms for a discrete subgroup of SL(2, R). The formulas (1.17) and (1.40) for z ∈ H and γ ∈ SL(2, R) determine right actions of SL(2, R) on R[[X]] and Ψ D(R), respectively. Given a discrete subgroup Γ of SL(2, R), the subspace Jλ (Γ ) ⊂ R[[X]] of Jacobi-like forms of weight λ and the subspace Ψ D(R)Γ ⊂ Ψ D(R) of automorphic pseudodifferential operators consist of the corresponding Γ -invariant elements as in Chapter 1. We also recall that Ψ D(R)β ⊂ Ψ D(R) with β ∈ Z is the subspace consisting of elements of the form Ψ (z) =

∞ 

ψk (z)∂ β−k

k=0

with ψk ∈ R for each k ≥ 0 and that R[[X]]α = X α R[[X]] for α ≥ 0. Then we see that the formulas (1.46) and (1.47) determine C-linear isomorphisms ∂ : R[[X]]δ → Ψ D(R)−δ−ξ , Iξ,δ

X Iξ,ε : Ψ D(R)−ε → R[[X]]ε−ξ

for integers δ, ε > 0 and ξ ≥ 0, and the diagram 0 −−−−→

R[[X]]δ+1 ⏐ ⏐ ∂ Iξ,δ+1 

−−−−→

R[[X]]δ ⏐ ∂ ⏐ Iξ,δ 

S

−−−δ−→ R −−−−→ 0   (4.23)  ∂ S−δ−ξ

0 −−−−→ Ψ D(R)−δ−1−ξ −−−−→ Ψ D(R)−δ−ξ −−−−→ R −−−−→ 0 ∂ are as in (1.37) and (4.16). Similarly, as in commutes, where Sδ and S−δ−ξ Theorem 4.1, the formula (4.4) determines the bilinear map

[ , ]X : R[[X]]δ × R[[X]]ε → R[[X]]δ+ε satisfying

(4.24)

88

4 Lie Algebras

  ∂ ∂ ∂ I2ξ,δ+ε [F (z, X), G(z, X)]X = [Iξ,δ (F (z, X)), Iξ,ε (G(z, X))]∂

(4.25)

for F (z, X) ∈ R[[X]]δ and G(z, X) ∈ R[[X]]ε . Thus we see easily that [ , ]X is a Lie bracket on R[[X]]; hence R[[X]] is a complex Lie algebra. Proposition 4.6 The space J2ξ (Γ ) of Jacobi-like forms for Γ of weight 2ξ is a complex Lie algebra with respect to the bracket operation in (4.24) whose formula is given by (4.4). Proof. We note that by Corollary 1.16 there are isomorphisms ∂ Iξ,δ : J2ξ (Γ )δ → Ψ DOΓ−δ−ξ ,

X Iξ,ε : Ψ DOΓ−ε → J2ξ (Γ )ε−ξ

(4.26)

for nonnegative integers δ and ε ≥ ξ. We consider elements F (z, X), G(z, X) ∈ J2ξ (Γ )δ , so that ∂ ∂ (F (z, X)), Iξ,δ (G(z, X)) ∈ Ψ D(R)Γ−δ−ξ . Iξ,δ

However, from (4.25) we see that X ∂ ∂ )[Iξ,δ (F (z, X)), Iξ,δ (G(z, X))]∂ . [F (z, X), G(z, X)]X = (I2ξ,2δ

Thus it follows that [F (z, X), G(z, X)]X ∈ J2ξ (Γ )2δ ; hence by restriction the Lie bracket [ , ]X on R[[X]] determines a Lie bracket  on J2ξ (Γ ). By taking the Γ -invariant elements of the terms of the two short exact sequences in the diagram (4.23) we obtain the commutative diagram 0 −−−−→

J2ξ (Γ )δ+1 ⏐ ⏐ ∂ Iξ,δ+1 

−−−−→

J2ξ (Γ )δ ⏐ ∂ ⏐ Iξ,δ 

S

−−−δ−→ M2δ+2ξ (Γ ) −−−−→ 0    ∂ S−δ−ξ

0 −−−−→ Ψ D(R)Γ−δ−1−ξ −−−−→ Ψ D(R)Γ−δ−ξ −−−−→ M2δ+2ξ (Γ ) −−−−→ 0. (4.27) If η : R → Ψ D(R)−ε is the linear map given by (4.19), then it satisfies ∂ (ηf ) = f, S−ε

η(A(z)f ) = A(v)(ηf )

for all f ∈ R and v ∈ sl(2, R), which implies that η(f |2ε γ) = (ηf ) ◦ γ for all γ ∈ SL(2, R). Hence, if f ∈ M2ε (Γ ), we see that ηf is an automorphic pseudodifferential operator belonging to Ψ D(R)Γ−ε , so that the resulting map

4.4 Lie Algebras of Jacobi-like Forms

89

η : M2ε (Γ ) → Ψ D(R)Γ−ε is a splitting for the short exact sequence in the second row of the diagram (4.27) for δ = ε − ξ. Thus η may be regarded as a lifting map from modular forms to automorphic pseudodifferential operators considered in Section 1.4. Indeed, using the notation in (1.54), we see that   2ε − 2 ∂ ε L0,ε . η = (−1)ε ε ε Using the liftings of two modular forms via η ∂ and the fact that the product of two Γ -automorphic pseudodifferential operators are Γ -automorphic, we obtain noncommutative products of modular forms, which are Rankin–Cohen brackets considered in Section 2.5. The Rankin–Cohen brackets can also be interpreted in terms of classical transvectants (see e.g. [95], [96]). X : Ψ D(R)−ε → J2ξ (Γ )ε−ξ to obtain the We now use the isomorphism Iξ,ε lifting X ◦ η : M2ξ (Γ ) → J2ξ (Γ )ε−ξ η X = Iξ,ε from modular forms to Jacobi-like forms satisfying (Sε−ξ ◦ η X )h = h for all h ∈ M2ξ (Γ ). Given f ∈ M2ξ (Γ ), from (1.47) and (4.19) we obtain ∞ 

(−1)k+ε (k + ε)!(k + ε − 1)! k=0   2ε − 2 (−1)k (k + ε)!(k + ε − 1)! (k) ×ε f (z)X k+ε−ξ ε k!(k + 2ε − 1)!  ∞  2ε − 2  f (k) (z) ε X k+ε−ξ . = (−1) ε ε k!(k + 2ε − 1)!

(η X f )(z, X) =

k=0

From this and (1.36) we see that  2ε − 2 ε−ξ L2ξ,ε−ξ . = (−1) ε ε 

η

X

ε

Chapter 5

Heat Operators

In this chapter we consider a differential operator DμX associated to a real number μ acting on the space of formal power series, which may be regarded as the heat operator with respect to the radial coordinate in the 2μ-dimensional space for a positive integer μ. If λ is an integer, we show that DλX carries Jacobi-like forms of weight λ to ones of weight λ + 2 and obtain the formula for the m-fold composite (DλX )[m] of such operators. We then determine the corresponding operators on modular series and as well as on automorphic pseudodifferential operators (cf. [72]).

5.1 Radial Heat Operators In this section we introduce differential operators called radial heat operators on the space R[[X]] of formal power series over R. We show that certain types of such operators carry Jacobi-like forms to Jacobi-like forms and determine the image of a Jacobi-like form under the composite of a finite number of such operators. Similar operators were considered for Jacobi forms by Eichler and Zagier [37] and others (see e.g. [15], [19]). Given a positive integer , let x1 , . . . , x denote the usual coordinate functions on the Euclidean space Rm . Then we recall that the Laplace operator Δ acting on the space of differentiable functions on Rm is given by Δ=

  ∂2 . ∂x2i i=1

(5.1)

If an additional coordinate function t is introduced, then the associated heat operator D can be written as

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_5

91

92

5 Heat Operators

 ∂2 ∂ ∂ −Δ= − . ∂t ∂t i=1 ∂x2i 

D=

(5.2)

Remark 5.1 If t represents time and T = T (t, x1 , . . . , x ) denotes the temperature as a function of time and space for heat propagation in an isotropic and homogeneous medium in the -dimensional space R , the function T satisfies the heat equation   ∂2T ∂T −κ = 0, ∂t ∂x2i i=1 where the constant κ depends on the thermal conductivity, the density and the heat capacity of the medium. By rescaling t we see that the above heat equation can be written as DT = 0, where D is as in (5.2). Lemma 5.2 Let r = (x21 + · · · + x2 )1/2 be the radial coordinate function on  of the Laplace operator Δ in the Euclidean space R . Then the radial part Δ (5.1) can be written in the form 2 = −1 ∂ + ∂ . Δ r ∂r ∂r2

(5.3)

Proof. For each i ∈ {1, . . . , } we have ∂ xi ∂ ∂r ∂ = , = ∂xi ∂xi ∂r r ∂r ∂2 ∂  xi ∂  = ∂x2i ∂xi r ∂r 1 ∂ x2 ∂  1 ∂  = + i r ∂r r ∂r r ∂r 1 ∂ x2i ∂ x2 ∂ 2 = − 3 + 2i 2 . r ∂r r ∂r r ∂r Thus it follows that     ∂2 1  2  ∂ 1  2  ∂ 2  ∂ − + = x x i ∂x2i r ∂r r3 i=1 ∂r r2 i=1 i ∂r2 i=1

=

∂2 −1 ∂  + 2 = Δ; r ∂r ∂r

hence the lemma follows.



 in (5.3) may be called the radial Laplace operator on the The operator Δ -dimensional space R . Similarly, the associated differential operator

5.1 Radial Heat Operators

93

2 = ∂ − −1 ∂ − ∂  = ∂ −Δ D ∂t ∂t r ∂r ∂r2

(5.4)

is the radial heat operator on R . We are now interested in such operators on R[[X]], where R[[X]] is the complex algebra of formal power series in X with coefficients in the space R of holomorphic functions on H as in Chapter 1. Definition 5.3 Given a real number μ, the associated radial heat operator on R[[X]] is the formal differential operator DμX given by DμX =

∂ ∂2 ∂ −μ −X . ∂z ∂X ∂X 2

(5.5)

Remark 5.4 The name, radial heat operator, of DμX in Definition 5.3 can √ be justified as follows. We consider the variable w = 2 X, so that we have √ ∂ ∂ 1 ∂ = = X , ∂w ∂w/∂X ∂X ∂X √ ∂ √ ∂  1 ∂ ∂2 ∂2 = + X = X X . ∂w2 ∂X ∂X 2 ∂X ∂X 2 Hence we obtain  ∂2 2μ ∂ ∂ 1 ∂  − − − ∂z w ∂w ∂w2 w ∂w ∂ 2μ − 1 ∂ ∂2 = − − . ∂z w ∂w ∂w2

DμX =

 in (5.4) we see that DX is the usual radial heat By comparing this with D μ operator on the 2μ-dimensional space, assuming that μ is a positive integer. Heat operators of this type were considered for Jacobi forms by Eichler and Zagier in [37]. As was noted by Zagier in [114], Jacobi-like forms obtained by lifting modular forms can be regarded as solutions of heat equations. To see this we now consider the solution of the formal differential equation DμX Φ = 0 with the initial condition Φ(z, 0) =

f (z) , δ!(δ + μ − 1)!

where f is a holomorphic function on H. If we set

(5.6)

94

5 Heat Operators

Φ(z, X) =

∞ 

an (z)X n+δ ,

n=0

then we have X

∞ 

(n + δ)(n + δ − 1)an (z)X n+δ−2

n=0

+μ

∞ 

(n + δ)an (z)X n+δ−1 −

n=0

∞ 

an (z)X n+δ = 0,

n=0

which implies that ∞ 

(n + δ)(n + δ + μ − 1)an (z)X n+δ−1 −

n=0

∞ 

an−1 (z)X n+δ−1 = 0.

n=1

Hence we obtain the recurrence relation (n + δ)(n + δ + μ − 1)an (z) − an−1 (z) = 0 for n ≥ 1. Thus we see that (n)

an =

a0 , (n + δ)!(n + δ + μ − 1)!

and, using the initial condition (5.6), we obtain Φ(z, X) =

∞ 

f (n) (z) X n+δ . (n + δ)!(n + δ + μ − 1)! n=0

(5.7)

If δ = 0 and if f is a modular form belonging to Mμ (Γ ), then the formal power series Φ(z, X) in (5.7) coincides with the lifting L0μ,0 (f ) of f to a Jacobi-like form belonging to Jμ (Γ )0 , where L0μ,0 is as in (1.36). Proposition 5.5 Given μ ∈ R, λ ∈ Z and a formal power series Φ(z, X) ∈ R[[X]], we have (DμX (Φ)|Jλ+2 γ)(z, X) = DμX (Φ|Jλ γ)(z, X) + (λ − μ)K(γ, z)(Φ|Jλ γ)(z, X) for all γ ∈ SL(2, R). Proof. Let γ be an element of SL(2, R) whose (2, 1)-entry is c, so that ∂ J(γ, z) = c = J(γ, z)K(γ, z), ∂z

(5.8)

5.1 Radial Heat Operators

95

where J(γ, z) and K(γ, z) are as in (1.12). Given a formal power series Φ(z, X) ∈ R[[X]], using (1.17), we see that ∂ (Φ |Jλ γ)(z, X) = −λcJ(γ, z)−λ−1 e−K(γ,z)X Φ(γz, J(γ, z)−2 X) ∂z + J(γ, z)−λ K(γ, z)2 Xe−K(γ,z)X Φ(γz, J(γ, z)−2 X) ∂Φ (γz, J(γ, z)−2 X) + J(γ, z)−λ e−K(γ,z)X J(γ, z)−2 ∂z + J(γ, z)−λ e−K(γ,z)X (−2c)J(γ, z)−3 ∂Φ (γz, J(γ, z)−2 X), ×X ∂X ∂ (Φ |Jλ γ)(z, X) = −J(γ, z)−λ K(γ, z)e−K(γ,z)X Φ(γz, J(γ, z)−2 X) ∂X + J(γ, z)−λ e−K(γ,z)X J(γ, z)−2 ∂Φ (γz, J(γ, z)−2 X) × ∂X = J(γ, z)−λ−2 e−K(γ,z)X  × −cJ(γ, z)Φ(γz, J(γ, z)−2 X) +

 ∂Φ (γz, J(γ, z)−2 X) , ∂X

 ∂2 J −λ−2 −K(γ,z)X c2 Φ(γz, J(γ, z)−2 X) (Φ |λ γ)(z, X) = J(γ, z) e ∂X 2 ∂Φ (γz, J(γ, z)−2 X) − 2K(γ, z) ∂X  2 −2 ∂ Φ −2 + J(γ, z) (γz, J(γ, z) X) . ∂X 2 From these relations and (5.5), we obtain ∂ ∂ (Φ |Jλ γ)(z, X) − μ (Φ |Jλ γ)(z, X) ∂z ∂X ∂2 −X (Φ |Jλ γ)(z, X) ∂X 2 = J(γ, z)−λ−2 e−K(γ,z)X  ∂Φ × −λcJ(γ, z)Φ + J(γ, z)2 K(γ, z)2 XΦ + ∂z

DμX (Φ |Jλ γ)(z, X) =

− 2cJ(γ, z)−1 XΦ + μcJ(γ, z)Φ

96

5 Heat Operators

∂Φ ∂Φ − c2 XΦ + 2K(γ, z)X ∂X ∂X  ∂2Φ −2 − J(γ, z) X (γz, J(γ, z)−2 X) ∂X 2  ∂Φ −λ−2 −K(γ,z)X ∂Φ −μ e = J(γ, z) ∂z ∂X ∂2Φ − J(γ, z)−2 X 2  ∂X −μ

+ (μ − λ)cJ(γ, z)Φ (γz, J(γ, z)−2 X),

where we used the relation c = J(γ, z)K(γ, z). Thus it follows that (DμX (Φ |Jλ γ))(z, X) = J(γ, z)−λ−2 e−K(γ,z)X (DμX Φ)(γz, J(γ, z)−2 X) + (μ − λ)cJ(γ, z)−λ−1 e−K(γ,z)X Φ(γz, J(γ, z)−2 X) = (DμX (Φ) |Jλ+2 γ)(z, X) + (μ − λ)K(γ, z)(Φ |Jλ γ)(z, X), 

which verifies (5.8).

Corollary 5.6 Given an integer λ and a formal power series Φ(z, X) ∈ R[[X]], we have DλX (Φ |Jλ γ)(z, X) = (DλX (Φ) |Jλ+2 γ)(z, X)

(5.9)

for all γ ∈ SL(2, R) and z ∈ H, where Φ |Jλ γ is as in (1.17). In particular, we have (5.10) DλX (Jλ (Γ )) ⊂ Jλ+2 (Γ ), and therefore DλX induces the complex linear map DλX : Jλ (Γ ) → Jλ+2 (Γ )

(5.11)

of Jacobi-like forms. Proof. Given a discrete subgroup Γ ⊂ SL(2, R), if μ = λ, the relation (5.8) can be written in the form (DλX (Φ)|Jλ+2 γ)(z, X) = DλX (Φ|Jλ γ)(z, X). Thus we obtain (5.9).



Given a positive integer m, we denote by X X ◦ · · · ◦ Dλ+2 ◦ DλX : Jλ (Γ ) → Jλ+2m (Γ ) (DλX )[m] = Dλ+2m−2

(5.12)

the composite of m consecutive linear maps of the form (5.11). The next theorem determines an explicit formula for this map.

5.1 Radial Heat Operators

97

∞ Theorem 5.7 Let Φ(z, X) = k=0 φk (z)X k+δ be a Jacobi-like form belonging to Jλ (Γ )δ with δ ≥ 0. Then its image under the map (DλX )[m] in (5.12) can be written in the form (DλX )[m] (Φ(z, X))) =

∞ 

φm,k (z)X k+δ−m ,

(5.13)

k=0

where φm,k = 0 for k < m − δ and φm,k =

m 

(−1)j

j=0

  m (k + δ − m + j)!(k + δ + λ + j − 2)! (m−j) φk−m+j (5.14) j (k + δ − m)!(k + δ + λ − 2)!

for k ≥ m − δ; here we assume that φ = 0 if  < 0.

Proof. We shall verify the relation with φm,k as in (5.14) by using

(5.13) ∞ k+δ φ (z)X ∈ Jλ (Γ )δ be as given. induction on m. Let Φ(z, X) = k k=0 Then from (5.5) and (5.14) we obtain (DλX Φ)(z, X) =

∞ 

φk (z)X k+δ − λ

k=0

− = =

∞ 

(k + δ)φk (z)X k+δ−1

k=0 ∞ 

(k + δ)(k + δ − 1)φk (z)X k+δ−1

k=0

∞  

 φk−1 (z) − (k + δ)(k + δ + λ − 1)φk (z) X k+δ−1

k=0 ∞ 

φ1,k (z)X k+δ−1 ,

k=0

which proves the case for m = 1. We now assume that (5.13) and (5.14) hold X to (5.13) we see that the for a positive integer m. Then by applying Dλ+2m coefficient φm+1,k of the series (DλX )[m+1] (Φ(z, X)) =

∞ 

φm+1,k (z)X k+δ−m−1 ∈ Jλ+2m+2 (Γ )

k=0

can be written in the form φm+1,k = φm,k−1 − (k + δ − m)(k + δ − m + (λ + 2m) − 1)φm,k = φm,k−1 − (k + δ − m)(k + δ + λ + m − 1)φm,k for each k ≥ 0. From this and (5.14) we obtain

98

5 Heat Operators

φm+1,k   m  j m (k + δ − m + j − 1)! (k + δ + λ + j − 3)! (m−j+1) φk−m−1+j = (−1) j (k + δ − m − 1)! (k + δ + λ − 3)! j=0 − (k + δ − m)(k + δ + λ + m − 1)   m  m (k + δ − m + j)! (k + δ + λ + j − 2)! (m−j) × φk−m+j (−1)j j (k + δ − m)! (k + δ + λ − 2)! j=0   m  j m (k + δ − m + j − 1)! (k + δ + λ + j − 3)! (m−j+1) φk−m−1+j = (−1) j (k + δ − m − 1)! (k + δ + λ − 3)! j=0 +

m+1 

 (k + δ − m + j − 1)! (k + δ + λ + j − 3)! (m−j+1) m φk−m−1+j j−1 (k + δ − m − 1)! (k + δ + λ − 3)!



(−1)

j

j=1

+ (m + 1)

m+1 

 (k + δ − m + j − 1)! m j−1 (k + δ − m − 1)!

 (−1)

j=1

j

× Using this and the identities       m+1 m m , = + j j−1 j

(k + δ + λ + j − 3)! (m−j+1) φk−m−1+j . (k + δ + λ − 2)! 



m (m + 1) j−1

 m+1 , =j j 

we see that φm+1,k =

m+1 

(−1)j

j=0

 m + 1 (k + δ − m + j − 1)! j (k + δ − m − 1)!



  (k + δ + λ + j − 3)! (m−j+1) φk−m−1+j × (k + δ + λ − 2) + j (k + δ + λ − 2)!   m+1  m + 1 (k + δ − m + j − 1)! (k + δ + λ + j − 2)! (m−j+1) φk−m−1+j , = (−1)j j (k + δ − m − 1)! (k + δ + λ − 2)! j=0 which is simply (5.14) with m replaced by m + 1; hence the theorem follows by induction.  If δ ≥ m, from (5.13) we see that (DλX )[m] (Φ(z, X)) ∈ Jλ+2m (Γ )δ−m , so that we obtain the map

5.1 Radial Heat Operators

99

(DλX )[m] : Jλ (Γ )δ → Jλ+2m (Γ )δ−m .

(5.15)

On the other hand, if δ ≤ m, then (5.13) can be written in the form (DλX )[m] (Φ(z, X))

=

∞ 

φm,k (z)X k+δ−m .

(5.16)

k=m−δ

Since (DλX )[m] belongs to Jλ+2m (Γ ), we see that its initial coefficient φm,m−δ is a modular form belonging to Mλ+2m (Γ ). If Ξλ,δ : Jλ (Γ )δ → Mλ (Γ )δ is the isomorphism in (1.30) and Ξλ,δ (Φ(z, X)) =

∞ 

+δ φΞ  (z)X

(5.17)

=0

as in (1.7), then we note that for each m ≥ δ by Lemma 1.9 the coefficient φΞ m−δ is also a modular form belonging to M2m+λ (Γ ). The next corollary shows that it is a constant multiple of φm,m−δ . Corollary 5.8 Let φm,k and φΞ  be as in (5.16) and (5.17), respectively. Then we have φm,m−δ =

(−1)m m! φΞ (m + λ − 2)!(2m + λ − 1) m

for each positive integer m ≥ δ. Proof. Given m ≥ δ, from (1.7) we see that the modular form φΞ m−δ ∈ M2m+λ (Γ ) can be written as φΞ m = (2m + λ − 1)

m−δ  r=0

(−1)r

(2m + λ − r − 2)! (r) φm−δ−r . r!

(5.18)

On the other hand, by (5.14) the initial term φm,m−δ of (DλX )[m] (Φ(z, X)) in (5.16) is given by   m j!(m + λ + j − 2)! (m−j) φj−δ j (m + λ − 2)! j=0   m  m (m − r)!(2m + λ − r − 2)! (r) φm−δ−r , = (−1)m−r r (m + λ − 2)! r=0

φm,m−δ =

m 

(−1)j

where we changed the index from j to r = m − j. Using this and the fact that φm−δ−r = 0 for r > m − δ, we obtain φm,m−δ

m−δ (2m + λ − r − 2)! (r) (−1)m m!  φm−δ−r . = (−1)r (m + λ − 2)! r=0 r!

100

5 Heat Operators



Hence the corollary follows by comparing this with (5.18).

5.2 Modular Series

Let Mλ (Γ ) be the space of modular series in Definition 1.7, which is isomorphic to the space Jλ (Γ ) of Jacobi-like forms. In this section we discuss operators on Mλ (Γ ) that are compatible with the composite maps of heat operators on Jλ (Γ ) studied in Section 5.1. Let F (z, X) ∈ R[[X]]δ with δ ≥ 0 be a formal power series of the form F (z, X) =

∞ 

fk (z)X k+δ

(5.19)

k=0

with fk ∈ R for each k ≥ 0. Given a positive integer m, let (DλX )[m] : Jλ (Γ )δ → Jλ+2m (Γ )δ−m be as in (5.15), and set (DλM )[m] (F (z, X)) =

∞ 

f k (z)X k+δ−m

(5.20)

k=0

for all z ∈ H, where f k is an element of R such that   m  k u−m+j   m f k = (−1)j+k−u j 2k + λ + 2δ − 1 j=0 u=0

(5.21)

=0

×

(k + m + u + λ − 2)!(u + j)! !(k − u)!(2u − 2m + 2j + 2δ + λ −  − 1)! (u + m + λ + j − 2)! × (u + m + λ − 2)!(u + δ − m)! (k+δ−u−j+)

× fu+j−

for each k ≥ 0. Thus the formula (5.20) determines a linear map (DλM )[m] : R[[X]]δ → R[[X]]δ−m of the spaces of formal power series over R.

5.2 Modular Series

101

Theorem 5.9 If F (z, X) is a modular series belonging to Mλ (Γ )δ , then we have (DλM )[m] (F (z, X)) ∈ M2m+λ (Γ )δ−m for each m ≥ 1. Furthermore, if (DλM )[m] : Mλ (Γ )δ → M2m+λ (Γ ) is the induced linear map on modular series, the diagram (D X )[m]

λ Jλ (Γ )δ −−− −−→ J2m+λ (Γ )δ−m ⏐ ⏐ ⏐ ⏐Ξ Ξλ,δ   2m+λ,δ−m

(5.22)

M [m] (Dλ )

Mλ (Γ )δ −−−−−→ M2m+λ (Γ )δ−m commutes, where Ξλ,δ and Ξ2m+λ,δ are the isomorphisms in (1.30) and (DλX )[m] is as in (5.15). Proof. Let F (z, X) ∈ Mλ (Γ )δ be as in (5.19), and let Λλ,δ : Mλ (Γ )δ → Jλ (Γ )δ be as in (1.29). Using (1.4) and (1.6), we have Λλ,δ (F (z, X)) =

∞ 

φk (z)X k+δ ,

k=0

where φk =

k  r=0

1 (r) f r!(2k + 2δ + λ − r − 1)! k−r

(5.23)

for each k ≥ 0 by (1.6). Then we have ((DλX )[m] ◦ Λλ,δ (F (z, X)) =

∞ 

φm,k (z)X k+δ−m ,

k=0

where φm,k is given by (5.14) for k ≥ m − δ and φm,k = 0 for k < m − δ. If we set (Ξ2m+λ,δ−m ◦ (DλX )[m] ◦ Λλ,δ )(F (z, X)) =

∞ 

fk (z)X k+δ−m ,

k=0

then from (1.7) and (5.14), for each k ≥ 0, we obtain fk (2k + 2δ + λ − 1) =

k  r=0

(−1)r

(2k + λ + 2δ − r − 2)! (r) φm,k−r r!

102

5 Heat Operators

=

m k  

(2k + 2δ + λ − r − 2)!(k + δ − m − r + j)! r!(k − r)!   m (k + δ − r + λ + j − 2)! (m+r−j) φk−m−r+j , × j (k + δ − r + λ − 2)!

(−1)j+r

r=0 j=0

which is a modular form belonging to M2k+2δ+λ (Γ ). We now set (DλM )[m] (F (z, X)) = (Ξ2m+λ,δ−m ◦ (DλX )[m] ◦ Λλ,δ )(F (z, X)). Then clearly (DλM )[m] (F (z, X)) is a modular series belonging to M2m+λ (Γ )δ−m and the diagram (5.22) commutes. On the other hand, by using (5.23) we have (m+r−j)

φk−m−r+j =

k−m−r+j  =0

1 (m−j+r+) f . !(2k − 2m − 2r + 2j + 2δ + λ −  − 1)! k−m+j−r−

Hence we obtain m  k k−m−r+j   m fk = j (2k + 2δ + λ − 1) j=0 r=0 =0

×

(−1) (k − r + j)! !r!(k + δ − r − m)! (2k + 2m + λ − r − 2)! × (2k − 2m − 2r + 2j + δ + λ −  − 1)! (k + m − r + λ + j − 2)! (m−j+r+) × f . (k + m − r + λ − 2)! k+j−r− j+r

Changing the index r to u = k − r, we obtain the formula (5.21); hence the proof of the theorem is complete. 

5.3 Pseudodifferential Operators

In this section we use the correspondence between Jacobi-like forms and pseudodifferential operators and the heat operators in Section 1.4 to determine the heat operators acting on the space of pseudodifferential operators. If ε, ξ ∈ Z with ξ ≥ 0, we define the linear map ∂ D2ξ,ε : Ψ D(R)−ε → Ψ D(R)−ε

by

5.3 Pseudodifferential Operators ∂ D2ξ,ε (Ψ (z)) = −

103

∞  

 (k + ε)(k + ε − 1)ψk−1 (z)

k=0

for Ψ (z) = denote by

∞ k=0

(5.24)

 + (k + ε − ξ)(k + ε + ξ − 1)ψk ∂ −k−ε

ψk (z)∂ −k−ε ∈ Ψ D(R)−ε . If m is a positive integer, we also

∂ ∂ ∂ )[m] = D2ξ,ε ◦ · · · ◦ D2ξ,ε : Ψ D(R)−ε → Ψ D(R)−ε (D2ξ,ε ∂ the composite of m copies of D2ξ .

X : J2ξ (Γ )δ → J2ξ+2 (Γ )δ−1 be the radial heat operaTheorem 5.10 Let D2ξ tor given by (5.5), and let Φ(z, X) ∈ J2ξ (Γ ). Then we have ∂ ∂ ∂ X (Iξ,δ (Φ(z, X))) = Iξ+1,δ−1 (D2ξ (Φ(z, X))), D2ξ,δ+ξ

(5.25)

∂ where Iξ,δ : J2ξ (Γ )δ → Ψ D(R)Γ−δ−ξ is the isomorphism in Corollary 1.16. More generally, we have ∂ ∂ ∂ X [m] )[m] (Iξ,δ (Φ(z, X))) = Iξ+m,δ−m ((D2ξ ) (Φ(z, X))) (D2ξ,δ+ξ

(5.26)

X [m] for each positive integer m, where (D2ξ ) is as in (5.12).

Proof. Let Φ(z, X) = X D2ξ (Φ(z, X)) =

∞ k=0

φk (z)X k+δ ∈ J2ξ (Γ )δ , so that

∞  

 φk−1 (z) − (k + δ)(k + δ + 2ξ − 1)φk (z) X k+δ−1

k=0

with φ−1 = 0 is a Jacobi-like form belonging to J2ξ+2 (Γ ). From this and (1.47) we obtain ∂ X ◦ D2ξ )(Φ(z, X)) (Iξ+1,δ−1 ∞  (−1)k+δ+ξ (k + δ + ξ)!(k + δ + ξ − 1)! = k=0

  × φk−1 (z) − (k + δ)(k + δ + 2ξ − 1)φk (z) ∂ −k−δ−ξ .

On the other hand, using (1.46), we have ∂ Iξ,δ (Φ(z, X)) =

∞ 

(−1)k+δ+ξ (k + δ + ξ)!(k + δ + ξ − 1)!φk (z)∂ −k−δ−ξ .

k=0

Combining this with (5.24), we see that

104

5 Heat Operators ∂ ∂ (D2ξ,δ+ξ ◦ Iξ,δ )(Φ(z, X)) ∞   (−1)k+δ+ξ−1 (k + δ + ξ)!(k + δ + ξ − 1)!φk−1 (z) =− k=0

+ (−1)k+δ+ξ (k + δ)(k + δ + 2ξ − 1)  × (k + δ + ξ)!(k + δ + ξ − 1)!φk (z) ∂ −k−δ−ξ ∞ 

=

(−1)k+δ+ξ (k + δ + ξ)!(k + δ + ξ − 1)!

k=0

 × φk−1 (z) − (k + δ)(k + δ + 2ξ − 1)φk (z) ∂ −k−δ−ξ ; 

hence we obtain (5.25). Then (5.26) follows easily from this by induction.



From Theorem 5.10 it follows that the diagram J2ξ (Γ )δ ⏐ ∂ ⏐ Iξ,δ 

X [m] (D2ξ )

−−−−−→

J2m+2ξ (Γ )δ−m ⏐ ⏐I ∂  m+ξ,δ−m

∂ )[m] (D2ξ,δ+ξ

Ψ D(R)Γ−δ−ξ −−−−−−−−→

Ψ D(R)Γ−δ−ξ

is commutative for each positive integer m. Remark 5.11 In [23] the heat operator DλX for λ = 1/2 was studied. Note that this operator does not send Jacobi-like forms to Jacobi-like forms. In the same paper, an isomorphism L∂ : R[[X]] → Ψ D(R) was considered whose ∂ in (1.46) with λ = 0, and a linear formula is the same as the one for Iλ,0 X endomorphism of Ψ D(R) compatible with D1/2 under this isomorphism was constructed. Such a map can also be obtained for an integer λ by setting  ∂ = ∂ − λI, D λ where I : Ψ D(R) → Ψ D(R) is the formal integration operator with respect to the symbol ∂, that is, an operator given by I

 ∞ k=0

Then we have

φk (z)∂ −k−δ

 =

∞  k=0

φk (z) ∂ −k−δ+1 . 1−k−δ

 ∂ ◦ L∂ L∂ ◦ DλX = D λ

for each nonnegative integer

∞λ. Given a positive integer m and a pseudodifferential operator Ψ (z) = k=0 ψk (z)∂ −k−δ , it can also be shown that

5.3 Pseudodifferential Operators

105

 ∂ )[m] (Ψ (z)) = (D λ

∞ 

ψm,k (z)∂ −k−δ+m ,

(5.27)

k=0

 ∂ and  ∂ )[m] is the m-fold composite of D where (D λ λ ψm,k =

m    m (k + δ + λ + j − 2)! j=0

j

(k + δ + λ − 2)!

(k + δ − m − 1)! (m−j) ψ (k + δ − m + j − 1)! k−m+j

for each k ≥ 0.

Theorem 5.12 Let Ψ (z) be an automorphic pseudodifferential operator of the form ∞  ψk (z)∂ −k−ε ∈ Ψ D(R)Γ−ε (5.28) Ψ (z) = k=0

with ε ≥ 0. Given integers m ≥ 1 and ξ ≥ 0, we have ∂ )[m] (Ψ (z)) (D2ξ,ε

=

∞ 

ψm,k (z)∂ −k−ε ,

k=0

where ψm,k = 0 for k < m − ε + ξ and ψm,k =

m    m (k + ε − ξ − m + j)! (k + ε + ξ + j − 2)! j=0

j

(k + ε − ξ − m)!

(k + ε + ξ − 2)! (m−j)

×

(−1)m (k + ε)!(k + ε − 1)!ψk−m+j (k − m + j + ε)!(k − m + j + ε − 1)!

for k ≥ m − ε + ξ with ψ = 0 for  < 0.

Proof. Let Ψ (z) ∈ Ψ D(R)Γ−ε be as in (5.28). Then by (1.47) and Corollary X (Ψ (z)) ∈ J2ξ (Γ )ε−ξ is given by 1.16 the Jacobi-like form Iξ,ε X (Ψ (z)) = Iξ,ε

∞  k=0

(−1)k+ε ψk (z) X k+ε−ξ . (k + ε)!(k + ε − 1)!

Thus, using (5.13) and (5.14), we have X [m] X ) ◦ Iξ,ε )(Ψ (z))) = ((D2ξ

∞  k=0

where ψm,k = 0 for k < m − ε + ξ and

ψm,k (z)X k+ε−ξ−m ,

106

5 Heat Operators

ψm,k =

m  j=0

(−1)j

  m (k + ε − ξ − m + j)! (k + ε + ξ + j − 2)! j (k + ε − ξ − m)! (k + ε + ξ − 2)! (m−j)

×

(−1)k−m+j+ε ψk−m+j (k − m + j + ε)!(k − m + j + ε − 1)!

for k ≥ m − ε + ξ, assuming that ψ = 0 for  < 0. From this and (1.46) we obtain ∂ ∂ X [m] X )[m] (Ψ (z))) = (Iξ+m,ε−ξ−m ◦ (D2ξ ) ◦ Iξ,ε )(Ψ (z)) (D2ξ,ε ∞  (−1)k+ε (k + ε)!(k + ε − 1)!ψm,k (z)∂ −k−ε ; = k=0

hence the theorem follows.



Chapter 6

Group Cohomology

The well-known Eichler–Shimura isomorphism (cf. [36], [107]) provides us a correspondence between modular forms for a discrete subgroup Γ ⊂ SL(2, R) and cohomology classes of Γ with certain coefficients. Hecke operators can be introduced to the space of such cohomology classes that are compatible with the usual Hecke operators on modular forms under the Eichler–Shimura isomorphism. In this chapter we construct linear maps from the space of automorphic pseudodifferential operators and from the space of Jacobi-like forms for Γ to the cohomology space of the group Γ in such a way that they are equivariant with the respective Hecke operator actions (cf. [68]).

6.1 Cohomology of Groups The cohomology of groups can be constructed by using either homogeneous or nonhomogeneous cochains. We review the construction of such cohomology using both types of cochains and describe their correspondence. Let G be a group, and let M be a left G-module, meaning that M is an abelian group on which G acts on the left. Given a nonnegative integer q, the group Cq (G, M ) of homogeneous q-cochains is an abelian group generated by the maps φ : Gq+1 → M satisfying φ(σσ0 , . . . , σσq ) = σφ(σ0 , . . . , σq ) for all σ, σ0 , . . . , σq ∈ G, and the associated coboundary map δq : Cq (G, M ) → Cq+1 (G, M ) is defined by

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_6

107

108

6 Group Cohomology

(δq φ)(σ0 , . . . , σq+1 ) =

q+1 

(−1)i φ(σ0 , . . . , σi−1 , σi+1 , . . . , σq+1 )

(6.1)

i=0

for all φ ∈ Cq (G, M ) and (σ0 , . . . , σq+1 ) ∈ Gq+2 . Then we have δq ◦ δq−1 = 0 for each q ≥ 0 with δ−1 = 0, and the q-th cohomology group of G with coefficients in M is given by Hq (G, M ) = Zq (G, M )/Bq (G, M ),

(6.2)

where Zq (G, M ) and Bq (G, M ) denote the kernel of δq and the image of δq−1 , respectively. The cohomology of the group G can also be defined by using nonhomogeneous cochains. Indeed, the group C q (G, M ) of nonhomogeneous q-cochains consists of the maps f : Gq → M , and the associated coboundary map ∂q : C q (G, M ) → C q+1 (G, M ) is given by (∂q f )(σ1 , . . . , σq+1 ) = σ1 f (σ2 , . . . , σq+1 ) +

q 

(6.3)

(−1)i f (σ1 , . . . , σi−1 , σi σi+1 , . . . , σq+1 )

i=1

+ (−1)q+1 f (σ1 , . . . , σq ) for all f ∈ C q (G, M ) and (σ1 , . . . , σq+1 ) ∈ Gq+1 , which satisfies ∂q ◦ ∂q−1 = 0 for all q ≥ 0 with ∂−1 = 0. The associated q-th cohomology group of G with coefficients in M is given by H q (G, M ) = Z q (G, M )/B q (G, M ), where Z q (G, M ) and B q (G, M ) are the kernel of ∂q and the image of ∂q−1 , respectively. We now describe below a correspondence between homogeneous and nonhomogeneous cochains. Given f ∈ C q (G, M ) and φ ∈ Cq (G, M ), we consider the elements fH ∈ Cq (G, M ) and φN ∈ C q (G, M ) given by −1 σq ) fH (σ0 , . . . , σq ) = σ0 f (σ0−1 σ1 , σ1−1 σ2 , . . . , σq−1

(6.4)

φN (σ1 , . . . , σq ) = φ(1, σ1 , σ1 σ2 , . . . , σ1 σ2 · · · σq )

(6.5)

for all σ0 , σ1 , . . . , σq ∈ G. Then we see that (fH )N (σ1 , . . . , σq ) = fH (1, σ1 , σ1 σ2 , . . . , σ1 σ2 · · · σq ) = f (σ1 , . . . , σq ), −1 σq ) (φN )H (σ0 , . . . , σq ) = σ0 φN (σ0−1 σ1 , σ1−1 σ2 , . . . , σq−1

6.1 Cohomology of Groups

109

= σ0 φ(1, σ0−1 σ1 , σ0−1 σ2 , . . . , σ0−1 σq ) = φ(σ0 , . . . , σq ) for all f ∈ C q (G, M ) and φ ∈ Cq (G, M ). Thus, by extending linearly we obtain the linear maps (·)H : C q (G, M ) → Cq (G, M ),

(·)N : Cq (G, M ) → C q (G, M )

such that (·)H ◦ (·)N and (·)N ◦ (·)H are identity maps on Cq (G, M ) and C q (G, M ), respectively. The next lemma shows that this correspondence between homogeneous and nonhomogeneous cochains is compatible with the coboundary maps. Lemma 6.1 Given a nonnegative integer q, we have (∂q f )H = δq fH ,

(δq φ)N = ∂q φN

for all f ∈ C q (G, M ) and φ ∈ Cq (G, M ). Proof. For elements σ0 , σ1 , . . . , σq+1 ∈ G and f ∈ C q (G, M ), using (6.1), (6.3) and (6.4), we have (∂q f )H (σ0 , σ1 , . . . , σq+1 ) = σ0 (∂q f )(σ0−1 σ1 , σ1−1 σ2 , . . . , σq−1 σq+1 ) = σ0 σ0−1 σ1 f (σ1−1 σ2 , . . . , σq−1 σq+1 ) +

q 

−1 −1 (−1)i σ0 f (σ0−1 σ1 , . . . , σi−2 σi−1 , σi−1 σi+1 , . . . , σq−1 σq+1 )

i=1

+ (−1)q+1 σ0 f (σ0−1 σ1 , . . . , σq−1 σq ) = fH (σ1 , . . . , σq+1 ) +

q 

(−1)i fH (σ0 , . . . , σi−1 , σi+1 , . . . , σq+1 )

i=1

+ (−1)q+1 fH (σ0 , . . . , σq ) = (δq fH )(σ0 , σ1 , . . . , σq+1 ). On the other hand, if φ ∈ Cq (G, M ), by using (6.1), (6.3) and (6.5) we see that (δq φ)N (σ1 , . . . , σq+1 ) = (δq φ)(1, σ1 , σ1 σ2 , . . . , σ1 σ2 · · · σq+1 ) = φ(σ1 , σ1 σ2 , . . . , σ1 σ2 · · · σq+1 ) +

q 

(−1)i φ(1, σ1 , . . . , σ1 · · · σi−1 , σ1 · · · σi+1 , . . . , σ1 · · · σq+1 )

i=1

+ (−1)q+1 φ(1, σ1 , . . . , σ1 · · · σq+1 )

110

6 Group Cohomology

= σ1 φN (σ2 , . . . , σq+1 ) +

q 

(−1)i φN (σ1 , . . . , σi−1 , σi σi+1 , . . . , σq+1 )

i=1 q+1

+ (−1) = (∂q φN )(σ1 , . . . , σq+1 );

φN (σ1 , . . . , σq+1 )

hence the lemma follows. From Lemma 6.1 we see that the diagram C q (G, M ) ⏐ ⏐ ∂q 

(·)H

−−−−→

Cq (G, M ) ⏐ ⏐ δq 

(·)H

(·)N

−−−−→

C q (G, M ) ⏐ ⏐∂  q

(·)N

C q+1 (G, M ) −−−−→ Cq+1 (G, M ) −−−−→ C q+1 (G, M ) is commutative, which implies that the maps (·)H and (·)N induce the isomorphisms ≈ (·)H : H q (G, M ) − → Hq (G, M ), ≈

→ H q (G, M ), (·)N : Hq (G, M ) −

(6.6)

of cohomology spaces for each q ≥ 0. Example 6.2 We consider the case where G = GL(2, C). Let {e1 , . . . , en+1 } be the standard basis for the complex vector space Cn+1 , and for ( zz12 ) ∈ C2 set  n n  z1 = t (z1n , z1n−1 z2 , . . . , z1 z2n−1 , z2n ) = z1n−k z2k ek+1 ∈ Cn+1 , z2 k=0

where t (·) denotes the transpose of the matrix (·). Let ρn : GL(2, C) → GL(n + 1, C) be the n-th symmetric tensor representation of GL(2, C), which is given by  n   n z1 z1 = γ ρn (γ) z2 z2 for all γ ∈ GL(2, C) and ( zz12 ) ∈ C2 . We also define the map v n : H → Cn+1 by  n  n z = z n−k ek+1 (6.7) v n (z) = 1 k=0   for all z ∈ C. Then for γ = ac db ∈ GL(2, R) we see that

6.2 Hecke Operators

111

n  n  z az + b ρn (γ) vn (z) = ρn (γ) = 1 cz + d n  = (az + b)n−k (cz + d)k−n ek+1

(6.8)

k=0

= (cz + d)n = (cz + d)n

n  k=0 n 

(az + b)n−k (cz + d)k ek+1 (γz)n−k ek+1 = (cz + d)n v n (γz),

k=0

where γz = (az + b)(cz + d)−1 . We denote by Sn (C2 ) the complex vector space Cn+1 equipped with the structure of a GL(2, C)-module given by (γ, v) → (det γ)n/2 ρn (γ)v for γ ∈ GL(2, C) and v ∈ Cn+1 . Let Γ be a discrete subgroup of SL(2, R) ⊂ GL(2, C). Then by (6.3) its first cohomology group with coefficients in Sn (C2 ) can be described as follows. The set Z 1 (Γ, Sn (C2 )) of nonhomogeneous 1cocycles consists of all maps u : Γ → Cn+1 satisfying u(γγ  ) = u(γ) + ρn (γ)u(γ  )

(6.9)

for all γ, γ  ∈ Γ . Given an element v0 ∈ Cn+1 , the set B 1 (Γ, Sn (C2 )) of coboundaries consists of the maps v : Γ → Cn+1 such that v(γ) = (ρn (γ) − 1n+1 )v0 for all γ ∈ Γ , where 1n+1 is the identity map on Cn+1 . Then the first cohomology group of Γ with coefficients in Sn (C2 ) is given by H 1 (Γ, Sn (C2 )) =

Z 1 (Γ, Sn (C2 )) . B 1 (Γ, Sn (C2 ))

(6.10)

6.2 Hecke Operators In this section we review Hecke operators acting on group cohomology in terms of homogeneous cochains introduced by Rhie and Whaples [101]. We also describe these operators in terms of nonhomogeneous cochains, which we apply to the case of the cohomology of a discrete subgroup of SL(2, R) to obtain the usual Hecke operators on such cohomology (see e.g. [48] and [108]). Let G be a group as in Section 6.1. We recall that two subgroups Γ, Γ  ⊂ G are commensurable, or Γ ∼ Γ  , if Γ ∩ Γ  has finite index in both Γ and Γ 

112

6 Group Cohomology

and that the commensurator Γ of Γ is given by Γ = {α ∈ G | α−1 Γ α ∼ Γ }. If α ∈ Γ, then the corresponding double coset Γ αΓ can be written as a disjoint union of right cosets of Γ in G of the form Γ αΓ =

d 

Γ αi

(6.11)

i=1

for some α1 , . . . , αd ∈ Γ. If γ ∈ Γ , then the same double coset can be written as d  Γ αΓ = Γ αi γ, i=1

which follows from the fact that Γ αΓ γ = Γ αΓ . Thus for 1 ≤ i ≤ d, we see that αi γ = ξi (γ) · αi(γ) (6.12) for some element ξi (γ) ∈ Γ , where {α1(γ) , . . . , αd(γ) } is a permutation of {α1 , . . . , αd }. For each i and γ, γ  ∈ Γ we have (αi γ)γ  = ξi (γ) · αi(γ) γ  = ξi (γ) · ξi(γ) (γ  ) · αi(γ)(γ  ) . Comparing this with αi (γγ  ) = ξi (γγ  )αi(γγ  ) , we see that i(γγ  ) = i(γ)(γ  ),

ξi (γγ  ) = ξi (γ) · ξi(γ) (γ  )

(6.13)

for all γ, γ  ∈ Γ . Let M be a left Γ -module. Given a homogeneous cochain φ ∈ Cq (Γ, M ) with q ≥ 0 and a double coset Γ αΓ with α ∈ Γ that has a decomposition as in (6.11), we consider the associated map T(α)φ : Γ q+1 → M given by (T(α)φ)(γ0 , . . . , γq ) =

d 

αi−1 φ(ξi (γ0 ), . . . , ξi (γq )),

(6.14)

i=1

where the maps ξi : Γ → Γ are determined by (6.12). Then it is known that T(α)φ is an element of Cq (Γ, M ) and is independent of the choice of representatives of the coset decomposition of Γ αΓ modulo Γ (see [101]). Thus each double coset Γ αΓ with α ∈ Γ determines the C-linear map T(α) : Cq (Γ, M ) → Cq (Γ, M ) for each q ≥ 0. It can be shown that T(α) ◦ δq = δq ◦ T(α)

(6.15)

6.2 Hecke Operators

113

for q ≥ 0. Hence it follows that the map T(α) in (6.15) induces the homomorphism T(α) : Hq (Γ, M ) → Hq (Γ, M ), which is the Hecke operator on Hq (Γ, M ) corresponding to α introduced by Rhie and Whaples [101]. In order to consider Hecke operators by using nonhomogeneous cochains, given α ∈ Γ, we set (T (α)ψ)(γ1 , . . . , γq ) =

d 

(6.16)

αi−1 ψ(ξi (γ1 ), ξi(γ1 ) (γ2 ), ξi(γ1 γ2 ) (γ3 ), . . . , ξi(γ1 ···γq−1 ) (γq ))

i=1

for all ψ ∈ C q (Γ, M ) and γ1 , . . . , γq ∈ Γ . Then the resulting map T (α) : C q (Γ, M ) → C q (Γ, M ) can be shown to satisfy T (α) ◦ ∂q = ∂q ◦ T (α) for q ≥ 0, and therefore we obtain the nonhomogeneous version of the Hecke operator T (α) : H q (Γ, M ) → H q (Γ, M ), on H q (Γ, M ) corresponding to α. Proposition 6.3 Given α ∈ Γ, the map T (α)f : Γ q → M is an element of C q (Γ, M ) and satisfies T (α)f = (T(α)fH )N for all f ∈ C q (Γ, M ), where the maps (·)H : C q (G, M ) → Cq (G, M ),

(·)N : Cq (G, M ) → C q (G, M )

are as in (6.4) and (6.5). Proof. Given f ∈ C q (Γ, M ), by (6.4) we have −1 σq ) fH (σ0 , σ1 , . . . , σq ) = σ0 · f (σ0−1 σ1 , σ1−1 σ2 , . . . , σq−1

for all σ0 , σ1 , . . . , σq ∈ Γ . Thus for α ∈ Γ, using (6.14), we obtain (T(α)fH )(σ0 , σ1 , . . . , σq ) =

d  i=1

αi−1 fH (ξi (σ0 ), . . . , ξi (σq ))

114

6 Group Cohomology

=

d 

αi−1 ξi (σ0 )−1 f (ξi (σ0 )−1 ξi (σ1 ), ξi (σ1 )−1 ξi (σ2 ), . . .

i=1

. . . , ξi (σq−1 )−1 ξi (σq )). Hence by using (6.5) we have (T(α)fH )N (γ1 , . . . , γq ) = (T(α)fH )(1, γ1 , γ1 γ2 , . . . , γ1 γ2 · · · γq ) =

d 

αi−1 f (ξi (γ1 ), ξi (γ1 )−1 ξi (γ1 γ2 ), ξi (γ1 γ2 )−1 ξi (γ1 γ2 γ3 ), . . .

i=1

. . . , ξi (γ1 · · · γq−1 )−1 ξi (γ1 · · · γq−1 γq )) for all γ1 , . . . , γq ∈ Γ . However, it follows from (6.13) that ξi (γ1 · · · γk−1 )−1 ξi (γ1 · · · γk−1 γk ) = ξi(γ1 ···γk−1 ) (γk ) for 2 ≤ k ≤ n. Hence we obtain (T(α)fH )N (γ1 , . . . , γq ) =

d 

αi−1 f (ξi (γ1 ), ξi(γ1 ) (γ2 ), ξi(γ1 γ2 ) (γ3 ), . . .

i=1

. . . , ξi(γ1 ···γq−1 ) (γq )), and therefore the proposition follows from this and (6.16).



Using Proposition 6.3 and the isomorphism (6.6), we see that the diagram T(α)

Hq (Γ, M ) −−−−→ Hq (Γ, M ) ⏐ ⏐ ⏐ ⏐(·) (·)N   N T (α)

H q (Γ, M ) −−−−→ H q (Γ, M ) is commutative for each q ≥ 0, so that the Hecke operators T (α) and T(α) are compatible under the isomorphism (·)N . We now consider the case of the cohomology of a discrete subgroup Γ of SL(2, R) described in Example 6.2. To consider Hecke operators acting on the cohomology group H 1 (Γ, Sn (C2 )) in (6.10), we choose an element α ∈ Γ ⊂ GL(2, R) such that the corresponding double coset has a decomposition of the form s  Γ αΓ = Γ αi i=1

with α1 , . . . , αs ∈ Γ. Then from (6.16) we see that the Hecke operator TnH (α) on the space H 1 (Γ, Sn (C2 )) can be defined by

6.3 Jacobi-like Forms and Group Cohomology

(TnH (α)(φ))(γ) =

s 

115

(det αi )n/2 ρn (αi )φ(ξi (γ))

(6.17)

i=1

for each 1-cocycle φ and γ ∈ Γ , where ξi is as in (6.12).

6.3 Jacobi-like Forms and Group Cohomology Given ξ, δ ∈ Z with δ ≥ 0, let J2ξ (Γ )δ ⊂ R[[X]]δ be the corresponding space of Jacobi-like forms of weight 2ξ for a discrete subgroup Γ of SL(2, R), and let H 1 (Γ, Sn (C2 )) with n ≥ 1 be the first cohomology group of Γ , which is a complex vector space, in (6.10). In this section we construct a linear map from J2ξ (Γ )δ to H 1 (Γ, S2m (C2 )) for each positive integer m and show that it is compatible with respect to the Hecke operator actions. Let Φ(z, X) be an element of J2ξ (Γ )δ of the form Φ(z, X) =

∞ 

φk (z)X k+δ

(6.18)

k=0

for z ∈ H. Although the coefficients φk are not modular forms, they are in fact special types of quasimodular forms as will be noted in Example 7.5. We fix a base point z0 ∈ H and, in analogy to periods of modular forms, consider certain integrals over paths in H originating at z0 which may be regarded as periods of coefficients of the Jacobi-like form Φ(z, X). If m, r and  are integers with m ≥ 0 and 0 ≤ r,  ≤ m + 1 − ξ − δ, we set  γz0 (r) ξ,δ (γ) = φm+1−ξ−δ−r (z)z  dz (6.19) m,r, z0

for all γ ∈ Γ . Note that the integral is independent of the choice of the path z0 → γz0 because the coefficients φk (z) of Φ(z, X) are holomorphic. Let {e1 , . . . , e2m+1 } be the standard basis for C2m+1 , and set ξ,δ  m,r (γ) =

2m 

ξ,δ m,r, (γ)e2m−+1 ∈ C2m+1 .

(6.20)

=0 ξ,δ (Φ) : Γ → C2m+1 by We now define the map Nm

ξ,δ Nm (Φ)(γ) =

m+1−ξ−δ  r=0

for all γ ∈ Γ .

(−1)r

(2m − r)! ξ,δ  m,r (γ) r!

(6.21)

116

6 Group Cohomology

Proposition 6.4 Given a Jacobi-like form Φ(z, X) ∈ J2ξ (Γ )δ , the assoξ,δ (Φ) : Γ → C2m+1 given by (6.21) is a cocycle belongciated map Nm 1 2m ing to Z (Γ, S (C2 )) for each nonnegative integer m, where the Γ -module S2m (C2 ) is as in Section 6.2. Proof. Given Φ(z, X) ∈ J2ξ (Γ )δ and a nonnegative integer m, since a cocycle belonging to Z 1 (Γ, S2m (C2 )) must satisfy (6.9), we need to show that ξ,δ ξ,δ ξ,δ (Φ)(γγ  ) = Nm (Φ)(γ) + ρ2m (γ)Nm (Φ)(γ  ) Nm

(6.22)

for all γ, γ  ∈ Γ . Assuming that Φ(z, X) is as in (6.18), by using (6.19), (6.20) and (6.21), we obtain ξ,δ Nm (Φ)(γ) =

m+1−ξ−δ 2m   r=0



=0

2m γz0 m+1−ξ−δ  

=

z0

 =

(−1)r

r=0

(−1)r

=0

γz0 m+1−ξ−δ 

z0

(2m − r)! ξ,δ m,r, (γ)e+1 r!

(−1)r

r=0

(2m − r)! (r) φm+1−ξ−δ−r (z)z  e+1 dz r!

(2m − r)! (r) φm+1−ξ−δ−r (z)v2m (z) dz, r!

where v2m (z) is as in (6.7). If we set fm =

m+1−ξ−δ 

(−1)r

r=0

(2m − r)! (r) φm+1−ξ−δ−r r!

(6.23)

for m ≥ ξ + δ, then by (1.34) the function fm is a modular form belonging to M2m+2 (Γ ), and we have  γz0 ξ,δ Nm (Φ)(γ) = fm (z)v2m (z) dz. (6.24) z0

Thus for γ, γ  ∈ Γ we see that ξ,δ (Φ)(γγ  ) = Nm



γγ  z0

fm (z)v2m (z) dz 

z0



γz0

=

γγ  z0

fm (z)v2m (z) dz + z0

=

(6.25)

ξ,δ (Φ)(γ) Nm



fm (z)v2m (z) dz γz0

γ  z0

+

fm (γz)v2m (γz) d(γz). z0

However, using (6.8), we have v2m (γz) = J(γ, z)−2m ρ2m (γ)v2m (z).

6.3 Jacobi-like Forms and Group Cohomology

117

From this, the relations d(γz) = J(γ, z)−2 dz,

fm (γz) = J(γ, z)2m+2 f (z), and (6.24) we obtain 



γ  z0

γ  z0

fm (γz)v2m (γz) d(γz) = ρ2m (γ) z0

=

fm (z)v2m (z) z0 ξ,δ ρ2m (γ)Nm (Φ)(γ  );

d(z)



hence (6.22) follows from this and (6.25). By Proposition 6.4 for each m ≥ 1 there is a linear map ξ,δ : J2ξ (Γ )δ → H 1 (Γ, S2m (C2 )) Nm

(6.26)

sending a Jacobi-like form Φ(z, X) ∈ J2ξ (Γ )δ to the cohomology class of ξ,δ (Φ(z, X)) in H 1 (Γ, S2m (C2 )). Nm ξ,δ Theorem 6.5 Given a positive integer m, the linear map Nm in (6.26) satisfies ξ,δ J H ξ,δ ◦ T2ξ (α) = T2m (α) ◦ Nm Nm

for each α ∈ Γ, where the Hecke operators J (α) : J2ξ (Γ )δ → J2ξ (Γ )δ T2ξ H T2m (α) : H 1 (Γ, S2m (C2 )) → H 1 (Γ, S2m (C2 )),

are as in (3.10) and (6.17), respectively. Proof. Let α ∈ Γ, and assume that the associated double coset of Γ has a decomposition of the form Γ αΓ =

s 

Γ αi

i=1

with α1 , . . . , αs ∈ GL+ (2, R) as in (6.11). Let Φ(z, X) ∈ J2ξ (Γ )δ be as in J (6.18), and for each m ≥ ξ + δ let fm be as in (6.23). We write T2ξ (α)Φ in the form ∞  T J (α)Φ(z, X) = φk (z)X k+δ , 2ξ

k=0

and set fm =

m+1−ξ−δ  r=0

(−1)r

(2m − r)! (r) φm+1−ξ−δ−r r!

118

6 Group Cohomology

for m ≥ ξ + δ. Then fm is a modular form belonging to M2m+2 (Γ ), and by using (3.19) it can be shown easily that fm = T2m+2 (α)fm , where T2m+2 (α) is as in (3.7). From this and (6.24) we obtain ξ,δ J (T2ξ (α)Φ)(γ) Nm  γz0 (T2m+2 (α)fm )(z)v2m (z) dz = z0

= = =

s  i=1 s  i=1 s 



(det αi )m+1 J(αi , z)−2m−2 (det αi )

m+1

ρ2m (αi )

−1

γz0

fm (αi z)v2m (z) dz z0



γz0

fm (αi z)v2m (αi z) d(αi z) z0



αi γz0

(det αi )m+1

fm (z)v2m (z) dz αi z 0

i=1

for all γ ∈ Γ . Using the relation αi γ = ξi (γ) · αi(γ) in (6.12), we may write 



αi γz0

ξi (γ)αi(γ) z0

= αi z 0

 −

z0

αi z0 z0



ξi (γ)αi(γ) z0

= z0

 αi(γ) z0  −ρ2m (ξi (γ)) z0   αi(γ) z0  + ρ2m (ξi (γ)) − z0

αi z 0 

. z0

However, we see that 

ξi (γ)αi(γ) z0

 −ρ2m (ξi (γ))

z0

αi(γ) z0 

fm (z)v2m (z) dz z0



ξi (γ)αi(γ) z0

=

fm (z)v2m (z) dz −

z0



ξi (γ)z0

=



ξi (γ)αi(γ) z0

fm (z)v2m (z) dz ξi (γ)z0

ξ,δ fm (z)v2m (z) dz = Nm (Φ)(ξi (γ)),

z0

where we used (6.24), and

6.3 Jacobi-like Forms and Group Cohomology





αi(γ) z0

ρ2m (ξi (γ))

 −

119

αi z 0 

fm (z)v2m (z) dz  αi(γ) z0 −1 = ρ2m (αi )ρ2m (γ)ρ2m (αi(γ) ) fm (z)v2m (z) dz z0  αi z0 − fm (z)v2m (z) dz. z0

z0

z0

Using the above relations, (6.17), and the fact that det αi(γ) = det αi , we obtain ξ,δ J (T2ξ (α)Φ)(γ) Nm s  ξ,δ (det αi )m+1 ρ2m (αi )−1 Nm (Φ)(ξi (γ)) = i=1

+ ρ2m (γ)

s 

(det αi(γ) )

m+1

ρ2m (αi(γ) )

−1

i=1

+

s 

(det αi )m+1 ρ2m (αi )−1





αi(γ) z0

fm (z)v2m (z) dz z0 αi(γ) z0

fm (z)v2m (z) dz z0

i=1

H ξ,δ (α)Nm (Φ))(γ) + (ρ2m (γ) − 12m+1 )u, = (T2m

where 12m+1 is the identity map on C2m+1 , and u=

s 

(det αi )m+1 ρ2m (αi )−1



αi z0

fm (z)v2m (z) dz. z0

i=1

Hence the theorem follows.



X Corollary 6.6 Let Iξ,ε : Ψ D(R)Γ−ε → J2ξ (Γ )ε−ξ be the isomorphism in Corollary 1.16, and set ∂,ξ,ε ξ,ε−ξ X = Nm ◦ Iξ,ε : Ψ D(R)Γ−ε → H 1 (Γ, S2m (C2 )). Nm

Then we have ∂,ξ,ε Ψ H ∂,ξ,ε ◦ T−ε (α) = T2m (α) ◦ Nm Nm Ψ H for each m ≥ 1, where T−ε (α) and T2m (α) are as in (3.25) and (6.17), respectively.

Proof. This follows immediately from Theorem 6.5 and the commutativity of the diagram (3.30).  From Theorem 6.5, Corollary 6.6, and the commutative diagram (3.30) we see that the diagram

120

6 Group Cohomology Ψ (α) T−ε

Ψ D(R)Γ−ε ⏐ X ⏐ Iξ,ε 

−−−−−→

J2ξ (Γ )ε−ξ ⏐ ξ,ε−ξ ⏐ Nm 

−−−−→

J T2ξ (α)

Ψ D(R)Γ−ε ⏐ ⏐I X  ξ,ε J2ξ (Γ )ε−ξ ⏐ ⏐ ξ,ε−ξ N m

T H (α)

H 1 (Γ, S2m (C2 )) −−2m −−−→ H 1 (Γ, S2m (C2 )) is commutative, where the composite of the vertical maps on each side is ∂,ξ,ε equal to Nm .

Chapter 7

Quasimodular Forms

Quasimodular forms generalize classical modular forms and were introduced by Kaneko and Zagier in [54]. Since then they have been studied in connection with various topics not only in number theory but also in applied mathematics (see e.g. [12], [39], [56], [80], [84], [92], [97], [104]). Unlike modular forms, derivatives of quasimodular forms are also quasimodular forms. In this chapter we describe quasimodular forms for a discrete subgroup of SL(2, R) and introduce quasimodular polynomials, which can be identified with quasimodular forms. We consider connections between Jacobi-like forms and quasimodular polynomials and study Hecke operators on quasimodular polynomials that are compatible with those on modular and Jacobi-like forms. Throughout this chapter we assume that the discrete subgroup Γ of SL(2, R) considered earlier is commensurable with SL(2, Z) in order to avoid it being too small.

7.1 Quasimodular and Modular Forms In this section we describe quasimodular forms for a discrete subgroup of SL(2, R) introduced by Kaneko and Zagier (see [54]). We review some of their properties, including their connections with modular forms, and provide a number of examples. Let R be the ring of holomorphic functions on H as in Chapter 1, and let Γ be a discrete subgroup of SL(2, R) commensurable with SL(2, Z) as noted above. We also fix a nonnegative integer m. Definition 7.1 Let ξ be an integer, and let |ξ be the operation in (1.16). An element f ∈ R is a quasimodular form for Γ of weight ξ and depth at most m if there are functions f0 , . . . , fm ∈ R such that

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_7

121

122

7 Quasimodular Forms

(f |ξ γ)(z) =

m 

fr (z)K(γ, z)r

(7.1)

r=0

for all z ∈ H and γ ∈ Γ , where K(γ, z) is as in (1.12). We denote by QMξm (Γ ) the space of quasimodular forms for Γ of weight ξ and depth at most m. Lemma 7.2 If the functions g0 , g1 , . . . , gm ∈ R satisfy m 

gr (z)K(γ, z)r = 0

(7.2)

r=0

for all z ∈ H and γ ∈ Γ , then gr = 0 for 0 ≤ r ≤ m. Proof. Since Γ is commensurable with SL(2, Z), there are infinitely many   distinct elements γ = ac db ∈ Γ with c = 0. If we set μ = d/c, the relation (7.2) can be written as m 

gr (z)(z + μ)−r = 0.

r=0

Thus we have 0 = (μ + z)m g0 (z) + (μ + z)m−1 g1 (z) + · · · + (μ + z)gm−1 (z) + gm (z) = μm g0 (z) + μm−1 (mzg0 (z) + g1 (z)) 2 + μm−2 (( m 2 ) z g0 (z) + (m − 1)zg1 (z) + g2 ) + · · ·   m−2 m · · · + μ(( m−1 ) z m−1 g0 (z) + m−1 g1 (z) + · · · + gm−1 ) m−2 z

+ (z m g0 (z) + z m−1 g1 (z) + · · · + zgm−1 + gm ) for all such μ. Hence it follows that each gr is identically zero.



Remark 7.3 (i) If we set γ ∈ Γ in (7.1) to be the identity matrix, then K(γ, z) = 0; hence it follows that f = f0 for f ∈ QMξm (Γ ). (ii) If m = 0, from (7.1) we obtain f |ξ γ = f0 = f, which shows that QMξ0 (Γ ) = Mξ (Γ ). Thus quasimodular forms of depth at most 0 are modular forms. (iii) If (7.1) is satisfied for another set of functions g0 , . . . , gm ∈ R, then we have

7.1 Quasimodular and Modular Forms m 

123

(gr (z) − fr (z))K(γ, z)r = 0

r=0

for all γ ∈ Γ ; hence gr = fr for each r by Lemma 7.2. Thus we see that the quasimodular form f determines the associated functions f0 , . . . , fm ∈ R uniquely. Example 7.4 Let E2 be the Eisenstein series given by E2 (z) = 1 − 24

∞ 

σ1 (n)e2πinz

(7.3)

n=1

for z ∈ H, where σ1 (n) = satisfies the relation

d|n

d. Then it is well known that the function E2

(E2 |2 γ)(z) = E2 (z) +

6 K(γ, z) iπ

(7.4)

for all γ ∈ SL(2, Z); hence E2 is a quasimodular form belonging to QM21 (Γ ) with Γ = SL(2, Z). Example 7.5 Given ξ, δ ∈ Z with δ ≥ 0, let Φ(z, X) ∈ Jξ (Γ )δ be a Jacobilike form for a discrete subgroup Γ of SL(2, R) given by Φ(z, X) =

∞ 

φk (z)X k+δ .

k=0

Then, for m ≥ 0, by Theorem 1.11 the coefficient φm satisfies (φm |2m+2δ+ξ γ)(z) =

m  1 K(γ, z)r φm−r (z) r! r=0

for all z ∈ H. Thus it follows that φm is a quasimodular form belonging to m (Γ ). From this and the correspondence between Jacobi-like forms QM2m+2δ+ξ and automorphic pseudodifferential operators we also see that the coefficients of automorphic pseudodifferential operators are also quasimodular forms. Example 7.6 Let f be a modular form belonging to Mw (Γ ) with w ∈ Z, so that (7.5) f (γz) = J(γ, z)w f (z) for z ∈ H and γ ∈ Γ . Since side of (7.5) is given by

d dz (γz)

= J(γ, z)−2 , the derivative of the left-hand

d (f (γz)) = J(γ, z)−2 f  (γz). dz On the other hand, since

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7 Quasimodular Forms

d J(γ, z) = J(γ, z)K(γ, z) dz as in (1.22), the derivative of the right-hand side of (7.5) can be written as d (J(γ, z)w f (z)) = J(γ, z)w f  (z) + wK(γ, z)J(γ, z)w f (z). dz Thus we obtain (f  |w+2 γ)(z) = J(γ, z)−w−2 f  (γz) = f  (z) + wf (z)K(γ, z). Hence it follows that the derivative f  of the modular form f ∈ Mw (Γ ) is a 1 quasimodular form belonging to QMw+2 (Γ ). Although the derivative of a modular form is not a modular form, the previous example shows that the derivative of a modular form is a quasimodular form. On the other hand, the next lemma shows that the derivative of a quasimodular form is also a quasimodular form. Lemma 7.7 Let f be a quasimodular form belonging to QMwm (Γ ). Then its m+1 (Γ ). derivative f  is a quasimodular form belonging to QMw+2 Proof. Since f ∈ QMwm (Γ ), for z ∈ H and γ ∈ Γ we have (f |w γ)(z) = J(γ, z)−w f (γz) =

m 

fk (z)K(γ, z)k

k=0

for some holomorphic functions f0 , f1 , . . . , fm on H. By taking the derivative of this relation we obtain   d d J(γ, z) f (γz) + J(γ, z)−w f  (γz) (γz) −wJ(γ, z)−w−1 dz dz m   d  = K(γ, z) . (fk )(z)K(γ, z)k + kfk (z)K(γ, z)k−1 dz k=0

Using this, (1.22), and the identity d (γz) = J(γ, z)−2 , dz we see that −wJ(γ, z)−w K(γ, z)f (γz) + J(γ, z)−w−2 f  (γz) m    = fk (z)K(γ, z)k − kfk (z)K(γ, z)k+1 . k=0

Thus we have

7.1 Quasimodular and Modular Forms

125

(f  |w+2 γ)(z) = wK(γ, z)(f |w γ)(z) m    + fk (z)K(γ, z)k − kfk (z)K(γ, z)k+1 k=0

= wK(γ, z)

m 

fk (z)K(γ, z)k

k=0

+

m  

fk (z)K(γ, z)k − kfk (z)K(γ, z)k+1



k=0

=

m+1 

(w − k + 1)fk−1 (z)K(γ, z)k +

k=1

m 

fk (z)K(γ, z)k .

k=0

Hence we may write (f  |w+2 γ)(z) =

m+1 

hk (z)K(γ, z)k ,

(7.6)

k=0

where h0 = f0 , hm+1 = (w − m)fm , and hk = (w − k + 1)fk−1 + fk m+1 for 1 ≤ k ≤ m. Thus it follows that f  ∈ QMw+2 (Γ ).



Lemma 7.8 Let f be a quasimodular form belonging to QMwm (Γ ), and assume that m  (f |w γ)(z) = fk (z)K(γ, z)k (7.7) k=0

for all z ∈ H and γ ∈ Γ , where f0 , f1 , . . . , fm are holomorphic functions on H. Then for 0 ≤ k ≤ m the function fk is a quasimodular form belonging to m−k (Γ ). QMw−2k Proof. Given elements γ, γ  ∈ Γ , using (7.7), we have (f |w γγ  )(z) = ((f |w γ) |w γ  )(z) = J(γ  , z)−w (f |w γ)(γ  z) m  = J(γ  , z)−w fk (γ  z)K(γ, γ  z)k k=0

for all z ∈ H. From this and the relation K(γ, γ  z) = J(γ  , z)2 (K(γγ  , z) − K(γ  , z)) in (1.14) we obtain

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7 Quasimodular Forms

(f |w γγ  )(z) = J(γ  , z)−w

m 

fk (γ  z)J(γ  , z)2k

(7.8)

k=0

× (K(γγ  , z) − K(γ  , z))k

=

k m  

(−1)k−

k=0 =0

  k fk (γ  z)J(γ  , z)2k−w 

× K(γγ  , z) K(γ  , z)k−   m m   k K(γ  , z)k− J(γ  , z)2k−w = (−1)k−  =0 k=

× fk (γ  z)K(γγ  , z) . On the other hand, using (7.7) again, we see that (f |w γγ  )(z) =

m 

f (z)K(γγ  , z) .

=0

By comparing this with (7.8) we obtain f (z) =

m 

(−1)k−

k=

  k K(γ  , z)k− J(γ  , z)2k−w fk (γ  z) 

for 0 ≤  ≤ m. Using this and (1.15) and replacing z by (γ  )−1 z  with z  ∈ H, we have   m  k K(γ  , (γ  )−1 z  )k− J(γ  , (γ  )−1 z  )2k−w fk (z  ) f ((γ  )−1 z  ) = (−1)k−  k= m    k K((γ  )−1 , z  )k− J((γ  )−1 , z  )w−2 fk (z  ), =  k=

where we used the identities J(γ  , (γ  )−1 z  ) = J((γ  )−1 , z  )−1 , K(γ  , (γ  )−1 z  ) = −J((γ  )−1 , z  )2 K((γ  )−1 , z  ). Thus we obtain  −1

(f |w−2 (γ )



)(z ) =

m    k k=



K((γ  )−1 , z  )k− fk (z  ),

m− (Γ ). which implies that f belongs to QMw−2



7.2 Polynomials and Formal Power Series

127

7.2 Polynomials and Formal Power Series In this section we introduce an action of SL(2, R) on polynomials over the ring R of holomorphic functions on the Poincar´e upper half plane H. We then consider surjective maps from formal power series to polynomials that are SL(2, R)-equivariant. We fix a nonnegative integer m and consider the complex vector space Rm [X] of polynomials in X over R of degree at most m. If F (z, X) ∈ Rm [X] and λ ∈ Z, we set (F λ γ)(z, X) = J(γ, z)−λ F (γz, J(γ, z)2 (X − K(γ, z)))

(7.9)

for all z ∈ H and γ ∈ SL(2, R), Lemma 7.9 If F (z, X) ∈ Rm [X] and λ ∈ Z, then we have ((F λ γ) λ γ  )(z, X) = (F λ (γγ  ))(z, X) for all γ, γ  ∈ SL(2, R). Proof. Given γ, γ  ∈ SL(2, R), λ ∈ Z and F (z, X) ∈ Rm [X], using (7.9), we have ((F λ γ) λ γ  )(z, X)

(7.10)

= J(γ  , z)−λ (F λ γ)(γ  z, J(γ  , z)2 (X − K(γ  , z))) = J(γ  , z)−λ J(γ, γ  z)−λ F (γγ  z, X  ), where X  = J(γ, γ  z)2 J(γ  , z)2 (X − K(γ  , z)) − J(γ, γ  z)2 K(γ, γ  z). Thus using (1.13) and (1.14), we see that X  = J(γγ  , z)2 (X − K(γ  , z)) − J(γ, γ  z)2 J(γ  , z)2 (K(γγ  , z) − K(γ  , z)) = J(γγ  , z)2 (X − K(γγ  , z)). From this and (7.10), we obtain ((F λ γ) λ γ  )(z, X) = J(γγ  , z)−λ F (γγ  z, J(γγ  , z)2 (X − K(γγ  , z)) = (F λ (γγ  ))(z, X); hence the lemma follows.



From Lemma 7.9 we see that the operation λ with λ ∈ Z determines an action of SL(2, R) on Rm [X] on the right. If Φ(z, X) ∈ R[[X]]δ with δ ≥ 0 is given by

128

7 Quasimodular Forms

Φ(z, X) =

∞ 

φk (z)X k+δ ,

k=0

then we set δ (Πm Φ)(z, X) =

m  1 φm−r (z)X r , r! r=0

(7.11)

which determines the surjective complex linear map δ : R[[X]]δ → Rm [X]. Πm

(7.12)

In particular, for m = 0 we obtain the map Π0δ : R[[X]]δ → R

(7.13)

(Π0δ Φ)(z) = φ0 (z)

(7.14)

defined by for Φ(z, X) ∈ R[[X]]δ as above and z ∈ H. Furthermore, if 0 ≤  ≤ m and F (z, X) =

m 

fk (z)X k ∈ Rm [X],

k=0

we set (S F )(z) = f (z)

(7.15)

for all z ∈ H, so that we obtain another complex linear map S : Rm [X] → R.

(7.16)

Lemma 7.10 Given m, δ ≥ 0, the diagram Πδ

 ι

0 0 −−−−→ R[[X]]δ+1 −−−−→ R[[X]]δ −−−− → ⏐ ⏐ ⏐ δ ⏐Π δ+1 Πm  m−1

R −−−−→ 0 ⏐ ⏐ μm 

(7.17)

S

ι

0 −−−−→ Rm−1 [X] −−−−→ Rm [X] −−−m −→ R −−−−→ 0 commutes, where ι and ι are inclusion maps and μm is multiplication by (1/m!). Furthermore, the two rows in the diagram are short exact sequences. Proof. Let Φ(z, X) ∈ R[[X]]δ and Ψ (z, X) ∈ R[[X]]δ+1 be given by Φ(z, X) =

∞ 

φk (z)X k+δ ,

Ψ (z, X) =

k=0

∞ 

ψk (z)X k+δ+1 .

k=0

Then from (7.11) and (7.15) we see that δ )Φ)(z) = φ0 (z)/m! = (μm φ0 )(z). ((Sm ◦ Πm

7.2 Polynomials and Formal Power Series

129

Since φ0 = Π0δ Φ by (7.14), we obtain δ = μm ◦ Π0δ . Sm ◦ Πm

On the other hand, we have ( ιΨ )(z, X) =

∞ 

ψ k (z)X k+δ

k=0

with ψ 0 = 0 and ψ k = ψk−1 for k ≥ 1. Hence we see that δ ((Πm ◦ ι)Ψ )(z, X) =

m m−1   1 1 ψm−r (z)X r = ψm−1−r (z)X r r! r! r=0 r=0

δ+1 δ+1 = (Πm−1 Ψ )(z, X) = ((ι ◦ Πm−1 )Ψ )(z, X), δ+1 δ ◦ ι = ι ◦ Πm−1 . The two rows in the diagram are clearly which shows that Πm short exact sequences. 

We recall from Chapter 1 that SL(2, R) acts on R[[X]]δ on the right by the δ operation in (1.17). The next proposition shows that the surjective map Πm in (7.12) is equivariant with respect to this action and the action of SL(2, R) on Rm [X] given by (7.9). Proposition 7.11 If m, δ and λ are integers with m, δ ≥ 0, we have δ δ (Φ |Jλ γ) = Πm (Φ) λ+2m+2δ γ Πm

(7.18)

for all Φ(z, X) ∈ R[[X]]δ and γ ∈ SL(2, R). Proof. Given Φ(z, X) = (1.17), we have (Φ |Jλ γ)(z, X) =

∞ 

∞ k=0

φk (z)X k+δ ∈ R[[X]]δ and γ ∈ SL(2, R), using

J(γ, z)−λ

k=0

∞  (−1) =0

!

K(γ, z) X  φk (γz) × J(γ, z)−2k−2δ X k+δ

= =

∞ ∞   (−1) k=0 =0 ∞ 

!

J(γ, z)−2k−2δ−λ K(γ, z) φk (γz)X k++δ

φn (z, X)X n+δ

n=0

with φn (z, X) =

n  (−1) =0

!

J(γ, z)−2n+2−2δ−λ K(γ, z) φn− (γz).

130

7 Quasimodular Forms

From this and (7.11) we see that δ (Πm (Φ |Jλ γ))(z, X) =

=

m  1 φm−r (z)X r r! r=0

(7.19)

m m−r   (−1) J(γ, z)2+2r−2m−2δ−λ K(γ, z) r!! r=0 =0

× φm−−r (γz)X r . On the other hand, using (1.3) and (7.9), we have δ Φ) λ+2m+2δ γ)(z, X) ((Πm δ = J(γ, z)−λ−2m−2δ (Πm Φ)(γz, J(γ, z)2 (X − K(γ, z))) m  1 = J(γ, z)−λ−2m−2δ φm−j (γz)J(γ, z)2j (X − K(γ, z))j j! j=0

=

  j m   1 j φm−j (γz)J(γ, z)2j−λ−2m−2δ r j! j=0 r=0 × (−1)j−r K(γ, z)j−r X r

=

m m   (−1)j−r J(γ, z)2j−λ−2m−2δ K(γ, z)j−r φm−j (γz)X r . r!(j − r)! r=0 j=r

Changing the index for the second summation in the previous line from j to  = j − r and comparing this with (7.19), we obtain the relation (7.18). 

7.3 Quasimodular Polynomials To study quasimodular forms it is often convenient to consider the naturally associated polynomials known as quasimodular polynomials, whose coefficients are holomorphic functions on H. Quasimodular forms for Γ can be defined to be Γ -invariant elements under the action of SL(2, R) on Rm [X] considered in Section 7.2. In this section we discuss their connections with Jacobi-like forms. We fix a nonnegative integer m and denote by QMξm (Γ ) with ξ ∈ Z the space of quasimodular forms for Γ of weight ξ and depth at most m as in Section 7.1, where Γ is a discrete subgroup of SL(2, R) commensurable with SL(2, Z) as in the previous section. If f ∈ R is a quasimodular form belonging to QMξm (Γ ) satisfying (7.1), then we define the corresponding polynomial (Qm ξ f )(z, X) ∈ Rm [X] by

7.3 Quasimodular Polynomials

131

(Qm ξ f )(z, X) =

m 

fr (z)X r

(7.20)

r=0

for z ∈ H. We note that Qm ξ f is well defined due to Remark 7.3(iii). Thus we obtain the linear map m Qm ξ : QMξ (Γ ) → Rm [X]

for each ξ ∈ Z. Definition 7.12 A quasimodular polynomial for Γ of weight ξ and degree at most m is an element of Rm [X] that is invariant with respect to the right Γ -action in (7.9). We denote by QPξm (Γ ) the space of all quasimodular polynomials for Γ of weight ξ and degree at most m, that is, QPξm (Γ ) = {F (z, X) ∈ Rm [X] | F ξ γ = F for all γ ∈ Γ }, where ξ is as in (7.9). Lemma 7.13 Given F (z, X) = we have Sr (F ξ γ)(z) =

m 

(−1)

−r

=r

(Sr (F ξ γ

−1

m r=0

fr (z)X r ∈ Rm [X] and γ ∈ SL(2, R),

   f (γz)J(γ, z)2−ξ K(γ, z)−r , r

) |ξ−2r γ)(z) =

m     =r

r

f (z)K(γ, z)−r

(7.21)

(7.22)

for 0 ≤ r ≤ m, where Sr is as in (7.15). Proof. Using (7.9), we have m 

(F ξ γ)(z, X) = J(γ, z)−ξ

f (γz)J(γ, z)2 (X − K(γ, z))

=0

  f (γz)J(γ, z)2−ξ (−1)−r K(γ, z)−r X r r =0 r=0   m m   −r  f (γz)J(γ, z)2−ξ K(γ, z)−r X r , = (−1) r r=0 =

m    

=r

which verifies (7.21). Replacing γ by γ −1 and z by γz in (7.21), we have Sr (F ξ γ −1 )(γz) = J(γ −1 , γz)−ξ ×

m  =r

(−1)−r

   f (z)J(γ −1 , γz)2 K(γ −1 , γz)−r . r

132

7 Quasimodular Forms

However, from the identities J(γ −1 , γz) = J(γ, z)−1 ,

K(γ −1 , γz) = −J(γ, z)2 K(γ, z)

(7.23)

in (1.15) it follows that Sr (F ξ γ −1 )(γz) = J(γ, z)ξ

m 

(−1)−r

=r

= J(γ, z)ξ−2r

m   =r

   f (z)J(γ, z)−2 r



× (−1)−r J(γ, z)2−2r K(γ, z)−r

 f (z)K(γ, z)−r ; r 

hence we obtain (7.22). By setting r = 0 in (7.21) we obtain S0 (F ξ γ)(z) =

m 

(−1) J(γ, z)2−ξ K(γ, z) f (γz),

(7.24)

=0

S0 (F ξ γ −1 )(γz) = J(γ, z)ξ

m 

fr (z)K(γ, z)r ,

r=0

where we used (7.23). In particular, if F (z, X) ∈ QPξm (Γ ), we have (S0 F )(z) =

m 

(−1) J(γ, z)2−ξ K(γ, z) f (γz)

=0

for all γ ∈ Γ . The following corollary provides an alternative proof of Lemma 7.8 by using quasimodular polynomials.

m Corollary 7.14 Let F (z, X) = r=0 fr (z)X r ∈ Rm [X]. Then F (z, X) is a quasimodular polynomial belonging to QPξm (Γ ) if and only if for each r ∈ {0, 1, . . . , m} the coefficient fr satisfies (fr |ξ−2r γ)(z) =

m    

f (z)K(γ, z)−r r =r m−r   + r  f+r (z)K(γ, z) = r

(7.25)

=0

for all z ∈ H and γ ∈ Γ . In particular, fr is a quasimodular form belonging m−r to QMξ−2r (Γ ). Proof. A polynomial F (z, X) ∈ Rm [X] belongs to QPξm (Γ ) if and only if

7.3 Quasimodular Polynomials

133

F (z, X) = (F ξ γ)(z, X) for all γ ∈ Γ . Hence (7.25) follows from this and (7.22).



From the previous corollary we see that the map Sr in (7.16) determines the map m−r (Γ ) (7.26) Sr : QPξm (Γ ) → QMξ−2r for 0 ≤ r ≤ m. For 0 ≤ r ≤ m we note that (7.25) can be written in the form (Sr F |ξ−2r γ)(z) =

 +r (S+r F )(z)K(γ, z) r

m−r  =0

for F (z, X) ∈ QPξm (Γ ). In particular, we obtain Sm F |ξ−2m γ = Sm F for all γ ∈ Γ . Thus it follows that Sm F ∈ Mξ−2m (Γ )

(7.27)

if F (z, X) ∈ QPξm (Γ ). On the other hand, we also have (Qm−r ξ−2r (Sr F ))(z, X) =

m−r  =0

 +r m−r (S+r F )(z)X  ∈ QPξ−2r (Γ ), r

m where Qm−r ξ−2r is as in (7.20). Thus, in particular, we see that the map Qξ given by (7.20) determines the complex linear map m m Qm ξ : QMξ (Γ ) → QPξ (Γ ),

(7.28)

so that Qm ξ carries quasimodular forms to quasimodular polynomials. In fact, the next proposition shows that it is an isomorphism. Proposition 7.15 The restriction of the map Sr in (7.16) with r = 0 to QPξm (Γ ) determines an isomorphism S0 : QPξm (Γ ) → QMξm (Γ )

(7.29)

whose inverse is the map Qm ξ in (7.28). Proof. Let F (z, X) =

m r=0

fr (z)X r be an element of QPξm (Γ ), so that f0 (z) = (S0 F )(z)

for z ∈ H. Then from Corollary 7.14 we see that

134

7 Quasimodular Forms

(f0 |ξ γ)(z) =

m 

f (z)K(γ, z)

=0

for all z ∈ H and γ ∈ Γ . Thus we have m ((Qm ξ ◦ S0 )F )(z, X) = (Qξ f0 )(z, X) =

m 

fj (z)X j = F (z, X).

j=0

We now consider an element h ∈ QMξm (Γ ) satisfying (h |ξ γ)(z) =

m 

h (z)K(γ, z)

=0

for all z ∈ H and γ ∈ Γ , so that (Qm ξ h)(z, X) =

m 

h (z)X  .

=0

Then we see that (S0 ◦ Qm ξ )(h(z)) = h0 (z) = h(z) for all z ∈ H, and therefore the proposition follows.



Remark 7.16 Let f ∈ QMξm (Γ ) be a quasimodular form satisfying (7.1). In the proof of Theorem 1 in [97], it was indicated that the corresponding polynomial F (z, X) = (Qm ξ f )(z, X) ∈ Rm [X] satisfies J(γ, z)−ξ F (γz, X) = F (z, J(γ, z)−2 X + K(γ, z))

(7.30)

for all z ∈ H and γ ∈ Γ . It can be shown, however, that (7.30) is equivalent to the condition that F ξ γ = F for all γ ∈ Γ . Indeed, if F (z, X) ∈ Rm [X] satisfies (7.30), by replacing γ and z by γ −1 and γz, respectively, we have J(γ −1 , γz)−ξ F (z, X) = F (γz, J(γ −1 , γz)−2 X + K(γ −1 , γz)).

(7.31)

From (1.15) and (7.31) we obtain F (z, X) = J(γ, z)−ξ F (γz, J(γ, z)2 X − J(γ, z)2 K(γ, z)) = (F ξ γ)(z, X), where we used (7.9).

7.3 Quasimodular Polynomials

135

δ Proposition 7.17 The map Πm in (7.11) induces the complex linear map δ m : Jλ (Γ )δ → QPλ+2m+2δ (Γ ) Πm

(7.32)

for each δ ≥ 0 and λ ∈ Z. Proof. We need to show that δ m (Jλ (Γ )δ ) ⊂ QPλ+2m+2δ (Γ ). Πm

However, this follows immediately from (1.26), Proposition 7.11 and Definition 7.12.  Lemma 7.18 Let F (z, X) be a quasimodular polynomial belonging to QPξm (Γ ), and set   1 φF (z) = F z, z−z for all z ∈ H. Then the function φF : H → C satisfies φF (γz) = J(γ, z)ξ φF (z)

(7.33)

for all z ∈ H and γ ∈ Γ . Proof. If F (z, X) ∈ QPξm (Γ ), we have F (γz, J(γ, z)2 (X − K(γ, z))) = J(γ, z)ξ F (z, X) for z ∈ H and γ ∈ Γ . Replacing X by 1/(z − z), we obtain      1 1 2 ξ − K(γ, z) = J(γ, z) F z, . F γz, J(γ, z) z−z z−z   However, if γ = ac db , we have J(γ, z) 1 cz + d − c(z − z) − K(γ, z) = = z−z (z − z)(cz + d) (z − z)J(γ, z) |J(γ, z)|2 1 = = , 2 (z − z)J(γ, z) (γz − γz)J(γ, z)2 where we used the identity γz − γz =

(ad − bc)(z − z) (z − z) az + b az + b − = = . cz + d cz + d (cz + d)(cz + d) |J(γ, z)|2

Thus (7.34) can be written as     1 1 = J(γ, z)ξ F z, , F γz, γz − γz z−z

(7.34)

(7.35)

136

7 Quasimodular Forms



which verifies (7.33).

If y = Im z and F (z, X) =

m 

fr (z)X r ∈ QPξm (Γ ),

r=0

then we see that φF (z) =

m m   fr (z) fr (z) 1 = . r (z − z) (2i)r y r r=0 r=0

This leads us to the notion of nearly holomorphic modular forms introduced by Shimura in [110].

Definition 7.19 A function φ : H → C is a nearly holomorphic modular form of weight ξ and depth at most m for Γ if it satisfies φ |ξ γ = φ for all γ ∈ Γ and there are holomorphic functions f0 , f1 , . . . , fm on H such that fm (z) f1 (z) φ(z) = f0 (z) + + ··· + (7.36) z−z (z − z)m for all z ∈ H. We denote by N Mξ (Γ ) the space of such nearly holomorphic modular forms.

Let φ ∈ N Mξ (Γ ) be as in (7.36). Then we see that φ(z) =

m m   fr (z) fr (γz) −ξ = J(γ, z) r (z − z) (γz − γz)r r=0 r=0 r   m  1 − K(γ, z) = J(γ, z)−ξ fr (γz) J(γ, z)2 z−z r=0

for all z ∈ H and γ ∈ Γ , where we used (7.35). Thus, if we set Fφ (z, X) =

m 

fr (z)X r ,

r=0

it satisfies (7.34) and therefore is a quasimodular polynomial belonging to QPξm (Γ ).

7.4 Hecke Operators

137

7.4 Hecke Operators In this section we introduce Hecke operators on quasimodular polynomials and on quasimodular forms that are compatible with those on modular and Jacobi-like forms. Let J, K : GL+ (2, R) × H → C be the maps given by (3.1). For λ ∈ Z we extend the action of SL(2, R) in (7.9) to that of GL+ (2, R) by setting (F λ α)(z, X) = det(α)λ/2 J(α, z)−λ F (αz, det(α)−1

(7.37)

× J(α, z) (X − K(α, z))) 2

for all z ∈ H, α ∈ GL+ (2, R) and F (z, X) ∈ Rm [X]. This formula determines a right action of GL+ (2, R) on Rm [X], so that we have (F λ α) λ α = F λ (αα )

(7.38)

for all α, α ∈ GL+ (2, R). If α is an element of the commensurator Γ of Γ such that the corresponding double coset is as in (3.6) and if F (z, X) ∈ QPkm (Γ ), we set (TkP (α)F )(z, X)

=

s 

(F k αi )(z, X)

(7.39)

i=1

for all z ∈ H. Proposition 7.20 For each α ∈ Γ the polynomial given by (7.39) is independent of the choice of the coset representatives α1 , . . . , αs , and the map F → TλP (α)F determines the linear endomorphism TλP (α) : QPλm (Γ ) → QPλm (Γ ). Proof. This can be proved as in the case of modular forms by using (7.38) and (7.39).  We now extend the map F (z, X) → S0 (F ||ξ γ)(z) : Rm [X] → R for an element γ of SL(2, R) to an element of GL+ (2, R) by changing (7.24) to m  S0 (F ||ξ α)(z) = (−1)j (det α)ξ/2−j J(α, z)2j−ξ K(α, z)j fj (αz) j=0

for α ∈ GL+ (2, R). If f is a quasimodular form belonging to QMξm (Γ ) and if α ∈ Γ is an element whose double coset has a decomposition as in (3.6), we set

138

7 Quasimodular Forms

TξQ (α)f =

s 

S0 ((Qm ξ f )||ξ α).

(7.40)

i=1

In other words, TξQ (α) is deduced from TξP (α) by the composition TξQ (α) = S0 ◦ TξP (α) ◦ Qm ξ . So this justifies that TξQ (α)f ∈ QMξm (Γ ). Thus we obtain the linear endomorphism (7.41) TξQ (α) : QMξm (Γ ) → QMξm (Γ ) for each α ∈ Γ. Definition 7.21 The linear endomorphism (7.41) given by (7.40) is the Hecke operator on QMξm (Γ ) associated to α ∈ Γ. Remark 7.22 Special types of Hecke operators on quasimodular forms as in Definition 7.21 were considered by Movasati in [86] for Γ = SL(2, Z). Theorem 7.23 For each α ∈ Γ the diagram Qm ξ

S

−→ Mξ−2m (Γ ) QMξm (Γ ) −−−−→ QPξm (Γ ) −−−m ⏐ ⏐ ⏐ ⏐ ⏐ ⏐T TξQ (α) TξP (α)  ξ−2m (α) Qm ξ

(7.42)

S

QMξm (Γ ) −−−−→ QPξm (Γ ) −−−m −→ Mξ−2m (Γ ) Q P is commutative, where the maps Sm , Qm x i, Tξ (α), Tξ (α) and Tξ−2m (α) are as in (7.27), (7.28), (7.39), (7.40) and (3.7), respectively.

Proof. Let α be an element of Γ such that the corresponding double coset has a decomposition as in (3.6). Given f ∈ QMξm (Γ ), using (7.40), we have     s s Q m m S = ◦ T (α))f = Q (Q f )  α (Qm (Qm 0 ξ ξ ξ ξ f ) ξ αi , ξ i ξ i=1

i=1

−1 by Proposition 7.15. On the other hand, from (7.39) we since Qm ξ = S0 obtain s  (TξP (α) ◦ Qm (Qm ξ )f = ξ f ) ξ αi ; i=1

hence it follows that Q P m Qm ξ ◦ Tξ (α) = Tξ (α) ◦ Qξ .

We now consider a quasimodular polynomial

7.4 Hecke Operators

139

F (z, X) =

m 

fr (z)X r ∈ QPξm (Γ ).

r=0

Using (7.37) and (7.39), we have (Sm ◦ TξP (α))F   s (F ξ αi ) = Sm i=1

=

s 

(det αi )ξ/2 J(αi , z)−ξ

i=1

× Sm

 m

fr (αi z)(det αi )−r J(αi , z)2r (X − K(αi , z))r



r=0

=

s 

(det αi )ξ/2−m J(αi , z)2m−ξ fm (αi z)

i=1

=

s 

(Sm F ) |ξ−2m αi = (Tξ−2m (α) ◦ Sm )F.

i=1

Thus we obtain Sm ◦ TξP (α) = Tξ−2m (α) ◦ Sm , and therefore the proof of the theorem is complete. Theorem 7.24 If

δ Πm



is as in (7.11), for each α ∈ Γ the diagram Πδ

m −→ QPλ+2m+2δ (Γ ) Jλ (Γ )δ −−−m ⏐ ⏐ ⏐ ⏐T P TλJ (α)  λ+2m+2δ (α)

(7.43)

δ Πm

m Jλ (Γ )δ −−−−→ QPλ+2m+2δ (Γ ), δ is commutative, where Πm is as in (7.12) and the Hecke operators TλJ (α) and P Tλ+2m+2δ (α) are as in (3.10) and (7.39), respectively.

Proof. Let α be an element of Γ such that the corresponding double coset has a decomposition as in (3.6). Using (3.10) and (7.39) as well as the extension of (7.18) to GL+ (2, R), we obtain δ (Πm (TλJ (α)Φ))(z, X) =

s 

δ Πm (Φ |Jλ αi )(z, X)

i=1

=

s 

δ (Πm Φ) λ+2m+2δ αi )(z, X)

i=1 P δ (α)(Πm Φ))(z, X) = ((Tλ+2m+2δ

140

7 Quasimodular Forms

for all Φ(z, X) ∈ Jλ (Γ )δ , which verifies the commutativity of the given diagram.  Let ∂ = d/dz be the derivative operator on R. If h is a modular form belonging to Mk (Γ ), then from Lemma 7.7 we see that ∂  h with  ≥ 1 is a  (Γ ). quasimodular form belonging to QMk+2 Theorem 7.25 Let f be a modular form belonging to M2w+λ (Γ ) for some w ∈ Z, and let Lw λ,δ : M2w+λ (Γ ) → Jλ (Γ )δ be the Cohen–Kuznetsov lifting in (1.36). m−w+δ (Γ ) satisfies (i) The quasimodular form ∂ m−w+δ f ∈ QMλ+2m+2δ ∂ m−w+δ f δ = (S0 ◦ Πm ◦ Lw λ,δ )f. (m − w + δ)!(m + w + δ + λ − 1)!

(7.44)

(ii) If α ∈ Γ, then we have Q (α)∂ m−w+δ f, ∂ m−w+δ (T2w+λ (α)f ) = Tλ+2m+2δ

(7.45)

Q (α) are the Hecke operators in (3.7) and where T2w+λ (α) and Tλ+2m+2δ (7.40), respectively.

Proof. From (1.36) and (7.11) we see that the map δ m ◦ Lw Πm λ,δ : M2w+λ (Γ ) → QPλ+2m+2δ (Γ )

is given by δ ◦ Lw ((Πm λ,δ )f )(z, X) =

m  r=0

f (m−r+δ−w) (z) Xr r!(m − r − w + δ)!(m − r + w + δ + λ − 1)!

for all f ∈ M2w+λ (Γ ). Thus we obtain (7.44) by applying the map S0 to this relation. On the other hand, given α ∈ Γ and f ∈ M2w+λ (Γ ), from the commutative diagram (3.20) we see that J w Lw λ,δ (Tλ (α)f ) = Tλ (α)(Lλ,δ f ).

Hence (ii) follows from this and Theorem 7.24. Theorem 7.25 shows that the diagram ∂ m−w+δ

m−w+δ (Γ ) M2w+λ (Γ ) −−−−−→ QMλ+2m+2δ ⏐ ⏐ ⏐ ⏐T Q T2w+λ (α)  λ+2m+2δ (α) ∂ m−w+δ

m−w+δ M2w+λ (Γ ) −−−−−→ QMλ+2m+2δ (Γ )



7.5 Vector Bundles

141

is commutative for each α ∈ Γ. In particular, if w = δ, we have the following commutative diagram: ∂m

m M2w+λ (Γ ) −−−−→ QMλ+2m+2δ (Γ ) ⏐ ⏐ ⏐ ⏐ Q T2w+λ (α) (α) Tλ+2m+2δ ∂m

m M2w+λ (Γ ) −−−−→ QMλ+2m+2δ (Γ )

7.5 Vector Bundles Let Γ be a discrete subgroup of SL(2, R) commensurable with SL(2, Z) as in Section 7.1. In this section we construct vector bundles over the quotient space Γ \H whose sections may be identified with quasimodular polynomials and therefore with quasimodular forms for Γ . We also construct certain morphisms of vector bundles (cf. [71]). Given integers λ, k and r with 0 ≤ k ≤ r ≤ m, we consider a map Wλ,k : SL(2, R) × H → C r defined by Wλ,k r (γ, z) =

  k J(γ, z)λ−2r K(γ, z)k−r r

(7.46)

for γ ∈ SL(2, R) and z ∈ H. given by (7.46) satisfies Lemma 7.26 The map Wλ,k r Wλ,k r (γ1 γ, z) =

k 

λ,k Wλ, r (γ1 , γz)W (γ, z)

=r

for all γ1 , γ ∈ SL(2, R) and z ∈ H. Proof. If γ1 , γ ∈ SL(2, R) and z ∈ H, from (7.46) we obtain   k J(γ1 γ, z)λ−2r K(γ1 γ, z)k−r . (γ γ, z) = Wλ,k 1 r r However, using (1.13) and (1.14), we have J(γ1 γ, z)λ−2r = J(γ1 , γz)λ−2r J(γ, z)λ−2r , K(γ1 γ, z)k−r = (K(γ, z) + J(γ, z)−2 K(γ1 , γz))k−r

(7.47)

142

7 Quasimodular Forms

=

k−r  j=0

=

 k−r J(γ, z)−2j K(γ1 , γz)j K(γ, z)k−r−j j

 k   k−r =r

−r

J(γ, z)−2+2r K(γ1 , γz)−r K(γ, z)k− .

Hence we see that Wλ,k r (γ1 γ, z)

=

 k    k−r k −r

r

=r

J(γ1 , γz)λ−2r J(γ, z)λ−2

(7.48)

× K(γ1 , γz)−r K(γ, z)k− . On the other hand, we have λ,k Wλ, r (γ1 , γz)W (γ, z)

   J(γ1 , γz)λ−2r K(γ1 , γz)−r = (7.49) r   k J(γ, z)λ−2 K(γ, z)k− ×      k J(γ1 , γz)λ−2r J(γ, z)λ−2 = r  × K(γ1 , γz)−r K(γ, z)k− .

From (7.48), (7.49) and the relation        k−r k! k k , = = r −r  r r!( − r)!(k − )! 

the formula (7.47) follows.

We fix a nonnegative integer m, and denote by Cm [X] the ring of polynomials in X over C of degree at most m. Given a polynomial of the form F (X) =

m 

cr X r ∈ Cm [X]

(7.50)

r=0

with c0 , . . . , cm ∈ C and an integer λ, we now set  γ m λ (z, F (X)) =

γz,

m m  

 r ck Wλ,k r (γ, z)X

(7.51)

r=0 k=r

for all γ ∈ SL(2, R) and z ∈ H. Proposition 7.27 The formula (7.51) determines a left action of SL(2, R) on the Cartesian product H × Cm [X].

7.5 Vector Bundles

143

Proof. Given elements γ, γ1 ∈ SL(2, R), z ∈ H and a polynomial F (X) ∈ Cm [X] as in (7.50), using (7.51), we obtain m γ1  m λ (γ λ (z, F (X)))   m  m m   λ, r = γ1 γz, ck Wλ,k (γ, z)W (γ , γz)X 1 r  r=0 =r k=

 =

γ1 γz,

m  m  k 

(7.52)

 λ, r ck Wλ,k  (γ, z)Wr (γ1 , γz)X

.

r=0 k=r =r

On the other hand, we have  (γ1 γ)

m λ

(z, F (X)) =

γ1 γz,

m m  

 r ck Wλ,k  (γ1 γ, z)X

.

(7.53)

r=0 k=r

Combining (7.52) and (7.53) with (7.47), we obtain m m γ1 m λ (γ λ (z, F (X))) = (γ1 γ) λ (z, F (X));



hence the proposition follows.

Let Γ be a discrete subgroup of SL(2, R), and denote the quotient of the space H × Cm [X] by Γ with respect to the action shown in Proposition 7.27 by (7.54) [V]m λ = Γ \(H × Cm [X]). If we denote the modular curve associated to Γ by U = Γ \H, then the natural projection map H × Cm [X] → H induces a surjective map π : [V]m λ → U such that π −1 (x) is isomorphic to Cm [X] for each x ∈ U . Thus [V]m λ has the structure of a complex vector bundle over U whose fiber is the (m + 1)dimensional complex vector space Cm [X] of polynomials in X. We denote by m Γ 0 (U, [V]m λ ) the space of all holomorphic sections of [V]λ over U . Theorem 7.28 The space Γ 0 (U, [V]m λ ) is canonically isomorphic to the space QPλm (Γ ) of all quasimodular polynomials for Γ of weight λ and depth at most m. m Proof. Let s : U → [V]m λ be a holomorphic section of [V]λ over U = Γ \H, and denote by  : H → U the natural projection map. Given z ∈ H, we have

s((z)) =



  m ∈ Γ \(H × Cm [X]) cr,z X r z, r=0

for some c0,z , . . . , cm,z ∈ C, where [(·)] denotes the Γ -orbit of the element (·) s on H by of H × Cm [X]. We define the C-valued functions f0s , . . . , fm frs (z) = cr,z

(7.55)

144

7 Quasimodular Forms

for all z ∈ H and 0 ≤ r ≤ m. Given γ ∈ Γ , since for each z ∈ H the Γ -orbits of z and γz are the same, we have

 s((z)) = s((γz)) =

γz,

m 

 cr,γz X

r

.

(7.56)

r=0

On the other hand, since [γ m λ (z, f (X))] = [(z, f (X))] for each (z, f (X)) ∈ H × Cm [X], by using (7.51) we have

m λ

s((z)) = γ

r=0

 =

   m r cr,z X z,

γz,

m m  



r ck,z Wλ,k r (γ, z)X

.

r=0 k=r

Comparing this with (7.56) and using (7.46) and (7.55), we see that frs (γz)

= cr,γz = =

m 

ck,z Wλ,k r (γ, z)

k=r m   k=r

 k J(γ, z)λ−2r K(γ, z)k−r fks (z); r

hence we obtain (frs |λ−2r

γ)(z) =

m    k

r

k=r

K(γ, z)k fks (z)

for 0 ≤ r ≤ m. In particular, we have (f0s |λ−2r

γ)(z) =

m 

K(γ, z)k−r fks (z),

k=0

and therefore it follows from Corollary 7.14 that the polynomial F s (z, X) =

m 

frs (z)X r

r=0

is a quasimodular polynomial belonging to QPλm (Γ ). We now assume that

m G(z, X) = r=0 gr (z)X r is a quasimodular polynomial belonging to QPλm (Γ ), and define the map sG : U → [V]m λ by

7.5 Vector Bundles

145

sG ((z)) =



  m z, gr (z)X r r=0

for all z ∈ H. Then for each γ ∈ Γ , using (7.25), (7.46) and (7.51), we have

 sG ((γz)) =

γz,

m 

 gk (γz)X k+δ

k=0 m  m  

  k λ−2r k−r J(γ, z) K(γ, z) gk (z) = γz, r k=0 k=r

  m m   = γz, Wλ,k (γ, z)g (z) k r



= γ

k=0 k=r    m m r λ z, gr (z)X , r=0

and therefore sG is well defined. Since clearly π ◦ sG = 1U , it follows that sG is a holomorphic section of [V]m λ over U ; hence the proof of the theorem is complete. 

Remark 7.29 If m = 0, then the bundle [V]0λ becomes a line bundle and we obtain the isomorphism Γ 0 (U, [V]0λ ) ∼ = Mλ (Γ ) for each λ, which provides us the usual identification of modular forms with holomorphic sections of a line bundle.

Given a polynomial F (z, X) ∈ Rm [X] of the form F (z, X) =

m 

fr (z)X r

(7.57)

r=0

with f0 , . . . , fm ∈ R, we set (Δp F )(z, X) =

m−p  r=0

 r+p fr+p (z)X r p

(7.58)

for each integer p with 0 ≤ p ≤ m, so that we obtain the complex linear map Δp : Rm [X] → Rm−p [X].

(7.59)

146

7 Quasimodular Forms

Lemma 7.30 Given λ ∈ Z, we have m−p (Γ ) Δp (QPpm (Γ )) ⊂ QPλ−2p

for each p ∈ {0, 1, . . . m}. Proof. Let F (z, X) ∈ QPλm (Γ ) be a quasimodular polynomial of the form given by (7.57). Then, given p ∈ {0, 1, . . . m}, by Corollary 7.14 the coefficients of F (z, X) satisfies (fp |λ−2p γ)(z) =

m    

p

=p

=

m−p  r=0

f (z)K(γ, z)−p

 r+p fr+p (z)K(γ, z)r p

for all z ∈ H and γ ∈ Γ . From this we see that fp is a quasimodular form m−p belonging to QMλ−2p (Γ ) and that (Δp F )(z, X) is a quasimodular polynomial m−p  belonging to QPλ−2p (Γ ). For p ∈ {0, 1, . . . , m}, we now define the map

by

p : H × Cm [X] → H × Cm−p [X] Δ

(7.60)

p (z, f (X)) = (z, Δp f (X)) Δ

(7.61)

for all f (X) ∈ Cm [X], where Δp : Cm [X] → Cm−p [X] is the map obtained from (7.59) by restriction. We consider the vector bundles [V]m λ = Γ \(H × Cm [X]),

[V]m−p λ−2p = Γ \(H × Cm−p [X]),

(7.62)

where the first bundle is as in (7.54) and the second quotient is with respect to the operation m−p λ−2p in (7.51) of Γ on H × Cm−p [X]. p in (7.60) induces a morphism Theorem 7.31 If 0 ≤ p ≤ m, the map Δ m−p [V]m λ → [V]λ−2p

of vector bundles in (7.62) over U = Γ \H. Proof. Given λ ∈ Z and p ∈ {0, 1, . . . , m}, it suffices to prove that p (γ m (z, f (X)) = γ m−p Δ p (z, f (X)) Δ λ λ−2p for all z ∈ H, γ ∈ Γ and f (X) ∈ Cm [X]. If z ∈ H, γ ∈ Γ and f (X) =

m  r=0

cr X r ∈ Cm [X],

(7.63)

7.5 Vector Bundles

147

using (7.51), (7.58) and (7.61), we obtain p (γ m (z, f (X)) Δ λ   m   m   k p γz, J(γ, z)λ−2r K(γ, z)k−r ck X r =Δ r r=0 k=r    m−p m   r + p  k J(γ, z)λ−2r−2p K(γ, z)k−r−p ck X r = γz, p r+p r=0 k=r+p

  m−p  m−p  r + pk + p J(γ, z)λ−2r−2p K(γ, z)k−r ck+p X r . = γz, r+p p r=0 k=r

On the other hand, we have  γ m−p λ−2p Δp (z, f (X))  m−p   r + p  r c = γ m−p X z, r+p λ−2p p r=0   m−p  m−p  k + p λ−2p r Wr = γz, (γ, z)X p r=0 k=r   m−p  m−p  k + pk  λ−2r−2p k−r r J(γ, z) = γz, K(γ, z) ck+p X r p r=0 k=r   m−p  m−p  r + pk + p λ−2r−2p k−r r J(γ, z) = γz, K(γ, z) ck+p X ; r+p p r=0 k=r



hence the theorem follows. Remark 7.32 If p = m in (7.63), then we obtain the morphism 0 [V]m λ → [V]λ−2m

from a vector bundle to a line bundle, where the holomorphic sections of the line bundle [V]0λ−2m can be identified with modular forms as in Remark 7.29.

Chapter 8

Quasimodular and Modular Polynomials

In this chapter we construct Poincar´e series for quasimodular forms by using an automorphism of the space of polynomials that is equivariant under certain actions of SL(2, R). For this purpose, in addition to the SL(2, R)-action introduced in Chapter 7 which was used to define quasimodular polynomials, we need to consider another action. Given a discrete subgroup Γ ⊂ SL(2, R), invariant polynomials under Γ with respect to this new action are modular polynomials. We introduce a particular automorphism of the space of polynomials that is SL(2, R)-equivariant with respect to the above-mentioned actions. The resulting isomorphism between the space of quasimodular polynomials and that of modular polynomials determines a correspondence between quasimodular forms and some finite sequences of classical modular forms. We use this equivariant automorphism to construct Poincar´e series for quasimodular forms.

8.1 Correspondences of Polynomials In this section we introduce linear automorphisms of the space of polynomials whose coefficients are holomorphic functions on the upper half plane. Such automorphisms will be used in Section 8.2 to obtain modular polynomials corresponding to quasimodular polynomials. We fix a nonnegative integer m and, as in earlier chapters, denote by Rm [X] the space of polynomials in X of degree at most m over the ring R of holomorphic functions on H. If an element Φ(z, X) ∈ Rm [X] is a polynomial of the form m  Φ(z, X) = φr (z)X r (8.1) r=0

and if λ is an integer with λ > 2m, we introduce two other polynomials

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_8

149

150

8 Quasimodular and Modular Polynomials m (Λm λ Φ)(z, X), (Ξλ Φ)(z, X) ∈ Rm [X]

defined by (Λm λ Φ)(z, X) =

m 

r φA r (z)X ,

(Ξλm Φ)(z, X) =

r=0

m 

r φB r (z)X

(8.2)

r=0

where φA k =

m−k 1 1  (r) φ k! r=0 r!(λ − 2k − r − 1)! m−k−r

φB k = (λ + 2k − 2m − 1)

k  (−1)r r=0

r!

(8.3)

(m − k + r)!

(8.4) (r)

× (2k + λ − 2m − r − 2)!φm−k+r , for each k ∈ {0, 1, . . . , m}. m Proposition 8.1 The maps Λm λ , Ξλ : Rm [X] → Rm [X] given by (8.2) are complex linear isomorphisms with

(Ξλm )−1 = Λm λ for each λ > 2m. Proof. Given λ > 2m, we first consider a polynomial Φ(z, X) and its image (Λm λ Φ)(z, X) under Λλ as in (8.1) and (8.2), respectively. Then from (8.3), we obtain  1 1 () φ (m − k + r)! !(λ − 2m + 2k − 2r −  − 1)! k−−r k−r

φA m−k+r =

=0

for 0 ≤ r ≤ k ≤ m. Thus, if we set ((Ξλm ◦ Λm λ )Φ)(z, X) =

∞ 

φ r (z)X r ,

r=0

then from (8.4) we see that φ k λ + 2k − 2m − 1 k  (−1)r (r) = (m − k + r)!(λ + 2k − 2m − r − 2)!(φA m−k+r ) r! r=0 =

k k−r   (−1)r (λ + 2k − 2m − r − 2)! (+r) φ r!!(λ − 2m + 2k − 2r −  − 1)! k−−r r=0 =0

(8.5)

8.1 Correspondences of Polynomials

=

u k   u=0 r=0

=

151

(−1)r (λ + 2k − 2m − r − 2)! (u) φ r!(u − r)!(λ + 2k − 2m − r − u − 1)! k−u

(λ + 2k − 2m − 2)! φk (λ + 2k − 2m − 1)!

u k  1 (u)  (−1)r u!(λ + 2k − 2m − r − 2)! φk−u u r!(u − r)!(λ + 2k − 2m − r − u − 1)!(u − 1)! u=1 r=0  u   k  1 (u)  u λ + 2k − 2m − r − 2 φk . + φk−u = u−1 r λ + 2k − 2m − 1 u=1 u r=0

+

Since λ > 2m, if 1 ≤ u ≤ k, we have λ + 2k − 2m − 2 ≥ 2k − 1 ≥ k ≥ u. Thus, using Lemma 1.1, we see that  u    u λ + 2k − 2m − r − 2 r=0

u−1

r

=0

for 1 ≤ u ≤ k. Hence (8.5) can be written in the form φ k φk = , λ + 2k − 2m − 1 λ + 2k − 2m − 1 and therefore we obtain ((Ξλm ◦ Λm λ )Φ)(z, X) = Φ(z, X). We now assume that (Ξλm Φ)(z, X) is as in (8.2) and that m ((Λm λ ◦ Ξλ )Φ)(z, X) =

m 

φr (z)X r .

r=0

Thus, in particular, (8.4) is valid for 0 ≤ k ≤ m. We shall verify that φk = φk for 0 ≤ k ≤ m using induction. First, from (8.3) and (8.4) we see that φm =

1 φB m!(λ − 2m − 1)! 0

 with φB 0 = m!(λ − 2m − 1)!φm ; hence we have φm = φm . Given a nonnegative integer n < m, we next assume that 1 1  (φB )(r) φk = φk = k! r=0 r!(λ − 2k − r − 1)! m−k−r m

(8.6)

152

8 Quasimodular and Modular Polynomials

holds for each k ∈ {m − n, . . . , m}. Then from (8.4) we obtain φB n+1 = (2n + λ − 2m + 1)

n+1  r=0

(−1)r (2n + λ − 2m − r)! r! (r)

× (m − n − 1 + r)!φm−n−1+r = (2n + λ − 2m + 1)!(m − n − 1)!φm−n−1 n  (−1)r (2n + λ − 2m − r − 1)! − (2n + λ − 2m + 1) (r + 1)! r=0 (r+1)

× (m − n + r)!φm−n+r , which can be written in the form φm−n−1 =

φB n+1 (2n + λ − 2m + 1)!(m − n − 1)!

(8.7)

 (−1)r 1 (2n + λ − 2m)!(m − n − 1)! r=0 (r + 1)! n

+

(r+1)

× (2n + λ − 2m − r − 1)!(m − n + r)!φm−n+r . However, since (8.6) holds for k ∈ {m−n, . . . , m} by the induction hypothesis, we have n  (−1)r (r+1) (2n + λ − 2m − r − 1)!(m − n + r)!φm−n+r (r + 1)! r=0

=

n n−r   r=0 =0 w n  

(−1)r (2n + λ − 2m − r − 1)! (φB )(+r+1) (r + 1)!!(λ − 2m + 2n − 2r −  − 1)! n−−r

(−1)r (2n + λ − 2m − r − 1)! (w+1) (φB n−w ) (r + 1)!(w − r)!(λ − 2m + 2n − r − w − 1)! w=0 r=0    n w  2n + λ − 2m − r − 1 1  r w+1 (w+1) (φB = (−1) . n−w ) w r+1 w + 1 r=0 w=0 =

On the other hand, using Lemma 1.1, we see that w  r=0

 (−1)

r

w+1 r+1

    2n + λ − 2m 2n + λ − 2m − r − 1 . = w w

Thus (8.7) can now be written as

8.2 Modular and Quasimodular Polynomials

φm−n−1 =

=

=

153

φB n+1 (2n + λ − 2m + 1)!(m − n − 1)! 1 + (2n + λ − 2m)!(m − n − 1)!   n  2n + λ − 2m 1 (w+1) (φB × n−w ) w w + 1 w=0 φB n+1 (2n + λ − 2m + 1)!(m − n − 1)! 1 + (2n + λ − 2m)!(m − n − 1)! n+1  1 2n + λ − 2m (w) (φB × n+1−w ) w−1 w w=1 n+1  1 1 (φB )(w) (m − n − 1)! w=0 w!(2n + λ − 2m − w + 1)! n+1−w

= φm−n−1 . Hence it follows that φk = φk for all k ∈ {0, 1, . . . , m} by induction. Thus we obtain m ((Λm λ ◦ Ξλ )Φ)(z, X) = Φ(z, X), 

and the proof of the proposition is complete.

8.2 Modular and Quasimodular Polynomials In Chapter 7 we considered an action of SL(2, R) on the space of polynomials so that the polynomials invariant under a discrete subgroup of SL(2, R) are quasimodular polynomials. In this section we introduce another action and show that the automorphisms in Proposition 8.1 are equivariant with respect to these actions. Let the ring R and the space Rm [X] be as in Section 8.1. If γ ∈ SL(2, R), λ ∈ Z and m  Φ(z, X) = φr (z)X r ∈ Rm [X], (8.8) r=0

we set −λ (Φ |X Φ(γz, J(γ, z)−2 X) = λ γ)(z, X) = J(γ, z)

m 

(φr |λ+2r γ)(z)X r ,

r=0

(8.9)

154

8 Quasimodular and Modular Polynomials

for all z ∈ H, where J(γ, z) is as in (1.12) and |λ+2r is as in (1.16). If γ  is another element of SL(2, R), then it can be shown easily that  X X  (Φ |X λ (γγ ))(z, X) = ((Φ |λ γ) |λ γ )(z, X);

hence the above operation |X λ determines a right action of SL(2, R) on the complex vector space Rm [X]. Theorem 8.2 Given a polynomial Φ(z, X) ∈ Rm [X] and an integer λ, we have m X (8.10) ((Λm λ Φ) λ γ)(z, X) = Λλ (Φ |λ−2m γ)(z, X) m ((Ξλm Φ) |X λ−2m γ)(z, X) = Ξλ (Φ λ γ)(z, X)

for all γ ∈ SL(2, R), where 8.1 and λ is as in (7.9).

Λm λ

Ξλm

and

(8.11)

are the isomorphisms in Proposition

Proof. Let Φ(z, X) ∈ Rm [X] be given by (8.8), so that (Λm λ Φ)(z, X) =

m 

r φA r (z)X

r=0

for λ ∈ Z, where the coefficients φA r (z) are as in (8.3). If γ ∈ SL(2, R), from (7.9) we obtain ((Λm λ Φ) λ

γ)(z, X) = J(γ, z)

−λ

m 

2  φA  (γz)J(γ, z) (X − K(γ, z))

=0

=

   m   

r

=0 r=0

=

m m  

(−1)

2−λ φA (−1)−r K(γ, z)−r X r  (γz)J(γ, z)

−r

r=0 =r

   A φ (γz)J(γ, z)2−λ K(γ, z)−r X r . r 

Thus we may write ((Λm λ Φ) λ γ)(z, X) =

m 

ξrA (γ, z)X r ,

r=0

where ξrA (γ, z)

=

m−r  =0

 (−1)



 +r A φ+r (γz)J(γ, z)2+2r−λ K(γ, z) r

for each r ≥ 0. Using (8.3), we have

8.2 Modular and Quasimodular Polynomials

φA +r (γz) =

155

m−r−  1 1 (k) φ (γz), ( + r)! k!(λ − 2r − 2 − k − 1)! m−r−−k k=0

and therefore we obtain ξrA (γ, z) =

m−r  m−r−  =0

k=0

(−1) J(γ, z)2+2r−λ K(γ, z) (k) φ (γz). k!r!!(λ − 2r − 2 − k − 1)! m−r−−k

(8.12)

On the other hand, from (8.2), (8.3) and (8.9) we see that X Λm λ (Φ |λ−2m γ)(z, X) =

m 

ηrA (γ, z)X r ,

r=0

where ηrA (γ, z) =

m−r 1 1  (φm−r−k |λ−2r−2k γ)(k) (z) r! k!(λ − 2r − k − 1)! k=0

for r ≥ 0. Using (1.21), we have (φm−r−k |λ−2r−2k γ)

(k)

(z) =

k 

(−1)

=0

×



λ − 2r − k − 1 k− !

k− k!



K(γ, z)k− () φ (γz). J(γ, z)λ−2r−2k+2 m−r−k

Thus we obtain ηrA (γ, z) = =

k m−r  k=0 =0 m−r  m−r  =0 k=

=

(−1)k− K(γ, z)k− J(γ, z)−λ+2r+2k−2 () φm−r−k (γz) r!!(k − )!(λ − 2r − 2k +  − 1)! (−1)k− K(γ, z)k− J(γ, z)−λ+2r+2k−2 () φm−r−k (γz) r!!(k − )!(λ − 2r − 2k +  − 1)!

m−r  m−−r  =0

k=0

(−1)k K(γ, z)k J(γ, z)−λ+2r+2k () φ (γz). r!!k!(λ − 2r − 2k −  − 1)! m−r−k−

Comparing this with (8.12), we have ξrA (γ, z) = ηrA (γ, z) for each r ∈ {0, . . . , m}, which verifies (8.10). The relation (8.11) follows from this and Proposition 8.1. 

We assume that the discrete subgroup Γ of SL(2, R) considered earlier is a Fuchsian group of the first kind.

156

8 Quasimodular and Modular Polynomials

Definition 8.3 An element F (z, X) ∈ Rm [X] is a modular polynomial for Γ of weight λ and degree at most m if it satisfies F |X λ γ =F for all γ ∈ Γ . We use M Pλm (Γ ) to denote the space of modular polynomials for Γ of weight λ and degree at most m. If a polynomial F (z, X) ∈ Rm [X] of the form m  F (z, X) = fr (z)X r r=0

belongs to

M Pλm (Γ ),

from (8.9) and Definition 8.3 we see that fr ∈ Mλ+2r (Γ )

(8.13)

for 0 ≤ r ≤ m. m Proposition 8.4 The isomorphisms Λm λ and Ξλ in Proposition 8.1 induce the isomorphisms m m Λm λ : M Pλ−2m (Γ ) → QPλ (Γ ),

m Ξλm : QPλm (Γ ) → M Pλ−2m (Γ )

for each λ ∈ Z. Proof. This follows immediately from Theorem 8.2 and Definition 8.3.



Let φ ∈ QMλm (Γ ) be a quasimodular form satisfying (φ |λ γ)(z) =

m 

φr (z)K(γ, z)r

r=0

for all γ ∈ Γ and z ∈ H, so that (Qm λ φ)(z, X) =

m 

φr (z)X r ∈ QPλm (Γ )

r=0

by (7.20). We then set B B m+1 λm (φ) = (φB , Ξ 0 , φ1 , . . . , φ m ) ∈ R

(8.14)

where the functions φB r ∈ R are the coefficients of the modular polynomial m ((Ξλm ◦ Qm λ )φ)(z, X) ∈ M Pλ (Γ )

given by (8.4).

8.3 Poincar´ e Series

157

Proposition 8.5 The formula (8.14) determines an isomorphism λm : QMλm (Γ ) → Ξ

m

Mλ+2k−2m (Γ ) k=0

of complex vector spaces for λ > 2m. Proof. From (8.13) we see that φB r ∈ Mλ+2r (Γ ) for each r ∈ {0, 1, . . . , m}; hence the proposition follows from (8.13).



Example 8.6 We consider modular forms g ∈ M2 (Γ ) and p ∈ Mλ−2m (Γ ) with ξ ∈ Z. Then we see that g k p ∈ Mλ+2k−2m (Γ ) for 0 ≤ k ≤ m; hence we have m

(p, gp, . . . , g m p) ∈

Mλ+2k−2m (Γ ). k=0

Then it can be easily shown that m 

1 (g m−r p)(r) r!(λ − r − 1)! r=0   r m   r 1 (g m−r )() p(r−) =  r!(λ − r − 1)! r=0

 m )−1 (p, gp, . . . , g m p) = (Ξ λ

=0

=

r m   r=0 =0

(g m−r )() p(r−) , !(r − )!(λ − r − 1)!

m (Γ ). and it is a quasimodular form belonging to QMξ+2m

8.3 Poincar´ e Series In this section we study some of the properties of quasimodular forms including their correspondence with finite sequences of modular forms. We also construct Poincar´e series for quasimodular forms (cf. [70]). In order to study Poincar´e series later in this section, as in Section 8.2, we assume that Γ ⊂ SL(2, R) is a Fuchsian group of the first kind and denote by F the subring of R consisting of holomorphic functions f : H → C that are bounded at each cusp of Γ . We fix a nonnegative integer m and denote

158

8 Quasimodular and Modular Polynomials

by Fm [X] the subspace of Rm [X] consisting of polynomials in X over F of degree at most m. We modify the definitions of modular forms, modular polynomials and quasimodular polynomials using F instead of R, and denote the resulting spaces of modular forms, modular polynomials and quasimodular polynomials using F as superscripts. Thus we have F

Mλ (Γ ) = {f ∈ F | f |λ γ = f for all γ ∈ Γ } = Mλ (Γ ) ∩ F,

F

M Pλm (Γ ) = {Φ(z, X) ∈ Fm [X] | Φ |X λ γ = Φ for all γ ∈ Γ },

F

QPλm (Γ ) = {Φ(z, X) ∈ Fm [X] | Φ λ γ = Φ for all γ ∈ Γ }.

Similarly, we can consider quasimodular forms belonging to F and write F

QMλm (Γ ) = QMλm (Γ ) ∩ F,

whose elements are quasimodular forms for Γ of weight λ and depth at most m. If a polynomial of the form F (z, X) =

m 

fr (z)X r ∈ Fm [X]

r=0

belongs to

F

M Pλm (Γ ), from (8.13) we see that fr ∈ F Mλ+2r (Γ )

m for 0 ≤ r ≤ m. The automorphisms Λm λ and Ξλ in Proposition 8.4 induce the isomorphisms F m F m Λm λ : M Pλ−2m (Γ ) → QPλ (Γ ),

m Ξλm : F QPλm (Γ ) → F M Pλ−2m (Γ )

for each λ ∈ Z. If 0 ≤  ≤ m, by restricting the map (7.13) to Fm [X] we obtain the complex linear map S : Fm [X] → F, which satisfies S

 m

 fr (z)X

r

= f (z)

r=0

for all z ∈ H and induces the map m− S : F QPλm (Γ ) → F QMλ−2 (Γ )

(8.15)

for each  over F. By modifying (7.20) to the case of F we also obtain the map

8.3 Poincar´ e Series

159 F m F m Qm λ : QMλ (Γ ) → QPλ (Γ ) F

F

(8.16)

→ is an isomorfor each λ ∈ Z, so that the map S0 : . phism whose inverse is Qm λ Given a cusp x of Γ and integers w ≥ 3 and u ≥ 0, we recall from (2.24) x that the corresponding Poincar´e series Pw,u : H → C for modular forms F belonging to Mw (Γ ) can be written as  x,u (ηw |w γ)(z), (8.17) Pwx,u (z) = QPλm (Γ )

QMλm (Γ )

γ∈Γx \Γ x,u is as in (2.23). where ηw If ξ ∈ Z, we consider the polynomial

Gm 2ξ,u (z, X) =

m 

x,u η2(ξ−m+r) (z)X r ∈ Fm [X],

(8.18)

r=0

and set



x,m (z, X) = P 2ξ,u

m ((Λm 2ξ G2ξ,u ) 2ξ−2m γ)(z, X)

(8.19)

γ∈Γx \Γ

for z ∈ H, where x is a cusp of Γ as above and Λm 2ξ is as in (8.2). x,m (z, X) given by (8.19) is a quasimodular Theorem 8.7 (i) The series P 2ξ,u m (Γ ). polynomial belonging to F QP2ξ x,m (z, X) can be written in the form (ii) The series P 2ξ,u x,m (z, X) = P 2ξ,u

m m−r    

(−1)−j (2πiu)j ! hj j!r!(2ξ − 2m −  − 1)! γ∈Γx \Γ r=0 =0 j=0   K(σγ, z)−j 2ξ − 2r − m − 1 × eu/h (σγz)X r . −j J(σγ, z)2ξ−2r−2m+2j

Proof. Using (8.9), (8.17), (8.18), (8.19) and Theorem 8.2, we have  m X x,m (z, X) = Λm P 2ξ (G2ξ,u |2ξ−2m γ)(z, X) 2ξ,u γ∈Γx \Γ

=



Λm 2ξ

γ∈Γx \Γ

= Λm 2ξ

 m

 m

 x,u (η2(ξ−m+r) |2(ξ−m+r) γ)(z)X r

r=0

 x,u P2(ξ−m+r) (z)X r .

r=0 x,u ∈ F M2(ξ−m+r) (Γ ) for each r ≥ 0, from (8.9) and Definition Since P2(ξ−m+r) 8.3 we see that the sum

160

8 Quasimodular and Modular Polynomials m 

F (z, X) =

x,u P2(ξ−m+r) (z)X r

(8.20)

r=0 m (Γ ), and therefore by Propois a modular polynomial belonging to F M P2ξ−2m sition 8.4 its image x,m (z, X) = (Λm F )(z, X) (8.21) P 2ξ 2ξ,u F m QP2ξ (Γ ); under Λm 2ξ in (8.2) is a quasimodular polynomial belonging to hence (i) follows. On the other hand, using (2.22), (8.2), (8.20) and (8.21), we have   m  m m u/h r (Λ2ξ F )(z, X) = (8.22) Λ2ξ (e |2(ξ−m+r) σγ)(z)X γ∈Γx \Γ

=

r=0

m  

φ r (z)X r ,

γ∈Γx \Γ r=0

where m−r 1 1  φ r = (eu/h |2(ξ−r−m) σγ)() . r! (2ξ − 2r −  − 1)!

(8.23)

=0

However, from (1.21) we see that (eu/h |2(ξ−r−m) σγ)() (z) =

 

(−1)−j

j=0

  ! 2ξ − 2r − m − 1 −j j!

(8.24)

K(σγ, z)−j (eu/h )(j) (σγz) J(σγ, z)2ξ−2r−2m+2j     (−1)−j (2πiu)j ! 2ξ − 2r − m − 1 = −j hj j! j=0 ×

×

K(σγ, z)−j eu/h (σγz). J(σγ, z)2ξ−2r−2m+2j

Thus we obtain (8.21) by combining (8.22) (8.23) and (8.24), which proves (ii). 

x,0 (z) denotes the image of P x,m (z, X) under the isomorRemark 8.8 If P 2ξ,u 2ξ,u m m (Γ ) → F QM2ξ (Γ ) we obtain phism S0 : F QP2ξ x,0 (z) = P 2ξ,u

m     γ∈Γx \Γ =0 j=0

(−1)−j (2πiu)j ! hj j!(2ξ − 2m −  − 1)!

8.4 Heat Operators

161

 ×



K(σγ, z)−j 2ξ − m − 1 eu/h (σγz) −j J(σγ, z)2ξ−2m+2j

for z ∈ H, which may be regarded as a Poincar´e series for quasimodular m forms in F QM2ξ (Γ ).

8.4 Heat Operators In this section we consider a differential operator on quasimodular polynomials which corresponds to the derivative operator on quasimodular forms. We show that such operators are compatible with heat operators on Jacobi-like forms (cf. [73]). Given ξ ∈ Z, we consider the formal differential operator ξ : Rm [X] → Rm+1 [X] D defined by

  ξ = ∂ + X ξ − X ∂ . D ∂z ∂X

(8.25)

ξ with ξ ∈ Z be as in (8.25), and let ∂ = d/dz : R → Proposition 8.9 Let D R be the derivative operator on R. Then we have m  Qm+1 ξ+2 ◦ ∂ = Dξ ◦ Qξ ,

(8.26)

m+1 where Qm ξ and Qξ+2 are as in (7.28)

Proof. Let f be a quasimodular form belonging to QMξm (Γ ), so that there are holomorphic functions f0 , f1 , . . . , fm ∈ R such that (f |ξ γ)(z) = J(γ, z)−ξ f (γz) =

m 

fk (z)K(γ, z)k

k=0

for all z ∈ H and γ ∈ Γ . Then, as shown in the proof of Lemma 7.7, ∂f is a m+1 quasimodular form belonging to QMξ+2 (Γ ) satisfying ((∂f ) |ξ+2 γ)(z) =

m+1 

(ξ − k + 1)fk−1 (z)K(γ, z)k +

k=1

m 

fk (z)K(γ, z)k .

k=0

From this and (7.20), we obtain ((Qm+1 ξ+2 ◦ ∂)f )(z) =

m+1  k=0

hk (z)X k ,

(8.27)

162

8 Quasimodular and Modular Polynomials

where h0 = f0 ,

hk = (ξ − k + 1)fk−1 + fk

hm+1 = (ξ − m)fm ,

(8.28)

for 1 ≤ k ≤ m. On the other hand, since (Qm ξ f )(z, X) =

m 

fk (z)X k ∈ Rm [[X]],

k=0

by using (8.25) we have ξ ◦ Qm )f (z, X) = (D ξ

m 

fk (z)X k + ξ

k=0

=

m 

m 

fk (z)X k −

k=0

fk (z)X k +

m+1 

k=0

m 

kfk (z)X k

k=0

(ξ − k + 1)fk−1 (z)X k .

k=1

Hence we obtain (8.26) by comparing this with (8.27).



Given λ ∈ Z, we now consider the heat operator DλX : Jλ (Γ ) → Jλ+2 (Γ ) of Jacobi-like forms in (5.11) given by DμX =

∂ ∂2 ∂ −μ −X ∂z ∂X ∂X 2

(8.29)

as in (5.11). We also recall that there is a surjective linear map δ m : Jλ (Γ )δ → QPλ+2m+2δ (Γ ) Πm

(8.30)

for each δ ≥ 0 as in (7.32) given by (7.11). Theorem 8.10 Given λ, δ ∈ Z, the diagram Jλ (Γ )δ ⏐ X⏐ Dλ 

Πδ

−−−m −→

m QPλ+2m+2δ (Γ ) ⏐ ⏐ Dλ+2m+2δ

δ−1 Πm+1

m+1 Jλ+2 (Γ )δ−1 −−−−→ QPλ+2m+2δ+2 (Γ )

commutes if and only if δ = −m − 1 or δ = −m − λ. Proof. Let Φ(z, X) ∈ R[[X]]δ be given by Φ(z, X) =

∞  k=0

φk (z)X k+δ .

(8.31)

8.4 Heat Operators

163

Then from (8.29) we obtain DλX Φ(z, X) ∞ ∞   = φk (z)X k+δ − λ (k + δ)φk (z)X k+δ−1 k=0

k=0

−X =

(8.32)

∞ 

(k + δ)(k + δ − 1)φk (z)X k+δ−2

k=0

∞  

 φk−1 (z) − (k + δ)(k + δ − 1 + λ)φk (z) X k+δ−1 ∈ R[[X]]δ−1

k=0

with φ−1 = 0. Using this and (7.11), we have δ−1 ((Πm+1 ◦ DλX )Φ)(z, X) (8.33)   m+1  1 = φ (z) − (m + 1 − k + δ)(m − k + δ + λ)φm+1−k (z) X k . k! m−k k=0

On the other hand, from δ Φ)(z, X) = (Πm

m  1 φm−k (z)X k k!

k=0

and (8.25) we see that λ+2m+2δ ◦ Π δ )Φ)(z, X) ((D m m m   1  λ + 2m + 2δ k = φ φm−k (z)X k+1 (z)X + k! m−k k! k=0

(8.34)

k=0

−

m  k=0

1 φm−k (z)X k+1 (k − 1)!

m m+1   λ + 2m + 2δ − k + 1 1  φm−k (z)X k + φm−k+1 (z)X k = k! (k − 1)! k=0 k=1  m+1  1 φm−k (z) + k(λ + 2m + 2δ − k + 1)φm+1−k (z) X k . = k! k=0

Comparing (8.33) and (8.34), we have δ−1 λ+2m+2δ ◦ Π δ ◦ DλX = D Πm+1 m

if and only if (m + δ + 1)(m + δ + λ) = 0;

164

8 Quasimodular and Modular Polynomials



hence the theorem follows. δ is the map in (8.30), we consider the corresponding linear map If Πm

δ : Jλ (Γ )δ → QM m Π m λ+2m+2δ (Γ ) defined by

δ = S0 ◦ Π δ , Π m m

(8.35)

where S0 is as in (7.29). Then from (7.11) and (7.20) we see that δ Φ)(z) = φm (z) (Π m for Φ(z) =

∞ k=0

φk (z)X k+δ ∈ Jλ (Γ )δ .

Corollary 8.11 Given m, λ, δ ∈ Z with m ≥ 0, if δ = −m−1 or δ = −m−λ, the diagram Jλ (Γ )δ ⏐ X⏐ Dλ 

δ Π

−−−m −→

m QMλ+2m+2δ (Γ ) ⏐ ⏐ ∂

 δ−1 Π

m m+1 Jλ+2 (Γ )δ−1 −−− −→ QMλ+2m+2δ+2 (Γ )

is commutative. Proof. From the relation (8.26) and the commutativity of the diagram (8.31) we see that the diagram Jλ (Γ )δ ⏐ X⏐ Dλ 

Πδ

−−−m −→

δ−1 Πm+1

S

0 m QPλ+2m+2δ (Γ ) −−−− → ⏐ ⏐ Dλ+2m+2δ

m QMλ+2m+2δ (Γ ) ⏐ ⏐ ∂

S

0 m+1 m+1 Jλ+2 (Γ )δ−1 −−−−→ QPλ+2m+2δ+2 (Γ ) −−−− → QMλ+2m+2δ+2 (Γ )

commutes, assuming that δ = −m − 1 or δ = −m − λ. Thus the corollary follows from this and the relations δ−1 = S0 ◦ Π δ−1 Π m m+1 and (8.35).



Chapter 9

Liftings of Quasimodular Forms

In this chapter we study various lifting maps among modular, quasimodular and Jacobi-like forms and pseudodifferential operators as well as some applications of such liftings. More specifically, we construct liftings of modular forms to quasimodular forms and liftings of quasimodular forms to Jacobi-like forms. Using these liftings and the correspondence between pseudodifferential operators and Jacobi-like forms, we construct Lie brackets on the space of quasimodular polynomials. As another application of the liftings, we also study Rankin–Cohen brackets for quasimodular forms (cf. [25], [75]).

9.1 Liftings of Modular Forms to Quasimodular Forms There is a natural surjective map from quasimodular polynomials to modular forms sending a quasimodular polynomial to its leading coefficient. In this section we construct a lifting of a modular form to a quasimodular form whose leading coefficient coincides with the given modular form. If Π0δ : R[[X]] → R and Sm : Rm [X] → R are as in (7.13) and (7.16), respectively, from (7.27) and Definition 1.7 we see that Π0δ (Jλ (Γ )δ ) ⊂ Mλ+2δ (Γ ),

Sm (QPξm (Γ )) ⊂ Mξ−2m (Γ ).

Hence the rows in (7.17) induce the short exact sequences of the form  ι

Πδ

0 → Jλ (Γ )δ −−→ Mλ+2δ (Γ ) → 0, 0 → Jλ (Γ )δ+1 −

ι

S

0 → QPξm−1 (Γ ) − → QPξm (Γ ) −−m → Mξ−2m (Γ ) → 0 for integers ξ, λ and δ ≥ 0. Furthermore, combining these results with (7.32) and the isomorphism (7.29), we see that the diagram (7.17) induces the commutative diagram

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_9

165

166

9 Liftings of Quasimodular Forms

0 ⏐ ⏐ 

0 ⏐ ⏐ 

0 ⏐ ⏐ 

δ+1 Πm−1

S

Πδ

S

0 m−1 m−1 Jλ (Γ )δ+1 −−−−→ QPλ+2m+2δ (Γ ) −−−− → QMλ+2m+2δ (Γ ) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ι  ι 

Jλ (Γ )δ ⏐ ⏐ Π0δ 

0 m m −−−m −→ QPλ+2m+2δ (Γ ) −−−− → QMλ+2m+2δ (Γ ) ⏐ ⏐ ⏐ ⏐ Sm  

μm

Mλ+2δ (Γ ) −−−−→ ⏐ ⏐ 

Mλ+2δ (Γ ) ⏐ ⏐ 

Mλ+2δ (Γ ) ⏐ ⏐ 

0

0

0

(9.1)

in which the three columns are short exact sequences. The next proposition m provides us a lifting from Mλ+2δ (Γ ) to QMλ+2m+2δ (Γ ) which makes the second column split.

Proposition 9.1 Given integers δ and λ with δ ≥ 0 and a modular form h)(z, X) ∈ Rm [X] given by h ∈ Mλ+2δ (Γ ), the polynomial (Q Aλ+2δ m m 

h(m−r) (z) Xr r!(m − r)!(m − r + 2δ + λ − 1)! r=0 (9.2) m (Γ ). Furthermore, we is a quasimodular polynomial belonging to QPλ+2m+2δ have h) = h. (9.3) Sm (Q Aλ+2δ m (Q Aλ+2δ h)(z, X) = m!(λ + 2δ − 1)! m

Proof. Since h ∈ Mλ+2δ (Γ ), if Lδλ,δ is the Cohen–Kuznetsov lifting map in (1.36), from (1.35) we see that the formal power series (Lδλ,δ h)(z, X) =

∞  =0

h() X +δ !( + 2δ + λ − 1)!

(9.4)

is a Jacobi-like form belonging to Jλ (Γ )δ . From this, (7.11), (9.1) and (9.2) we obtain λ+2δ ((Πm ◦ Lδλ,δ )h)(z, X) =

m  r=0

=

h(m−r) (z) Xr r!(m − r)!(m − r + 2δ + λ − 1)!

1 (Q Aλ+2δ h)(z, X), m m!(λ + 2δ − 1)!

(9.5)

9.1 Liftings of Modular Forms to Quasimodular Forms

167

and by Proposition 7.17 or (9.1) it is a quasimodular polynomial belonging to m (Γ ). On the other hand, we see easily that the coefficient of X m QPλ+2m+2δ in this polynomial is equal to h, which shows that Sm (Q Aλ+2δ h) = h.  m

The formula (9.2) determines the linear map λ+2δ Q Am

m : Mλ+2δ (Γ ) → QPλ+2m+2δ (Γ ),

which may be regarded as a lifting from modular forms to quasimodular polynomials. Since this map satisfies (9.3), it follows that the short exact sequence in the second column of (9.1) splits. If we set L δλ,δ = (λ + 2δ − 1)!Lδλ,δ : Mλ+2δ (Γ ) → Jλ (Γ )δ

(9.6)

with Lδλ,δ being as in (9.4), then we have (L δλ,δ f )(z, X) = (λ + 2δ − 1)!

∞  =0

f () (z) X +δ !( + 2δ + λ − 1)!

(9.7)

for f ∈ Mλ+2δ (Γ ); hence we see easily that Π0δ ◦ L δλ,δ (f ) = f,

(9.8)

where Π0δ is as in (7.12). Thus the short exact sequence in the first column of (9.1) also splits. Furthermore, from (9.5) and (9.6) it follows that the diagram δ L λ,δ

Mλ+2δ (Γ ) −−−−→ ⏐ ⏐ μm  QA

Jλ (Γ )δ ⏐ ⏐ δ  Πm

λ+2δ

m m Mλ+2δ (Γ ) −−−− −→ QPλ+2m+2δ (Γ )

is commutative, where μm is multiplication by 1/m! as in (7.17). λ+2δ We note that the maps Lδλ,δ and Πm are Hecke equivariant by (3.20) and (7.43), respectively. Thus, using (9.5), we see that Q Aλ+2δ is also Hecke m equivariant, meaning that the diagram QA

λ+2δ

m m Mλ+2δ (Γ ) −−−− −→ QPλ+2m+2δ (Γ ) ⏐ ⏐ ⏐ ⏐ P Tλ+2δ (α) (α) Tλ+2m+2δ QA

λ+2δ

m m Mλ+2δ (Γ ) −−−− −→ QPλ+2m+2δ (Γ )

is commutative for each α ∈ Γ.

(9.9)

168

9 Liftings of Quasimodular Forms

Remark 9.2 Using the correspondence between quasimodular polynomials and forms via the isomorphism S0 in (7.29), we see that the map Q Aλ+2δ m given by (9.2) determines the lifting λ+2δ Q Λm

m = S0 ◦ Q Aλ+2δ : Mλ+2δ (Γ ) → QMλ+2m+2δ (Γ ) m

from modular forms to quasimodular forms given by λ+2δ (h) Q Λm

=

(λ + 2δ − 1)! h(m) (m + 2δ + λ − 1)!

for h ∈ Mλ+2δ (Γ ). Thus it follows that the lifting of a modular form to a quasimodular form described above is simply given by a constant multiple of a derivative of the given modular form.

9.2 Liftings of Quasimodular Forms to Jacobi-like Forms In this section we obtain an explicit formula for a lifting of a quasimodular polynomial to a Jacobi-like form (cf. [25], [75]). Since quasimodular forms correspond to quasimodular polynomials, the same formula also determines a lifting of a quasimodular form to a Jacobi-like form, which generalizes the Cohen–Kuznetsov lifting of a modular form. δ Let Πm : R[[X]]δ → Rm [X] be the surjective map in (7.12) given by (7.11), and consider another surjective map

δ : R[[X]]δ → Rm [X] Π m defined by δ m Φ)(z, X) = (Π

m 

φr (z)X r ,

(9.10)

r=0

for a formal power series of the form Φ(z, X) =

∞ 

φk (z)X k+δ ∈ R[[X]]δ .

k=0

Proposition 9.3 For each nonnegative integer λ the diagram R[[X]]δ ⏐ δ ⏐ Πm 

Ξλ,δ

−−−−→

m Ξλ+2δ+2m

R[[X]]δ ⏐ ⏐ δ Πm

Rm [X] −−−−−−−→ Rm [X]

(9.11)

9.2 Liftings of Quasimodular Forms to Jacobi-like Forms

169

m commutes, where Ξλ,δ and Ξλ+2δ+2m are isomorphisms in Proposition 1.2 and Proposition 8.1.

Proof. Given a formal power series Φ(z, x) ∈ R[[X]]δ as in (9.11), from (1.5) and (9.10) we obtain δ ◦ Ξλ,δ )Φ)(z, X) = ((Π m

m 

r φΞ r (z)X ,

r=0

where φΞ r = (2r + λ + 2δ − 1)

r 

(−1)j

j=0

(2r + λ + 2δ − j − 2)! (j) φr−j j!

for 0 ≤ r ≤ m. On the other hand, if we set 1 φ k = φm−k k! for 0 ≤ k ≤ m, from (8.2) and (7.11) we see that m δ ((Ξλ+2δ+2m ◦ Πm )Φ)(z, X) =

m 

r φ B r (z)X ,

r=0

where φ B r = (λ + 2δ + 2r − 1)

r  (−1)j j=0

(j) (m − r + j)!(2r + λ + 2δ − j − 2)!φ m−r+j

j!

with (j) φ m−r+j =

1 (j) φ (m − r + j)! r−j

for 0 ≤ j ≤ r ≤ m. Thus we have B φΞ r = φr for 0 ≤ r ≤ m, which implies that δ δ ◦ Ξλ,δ = Ξ m Π m λ+2δ+2m ◦ Πm ;



hence the proposition follows. We note that the map

δ Πm

(9.12)

satisfies

δ δ (Φ |Jλ γ) = Πm (Φ) λ+2m+2δ γ, Πm

for all Φ(z, X) ∈ R[[X]] and γ ∈ SL(2, R) by Proposition 7.11. On the other δ satisfies hand, we also see easily that the map Π m

170

9 Liftings of Quasimodular Forms

δ (Φ) |X δ (Φ |M γ) = Π Π m λ m λ+2δ γ. Thus, if Γ is a discrete subgroup of SL(2, R) commensurable with SL(2, Z) as before, these maps induce the complex linear maps δ : Mλ (Γ )δ → M P m (Γ ). Π m λ+2δ

δ m : Jλ (Γ )δ → QPλ+2m+2δ (Γ ), Πm

(9.13)

Hence we obtain the commutative diagram Ξλ,δ

−−−−→

Jλ (Γ )δ ⏐ δ ⏐ Πm 

Mλ (Γ )δ ⏐ ⏐ δ  Πm

m Ξλ+2δ+2m

m m (Γ ) QPλ+2δ+2m (Γ ) −−−−−−−→ M Pλ+2δ

for each nonnegative integer λ. We now consider the natural embedding Eδm : Rm [X] → R[[X]]δ defined by (Eδm Ψ )(z, X) =

∞ 

ψk (z)X k+δ

(9.14)

k=0

for

m 

Ψ (z, X) =

ψr (z)X r ∈ Rm [X],

(9.15)

r=0



where ψk =

ψk 0

for 0 ≤ k ≤ m; for k > m.

Then we see easily that it induces an embedding Eδm : M Pλm (Γ ) → Mλ−2δ (Γ )δ

(9.16)

of modular polynomials into modular series satisfying δ ◦ E m )Ψ (z, X) = Ψ (z, X) (Π m δ

(9.17)

for all Ψ (z, X) ∈ M Pλm (Γ ). Given λ > 2m, we now define the linear map m : Rm [X] → R[[X]]δ Kδ,λ

by setting m Ψ )(z, X) = (Kδ,λ

∞  k=0

ψk∗ (z)X k+δ

9.2 Liftings of Quasimodular Forms to Jacobi-like Forms

171

for Ψ (z, X) ∈ Rm [X] as in (9.15), where ψk∗

k 

=

k−r 

(λ + 2k − 2r − 2m − 1)

(9.18)

r=max(k−m,0) j=0

×

(−1)j (m − k + r + j)!(2k − 2r + λ − 2m − j − 2)! (j+r) ψm−k+r+j j!r!(2k + λ − 2m − r − 1)!

for each k ≥ 0. m Theorem 9.4 The map Kδ,λ induces a lifting m : QPλm (Γ ) → Jλ−2δ−2m (Γ )δ Kδ,λ

(9.19)

of quasimodular polynomials to Jacobi-like forms such that δ m ◦ Kδ,λ )Ψ )(z, X) = Ψ (z, X) ((Πm

(9.20)

for all Ψ (z, X) ∈ QPλm (Γ ). Proof. From (1.29), Proposition 8.4 and (9.16) we obtain the following sequence of maps: Ξm

m QPλm (Γ ) −−−λ−→ M Pλ−2m (Γ ) Em

Λλ−2δ−2m,δ

−−−δ−→ Mλ−2δ−2m (Γ )δ −−−−−−−−→ Jλ−2δ−2m (Γ )δ m We shall first show that the composite of these maps coincides with Kδ,λ . Given a quasimodular polynomial

Ψ (z, X) =

m 

ψr (z)X r ∈ QPλm (Γ ),

r=0

using (8.2) and (9.14), we have ((Eδm ◦ Ξλm )Ψ )(z, X) =

∞ 

ηk (z)X k+δ ∈ Mλ−2δ−2m (Γ )δ

k=0



with ηk =

ψkB 0

for 0 ≤ k ≤ m; for k > m,

where ψkB is as in (8.4). From this and (1.4) we see that ((Λλ−2δ−2m,δ ◦ Eδm ◦ Ξλm )Ψ )(z, X) =

∞  k=0

ηkΛ (z)X k+δ ∈ Jλ−2δ−2m (Γ )δ ,

172

9 Liftings of Quasimodular Forms

where ηkΛ =

k  r=0

1 (r) η r!(2k + λ − 2m − r − 1)! k−r k 

=

r=max(k−m,0)

1 (ψ B )(r) r!(2k + λ − 2m − r − 1)! k−r

for each k ≥ 0. Noting that B = (λ + 2k − 2r − 2m − 1) ψk−r

k−r  j=0

(−1)j (m − k + r + j)! j! (j)

× (2k − 2r + λ − 2m − j − 2)!ψm−k+r+j by (8.4), we obtain m = Λλ−2δ−2m,δ ◦ Eδm ◦ Ξλm . Kδ,λ

On the other hand, from (9.12) we see that δ ◦ Ξλ−2δ−2m,δ = Ξ m ◦ Π δ , Π m λ m which implies

δ δ Λm λ ◦ Πm ◦ Ξλ−2δ−2m,δ = Πm

by Proposition 8.1. Using this, (1.8) and (9.17), we have δ m ◦ Kδ,λ )Ψ )(z, X) (Πm m δ = (Λm λ ◦ Πm ◦ Ξλ−2δ−2m,δ ◦ Kδ,λ )Ψ )(z, X) m m δ = ((Λm λ ◦ Πm ◦ Ξλ−2δ−2m,δ ◦ Λλ−2δ−2m,δ ◦ Eδ ◦ Ξλ )Ψ )(z, X)

= Ψ (z, X) for all Ψ (z, X) ∈ QPλm (Γ ); hence the proof of the theorem is complete.



In order to describe the lifting in the previous theorem in terms of quasimodular forms, we consider the map δ : Jλ (Γ )δ → R πm

for each nonnegative integer m defined by δ (Φ) = φm πm

∞ δ if Φ(z, X) = k=0 φk (z)X k+δ ∈ Jλ (Γ )δ . Then πm (Φ) is the constant term δ m Φ)(z, X) ∈ QPλ+2δ+2m (Γ ) in (7.11) in the quasimodular polynomial (Πm

9.2 Liftings of Quasimodular Forms to Jacobi-like Forms

173

m δ and therefore is a quasimodular form belonging to QMλ+2δ+2m (Γ ). Thus πm determines the map δ m : Jλ (Γ )δ → QMλ+2δ+2m (Γ ). πm

 m defined by We now introduce the map K δ,λ  m = Km ◦ Qm : QM m (Γ ) → R[[X]]δ , K δ,λ δ,λ λ λ where Qm λ is the isomorphism in (7.28).  m induces a lifting Corollary 9.5 The map K δ,λ  m : QM m (Γ ) → Jλ−2δ−2m (Γ )δ K δ,λ λ of quasimodular forms to Jacobi-like forms such that δ  m )(ψ) = ψ ◦K (πm δ,λ

for all ψ ∈ QMλm (Γ ). Proof. This follows from Theorem 9.4 and the fact that the coefficient of X m in the Jacobi-like form  m ψ)(z, X) = ((Km ◦ Qm )ψ)(z, X) ∈ Jλ−2δ−2m (Γ )δ (K δ,λ δ,λ λ coincides with the constant term in the quasimodular polynomial m (Qm λ ψ)(z, X) ∈ QPλ (Γ ),



which is equal to ψ(z).

Example 9.6 (i) We consider the lifting (9.19) for m = 0. First, we note that QPλ0 (Γ ) can be identified with Mλ (Γ ). Thus we have Ψ (z, X) = ψ0 (z) with ψ0 ∈ Mλ (Γ ), and we see in the formula (9.18) that r = k and j = 0, so that (λ − 1)! (k) ψk∗ = ψ , k!(k + λ − 1)! 0 0 Ψ )(z, X) (Kδ,λ

= (λ − 1)!

∞  k=0

(k)

ψ0 (z) X k+δ ∈ Jλ−2δ (Γ )δ . k!(k + λ − 1)!

(9.21)

0 Ψ )(z, X)/(λ − 1)! is the well-known Cohen– Thus the Jacobi-like form (Kδ,λ Kuznetsov lifting of the modular form f0 in (1.35) (see e.g. [31]). (ii) We now consider the case of m = 1. First, for k = 0 we have r = j = 0 in the sum in (9.19), and therefore

ψ0∗ = (λ − 3)

(λ − 4)! ψ1 = ψ1 . (λ − 3)!

174

9 Liftings of Quasimodular Forms

On the other hand, using (9.19) for k ≥ 1, we obtain ψk∗

=

k−r k  

(λ + 2k − 2r − 3)

r=k−1 j=0

(−1)j (1 − k + r + j)!(2k − 2r + λ − j − 4)! (j+r) ψ1−k+r+j j!r!(2k + λ − r − 3)! (λ − 2)! (λ − 3)! (k−1) (k) ψ ψ = (λ − 1) − (λ − 1) (k − 1)!(k + λ − 2)! 0 (k − 1)!(k + λ − 2)! 1 (λ − 4)! (k) ψ + (λ − 3) k!(k + λ − 3)! 1 (λ − 2)! (λ − 2)!(k − 1) (k) (k−1) ψ ψ = (λ − 1) − (k − 1)!(k + λ − 2)! 0 k!(k + λ − 2)! 1  (λ − 2)!  (k−1) (k) (λ − 1)kψ0 = − (k − 1)ψ1 . k!(k + λ − 2)! ×

Thus it follows that 1 (Kδ,λ Ψ )(z, X) = ψ1 (z)X δ +

∞ 

(λ − 2)! (9.22) k!(k + λ − 2)! k=1   (k−1) (k) (z) − (k − 1)ψ1 (z) X k+δ , × (λ − 1)kψ0

which belongs to Jλ−2δ−2 (Γ )δ . (iii) Let f be a modular form belonging to Mw (Γ ). Then it can be shown 1 (Γ ) which satisfies that f  is a quasimodular form belonging to QMw+2 (f  |w+2 γ)(z) = f  (z) + wf (z)K(γ, z) for all z ∈ H and γ ∈ Γ . Thus, if we set Ψ (z, X) = (Q1w+2 (f  ))(z, X) = ψ0 (z) + ψ1 (z)X with Q1w+2 as in (7.20), by using (9.22) for λ = w + 2 we obtain ψ0 = f  , (k−1)

(λ − 1)kψ0

(k)

− (k − 1)ψ1

ψ1 = wf,

= (w + 1)kf (k) − (k − 1)wf (k) = (k + w)f (k) .

Thus we have 1 (Kδ,w+2 Ψ )(z, X) = wf (z)X δ + w!

∞  (k + w)f (k) k=1

= w!

∞  k=0

(k)

k!(k + w)!

f X k+δ , k!(k + w − 1)!

X k+δ

9.2 Liftings of Quasimodular Forms to Jacobi-like Forms

175

which belongs to Jw−2δ (Γ )δ . Comparing this with (9.21), we see that the lifting of the quasimodular polynomial (Q1w+2 (f  ))(z, X) corresponding to the quasimodular form f  is equal to the lifting of the modular form f times the weight of f .

δ Remark 9.7 (i) We see easily that the kernel of the complex linear map Πm in (9.13) is equal to Jλ (Γ )δ+m+1 . On the other hand, by Theorem 9.4 the same map is surjective. Thus we obtain the short exact sequence of the form Πδ

m 0 −−−−→ Jλ (Γ )δ+m+1 −−−−→ Jλ (Γ )δ −−−m −→ QPλ+2m+2δ (Γ ) −−−−→ 0, (9.23) where the second arrow represents the inclusion map. Furthermore, from (9.19) we obtain the map m m Kδ,λ+2δ+2m : QPλ+2δ+2m (Γ ) → Jλ (Γ )δ

satisfying δ m ◦ Kδ,λ+2δ+2m )Ψ = Ψ (Πm m (Γ ). Thus it follows that the short exact sequence for all Ψ (z, X) ∈ QPλ+2δ+2m (9.23) splits.

r = Sr ◦ Qm with Sr and Qm as in (7.26) and (7.28), (ii) If we set S λ λ respectively, then we obtain the complex linear map r : QM m (Γ ) → QM m−r (Γ ) S λ λ−2r for 0 ≤ r ≤ m. Using this map, the formula for the lifting map  m : QM m (Γ ) → Jλ−2δ−2m (Γ )δ K δ,λ λ in Corollary 9.5 can be written as m  δ,λ ψ)(z, X) = (K

∞ 

ψk# (z)X k+δ

k=0

for ψ ∈ QMλm (Γ ), where ψk# =

k 

k−r 

r=max(k−m,0) j=0

(−1)j (m − k + r + j)!(2k − 2r + λ − 2m − j − 2)! j!r!(2k + λ − 2m − r − 1)!

× (λ + 2k − 2r − 2m − 1)(Sm−k+r+j ψ)(j+r) for each k ≥ 0.

176

9 Liftings of Quasimodular Forms

9.3 Lie Brackets As was discussed in Section 4.4, the space of Jacobi-like forms has the structure of a Lie algebra whose Lie bracket is compatible with the Lie bracket for pseudodifferential operators associated to the noncommutative product. In this section we study Lie algebra structures for quasimodular forms. If Ψ (z) ∈ Ψ D(R)−ε is of the form Ψ (z) =

∞ 

ψk (z)∂ −k−ε ∈ Ψ D(R)−ε

(9.24)

m  ψm−r (z) r X , r!Cm−r+ε r=0

(9.25)

k=0

with ε > 0, we set −ε Ψ )(z, X) = (∂ Πm

where Cm−r+ε = (−1)m−r+ε (m − r + ε)!(m − r + ε − 1)! as in (1.48). Lemma 9.8 If Ψ (z) ∈ Ψ D(R)Γ−ε and F (z, X) ∈ J2ξ (Γ )ε−ξ , then we have ∂

−ε ε−ξ X Πm Ψ = Πm (Iξ,ε Ψ ),

ε−ξ −ε ∂ Πm F = ∂ Πm (Iξ,ε−ξ F)

(9.26)

for all ξ ∈ Z, where the maps ∂ : J2ξ (Γ )ε−ξ → Ψ D(R)Γ−ε , Iξ,ε−ξ X : Ψ D(R)Γ−ε → J2ξ (Γ )ε−ξ , Iξ,ε ε−ξ m Πm : J2ξ (Γ )ε−ξ → QP2m+2ε (Γ ), −ε Ψ )(z, X) is a quasiare as in (1.46), (1.47) and (9.13). In particular, (∂ Πm m modular polynomial belonging to QP2m+2ε (Γ ). X Ψ )(z, X) is given by Proof. Let Ψ (z) ∈ Ψ D(R)−ε be as in (9.24), so that (Iξ,ε (1.47). From this and (7.11) we obtain

ε−ξ X (Πm (Iξ,ε Ψ ))(z, X) =

m  1 −1 −ε Cm−r+ε ψm−r (z)X r = (∂ Πm Ψ )(z, X). r! r=0

X Ψ )(z, X) ∈ J2ξ (Γ )ε−ξ by Corollary 1.16, we see that Since (Iξ,ε ε−ξ X m (Πm (Iξ,ε Ψ ))(z, X) belongs to QP2m+2ε (Γ ) by Proposition 7.17 and that the first relation in (9.26) holds. The second relation follows from this and Corollary 1.16. 

By Lemma 9.8 there is a linear map

9.3 Lie Brackets

177 ∂

−ε m Πm : Ψ D(R)Γ−ε → QP2m+2ε (Γ )

such that the diagram Ψ D(R)Γ−ε ⏐ ∂ −ε ⏐ Πm 

X Iξ,ε

−−−−→

J2ξ (Γ )ε−ξ ⏐ ⏐ ε−ξ Πm

id

m m QP2m+2ε (Γ ) −−−−→ QP2m+2ε (Γ )

is commutative. We also see that there are short exact sequences of the form ∂

Π −ε

m 0 → Ψ D(R)Γm−ε → Ψ D(R)Γ−ε −−−m −→ QP2m+2ε (Γ ) → 0, ∂

 −ε Π

m −→ QM2m+2ε (Γ ) → 0, 0 → Ψ D(R)Γm−ε → Ψ D(R)Γ−ε −−−m

(9.27) (9.28)

 δ = S0 ◦ ∂ Π δ with S0 being in (7.29). If the map where ∂ Π m m m m : QP2m+2ε (Γ ) → J2ε−2δ (Γ )δ Kδ,2m+2ε

is as in (9.19), we set ∂

m ∂ m m Kδ,2m+2ε = Iε−δ ◦ Kδ,2m+2ε : QP2m+2ε (Γ ) → Ψ D(R)Γ−ε .

(9.29)

Using (9.20), (9.26) and (9.29) with ξ = ε − δ and Corollary 1.16, we have −ε ∂ m δ X ∂ m ◦ Kδ,2m+2ε )F = (Πm ◦ Iε−δ,ε ) ◦ (Iε−δ,δ ◦ Kδ,2m+2ε )F (∂ Π m δ m = (Πm )F = F ◦ Kδ,2m+2ε m m for all F (z, X) ∈ QP2m+2ε (Γ ). Thus the map ∂ Kδ,2m+2ε is a lifting of quasimodular forms to automorphic pseudodifferential operators, and both of the short exact sequences (9.27) and (9.28) split.

As was described in (4.1), the noncommutative multiplication operation in Ψ D(R) defined by the Leibniz rule determines the Lie bracket [ , ]∂ : Ψ D(R) × Ψ D(R) → Ψ D(R), on the same space given by [Ψ (z), Φ(z)]∂ = Ψ (z)Φ(z) − Φ(z)Ψ (z)

(9.30)

for all Ψ (z), Φ(z) ∈ Ψ D(R), which provides Ψ D(R) with a structure of a complex Lie algebra.

178

9 Liftings of Quasimodular Forms

Proposition 9.9 If [·, ·]X and [·, ·]∂ are as in (4.5) and (9.30), then we have   (9.31) IξX1 +ξ2 ,ε1 +ε2 [Ψ (z), Φ(z)]∂ = [IξX1 ,ε1 (Ψ (z)), IξX2 ,ε2 (Φ(z))]X for Ψ (z) ∈ Ψ D(R)−ε1 , Φ ∈ Ψ D(R)−ε2 and ξ1 , ξ2 ≥ 0. Proof. Given Ψ (z) ∈ Ψ D(R)−ε1 and Φ ∈ Ψ D(R)−ε2 , using (1.49), (1.50) and (4.5), we have [Ψ (z), Φ(z)]∂ = [(Iξ∂1 ,ε1 −ξ1 ◦ IξX1 ,ε1 )(Ψ (z)), (Iξ∂2 ,ε2 −ξ2 ◦ IξX2 ,ε2 )(Φ(z))]∂ = Iξ∂1 +ξ2 ,ε1 +ε2 −ξ1 −ξ2 [IξX1 ,ε1 (Ψ (z)), IξX2 ,ε2 (Φ(z))]X . 

Thus we obtain (9.31) by applying IξX1 +ξ2 ,ε1 +ε2 to this relation. For each i ∈ {1, 2} we consider a quasimodular polynomial given by Fi (z, X) =

m 

m fi,r (z)X r ∈ QP2ξ (Γ ), i

(9.32)

r=0

and set [F1 (z, X), F2 (z, X)]Q =

p m m−r   (r + p)!(m − p + q)!

r!

r=0 p=0 q=0

(9.33)

 (q) f1,r+p (z)f2,m−p+q (z) × Ξδm−r,p,q 1 +ξ1 ,δ2 +ξ2

 (q) f (z)f (z) Xr, − Ξδm−r,p,q 2,r+p 1,m−p+q 2 +ξ2 ,δ1 +ξ1

∗,∗,∗ m be the graded complex are as in (4.2). Let R where the coefficients Ξ∗,∗ vector space m = QPm (Γ ) R ≥0

m has the structure of a complex for m ≥ 0. The next theorem shows that R Lie algebra. Theorem 9.10 The bilinear map [·, ·]Q given by (9.33) is a Lie bracket on m compatible with the Lie bracket [·, ·]X on Jacobi-like forms given the space R by (4.4), meaning that the following diagram is commutative [·,·]X

J2(ξ1 −m−δ1 ) (Γ )δ1 × J2(ξ2 −m−δ2 ) (Γ )δ2 −−−−→ J2(ξ1 +ξ2 −2m−δ1 −δ2 ) (Γ )δ1 +δ2 ⏐ ⏐ ⏐ δ1 +δ2 δ1 δ1 ⏐ (Πm ,Πm )  Πm m m QP2ξ (Γ ) × QP2ξ (Γ ) 1 2

[·,·]Q

−−−−→

m QP2ξ (Γ ). 1 +2ξ2 −2m

9.3 Lie Brackets

179

Proof. Let Fi (z, X) with 1 ≤ i ≤ 2 be as in (9.32), and assume that Kδmi ,2ξi Fi (z, X) = Φi (z, X) =

∞ 

φi,k (z)X k+δi ∈ Jλi −2m−2δi (Γ )δi

k=0 δi with Πm (Kδmi ,2ξi Fi ) = Fi by (9.20), where Kδmi ,2ξi is as in (9.19). Then we have (9.34) φi,r = (m − r)!fi,m−r

for each r ∈ {0, 1, . . . m}. From (4.4) we see that [Φ1 (z, X), Φ2 (z, X)]X =

p  r  ∞  

(q)

Ξδr,p,q φ1,r−p (z)φ2,p−q (z) 1 +ξ1 ,δ2 +ξ2

r=0 p=0 q=0

 (q) φ (z)φ (z) X r+δ1 +δ2 , − Ξδr,p,q 2,r−p 1,p−q 2 +ξ2 ,δ1 +ξ1

which is a Jacobi-like form belonging to J2ξ1 +2ξ2 (Γ )δ1 +δ2 . Thus, using (9.34), we obtain δ1 +δ2 [Φ1 (z, X), Φ2 (z, X)]X Πm  p m m−r   1 (q) Ξδm−r,p,q = φ1,m−r−p (z)φ2,p−q (z) 1 +ξ1 ,δ2 +ξ2 r! r=0 p=0 q=0  (q) φ (z)φ (z) Xr − Ξδm−r,p,q 2,m−r−p 1,p−q 2 +ξ2 ,δ1 +ξ1

= [F1 (z, X), F2 (z, X)]Q , m (Γ ). In parand it is a quasimodular polynomial belonging to QP2ξ 1 +2ξ2 −2m ticular, we have δ1 +δ2 [Kδm1 ,2ξ1 F1 (z, X), Kδm2 ,2ξ2 F2 (z, X)]X . [F1 (z, X), F2 (z, X)]Q = Πm

We now consider an element Ψi (z, X) =

∞ 

ψi,k (z)X k+δi ∈ Jλi −2m−2δi (Γ )δi

k=0

for each i ∈ {1, 2}, and let δi Ψi (z, X). Gi (z, X) = Πm

If we set [Kδm1 ,2ξ1 G1 (z, X), Kδm2 ,2ξ2 G2 (z, X)]X =

∞  k=0

αk (z)X k+δi +δi ,

180

9 Liftings of Quasimodular Forms

[Ψ1 (z, X), Ψ2 (z, X)]X =

∞ 

βk (z)X k+δi +δi ,

k=0

then from (4.4) we see that αk = βk for each k ∈ {0, 1, . . . , m}; hence we obtain δ1 +δ2 δ1 +δ2 [Kδm1 ,2ξ1 G1 (z, X), Kδm2 ,2ξ2 G2 (z, X)]X = Πm [Ψ1 (z, X), Ψ2 (z, X)]X . Πm

Thus we have δ1 δ2 δ1 +δ2 Ψ1 (z, X), Πm Ψ2 (z, X)]Q = Πm [Ψ1 (z, X), Ψ2 (z, X)]X , [Πm



which proves the theorem.

Remark 9.11 From (9.33) we see that the coefficient of X m in the polynomial [F1 (z, X), F2 (z, X)]Q is given by Sm [F1 (z, X), F2 (z, X)]Q = Ξδ0,0,0 f1,m (z)f2,m (z) − Ξδ0,0,0 f2,m (z)f1,m (z) = 0; 1 +ξ1 ,δ2 +ξ2 2 +ξ2 ,δ1 +ξ1 hence it follows that m−1 (Γ ). [F1 (z, X), F2 (z, X)]Q ∈ QP2ξ 1 +2ξ2 −2m

On the other hand, the coefficient of X m−1 is equal to Sm−1 [F1 (z, X), F2 (z, X)]Q  = m! Ξδ1,0,0 f1,m−1 (z)f2,m (z) 1 +ξ1 ,δ2 +ξ2 f2,m−1 (z)f1,m (z) − Ξδ1,0,0 2 +ξ2 ,δ1 +ξ1

 f1,m (z)f2,m−1 (z) + m! Ξδ1,1,0 1 +ξ1 ,δ2 +ξ2

f2,m (z)f1,m−1 (z) − Ξδ1,1,0 2 +ξ2 ,δ1 +ξ1

  f1,m (z)f2,m (z) + m(m!) Ξδ1,1,1 1 +ξ1 ,δ2 +ξ2

 f2,m (z)f1,m (z) − Ξδ1,1,1 2 +ξ2 ,δ1 +ξ1







  f1,m (z)f2,m (z) = m(m!) Ξδ1,1,1 1 +ξ1 ,δ2 +ξ2

  f2,m (z)f1,m (z) , − Ξδ1,1,1 2 +ξ2 ,δ1 +ξ1

and it is a modular form belonging to M2ξ1 +2ξ2 −4m+2 (Γ ) by (7.27).

9.4 Rankin–Cohen Brackets on Quasimodular Forms

181

m given by (9.33) is compatible Corollary 9.12 The Lie bracket [·, ·]Q on R ∂ with the Lie bracket [·, ·] on Ψ D(R) given by (9.30), so that [·,·]∂

Ψ D(R)Γ−ξ1 +m × Ψ D(R)Γ−ξ2 +m −−−−→ Ψ D(R)Γ−ξ1 −ξ2 +2m ⏐ ⏐ ⏐∂ 2m−ξ1 −ξ2 m−ξ1 ∂ m−ξ2 ⏐ (∂ Π m , Πm )  Πm m m QP2ξ (Γ ) × QP2ξ (Γ ) 1 2

[·,·]Q

m −−−−→ QP2ξ (Γ ) 1 +2ξ2 −2m

∗ is commutative, where ∂ Πm is as in (9.25).



Proof. This follows from (9.26), (9.31) and Theorem 9.10,

9.4 Rankin–Cohen Brackets on Quasimodular Forms Rankin–Cohen brackets for modular forms are well known (cf. [31]), and similar brackets for Jacobi-like forms were introduced in [23] by using the heat operator (see also [15], [17], [16], [20], [21], [22], [28], [27]). Rankin–Cohen brackets on quasimodular forms were also studied by Martin and Royer in [84] for congruence subgroups of SL(2, Z). In this section we construct Rankin– Cohen brackets on quasimodular forms for more general discrete subgroups of SL(2, R) that are compatible with Rankin–Cohen brackets on Jacobi-like forms (cf. [24]). Given μ ∈ R, we now recall that, as in (5.5), the formal differential operator DμX on R[[X]] is given by DμX =

∂ ∂2 ∂ −μ −X , ∂z ∂X ∂X 2

(9.35)

which is the radial heat operator studied in Section 5.1. If  is a positive integer, we extend (5.12) to the case of DμX with μ ∈ R to obtain X X (DμX )[] = Dμ+2−2 ◦ · · · ◦ Dμ+2 ◦ DμX .

As in Theorem 5.7, it can be shown that ((DμX )[] Φ)(z, X) =

∞ 

k+δ [Φ],k μ,δ (z)X

k=0

for Φ(z, X) =

∞ k=0

φk (z)X k+δ ∈ R[[X]]δ with δ ≥ 0, where

(9.36)

182

9 Liftings of Quasimodular Forms

[Φ],k μ,δ (z)

=

 

(−1)

−j

j=0

×

   j

(9.37)

(k + δ +  − j)!(k + δ + μ + 2 − j − 2)! (j) φk+−j (z) (k + δ)!(k +  + δ + μ − 2)!

for all z ∈ H and k ≥ 0. We note that the case where μ = λ ∈ Z is given by (5.14).

We now introduce bilinear maps on the space R[[X]] of formal power series by using the operators of the form DμX with μ ∈ 12 Z. Given a nonnegative integer n, elements μ1 , μ2 ∈ 12 Z and formal power series Φ1 (z, X) ∈ Jλ1 (Γ )δ1 ,

Φ2 (z, X) ∈ Jλ2 (Γ )δ2 ,

we define the associated formal power series [Φ1 , Φ2 ]J(μ1 ,μ2 ),n (z, X) belonging to ∈ R[[X]]δ1 +δ2 by [Φ1 , Φ2 ]J(μ1 ,μ2 ),n (z, X)

=

n 

 (−1)



=0

  n + λ 2 − μ2 − 1 

n + λ 1 − μ1 − 1 n−

× ((DμX1 )[] (Φ1 (z, X)))((DμX2 )[n−] (Φ2 (z, X))), where we used the notation in (9.37).

Thus, using (9.36), we may write [Φ1 , Φ2 ]J(μ1 ,μ2 ),n (z, X) =

∞ 

ξμ1 ,μ2 ,u (z)X u+δ1 +δ2

(9.38)

u=0

with ξ(μ1 ,μ2 ),u (z) =

n u   t=0 =0

 (−1)

  n + λ 2 − μ2 − 1 

n + λ 1 − μ1 − 1 n−

n−,u−t × [Φ1 ],t (z) μ1 ,δ1 (z)[Φ2 ]μ2 ,δ2 n−,u−t for all z ∈ H and u ≥ 0, where [Φ1 ],t μ1 ,δ1 and [Φ2 ]μ2 ,δ2  are as in (9.37). Here r 1 we note  that the binomial coefficients of the form  with r ∈ 2 Z is given r by 0 = 1 and   r r(r − 1) · · · (r −  + 1) =  !

for  ≥ 1.

9.4 Rankin–Cohen Brackets on Quasimodular Forms

183

Proposition 9.13 If Φ1 (z, X) and Φ2 (z, X) are Jacobi-like forms with Φ1 (z, X) ∈ Jλ1 (Γ )δ1 ,

Φ2 (z, X) ∈ Jλ2 (Γ )δ2 ,

then the formal power series [Φ1 , Φ2 ]J(μ1 ,μ2 ),n (z, X) is a Jacobi-like form belonging to Jλ1 +λ2 +2n (Γ )δ1 +δ2 .

Proof. This was proved in [23] in the case where μ1 = μ2 = 1/2. The general case can be proved in a similar manner by using (9.37). 

From Proposition 9.13 we obtain the bilinear map [ , ]J(μ1 ,μ2 ),n : Jλ1 (Γ )δ1 × Jλ2 (Γ )δ2 → Jλ1 +λ2 (Γ )δ1 +δ2 ,

(9.39)

which may be regarded as the n-th Rankin–Cohen bracket for Jacobi-like forms (see [23]). Given integers λ1 , λ2 , nonnegative integers δ1 , δ2 , m1 , m2 , n, and quasimodular polynomials F1 (z, X) ∈ QPλm1 1+2(m1 +δ1 ) (Γ ) ⊂ Rm1 [X],

(9.40)

F2 (z, X) ∈ QPλm2 2+2(m2 +δ2 ) (Γ ) ⊂ Rm2 [X],

(9.41)

using the lifting map in (9.19), we obtain the Jacobi-like forms Kδm11,λ1 +2(m1 +δ1 ) F1 (z, X) ∈ Jλ1 (Γ )δ1 , Kδm22,λ2 +2(m2 +δ2 ) F2 (z, X) ∈ Jλ2 (Γ )δ2 . If (μ1 , μ2 ) ∈ 12 Z, we define the bilinear map ,P [ , ]λδ11,δ,λ22,(μ : QPλm1 1+2(m1 +δ1 ) (Γ ) × QPλm2 2+2(m2 +δ2 ) (Γ ) → Rm1 +m2 [X] 1 ,μ2 ),n

of polynomials by ,P (z, X) [F1 , F2 ]δλ11,δ,λ22,(μ 1 ,μ2 ),n

=

n m 1 +m2 m1 +m  2 −r =0

r=0

t=0

(−1) r!

 

(9.42)   n + λ 1 − μ2 − 1 n + λ 2 − μ2 − 1 n− 

× [Kδm11,λ1 +2(m1 +δ1 ) F1 ],t μ1 ,δ1 (z) 1 +m2 −r−t × [Kδm22,μ2 +2(m2 +δ2 ) F2 ]μn−,m (z)X r , 2 ,δ2

where the square brackets on the right-hand side are as in (9.37).

184

9 Liftings of Quasimodular Forms

Proposition 9.14 The formula (9.42) determines a bilinear map ,P : QPλm1 1+2(m1 +δ1 ) (Γ ) × QPλm2 2+2(m2 +δ2 ) (Γ ) [ , ]λδ11,δ,λ22,(μ 1 ,μ2 ),n

(9.43)

+m2 −→ QPλm1 1+λ (Γ ) 2 +2(m1 +m2 +n+δ1 +δ2 )

of quasimodular polynomials. Proof. Let F1 (z, X) and F2 (z, X) be quasimodular polynomials as in (9.40) and (9.41). If [ , ]J(μ1 ,μ2 ),n is as in (9.39), then from (9.38) and (9.39) we obtain [Kδm11,λ1 +2(m1 +δ1 ) F1 , Kδm22,λ2 +2(m2 +δ2 ) F2 ]J(μ1 ,μ2 ),n (z, X) =

∞ 

(9.44)

ξu (z)X u+δ1 +δ2 ∈ Jλ1 +λ2 (Γ )δ1 +δ2 ,

u=0

where ξu (z) =

n u  

 (−1)

t=0 =0

  n + λ 2 − μ2 − 1 

n + λ 1 − μ1 − 1 n−

(9.45)

× [Kδm11,λ1 +2(m1 +δ1 ) F1 ],t μ1 ,δ1 (z) × [Kδm22,λ2 +2(m2 +δ2 ) F2 ]n−,u−t (z). μ2 ,δ2 Using (9.20), (9.42), (9.44) and (9.45), we see that ,P [F1 , F2 ]δλ11,δ,λ22,(μ (z, X) 1 ,μ2 ),n m 1 +m2

1 ξm1 +m2 −r (z)X r r! r=0   δ1 +δ2 m1 m2 J = Πm F , K F ] (z, X) , [K 1 2 (μ1 ,μ2 ),n δ1 ,λ1 +2(m1 +δ1 ) δ2 ,λ2 +2(m2 +δ2 ) 1 +m2

=

which is a quasimodular polynomial belonging to +m2 QPλm1 1+λ (Γ ); 2 +2(m1 +m2 +n+δ1 +δ2 )



hence the theorem follows. From the previous proposition we see that the diagram K

Jλ1 (Γ )δ1 × Jλ2 (Γ )δ2 ←−−−− QPλm1 1+2(m1 +δ1 ) (Γ ) × QPλm2 2+2(m2 +δ2 ) (Γ ) ⏐ ⏐ ⏐ ⏐ λ ,λ ,P [ , ]J [ , ]δ 1,δ 2,(μ ,μ ),n  (μ1 ,μ2 ),n  1 2 1 2 Π δ1 +δ2

m −−→ Jλ1 +λ2 +2n (Γ )δ1 +δ2 −−−

QPλm1 +λ2 +2(m+n+δ1 +δ2 ) (Γ ) (9.46)

is commutative, where K = [Kδm11,λ1 +2(m1 +δ1 ) , Kδm22,λ2 +2(m2 +δ2 ) ].

Chapter 10

Quasimodular Forms and Vector-valued Modular Forms

In [60] Kuga and Shimura determined all holomorphic vector differential forms ω satisfying ω ◦ γ = ρ(γ)ω (10.1) for all γ ∈ Γ , where ρ is a symmetric tensor representation of a discrete subgroup Γ of SL(2, R). They constructed such a form corresponding to each modular form of weight ≤ n + 2 and showed that any holomorphic vector form satisfying (10.1) can be written as a sum of the vector forms associated to modular forms of weight ≤ n + 2. As a result, they obtained a correspondence between vector-valued differential forms, or equivalently vector-valued modular forms of weight two, and certain finite sequences of scalar-valued modular forms. In this chapter we discuss an extension of the correspondence of Kuga and Shimura in [60] to the case of vector-valued modular forms of arbitrary weight k and study its connection with quasimodular forms (cf. [26]). Some of the results described in this chapter were also studied by S. Zemel in [116].

10.1 Vector-valued Forms Let Γ be a discrete subgroup of SL(2, R) commensurable with SL(2, Z) as in earlier sections. In this section we introduce vector-valued versions of modular forms, quasimodular forms, and Jacobi-like forms associated to a representation of Γ . Throughout this chapter we fix a positive integer n and denote by Rn the space of Cn -valued functions on H whose components belong to the ring R of holomorphic functions on H. We then denote by Rn [[X]] the space of formal power series in X with coefficients in Rn . We shall use t (·) to denote the transpose of the matrix (·). In particular, if x1 , . . . , x ∈ C, the corresponding column vector belonging to C can be written as t (x1 , . . . , x ).

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_10

185

186

10 Quasimodular Forms and Vector-valued Modular Forms

Let ρ : Γ → GL(n, C) be a representation of Γ , and let ρ∗ : Γ → GL(n, C) be the contragredient, or the inverse transpose, of ρ, so that ρ∗ (γ) = t ρ(γ)−1 for all γ ∈ Γ . Definition 10.1 (i) An element f ∈ Rn is a vector-valued modular form of weight λ for Γ with respect to ρ if it satisfies f (γz) = ρ(γ)J(γ, z)λ f (z) for all z ∈ H and γ ∈ Γ and each component of f is bounded at every cusp of Γ . (ii) A vector-valued Jacobi-like form of weight λ for Γ with respect to ρ X) ∈ Rn [[X]] satisfying is a formal power series Φ(z, X) Φ(γz, J(γ, z)−2 X) = J(γ, z)λ eK(γ,z)X ρ(γ)Φ(z,

(10.2)

for all z ∈ H and γ ∈ Γ such that the component functions of its coefficients are bounded at each cusp of Γ . !λ (Γ, ρ) and J λ (Γ, ρ) the spaces of vector-valued modWe shall denote by M ular forms and Jacobi-like forms of the type described in the above definition.

Lemma 10.2 Given nonnegative integers λ1 , λ2 , w and representations ρ, ρ1 , ρ2 : Γ → GL(n, C) of Γ , there are bilinear maps (λ1 ,λ2 ) ! !λ (Γ, ρ2 ) → M !λ +λ +2w (Γ, ρ1 ⊗ ρ2 ), : Mλ1 (Γ, ρ1 ) × M [ , ]w 2 1 2 (λ1 ,λ2 ) ! !λ (Γ, ρ∗ ) → Mλ +λ +2w (Γ ) : Mλ1 (Γ, ρ) × M [[ , ]]w 2 1 2

defined by (λ1 ,λ2 ) = [φ1 , φ2 ]w

w 

λ1 + w − 1 w−i

  λ2 + w − 1 (i) (w−i) φ1 ⊗ φ2 i

(10.3)

  λ1 + w − 1 λ2 + w − 1 t (i) (w−i) (φ )ψ w−i i

(10.4)

 (−1)i

i=0

(λ1 ,λ2 ) [[φ, ψ]]w =

w  i=0

 (−1)i

!λ (Γ, ρ1 ), φ2 ∈ M !λ (Γ, ρ2 ), φ ∈ M !λ (Γ, ρ) and ψ ∈ M !λ (Γ, ρ∗ ), for all φ1 ∈ M 1 2 1 2 where () () () () φj = t (φj,1 , φj,1 , . . . , φj,n )

10.1 Vector-valued Forms

187

for j = 1, 2 and similarly for φ(i) and ψ (i) . Proof. We sketch below the idea of the proof. The details can be found in [67]. Given a real number μ, vector-valued Jacobi-like forms of index μ can considered by modifying the relation (10.2) in Definition 10.1 to X), Φ(γz, J(γ, z)−2 X) = J(γ, z)λ eμK(γ,z)X ρ(γ)Φ(z, and the Cohen–Kuznetsov liftings considered in Example 1.12 can be extended to such Jacobi-like forms. Thus, if J λ,μ (Γ, ρ) denotes the space of such Jacobi-like forms, there exist liftings Φμφ1 (z, X) ∈ J λ1 ,μ (Γ, ρ1 ) and Φ−μ (z, X) ∈ J λ ,−μ (Γ, ρ2 ) of φ1 and φ2 . Then the tensor product Φμ (z, X)⊗ 2

φ1

φ1

Φ−μ φ1 (z, X) is a vector-valued Jacobi-like form of index 0, or a vector-valued modular series, whose coefficients are vector-valued modular forms. From (λ ,λ ) !λ +λ +2w (Γ, ρ1 ⊗ ρ2 ). this it can be shown that [φ1 , φ2 ]w 1 2 belongs to M 1 2 μ On the other hand, we can also consider the liftings Φφ (z, X) ∈ J λ,μ (Γ, ρ) and Φ−μ (z, X) ∈ J λ,μ (Γ, ρ) of φ and ψ. Then their product t Φμ (z, X)Φ−μ (z, X) ψ

φ

ψ (λ ,λ2 )

is a modular series for Γ . This can be used to show that [[φ, ψ]]w 1 modular form belonging to Mλ1 +λ2 +2w (Γ ). (λ ,λ )

is a 

(λ ,λ )

The two bilinear maps [ , ]w 1 2 and [[ , ]]w 1 2 in Lemma 10.2 may be regarded as Rankin–Cohen brackets for vector-valued modular forms. We can consider the bracket given by (10.3) when one of the modular forms is scalar valued by considering it as a 1-dimensional vector-valued modular form with respect to the trivial representation. Thus, for example, we have a bilinear map of the form (λ1 ,λ2 ) !λ (Γ, ρ) → M !λ +λ +2w (Γ, ρ), : Mλ1 (Γ ) × M [ , ]w 2 1 2

given by (λ1 ,λ2 ) [f, φ]w

=

w  i=0

 (−1)

i

  λ2 + w − 1 (i) f (z)φ(w−i) i

λ1 + w − 1 w−i

(10.5)

!λ (Γ, ρ). for f ∈ Mλ1 (Γ ) and φ ∈ M 2 Remark 10.3 Although the proof of Lemma 10.2 in [67] was given for vector-valued modular forms of nonnegative weight, we shall use the same formulas to consider brackets for modular forms of negative weight by using (1.20). Similarly, the bracket in (10.4) can also be extended to the negative weight case, and such brackets are used in Theorem 10.14. n+1 [X] be the space of polynomials Given a nonnegative integer m, let Rm of the form m  F (z, X) = fr (z)X r (10.6) r=0

188

10 Quasimodular Forms and Vector-valued Modular Forms

with fr ∈ Rn+1 for 0 ≤ r ≤ m. We consider two surjective maps n+1 [X] πm , Πm : Rn+1 [[X]] → Rm

defined by (πm Φ)(z, X) =

m 

φr (z)X r ,

r=0

(Πm Φ)(z, X) =

m  r=0

for Φ(z, X) =

∞ 

1 φm−r (z)X r r!

φk (z)X k ∈ Rn+1 [[X]].

k=0

We also define the complex linear endomorphism n+1 n+1 Qm : Rm [X] → Rm [X]

by setting (Qm F )(z, X) =

m  1 fm−r (z)X r r! r=0

(10.7)

n+1 [X] as in (10.6). Then Qm is clearly bijective and satisfies for F (z, X) ∈ Rm

Π m = Qm ◦ π m . Definition 10.4 Given λ ∈ Z, a vector-valued quasimodular polynomial for Γ of weight λ and degree at most m with respect to ρ is an element n+1 F (z, X) ∈ Rm [X] satisfying F (γz, J(γ, z)2 (X − K(γ, z))) = J(γ, z)λ ρ(γ)(F (z, X)) for all z ∈ H and γ ∈ Γ such that its coefficient functions are bounded at m (Γ, ρ) the space of such polynomials. each cusp of Γ . We denote by QP λ Proposition 10.5 Let λ and m be integers with m ≥ 0. n+1 [X] satisfies (i) If F (z, X) ∈ Rm   F (γz, J(γ, z)−2 X) = J(γ, z)λ πm eK(γ,z)X ρ(γ)(F (z, X))

(10.8)

for all z ∈ H and γ ∈ Γ , then (Qm F )(z, X) in (10.7) is a vector-valued m (Γ, ρ). quasimodular polynomial belonging to QP λ+2m m (ii) Given Ψ (z, X) ∈ QPλ+2m (Γ, ρ), there is a vector-valued Jacobi-like form Ψ(z, X) ∈ J λ (Γ, ρ) such that Πm (Ψ(z, X)) = Ψ (z, X).

10.2 Symmetric Tensor Representations

189

n+1 Proof. If F (z, X) ∈ Rm [X] satisfies (10.8), it is the image under πm of a Jacobi-like form belonging to J λ (Γ, ρ). Thus (ii) can obtained easily by extending the result in (7.32) for the scalar-valued to the vector-valued case. The scalar-valued version of (ii) was proved in Theorem 9.4, and it can be extended to the vector-valued case in a straightforward manner. 

10.2 Symmetric Tensor Representations In this section we introduce certain vector-valued modular forms with respect to a symmetric tensor representation and discuss some of their properties. As was discussed in Example 6.2, we recall that the n-th symmetric tensor representation ρn : SL(2, R) → GL(n + 1, C) is defined by

 n   n z1 z1 = γ ρn (γ) z2 z2

(10.9)

for all γ ∈ SL(2, R), where  n z1 = t (z1n , z1n−1 z2 , . . . , z1 z2n−1 , z2n ) ∈ Cn+1 z2 with ( zz12 ) ∈ C2 . Proposition 10.6 The vector valued functions v n , u n ∈ Rn+1 given by v n (z) = t (z n , z n−1 , . . . , z, 1),

u n (z) = t (1, n(−z), . . . , ( nr ) (−z)r , . . . , (−z)n )

for z ∈ H satisfy vn , v n |−n γ = ρn (γ)

u n |−n γ = ρ∗n (γ) un

(10.10)

for all γ ∈ Γ , where ρ∗n : SL(2, R) → GL(n + 1, C) is the contragredient of ρn . Proof. For γ = that

a b c d

∈ Γ , noting that v n (z) =

 z n 1

and using (10.9), we see

 n   n z z ρn (γ) vn (z) = ρn (γ) = γ 1 1 n   n az + b γz = = (cz + d)n cz + d 1 = J(γ, z)n v n (γz) = ( vn |−n γ)(z)

190

10 Quasimodular Forms and Vector-valued Modular Forms

for all z ∈ H. To verify the transformation formula for u n we consider complex numbers ξ and η. Then we have  n ξ n t n (γz)ρn (γ) J(γ, z) · u η  n aξ + bη n r n n = (cz + d) (1, n(−γz), . . . , ( r ) (−γz) , . . . , (−γz) ) cξ + dη ⎛ ⎞ (aξ + bη)n ⎜(aξ + bη)n−1 (cξ + dη)⎟ ⎜ ⎟ n r n ⎜ n ⎟ ··· = (cz + d) (1, n(−γz), . . . , ( r ) (−γz) , . . . , (−γz) ) ⎜ ⎟ ⎝(aξ + bη)(cξ + dη)n−1 ⎠ (cξ + dη)n = (cz + d)n (aξ + bη − γz(cξ + dη))n = ((cz + d)(aξ + bη) − (az + b)(cξ + dη))n = (bcηz + adξ − adηz − bcξ)n = (ξ − ηz)n for z ∈ H. On the other hand, we have ⎛ t

ξn

⎞

⎜ξ n−1 η ⎟  n ⎜ ⎟ ξ r n ⎜ n n u n (z) = (1, n(−z), . . . , ( r ) (−z) , . . . , (−z) ) ⎜ · · · ⎟ ⎟ = (ξ − ηz) . η n−1 ⎝ξη ⎠ ηn

From these relations and the fact that ξ and η are arbitrary complex numbers, we obtain J(γ, z)n · t u n (γz)ρn (γ) = t u n (z), and therefore t

u n (γz) = J(γ, z)−n · t u n (z)ρn (γ)−1 .

By taking the transpose of each side of this relation, we obtain n (z) = J(γ, z)−n ρ∗n (γ) un (z); u n (γz) = J(γ, z)−n · t ρn (γ)−1 u hence the proposition follows.



The following lemma will be used in Section 10.3 in the proof of Theorem 10.14(ii). Lemma 10.7 Let α and β be integers with 0 ≤ α, β ≤ n. Then we have 0 if α + β < n; t (α) (β) u n (z) vn (z) = (10.11) α (−1) n! if α + β = n for all z ∈ H.

10.2 Symmetric Tensor Representations

191 (β)

Proof. For each r ∈ Z with 0 ≤ r ≤ n the (r + 1)-th entry of v n (z) is equal to   n−r dβ n−r β!z n−r−β , z = β dz β (α)

if r + β ≤ n. On the other hand, the (r + 1)-th entry of u n (z) is given by   α    r n d r r n α!z r−α z = (−1) (−1)r α r dz α r n! z r−α = (−1)r (n − r)!(r − α)!    n−α n α!z r−α , = (−1)r α r−α (α)

assuming that r ≥ α. We note that the (r + 1)-th entry of u n (z) (resp. (β) v n (z)) is zero for r < α (resp. r > β). Thus, if α + β < n, we see that t (α) u n (z) vn(β) (z)

=

n−β 

 (−1)r

r=α

n−α r−α

    n n−r α!z r−α β!z n−r−β α β

  n−β  n (n − α)! (n − r)! α!β!z n−α−β = (−1)r α (n − r)!(r − α)! (n − r − β)!β! r=α      n−β  n n−α n−α−β α!β!z n−α−β = (−1)r β α r−α r=α      n−α−β  n−α α n n−α−β r n−α−β α!β!z = (−1) (−1) β α r r=0    n n−α α!β!z n−α−β (1 − 1)n−α−β = 0. = (−1)α β α If α + β = n, from the same computation we obtain t (α) u n (z) vn(β) (z)

= (−1)α n!; 

hence the lemma follows.

We now consider the matrix-valued function Ln : H → GL(n + 1, C) on H defined by

192

10 Quasimodular Forms and Vector-valued Modular Forms

⎛

1   ⎜0 ⎜ 1z Ln (z) = ρn =⎜ ⎜0 01 ⎝ 0

⎞ 2 n nz n(n−1) z · · · z 2 1 (n − 1)z · · · z n−1 ⎟ ⎟ 0 1 · · · z n−2 ⎟ ⎟ ⎠ ............... 0 0 ··· 1

  for all z ∈ H. Then it is known (see p. 266 in [60]) that, for γ = ac db ∈ SL(2, R),  −1  J 0 −1 = A, (10.12) Ln (γz) ρn (γ)Ln (z) = ρn c J where A is the (n + 1) × (n + 1) lower triangular matrix whose (r + 1)-th row is of the form   r ) cJ−n+2r−1 , J−n+2r , 0, . . . , 0 cr J−n+r , ( 1r ) cr−1 J−n+r+1 , . . . , ( r−1 for 0 ≤ r ≤ n with J = J(γ, z). !k (Γ, ρn ) be a vector-valued modular form of the Lemma 10.8 Let ω ∈ M form (10.13) ω(z) = Ln (z) · t (f0 (z), f1 (z), . . . , fn (z)) for all z ∈ H, where f0 , . . . , fn ∈ R. If r is an integer with 0 ≤ r ≤ n such that f ≡ 0 for all  with 0 ≤  < r, then fr is a modular form belonging to Mk−n+2r (Γ ). !k (Γ, ρn ) as in (10.13) and an element γ ∈ Γ , using (10.12) Proof. Given ω ∈ M and the relation ω |k γ = ρn (γ)ω, we see that J(γ, z)−k Ln (γz) · t (f0 (γz), f1 (γz), . . . , fn (γz)) = ρn (γ)Ln (z) · t (f0 (z), f1 (z), . . . , fn (z)) = Ln (γz)A · t (f0 (z), f1 (z), . . . , fn (z)); hence we have the identity J(γ, z)−k · t (f0 (γz), f1 (γz), . . . , fn (γz)) = A · t (f0 (z), f1 (z), . . . , fn (z)) of vectors in Cn+1 . Thus by comparing the (r + 1)-th entries we obtain J(γ, z)−k fr (γz) = J(γ, z)−n+2r fr (z), which shows that fr satisfies the transformation formula fr |k−n+2r γ = fr .

10.2 Symmetric Tensor Representations

193

On the other hand, since ω is a vector-valued modular form belonging to !k (Γ, ρn ), its components M f0 + nzf1 + · · · + z n fn , f1 + (n − 1)zf2 + · · · + z n−1 fn , . . . , fn−1 + zfn , fn are bounded at each cusp of Γ . Thus it follows that the functions f0 , f1 , . . . , fn are also bounded at each cusp of Γ , and therefore fr belongs to Mk−n+2r (Γ ).  For λ, w, r ∈ Z with 0 ≤ r ≤ w we set    λ+w−1 w−n−1 λ , αw,r = (−1)r r w−r

(10.14)

where the binomial coefficients are computed according to Remark 10.3. Then by extending the Rankin–Cohen bracket in (10.5) to modular forms without the boundedness condition at the cusps we may write [f, v n ](λ,−n) = w

w 

λ αw,r f (r) v n(w−r)

(10.15)

r=0

for f ∈ Mλ (Γ ). Lemma 10.9 If gj ∈ Mk−n+2j (Γ ) with 0 ≤ j ≤ n, we have (k−n+2j,−n)

[gj , v n ]n−j

(z) = Ln (z) · t (g0L (z), g1L (z), . . . , gnL (z))

for all z ∈ H, where gL =

0 k−n+2j (−j) (n − )!αn−j,−j gj

for 0 ≤  ≤ j − 1; for j ≤  ≤ n

k−n+2j with αn−j,−j being as in (10.14).

Proof. A direct computation shows that ( vn(n) (z), v n(n−1) (z), . . . , v n (z)) = Ln (z) · diag [n!, (n − 1)!, . . . , 1], where diag [· · · ] denotes the (n + 1) × (n + 1) diagonal matrix having [· · · ] as its diagonal entries. Thus we see that Ln (z) · t (g0L (z), g1L (z), . . . , gnL (z)) = ( vn(n) (z), v n(n−1) (z), . . . , v n ) · diag [n!, (n − 1)!, . . . , 1]−1 × t (g0L (z), g1L (z), . . . , gnL (z)) =

n  i=0

1 v (n−i) (z)giL (z) (n − i)! n

194

10 Quasimodular Forms and Vector-valued Modular Forms

=

n 

(i−j)

k−n+2j (n−i) αn−j,i−j v n (z)gj

(z)

i=j

=

n−j 

()

k−n+2j (n−−j) αn−j, v n (z)gj (z)

=0

for all z ∈ H. On the other hand, from (10.15) we obtain (k−n+2j,−n)

[gj , v n ]n−j

(z) =

n−j 

()

k−n+2j αn−j, gj (z) vn(n−−j) (z);

=0



hence the lemma follows.

10.3 Scalar- and Vector-valued Modular Forms In this section we discuss a correspondence between vector-valued modular forms associated to symmetric tensor representations and finite sequences of scalar-valued modular forms. We note that by Proposition 10.6 the vector-valued function v n ∈ Rn+1 given by  n z v n (z) = = t (z n , z n−1 , . . . , z, 1) (10.16) 1 for z ∈ H satisfies the transformation formula for vector-valued modular forms of weight (−n) with respect to the representation ρn . The following proposition provides an analogue of the Cohen–Kuznetsov lifting of v n . Proposition 10.10 Let Φ∗vn (z, X) ∈ Rn+1 [[X]] be the formal power series associated to v n in (10.16) given by Φ∗vn (z, X)

=

=

∞ (k)  (−1)k (n − k)! vn (z) k=0 n  k=0

k!

Xk

(10.17)

(k)

(−1)k (n − k)! vn (z) k X . k!

Then it satisfies the transformation formula for a vector-valued Jacobi-like form in J −n (Γ, ρn ), that is, Φ∗vn (γz, J(γ, z)−2 X) = J(γ, z)−n eK(γ,z)X ρn (γ)(Φ∗vn (z, X))

(10.18)

for all γ ∈ Γ and z ∈ H. Proof. Given α ∈ SL(2, R), it can be shown by induction on ν ≥ 0 that

10.3 Scalar- and Vector-valued Modular Forms

( vn(ν) |−n+2ν α)(z) =

195

ν  (−1)ν− ν!(n − )! =0

!(ν − )!(n − ν)!

K(α, z)ν−

d (( vn |−n α)(z)) dz 

for z ∈ H. Using this (10.10) and (10.17), for γ ∈ Γ we have Φ∗vn (γz, J(γ, z)−2 X) ∞  (−1)j (n − j)! (j) = v n (γz)J(γ, z)−2j X j j! j=0 =

∞  (−1)j (n − j)!

j!

j=0

=

∞  (−1)j (n − j)!

j!

j=0

×

( vn(j) |−n+2j γ)(z)J(γ, z)−n X j J(γ, z)−n X j

j  (−1)j− j!(n − )! =0

= J(γ, z)−n

!(j − )!(n − j)!

K(γ, z)j−

j ∞   (−1) (n − )! j=0 =0

!(j − )!

d (( vn |−n γ)(z)) dz 

K(γ, z)j− ρn (γ)( vn() (z))X j .

On the other hand, we see that J(γ, z)−n eK(γ,z)X Φ∗vn (z, X) = J(γ, z)

−n

∞  ∞ ()  (−1) (n − )! vn (z) k=0 =0

= J(γ, z)−n

!k!

j ∞  ()  (−1) (n − )! vn (z) j=0 =0

!(j − )!

K(γ, z)k X +k K(γ, z)j− X j , 

and therefore we obtain (10.18).

Theorem 10.11 Let k and n be integers with k ≡ n (mod 2) and k > n ≥ 0, and let v n (z) = t (z n , . . . , z, 1) ∈ Cn+1 for all z ∈ H. Then for each integer  with 0 ≤  ≤ n, there exists a complex linear injective map !k (Γ, ρn ) Vk,n, : Mk−n+2 (Γ ) → M

(10.19)

defined by (k−n+2,−n)

Vk,n, (g ) = [g , v n ]n−

(10.20)

196

10 Quasimodular Forms and Vector-valued Modular Forms (k−n+2,−n)

for all g ∈ Mk−n+2 (Γ ), where [g , v n ]n− denotes the (n − )-th Rankin–Cohen bracket of g and v n given by (10.5).

Proof. We consider an element n

g = (g0 , g1 , . . . , gn ) ∈

Mk−n+2 (Γ ) =0

with g ∈ Mk−n+2 for each  ∈ {0, 1, . . . , n}. If Lw λ,δ is the Cohen–Kuznetsov lifting map in (1.36), we set g (z, X) = (L0k−n+2,0 g )(z, X) =

(10.21)

∞ 

(j) g (z)

j=0

j!(j + k − n + 2 − 1)!

X j ∈ Jk−n+2 (Γ )0 .

Let Φ∗vn (z, X) be the power series given by (10.18), and let F (z, X) ∈ Rn+1 [[X]] be the vector-valued formal power series defined by F (z, X) = g (z, −X)Φ∗vn (z, X). Then, using (10.18) and the fact that g (z, X) belongs to Jk−n+2 (Γ )0 , we have F (γz, J(γ, z)−2 X) = J(γ, z)k−n+2 e−K(γ,z)X g (z, −X) × J(γ, z) = J(γ, z)

k−2n+2

−n K(γ,z)X

e

(10.22)

ρn (γ)(Φ∗vn (z, X))

ρn (γ)(F (z, X))

for all γ ∈ Γ . Hence, if we write F (z, X) =

∞ 

η jg (z)X j ,

j=0

from (10.22) we see that !k−2n+2+2j (Γ, ρn ) η jg ∈ M

(10.23)

for each j ≥ 0. On the other hand, if we set φ∗k =

(k)

vn (−1)k (n − k)! k!

for k ≥ 0, from (10.17) and (10.21) we obtain

(10.24)

10.3 Scalar- and Vector-valued Modular Forms

F (z, X) = g (z, −X)

∞ 

197

φ∗k (z)X k

k=0

=

∞ ∞   r=0 s=0

(−1)r g (z)φ∗s (z) X r+s r!(r + k − n + 2 − 1)! (r)

(r) j ∞   (−1)r g (z)φ∗j−r (z) Xj; = r!(r + k − n + 2 − 1)! j=0 r=0

hence we have η jg =

j  r=0

(−1)r g φ∗j−r r!(r + k − n + 2 − 1)! (r)

for all j ≥ 0. In particular, from (10.24) we see that η jg =

j (r) (j−r)  (−1)j (n − j + r)!g v n r!(j − r)!(r + k − n + 2 − 1)! r=0

for 0 ≤ j ≤ n. Now, using (10.5), for 0 ≤ j,  ≤ n we have (k−n+2,−n) [g , v n ]j

=

j 

 (−1)

r

r=0

=

j 

(−1)r

r=0

k − n + 2 + j − 1 j−r

(k − n + 2 + j − 1)! (j − r)!(k − n + 2 + r − 1)! ×

=

j  r=0

  −n + j − 1 (r) (j−r) g v n r

(−n + j − 1) · · · (−n + j − r) (r) (j−r) g v n r!

(k − n + 2 + j − 1)!(n + r − j)! (r) g v (j−r) (j − r)!(k − n + 2 + r − 1)!r!(n − j)!  n

= (−1)j

(k − n + 2 + j − 1)! g η j , (n − j)!

!k−2n+2+2j (Γ, ρn ) by (10.23). Hence it follows that which belongs to M (k−n+2,−n)

[g , v n ]n−

!k (Γ, ρn ), ∈M

and therefore we obtain the map Vk,n, in (10.19) is well-defined. To prove the injectivity of this map, given  ∈ {0, 1, . . . , n}, we assume that Vk,n, (g) = Vk,n, (h) with g, h ∈ Mk−n+2 (Γ ). Then from (10.20) and Lemma 10.9 we see that

198

10 Quasimodular Forms and Vector-valued Modular Forms (k−n+2,−n)

0 = [g − h, v n ]n−

(z) = Ln (z) · t (f0 (z), f1 (z), . . . , fn (z))

for all z ∈ H, where 0 fj = k−n+2 (n − j)!αn−,j− (g (j−) − h(j−) )

for 0 ≤ j ≤  − 1; for  ≤ j ≤ n.

Thus we have k−n+2 0 = f = (n − )!αn−,0 (g − h),



and therefore g = h, which shows that Vk,n, is injective.

Theorem 10.12 If Im (Vk,n, ) denotes the image of the map (10.19) for 0 ≤  ≤ n, then there is an isomorphism !k (Γ, ρn ) = M

n

Im (Vk,n, ). =0

!k (Γ, ρn ), and assume Proof. We consider a vector-valued modular form F ∈ M that Ln (z)−1 F (z) = t (f0 (z), f1 (z), . . . , fn (z)) for all z ∈ H with f0 , . . . , fn ∈ R. Let t be the first positive integer such that ft is not identically zero. Then by Lemma 10.8 the function ft is a modular form belonging to Mk−n+2t (Γ ). Here, we note that k − n + 2t > 0 because of our assumption k > n ≥ 0. Using Lemma 10.9, we see that the first nonzero entry of the vector Ln (z)−1 Vk,n,t (ft )(z) = Ln (z)−1 [ft , v n ]n−t

(k−n+2t,−n)

(z)

is the t-th component which is equal to k−n+2t ft (z). (n − t)!αn−t,0

Thus, if we set ξ0,t =

1 V (f ) ∈ Im(Vk,n,t ), k−n+2t k,n,t t (n − t)!αn−t,0

the first (t + 1) entries of the vector Ln (z)−1 (F (z) − ξ0,t (z)) !k (Γ, ρn ). Applying the same argument to the vectorare zero with F −ξ0,t ∈ M valued modular form F − ξ0,t , we can find an element ξ1,t+1 ∈ Im Vk,n,t+1 such that the first (t + 1) components of the vector Ln (z)−1 (F (z) − ξ0,t (z) − ξ1,t+1 (z))

10.3 Scalar- and Vector-valued Modular Forms

199

are all zero. By repeating this process we obtain the expression F =

n−t 

ξ,t+ ∈

=0

which implies that !k (Γ, ρn ) = M

n−t 

Im(Vk,n,t+ ),

=0

n 

Im(Vk,n, ).

(10.25)

=0

To show that the sum is direct we consider an element η ∈ Im (Vk,n, ) ∩ Im (Vk,n,j ) with  > j ≥ 0. Then we have (k−n+2,−n)

η = [g, v n ]n−

(k−n+2j,−n)

= [h, v n ]n−j

for some g ∈ Mk−n+2 (Γ ) and h ∈ Mk−n+2j (Γ ). Thus we obtain n− 

 (−1)

r

r=0

k+−1 n−−r

  − − 1 (r) (n−−r) g v n r    n−j  −j − 1 (r) (n−j−r) r k+j −1 h v n = (−1) . r n−j−r r=0

Noting that v n(n−q−r) = t (Cn z q+r , Cn−1 z q+r−1 , . . . , Cn−q−r+1 z, Cn−q−r , 0, . . . , 0) with Cp = p(p − 1) · · · (p − n + q + r − 1), (k−n+2j,−n)

we see that the (j + r + 1)-th entry of the vector [h, v n ]n−j to   k+j−1 (−j − 1)Cn−j−r h. n−j

is equal

On the other hand, since  > j, the (j + r + 1)-th entry of the vector (k−n+2,−n) [g, v n ]n− is zero. Thus it follows that h = 0 and therefore η is the zero vector in Cn+1 . Hence the sum in (10.25) is direct, and the proof of (ii) is complete.  Corollary 10.13 The map (10.19) determines an isomorphism Mk−n+2 (Γ ) ∼ = Im (Vk,n, ) for each  ∈ {0, 1, . . . , n}; hence we obtain a canonical isomorphism

200

10 Quasimodular Forms and Vector-valued Modular Forms

!k (Γ, ρn ) ∼ M =

n

Mk−n+2 (Γ )

(10.26)

=0

between vector-valued modular forms with respect to ρn and finite sequences of modular forms. Theorem 10.14 Let k and n be integers with k ≡ n (mod 2) and k > n ≥ 0, and let u n (z) = t (1, n(−z), . . . , ( nr ) (−z)r , . . . , (−z)n ) ∈ Cn+1 for all z ∈ H. (i) There exists a complex linear map !k (Γ, ρn ) → Wk,n : M

n

Mk−n+2 (Γ )

(10.27)

=0

defined by (−n,k)

un , F ]]0 F → Wk,n (F ) = ([[ (−n,k)

where [[ un , F ]]r for 0 ≤ r ≤ n.

(−n,k)

, [[ un , F ]]1

, . . . , [[ un , F ]](−n,k) ), (10.28) n

is the r-th Rankin–Cohen bracket of u n and F in (10.4)

(ii) If f is a modular form belonging to Mk−n+2p (Γ ) with 0 ≤ p ≤ n, then we have (10.29) (Wk,n ◦ Vk,n,p )(f ) = Ck,n,p f, where Ck,n,p is a constant given by Ck,n,p =

n!(k + p − 1)! (10.30) (n − p)!(k + 2p − n − 1)! p  (k + p − 1)!(−n + p − 1)(−n + p − 2) · · · (−n + s) × . s!(p − s)!(k + p − 1 − s)! s=0

In particular, we obtain (Wk,n ◦ Vk,n, )(Mk−n+2 (Γ )) ⊂ Mk−n+2 (Γ ) for each  with 0 ≤  ≤ n. Proof. Given integers k and n with k > n ≥ 0 and k ≡ n (mod 2), using the un for γ ∈ Γ in (10.10), we see easily that the map identity u n |−n γ = ρ∗n (γ) Wk,n is well-defined; hence (i) follows. To prove (ii) we consider a modular form f ∈ Mk−n+2p (Γ ). Then we have

10.3 Scalar- and Vector-valued Modular Forms

201

(k−n+2p,−n)

Vk,n,p (f ) = [f, v n ]n−p    n−p  k − n + 2p + n − p − 1 −n + n − p − 1 (r) (n−p−r) f v n = (−1)r r n−p−r r=0    n−p  −p − 1 (r) (n−p−r) r k+p−1 f v n = (−1) . r n−p−r r=0 On the other hand, if we set φ = Vk,n,p (f ), from (10.4) and Remark 10.3 we obtain    q  k + q − 1 t (s) (q−s) (−n,k) s −n + q − 1 = (−1) u n φ [[ un , φ]]q s q−s s=0 for 0 ≤ q ≤ n, where φ(q−s) =

n−p 

 (−1)r

r=0

=

n−p q−s 

k+p−1 n−p−r 

(−1)r

r=0 j=0

 q−s    −p − 1  q − s (f (r) )(q−s−j) ( vn(n−p−r) )(j) r j j=0

k+p−1 n−p−r

   −p − 1 q − s (r+q−s−j) (n−p−r+j) f v n . j r

Hence we see that [[ un , φ]](−n,k) = q

q n−p q−s  

  −n + q − 1 k + q − 1 s q−s     k + p − 1 −p − 1 q − s × j r n−p−r 

(−1)r+s

s=0 r=0 j=0

× f (r+q−s−j)t u (s) n(n−p−r+j) . n v However, by Lemma 10.7 the only nonzero terms in the previous sum occur when the indices satisfy s + n − p − r + j = n, in which case we have j = p + r − s,

r + q − s − j = q − p.

Since 0 ≤ j ≤ q − s, we see that s − p ≤ r ≤ q − p. Thus we obtain (−n,k) = 0 for q < p and [[ un , φ]]q

202

10 Quasimodular Forms and Vector-valued Modular Forms

[[ un , φ]](−n,k) q

  −n + q − 1 k + q − 1 = (−1) s q−s s=0 r=max{0,s−p}     q−s k + p − 1 −p − 1 × p+r−s r n−p−r q 



q−p 

r+s

× f (q−p) (−1)s n! for q ≥ p, where we used (10.11). Thus we see that p,q (q−p) = Ck,n f [[ un , φ]](−n,k) q

for q ≥ p with p,q Ck,n

  −n + q − 1 k + q − 1 = n! (−1) s q−s s=0 r=max{0,s−p}     q−s k + p − 1 −p − 1 . × p+r−s r n−p−r q 

q−p 



r

(−n,k)

However, since [[ un , φ]]q is a modular form belonging to Mk−n+2q (Γ ) and f (q−p) is not a modular form for q > p (see p. 59 in [114]), the sum p,q must be zero for q > p. Hence it follows that representing Ck,n p,p f. (Wk,n ◦ Vk,n,p )(f ) = Ck,n p,p coincides with Ck,n,p in (10.30), the proof of (ii) Since we see easily that Ck,n is complete. 

10.4 Quasimodular Polynomials and Vector-valued Modular Forms In the previous section we obtained a correspondence between vector-valued modular forms and certain finite sequences of modular forms. On the other hand, as was described in Section 8.2, such sequences of modular forms correspond to quasimodular polynomials, which allows us to establish a correspondence between vector-valued modular forms and quasimodular polynomials. In this section we discuss an explicit description of this correspondence.

Theorem 10.15 If k and n are integers with k ≡ n (mod 2) and k > n ≥ 0, then there is a canonical isomorphism n !k (Γ, ρn ) ∼ (Γ ) M = QPk+n

(10.31)

10.4 Quasimodular Polynomials and Vector-valued Modular Forms

203

between vector-valued modular forms and quasimodular polynomials. This result implies the existence of an isomorphism n !k (Γ, ρn ) ∼ (Γ ) M = QMk+n

(10.32)

between vector-valued modular forms and quasimodular forms. Proof. We first note that an element (f0 , f1 , . . . , fn ) of the direct sum (n M k−n+2 (Γ ) in (10.26) determines a modular polynomial of degree at =0 most n of weight k − n in Definition 8.3. Denoting the space of such modular n (Γ ) as in Section 8.2, we obtain the isomorphism polynomials by M Pk−n n

n Mk−n+2 (Γ ) ∼ (Γ ). = M Pk−n

=0

On the other hand, using this, (10.26), and the isomorphism n n (Γ ) ∼ (Γ ) M Pk−n = QPk+n

in Proposition 8.4, we obtain (10.31). Then the isomorphism (10.32) can be obtained by combining (10.31) with the map (7.29).  The next theorem gives an expression of the formula for the vector-valued function corresponding to a quasimodular polynomial via the isomorphism (10.31) in terms of derivatives of v n and the coefficients of the given quasimodular polynomial. Theorem 10.16 Let k and n be integers with k ≡ n (mod 2) and k > n ≥ 0, and let v n be as in Proposition 10.6. Then there exists an isomorphism ≈ ! n Un : QPk+n (Γ ) − → Mk (Γ, ρn )

defined by F (z, X) =

n 

fr (z)X r → Un (F ) =

r=0 ()

with v n =

n 

(n − )! vn() f

(10.33)

=0

d v . dz  n

n (Γ ) with Proof. We consider a quasimodular polynomial F (z, X) ∈ QPk+n k ∈ Z of the form n  F (z, X) = fr (z)X r . (10.34) r=0 n !k (Γ, ρn ) ∼ (Γ ) in (10.31), it Since we already have the isomorphism M = QPk+n !k (Γ, ρn ) and that the resulting map suffices to show that Un (F ) belongs to M

204

10 Quasimodular Forms and Vector-valued Modular Forms n !k (Γ, ρn ) Un : QPk+n (Γ ) → M

(10.35)

is injective. First, using the scalar-valued version of Proposition 10.5(ii), we see that F (z, X) can be lifted to a Jacobi-like form ΦF (z, X) =

∞ 

φ (z)X  ∈ Jk−n (Γ )

=0

with fr =

1 φn−r r!

for 0 ≤ r ≤ n satisfying ΦF (γz, J(γ, z)−2 X) = J(γ, z)k−n eK(γ,z)X ΦF (z, X)

(10.36)

for all γ ∈ Γ . If Φ∗vn (z, X) ∈ J −n (Γ, ρn ) is as in (10.17), we set Ψ (z, X) = ΦF (z, −X)Φ∗vn (z, X) ∞  n  (−1)j+ (n − j)! (j) = v n (z)φ (z)X j+ j! j=0 =0

=

r ∞   (−1)r (n − r + )! r=0 =0

(r − )!

Thus we may write Ψ (z, X) =

∞ 

v n(r−) (z)φ (z)X r .

ψr (z)X r ,

(10.37)

r=0

where ψr =

r  (−1)r (n − r + )! =0

=

v n(r−) φ

r  (−1)r (n − r + )!(n − )! =0

=

(r − )! (r − )!

r  (−1)r (2n − r − )!! =0

(r − n + )!

v n(r−) fn−

v n(r−n+) f

for r ≥ 0. Using (10.18) and (10.36), we see that Ψ (γz, J(γ, z)−2 X) = J(γ, z)k−2n ρn (γ)(Ψ (z, X)) for γ ∈ Γ . From this and (10.37) it follows that !k−2n+2r (Γ, ρn ) ψr ∈ M

10.4 Quasimodular Polynomials and Vector-valued Modular Forms

205

for each r ≥ 0. In particular, we obtain ψn =

n 

!k (Γ, ρn ). (−1)n (n − )! vn() f ∈ M

=0

From this and the identity (Un (F ))(z) = (−1)n ψn (z) we see that Un determines the complex linear map (10.35). To prove the injectivity of this map we assume that (Un (F ))(z) with F (z, X) given by (10.34) is equal to the zero vector in Cn+1 . Then we have t

(0, 0, . . . , 0) = n! vn f0 + (n − 1)! vn(1) f1 + · · · + v n(n) fn = B · t (f0 , f1 , . . . , fn ),

where B is a triangular matrix of the form ⎛ n!z n n!z n−1 · · · n!z ⎜n!z n−1 (n − 1)!(n − 1)z n−2 · · · (n − 1)! ⎜ ............................... B=⎜ ⎜ ⎝ n!z (n − 1)! ··· 0 n! 0 ··· 0

⎞ n! 0⎟ ⎟ ⎟. ⎟ 0⎠ 0

Thus we see easily that f0 = f1 = · · · = fn = 0, 

which show that Un is injective. Example 10.17 We consider the weight 12 cusp form Δ(z) = q

∞ 

(1 − q n )24

(10.38)

n=1

for Γ = Γ (1) = SL(2, Z) with q = e2πiz and z ∈ H. (i) Using (10.20), we obtain the vector-valued modular forms !14 (Γ (1), ρ2 ), V14,2,0 (Δ) = Δ(2) v 2 + 13Δ(1) v 2 + 78Δ v2 ∈ M (1)

V12,2,1 (Δ) = Δ(1) v 2 +

(1) 12Δ v2

(2)

!12 (Γ (1), ρ2 ), ∈M

!10 (Γ (1), ρ2 ) v2 ∈ M V10,2,2 (Δ) = Δ (1)

with v 2 , v 2

(2)

and v 2

being vector-valued functions given by ⎛ 2⎞ ⎛ ⎞ ⎛ ⎞ z 2z 2 (1) (2) v 2 (z) = ⎝ z ⎠ , v 2 (z) = ⎝ 1 ⎠ , v 2 (z) = ⎝0⎠ 1 0 0

(10.39)

206

10 Quasimodular Forms and Vector-valued Modular Forms

for z ∈ H. (ii) For n = 2 the formula (10.33) determines a vector-valued modular form (1) (2) !k (Γ (1), ρ2 ) v2 f0 + v 2 f1 + v 2 f2 ∈ M (10.40) U2 (F ) = 2 corresponding to a quasimodular polynomial 2 (Γ (1)) F (z, X) = f0 (z) + f1 (z)X + f2 (z)X 2 ∈ QPk+2

for k = 10, 12, 14. By comparing (10.39) and (10.40) we obtain the quasimodular polynomials 1 (2) 2 Δ (z) + 13Δ(1) (z)X + 78Δ(z)X 2 ∈ QP16 (Γ (1)), 2 1 2 (Γ (1)) ⊂ QP14 (Γ (1)), (U2−1 ◦ V12,2,1 )(Δ))(z, X) = Δ(1) (z) + 12Δ(z)X ∈ QP14 (U2−1 ◦ V14,2,0 )(Δ))(z, X) =

0 2 (Γ (1)) = M12 (Γ (1)) ⊂ QP12 (Γ (1)). (U2−1 ◦ V10,2,2 )(Δ))(z, X) = Δ(z) ∈ QP12

(iii) We now apply the map (10.27) to the vector-valued modular forms in (10.39). Then a direct computation shows that W14,2 (V14,2,0 (Δ)) = 156Δ, W12,2 (V12,2,1 (Δ)) = 240Δ, W10,2 (V10,2,2 (Δ)) = 90Δ. On the other hand, using (10.30), we obtain C14,2,0 = 156,

C12,2,1 = 240,

which confirms the formula (10.29) in this case.

C10,2,2 = 90,

Chapter 11

Differential Operators on Modular Forms

In this chapter, we assume that Γ is a congruence subgroup of SL(2, Z), and consider a character χ on Γ . Given a quasimodular form φ ∈ QMλm (Γ ) and nonnegative integers α and ν with ν ≤ m, we introduce a linear differential operator of order ν of the form Dφα,ν =

ν  p=0

ξp (z)

dp dz p

k φ, which with coefficients ξp given in terms of the quasimodular forms S determines a complex linear map Dφα,ν : Mα (Γ ) → Mα+λ−2m+2ν (Γ, χ) from modular forms to modular forms with character χ. We then show that Rankin–Cohen brackets of modular forms can be expressed in terms of such operators. As an application, we obtain differential operators associated to certain theta series studied by Dong and Mason in [34]. These results were obtained in [77] and [76].

11.1 Jacobi-like Forms and Quasimodular Forms with Character Given an integer λ, we can consider right actions |λ and |Jλ of SL(2, R) on R and R[[X]] given by (1.16) and (1.17). Then the associated modular forms and Jacobi-like forms of weight λ with character can be described as follows.

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_11

207

208

11 Differential Operators on Modular Forms

Definition 11.1 (i) A modular form of weight λ for Γ with character χ is a holomorphic function f ∈ R satisfying f |λ γ = χ(γ)f for all γ ∈ Γ . (ii) A formal power series Φ(z, X) ∈ R[[X]] is a Jacobi-like form of weight λ for Γ with character χ if it satisfies (Φ |Jλ γ)(z, X) = χ(γ)Φ(z, X) for all z ∈ H and γ ∈ Γ . (iii) A formal power series Φ(z, X) ∈ F[[X]] is a modular series of weight λ for Γ with character χ if it satisfies (Φ |M λ γ)(z, X) = χ(γ)Φ(z, X) for all z ∈ H and γ ∈ Γ . We denote by Mλ (Γ, χ), Jλ (Γ, χ) and Mλ (Γ, χ) the spaces of modular forms, Jacobi-like forms and modular series, respectively, of the type described in the previous definition, and set Jλ (Γ, χ)δ = Jλ (Γ, χ) ∩ R[[X]]δ ,

Jλ (Γ, χ)δ = Mλ (Γ, χ) ∩ R[[X]]δ

with R[[X]]δ being as in (1.2). Theorem 11.2 The automorphisms Λλ,δ , Ξλ,δ : R[[X]]δ → R[[X]]δ in Proposition 1.2 induce isomorphisms Λλ,δ : Mλ (Γ, χ)δ → Jλ (Γ, χ)δ , Ξλ,δ : Jλ (Γ, χ)δ → Mλ (Γ, χ)δ for λ, δ ∈ Z with δ ≥ 0 and λ > −2δ. Proof. This theorem can be proved by slightly modifying the proof of the corresponding results for the case of the trivial character in Proposition 1.10.  Corollary 11.3 If a formal power series Φ(z, X) ∈ R[[X]]δ given by Φ(z, X) =

∞ 

φk (z)X k+δ

(11.1)

k=0

is a Jacobi-like form belonging to Jλ (Γ, χ)δ , then the function hk : H → C given by k  (2k + 2δ + λ − r − 2)! (r) φk−r (−1)r (11.2) hk = r! r=0

11.1 Jacobi-like Forms and Quasimodular Forms with Character

209

is a modular form belonging to M2k+2δ+λ (Γ, χ) for each k ≥ 0. Proof. Given Φ(z, X) ∈ Jλ (Γ, χ)δ and hk as in (11.1) and (11.2), using (1.5), we obtain (Ξλ,δ Φ)(z, X) =

∞ 

(2k + 2δ + λ − 1)hk (z)X k+δ ,

k=0

which is a modular series belonging to Mλ (Γ, χ)δ by Theorem 11.2. Then the corollary follows easily from this as in the case of the trivial character in Theorem 1.11.  Let Rm [X] with m ≥ 0 be as in Section 7.2, and let the right action λ of SL(2, R) on Rm [X] be as in (7.9). Definition 11.4 (i) An element Φ(z, X) ∈ Rm [X] is a quasimodular polynomial with character χ for Γ of weight λ and degree at most m if it satisfies Φ λ γ = χ(γ)Φ for all γ ∈ Γ . (ii) A holomorphic function ψ ∈ R is a quasimodular form for Γ with character χ of weight λ and depth at most m if there are functions ψ0 , ψ1 , . . . , ψm ∈ R satisfying m  ψr (z)K(γ, z)r (11.3) (ψ |λ γ)(z) = χ(γ) r=0

for all z ∈ H and γ ∈ Γ , where K(γ, z) is as in (1.12). We denote by QPλm (Γ, χ) and QMλm (Γ, χ) the spaces of quasimodular polynomials and quasimodular forms, respectively, of the type described in Definition 11.4. If ψ ∈ QMλm (Γ, χ) satisfies (11.3), the functions ψr are uniquely determined and therefore we can consider the map m Qm λ : QMλ (Γ, χ) → Rm [X]

defined by (Qm λ ψ)(z, X) =

m 

ψr (z)X r .

r=0

As in Proposition 7.15, it can be shown that Qm λ determines the complex linear map m m (11.4) Qm λ : QMλ (Γ, χ) → QPλ (Γ, χ). Lemma 11.5 Let Φ(z, X) be a quasimodular polynomial belonging to QPλm (Γ, χ), and let Sr with 0 ≤ r ≤ m be as in (7.15). (i) The image Sr Φ of Φ under the map Sr in (7.16) is a quasimodular m−r (Γ, χ) for 0 ≤ r ≤ m. form belonging to QMλ−2r

210

11 Differential Operators on Modular Forms

(ii) The map Φ → S0 (Φ) determines an isomorphism S0 : QPλm (Γ, χ) → QMλm (Γ, χ),

(11.5)

whose inverse is the map Qm λ in (11.4). (iii) The function given by Λr (Φ) =

r  (−1)j j=0

j!

(m − r + j)!(2r + λ − 2m − j − 2)!(Sm−r+j Φ)(j) (11.6)

is a modular form belonging to Mλ−2m+2r (Γ, χ) for 0 ≤ r ≤ m. Proof. The results in (i) and (ii) can be proved by slightly modifying the proofs of the corresponding results for the trivial character in (7.26) and Proposition 7.15. On the other hand, for the trivial character (iii) follows from Proposition 8.4, and the nontrivial case can be proved by modifying its proof.  r = Sr ◦ Qm = Sr ◦ S−1 with Qm and S0 If 0 ≤ r ≤ m, by setting S 0 λ λ being the isomorphisms in (11.4) and (11.5), we obtain the linear map r : QM m (Γ, χ) → QM m−r (Γ, χ) S λ λ−2r satisfying (φ |λ γ)(z) = χ(γ)

m 

(11.7)

r φ)(z)K(γ, z)r (S

r=0

for φ ∈ QMλm (Γ, χ), γ ∈ Γ and z ∈ H. As in the case where χ is trivial, each coefficient of a Jacobi-like form is a quasimodular form. To be more precise, δ is as in (7.32), there is a complex linear map if Πm δ m : Jλ (Γ, χ)δ → QMλ+2m+2δ (Γ, χ) S0 ◦ Π m

(11.8)

given by δ )Φ = φm (S0 ◦ Πm

for a Jacobi-like form Φ(z, X) =

∞ 

φk (z)X k+δ ∈ Jλ (Γ, χ)δ ,

k=0

which satisfies r ◦ S0 ◦ Π δ )Φ = S r (φm ) = (S m for 0 ≤ r ≤ m.

1 φm−r r!

(11.9)

11.2 Differential Operators on Modular Forms

211

11.2 Differential Operators on Modular Forms Given a quasimodular form φ ∈ QMλm (Γ, χ) and nonnegative integers α and ν with ν ≤ m, we consider the linear differential operator Dφφ,α of order ν on R defined by Dφα,ν =

ν ν   (−1)j (m − ν + j)!(2ν + α + λ − 2m − j − 2)!

p!(j − p)!

p=0 j=p

(11.10)

p m−ν+j φ)(j−p) (z) d × (S dz p

m−ν+j as in (11.7). with z ∈ H and S

Theorem 11.6 The formula (11.10) determines a differential operator Dφα,ν : Mα (Γ ) → Mα+λ−2m+2ν (Γ, χ) carrying modular forms with trivial character to modular forms with character χ.

r φ = φr for 0 ≤ r ≤ m, we have Proof. If f ∈ Mα (Γ ) and S J(γ, z)−α f (γz) = f (z),

J(γ, z)−λ φ(γz) =

m 

χ(γ)φr (z)K(γ, z)r ,

r=0

so that J(γ, z)−α−λ (f φ)(γz) =

m 

χ(γ)f (z)φr (z)K(γ, z)r

r=0

for all γ ∈ Γ and z ∈ H. Thus it follows that m (Γ, χ), f φ ∈ QMα+λ

and the corresponding quasimodular polynomial can be written as Qm α+λ (f φ)(z, X) =

m 

m f (z)φ (z)X  ∈ QPα+λ (Γ, χ).

=0

Using this and (11.6), we see that the function Λν (Qm α+λ (f φ)) given by

212

11 Differential Operators on Modular Forms

Λν (Qm α+λ (f φ)) =

ν  (−1)j j=0

j!

(m − ν + j)!

× (2ν + α + λ − 2m − j − 2)!(f φm−ν+j )(j)   j ν   (−1)j j (m − ν + j)! = p j! j=0 p=0 (j−p)

× (2ν + α + λ − 2m − j − 2)!f (p) φm−ν+j is a modular form belonging to Mα+λ−2m+2ν (Γ, χ). By changing the order of summation we obtain (Dφα,ν f )(z) = Λν (Qm α+λ (f φ))(z) for z ∈ H; hence the theorem follows.



Example 11.7 For ν = 1 and ν = 2 the formula (11.10) determines the differential operators Dξα,1 : Mα (Γ ) → Mα+λ−2m+2 (Γ, χ), Dηα,2 : Mα (Γ ) → Mα+λ−2m+4 (Γ, χ) given by Dφα,1 = (m − 1)!(α + λ − 2m − 1)!   m φ) (z) − m(S m φ)(z) d , m−1 φ)(z) − m(S × (α + λ − 2m)(S dz   2 m!(α + λ − 2m)! d d Dφα,2 = ξ1 (z) 2 + ξ2 (z) + ξ3 (z) , 2 dz dz where m φ, ξ1 = S m−1 φ), m φ) − 2(1 + α + λ − 2m) (S ξ2 = 2(S m m−1 φ) m φ) − 2(1 + α + λ − 2m) (S ξ 3 = (S m 2(2 + α + λ − 2m)(1 + α + λ − 2m) (Sm−2 φ). + m(m − 1) In particular, if ξ ∈ Q1λ (Γ, χ) and η ∈ Q2λ (Γ, χ), we obtain the operators Dφξ,1 , Dφη,2 : Mα (Γ ) → Mα+λ (Γ, χ) given by

11.2 Differential Operators on Modular Forms

213

  1 φ) (z) − (S 1 ξ)(z) d , (11.11) Dξα,1 = (α + λ − 3)! (α + λ − 2)ξ(z) − (S dz 2 d 2 η) Dηα,2 = (α + λ − 4)!(S (11.12) dz 2 d  1 η) 2 η) − (α + λ − 3)!(S + 2(α + λ − 4)!(S dz 2 η) − (α + λ − 3)!(S 1 η) + (α + λ − 4)!(S 0 η). + (α + λ − 2)!(S We now want to relate the operator in (11.10) with Rankin–Cohen brackets on modular forms. We recall that the Rankin–Cohen bracket of order ν ≥ 1 associated to nonnegative integers μ and α is the bilinear map [ , ]μ,α : Mμ (Γ ) × Mα (Γ ) → Mμ+α+2ν (Γ ) ν

(11.13)

given by = [f, g]μ,α ν

ν 

  μ + ν − 1 α + ν − 1 (j) (ν−j) f g j ν−j

 (−1)j

j=0

(11.14)

for f ∈ Mμ (Γ ) and g ∈ Mα (Γ ) (see e.g. [31]). It is known that this Rankin– Cohen bracket is a unique bilinear differential operator Mμ (Γ ) × Mα (Γ ) → Mμ+α+2ν (Γ ) up to constant multiples. From Example 7.6 and Lemma 7.7 we see that the ν-th derivative f (ν) of a ν (Γ ). modular form f ∈ Mμ (Γ ) is a quasimodular form belonging to QMμ+2ν The next theorem provides a relation between the differential operators (11.10) associated to the quasimodular form f (ν) and the Rankin–Cohen brackets of order ν described above. Theorem 11.8 Given modular forms f ∈ Mμ (Γ ) and g ∈ Mα (Γ ), we have = [f, g]μ,α ν

(−1)ν Dα,ν g ν!(α + μ + ν − 2)! f (ν)

for each positive integer ν. Proof. For f ∈ Mμ (Γ ), g ∈ Mα (Γ ) and ν ≥ 1, using (11.10) with m = ν and λ = μ + 2ν, we obtain Dfα,ν (ν) g =

ν ν   (−1)j j!(α + μ + 2ν − j − 2)! p=0 j=p

p!(j − p)!

1 (Γ ) with We note that f  belongs to QMμ+2

j f (ν) )(j−p) g (p) . (S

(11.15)

214

11 Differential Operators on Modular Forms

0 (f  ) = f  , S

1 (f  ) = μf. S

(11.16)

n+1 (Γ ) with On the other hand, if η ∈ QMβn (Γ ), then η  ∈ QMβ+2

0 η = η , S

n+1 η  = (β − n)S n η, S

(11.17)

k η  = (β − k + 1)(S k−1 η) + (S k η) S j (f (ν) ) is a linear combination for 1 ≤ k ≤ n. Hence by induction we see that S of derivatives of f for 0 ≤ j ≤ ν. From this and (11.15) it follows that the map (f, g) → Dfα,ν (ν) g is a bilinear differential operator Mμ (Γ ) × Mα (Γ ) → Mμ+α+2ν (Γ ), and therefore it is a constant multiple of the map (11.13) by the uniqueness of Rankin–Cohen brackets. Using (11.16), (11.17) and induction, we have   ν f (ν) = μ(μ + 1) · · · (μ + ν − 1)f = μ + ν − 1 (ν!)f, S ν so that the coefficient of f g (ν) in the sum in (11.15) is equal to   (−1)ν ν!(α + μ + 2ν − ν − 2)! μ + ν − 1 (ν!) ν ν!   μ+ν−1 ν = (−1) ν!(α + μ + ν − 2)! . ν Since the coefficient of f g (ν) in (11.14) is   μ+ν−1 , ν the theorem follows by comparing these coefficients.



Example 11.9 Given f ∈ Mμ (Γ ) and g ∈ Mα (Γ ), using ξ = f  ∈ 1 0ξ = f , S 1 ξ = μf and λ = μ + 2, we have QMμ+2 (Γ ) in (11.11) with S μ,α   . Dfα,1  g = (α + μ − 1)!(αf g − μf g ) = −(α + μ − 1)![f, g]1 2 0 η = f  , (Γ ) in (11.12) with S On the other hand, using η = f  ∈ QMμ+4 1 η = 2(μ + 1)f  , S 2 η = μ(μ + 1)f and λ = μ + 4, we see that S    − 2(μ + 1)(α + 1)f  g  + α(α + 1)f  g Dfα,2  g = (α + μ)! μ(μ + 1)f g

= 2(α + μ)![f, g]μ,α 2 .

11.3 Theta Functions

215

11.3 Theta Functions In [34] Dong and Mason considered a certain family of theta functions associated to symmetric positive definite matrices. In this section we construct modular forms by modifying the formulas for such theta series (cf. [78]). We fix a positive integer σ, an element v of C2σ regarded as a column vector, and a symmetric positive definite integral 2σ × 2σ matrix A whose diagonal entries are even. For each nonnegative integer k, we consider the theta function θk : H → C defined by  t θk (z) = (v t A)k eπi( A)z (11.18) ∈Z2σ

for all z ∈ H, which was studied by Dong and Mason in [34]. Let N be the smallest positive integer such that N (2A)−1 is an integral matrix with even diagonal entries, and let Γ0 (N ) ⊂ SL(2, Z) be the associated congruence subgroup given by  )  *  ab Γ0 (N ) = ∈ SL(2, Z)  c ≡ 0 (mod N ) . (11.19) cd We consider a function ε defined on the set of nonzero integers defined by ε(n) =

 (−1)σ det A  n

for n > 0 and ε(n) = (−1)σ for n < 0, where ( character

• •

 (−1)σ det A  −n

) is the Jacobi symbol. We then define the associated χ : Γ0 (N ) → C×

on Γ0 (N ) by setting χ

  ab = ε(d) cd

(11.20)

  for ac db ∈ Γ0 (N ). Then it is known that the theta series θk in (11.18) for k = 0 is a modular form belonging to Mσ (Γ0 (N ), χ) (cf. [106]). We now modify the formula (11.18) by setting  t ϑk (z) = ξk eπi( A)z ∈Z2σ

for k ≥ 1, where

(11.21)

216

11 Differential Operators on Modular Forms

ξk =

k  (−1)r (2k + σ − r − 2)! r=0

4r r!(2k − 2r)!

(v t A)2k−2r (t A)r (v t Av)r .

Theorem 11.10 The function ϑk : H → C given by (11.21) is a modular form belonging to M2k+σ (Γ0 (N ), χ) for each positive integer k. Proof. We define the formal power series Θ(z, X) ∈ F[[X]] associated to the sequence {θn }∞ n=0 of theta functions in (11.18) by ∞  2n (2πi)n Θ(z, X) = θ2n (z)X n (2n)! n=0

(11.22)

for all z ∈ H. If Γ0 (N ) is as in (11.19) and χ : Γ0 (N ) → C× is the character given by (11.20), then in [34] Dong and Mason proved that the power series (11.22) satisfies the transformation property Θ(γz, J(γ, z)−2 X) = χ(γ)J(γ, z)σ exp[v t AvK(γ, z)X] · Θ(z, X)

(11.23)

for all z ∈ H and γ ∈ Γ0 (N ). Thus we see easily that the formal power series ϑ(z, X) = Θ(z, X/(v t Av)) =

∞ 

(4πi)n θ (z)X n t Av)n 2n (2n)!(v n=0

(11.24)

is a Jacobi-like form belonging to Jσ (Γ0 (N ), χ)0 . If we set ψn =

(4πi)n θ2n (2n)!(v t Av)n

(11.25)

for n ≥ 0, then by Corollary 11.3 the function hk =

k 

(−1)r

r=0

(2k + σ − r − 2)! (r) ψk−r r!

is a modular form belonging to M2k+σ (Γ0 (N ), χ) for k ≥ 1. Using (11.18) and (11.25), we have (r)

(4πi)k−r (r) θ (z) (2k − 2r)!(v t Av)k−r 2k−2r  t (4πi)k−r = (v t A)2k−2r (πi)r (t A)r eπi( A)z t k−r (2k − 2r)!(v Av) 2σ

ψk−r (z) =

∈Z

 t (πi) 2 (v t A)2k−2r (t A)r eπi( A)z t k−r (2k − 2r)!(v Av) 2σ 2k−2r

=

k

∈Z

for 0 ≤ r ≤ k. From this and (11.21) we obtain

11.3 Theta Functions

217

hk (z) = (πi)k

k   (−1)r 22k−2r (2k + σ − r − 2)! r!(2k − 2r)!(v t Av)k−r 2σ r=0

∈Z

× (v t A)2k−2r (t A)r eπi(

∈Z2σ r=0

22r r!(2k − 2r)! ×

=

A)z

k   (−1)r (2k + σ − r − 2)!

= (4πi)k



t

4πi v t Av

k

(v t A)2k−2r (t A)r πi(t A)z e (v t Av)k−r

ϑk (z)

for z ∈ H. Hence it follows that ϑk is a modular form belonging to  M2k+σ (Γ0 (N ), χ), and the proof of the theorem is complete. Example 11.11 Using (11.8) for k = 1, 2, 3, we obtain modular forms f1 ∈ Mσ+2 (Γ0 (N ), χ),

f2 ∈ Mσ+4 (Γ0 (N ), χ),

f3 ∈ Mσ+6 (Γ0 (N ), χ)

given by 2 ϑ1 (z) σ!    t 1 t ( A)(v t Av) eπi( A)z (v t A)2 − = 2σ 2σ

f1 (z) =

∈Z

4! ϑ2 (z) (σ + 2)!   (v t A)4 − =

f2 (z) =

∈Z2σ

+ 6! ϑ3 (z) (σ + 4)!   (v t A)6 − =

3 (v t A)2 (t A)(v t Av) σ+2  t 3 (t A)2 (v t Av)2 eπi( A)z 4(σ + 2)(σ + 1)

f3 (z) =

∈Z2σ

15 (v t A)4 (t A)(v t Av) 2(σ + 4)

45 (v t A)2 (t A)2 (v t Av)2 4(σ + 4)(σ + 3)  t 15 (t A)3 (v t Av)3 eπi( A)z − 8(σ + 4)(σ + 3)(σ + 2)

+

1 for z ∈ H. In particular,   if we consider the case for σ = 1, v = ( 1 ) and A = ( 20 02 ), using  = 12 ∈ Z2 , we have

218

11 Differential Operators on Modular Forms

t A = 2(21 + 22 ),

v t Av = 4,

v t A = 2(1 + 2 ).

Thus we obtain modular forms g1 ∈ M3 (Γ0 (N ), χ),

g2 ∈ M5 (Γ0 (N ), χ),

g3 ∈ M7 (Γ0 (N ), χ)

given by g1 (z) =



1 2 eπi(

t

A)z

,

∈Z2σ

g2 (z) =



(421 22 − (41 + 42 ))eπi(

t

A)z

,

∈Z2σ

g3 (z) =



4(1 + 2 )4 (21 22 − 21 − 22 )

∈Z2σ

 t + (21 + 22 )2 (91 2 − 421 − 822 ) eπi( A)z

for all z ∈ H.

11.4 Differential Operators Associated to Theta Functions In this section we construct differential operators on modular forms associated to the functions θk in (11.18) by using the fact that they are quasimodular forms. Let ϑ(z, X) ∈ Jσ (Γ0 (N ), χ)0 be the Jacobi-like form in (11.24), and let 0 m : Jσ (Γ0 (N ), χ)0 → QPσ+2m (Γ0 (N ), χ) Πm

be the extension of the map in (7.32) to the case of a nontrivial character. Then we see that the polynomial 0 ϑ(z, X) = Πm

m  r=0

(4πi)m−r θ2m−2r (z)X r r!(2m − 2r)!(v t Av)m−r

m is a quasimodular polynomial belonging to QPσ+2m (Γ0 (N ), χ). Thus, if we set = (S0 ◦ Π 0 )ϑ(z, X), ϑ(z) m

we have ϑ = with

(4πi)m m θ2m ∈ QMσ+2m (Γ0 (N ), χ) (2m)!(v t Av)m

11.4 Differential Operators Associated to Theta Functions

r ϑ = S

219

(4πi)m−r θ2m−2r r!(2m − 2r)!(v t Av)m−r

for 0 ≤ r ≤ m, where we used (11.9). From this, (11.10) and Theorem 11.6 we obtain the map Dα,ν  : Mα (Γ0 (N )) → Mα+σ+2ν (Γ0 (N ), χ) ϑ

given by Dα,ν  =

ν ν   (−1)j (2ν + α + σ − j − 2)!(4πi)ν−j

p!(j − p)!(2ν −

ϑ

p=0 j=p

2j)!(v t Av)ν−j

(j−p)

θ2ν−2j (z)

dp . (11.26) dz p

Example 11.12 Using (11.26) for ν = 1, 2, we obtain the operators Dα,1  : Mα (Γ0 (N )) → Mα+σ+2 (Γ0 (N ), χ), ϑ

Dα,2  : Mα (Γ0 (N )) → Mα+σ+4 (Γ0 (N ), χ) ϑ

given by   d 1 2πi Lα,1 = θ0 + θ0 − (α + σ) t θ2 , (α + σ − 1)! dz v Av 2 Lα,2 = (α + σ)!   4πi   d d2  θ2 = θ0 2 + 2θ0 − (α + σ + 1) t dz v Av dz   4πi  θ + θ0 − (α + σ + 1) t v Av 2  4πi 2  1 + (α + σ + 2)(α + σ + 1) t θ4 . 12 v Av

Dα,1  =− ϑ

Dα,2  ϑ

We next consider differential operators carrying modular forms to modular forms with trivial character. By (11.7) there is a complex linear map k : QP m (Γ ) → QM m−k (Γ ) S λ λ−2k from quasimodular polynomials to quasimodular forms for 0 ≤ k ≤ m. In m η is a modular form belonging to Mλ−2m (Γ ) = QM 0 particular, S λ−2m (Γ ) for each η ∈ QPλm (Γ ). Lemma 11.13 Given quasimodular forms φ ∈ QMλm (Γ ), the function

ψ ∈ QMμm (Γ ),

220

11 Differential Operators on Modular Forms

m ψ)(S m−1 φ) − (S m φ)(S m−1 ψ) (S is a modular form belonging to Mλ+μ−4m+2 (Γ ). m ψ are modular forms with m φ and S Proof. Since S m φ ∈ Mλ−2m (Γ ), S

m ψ ∈ Mμ−2m (Γ ), S

we see that m φ)ψ) |λ+μ−2m γ)(z) m ψ)φ − (S (((S m ψ)(γz)φ(γz) − (S m φ)(γz)ψ(γz)) = J(γ, z)−λ−μ+2m ((S m ψ)(z)(φ |λ γ)(z) − (S m φ)(z)(ψ |μ γ)(z) = (S m−1   r φ)(z) − (S m φ)(z)(S r ψ)(z) K(γ, z)r m ψ)(z)(S = (S r=0

for all γ ∈ Γ and z ∈ H. In particular, m φ)ψ m ψ)φ − (S (S m−1 (Γ ); hence it follows that is a quasimodular form belonging to QMλ+μ−2m

m ψ)φ − (S m φ)ψ) = (S m ψ)(S m−1 φ) − (S m φ)(S m−1 ψ) m−1 ((S S is an element of Mλ+μ−4m+2 (Γ ). We now consider the theta function given by (11.18) for k = 0, 2 associated to the vector v ∈ C2σ and the symmetric integral matrix A with v t Av = 0, and set d 2πiα − t θ0 θ2 (11.27) Lα = θ02 dz v Av for a nonnegative integer α. Theorem 11.14 The differential operator given by (11.27) determines the complex linear map Lα : Mα (Γ0 (N )) → Mα+2σ+2 (Γ0 (N )) for each α ≥ 0. Proof. By taking the square of Θ(z, X) in (11.22) and using (11.23), it can be shown easily that the formal power series ϑ(z, X) ∈ R[[X]] given by ϑ(z, X) =

n  ∞   2πi n θ2 (z)θ2n−2 (z) n X v t Av (2)!(2n − 2)! n=0 =0

is a Jacobi-like form belonging to J2σ (Γ0 (N ))0 . Thus, if we set

11.4 Differential Operators Associated to Theta Functions

ξn =

221

n   2πi n θ2 θ2n−2 v t Av (2)!(2n − 2)! =0

for n ≥ 0 and use the linear map (11.8) for λ = 2σ and δ = 0, we see that 0 m ϑ(z, X) = ξm ∈ QM2σ+2m (Γ0 (N )) Πm

with  m−r m−r  θ2 θ2m−2r−2 r (ξm ) = 1 ξm−r = 1 2πi S t r! r! v Av (2)!(2m − 2r − 2)! =0

for 0 ≤ r ≤ m. In particular, we have m (ξm ) = 1 θ2 , S m! 0

(11.28)

 2πi   θ θ 1 2 2−2 (m − 1)! v t Av (2)!(2 − 2)! =0  2πi  1 θ0 θ 2 . = (m − 1)! v t Av 1

m−1 (ξm ) = S

(11.29)

We now consider a modular form f ∈ Mα (Γ0 (N )), satisfying f (γz) = J(γ, z)α f (z) for all γ ∈ Γ0 (N ) and z ∈ H. By taking its derivative with respect to z and using the identity d J(γ, z) = J(γ, z)K(γ, z), dz we obtain J(γ, z)−2 f  (γz) = αJ(γ, z)α K(γ, z)f (z) + J(γ, z)α f  (z). Thus, if we set η=

f , f

we have J(γ, z)2 (αJ(γ, z)α K(γ, z)f (z) + J(γ, z)α f  (z)) J(γ, z)α f (z)    f (z) + αK(γ, z) = J(γ, z)2 f (z)

η(γz) =

222

11 Differential Operators on Modular Forms

 = J(γ, z)

2

 η(z) + αK(γ, z)

for all γ ∈ Γ0 (N ) and z ∈ H; hence we see that (η |2 γ)(z) = η(z) + αK(γ, z). Thus the m-th power η m of η for a positive integer m satisfies (η

m

|2m

   m−r f (z) m γ)(z) = α K(γ, z)r r f (z) r=0 m 

r

for all γ ∈ Γ0 (N ) and z ∈ H. In particular, η m is a quasimodular form m (Γ0 (N )) with belonging to QM2m m (η m ) = αm , S

m−1 (η m ) = mαm−1 f  /f. S

From this, (11.28) and (11.29) we obtain m−1 (η m ) − S m (η m )S m−1 (ξm ) m (ξm )S S    2πi  f 1 θ02 mαm−1 − m t θ0 θ 2 α m = m! f v Av   m−1 α 2πiα = θ02 f  − t θ0 θ2 f , (m − 1)!f v Av which belongs to Mα+2σ+2 (Γ0 (N )) by Lemma 11.13; hence it follows that  Lα ∈ Mα+2σ+2 (Γ0 (N )).

Chapter 12

Half-integral Weight Forms

A Shimura correspondence is a Hecke equivariant map from half-integral to integral weight modular forms, and a Shintani lifting provides a similar map from integral to half-integral weight forms. In this chapter we extend the notion of quasimodular and Jacobi-like forms to the cases of half-integral weights and study quasimodular analogs of the Shimura and Shintani correspondences.

12.1 Quasimodular Forms of Half-integral Weight In this section we modify the definitions of Jacobi-like forms and quasimodular forms to include the forms of half-integral weights. We also describe Hecke operators on spaces of such forms. We fix a positive integer N and assume that Γ = Γ0 (4N ) is the congruence subgroup of SL(2, Z) of level 4N using the notation in (11.19). Let θ(z) be the theta series given by θ(z) =

∞ 

e2πin

2

z

n=−∞

for z ∈ H, and set d z) = 2 K(γ, J(γ, z)−1 J(γ, z) dz   for all γ ∈ Γ and z ∈ H. If γ = ac db ∈ Γ , it is known that θ(γz) , J(γ, z) = θ(z)

(12.1)

 −1  (cz + d), J(γ, z)2 = d

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_12

223

224

12 Half-integral Weight Forms

where ( −1 d ) is the Legendre symbol (see e.g. [57]). Furthermore, the resulting : Γ × H → C can be shown to satisfy maps J, K J(γ, γ  z) J(γ  , z), J(γγ  , z) =

(12.2)

  , z) + γ  z) , z) = K(γ J(γ  , z)−2 K(γ, K(γγ

(12.3)

for all γ, γ  ∈ Γ and z ∈ H. We recall that R denotes the ring of holomorphic functions on H and that R[[X]] is the complex algebra of formal power series in X with coefficients in R. If m is a nonnegative integer, we also note that Rm [X] is the complex vector space of polynomials in X over R of degree at most m. In the remainder of this section we fix a half integer λ, so that 2λ is an odd integer. Given elements f ∈ R, Φ(z, X) ∈ R[[X]], F (z, X) ∈ Rm [X], and γ ∈ Γ , we set (f |λ γ)(z) = J(γ, z)−2λ f (z),

(12.4)

 (Φ |Jλ γ)(z, X) = J(γ, z)−2λ e−K(γ,z)X Φ(γz, J(γ, z)−2 X),

(12.5)

J(γ, z) (F λ γ)(z, X) =

(12.6)

−2λ

z))) F (γz, J(γ, z) (X − K(γ, 2

for all z ∈ H. If γ  is another element of Γ , then by using (12.2) and (12.3) it can be shown that f |λ (γγ  ) = (f |λ γ) |λ γ  , Φ |Jλ (γγ  ) = (Φ |Jλ γ) |Jλ γ  , F λ (γγ  ) = (F λ γ) λ γ  so that the operations |λ , |Jλ and λ determine right actions of Γ on R, R[[X]] and Rm [X], respectively. Let χ be a Dirichlet character of level 4N , that is, an arithmetic function χ : Z → C which factors through a homomorphism (Z/4N Z)× → C. The same notation will be used to denote the associated character χ : Γ → C× of Γ given by for γ =

a b c d

χ(γ) = χ(d) ∈ Γ.

Definition 12.1 Given a half integer λ and a nonnegative integer m, an element f ∈ R is a quasimodular form for Γ of weight λ and depth at most m with character χ if there are functions f0 , . . . , fm ∈ R such that (f |λ γ)(z) = χ(γ)

m  r=0

z)r fr (z)K(γ,

(12.7)

12.1 Quasimodular Forms of Half-integral Weight

225

z) is as in (12.1) and |λ is the operation for all z ∈ H and γ ∈ Γ , where K(γ, m in (12.4). We denote by QMλ (Γ, χ) the space of quasimodular forms for Γ of weight λ and depth at most m with character χ. If we denote by Mλ (Γ, χ) the space of modular forms for Γ of weight λ and character χ, then we see that QMλ0 (Γ, χ) = Mλ (Γ, χ). We also note that f = f0 if f satisfies (12.7). Definition 12.2 (i) A quasimodular polynomial for Γ of weight λ and degree at most m is an element of Rm [X] that is Γ -invariant with respect to the right Γ -action in (12.6). (ii) A formal power series Φ(z, X) belonging to R[[X]] is a Jacobi-like form for Γ of weight λ with character χ if it satisfies (Φ |Jλ γ)(z, X) = χ(γ)Φ(z, X) for all z ∈ H and γ ∈ Γ , where |Jλ is as in (12.5). We denote by QPλm (Γ, χ) the space of all quasimodular polynomials for Γ of weight λ and degree at most m with character χ. We also denote by Jλ (Γ, χ) the space of all Jacobi-like forms for Γ of weight λ with character χ, and set Jλ (Γ, χ)δ = Jλ (Γ, χ) ∩ R[[X]]δ for each nonnegative integer δ. Let f ∈ R be a quasimodular form belonging to QMλm (Γ, χ) and satisfying (12.7). Then we define the corresponding polynomial (Qm λ f )(z, X) ∈ Rm [X] by m  f )(z, X) = fr (z)X r , (12.8) (Qm λ r=0

so that we obtain the linear map m Qm λ : QMλ (Γ, χ) → Rm [X]

for each nonnegative integer m. Then it induces the isomorphism m m Qm λ : QMλ (Γ, χ) → QPλ (Γ, χ),

which can be easily shown by modifying the proof of Proposition 7.15. We now extend the notion of Hecke operators to the half-integral cases. We first recall that GL+ (2, R) acts on H by linear fractional transformations, and set G = {(α, det(α)−1/2 J(α, z)) | α ∈ GL+ (2, R), z ∈ H}.

226

12 Half-integral Weight Forms

Then G is a group with respect to the multiplication given by (α, det(α)−1/2 J(α, z)) · (β, det(β)−1/2 J(β, z)) = (αβ, det(αβ)−1/2 J(α, β(z)) J(β, z)). We shall write

J(α, z)) ∈ G. α = (α, det(α)−1/2

Given λ with 2λ ∈ Z odd, we extend the actions of SL(2, R) in (1.16), (1.17) and (7.9) to those of G by setting )(z) = det(α)λ/2−1 J(α, z)−2λ f (αz), (f |λ α 

(Φ |Jλ α )(z, X) = (det α)λ/2−1 J(α, z)−2λ e−K(α,z) × Φ(αz, (det α) J(α, z)−2 X),

(12.9)

(F λ α )(z, X) = (det α)λ/2−1 J(α, z)−2λ × F (αz, (det α)−1 z))) J(α, z)2 (X − K(α, for all z ∈ H, α ∈ G, f ∈ R, Φ(z, X) ∈ R[[X]] and F (z, X) ∈ Rm [X]. Then we have ) |λ α  = f |λ ( αα  ), (f |λ α ) |Jλ α  = Φ |Jλ ( αα  ), (Φ |Jλ α ) λ α  = F λ ( αα  ) (F λ α for all α , α  ∈ G, and therefore G acts on R, R[[X]] and Rm [X] on the right. Given a discrete subgroup Γ of SL(2, R), we set Γ = {(α, J(α, z)) | α ∈ Γ, z ∈ H}. Let Γ be the commensurator of Γ , and consider the subgroup GΓ of G defined by GΓ = {(α, det(α)−1/2 J(α, z)) | α ∈ Γ, z ∈ H}. If α ∈ GΓ , the corresponding double coset has a decomposition of the form Γ α Γ =

s 

Γα i

(12.10)

i=1

∈ GΓ ⊂ G be an element for some elements α i ∈ G with 1 ≤ i ≤ s. Let α whose double coset is as in (12.10). Then the associated Hecke operator

12.2 Shimura Correspondences

227

Tλ ( α) : Mλ (Γ, χ) → Mλ (Γ, χ) is given by α)f )(z) = det(α)−λ/2−1 (Tλ (

s 

(12.11)

(f |λ α i )(z)

i=1

for all f ∈ Mλ (Γ, χ) and z ∈ H, where α and αi are elements of Γ corresponding to α and α i , respectively. Similarly, given a Jacobi-like form Φ(z, X) ∈ Jλ (Γ, χ) and a quasimodular polynomial F (z, X) ∈ QPλm (Γ, χ), we set (TλJ ( α)Φ)(z, X) = det(α)−λ/2−1 (TλP ( α)F )(z, X) = det(α)−λ/2−1

s  i=1 s 

(Φ |Jλ α i )(z, X),

(12.12)

(F λ α i )(z, X)

(12.13)

i=1

for all z ∈ H. α)Φ)(z, X) Proposition 12.3 For each α ∈ GΓ the formal power series (TλJ ( P α)F )(z, X) given by (12.12) and (12.13), respecand the polynomial (Tλ ( tively, are independent of the choice of the coset representatives α1 , . . . , αs , α)Φ and F → TλP ( α)F determine linear endomorand the maps Φ → TλJ ( phisms α) : Jλ (Γ, χ) → Jλ (Γ, χ), TλJ (

TλP ( α) : QPλm (Γ, χ) → QPλm (Γ, χ). (12.14)

Proof. This can be proved as in the case of the usual Hecke operators for modular forms.  α) and TλP ( α) are Hecke operators on Jλ (Γ, χ) The endomorphisms TλJ ( m and QPλ (Γ, χ), respectively, for half-integral weight Jacobi-like forms and quasimodular polynomials with character.

12.2 Shimura Correspondences In the classical theory of modular forms, a Shimura correspondence is a map from half-integral weight modular forms to integral weight modular forms that is Hecke equivariant. In this section we introduce a similar map for quasimodular forms. First, we review Shimura’s construction of a Hecke-equivariant map from half integral weight cusp forms to integral weight cusp forms (see [109]). Given a Dirichlet character χ : Z → C of level N , the associated L-function is given by

228

12 Half-integral Weight Forms

L(s, χ) =

∞  χ(m) . ms m=1

If λ is a half integer λ with 2λ odd, let Sλ (Γ0 (N ), χ) be the space of cusp forms for Γ0 (N ) of weight λ with character χ. We consider an element f ∈ Sλ (Γ0 (N ), χ), so that it satisfies (f |λ γ )(z) = det(γ)λ/2−1 J(γ, z)−2λ f (γz) = χ(γ)f (z) for all γ ∈ Γ0 (N ) and z ∈ H. The following theorem is the main result obtained by Shimura in [109]. Theorem 12.4 Suppose that g(z) =

∞ 

bk+1/2 (g, n)q n ∈ Sk+1/2 (Γ0 (4N ), χ)

n=1

with q = e2πiz is a half-integral weight cusp form with k ∈ N. Let t be a positive square-free integer, and define the Dirichlet character ψt by ψt (n) = χ(n)

 −1 k  t  n

n

.

If the complex numbers At (n) are defined by ∞ ∞   bk+1/2 (g, n2 t) At (n) = L(s − k + 1, ψ ) , t s n ns n=1 n=1

then the function St,k (g) given by (St,k (g))(z) =

∞ 

At (n)q n

n=1

is a weight 2k modular form belonging to M2k (Γ0 (2N ), χ2 ). If k ≥ 2 then St,k (g) is a cusp form. Furthermore, for k = 1, St,1 (g) is a cusp form if g is in the orthogonal complement of the subspace of S3/2 (Γ0 (4N ), χ) spanned by single variable theta functions. From Theorem 12.4 we obtain the Hecke equivariant Shimura map Shλ,t,χ : Sλ (Γ0 (4N ), χ) → M2λ−1 (Γ0 (2N ), χ2 ) defined by (Shλ,t,χ (g))(z) =

∞  n=1

for

At (n)q n

12.2 Shimura Correspondences

229

g(z) =

∞ 

bλ (g, n)q n ,

n=1

where At (n) = L(s − λ + 3/2, ψt )bλ (f, n2 t) for all n ≥ 1. In order to discuss the quasimodular analog of the Shimura map, given m, r ∈ Z with m ≥ r ≥ 0, we consider a quasimodular polynomial F (z, X) ∈ m−r (Γ0 (4N ), χ) of the form QPλ+2m F (z, X) =

m−r 

fu (z)X u

(12.15)

u=0

for z ∈ H, and set  ,r (Q Shm,m λ,t,χ

F )(z, X) =

 m −r 

=0

Γ (m − r + 1)Γ (2λ + 4r) Γ ( + 1)Γ (m − r −  + 1)

(12.16) 

×

(Shλ+2r,t,χ (fm−r ))(m −r−) (z) r X , Γ (m −  + 2λ + 3r)

which is an element of Rm −r [X] with m ≥ r ≥ 0. Proposition 12.5 The formula (12.16) determines a Hecke equivariant linear map m,m ,r Q Shλ,t,χ



m−r m −r 2 : QPλ+2m (Γ0 (4N ), χ) → QP2(λ+m  +r)−1 (Γ0 (2N ), χ )

for m, m ≥ 0 and 0 ≤ r ≤ min{m, m }. Proof. We first note that the map Sm in (7.26) can be extended to the halfintegral case, so that Sm (QPλm (Γ, χ)) ⊂ Mλ−2m (Γ, χ). The lifting formula (9.2) can also be modified so that for each h ∈ Mμ (Γ, χ) with μ ∈ 12 Z the polynomial (Q Aμm h)(z, X) =

m  =0

Γ (m + 1)Γ (μ)h(m−) (z) X Γ ( + 1)Γ (m −  + 1)Γ (m −  + μ)

(12.17)

m (Γ, χ) and satisfies is a quasimodular polynomial belonging to QPμ+2m

Sm (Q Aμm h) = h.

230

12 Half-integral Weight Forms

m−r We now consider a quasimodular polynomial F (z, X) ∈ QPλ+2m (Γ0 (4N ), χ) given by (12.15), so that

Sm−r (F ) ∈ Mλ+2r (Γ0 (4N ), χ), Shλ+2r,t,χ (Sm−r F ) ∈ M2(λ+2r)−1 (Γ0 (2N ), χ2 ). Thus, using (12.16) and (12.17), we obtain 2(λ+2r)−1 (Shλ+2r,t,χ (Sm−r F )) Q Am −r

(12.18)

(m −r−) m −r  Γ (m − r + 1)Γ (2λ + 4r − 1) Shλ+2r,t,χ (Sm−r F ))  X Γ ( + 1)Γ (m − r −  + 1)Γ (m −  + 2λ + 3r) =0 

=



,r = Q Shm,m λ,t,χ (F );

hence it follows that m,m ,r Q Shλ,t,χ (F )



m −r 2 ∈ QP2(λ+m  +r)−1 (Γ0 (2N ), χ ). 2(λ+2r)−1

are Hecke equivariant On the other hand, the maps Sm−r and Q Am −r by the commutativity of the diagrams (7.42) and (9.9), respectively; hence  ,r follows from the same property for the the Hecke equivariance of Q Shm,m λ,t,χ  Shimura map Shλ+2r,t,χ of modular forms. From (12.18) we obtain the relation m,m ,r Q Shλ,t,χ

2(λ+2r)−1

= Q Am −r

◦ Shλ+2r,t,χ ◦Sm−r

and therefore the Hecke equivariant commutative diagram m,m ,r Q Shλ,t,χ



m−r m −r  (Γ0 (4N ), χ) −−−−−−−→ QP2(λ+r+m QPλ+2m  )−1 (Γ0 (2N ), χ ) ⏐  ⏐ ⏐ A2(λ+2r)−1 Sm−r  ⏐Q m −r

Sλ+2r (Γ0 (4N ), χ)

Shλ+2r,t,χ

−−−−−−−→

S2(λ+2r)−1 (Γ0 (2N ), χ2 )

for each r ∈ {0, . . . , m}. Given a quasimodular polynomial m (Γ0 (4N ), χ), G(z, X) ∈ QPλ+2m

for each r ∈ {0, 1, . . . , m} we consider the polynomial m−r (Γ0 (4N ), χ) Gr (z, X) ∈ QPλ+2m

defined recursively by G0 = G and

12.3 Shintani Liftings

231

G+1 = G − (m − )!(Πm− ◦ L 0λ,0 ◦ Sm− )G for 0 ≤  ≤ m − 1, where L 0λ,0 is as in (9.7). Then Gr (z, X) is a quasimodular m−r polynomial belonging to QPλ+2m (Γ0 (4N ), χ). For m ≥ m, if we set m

m,m ,r Q Shλ,t,χ (G)







,0 m,m ,1 m,m ,m = (Q Shm,m Gm ) λ,t,χ G0 , Q Shλ,t,χ G1 , . . . , Q Shλ,t,χ

r=0

we obtain the complex linear map m

m,m ,r Q Shλ,t,χ

m

r=0



m −r 2 QP2(λ+r+m  )−1 (Γ0 (2N ), χ ),

m : QPλ+2m (Γ0 (4N ), χ) → r=0

which is Hecke equivariant.

12.3 Shintani Liftings It was Shintani (cf. [111]) who constructed Hecke equivariant maps from integral weight λ cusp forms to half-integral weight cusp forms, which may be regarded as adjoints of Shimura maps with respect to the Petersson inner product. In this section we study the quasimodular version of Shintani maps. We first review the construction of Shintani maps for modular forms. Let Q be the space of integral indefinite binary quadratic forms of the form Q = Q(X, Y ) = aX 2 + bXY + cY 2 = [a, b, c] with Disc(Q) = b2 − 4ac > 0. Given a positive integer M , we set QM = {Q(X, Y ) = [a, b, c] ∈ Q | (a, M ) = 1, b ≡ c ≡ 0 (mod M )} if M is odd, and QM = {Q(X, Y ) = [a, b, c] ∈ Q | (a, M ) = 1, b ≡ 0 (mod 2M ), c ≡ 0 (mod M )} if M is even. Then the congruence subgroup Γ0 (M ) acts on QM on the left by (γ · Q)(X, Y ) = Q((X, Y )γ −1 ) for all γ ∈ Γ0 (M ) and Q ∈ Q. Following [111], to each integral indefinite binary form Q ∈ QM we associate a pair of points ωQ , ω  Q ∈ P1 (R) = R ∪ {i∞} given by

232

12 Half-integral Weight Forms

⎧ √ √  b+ Disc(Q) b− Disc(Q) ⎪ ⎪ , ⎨ 2c 2c



(ωQ , ω Q ) =

a

(i∞, b ) ⎪ ⎪ ⎩ a ( b , i∞)

if c = 0; c = 0 and b > 0; c = 0 and b < 0.

Given Q ∈ Q, we denote by γQ the unique generator of the stabilizer of Q ∈ QM in the congruence group Γ0 (M ). We then consider the path CQ in H defined by  ) if Disc(Q) is a perfect square; (ωQ , ωQ CQ = (ω, γQ ω) otherwise, where ω is an arbitrary point in P1 (Q) and (·, ·) denotes the oriented geodesic path joining the given pair of points. We write χ(Q) = χ(a) if Q = [a, b, c] ∈ QM . Given f ∈ S2λ−1 (Γ0 (M ), χ2 ), we also set Disc(Q) M if M is odd; Q∈Q/Γ0 (M ) Iλ,χ q Θλ,χ (f ) = Disc(Q) 4M if M is even, Q∈Q/Γ0 (M ) Iλ,χ q 

and

3

f (τ )Q(1, −τ )λ− 2 dτ.

Iλ,χ (f, Q) = χ(Q) CQ

If χ is a Dirichlet character defined modulo M , we define the associated Nebentype character χ modulo M by   (−1)2λ+1 M χ (d) = χ(d) d for all d ∈ (Z/4M Z)× . Let Sλ (Γ0 (4M ), χ ) be the space of cusp forms of level 4M , half integral weight λ and Nebentype character χ . The following result was obtained by Shintani. Theorem 12.6 Let χ be a Dirichlet character defined modulo M . Then for each f ∈ S2λ−1 (Γ0 (M ), χ2 ), the series Θλ,χ (f ) is the q-expansion of a halfintegral weight cusp form in Sλ (Γ0 (4M ), χ ). Moreover, the map Θλ,χ : S2λ−1 (Γ0 (M ), χ2 ) → Sλ (Γ0 (4M ), χ ) is a Hecke equivariant C-linear map. Proof. See [111].



To introduce a quasimodular version of Shintani’s result, we consider a m−r (Γ0 (4M ), χ2 ) of the form quasimodular polynomial F (z, X) ∈ QP2λ−1+2m

12.3 Shintani Liftings

233

F (z, X) =

m−r 

fu (z)X u

u=0

for z ∈ H, and set  ,r (Q Θm,m λ,χ

F )(z, X) =

 m −r 

=0

Γ (m − r + 1)Γ (λ + r) Γ ( + 1)Γ (m − r −  + 1)

(12.19) 

(Θλ+r,χ (fm−r ))(m −r−) (z)  X , × Γ (m −  + λ) which is an element of Rm −r [X] with m ≥ r ≥ 0. Proposition 12.7 Given λ ∈ 12 Z with λ > 0, the formula (12.19) determines the Hecke equivariant linear map m,m ,r Q Θλ,χ



m−r m −r  : QP2(λ+m)−1 (Γ0 (M ), χ2 ) → QPλ+m  (Γ0 (4M ), χ )

for m, m ≥ r and 0 ≤ r ≤ min{m, m }. Proof. Given a quasimodular polynomial F (z, X) = m−r (Γ0 (M ), χ2 ), we have QP2λ−1+2m

m−r r=0

fr (z)X r in

Sm−r F ∈ S2(λ+r)−1 (Γ0 (M ), χ2 ), Θλ+r,χ (Sm−r F ) ∈ Sλ+r (Γ0 (4M ), χ ). Thus, using (12.17) and (12.19), we see that λ+r Q Am −r (Θλ+r,χ (Sm−r F )) 

=

m −r  =0

(12.20) 

Γ (m − r + 1)Γ (λ + r)(Θλ+r,χ (fm−r ))(m −r−) (z)  X Γ ( + 1)Γ (m − r −  + 1)Γ (m −  + λ) 

,r = Q Θm,m (F ); λ,χ

hence it follows that m,m ,r (F ) Q Θλ,χ



m −r  ∈ QPλ+m  (Γ0 (4M ), χ ). 2(λ+r)

On the other hand, the maps Sm−r and Q Am −r are Hecke equivariant by the commutativity of the diagrams (7.42) and (9.9), respectively; hence  ,r the Hecke equivariance of Q Θm,m follows from the same property for the λ,χ  Shintani map Θλ+r,χ of modular forms. From (12.20) we obtain the relation m,m ,r Q Θλ,χ

2(λ+r)

= Q Am −r ◦ Θλ+r,χ ◦ Sm−r

234

12 Half-integral Weight Forms

as well as the Hecke equivariant commutative diagram m,m ,r Q Θλ,χ



m−r m −r  (Γ0 (M ), χ2 ) −−−−−−→ QPλ+m QP2(λ+m)−1  (Γ0 (4M ), χ ) ⏐ ⏐ ⏐ ⏐S  Sm−r   m −r Θλ+r,χ

Sλ+r (Γ0 (4M ), χ )

−−−−−→

S2(λ+r)−1 (Γ0 (M ), χ2 )

for each r ∈ {0, 1, . . . , m}. We now consider a quasimodular polynomial m−r G(z, X) ∈ QP2(λ+m)−1 (Γ0 (M ), χ2 )

and define the associated quasimodular polynomials m−r (Γ0 (M ), χ2 ) Gr (z, X) ∈ QP2(λ+m)−1

with m ≥ m for each 0 ≤ r ≤ m by G0 = G,

m− G+1 = G − (m − )!(Πm− ◦ L 0λ,0 ◦ Sm− )G ∈ QP2(λ+m)−1

for 0 ≤  ≤ m − 1, where L 0λ,0 is as in (9.6). Then by setting m

m,m ,r (G) Q Θλ,χ







,0 ,1 ,m = (Q Θm,m G0 , Q Θm,m G1 , . . . , Q Θm,m Gm ) λ,χ λ,χ λ,χ

r=0

we obtain the C-linear map m

m,m ,r Q Θλ,χ

m

:

m QP2(λ+m)−1 (Γ0 (M ), χ2 )

r=0

which is Hecke equivariant.



m −r  QPλ+m  (Γ0 (4M ), χ ),

→ r=0

Chapter 13

Projective Structures

It is well known that modular forms for a discrete subgroup Γ ⊂ SL(2, R) can be interpreted geometrically as sections of a sheaf or a line bundle over the Riemann surface X = Γ \H corresponding to Γ . On the other hand, as we discussed in Section 2.3, automorphic pseudodifferential operators for Γ can be identified with sections of a sheaf or a vector bundle over X. Consequently, we can consider a morphism of sheaves over X corresponding to the lifting from modular forms to automorphic pseudodifferential operators described in (1.53). This lifting morphism depends on the structure of the Riemann surface X. A projective structure on a Riemann surface is a maximal atlas of charts in which the transition functions are linear fractional transformations, and it is an important object in the theory of Riemann surfaces. Since each automorphism of the Poincar´e upper half plane H can be given by a linear fractional transformation, the above Riemann surface X has a natural projective struc is another projective structure on X, then we are interested in ture P. If P the difference between the two lifting morphisms of sheaves corresponding  One of the questions asked by Cohen, to the projective structures P and P. Manin, and Zagier in [31] is whether the difference in the images of a local section of a sheaf under those two lifting sheaf morphisms described above can be expressed in terms of certain Schwarzian derivatives. In this chapter we discuss this question providing a positive answer for some special cases (cf. [74]).

13.1 Projective Structures on Riemann Surfaces There are two important structures on each Riemann surface that are finer than the complex structure, namely, the affine and projective structures. The projective structure is closely linked to the differential operator known as the Schwarzian derivative. In this section we review some basic properties of the

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_13

235

236

13 Projective Structures

projective structure and its relation with the Schwarzian derivative (see, e.g., [45, Section 9] for more details). We also describe one of the questions asked by Cohen, Manin, and Zagier in [31]. Definition 13.1 Let U and V be open subdomains of C, and let f : U → V be a complex analytic local homeomorphism. The Schwarzian derivative of f is the function Sf defined by Sf =

 f   f

−

1  f  2 2f  f  − 3(f  )2 = .  2 f 2(f  )2

(13.1)

In the above definition, since f is a local homeomorphism, we see that f  (z) = 0 for all z ∈ U ; hence Sf is holomorphic on U . Lemma 13.2 Let f : U → V be a complex analytic local homeomorphism of open subdomains U and V of C. Then Sf = 0 if and only if f is a linear fractional transformation. Proof. If f : U → V is a linear fractional transformation given by f (z) = with

a b c d

az + b cz + d

∈ SL(2, C), then we have f  (z) = −2c(cz + d)−1 ; f  (z)

hence we see that 1 (Sf )(z) = 2c2 (cz + d)−2 − (−2c(cz + d)−1 )2 = 0 2 for all z ∈ U . To verify the converse, we now assume that f satisfies Sf = 0. Then, using the fact that Sf (z) can be written in the form (Sf )(z) = −2f  (z)1/2 we obtain

d2   −1/2  f (z) , dz 2

d2  −1/2 f (z) = 0. dz 2

This shows that f  (z)−1/2 = cz + d,

f  (z) = (cz + d)−2 ,

f (z) =

az + b cz + d

for some a, b, c, d ∈ C. Since f is a local homeomorphism, f  (z) = 0 for each z ∈ U , which implies that ad − bc = 0; hence f is a linear fractional transformation. 

13.1 Projective Structures on Riemann Surfaces

237

If f = h ◦ g is the composite of complex analytic local homeomorphisms g : U → V and h : V → W for some open subdomains U , V and W of C, then it can be easily shown that (Sf )(z) = (Sh)(g(z)) · g  (z)2 + (Sg)(z)

(13.2)

for all z ∈ U . In particular, if g is a linear fractional transformation of the form az + b g(z) = γz = cz + d a b for some γ = c d ∈ SL(2, C), then we have (Sf )(z) = (Sh)(g(z)) · g  (z)2 = (Sh)(γz)(cz + d)−4 . Using this and the relation d(γz) = (cz + d)−2 dz, we obtain (Sh)(γz)d(γz)2 = (Sh)(z)dz 2 .

(13.3)

Thus it follows that the quadratic differential (Sh)(z)dz 2 is invariant under linear fractional transformations. Definition 13.3 A projective structure on a Riemann surface X is a maximal atlas of charts on X such that each transition function is a linear fractional transformation. Remark 13.4 Let X be a Riemann surface equipped with a projective structure. Then by Lemma 13.2 the Schwarzian derivative of each transition function is equal to zero. Furthermore, by (13.2) the Schwarzian derivative of the composite of two transition functions is also zero. In fact, a projective structure can also be defined to be a maximal atlas of charts in which the Schwarzian derivative of each transition function is equal to zero. Example 13.5 Let Γ be a discrete subgroup of SL(2, R), which acts on H by linear fractional transformations. If f is a holomorphic function on H, then by (13.3) the associated quadratic differential Sf (z)dz 2 on H is Γ -invariant; hence it induces a quadratic differential form on the Riemann surface X = Γ \H. On the other hand, given a holomorphic quadratic differential hdz 2 on X, by solving locally the differential equation Sf = h in f we can recover the charts of a projective structure on X. Here we need to use the fact that any two local solutions of Sf (z) = h differ by composition with a linear fractional transformation (see [58, Proposition 1]). Let OX be the sheaf of germs of holomorphic functions on a Riemann surface X. If an open subset U of X has a local coordinate z and if m is an integer, we set EXm (U ) =

∞ / k=0

 0  hk (z)∂zm−k  hk ∈ OX (U ) for each k ≥ 1 ,

(13.4)

238

13 Projective Structures

where ∂z = ∂/∂z and OX (U ) is the ring of holomorphic functions on U . We denote by EXm the sheaf on X associated to the presheaf U → EXm (U ). Let ωX be the sheaf of holomorphic differentials on X. Then it is an invertible sheaf, ⊗m for each integer m. If (U, z) is and we can consider its tensor power ωX ⊗m ⊗m , then an an open chart and if ωX (U ) denotes the space of sections of ωX ⊗m m element of ωX (U ) can be written in the form f (dz) for some f ∈ OX (U ). We now consider the case where X = Γ \H is the quotient of H by a discrete subgroup Γ of SL(2, R), where the quotient is taken with respect to the linear fractional action of Γ on H. If Uα is an open subset of X, then α of Uα onto an open subset U α of H there is an isomorphism zα : Uα → U such that (Uα , zα ) is a chart. Using this and the fact that automorphisms of H are given by linear fractional transformations, we can consider a natural maximal atlas (Uα , zα )α∈I whose transition functions are linear fractional. We shall call this the natural projective structure on X and denote it by P. ⊗m are Γ -invariant forms of the form f (dz)m with Sections of the sheaf ωX f ∈ OX (H). Since d(γz) = (cz + d)−2 dz for γ = ac db ∈ Γ , we see that ⊗m sections of ωX on X can be identified with modular forms belonging to M2m (Γ ). On the other hand, by (13.4) sections of the sheaf EXm may be regarded as automorphic pseudodifferential operators belonging to Ψ DOΓm . Thus the sequence (1.55) induces the short exact sequence of the type ⊗m → 0. 0 → EX−m−1 → EX−m → ωX

(13.5)

In particular, there is a canonical isomorphism ⊗m EX−m /EX−m−1 ∼ = ωX

of sheaves on X for each m ≥ 0.

13.2 Pseudodifferential Operators be the ring of holomorphic functions on C on which SL(2, C) acts Let R as in (1.40). We extend the notion of pseudodifferential operators over R introduced in Section 1.4 by considering pseudodifferential operators over R. We denote by Ψ DO the space of all pseudodifferential operators over R, and for m ∈ Z we set  ) * ∞  m−k  Ψ DOm = ξk (z)∂  ξk ∈ R ⊂ Ψ DO. k=0

If m ≥ 0, then Ψ DOm contains the subspace

13.2 Pseudodifferential Operators

DOm =

239

) m

 *  ξk (z)∂ m−k  ξk ∈ R

k=0

consisting of differential operators of order at most m. we set Given a positive integer p and an element f ∈ R, Lp (f ) =

∞ 

(−1)

=0

L−p (f ) =

p−1  =0

( + p)!( + p − 1)! () −p− f ∂ , !( + 2p − 1)!

(2p − )! f () ∂ p− . !(p − )!(p −  − 1)!

Then the maps f → Lp (f ) and f → L−p (f ) determine the linear maps → Ψ DO−p , Lp : R

→ DOp L−p : R

by L0 , we obtain a for each p ≥ 1. Thus, if we denote the identity map on R complex linear map → Ψ DO−w (13.6) Lw : R for each integer w. → Ψ DO−w in (13.6) satisfies Proposition 13.6 Each linear map Lw : R the condition Lw (f |2w γ) = Lw (f ) ◦ γ and γ ∈ SL(2, C). for all f ∈ R Proof. See Proposition 1 in [31]. If (Uα , zα ) is a chart on X belonging to the projective structure P, we set ΛP 0 (f ) = f, ∞  ( + k)!( + k − 1)! () −k− k ΛP f ∂z α , (f (dz ) ) = (−1) α k !( + 2k − 1)!

(13.7) (13.8)

=0

−k )= ΛP −k (f (dzα )

k−1  =0

(2k − )! f () ∂zk− α !(k − )!(k −  − 1)!

(13.9)

for each positive integer k and an element f ∈ OX (Uα ), where ∂zα = ∂/∂zα . The next lemma shows that the short exact sequence (13.5) splits. Lemma 13.7 Let P be the natural projective structure on X described above. Then the formulas (13.7), (13.8) and (13.9) determine a morphism ⊗m → EX−m ΛP m : ωX

of sheaves on X for each m ∈ Z.

(13.10)

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13 Projective Structures

Proof. First, to see that ΛP m is well-defined we consider two charts (Uα , zα ) are linear fractional transformations, and (Uβ , zβ ). Since transition  functions  zα = γαβ zβ for some γαβ = ac db ∈ SL(2, R), so that dzα = (czβ + d)−2 dzβ . Hence, if f ∈ OX (Uα ), we have f (dzα )m = (f |2m γαβ )(dzβ )m

(13.11)

on Uα ∩ Uβ . We note that m ΛP m (f (dzα ) ) = Lm (f ),

(13.12)

where Lm is as in (13.6). Thus the fact that ΛP m is well-defined follows from (13.11), (13.12) and Proposition 13.6. By (13.7) the map ΛP 0 is simply the 0 into EX0 . On the other hand, given a positive inclusion morphism of OX = ωX integer k and a chart (Uα , zα ) in the projective structure P on X, the sections ⊗(−k) ⊗k of the sheaves ωX and ωX over Uα are generated by elements of the k −k form f (dzα ) and h(dzα ) , respectively, for some f, h ∈ OX (Uα ). Hence the lemma follows from this and (13.4).   corresponding to a local We now consider another projective structure P coordinate z on X, and set d z . (13.13) J= dz Then by (13.1) the Schwarzian derivative of the coordinate transformation map z → z can be written as Szz =

2JJ  − 3(J  )2 3 = J −1 J  − J −2 (J  )2 . 2J 2 2

(13.14)

One of the questions asked by Cohen, Manin, and Zagier in [31, Section 8] may be paraphrased as follows.  , z) be charts on X with U ∩ U  = ∅ belonging Question 13.8 Let (U, z) and (U  and P, respectively, with U ∩ U  = ∅, and let to the projective structures P  ). Is it true that f ∈ OX (U ∩ U 

P m (ΛP m − Λm )(f (dz) )

can be expressed in terms of f and the Schwarzian derivative Szz for each m ∈ Z? It was pointed out by Cohen, Manin, and Zagier in [31, Section 8] that 

P ΛP m − Λm = 0

for m = 1, 0, −1, −2. To see the cases of m = 0, −1, −2, note that

13.2 Pseudodifferential Operators

∂z =

241

1 d d = = J −1 ∂, d z d z /dz dz

(13.15)

∂z2 = J −1 ∂(J −1 ∂) = J −1 (−J −2 J  ∂ + J −1 ∂ 2 ) = J −2 ∂ 2 − J  J −3 ∂. On the other hand, since d z = Jdz, we have z )m = J −m f (d z )m , f (dz)m = f (J −1 d

(13.16)

so that 



P P m −m m f (d z )m ) − ΛP (ΛP m − Λm )(f (dz) ) = Λm (J m (f (dz) )

(13.17)

for each m ∈ Z. For m = 0, we have 

0 z )0 ) = Λ P ΛP 0 (f (d 0 (f (dz) ) = f ;

hence we see that



P (ΛP 0 − Λ0 )(f ) = 0.

For m = −1, from (13.9) we obtain −1 )= ΛP −1 (f (dz)

0 

(2 − n)! f (n) ∂ 1−n = 2f ∂. n!(1 − n)!(−n)! n=0

Thus we have 

P −1 (ΛP ) = 2(Jf (z)∂z − f ∂) = 2(Jf (z)J −1 ∂ − f (z)) = 0. −1 − Λ−1 )(f (dz)

For m = −2, we see from (13.9) that −2 )= ΛP −2 (f (dz)

1 

(4 − n)! f (n) (z)∂ 2−n = 12f (z)∂ 2 + 6f  (z)∂, n!(2 − n)!(1 − n)! n=0

and therefore we have  (ΛP −2

−

−2 ΛP ) −2 )(f (dz)



= 12(J

2

f ∂z2

 d 2  (J f )∂z − f (z)∂ , − f∂ ) + 6 d z 2

where d 2 d (J f ) = J −1 (J 2 f ) = J −1 (2JJ  f + J 2 f  ) = 2J  f + Jf  . d z dz From this and (13.15) it follows that 

P −2 (ΛP ) = 12(J 2 f (J −2 ∂ 2 − J  J −3 ) − f ∂ 2 ) −2 − Λ−2 )(f (dz)

+ 6((J −1 (2J  f + Jf  − f  ∂)J −1 ∂ − f  ∂) = 0.

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13 Projective Structures 

P Cohen, Manin, and Zagier also gave a formula for ΛP −m − Λ−m in the case of m = −3, which we show in the next lemma using our notation.

 , z) belonging to the projective Lemma 13.9 Given charts (U, z) and (U  and P, respectively, on X with U ∩ U  = ∅, we have structures P 

P −3 ) = −24Szzf ∂ (ΛP −3 − Λ−3 )(f (dz)

(13.18)

 ). for f ∈ OX (U ∩ U Proof. Using (13.9), we obtain −3 )= ΛP −3 (f (dz)

2 

(6 − n)! f (n) ∂ 3−n n!(3 − n)!(2 − n)! n=0

= 60f ∂ 3 + 60f  ∂ 2 + 12f  ∂; hence from (13.17) we see that  (ΛP −3

−

−3 ΛP ) −3 )(f (dz)



= 60(J

3

 d 3 2  2 (J f )∂z − f ∂ ) − f ∂ ) + 60 d z (13.19)  2  d + 12 (J 3 f )∂z − f  (z)∂ . d z2

f ∂z3

3

Here ∂z and ∂z2 satisfy (13.15), and ∂z3 = J −1 ∂(J −2 ∂ 2 − J  J −3 ∂) =J

−1

=J

−3 3

(−2J

−3

 2

J ∂ +J 

∂ − 3J J

−3 2

(13.20)

−2 3



∂ − (J J  2

−3 

 2

− 3(J ) J

∂ + (3(J ) − JJ )J

−5

−4



)∂ − J J

−3 2

∂ )

∂.

On the other hand, we have d 3 (J f ) = J −1 (J 3 f ) d z = J −1 (3J 2 J  f + J 3 f  ) = 3JJ  f + J 2 f  , d2 3 (J f ) = J −1 (3JJ  f + J 2 f  ) d z2 = J −1 (3(J  )2 + 3JJ  )f + 3JJ  f  + 2JJ  f  + J 2 f  ) = (3(J  )2 J −1 + 3J  )f + 5J  f  + Jf  . Thus we obtain 1 P d 1 d2 3  Λ−3 (f (dz)−3 ) = J 3 f ∂z3 + (J 3 f )∂z2 + (J f )∂z 60 d z 5 d z2

13.2 Pseudodifferential Operators

243

= J 3 f (J −3 ∂ 3 − 3J  J −4 ∂ 2 + (3(J  )2 − JJ  )J −5 ∂) + (3JJ  f + J 2 f  )(J −2 ∂ 2 − J  J −3 ∂) 1 + ((3(J  )2 J −1 + 3J  )f + 5J  f  + Jf  )J −1 ∂ 5 = f ∂ 3 − 3J  J −1 f ∂ 2 + (3(J  )2 J −2 − J  J −1 f )∂ + (3J  J −1 f + f  )∂ 2 − (3(J  )2 J −2 f + J  J −1 f  )∂ 1 + (3(J  )2 J −2 f + 3J  J −1 f + 5J  J −1 f  + f  )∂ 5 1 3  2 = (f ∂ + f ∂ + f  ∂) + (3(J  )2 J −2 − 2J  J −1 )f ∂, 5 which implies that 1 P 1  −3 (Λ − ΛP ) = (3(J  )2 J −2 − 2J  J −1 )f ∂. −3 )(f (dz) 60 −3 5 

Thus (13.18) follows from this and (13.14).

Remark 13.10 The formula given by Cohen, Manin and Zagier in [31] has the right-hand side of (13.18) in the form   1 J  (J  )2 1 2 − 3 2 f ∂ = Szzf ∂ 5 J J 5 in our notation. The right-hand side of (13.18) would coincide with this if the formula (13.19) is normalized by dividing its right-hand side by the coefficient of f ∂ 3 . We now state a theorem that provides a positive answer to Question 13.8 for some special cases. The proof of this theorem will be given in the next two sections.  , z) be charts belonging to the projective Theorem 13.11 Let (U, z) and (U   = ∅, and let f ∈ OX (U ∩ U  ). structures P and P, respectively, on X with U ∩ U Then we have 

P −4 ) = −800Szzf ∂ 2 − 400(Szzf ) ∂, (ΛP −4 − Λ−4 )(f (dz)

(13.21)

 (ΛP −5

(13.22)

−5 − ΛP ) −5 )(f (dz)

=

−210Szzf ∂ 3 −

− 210(Szzf ) ∂ 2  30 37(Szz) f + 16(Szz)2 f

 + 65(Szz) f  + 30Szzf  ∂

  ), where ΛP and ΛP for f ∈ OX (U ∩ U m for m = −4, −5 are morphisms of m sheaves on X in (13.10) and Szz is the Schwarzian derivative of the coordinate transformation map z → z given by (13.14).

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13 Projective Structures

13.3 The Degree Four Case

In this section we prove the relation (13.21) in Theorem 13.11. Let f ∈  ), where (U, z) and (U  , z) with U ∩ U  = ∅ are charts on X belonging OX (U ∩ U  to the projective structures P and P, respectively. Then, using (13.9), we have −4 )= ΛP −4 (f (dz)

3 

(8 − n)! f (n) ∂ 4−n n!(4 − n)!(3 − n)! n=0

(13.23)

7!  3 6!  2 5!  8! f ∂4 + f∂ + f ∂ + f ∂ 4!3! 3!2! 2!2! 3! = 280f ∂ 4 + 420f  ∂ 3 + 180f  ∂ 2 + 20f  ∂.

=

From this and (13.16) we obtain 



−4 4 ΛP ) = ΛP z )−4 ) −4 (f (dz) −4 (J f (d

= 280J 4 f ∂z4 + 420

(13.24)

d 4 (J f )∂z3 d z d2 d3 + 180 2 (J 4 f )∂z2 + 20 3 (J 4 f )∂z, d z d z

where J is as in (13.13). Since ∂z = J −1 ∂, using (13.20), we see that (13.25) ∂z4 = J −1 ∂(J −3 ∂ 3 − 3J  J −4 ∂ 2 + (3(J  )2 J −5 − J  J −4 )∂)  −1 −4  3 −3 4  −4  2 −5 2  −4 3 =J −3J J ∂ + J ∂ − 3(J J − 4(J ) J )∂ − 3J J ∂ + (6J  J  J −5 − 15(J  )3 J −6 − J  J −4 + 4J  J  J −5 )∂  + (3(J  )2 J −5 − J  J −4 )∂ 2 = −3J −5 J  ∂ 3 + J −4 ∂ 4 − 3(J  J −5 − 4(J  )2 J −6 )∂ 2 − 3J  J −5 ∂ 3 + (10J  J  J −6 − 15(J  )3 J −7 − J  J −5 )∂ + (3(J  )2 J −6 − J  J −5 )∂ 2  = J −4 ∂ 4 − 6J  J −5 ∂ 3 + 15(J  )2 J −6 − 4J  J −5 ∂ 2   + 10J  J  J −6 − 15(J  )3 J −7 − J  J −5 ∂. 

On the other hand, we have d d 4 (J f ) = J −1 (J 4 f ) d z dz = J −1 (4J 3 J  f + J 4 f  ) = 4J 2 J  f + J 3 f  , d2 4 (J f ) = J −1 (4J 2 J  f + J 3 f  ) d z2   = J −1 (8J(J  )2 + 4J 2 J  )f + 4J 2 J  f  + 3J 2 J  f  + J 3 f 

13.3 The Degree Four Case

245

= (8(J  )2 + 4J  J)f + 7J  Jf  + J 2 f  ,   d3 4 (J f ) = J −1 (8(J  )2 + 4J  J)f + 7J  Jf  + J 2 f  3 d z  = J −1 (16J  J  + 4J  J + 4J  J  )f + (8(J  )2 + 4J  J)f  + 7(J  J + (J  )2 )f  + 7J  Jf  + 2J  Jf  + J 2 f 



= (20J  J  J −1 + 4J  )f + (8(J  )2 J −1 + 4J  )f  + 7(J  + (J  )2 J −1 )f  + 7J  f  + 2J  f  + Jf  = (20J  J  J −1 + 4J  )f + (15(J  )2 J −1 + 11J  )f  + 9J  f  + Jf  . From these, (13.15), (13.20), (13.24) and (13.25) we obtain 1 P  Λ (f (dz)−4 ) 20 −4 d d2 d3 = 14f ( z )∂z4 + 21 f ( z )∂z3 + 9 2 f ( z )∂z2 + 3 f ( z )∂z d z d z d z    = 14J 4 f J −4 ∂ 4 − 6J  J −5 ∂ 3 + 15(J  )2 J −6 − 4J  J −5 ∂ 2    + 10J  J  J −6 − 15(J  )3 J −7 − J  J −5 ∂ + 21(4J 2 J  f + J 3 f  )(J −3 ∂ 3 − 3J  J −4 ∂ 2 + (3(J  )2 − JJ  )J −5 ∂) + 9((8(J  )2 + 4J  J)f + 7J  Jf  + J 2 f  )(J −2 ∂ 2 − J  J −3 ∂)  + (20J  J  J −1 + 4J  )f  + (15(J  )2 J −1 + 11J  )f  + 9J  f  + Jf  (J −1 ∂)   = 14f ∂ 4 − 84J  J −1 f ∂ 3 + 210(J  )2 J −2 f − 56J  J −1 f ∂ 2   + 140J  J  J −2 f − 210(J  )3 J −3 f − 14J  J −1 f ∂     + 84J  J −1 f + 21f  ∂ 3 − 252(J  )2 J −2 f + 63J  J −1 f  ∂ 2   + 252(J  )3 J −3 f − 84J  J  J −2 f + 63(J  )2 J −2 f  − 21J  J −1 f  ∂   + 72(J  )2 J −2 f + 36J  J −1 f + 63J  J −1 f  + 9f  ∂ 2   − 72(J  )3 J −3 f + 36J  J  J −2 f + 63(J  )2 J −2 f  + 9J  J −1 f  ∂  + 20J  J  J −2 f + 4J  J −1 f  + 15(J  )2 J −2 f  + 11J  J −1 f  + 9J  J −1 f  + f  ∂. Using this and (13.23), we have 1 P  −4 (Λ − ΛP ) −4 )(f (dz) 20 −4     = 10 3(J  )2 J −2 − 2J  J −1 f ∂ 2 + 5 3(J  )2 J −2 − 2J  J −1 f  ∂   + 10 4J  J  J −2 − 3(J  )3 J −3 − J  J −1 f ∂.

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13 Projective Structures

If Szz is as in (13.14), its derivative with respect to z is given by 3 (Szz) = J  J −1 − J  J  J −2 − (2J  J  J −2 − 2(J  )3 J −3 ) 2 = J  J −1 + 3(J  )3 J −3 − 4J  J  J −2 .

(13.26)

Thus we have −

1  −4 (ΛP −ΛP ) = 2Szzf ∂ 2 +(Szzf  +(Szz) f )∂ = 2Szzf ∂ 2 +(Szzf ) ∂, −4 )(f (dz) 400 −4

which verifies the relation (13.21) in Theorem 13.11.

13.4 The Degree Five Case In this section we prove the relation (13.22) in Theorem 13.11. If  ) is as in Section 13.3, using (13.9) and (13.16), we have f ∈ OX (U ∩ U −5 ΛP )= −5 (f (dz)

4 

(10 − n)! f (n) ∂ 5−n n!(5 − n)!(4 − n)! n=0

(13.27)

9!  4 8!  3 7!  2 6! (4) 10! 5 f∂ + f∂ + f ∂ + f ∂ + f ∂ 5!4! 4!3! 2!3!2! 3!2! 4!  5  4  3  2 (4) = 30 42f ∂ + 84f ∂ + 56f ∂ + 14f ∂ + f ∂ ,

=





−5 5 ΛP ) = ΛP z )−5 ) (13.28) −5 (f (dz) −5 (J f (d  2 d d = 30 42J 5 f ∂z5 + 84 (J 5 f )∂z4 + 56 2 (J 5 f )∂z3 d z d z  3 d d4 + 14 3 (J 5 f )∂z2 + 4 (J 5 f )∂z . d z d z

Since ∂z5 = J −1 ∂∂z4 , by using (13.25) we see that  J∂z5 = ∂ J −4 ∂ 4 − 6J  J −5 ∂ 3 + (15(J  )2 J −6 − 4J  J −5 )∂ 2 + (10J  J  J −6 − 15(J  )3 J −7 − J  J −5 )∂



= −4J −5 J  ∂ 4 + J −4 ∂ 5 − 6(J  J −5 ∂ 3 − 5(J  )2 J −6 ∂ 3 + J  J −5 ∂ 4 ) + 15(2J  J  J −6 ∂ 2 − 6(J  )3 J −7 ∂ 2 + (J  )2 J −6 ∂ 3 ) − 4(J  J −5 ∂ 2 − 5J  J  J −6 ∂ 2 + J  J −5 ∂ 3 ) + 10(J  J  J −6 ∂ + (J  )2 J −6 ∂ − 6J  (J  )2 J −7 ∂ + J  J  J −6 ∂ 2 ) − 15(3J  (J  )2 J −7 ∂ − 7(J  )4 J −8 ∂ + (J  )3 J −7 ∂ 2 ) − (J (4) J −5 ∂ − 5J  J  J −6 ∂ + J  J −5 ∂ 2 );

13.4 The Degree Five Case

247

hence we obtain ∂z5 = J −5 ∂ 5 − 10J  J −6 ∂ 4 + (−10J  J −6 + 45(J  )2 J −7 )∂ 3 



+ (60J J J 

−7

 3

− 105(J ) J

 2

+ (−105J (J ) J

−8

−8



− 5J J

 4

+ 105(J ) J

−9

−6

)∂

(13.29)

2

+ 15J  J  J −7

+ 10(J  )2 J −7 − J (4) J −6 )∂. On the other hand, we have d 5 (J f ) = J −1 (J 5 f ) = J −1 (5J 4 J  f + J 5 f  ) = 5J 3 J  f + J 4 f  , d z d2 5 (J f ) = J −1 (5J 3 J  f + J 4 f  ) d z2   = J −1 (15J 2 (J  )2 + 5J 3 J  )f + 5J 3 J  f  + 4J 3 J  f  + J 4 f  = (15(J  )2 J + 5J  J 2 )f + 9J  J 2 f  + J 3 f  ,   d3 5 (J f ) = J −1 (15(J  )2 J + 5J  J 2 )f + 9J  J 2 f  + J 3 f  3 d z  = J −1 (30J  J  J + 15(J  )3 + 5J  J 2 + 10J  J  J)f + (15(J  )2 J + 5J  J 2 )f  + (9J  J 2 + 18(J  )2 J)f   + 9J  J 2 f  + 3J  J 2 f  + J 3 f  = (40J  J  + 15(J  )3 J −1 + 5J  J)f + (33(J  )2 + 14J  J)f  + 12J  Jf  + J 2 f  ,  d4 5 (J f ) = J −1 (40J  J  + 15(J  )3 J −1 + 5J  J)f 4 d z  + (33(J  )2 + 14J  J)f  + 12J  Jf  + J 2 f   = J −1 (40J  J  + 40(J  )2 + 45J  (J  )2 J −1 − 15(J  )4 J −2 + 5J (4) J + 5J  J  )f + (40J  J  + 15(J  )3 J −1 + 5J  J)f  + (66J  J  + 14J  J + 14J  J  )f  + (33(J  )2 + 14J  J)f   + (12J  J + 12(J  )2 )f  + 12J  Jf  + 2J  Jf  + J 2 f (4) = (45J  J  J −1 + 40(J  )2 J −1 + 45J  (J  )2 J −2 − 15(J  )4 J −3 + 5J (4) )f + (120J  J  J −1 + 15(J  )3 J −2 + 19J  )f  + (45(J  )2 J −1 + 26J  )f  + 14J  f  + Jf (4) . Combining these relations with (13.27), (13.28) and (13.29) as well as (13.15), (13.20) and (13.25), we obtain

248

13 Projective Structures

1 P  −5 (Λ − ΛP ) −5 )(f (dz) 30 −5 = 42J 5 f (J −5 ∂ 5 − 10J  J −6 ∂ 4 + (−10J  J −6 + 45(J  )2 J −7 )∂ 3 + (60J  J  J −7 − 105(J  )3 J −8 − 5J  J −6 )∂ 2 + (−105J  (J  )2 J −8 + 105(J  )4 J −9 + 15J  J  J −7 + 10(J  )2 J −7 − J (4) J −6 )∂   + 84(5J 3 J  f + J 4 f  )(J −4 ∂ 4 − 6J  J −5 ∂ 3 + 15(J  )2 J −6 − 4J  J −5 ∂ 2   + 10J  J  J −6 − 15(J  )3 J −7 − J  J −5 ∂) + 56((15(J  )2 J + 5J  J 2 )f + 9J  J 2 f  + J 3 f  ) × (J −3 ∂ 3 − 3J  J −4 ∂ 2 + (3(J  )2 − JJ  )J −5 ∂) + 14((40J  J  + 15(J  )3 J −1 + 5J  J)f + (33(J  )2 + 14J  J)f  + 12J  Jf  + J 2 f  )(J −2 ∂ 2 − J  J −3 ∂) + ((45J  J  J −1 + 40(J  )2 J −1 + 45J  (J  )2 J −2 − 15(J  )4 J −3 + 5J (4) )f + (120J  J  J −1 + 15(J  )3 J −2 + 19J  )f  + (45(J  )2 J −1 + 26J  )f  + 14J  f  + Jf (4) )(J −1 ∂). Thus, if we write 1 P  −5 (Λ − ΛP ) = C5 ∂ 5 + C4 ∂ 4 + C3 ∂ 3 + C2 ∂ 2 + C1 ∂, −5 )(f (dz) 30 −5 then the coefficients C1 , . . . , C5 can be written in the form C5 = 42f − 42f = 0, C4 = −420J  J −1 f + 420J  + 84f  − 84f = 0, C3 = −420J  J −1 f + 1890(J  )2 J −2 f − 2520(J  )2 J −2 f − 504J  J −1 f  + 840(J  )2 J −2 f + 280J  J −1 f + 504J  J −1 f  + 56f  − 56f  = 210(J  )2 J −2 f − 140J  J −1 f = 70Szzf, C2 = 2520J  J  J −2 f − 4410(J  )3 J −3 f − 210J  J −1 f + 6300(J  )3 J −3 f + 1260(J  )2 J −2 f  − 1680J  J  J −2 f − 336J  J −1 f  − 2520(J  )3 J −3 f − 840J  J  J −2 f − 1512(J  )2 J −2 f  − 168J  J −1 f  + 560J  J  J −2 f + 210(J  )3 J −3 f + 70J  J −1 f + 462(J  )2 J −2 f  + 196J  J −1 f  + 168J  J −2 f  + 14f  − 14f  = 560J  J  J −2 f − 420(J  )3 J −3 f − 140J  J −1 f + 210(J  )2 J −2 f  − 140J  J −1 f  = 70((Szz) f + Szzf  ) = 70(Szzf ) ,

13.4 The Degree Five Case

249

C1 = −4410J  (J  )2 J −3 f + 4410(J  )4 J −4 f + 630J  J  J −2 f + 420(J  )2 J −2 f − 42J (4) J −1 f + 4200J  (J  )2 J −3 f − 6300(J  )4 J −4 f − 420J  J  J −2 f + 840J  J  J −2 f  − 1260(J  )3 J −3 f  − 84J  J −1 f  + 2520(J  )4 J −4 f + 840J  (J  )2 J −3 f + 1512(J  )3 J −3 f  + 168(J  )2 J −2 f  − 840J  (J  )2 J −3 f − 280(J  )2 J −2 f − 504J  J  J −2 f  − 56J  J −1 f  − 560J  (J  )2 J −3 f − 210(J  )4 J −4 f − 70J  J  J −2 f − 462(J  )3 J −3 f  − 196J  J  J −2 f  − 168(J  )2 J −2 f  − 14J  J −1 f  + 45J  J  J −2 f + 40(J  )2 J −2 f + 45J  (J  )2 J −3 f − 15(J  )4 J −4 f + 5J (4) J −1 f + 120J  J  J −2 f  + 15(J  )3 J −3 f  + 19J  J −1 f  + 45(J  )2 J −2 f  + 26J  J −1 f  + 14J  J −1 f  + f (4) − f (4)  = −725J  (J  )2 J −3 + 405(J  )4 J −4

 + 185J  J  J −2 + 180(J  )2 J −2 − 37J (4) J −1 f   + 260J  J  J −2 − 195(J  )3 J −3 − 65J  J −1 f    + 45(J  )2 J −2 − 30J  J −1 f    = 37 5J  J  J −2 + 4(J  )2 J −2 − 17J  (J  )2 J −3 + 9(J  )4 J −4 − J (4) J −1 f   + 8 9(J  )4 J −4 − 12J  (J  )2 J −3 + 4(J  )2 J −2 f     + 65 4J  J  J −2 − 3(J  )3 J −3 − J  J −1 f  + 15 3(J  )2 J −2 − 2J  J −1 f  .

If Szz is as in (13.14), we have 1 (3(J  )2 J −2 − 2J  J −1 )2 4 1 = (9(J  )4 J −4 − 12J  (J  )2 J −3 + 4(J  )2 J −2 ). 4

(Szz)2 =

On the other hand, using (13.26), we obtain (Szz) = J (4) J −1 − J  J  J −2 + 9J  (J  )2 J −3 − 9(J  )4 J −4 − 4J  J  J −2 − 4(J  )2 J −2 + 8J  (J  )2 J −3 = −5J  J  J −2 − 4(J  )2 J −2 + 17J  (J  )2 J −3 − 9(J  )4 J −4 + J (4) J −1 ). Thus it follows that −

1 P  −5 (Λ − ΛP ) = 70Szzf ∂ 3 + 70(Szzf ) ∂ 2 −5 )(f (dz) 30 −5 + (37(Szz) f + 16(Szz)2 f + 65(Szz) f  + 30Szzf  )∂.

Hence we obtain the formula (13.22), and the proof of Theorem 13.11 is complete.

Chapter 14

Applications of Quasimodular Forms

Connections of quasimodular forms with geometry were first considered by Dijkgraaf in [33]. More specifically, he showed that the generating function, given by a power series in q = e2πiz with z ∈ H, counting the genus g covers of an elliptic curve is a quasimodular form of weight 6g − 6 for the full modular group SL(2, Z), although a more rigorous proof was given by Kaneko and Zagier in [54]. Since then the quasimodularity of generating functions of various other objects has been investigated in many papers. In this chapter we discuss some of those results. In the first section we review the structure of the ring of quasimodular forms for the full modular group SL(2, Z) keeping in mind that most of the applications in this chapter involve such quasimodular forms. In Section 2, we present the work of Kaneko and Zagier [54], who proved that the coefficient functions of a power series associated to a generalized classical Jacobi theta function are also quasimodular forms for SL(2, Z). This provides, in particular, a direct proof of the theorem of Dijkgraaf on the quasimodularity of a generating function for counting covers of an elliptic curve, which is the subject of Section 3. Section 4 is concerned with the result of Bloch and Dijkgraaf [10] about the quasimodularity of functions defined by sums over partitions. For the proof of this result we follow the one given by Zagier in [115]. In Section 5 we discuss modular linear differential equations whose coefficients are quasimodular forms, and Section 6 treats holomorphic anomaly equations and the quasimodularity of their solutions. Reviews of some additional applications are contained in Section 7.

14.1 Quasimodular Forms for the Full Modular Group In this section we discuss the structure of the graded ring of quasimodular forms for the full modular group Γ1 = SL(2, Z), which is contained in the paper of Kaneko and Zagier [54]. We shall also follow some of the descrip-

© Springer Nature Switzerland AG 2019 Y. Choie and M. H. Lee, Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-29123-5_14

251

252

14 Applications of Quasimodular Forms

tions in the paper [105] by E. Royer. We note that most of the applications contained in this chapter are concerned with quasimodular forms for Γ1 . If k is a positive even integer with k ≥ 2, we recall that the corresponding Eisenstein series Ek of weight k is given by Ek (z) = 1 −

∞ 2k   k−1  k q d Bk n=1

(14.1)

d|n

for z ∈ H, where Bk is the k-th Bernoulli number. Then, as is well known, Ek is a modular form (or quasimodular form of depth 0) of weight k for the full modular group Γ1 = SL(2, Z) if k ≥ 4, and E2 is a quasimodular form belonging to QM21 (Γ1 ) satisfying 6 K(γ, z) πi

(E2 |2 γ)(z) = E2 (z) +

(14.2)

for all z ∈ H and γ ∈ Γ1 as in Example 7.4. Proposition 14.1 For each quasimodular form φ ∈ QMλm (Γ1 ) with m ≥ 0 there are modular forms ψr ∈ Mλ−2r (Γ1 ) for 0 ≤ r ≤ m such that φ=

m 

ψr E2r .

r=0

Proof. The proof will be carried out by induction on r. Given φ ∈ QMλm (Γ1 ), we set  πi m (Sm φ)E2m , η =φ− 6 where 0 Sm : QMλm (Γ1 ) → QMλ−2m (Γ1 ) = Mλ−2m (Γ1 ) is as in (7.15). Then for γ ∈ Γ1 we have (η |λ γ)(z) = (φ |λ γ)(z) − =

m 

 πi m 6

J(γ, z)−λ (Sm φ)(γz)E2m (γz)

(Sr φ)(z)K(γ, z)r −

r=0

=

m 

 πi m 6

J(γ, z)−λ (Sm φ)(γz)E2m (γz)

(Sr φ)(z)K(γ, z)r

r=0

−

 πi m 6

J(γ, z)

−2m+2



6 (Sm φ)(z) E2 (z) + K(γ, z) πi

m

for all z ∈ H, where we used (14.2) and the fact that Sm φ belongs to Mλ−2m (Γ1 ). Thus we obtain

14.1 Quasimodular Forms for the Full Modular Group

(η |λ γ)(z) =

m 

(Sr φ)(z)K(γ, z)r

r=0

−

=

253

 πi m 6

m−1  r=0

m    6 j m (Sm φ)(z) E2 (z)m−j K(γ, z)j j πi j=0

    m πi m−r m−r (Sm φ)(z)E2 (z) (Sr φ)(z) − K(γ, z)r . r 6

Hence we see that η ∈ QMλm−1 (Γ1 ), and therefore the proposition follows by induction.  Corollary 14.2 There is an isomorphism QMλm (Γ1 ) ∼ =

m

Mλ−2r (Γ1 )E2r . r=0

Proof. This follows immediately from Proposition 14.1.



If η ∈ QMνp (Γ1 ) with p and ν being nonnegative integers, as in (7.6), its p+1 derivative ∂η ∈ QMν+2 (Γ1 ) satisfies S0 (∂η) = ∂(S0 η) = ∂η,

Sp+1 (∂η) = (ν − p)(Sp η),

Sk (∂η) = (ν − k + 1)(Sk−1 η) + ∂(Sk η)

(14.3)

for 1 ≤ k ≤ p. In particular, if ψ ∈ Mλ−2m (Γ1 ), then ∂ m−1 ψ is a quasimodm−1 (Γ1 ), and hence we have ular form belonging to QMλ−2 Sm (∂ m ψ) = Sm (∂∂ m−1 ψ)

(14.4)

= (λ − 2 − (m − 1))(Sm−1 (∂ m−1 ψ)) = (λ − m − 1)(Sm−1 (∂ m−1 ψ)) = (λ − m − 1)(λ − m − 2)(Sm−2 (∂ m−2 ψ)) = (λ − m − 1)(λ − m − 2) · · · (λ − 2m)(S0 (∂ 0 ψ))   λ−m−1 ψ. = m! m Proposition 14.3 Given a quasimodular form φ ∈ QMλm (Γ1 ) with λ being a nonnegative even integer, we have ( λ/2−1 i ∂ Mλ−2i (Γ1 ) for m < λ/2; QMλm (Γ1 ) = (i=0 λ/2−1 i λ/2−1 ∂ Mλ−2i (Γ1 ) ⊕ C∂ E2 for m = λ/2. i=0 Proof. Let φ ∈ QMλm (Γ1 ) with λ nonnegative even. If 0 ≤ m < λ/2, then the function

254

14 Applications of Quasimodular Forms

ξ=

−1  1 λ−m−1 (Sm φ) m m!

is a modular form belonging to Mλ−2m (Γ1 ). Using this and (14.4), we have   λ−m−1 ξ = Sm φ; Sm (∂ m ξ) = m! m hence we see that φ − ∂ m ξ ∈ QMλm−1 (Γ1 ). Thus we obtain φ ∈ ∂ m Mλ−2m (Γ1 ) ⊕ QMλm−1 (Γ1 ), and therefore by induction on m it follows that m

φ∈

∂ r Mλ−2r (Γ1 ). r=0 λ/2

We now consider the case where φ ∈ QMλ (Γ1 ). Since ∂ λ/2−2 E2 belongs to λ/2−1 QMλ−2 (Γ1 ), using (14.2) and (14.3), we have Sλ/2 (∂ λ/2−1 E2 ) = Sλ/2 (∂∂ λ/2−2 E2 ) = (λ − 2 − (λ/2 − 1))Sλ/2−1 (∂ λ/2−2 E2 ) = (λ/2 − 1)Sλ/2−1 (∂ λ/2−2 E2 ) = (λ/2 − 1)(λ/2 − 2)Sλ/2−2 (∂ λ/2−3 E2 ) = (λ/2 − 1)(λ/2 − 2) · · · (λ/2 − (λ/2 − 1))S1 ∂ 0 E2 = (λ/2 − 1)!S1 E2 6 . = (λ/2 − 1)! πi Noting that Sλ/2 φ ∈ QM00 (Γ1 ) = M0 (Γ1 ) = C, we set κ=

πi (Sλ/2 φ), 6(λ/2 − 1)!

so that 6 (λ/2 − 1)! πi

0 = Sλ/2 φ − κ

  = Sλ/2 φ − κSλ/2 (∂ λ/2−1 E2 ) = Sλ/2 φ − κ(∂ λ/2−1 E2 ) .

Thus we have λ/2−1

φ − κ∂ λ/2−1 E2 ∈ QMλ

(Γ1 ),

14.1 Quasimodular Forms for the Full Modular Group

255

hence it follows that λ/2−1 λ/2

∂ i Mλ−2i (Γ1 ) ⊕ C∂ λ/2−1 E2 ,

QMλ (Γ1 ) = i=0



and therefore the proposition follows. We now set D=

1 d d 1 ∂= =q , 2πi 2πi dz dq

(14.5)

∞

QMλm (Γ1 )

QMλ (Γ1 ) = m=0

for λ ≥ 0, and consider a sequence {Hn }∞ n=0 of quasimodular forms with Hn ∈ QMn (Γ1 ) defined recursively by H0 = 1,

H1 = 0,

4n(n + 1)Hn = 8D(Hn−2 ) + E2 Hn−2

for n ≥ 2. In closed form we have  1 r 1 2D + E2 (1), H2r = (2r + 1)! 4

H2r+1 = 0

for r ≥ 0. Let ϑ(ζ) ∈ Q[[q]][[ζ]] be the Jacobi theta series  2 (−1)[ν] eνζ q ν /2 , ϑ(ζ) =

(14.6)

(14.7)

ν∈F

with ν → [ν] being the greatest integer function and F = Z + 1/2. The next lemma will be used in the proof of Theorem 14.19. Lemma 14.4 The Taylor expansion of ϑ(ζ)/ϑ (0) can be written in the form ∞  ϑ(ζ) = Hn (z)ζ n+1 ϑ (0) n=0

for z ∈ H. Proof. Given a nonnegative integer r, from (14.7) we see that Dr ϑ (ζ) =

 ν∈F

(−1)[ν] νeνζ q

2 ν 2r+1 d ν 2 /2  q = (−1)[ν] r eνζ q ν /2 . dq 2

ν∈F

(14.8)

256

14 Applications of Quasimodular Forms

In particular, we have Dr ϑ (0) =



(−1)[ν]

ν∈F

ν 2r+1 ν 2 /2 q . 2r

(14.9)

On the other hand, assuming that (14.8) holds and using (14.7), we have ϑ (0)

∞ 

Hn (z)ζ n+1 =

n=0

∞    2 ν 2 (−1)[ν] eνζ q ν /2 = (−1)[ν] q ν /2 ζ  . ! ν∈F

=0 ν∈F

Comparing the coefficient of ζ 2r+1 and using (14.9), we obtain  2 (−1)[ν] ν 2r+1 q ν /2 = (2D)r ϑ (0). (2r + 1)!ϑ (0)H2r (z) =

(14.10)

ν∈F

We note that the Dedekind eta function η given by  η(z) = q 1/24 (1 − q n )

(14.11)

n>0

for z ∈ H satisfies η 3 = ϑ (0),

Dη =

1 E2 η. 24

It can be easily shown that (2D)r ϑ (0) = (2D)r η 3 = ((2D + E2 /4)r (1))η 3 ; hence the formula (14.6) follows from this and (14.10).



14.2 The Work of Kaneko–Zagier In this section we describe main results contained in the paper of Kaneko and Zagier [54], which will be used in Section 14.3 to prove the theorem of Dijkgraaf. We consider a formal power series in X and Y with z ∈ H, which may be regarded as a generalized classical Jacobi theta function, given by the triple product   2 2 Θ(X, z, Y ) = (1 − q n ) (1 − en X/8 q n/2 ζ)(1 − e−n X/8 q n/2 ζ −1 ), n>0

n>0,n:odd

(14.12) where ζ = eY and q = e2πiz with z ∈ H. Let η be the Dedekind eta function given by (14.11), and let θ be the theta series given by

14.2 The Work of Kaneko–Zagier

257

θ(z) =



(−1)n q n

2

/2

(14.13)

n∈Z

for z ∈ H. Then it is well known that these functions satisfy the identity θ(z)η(z) = η(z/2)2

(14.14)

for all z ∈ H. Given an even integer k ≥ 2, if Ek is the Eisenstein series in (14.1), we set Gk = −

Bk Ek , 2k

and define the functions (1)

(2)

Fk , Fk

:H→C

(14.15)

by setting (1)

Fk (z) = Gk (z/2) − Gk (z),

(2)

Fk (z) = Gk (z/2) − 2k−1 Gk (z)

(14.16)

for all z ∈ H. Lemma 14.5 The functions in (14.15) are quasimodular forms of weight k for the congruence group   Γ 0 (2) = { ac db ∈ Γ1 | b ≡ 0 (mod 2)}, and they can written as (1) Fk (z)

∞    k−1   n = q n/2 , d n=1

(14.17)

d|n,2d ∞

(2) Fk (z)

k−1

= (2

Bk  + − 1) 2k n=1

 

 d

k−1

q n/2

(14.18)

d|n,2d

for z ∈ H. Proof. This follows easily from (14.1), (14.15) and (14.16) and the fact that Gk is a modular or quasimodular form for Γ1 of weight k.  Proposition 14.6 The formal power series Θ(X, z, Y ) in (14.12) can be expanded in the form Θ(X, z, Y ) = θ(z)

 j,≥0

fj, (z)

Xj Y  , j! !

(14.19)

where f0,0 (z) = 1 for all z ∈ H and fj, is a quasimodular form of weight 3j +  for Γ 0 (2).

258

14 Applications of Quasimodular Forms

Proof. Using (14.11), (14.12), (14.13) and (14.14), we see that Θ(X, z, Y ) Θ(X, z, Y )η(z) = θ(z) η(z/2)2  (1 − q n )2 = (1 − q n/2 )2 n>0



(1 − en

2

X/8 n/2

q

ζ)(1 − e−n

2

X/8 n/2 −1

q

ζ

).

n>0,n:odd

Thus we have log

 Θ(X, z, Y )  = 2 log(1 − q n ) − 2 log(1 − q n/2 ) θ(z) n>0 n>0    n2 X/8 n/2 −n2 X/8 n/2 −1 q ζ) + log(1 − e q ζ ) . log(1 − e + n>0,n:odd

Note, however, that  log(1 − q n/2 ) = n>0

 n>0,n:even

=





log(1 − q n/2 ) + 

log(1 − q ) + n

n>0

log(1 − q n/2 )

n>0,n:odd

log(1 − q n/2 ).

n>0,n:odd

Hence we obtain log

Θ(X, z, Y ) = θ(z)





log(1 − en

2

X/8 n/2

q

ζ)

n>0,n:odd



−n2 X/8 n/2 −1



=−

n,r>0,n:odd

q ζ ) − 2 log(1 − q + log(1 − e   2 1 n2 rX/8 r e ζ + e−n rX/8 ζ −r − 2 q nr/2 . r

n/2

)

Using the relation ζ = eY , we have    1 n2 rX/8+rY Θ(X, z, Y ) −n2 rX/8−rY log =− e +e − 2 q nr/2 θ(z) r n,r>0,n:odd

=−

 n,r>0,n:odd

q

nr/2

r



(14.20) j

n2j rj+

j,≥0

+ (−1)



(X/8) Y j! !

j+ 2j j+ (X/8)

n r

j!

j

Y !



 −2

14.2 The Work of Kaneko–Zagier



=−

n,r>0,n:odd

259

q nr/2 r



= −2

 j,≥0,j+>0 j≡ (mod 2)

ξj, (z)

j,≥0,j+>0 j≡ (mod 2)

2n2j rj+

(X/8)j Y  j! !

(X/8)j Y  , j! !

where 

ξj, (z) = =

n2j rj+−1 q nr/2

n,r>0,n:odd ∞  

n2j

 m j+−1

m=1 n>0,n:odd,n|m

n

q m/2

∞   nj+−1 n/2 = q . d−j−1 n=1 d|n,2d

If D is as in (14.5) and if  > j, using (14.17), we have (1)

22j D2j F−j (z) = 22j

∞   −j−1  2j  n n q n/2 = ξj, (z). d 2 n=1 d|n,2d

On the other hand, if j ≥ , from (14.18) we see that (2)

2j+−1 Dj+−1 Fj−+2 (z) = 2j+−1

∞  

dj−+1

n=1 d|n,2d

 n j+−1 2

q n/2 = ξj, (z).

Thus we may write ξj, =

(1)

22j D2j F−j (2) 2j+−1 Dj+−1 Fj−+2

for j < ; for j ≥ .

Using Lemma 14.5 and the fact that the derivative of a quasimodular form (1) of weight k is a quasimodular form of weight k + 2, we see that both D2j F−j (2)

and Dj+−1 Fj−+2 are quasimodular forms of weight 3j +  for Γ 0 (2); hence the same is true for ξj, . Using this and (14.20), we may write the function Θ(X, z, Y ) in the form    ξr,s (z) r s X Y Θ(X, z, Y ) = θ(z) exp −2 8r r!s! r,s≥0,r+s>0 r≡s (mod 2)

260

14 Applications of Quasimodular Forms

= θ(z)



fj, (z)

j,≥0

Xj Y  , j! !

where fj, (z) is a linear combination of expressions of the form (ξr,s (z))p with j = pr and  = ps and f0,0 (z) = 1. Since the function z → (ξr,s (z))p is a quasimodular form for Γ 0 (2) of weight (3r + s)p = 3j + , the same is true  for fj, ; hence the proof of the proposition is complete. We next consider another formal power series H(X, z, Y ) in X and Y given by the infinite product  2 2 H(X, z, Y ) = q −1/24 (1 − wn /8 q n/2 ζ)(1 − w−n /8 q n/2 ζ −1 ), (14.21) n>0,n:odd

 to denote the same function with ζ = eY as before and w = eX . We use H in terms of the variables w, q and ζ, so that  H(w, q, ζ) = H(X, z, Y ).

(14.22)

 0 (w, q) be the coefficient of ζ 0 in the Laurent series expansion of Let H   0 (w, q). H(w, q, ζ) in ζ, and let H0 (X, z) = H Proposition 14.7 The Laurent series expansion of H(X, z, Y ) in ζ is given by    0 (w, wr q)wr3 /6 q r2 /2 ζ r . H(X, z, Y ) = H(w, q, ζ) = (−1)r H (14.23) r∈Z

Proof. From (14.21) and (14.22) we obtain  2  (1 − w(n+2) /8 q (n+2)/2 ζ) H(w, wq, w1/2 qζ) = (wq)−1/24 n>0,n:odd 2

× (1 − w−(n−2) Using this and the identities  2 (1 − wn /8 q n/2 ζ) = (1 − w1/8 q 1/2 ζ) n>0,n:odd



/8 (n−2)/2 −1

q

ζ

(1 − w(n+2)

2

/8 (n+2)/2

q

n>0,n:odd

and 

2

(1 − w−(n−2)

/8 (n−2)/2 −1

q

ζ

n>0,n:odd

= (1 − w−1/8 q −1/2 ζ −1 )

) 

n>0,n:odd

we have

2

(1 − w−n

/8 n/2 −1

q

).

ζ

),

ζ)

14.2 The Work of Kaneko–Zagier

261

1 − w−1/8 q −1/2 ζ −1   H(w, q, ζ) H(w, wq, w1/2 qζ) = w−1/24 1 − w1/8 q 1/2 ζ  = w−1/24 (−w−1/8 q −1/2 ζ −1 )H(w, q, ζ)

(14.24)

 = −w−1/6 q −1/2 ζ −1 H(w, q, ζ).  We now write the Laurent series expansion of H(w, q, ζ) in ζ in the form   r (w, q)wr3 /6 q r2 /2 ζ r .  (−1)r H (14.25) H(w, q, ζ) = r∈Z

Then we have  H(w, wq, w1/2 qζ) =



 r (w, wq)wr (−1)r H

3

/6

(wq)r

2

/2

(w1/2 qζ)r

r∈Z

  r (w, wq)w(r3 +3r2 +3r)/6 q (r2 +2r)/2 ζ r = (−1)r H r∈Z

= w−1/6 q −1/2



 r (w, wq)w(r+1) (−1)r H

3

/6 (r+1)2 /2 r

q

ζ .

r∈Z

Comparing this with (14.24), we have   r (w, wq)w(r+1)3 /6 q (r+1)2 /2 ζ r .  (−1)r H H(w, q, ζ) = −ζ r∈Z

From this and the expansion (14.25) written in the form   r+1 (w, q)w(r+1)3 /6 q (r+1)2 /2 ζ r ,  (−1)r H H(w, q, ζ) = −ζ r∈Z

we obtain

 r+1 (w, q) = H  r (w, wq). H

Thus by induction we have  0 (w, wr q)  r (w, q) = H H for all r ∈ Z, and therefore the expansion in (14.25) can be written as in (14.23).  Let Θ0 (X, z) ∈ Q[[q, X]] be the coefficient of ζ 0 for the Laurent series in ζ of Θ(X, z, Y ) given by (14.12), and consider its Taylor series of the form Θ0 (X, z) =

∞ 

An (z)X 2n ,

n=0

where An (z) is a power series in q belonging to Q[[q]] for each n ≥ 0.

(14.26)

262

14 Applications of Quasimodular Forms

Theorem 14.8 (Kaneko–Zagier) For each nonnegative integer n the coefficient function An in (14.26) is a quasimodular form of weight 6n for Γ1 . Proof. From (14.12), (14.13), (14.19), and (14.21) we see that H(X, z, Y ) =

1 θ(z)  Xj Y  Θ(X, z, Y ) = . fj, (z) η(z) η(z) j! !

(14.27)

j,≥0

On the other hand, if we set  0 (w, q) = H  0 (eX , e2πiz ), H0 (X, z) = H the relation (14.23) can be written as  3 2 (−1)r H0 (X, z + rX/(2πi))er X/6+rY q r /2 . H(X, z, Y ) = r∈Z

Noting that H0 (X, z) =

∞  An 1 Θ0 (X, z) = (z)X 2n , η(z) η n=0

by (14.26) and using the Taylor series expansions er

3

X/6+rY

=

 r3k+ X k Y , 6k k!!

k,≥0

    

An 1 dm An rX rX z+ = z + Xm η 2πi m! dX m η 2πi X=0 m≥0    rm m An D (z) X m = m! η m≥0

with D being as in (14.5), we obtain    rX r3 X/6+Y r2 /2 2n r An z+ e (−1) q X H(X, z, Y ) = η 2πi r∈Z,n≥0    2 r3k++m m An D (z) q r /2 X 2n+m+k Y  = (−1)r k 6 k!!m! η r∈Z,n,m,k,≥0     2 j! Xj Y  m An D (z) . = (−1)r r3k++m q r /2 k 6 k!m! η j! ! j,≥0

m,n,k≥0 2n+m+k=j

Comparing this with (14.27), we have

r∈Z

14.3 Covers of Elliptic Curves

θ(z) fj, (z) = η(z)

263



 m,n,k≥0 k+2n+m=j

An j! Dm (z) 6k k!m! η



(−1)r r3k++m q r

2

/2

.

r∈Z

We now use (14.13) and the identity Ds qr

2

/2

= r2s q r

2

/2

/2s

to obtain θ(z) fj, (z) = η(z)

 m,n,s≥0 2m+2s+6n=3j+

j! 6j−2n−m (j − 2n − m)!m!  2 An (z) (−1)r r2s q r /2 η r∈Z   j! m An D (z) Ds θ(z) 6j−2n−m (j − 2n − m)!m! η 

× Dm

=

 m,n,s≥0 2m+2s+6n=3j+

=

 m,n,s≥0 2m+2s+6n=3j+

   j 2s (2n)! 2n + m 2n + m m 6j−2n−m  × Dm

 An (z) Ds θ(z). η

We note that θ and η are modular forms of weight 1/2 with character and fj, is a quasimodular form of weight 3j +  for Γ 0 (2) by Proposition 14.6. Combining this with the fact that the application of D to a quasimodular form raises its weight by 2 and using induction, we see that An is a quasimodular form of weight 6n for Γ 0 (2). Since Γ1 is generated by Γ 0 (2) and ( 10 11 ), a quasimodular form for Γ 0 (2) is also a quasimodular form for Γ1 ; hence it follows that An is a quasimodular form of weight 6n for Γ1 . 

14.3 Covers of Elliptic Curves In this section we discuss the problem of counting covers of an elliptic curve in connection with quasimodular forms studied by Dijkgraaf (cf. [33]). More details can be found in the expository paper [103] by M. Roth on Dijkgraaf’s work in addition to the original paper [33] of Dijkgraaf. We first recall some properties of holomorphic maps of Riemann surfaces (see [42] and [43] for details). Let M and N be Riemann surfaces, and let f : M → N be a nonconstant holomorphic map. Given a point P ∈ M , we choose local coordinates τ on M vanishing at P and ζ on N vanishing at

264

14 Applications of Quasimodular Forms

f (P ). Then there is a positive integer n and complex numbers ak for k ≥ n with an = 0 such that ∞  ak τ k . ζ = f (τ ) = k=n

Thus we may write ζ = τ n (h(τ ))n = (τ h(τ ))n for some holomorphic function τ → h(τ ) with h(0) = 0. If we use z for the local coordinate τ → τ h(τ ), we have ζ = zn. The positive integer n is the ramification number or multiplicity vf (P ) of f at P , and bf (P ) = vf (P ) − 1 is the branching order of f at P . If bf (P ) ≥ 1, then P is called a branch point of order bf (P ) or a ramification point of order vf (P ). If bf (P ) = 0, or equivalently vf (P ) = 1, then f is said to be unramified at P . We now assume that f : M → N is a nonconstant holomorphic map of compact Riemann surfaces M and N . Then there is a positive integer m such that     m= vf (P ) bf (P ) + 1 = P ∈f −1 (Q)

P ∈f −1 (Q)

for each point Q ∈ N . In this case, the number m is called the degree of f , which we denote by deg(f ), and f is a branched m-sheeted cover of N by M . Since M is compact, there are only a finite number of points P ∈ M such that bf (P ) = 0, and therefore the number  b(f ) = bf (P ), P ∈M

called the total branching order of f , is well-defined. If g is the genus of M , g  is the genus of N , then the m-sheeted cover f satisfies the relation g = m(g  − 1) + 1 + b(f )/2, which is known as the Riemann–Hurwitz formula (see e.g. [42] and [43]). To discuss covers of elliptic curves we fix an elliptic curve E over C, and consider a set S = {α1 , . . . , αb } of b distinct points in E. Definition 14.9 (i) A cover of E of genus g and degree d is an irreducible smooth complex curve C of genus g together with a holomorphic map p : C → E of degree d that is simply branched at each point of S. (ii) Two such covers p1 : C1 → E and p2 : C2 → E are equivalent if there is an isomorphism μ : C1 → C2 such that p1 = p2 ◦ μ.

14.3 Covers of Elliptic Curves

265

We first consider the case of covers of genus greater than one. A holomorphic map of degree d from a curve Cg of genus g > 1 to an elliptic curve E is a d-sheeted connected cover of E, and counting problems of such maps involve combinatorics of the symmetric group Symd of a d-element set. Let Xg,d be the set of simple branched covers of genus g and degree d, so that the branching number of each cover belonging to Xg,d is one or its ramification number is two. If π1b denotes the fundamental group of the b-punctured curve E − S, then we may write Xg,d = Hom (π1b , Symd )/Symd , where the prime indicates that the holonomy around all punctures αi lies in the conjugacy class of single transpositions in Symd and that the resulting cover is a connected curve. The group Symd acts on the homomorphisms by conjugation, and by the Riemann–Hurwitz theorem the number b of branch points is given by b = 2g − 2, which does not depend on the degree d of the map. Given a cover p : C → E, we denote by Autp (C) the group of automorphisms of C over E, that is, Autp (C) = {ψ ∈ Aut(C) | p ◦ ψ = p}. Let Cov(E)Sg,d be the set of equivalence classes of covers of E of genus g and degree d. If (C, p) and (C  , p ) are equivalent covers of E, then we have |Autp (C)| = |Autp (C  )|. We define the weight of each element of Cov(E)Sg,d to be 1/|Autp (C)| when (C, p) is a representative of the given element, and denote by Ng,d the weighted count of Cov(E)Sg,d . The number Ng,d does not depend on the choice of E or S, and it can also be written as Ng,d =

 ξ∈Xg,d

1 , |Autξ|

where the group Autξ of automorphisms of ξ is the product of the centralizer of the image ξ(π1b ) ⊂ Symd and the group Symb permutations of the branch points. The generating function Fg with g > 1 for the numbers Ng,d can now be written as ∞  Fg (q) = Ng,d q d d=1

with q = e2πiz for z ∈ H as before. The case for g = 1 should be treated separately, because in this case the covers are unbranched and therefore there

266

14 Applications of Quasimodular Forms

is a contribution of degree zero maps, namely constant maps. We have ∞

F1 (q) = −

 1 log q + N1,d q d . 24 d=1

We consider a generating function Z(q, λ) for Fg (q) given by a formal power series in λ of the form Z(q, λ) =

=

∞  Fg (q) 2g−2 λ (2g − 2)! g=1 ∞ ∞   g=1 d=1

We also set λ) = Z(q,

∞ ∞   g=1 d=1

Ng,d q d λ2g−2 . (2g − 2)! g,d N q d λ2g−2 , (2g − 2)!

g,d is the weighted number of all, disconnected (i.e., not necessarily where N connected) covers of E with Euler number 2 − 2g and degree d. In fact, we have g,d = 1 |Hom(π 2g−2 , Symd )|, N 1 d! where the holonomy around the b punctures is a cycle of length two in Symd . λ) satisfy the relation Lemma 14.10 The series Z(q, λ) and Z(q, λ) = exp(Z(q, λ)) − 1. Z(q,

(14.28)

Proof. We sketch below the proof, whose details can be found in [103]. If a disconnected cover of genus g and degree d consists of kj components of genus gj and degree dj for 1 ≤ j ≤ r, then we have r  j=1

kj dj = d,

r 

kj (2gj − 2) = 2g − 2.

j=1

We need to compute the weighted count of disconnected covers of genus g and degree d of this type, and this number should be of the form αNgk11d1 Ngk22d2 · · · Ngkrrdr for some constant α, which depends on the choices of branch points taking into account the automorphisms among components of genus 1. We note first that the number of ways of splitting up 2g − 2 branch points into kj sets of 2gj − 2 branch points for 1 ≤ j ≤ r is equal to

14.3 Covers of Elliptic Curves



267

2g − 2 (2g1 − 2)[k1 ] , . . . , (2gr − 2)[kr ]



= (2g − 2)!

 kj with gj >1

r 

1 kj ! 

(2gj − 2)−kj

j=1

kj with gj >1

1 , kj !

where (2gj − 2)[kj ] in the multinomial coefficient is the kj -fold list (2gj − 2)[kj ] = 2gj − 2, . . . , 2gj − 2 of 2gj − 2 for each j ∈ {1, . . . , r}.



Lemma 14.11 If Θ0 (X, z) is as in (14.26), we have Θ0 (X, z) =

∞ 

X) + 1) (1 − q n )(Z(q,

(14.29)

n=0

for all z ∈ H. Proof. Again we shall sketch below the idea of the proof given by M. Roth in [103]. Given a positive integer d, let Md be the square matrix whose rows and columns are indexed by the conjugacy classes of the symmetric group Symd such that the entry (Md )c,c indexed by the conjugacy classes c and c is determined as follows. Let σ2 be a representative of the conjugacy class c. Then (Md )c,c is the number of transpositions τ ∈ Symd such that τ σ2 ∈ c . If π(d) is the number of partitions of d, then π(d) is equal to the number of conjugacy classes in Symd , so that Md is a π(d) × π(d)-matrix. Then it can be shown that ∞  λ) = Z(q, Tr(exp(Md · λ))q d . d=1

If {μ1,d , . . . , μπ(d),d } is the set of eigenvalues of Md , we can write λ) = Z(q,

∞ π(d)  

exp(μi,d λ).

d=1 i=1

If σ is the conjugacy class of a transposition in Symd , the eigenvalues {μ1,d , . . . , μπ(d),d } can be given by   d 1 χ(σ) dim(χ) 2 with χ running through the irreducible characters of Symd . Using this and a formula of Frobenius, it can be shown that the coefficient of ζ 0 in the expansion of

268

14 Applications of Quasimodular Forms



(1 − en

2

X/8 n/2

q

2

ζ)(1 − e−n

X/8 n/2 −1

q

ζ

)

n>0,n:odd

in ζ is equal to

λ) + 1. Z(q,

Then the lemma follows from this and the fact that Θ0 (X, z) is the coefficient of ζ 0 in the Laurent series in ζ of the function Θ(X, z, Y ) given by (14.12).  Theorem 14.12 (Dijkgraaf ) Given g ≥ 2, the function z → Fg (q) is a quasimodular form for Γ1 = SL(2, Z) of weight 6g − 6, and it belongs to Q[E2 , E4 , E6 ], where E2 , E4 and E6 are Eisenstein series in (14.1) Proof. By Kaneko–Zagier’s theorem the coefficient An of X 2n in its power series expansion of Θ0 (X, z) in (14.26) is a quasimodular form for Γ1 of weight 6n, and we see that the same is true for the power series expansion of log(Θ0 (X, z)). Using (14.28) and (14.29), we have log(Θ0 (X, z)) = =

∞  n=0 ∞ 

X) + 1) (1 − q n ) + log(Z(q, (1 − q n ) + Z(q, X).

n=0

Hence the coefficient of X 2g−2 in the power series expansion of log(Θ0 (X, z))  is equal to Fg (q)/(2g − 2)!; hence the theorem follows.

14.4 Partitions In this section we review connections of quasimodular forms with partitions of positive integers studied by Bloch and Okounkov in [10]. The main result of Bloch and Okounkov was proving the quasimodularity of a certain function defined in terms of partitions. A shorter proof of this result was given by Zagier in [115], and we closely follow the exposition of Zagier in this section. Let P be the set of partitions λ = (λ1 , λ2 , . . .) given by nonnegative integers λj with λ1 ≥ λ2 ≥ · · · and λj = 0 for all but finitely many j. The same partition can be represented by a Young diagram Yλ whose j-th row consists of λj boxes. For a partition λ of this form we set  |λ| = λj . j≥1

Given λ ∈ P, we introduce the corresponding sequence of numbers Pk (λ) ∈ Z[1/2] for k ≥ 0, which can be defined by using the Young diagram Yλ as

14.4 Partitions

269

follows. We first consider the Frobenius coordinates of λ given by (r; a1 , . . . , ar ; b1 , . . . , br ), where r is the length of the longest principal diagonal contained in the Young diagram Yλ of λ and the numbers a1 > · · · > ar > 0,

b1 > · · · > br > 0

are the number of cells to the right and below, respectively, of the cells on this diagonal. We then set Cλ = {−b1 − 1/2, . . . , −br − 1/2, ar + 1/2, . . . , a1 + 1/2} ⊂ Z1/2 = Z + 1/2, Pk (λ) =



sgn(c)ck =

c∈Cλ

r  

(ai + 1/2)k − (−bi − 1/2)k



i=1

for k ≥ 0. In particular, we see that P0 (λ) =



sgn(c) = 0,

c∈Cλ

P1 (λ) =



|c| =

c∈Cλ

r 

(ai + bi + 1) = |λ|. (14.30)

i=1

Example 14.13 Let λ = (6, 5, 5, 3, 1, 0, 0, . . .) ∈ P, so that its Young diagram Yλ is as follows: • • •

Then its Frobenius coordinates are given by (3; 5, 3, 2; 4, 2, 1), and we have Cλ = (−9/2, −5/2, −3/2, 5/2, 7/2, 11/2), (P0 , P1 , P2 , . . .) = (0, 20, 20, . . .). Given a partition λ = (λ1 , λ2 , . . .) ∈ P, we set Xλ = {λj − j + 1/2 | j ≥ 1} ⊂ Z1/2 = Z + 1/2. Then we see that + Z+ 1/2 ∩ Xλ = Cλ = {ar + 1/2, . . . , a1 + 1/2}, − Z− 1/2 \ Xλ = Cλ = {−b1 − 1/2, . . . , −br − 1/2}

270

14 Applications of Quasimodular Forms

− with Z+ 1/2 = {a ∈ Z1/2 | a > 0} and Z1/2 = {a ∈ Z1/2 | a < 0}. Let XZ1/2 be the set of all subsets X ⊂ Z1/2 that are bounded from above and whose complements are bounded from below, so that an element X of XZ1/2 is of the form

X = {cm + 1/2, cm−1 + 1/2, . . . , c0 + 1/2, c0 − 1/2, c0 − 3/2, · · · } (14.31) for some c0 , c1 , . . . , cm ∈ Z with c0 < c1 < · · · < cm . Let XZ01/2 be a subset of XZ1/2 given by − XZ01/2 = {X ∈ XZ1/2 | |Z+ 1/2 ∩ Xλ | = |Z1/2 \ Xλ |}.

Then the map λ → Xλ determines a bijection between P and XZ01/2 . Given X ∈ XZ1/2 , we consider the generating formal series wX (T ) =



T x ∈ T 1/2 Z[[T, T −1 ]].

x∈X

We consider the corresponding meromorphic function  WX (z) = wX (ez ) = ezx

(14.32)

x∈X

for C \ 2πiZ, and define the sequence {Qk (X)}k≥0 by the series WX (z) =

∞ 

Qk (X)z k−1

(14.33)

k=0

for 0 < |z| < 2π. If X is as in (14.31), then we have WX (z) = ez/2 (ecm z + · · · + ec1 z ) + e(c0 +1/2)z (1 − e−z )−1 −1  z2 z3 z z/2 cm z c1 z (c0 +1/2)z −1 − + ··· 1− + = e (e + ··· + e ) + e z 2! 3! 4! = ez/2 (ecm z + · · · + ec1 z ) + (1 + (c0 + 1/2)z + · · · )z −1 (1 + z/2 + · · · ). Thus we see that Q0 (X) = 1 and Q1 (X) = m + c0 + 1.

(14.34)

In fact, it can be shown that − Q1 (X) = |Z+ 1/2 ∩ X| − |Z1/2 \ X|.

(14.35)

For example, if ck + 1/2 > 0 > ck−1 + 1/2 for some k with 1 ≤ k ≤ m − 1 in (14.31), we have

14.4 Partitions

271

Z+ 1/2 ∩ X = {cm + 1/2, cm−1 + 1/2, . . . , ck + 1/2}, Z− 1/2 \ X = {−1/2, . . . , c0 + 3/2} \ {ck−1 + 1/2, . . . , c1 + 1/2}, so that |Z− 1/2 \ X| = (|c0 | − 1) − (k − 1) = −c0 − k;

|Z+ 1/2 ∩ X| = m − k + 1,

hence the identity (14.35) follows from this and (14.34). Thus Q1 (X) = 0 if and only if X ∈ XZ01/2 . For each λ ∈ P we set Qk (λ) = Qk (Xλ ),

Wλ (z) = WXλ (z) =

∞ 

Qk (λ)z k−1 ,

(14.36)

k=0

and consider a sequence {βk }∞ k=0 of numbers defined by the power series expansion ∞  z/2 = βk z k . (14.37) sinh(z/2) k=0

Lemma 14.14 The functions Pk and Qk satisfy the relation Qk (λ) =

Pk−1 (λ) + βk (k − 1)!

(14.38)

for each k ≥ 0. Proof. By (14.36) it suffices to show that WXλ (z) =

∞  Pk−1 (λ) k=0

(k − 1)!

z k−1 +

∞ 

βk z k−1 .

k=0

Using (14.30) and (14.37), we have ∞ 

ez/2 1/2 = z , sinh(z/2) e −1

βk z k−1 =

k=0

∞  Pk (λ) k=0

k!

zk = =

r  ∞  

(ai + 1/2)k − (−bi − 1/2)k

i=1 k=0 r   (ai +1/2)z

e

 zk k!

 − e(−bi −1/2)z .

i=1

On the other hand, noting that Xλ = ({ar + 1/2, . . . , a1 + 1/2} ∪ Z− 1/2 ) \ {−b1 − 1/2, . . . , −br − 1/2},

272

14 Applications of Quasimodular Forms

we obtain Wλ (z) =



exz =

r  

∞   e(−k−1/2)z . e(ai +1/2)z − e(−bi −1/2)z +

i=1

x∈Xλ

k=0

However, we have ∞ 

e(−k−1/2)z =

k=0

e−z/2 ez/2 ; = 1 − e−z ez − 1 

hence the lemma follows

Example 14.15 Given λ ∈ P, noting that Xλ ∈ XZ01/2 and using (14.30) and (14.38), we have Q0 (λ) = Q0 (Xλ ) = 1,

(14.39)

Q1 (λ) = Q1 (Xλ ) = 0, Q2 (λ) = Q2 (Xλ ) = P1 (λ) + β2 = |λ| − 1/24, where β2 is determined by the relation (14.37). Let R = Q[Q1 , Q2 , . . .] be the Q-algebra generated by the Qi . If f ∈ R, we use the same letter to denote the functions f : XZ1/2 → Q,

f :P→Q

given by f (X) = f (Q1 (X), Q2 (X), . . .),

f (λ) = f (Xλ ) = f (Q1 (λ), Q2 (λ), . . .)

for X ∈ XZ1/2 and λ ∈ P. We can also define a map f : XZ → Q in a similar manner, where XZ is defined as in XZ1/2 using Z instead of Z1/2 . Let ∂ : R → R be the derivation defined on the generators by k ) = Qk−1 ∂(Q for k ≥ 1 with Q0 = 1. Lemma 14.16 Given ν ∈ Z1/2 , the map 

f → eν ∂ f : R → R is a ring homomorphism. 

Proof. The map f → eν ∂ f clearly preserves addition. Given generators Qa and Qb of R, we have

14.4 Partitions

273 

p ∞  

νp (∂ p−r Qa )(∂ r Qb ) (p − r)!r! p=0 r=0  p    ∞  ν p  p p−r (∂ = Qa )(∂ r Qb ) p! r=0 r p=0



(eν ∂ Qa )(eν ∂ Qb ) =

=

∞  νp

p!

p=0

 ∂ p (Qa Qb ) = eν ∂ (Qa Qb );



hence the same map preserves multiplication by induction. We now consider a map Ψ : Z1/2 × P → XZ defined by Ψ (ν, λ) = Xλ + ν for ν ∈ Z1/2 and λ ∈ P.

Proposition 14.17 The map Ψ determines an isomorphism Z1/2 × P ∼ = XZ such that Q2 (Ψ (ν, λ)) = |λ| − 1/24 + ν 2 /2,

Q1 (Ψ (ν, λ)) = ν,



f (Ψ (ν, λ)) = (eν ∂ f )(λ)

(14.40)

for all f ∈ R and (ν, λ) ∈ Z1/2 × P. Proof. Using (14.32) and (14.33), we have ∞ 

Qk (X + ν)z k−1 = wX+ν (ez ) = eνz wX (ez )

k=0

=

∞ ∞   νj =0 j=0

j!

Q (X)z j+−1 =

k ∞   νj k=0 j=0

j!

Qk−j (X)z k−1

for X ∈ XZ1/2 and ν ∈ Z1/2 . Thus we obtain Qk (X + ν) =

k  νj j=0

j!



Qk−j (X) = (eν ∂ Qk )(X).

(14.41)

Using this and (14.39), we have Q1 (Ψ (ν, λ)) = Q1 (Xλ + ν) = Q1 (Xλ ) + ν = ν, Q2 (Ψ (ν, λ)) = Q2 (Xλ + ν) = Q2 (Xλ ) + Q1 (Xλ )ν + Q0 (Xλ )ν 2 /2 = |λ| − 1/24 + ν 2 /2.

274

14 Applications of Quasimodular Forms

In particular, if Ψ (ν, λ) = Ψ (ν  , λ ) we have ν = Q1 (Ψ (ν, λ)) = Q1 (Ψ (ν  , λ )) = ν  , and therefore Ψ : Z1/2 × P → XZ is a bijection (note that the map λ → Xλ determines a bijection P ∼ = XZ01/2 .) On the other hand, (14.40) follows from 

(14.41) and the fact that R is generated by the Qk and the map f → eν ∂ f is a ring endomorphism of R by Lemma 14.16.  If f : P → Q is a function on the set P of all partitions, its q-bracket is given by

|λ| λ∈P f (λ)q f q = ∈ Q[[q]]. (14.42) |λ| λ∈P q Assuming that f has at most polynomial growth in |λ|, the series in the numerator and denominator converge for all q ∈ C. Using the Dedekind etafunction ∞  η(z) = q 1/24 (1 − q n ), n=1

the q-bracket in (14.42) can be written in the form  1 f q = f (λ)q |λ|−1/24 η(z)

(14.43)

λ∈P

for z ∈ H. = Q[Q2 , Q3 , . . .] of R and Theorem 14.18 If f belongs to the subalgebra R if ϑ is the Jacobi theta series given by (14.7), then we have q = 0, ϑ(∂)f where 

(14.44)

q : Q[[q]]P → Q[[q]] is defined by linearity.

Proof. Given f ∈ R, using (14.43), (14.7) and Proposition 14.17, we have  1 q = )(λ)q |λ|−1/24 ϑ(∂)f (ϑ(∂)f η(z) λ∈P   2  (−1)[ν] (eν ∂ f )(λ)q |λ|−1/24+ν /2 = ν∈Z1/2 λ∈P

=

 

(−1)[Q1 (Ψ (ν,λ))] f (λ)q Q2 (Ψ (ν,λ))

ν∈Z1/2 λ∈P

=



(−1)[Q1 (X)] f (λ)q Q2 (X) .

X∈XZ

We now consider an involution ( )∗ : XZ → XZ defined by

(14.45)

14.4 Partitions

275

X∗ =

X \ {0} if 0 ∈ X; X ∪ {0} if 0 ∈ /X

for X ∈ XZ . Then we have wX ∗ (T ) = wX (T ) ± 1,

∞ 

Qk (X ∗ )z k−1 =

k=0

∞ 

Qk (X)z k−1 ± 1;

k=0

hence we see that Q1 (X ∗ ) = Q1 (X) ± 1,

[Q1 (X ∗ )] = −[Q1 (X)],

Qk (X ∗ ) = Qk (X) Then we have f (X ∗ ) = f (X), and for k ≥ 2. We now assume that f ∈ R. therefore all the terms in the last sum in (14.45) cancel in pairs; hence we q = 0.  obtain ϑ(∂)f

We note that the Q-algebra R = Q[Q1 , Q2 , · · · ] is a graded algebra, which can be written in the form ∞

R=

Rk k=0

with Rk consisting of homogeneous elements of weight k. Similarly, the sub = Q[Q2 , Q3 , . . .] of R considered in Theorem 14.18 is also a graded algebra R k = Rk ∩ R. algebra with R Theorem 14.19 (Bloch–Okounkov) If f ∈ Rk is a homogeneous element of weight k ≥ 0, then its q-bracket f q is a quasimodular form belonging to QMk (Γ1 ). Proof. For k = 0 we have R0 = Q · 1 and 1q = 1. On the other hand, for k = 1 we have R1 = Q1 R. Noting that Q1 (λ) = 0 for λ ∈ P, we see that f (λ) = 0 for each f ∈ R1 . Hence the statement is true for k = 0 and k = 1. does not contain Q1 , we see We now consider f ∈ Rk with k > 1. Since R that Rk has a decomposition of the form R k+1 ). Rk = Q1 Rk−1 ⊕ ∂( for some h ∈ R k+1 . Using (14.8) and Thus we may assume that f = ∂h (14.44), we have 0=

∞  n=0

Hn (z)∂ n+1 (h)q =

∞ 

Hn (z)∂ n (f )q

n=0

= f q + H2 (z)∂ 2 (f )q + H4 (z)∂ 4 (f )q + · · · .

276

14 Applications of Quasimodular Forms

Since Hn is a quasimodular form belonging to QMn (Γ1 ) for each n ≥ 0, and the weight of ∂ n (f ) is k − n < k, we obtain the assertion by induction. 

14.5 Modular Linear Differential Equations In this section we discuss modular linear differential equations (MLDEs), which are linear differential equations with quasimodular coefficients. Solutions of such equations can occur as components of vector-valued modular forms (cf. [44]). Quasimodular solutions can also be found in some cases (see e.g. [53]). On the other hand, solutions of certain types of MLDEs are closely linked to characters of conformal field theories in physics and vertex operator algebras (see e.g. [4] and [47]). If k is a nonnegative integer and E2 is the Eisenstein series in (14.1), we consider the modular derivative operator Dk = q with q = e2πiz . For γ = f : H → C we have

a b c d

k d − E2 dq 12

(14.46)

∈ Γ1 = SL(2, Z) and a holomorphic function

∂(f |k γ)(z) = ∂(J(γ, z)−k f (γz)) = −kcJ(γ, z)−k−1 f (γz) + (∂f )(γz)J(γ, z)−k−2 = −kK(γ, z)(f |k γ)(z) + ((∂f ) |k+2 γ)(z) for z ∈ H, where J(γ, z) and K(γ, z) are as in (1.12); hence we obtain (∂f ) |k+2 γ = ∂(f |k γ) + kK(γ, z)(f |k γ). Using this and the relation (E2 |2 γ)(z) = E2 (z) +

6 K(γ, z) πi

in (14.2), we see that   1 k  ∂f − E2 f  γ (z) 2πi 12 k+2 k 1 ((∂f ) |k+2 γ)(z) − (E2 |2 γ)(z)(f |k γ)(z) = 2πi  12  1 ∂(f |k γ)(z) + kK(γ, z)(f |k γ)(z) = 2πi   6 k E2 (z) + K(γ, z) (f |k γ)(z) − 12 iπ

((Dk f ) |k+2 γ)(z) =



14.5 Modular Linear Differential Equations

277

1 k ∂(f |k γ)(z) − E2 (z)(f |k γ)(z) 2πi 12 = Dk ((f |k γ)(z)). =

Thus we have (Dk f ) |k+2 γ = Dk (f |k γ)

(14.47)

for all γ ∈ Γ1 . From this and by checking the growth condition at the cusps we see that Dk carries modular forms of weight k to modular forms of weight k + 2 for Γ1 . Given a positive integer n, we set (n)

Dk

= Dk+2(n−1) ◦ Dk+2(n−2) ◦ · · · ◦ Dk .

(14.48)

Then it is well known that D2 (E2 ) = −

E4 , 12

D4 (E4 ) = −

E6 , 3

D6 (E6 ) = −

E42 , 2

where E4 and E6 are Eisenstein series in (14.1) that are modular forms for Γ1 of weight 4 and 6, respectively. If we set D=

1 1 d ∂= 2πi 2πi dz

(14.49)

as in (14.5), so that Dk = D −

k E2 , 12

then it can be shown easily that D(E2 ) =

1 (E 2 − E4 ). 12 2

(14.50)

Given nonnegative integers w, k and m, as was described in [44], a modular linear differential equation (MLDE) of order m and weight (w, k) is a linear differential equation of the form m 

(j)

Pw+2(m−j) (E4 , E6 )Dk f = 0,

(14.51)

j=0

where Pd (E4 , E6 ) is a homogeneous polynomial in E4 and E6 of degree d with E4 and E6 regarded as having degrees 4 and 6, respectively. For example, for n = 3 and w = 10 the differential operator in (14.56) is of the form (3)

(2)

c1 E4 E6 Dk + (c2 E62 + c3 E43 )Dk + c4 E42 E6 Dk + (c5 E44 + c6 E4 E62 ) for some c1 , . . . , c6 ∈ C.

278

14 Applications of Quasimodular Forms

Since E4 and E6 are modular forms of weight 4 and 6, respectively, the (j) coefficient Pw+2(m−j) (E4 , E6 ) of Dk in (14.56) is a modular form of weight w + 2(m − j) for Γ1 . On the other hand, using (14.47) and (14.48), we have (n)

(n)

Dk (f |k γ) = (Dk+2(j−1) ◦ Dk+2(j−2) ◦ · · · ◦ Dk )(f |k γ) = (Dk f ) |k+2j γ for γ ∈ Γ1 and a holomorphic function on H; hence we see that (n)

(n)

(Pw+2(m−j) (E4 , E6 )Dk )(f |k γ) = (Pw+2(m−j) (E4 , E6 )Dk f ) |2m+w+k γ for 0 ≤ j ≤ m. Thus it follows that the differential operator in (14.56) carries modular forms of weight k to modular forms of weight 2m + w + k for Γ1 . Example 14.20 We consider an MLDE of order 2 and weight (0, k) given by k(k + 2) (2) E4 (z)f = 0. Dk f − (14.52) 144 This equation can be written in terms of the operator D in (14.49) as follows. Using (14.46) and (14.48), we have   k+2   k (2) E 2 ◦ D − E2 Dk = Dk+2 ◦ Dk = D − 12 12 k+2 k k(k + 2) 2 2 =D − E2 D − (D(E2 ) + E2 D) + E2 . 12 12 144 From this and the identity (14.50) we obtain (2)

k+2 E2 D − 12 k+1 E2 D + = D2 − 6 k+1 E2 D + = D2 − 6 k+1 = D2 − E2 D + 6

Dk = D2 −

k k k(k + 2) 2 (E 2 − E4 ) − E2 D + E2 144 2 12 144 k (E4 + (k + 1)E22 ) 144 k (E4 + (k + 1)(12D(E2 ) + E4 )) 144 k(k + 1) k(k + 2)E4 (DE2 ) + . 12 144

Thus the MLDE in (14.52) can be written as D2 f −

k+1 k(k + 1)  E2 (z)Df + E2 (z)f = 0 6 12

(14.53)

with E2 = D(E2 ). Modular solutions of this equation were studied by Kaneko and Zagier in [55] in connection with liftings of supersingular j-invariants of elliptic curves. To be more specific, if k = p − 1 for some prime p ≥ 5, this equation has a modular solution Fp−1 that is unique up to a constant multiple. If z0 is a zero of Fp−1 , then the value of the j-function at z0 is algebraic and its reduction mod p is a supersingular j-invariant of characteristic p. Conversely,

14.5 Modular Linear Differential Equations

279

all supersingular j-invariants can be obtained in this manner from a solution Fp−1 for certain choices of z0 . For various values of k in (14.53), solutions of the equation were obtained by Kaneko and Koike in [52] that are modular forms for Γ1 and some of its subgroups. Each modular solution was also expressed in terms of a hypergeometric polynomial in a modular function. In addition, in the same paper some quasimodular solutions of the equation were also obtained. Example 14.21 We consider an MLDE of order 3 and weight (0, k) given by   3k 2 + 12k + 8 3 E4 (z)Dk f Dk f + c1 − (14.54) 144   k k 2 (k + 3) + c2 + c1 − E6 (z)f = 0 12 864 for some c1 , c2 ∈ C. Using (14.50) and the identities DE4 = E2 E4 − E6 ,

DE6 =

1 (E2 E6 − nE42 ), 2

the equation (14.54) can be written as   (k + 1)(k + 2)  k+2 3 2 E2 (z)D f + E2 (z) + c1 E4 (z) Df D f− 4 4   k(k + 1)(k + 2)  k E2 (z) + c1 E4 (z) − c2 E6 (z) f = 0 − 24 4 with E2 = DE2 , E4 = DE4 and E2 = D2 E2 . Various aspects of this equation were studied by Kaneko, Nagatomo and Sakai in [81] including its modular and quasimodular solutions. MLDEs also have applications in physics. A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space that is invariant under conformal transformations. Rational conformal field theories are characterized by the property of having a finite number of primary fields. Two-dimensional rational conformal field theories (RCFTs) play an important role in various areas of physics such as string theory, particle physics, statistical mechanics and condensed matter physics. The characters χi of two-dimensional RCFTs are defined as χi (q) = TrHi q −c/24+L0 , where the trace is taken over the Hilbert space Hi of chiral states above the i-th primary state (cf. [46]), and they correspond to primary fields of the full chiral algebra of the full theory. The characters are known to be modular forms that are solutions to MLDEs, and this property can be used, for

280

14 Applications of Quasimodular Forms

example, to prove that the central charge and conformal weights of primary fields in an RCFT are rational numbers. Given an RCFT, the existence of null states in representations of the chiral algebra can be used to obtain an MLDE satisfying the characters of the given RCFT. On the other hand, new RCFTs may be discovered by looking for solutions of MLDEs for their characters (see e.g. [7], [46], [47]).

Example 14.22 For m = 2 and w = k = 0 the MLDE in (14.56) can be written in the form (2) (D0 + cE4 )f = 0 for some c ∈ C. In terms of the operator D the same equation can be written as 1 (14.55) D2 f − E2 Df + cE4 f = 0. 6 For m = 3 and w = k = 0 the operator in (14.56) can be written in the form D20 + c1 E4 D0 + c2 E6 for some c1 , c2 ∈ C. We note that D20 = D4 ◦ D2 ◦ D0 = (D − (1/3)E2 )(D2 − (1/6)E2 D) 1 1 1 = D3 − D(E2 D) − E2 D2 + E22 D 6 3 18  1 1 1 E2 D2 + (DE2 )D − E2 D2 + E22 D = D3 − 6 3 18 1  1 1 E22 − (DE2 ) D. = D 3 − E2 D 2 + 2 18 6 Using this and the identity E22 = 12(DE2 ) + E4 , we have  1 1 1 E22 − (DE2 ) + c1 E4 D + c2 E6 D20 + c1 E4 D0 + c2 E6 = D3 − E2 D2 + 2 18 6   1 1 1 3 2 (DE2 ) + + c 1 E 4 D + c 2 E6 . = D − E2 D + 2 2 18 When c1 = −μ1 /4 and c2 = −μ2 /8, the corresponding MLDE equation is given by 1 1 μ1   1 μ2 D 3 f − E2 D 2 f + (DE2 ) + − E4 Df − E6 f = 0. 2 2 18 4 8

(14.56)

Various properties of solutions of (14.55) and (14.56) in connection with RCFTs have been studied in a number papers (see e.g. [47]).

14.6 Holomorphic Anomaly Equations

281

14.6 Holomorphic Anomaly Equations Quasimodular forms are also related to holomorphic anomaly equations, which generalize the classical genus 0 mirror symmetry. The genus 0 mirror symmetry provides a method of counting the rational curves in a Calabi–Yau threefold (A-model) by studying the variation of Hodge structures of its mirror Calabi–Yau threefold (B-model). Higher genus mirror symmetry is related to counting higher genus curves in a Calabi–Yau threefold. As a candidate for a higher genus B-model, in [8] and [9], Bershadsky, Cecotti, Ooguri and Vafa introduced holomorphic anomaly equations. These equations describe the antiholomorphicity of the topological string amplitudes and recursively relate the genus g topological string amplitude to those of lower genera. This means that in the higher genus mirror symmetry the theory is no longer governed by holomorphic objects but by a combination of holomorphic and antiholomorphic objects. Numerous papers devoted to the study of holomorphic anomaly equations have appeared in recent years (see e.g. [49], [50], [51], [90]). In many cases solutions of holomorphic anomaly equations occur as generating functions of Gromov–Witten invariants, and they are quasimodular forms. In this section we describe a couple of cases, where solutions of holomorphic anomaly equations possess quasimodular properties. We first review some results obtained by Hosono, Saito and Takahashi in [50] (see also [49]). Let S be a rational elliptic surface, which is obtained by blowing up nine base points of two generic cubics in P2 , and let X be a Calabi–Yau threefold that contains S. By using the moduli of stable maps from genus g curves to S, we can consider the Gromov–Witten invariant Ng (β) associated to a cycle β ∈ H2 (S, Z). Let F and σ be the fiber class and a section of the elliptic fibration, respectively, in H 2 (S, Z). Given positive integers n and d, we set  Ng (d, n) = Ng (β), β·σ=d, β·F =n

and consider the corresponding generating function defined by Zg,n (q) =

∞ 

Ng (d, n)q d

d=0

with q = e2πiz . We note that the generating function Fg (q, p) =

∞ 

Zg,n (q)pn

n=0

represents the genus g prepotential in topological string theory. One of the results obtained in [50] is that this function can be written in the form

282

14 Applications of Quasimodular Forms

Zg,n (q) = Pg,n (E2 , E4 , E6 ) where η(q) = q 1/24

∞ 

q n/2 , η(q)12n

(1 − q n ),

n=1

and Pg,n is a homogeneous polynomial of degree 2g + 6n − 2 with E2 , E4 and E6 regarded as having degrees 2, 4 and 6, respectively. In particular, Zg,n (q) is a quasimodular form for the full modular group Γ1 . In addition, the same function satisfies the holomorphic anomaly equation given by n−1 ∂ 1   n(n + 1) Zg−1,n Zg,n = r(n − r)Za,r Zb,n−r + ∂E2 24 24 r=1 a+b=g

with the initial condition Z0,1 (q) =

q 1/2 E4 (q) . η(q)12

We now describe the case of cycle-valued quasimodular solutions of holomorphic anomaly equations studied by Oberdieck and Pixton in [90]. Given a nonsingular elliptic curve, we denote by M g,n (E, d) the moduli space of degree d stable maps of connected curves of genus g to E with n markings. Then there is a natural forgetful morphism π : M g,n (E, d) → M g,n from M g,n (E, d) to moduli space M g,n of stable genus g curves with n markings as well as the map ev ×···×ev

1 n M g,n (E, d) −−− −−−−−→ En

defined by the evaluation maps ev1 , . . . , evn at the n markings. The Gromov– Witten classes of E are then defined by the action of the virtual fundamental class [M g,n (E, d)]vir ∈ H∗ (M g,n (E, d)) on cohomology classes belonging to H ∗ (E). Indeed, given cohomology classes γ1 , . . . , γn ∈ H ∗ (E), if the application of Poincar´e duality on M g,n is suppressed, the corresponding Gromov–Witten class can be written as  Cg,d (γ1 , . . . , γn ) = π∗ [M g,n (E, d)]

vir

n 

ev∗i (γi )



∈ H ∗ (M g,n ).

i=1

Using such classes for various values of d ≥ 0, we can consider the generating series

14.6 Holomorphic Anomaly Equations

Cg (γ1 , . . . , γn ) =

∞ 

283

Cg,d (γ1 , . . . , γn )q d ∈ H ∗ (M g,n ) ⊗ Q[[q]]

(14.57)

d=0

with q = e2πiz . If E2k is the Eisenstein series for a positive even integer 2k in (14.1), we set B2k E2k C2k = − 2k(2k)! with B2k being the Bernoulli number for the index 2k. Then one of the results about the generating series (14.57) obtained by Oberdieck and Pixton in [90] is as follows. Theorem 14.23 Given cohomology classes γ1 , . . . , γn ∈ H ∗ (E), we have Cg (γ1 , . . . , γn ) ∈ H ∗ (M g,n ) ⊗ Q[C2 , C4 , C6 ]. In particular, the generating series in (14.57) is a cycle-valued quasimodular form for the full modular group Γ1 . The Gromov–Witten invariants of E are obtained from the Gromov– Witten classes by integration against the cotangent line classes ψi ∈ H 2 (M g,n ), that is,  E τk1 (γ1 ) · · · τkn (γn )g,d = ψ1k1 · · · ψnkn · Cg,d (γ1 , . . . , γn ), M g,n

so that the corresponding generating series is given by ∞  d=0

 d τk1 (γ1 ) · · · τkn (γn )E g,d q =

ψ1k1 · · · ψnkn · Cg (γ1 , . . . , γn ). M g,n

Thus Theorem 14.23 generalizes the quasimodularity of the Gromov–Witten invariants of elliptic curves proved by Okounkov and Pandharipande in [92] and [93]. In order to discuss the holomorphic anomaly equations we denote by ι : M g−1,n+2 → M g,n the gluing map along the last two marked points. Given positive integers g1 and g2 with g = g1 + g2 and a disjoint union {1, . . . , n} = S1 S2 , let j : M g1 ,S1 {•} × M g2 ,S2 {•} → M g,n be the map which glues the points marked by •, where M gi ,Si is the moduli space of stable curves with markings in the set Si . Then the following theorem by Oberdieck and Pixton in [90] shows that the generating series in (14.57) satisfies holomorphic anomaly equations.

284

14 Applications of Quasimodular Forms

Theorem 14.24 If the generating series Cg (γ1 , . . . , γn ) given by (14.57) is regarded as a polynomial in C2 , C4 , C6 with coefficients in H ∗ (M g,n ), then we have d Cg (γ1 , . . . , γn ) = ι∗ Cg−1 (γ1 , . . . , γn , 1, 1) dC2  + j∗ (Cg1 (γS1 , 1)  (Cg2 , (γS2 , 1)  n   −2 γi ψi · Cg (γ1 , . . . , γi−1 , 1, γi+1 , . . . , γn ), i=1

E

where γSi = (γk )k∈Si .

14.7 Other Applications In Section 14.4 we discussed the work of Bloch and Okounkov on partitions. In this section we describe some additional work by Okounkov and his collaborators related to quasimodular forms as well as some other applications of quasimodular forms.

14.7.1 Gromov–Witten Invariants Although Gromov–Witten invariants were used in Section 14.6 to generate solutions of holomorphic anomaly equations, in this subsection we consider Gromov–Witten invariants for more general varieties over C. Let V be a nonsingular projective variety over C, and β ∈ H2 (V ). Then the corresponding compact moduli space M g,n (V, β) of stable maps of class β is the space of isomorphism classes of data (C, p1 , . . . , pn , f ), where C is a complex projective curve of genus g with n marked points p1 , . . . , pn and f : C → V is a stable map such that [f (C)] = β. Here stable means that components of C that are preimages of points under f have finite automorphism groups. For 1 ≤ j ≤ n we consider the evaluation map evj : M g,n (V, β) → V given by evj : (C, p1 , . . . , pn , f ) → f (pj ). Let Lj be the line bundle on M g,n (V, β) whose fiber at (C, p1 , . . . , pn , f ) is the cotangent space Tp∗j C to C at pj , and let ψj = c1 (Lj ) ∈ H 2 (M g,n (V, β))

14.7 Other Applications

285

be the first Chern class of Lj for 1 ≤ j ≤ n. Then the Gromov–Witten invariants are given by the intersection numbers  τk1 (α1 ) · · · τkn (αn )Vβ,g =

k

M g,n (V, β)ψj j

n 

ev∗j αj ,

j=1

where ev∗j αj ∈ H ∗ (M g,n (V, β)) is the cohomology class obtained by pulling back αj ∈ H ∗ (V ) via the evaluation map evj . In a series of papers [92], [93] and [94] Okounkov and Pandharipande studied Gromov–Witten invariants when V is a curve. When each αj is the Poincar´e dual ω of a point and β = d · [V ], they obtain the explicit formula τk1 (ω) · · · τkn (ω)•V β,g

n   dim λ 2−2g(V )  pk

=

|λ|=d

d!

i=1

(λ) , (ki + 1)! i +1

where g(V ) is the genus of V , the summation is over all partitions λ of d, and dim λ is the dimension of the corresponding irreducible representation of the symmetric group Symd . The bullet in this formula indicates including stable maps with possibly disconnected domains, and the functions pk (λ) on partitions are as described below. We first consider the Hurwitz numbers for the curve V of degree d, which are given by a formula of Burnside involving representations of Symd . If the covering map at pi is of the form z → z ki +1 , then the corresponding Hurwitz number is given by HdV (k1 + 1, . . . , kn + 1) =

n   dim λ 2−2g(V )  |λ|=d

d!

fk+i+1 (λ).

i=1

Thus a correspondence between Gromov–Witten invariants and Hurwitz numbers can be obtained by replacing pki +1 (λ)/(ki + 1)! by fki +1 (λ). The functions fk and pk are examples of shifted symmetric functions, and we have   pk (λ) = (λj − j + 1/2)k − (−j + 1/2)k + (1 − 2−k )ζ(−k), j≥1

where ζ is the Riemann zeta function. We consider the generating function for Gromov–Witten invariants given by G(q) =

∞ 

q d τk1 (ω) · · · τkn (ω)•V β,g ,

d=1

where q = e2πiz with z ∈ H as before. The next theorem provides the quasimodularity of this function, whose proof can be found in [92].

286

14 Applications of Quasimodular Forms

Theorem 14.25 (Okounkov–Pandharipande) The function z → G(q) belongs to the ring Q[E2 , E4 , E6 ]. In particular, it is a quasimodular form for Γ1 = SL(2, Z).

14.7.2 Covers of Pillowcases We now discuss pillowcase covers studied by Eskin and Okounkov [40]. Let T2 = C/L be the complex torus corresponding to a lattice L in C. Then the associated pillowcase orbifold is given by the quotient P = T2 /± of T2 by the automorphism z → −z, which is a sphere with four (Z/2)orbifold points. We note that the natural projection map T2 → P is essentially the Weierstrass ℘-function and that the quadratic differential (dz)2 on T2 descends to a quadratic differential on P with simple poles at the corner points. , λ2 , . . .) with λ1 ≥ λ2 ≥ · · · > 0, we recall Given a partition λ = (λ1 ∞ that its size is given by |λ| = i=1 λi . On the other hand, its length (λ) is defined to be the number of parts, or the largest index k with λk > 0. We can also consider partitions with multiplicities as follows. Let {r1 , r2 , r3 , . . .} be the set of nonnegative integers consisting of only a finite number of positive integers. Then we can consider the corresponding partition λ = 1r1 2r2 3r3 · · · such that rm = |{i ≥ 1 | λi = m}|. In this case we have |λ| =

∞ 

mrm ,

(λ) =

m=1

∞ 

rm .

m=1

We consider covers π:C→P of P of degree 2d by curves C. Given such a cover, we note that a ramification profile over a point in P is given by a finite sequence Π = (μ(1) , . . . , μ(n) ) of (i) (i) partitions μ(i) = (μ1 , μ2 , . . .) for 1 ≤ i ≤ n such that n  ∞ 

(i)

(μj − 1) = 2g − 2,

i=1 j=1

where g is the genus of C. We now choose a cover of the above type satisfying the following ramification data. Let μ be a partition, and let ν be a partition of an even number

14.7 Other Applications

287

into odd parts. When π is regarded as a map to the sphere, it has profile (ν, 2d−|ν|/2 ) over 0 ∈ P and profile (2d ) over the other three corners of P. The map π is also required to have profile (μi , 12d−μi ) over some (μ) given points of P and unramified elsewhere. Then the genus g(C) of C is given by g(C) = (μ) + (ν) − |μ| − |ν|/2. We consider the generating function Z(μ, ν; q) =

 q deg π , |Aut(π)| π

where π runs through all inequivalent covers with ramification data μ and ν as above. If μ = ν = ∅, then the connected cover π is the composition of the maps of the form π T2 →0 T2 → T2 /± with π0 being unramified. In this case we have |Aut(π)| = 2|Aut(π0 )| corresponding to the lifting of ± and  Z(∅, ∅; q) = (1 − q 2n )−1/2 . n

We set

ν; q) = Z(μ, ν; q) . Z(μ, Z(∅, ∅; q)

Then it enumerates covers without unramified connected components, and it is a polynomial in E2 (q 2 ), E2 (q 4 ) and E4 (q 4 ). In particular, it is a quasimodular form. ν; q) is a polynomial Theorem 14.26 (Eskin–Okounkov) The series Z(μ, 2 4 4 in E2 (q ), E2 (q ) and E4 (q ) of weight |μ| + (μ) + |ν|/2. In particular, ν; q) is a quasimodular form for the congruence group Γ0 (4). Z(μ,

14.7.3 Square-tiled Surfaces We now describe the work of Leli`evre and Royer [80] relating quasimodular forms and square-tiled surfaces. A square-tiled surface consists of unit squares with identifications of the opposite sides, so that each top side is identified with the bottom side and each right side is identified with the left side. Assuming that the resulting surface is connected, a square-tiled surface tiled by n squares is a degree n branched cover of the torus

288

14 Applications of Quasimodular Forms

T = C/(Z + iZ) with a single branch point. Each vertex of a square-tiled surface is associated with an angle which is a multiple of 2π. If the number of vertices of the surface is s and if (kj + 1)2π is the angle corresponding to vertex j, we have s 

kj = 2g − 2

j=1

with g being the genus of the surface, and this surface is denoted by H(k1 , . . . , ks ). In this subsection we consider H(2), which is the moduli space of holomorphic 1-forms with a double zero on a compact Riemann surface of genus 2. Then there is a natural action of GL(2, R) on H(2) whose orbits are known as Teichm¨ uller discs. The space H(2) can be tiled by n squares such that each square is a cover of T , and it is a degree n branched cover of T with one double ramification point. The notion of primitivity can be introduced for square-tiled surfaces in such a way that the action of Γ1 on such a surface preserves primitivity as well as the number n of square tiles. Eskin, Masur and Schmoll [38] obtained the formula   3 1 2 (n − 2)n 1− 2 8 p p|n

for the number of primitive n-square tiled surfaces. On the other hand, if n ≥ 5 is prime, it is known that the set of n-square tiled surfaces in H(2) P decomposes into two Γ1 -orbits. If aP n and bn denote the number of n-square tiled surface primitive orbits in each orbit, then Leli`evre and Royer [80] proved the formulas     3 3 1 1 2 P 2 (n − 1)n (n − 3)n = = 1 − , b 1 − , aP n n 16 p2 16 p2 p|n

p|n

which had been conjectured by Hubert and Leli`evre. If n is odd, the surfaces tiled  by0 n squares in H(2) can be divided into two types as follows. The matrix −1 0 −1 determines an involution on H(2) which has six fixed points, known as Weierstrass points of the surface. For squaretiled surface the coordinates of these points belong to the set 12 Z. A surface is of type A (resp. type B) if it has one (resp. three) integer Weierstrass points. For each odd positive integer n, let an be the number of square-tiled surfaces of type A in H(2). We consider the generating series Ψ (z) =

∞  n=0

an q n

14.7 Other Applications

289

for z ∈ H with q = e2πiz as before. Then the following result can be obtained P by using the above-mentioned formulas for aP n and bn . Theorem 14.27 (Leli` evre–Royer) The function Ψ : H → C can be written in the form   1 5 d Ψ (z) = E4 (z) + E2 (z) 1280 πi dz for z ∈ H. In particular, Ψ is a quasimodular form belonging to QM42 (Γ1 ).

14.7.4 Curves on Abelian Surfaces Given a positive integer k, we first note that the sum of divisors function σk (n) is given by  dk σk (n) = d|n

for each positive integer n. For k = 1, the generating function A1 (q) of σ1 can be written in the form A1 (q) =

∞ 

σ1 (n)q n =

k=1

∞  k=1

qk (1 − q k )2

with q = e2πiz . This function was generalized by P. A. MacMahon [82] by considering the generating functions Ak (q) =

 0