Spectral Means of Central Values of Automorphic L-functions for Gl(2) 1470410192, 9781470410193

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EMOIRS M of the American Mathematical Society

Volume 235 • Number 1110 (fifth of 5 numbers) • May 2015

Spectral Means of Central Values of Automorphic L-Functions for GL(2) Masao Tsuzuki

ISSN 0065-9266 (print)

ISSN 1947-6221 (online)

American Mathematical Society

EMOIRS M of the American Mathematical Society

Volume 235 • Number 1110 (fifth of 5 numbers) • May 2015

Spectral Means of Central Values of Automorphic L-Functions for GL(2) Masao Tsuzuki

ISSN 0065-9266 (print)

ISSN 1947-6221 (online)

American Mathematical Society Providence, Rhode Island

Library of Congress Cataloging-in-Publication Data Tsuzuki, Masao, 1968Spectral means of central values of automorphic L-functions for GL(2) / Masao Tsuzuki. pages cm. – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; volume 235, number 1110) Includes bibliographical references. ISBN 978-1-4704-1019-3 (alk. paper) 1. Automorphic functions. I. Title. QA353.A9T78 2015 2014049959 515.9–dc23 DOI: http://dx.doi.org/10.1090/memo/1110

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Dedicated to Professor Takayuki Oda on his 60th birthday

Contents Chapter 1. Introduction

1

Chapter 2. Preliminaries

9

Chapter 3. Preliminary analysis

25

Chapter 4. Green’s functions on GL(2, R)

33

Chapter 5. Green’s functions on GL(2, Fv ) with v a non archimedean place

37

Chapter 6. Kernel functions

41

Chapter 7. Regularized periods

47

Chapter 8. Automorphic Green’s functions

55

Chapter 9. Automorphic smoothed kernels

59

Chapter 10. Periods of regularized automorphic smoothed kernels: the spectral side

67

Chapter 11. A geometric expression of automorphic smoothed kernels

75

Chapter 12. Periods of regularized automorphic smoothed kernels: the geometric side

91

Chapter 13. Asymptotic formulas

101

Chapter 14. An error term estimate in the Weyl type asymptotic law

111

Chapter 15. Appendix

123

Bibliography

127

v

Abstract Starting with Green’s functions on adele points of GL(2) considered over a totally real number field, we elaborate an explicit version of the relative trace formula, whose spectral side encodes the informaton on period integrals of cuspidal waveforms along a maximal split torus. As an application, we prove two kinds of asymptotic mean formula for certain central L-values attached to cuspidal waveforms with square-free level.

Received by the editor April 8, 2010 and, in revised form, April 11, 2013. Article electronically published on October 24, 2014. DOI: http://dx.doi.org/10.1090/memo/1110 2010 Mathematics Subject Classification. Primary 11F72; Secondary 11F67, 11F36. c 2014 American Mathematical Society

vii

CHAPTER 1

Introduction 1.1 Let φ be a weight k holomorphic cuspform on the Poincar´e upper half plane H with respect to the modular group Γ0 (N ) of prime level N . In [41], Ramakrishnan and Rogawski considered the average of central values for a product of the modular L-series L(s, φ) and its quadratic twist L(s, φ ⊗ η) divided by the Petersson norm squared  dxdy φ2 = [SL2 (Z) : Γ0 (N )]−1 |φ(τ )|2 y k 2 y Γ0 (N )\H for such φ’s : (1.1)

 φ∈Fk (N )new ap (φ)∈J

L(1/2, φ) L(1/2, φ ⊗ η) , φ2

where Fk (N )new is an orthogonal basis of the space of weight k( 4) Hecke-eigen holomorphic cuspidal newforms for Γ0 (N ), and the summation is taken for those φ such that its p-th normalized Hecke eigenvalue ap (φ) belongs to a given interval J ⊂ [−2, 2]. When the level N , the prime p and the conductor of the quadratic Dirichlet character η are mutually co-prime, they obtained an asymptotic formula as N → +∞ of the average (1.1), observing that the Sato-Tate measure of J showed up in the main term and explicating an error term. Moreover, they suggested a similar kind of asymptotic formula for Maass cusp forms should be true. One of our aims in this article is to realize this expectation in a setting of automorphic representations of GL(2) over a totally real algebraic number field. We remark that the result of [41] is generalized to holomorphic Hilbert modular case by [10]. To explain the main result about this theme, let us introduce some notation which will be used in this article throughout. Let F be a totally real number field, dF = [F : Q] its degree and oF its maximal order. Let Σfin and Σ∞ be the set of finite places of F and the set of infinite places of F , respectively. The completion of F at a place v ∈ Σfin ∪ Σ∞ is denoted by Fv . If v ∈ Σfin , Fv is a non-archimedean local field, whose maximal order is denoted by ov ; we fix a uniformizer v of ov once and for all, and designate by qv the order of the residue field ov /pv , where pv = v ov is the maximal ideal of ov . Let A and Afin be the adele ring of F and the finite adele ring of F , respectively. For an oF -ideal a, let us designate by S(a) the set of all v ∈ Σfin such that aov ⊂ pv .  Let S be a finite set of places containing Σ∞ , and η = v∈Σ∞ ∪Σfin ηv an idele-class character of order 2 with conductor f such that η is unramified over Sfin = S ∩ Σfin and such that ηv is trivial for any v ∈ Σ∞ . Let I∗S,η be the set of all the square free oF -ideals n with the following properties. 1

2

1. INTRODUCTION

• The set S(n) is disjoint from S(f) ∪ Sfin and has the even cardinality. • ηv (v ) = −1 for any v ∈ S(n). For any ideal n satisfying these conditions, which are sufficient for the sign of the relevant global -factor to be +1 (Lemma 2.3), let Πcus (n) be the set of all the irreducible cuspidal representations π of GL(2, A) with trivial central characters having non zero K∞ K0 (n)-fixed vectors, where K∞ is the standard maximal compact subgroup of GL(2, F ⊗Q R) and    { acvv dbvv ∈ GL(2, ov )| cv ∈ nov }. K0 (n) = v∈Σfin

Let π ∈ Πcus (n) with n ∈ I∗S,η . Then, there exists a family of irreducible smooth representations πv of GL(2, Fv ) forall places v such that πv is unramified for almost all v ∈ Σfin and such that π ∼ = v πv . When v ∈ Σ∞ or v ∈ Σfin prime to n, ν /2 then πv is isomorphic to an unramified principal series representation Iv (| |vv ) 0+ 0+ (see 2.5.1) with the parameter νv belonging to the space Xv , where Xv denotes iR+ ∪(0, 1) or i[0, 2π(log qv )−1 ]∪({0, 2πi(log qv )−1 }+(0, 1)) according to v ∈ Σ∞ or v ∈ Σfin , respectively. Note that the set of Kv -spherical unitary dual of PGL(2, Fv ) coincides with the union of {Iv (| |v )ν/2 | ν ∈ X0+ v } and the set of unitary characters of PGL(2, Fv ) trivial on Kv , where Kv is the standard maximal compact subgroup of PGL(2, Fv ). The spectral parameters at S of π is defined to be the point νπ,S =  0+ 0+ (νv )v∈S lying in the product space X0+ S = v∈S Xv . We endow the set XS with the induced topology from CS . On the one hand, the set Πcus (n) defines a Radon measure ληS (n) on the space X0+ S by  L(1/2, π) L(1/2, π ⊗ η) ληS (n), f = f ∈ Cc0 (X0+ f (νπ,S ), S ), N(n) L(1, π; Ad) ∗ π∈Πcus (n)

Π∗cus (n)

where is the set of all π ∈ Πcus (n) with conductor n, L(s, π) is the standard L-function of π ([27]) and L(s, π; Ad) the adjoint square L-function of π ([11]). We remark that, by the Rankin-Selberg method, the special value L(1, π; Ad) is of π up to an identified with the Petersson norm squared of the newform ϕnew π elementary positive constant. Thus, by non-negativity of the central L-values ([17], [56]), the measure ληS (n) is non-negative. On the other hand, the space X0+ S carries  3/2 a measure dληS = 4DF L(1, η) v∈S dληvv , where DF is the absolute discriminant of F , L(1, η) is the completed Hecke L-series of η and dληvv is a measure on X0+ v supported on the purely imaginary points, on which it is given by iy/2

(1.2) with

(1.3)

dληvv (iy) =

L(1/2, Iv (| |v

iy/2

)) L(1/2, Iv (| |v

iy/2 L(1, Iv (| |v ), Ad)

) ⊗ ηv ) ζF,v (2) dPv (y) L(1, ηv )

⎧

1 − q −iy 2 −1 ⎪ (1 + q ) log q

⎪ v v v ⎪ ⎨

dy,

1 − qv−(1+iy) 4π dPv (y) =

⎪ ⎪ 1 Γ((1 + iy)/2) 2 ⎪

dy, ⎩ 4π Γ(iy/2)

v ∈ Sfin , v ∈ Σ∞

the Kv -spherical Plancherel measure of PGL(2, Fv ). Here, L(s, πv ) denotes the local L-factor of an irreducible smooth representation πv of GL(2, Fv ) and L(1, ηv ) denotes Tates’ local L-factor of ηv .

1.2

3

Then, our first main result is stated as follows. Theorem 1.1. As N(n) → +∞ with n ∈ I∗S,η , the measure ληS (n) converges ∗-wealky to the measure ληS , i.e., ληS (n), f → ληS , f

for any

f ∈ Cc0 (X0+ S ).

As a corollary to this theorem, we have Corollary 1.2. Let {Jv }v∈S be a family of intervals such that Jv ⊂ [1/4, +∞) if v ∈ Σ∞ and Jv ⊂ [−2, 2] if v ∈ Sfin . Then, for any δ > 0, there exists an irreducible cuspidal automorphic representation π with trivial central character having the following properties. (1) The conductor fπ of π belongs to I∗S,η and N(fπ ) > δ. (2) L(1/2, π) = 0 and L(1/2, π ⊗ η) = 0. (3) The spectral parameters νπ,S = (νv )v∈S at S of π satisfies (1 − νv2 )/4 ∈ Jv −ν /2 ν /2 for any v ∈ Σ∞ and qv v + qv v ∈ Jv for any v ∈ Sfin . If we take a function φ on HdF corresponding to the new vector ϕnew on π GL(2, A), then (1 − νv2 )/4 coincides with the v-th Laplace eigenvalue for φ and ν /2 −ν /2 qv v + qv v coincides with the v-th Hecke eigenvalue for φ. Thus, Corollary 1.2 is regarded as an analogue of [41, Corollary B]. We deduce Theorem 1.1 and Corollary 1.2 in the last part of §13 from a more general result Theorem 13.17. Here are some comments on existing works related to our result. Theorem 1.1 means that the Satake parameters of automorphic representations weighted by the central L-values are equidistributed in the parameter space. This kind of equidistribution phenomenon of the Satake parameters (or Hecke eigenvalues) of automorphic forms is first observed by Serre ([46]) and independently by Corney-Duke-Farmer [7] for the elliptic modular forms without the weighting by L-values. A convergence result for spectral average with L-value weighting slightly different from ours was established by Royer ([45]) for the elliptic modular case, and by [41] and [10] for the holomorphic Hilbert modular case with the same L-value weighting as ours. Recently, Shin ([47]) far extended the Serre’s result to the cuspidal automorphic representations whose archimedean components belong to a fixed discrete series Lpacket, working on a general reductive group by means of Arthur’s trace formula applied to the Euler-Poincar´e functions. In somewhat different but related context, Trotabas ([52]) studied the first two mollified moments of L-values L(1/2, π) for cuspidal representations π corresponding to holomorphic Hilbert modular forms. In this work, the central L-values are captured by the Dirichlet series expression (rather than the Euler product expression as we do) and the main tool of investigation is Petersson’s formula for Fourier coefficients extended to the general Hilbert modular setting. 1.2 To explain our second result, we start with the classical situation again. Let 2 2 hyperbolic Laplacian acting on   = −y 2 (∂ 2 /∂x2 +  ∂ /∂x ) be the 2 −2 L SL2 (Z)\H; y dxdy , and {λj = (1−νj2 )/4| j  1 } the non-decreasing sequence of the cuspidal eigenvalues of , counted with multiplicity. Fix an orthonormal system of Maass cusp forms {uj }∞ j=1 such that  uj = λj uj and uj (x + iy) =

4

1. INTRODUCTION

j uj (−x + iy) with some j ∈ {±1} for each j. Then, the L-series of uj is defined to be the absolutely convergent series Lj (s) =

∞  cj (n) , ns n=1

Re(s)  0,

 where n∈Z cj (n) y 1/2 Kνj /2 (2π|n|y) e2πinx is the Fourier expansion of uj (x + iy) ˆ j (s) = ΓR (s − νj ) ΓR (s + at the cusp i∞. As is well-known, the completed L-series L ˆ j (1−s) = νj ) Lj (s) is continued meromorphically to C with the functional equation L ˆ j (s). The asymptotic behaviours of various spectral means of the central values j L Lj (1/2) are extensively studied by Motohashi [37] by means of Kuznetsov’s formula. Among other things, the asymptotic formula of the square mean values (1.4)  |Lj (1/2)|2 κj t

cosh(πκj )

=

2 2 t (log t + CEuler − 1/2 − log(2π)) + O(t(log t)6 ), π2

t → +∞

is obtained ([37, Theorem 2]), where κj = iνj /2 and CEuler is the Euler constant. In this paper, we prove an analogous asymptotic formula not for the mean ˆ j (1/2)|2 ’s in the context of |Lj (1/2)|2 /cosh(πκj )’s, but rather for the mean of |L of automorphic representations of GL(2, F ). This formula also can be viewed as an analogue of multidimensional Weyl’s law for tempered cuspidal multiplicities of automorphic representations ([9], [30]). Theorem 1.3. Let n be any square free ideal. Let η be a real-valued idele × class  character of F unramified 0over n such that ηι (−1) = 1 for all ι ∈ Σ∞ and v∈S(n) ηv (v ) = 1. Let J ⊂ XΣ∞ be a compact subset with smooth boundary, which is “positive”, i.e., Im(νι ) > 0 for any ν ∈ J and for any ι ∈ Σ∞ . Then, for any  > 0, 

3/2

wnη (π)

π∈Πcus (n) νπ,Σ∞ ∈tJ

L(1/2, π) L(1/2, π ⊗ η) 4(1 + δn,oF ) DF = N(n) L(1, π; Ad) (2π)dF

vol(J)

tdF (dF R(η) log t + Cη (F, n))

    + O tdF −1 (log t)3 + O tdF (1+4θ)+ ,

t → +∞,

where tJ = {tν| ν ∈ J },  1 + ηv (v ) N(nf−1 π ) , νv /2 −νv /2 1/2 −1/2 [K0 (fπ ) : K0 (n)] + q )/(q + q ) −1 1 + (qv v v v v∈S(nfπ )   dF (CEuler + 2 log 2 − log π) + log(N(n)1/2 DF ) R(η) Cη (F, n) = CTs=1 L(s, η) + 2  with L(s, η) = ΓR (s)dF v∈Σfin (1 − ηv (v ) qv−s )−1 the completed L-function of η, R(η) = Ress=1 L(s, η) and θ ∈ R is any constant satisfying  (1.5) (1 + |t + b(χι )|)1/4+θ ), t ∈ R |Lfin (1/2 + it, χ)| = O( wnη (π) =

ι∈Σ∞

uniformly for all the unramified idele-class characters χ of  F × with the archimedean ib(χι ) (ι ∈ Σ∞ ). Here, Lfin (s, χ) = v∈Σfin (1−χv (v ) qv−s )−1 components χι (x) = |x| is the L-series of χ. 

1.3

5

We remark that a recent work of Michel and Venkatesh ([32]) provides subconvexity bounds (in any aspect) for a class of automorphic L-functions of GL(1) and GL(2) over an arbitrary number field in a uniform way. In particular, not only a bound (1.5) with θ < 0 but also a subconvexity bound for Lfin (1/2, π) in Laplace eigenvalues aspect also follows from their work. However, we remark here that a subconvex bound for Lfin (1/2, π) in Laplace eigenvalues aspect depending on θ < 0 as above is obtained in the course of the proof of Theorem 1.3: Corollary 1.4. Let n be a square free ideal. Let θ ∈ R be a constant such that (1.5) holds uniformly for all unramified idele-class characters χ of F × . Let J ⊂ X0Σ∞ be a closed cone, which is “positive”, i.e., Im(νι ) > 0 for any ν ∈ J and for any ι ∈ Σ∞ . Then, for any  > 0,   π ∈ Πcus (n)J , |Lfin (1/2, π)| = O (1 + νπ,Σ∞ )dF /2+sup(2dF θ,−1/2)+ , where Πcus (n)J = {π ∈ Πcus (n)| νπ,Σ∞ ∈ J }. If θ < 0, this breaks the convexity bound when π varies over Πcus (n)J for any J in the theorem. 1.3 From now on we set G = GL(2). In order to prove Theorem 1.1 and Theorem 1.3, as in [41], we develop a version of the relative trace formula, which encodes information on toral periods of automorphic forms on G(A) in the spectral side. This kind of formula was invented by H. Jacquet as an apparatus to establish functorial lifts of automorphic representations on different groups in terms of periods of automorphic forms ([23], [25]). Through many works, its importance in the study of L-values is now evident ([23], [24], [17], [26], [1], [31]). A common feature of these works is that several relative trace formulas on different groups for a family of “matching” test functions are considerd simultaneously to be compared. Contrary to these, in [41], a variant of the relative trace formula on a single group G(AQ ) was calculated quite explicitly for a particular test function, whose archimedean component is a matrix coefficient of an integrable discrete series representation. The deduction of the relative trace formula in [41] proceeds like this: one starts with a smooth test function on G(A), makes the kernel function on G(A) × G(A) and then integrates the kernel function along the product of diagonal maximal split torus H(A) against a character, circumventing divergence of the integral by truncating the integration domain. Contrary to this, we start with an adelic Green function which is a continuous but nonsmooth function on G(A) having left H(A)equivariance. Then, at least morally, we deduce our version of relative trace formula by making the Poincar´e series of the adelic Green function and then by integrating the Poincar´e series along an orbit of the split torus H(A) against a character. Actually, we have to avoid various divergence pertaining to these process. First of all, inspired by [48], we regularize the period integral along H(F )\H(A) for non cuspidal automorphic forms as explained in 7.1 without truncating the integration domain. In accordance with this, the adelic Green functions should also be regularized as in 6.4. The resulting relative trace formula (13.1) (see also Lemmas 10.5 and 12.1) is an identity between linear functionals for test functions on the space of spectral parameters. Thus, it looks quite different from those elaborated, for example, in [23] or [41], where the test functions are taken from the space of compactly

6

1. INTRODUCTION

supported smooth functions on the group G(A). This feature of our formula has a novelty for our purpose, because the limiting measure ληS in Theorem 1.1 is almost evident from the term Jηu (n|α) + Jηu¯ (n|α) occurring in the geometric side given in Lemma 12.1. We should remark that, contrary to [10], our framework includes the case when η is trivial. Here is a brief explanation of the structure of this article. In §2, after introducing notation for fundamental objects like Haar measures, characters and representations for various local or adelic groups, we study period integrals of automorphic forms on G(A) in connection with the L-functions of automorphic representations. In the final part of §2, we review a basic theory of Eisenstein series including a computation of intertwining operators for several special vectors of the principal series. In §3, we prepare lemmas necessary to discuss the convergence of series and integrals for functions on G(A) with left H(F )-invariance. The subsection 3.3 is devoted to the study of a space consisting of moderate growth functions on the discrete quotient of G(A); in particular, we establish a density of compactly supported functions (Lemma 3.8), which plays a crucial role in the proof of Lemma 8.2. In §4 and §5, we introduce Green’s functions on G = GL(2) over a local field. When the field Fv is archimedean, such functions were already studied in [54]. (z) Recall that they form a family of continuous functions Ψv (s; gv ) depending on two (z) complex parameters (z, s) and satisfying the equivariance condition Ψv (s; hgk) = (z) χz (h)Ψv (g) for h ∈ H(Fv ) and k ∈ Kv with χz being the quasicharacter of H(Fv ) (z) defined by χz (diag(t1 , t2 )) = |t1 /t2 |z , and most importantly, each function Ψv (s), when regarded as a distribution on G(Fv ), satisfies the differential equation (Green’s equation)   (z) (s) ∗ Ω − λ (s)Ψ (s), f

= f (hk)χz (h)−1 dh dk, f ∈ Cc∞ (G(Fv )) Ψ(z) v v v v H

Kv

with λv (s) = (1 − s )/4, where Ωv is the Casimir element of G(Fv ) (see §4). This kind of function on the upper half plane have been used classically to construct the resolvent of the hyperbolic Laplacian for automorphic forms ([18]), and its higher dimensional analogue for non Riemannian symmetric spaces was studied by [39] and [40] to obtain the automorphic Green current for arithmetic cycles on locally symmetric spaces of unitary groups. Over a non archimedean field Fv , we introduce (z) a similar function Ψv (s; gv ) in analogy with the archimedean counterpart as a function on G(Fv ) with the same (H(Fv ), Kv )-equivariance as above such that it satisfies the inhomogeneous equation 2

(z)

(z) Ψ(z) v (s) ∗ Tv − λv (s) Ψv (s) = Φov ,v , (1+s)/2

(1−s)/2

with λv (s) = qv + qv , where Tv is the Hecke operator corresponding to (z) the double coset Kv diag(v , 1)Kv , and, for an ov -ideal a, Φa,v denotes the unique (z) function on G(Fv ) supported on H(Fv )K0 (a)v and satisfying Φa,v (hk) = χz (h) for any h ∈ H(Fv ) and k ∈ K0 (a)v (Lemma 5.2). In §6, we introduce various kernel functions on G(A) such as the adelic Green function and its smoothing. Given a finite set of places S containing Σ∞ and a family of complex numbers (z, s = {sv }v∈S ), the adelic Green function Ψ(z) (n|s; g) (z) is defined to be the product of the Green functions Ψv (sv ; gv ) over the places

1.3

7

(z)

v in S and the functions Ψnov ,v (gv ) over the places v outside S. Unfortunately, the  adelic Green(z)function behaves too badly along H(A) for the Poincar´e series (n|s; γg) to be convergent absolutely. To compensate this defect, γ∈H(F )\G(F ) Ψ we define its smoothing Ψβ,λ (n|s, g) by auxiliary introducing an even entire function β(z) with fast decay as |Im(z)| → ∞ together with a complex parameter λ, and by taking the contour integral of Ψ(z) (n|s; g) against β(z)/(λ + z) along a vertical contour Re(z) = σ (see 6.4.1). For our purpose, the most important variant of Green’s functions is the regularized smoothed kernel function defined as  #S  1 ˆ Ψβ,λ (n|α; g) = Ψβ,λ (n|s; g) α(s) dμS (s), 2πi Re(s)=c where α(s) is a certain even entire  functionin s, dμS (s) is a holomorphic form on the complex manifold XS := v∈Σ∞ C × v∈S−Σ∞ (C/4πi(log qv )−1 Z) and the integration is the multidimensional contour integral (see 6.4.2). For a cusp form ϕ on G(A) with trivial central character, the integral  (1.6) ϕ(h) η(h) dh Z(A)H(F )\H(A)

is often called the (H, η)-period of ϕ, where Z denotes the center of G. In §7, we introduce a regularization procedure for the (H, η)-period for ϕ not necessarily cuspidal in such a way that the (H, η)-regularized period Pηreg (ϕ) coincides with the absolutely convergent integral (1.6) when ϕ is happend to be cuspidal. We explicitly compute the regularized periods for basic automorphic forms constructed in §2 in terms of the associated L-functions (see Lemmas 7.4, 7.5 and 7.8). Although the computation is quite standard for cuspidal new forms ([27], [5]), some extra work is necessary to treat old forms simultaneously. In §8 and §9, we make the average of the kernel functions constructed in §6 over the discrete orbit space H(F )\G(F ) to obtain the associated automorphic kernel functions. The convergence of the infinite series pertaining to this procedure and the square integrability of the resulting automorphic kernel functions are discussed here. The crucial property of the automorphic Green function  Ψβ,λ (n|s; γg), g ∈ G(A), s = (sv )v∈S ∈ XS Ψβ,λ (n|s; g) = γ∈H(F )\G(F )

with sufficiently large Re(sv ) for all v ∈ S is established in Lemma 8.2, which is a key to deduce the spectral expansion of the automorphic regularized smoothed kernel  ˆ β,λ (n|α; γg), g ∈ G(A) ˆ β,λ (n|α; g) = Ψ Ψ (1.7) γ∈H(F )\G(F )

in 9.1. These functions depend holomorphically on an auxiliary complex parameter λ, which should be kept large for the defining series to be convergent. (Other than λ, the latter function also depends on an ideal n ∈ I∗S,η , an auxiliary entire function β and a test function α on the space XS .) We continue the holomorphic function (1.7) in λ meromorphically to a neighborhood of the point λ = 0, showing that ˆ reg (n|α) to be the constant term at λ = 0 is proportional to β(0); then we define Ψ the proportionality constant. The required analysis is carried out in the last part of §9. In the course (Lemmas 9.7 and 9.9), we need a (weak) polynomial bound of automorphic forms with both the spatial parameter and the spectral parameter

8

1. INTRODUCTION

varied. Fortunatly, such a bound is established in a quite general setting in [13] (for maximal cuspidal Eisenstein series). We quote the result in our setting in the Appendix (Propositions 15.2 and 15.1). ˆ reg (n|α)) using the In §10, we compute the regularized (H, η)-period Pηreg (Ψ ˆ spectral expansion of Ψreg (n|α) given by (9.14). The final outcome of §10 is Theorem 10.5. ˆ reg (n|α)) with the defining series (1.7) subdiIn §11 and §12, we compute Pηreg (Ψ vided according to the double cosets H(F )γH(F ). Having convergence results to be established in §11 which is the most technical section in this article, we explicitly ˆ reg (n|α) according to the above mencompute the regularized (H, η)-period of Ψ tioned subdivision by double cosets. The final result is given in Theorem 12.1. By ˆ reg (n|α)) given by Theorems 10.5 equating the two different expressions of Pηreg (Ψ and 12.1, we arrive at the relative trace formula (13.1). In §13, we prove Theorem 1.1 and Corollary 1.2 under a slightly weaker assumption on η. In §14, imitating the technique used in [37], [9] and [30], we prove Theorem 14.1, from which Theorem 1.3 follows immediately. Acknowledgement The author would like to thank Shingo Sugiyama, who read the manuscript very carefully pointing out various mistakes and inaccuracies therein. Thanks are also due to Masatoshi Suzuki for informing the author of the work [32] when it was a preprint. Notation : The number 0 is included in the set of natural numbers: N = {0, 1, 2, . . . }. We set N∗ = N − {0}. For two non-negative real-valued functions f (x) and g(x) on a set X, we write f (x)  g(x), x ∈ X (or f (x) = O(g(x))) if there exists a positive constant C such that the inequality f (x)  C g(x) holds for any x ∈ X. If f (x)  g(x) and g(x)  f (x), we write f (x)  g(x). If a condition P is given, the Kronecker symbol δ(P) in a generalized sense is defined by requiring that δ(P) is 1 if P is true and is 0 otherwise. For any set X and its subset S, the characteristic function of S is denoted by χS ; thus χS (x) = δ(x ∈ S) for any x ∈ X. For c ∈ R, let Lc denote the vertical line {s ∈ C| Re(s) = c } on the complex plane; when we regard it as a contour, we give it the direction with increasing imaginary part.

CHAPTER 2

Preliminaries 2.1 For v ∈ Σfin , let dv be the local differential exponent of Fv over Qp , where p is the characteristic of ov /pv : v p−d = {x ∈ Fv | trFv /Qp (xov ) ⊂ Zp }. v

Then the global different dF/Q is the ideal of oF such that dF/Q ov = pdvv for all v ∈ Σfin ; the discriminant DF of F/Q is defined to be the absolute norm N(dF/Q ). Then, S(dF/Q ) coincides with the set of ramified places of F/Q. Let ψ be the additive √ character of the adele ring AQ of Q with archimedean component x → exp(2π −1x), x ∈ R. Then, ψF = ψ ◦ trF/Q is a non-trivial additive character of the adele ring A of F , which is decomposed to a product of v = 1, additive characters ψF,v of Fv over all places v of F . We note that ψF,v |p−d v −dv −1 ψF,v |pv = 1. The R-algebra F ⊗Q R is denoted by F∞ , which is a direct product of Fv ∼ =R over all v ∈ Σ∞ . 2.2 Let G be the F -algebraic group GL(2). For any F -subgroup M of G and for a place v of F , the Fv -points of M is denoted by Mv . The points of finite adeles and the F∞ -points of M are denoted by Mfin and M∞ , respectively. Then, MA = Mfin M∞ . The points of finite adeles Gfin of G is realized as a restricted direct product of the local groups Gv = GL(2, Fv ) with respect to the maximal compact subgroups Kv = GL(2, ov ) over all v ∈ Σfin . The direct product  Kfin = v∈Σfin Kv is a maximal compact subgroup of Gfin . The Lie group G∞  is isomorphic to v∈Σ∞ GL(2, Fv ). For each v ∈ Σ∞ , let  Kv be the image of O(2, R) by the isomorphism GL(2, R) ∼ = Gv . Then, K∞ = v∈Σ∞ Kv is a maximal compact subgroup of G∞ and K = Kfin K∞ is a maximal compact subgroup of GA = Gfin G∞ . Let Z be the center of G, which coincides with the scalar matrices in G. Let H be the F -split torus of G consisting of all the diagonal matrices and N the F subgroup of G consisting of all the upper triangular unipotent matrices. Then B = HN is a Borel subgroup of G consisting of all the upper triangular matrices in G. For any place v of F , we have the Iwasawa decomposition Gv = Bv Kv = Hv Nv Kv . 2.3. Haar measures For any place v, we take the self-dual Haar measure dxv of the additive group −d /2 Fv with respect to the duality defined by ψF,v . If v ∈ Σfin , then vol(ov ) = qv v . If v ∈ Σ∞ , then dxv is the Lebesgue measure of Fv = R. Fix a multiplicative 9

10

2. PRELIMINARIES

Haar measure d× xv on Fv× by d× xv = cv dxv /|xv |v , where cv = 1 if v ∈ Σ∞ and − qv−1 )−1 if v ∈ Σfin . We fix a Haar measure of the idele group A× by cv = (1 × d x = v d× xv . For y > 0, let y ∈ A× be the idele such that y ι = y 1/dF for all ι ∈ Σ∞ and y v = 1 for all v ∈ Σfin . Then, y → y is a section of the idele norm | |A : A× → R∗+ , which allows us to identify A× with the direct product of 1 × R× + = {y| y > 0 } and the norm one subgroup A = {x ∈ A | |x|A = 1 }. We fix a 1 1 Haar measure d u on A by requiring   +∞  f (x) d× x = f (y u) d× y d1 u, f ∈ L1 (A× ). A×

A1

0

We fix Haar measures dhv , dnv , dkv and dgv on groups Hv , Nv , Kv and Gv respectively by requiring vol(Kv ; dkv ) = 1 and   t 0 × dhv = d× t1,v d× t2,v if hv = 1,v 0 t2,v , t1,v , t2,v ∈ Fv ,   dnv = dxv , if nv = 10 x1v , xv ∈ Fv , dgv = dhv dnv dkv

if

gv = hv nv kv (hv ∈ Hv , nv ∈ Nv , kv ∈ Kv ).

The adele group GA is decomposed to a direct product of G1A = {g ∈ GA | | det g|A = 1 } and A = {y 12 | y > 0 }. By taking the tensor product of measures dgv on Gv , we fix a Haar measure dg on GA ; then the decomposition GA = G1A A yields the Haar measure d1 g on G1A such that dg = d1 g d× y. By virtue of the Iwasawa decomposition, (2.1)   ϕ(g) d1 g = G1 A

0

+∞





 ϕ

x∈A

(u1 ,u2 )∈(A1 )2

k∈K



1x 0 1

  tu1 0

0 t−1 u2

 k t−2 d× t d1 u1 d1 u2 dx dk,

where dk is the Haar measure on K with total mass 1. 2.3.1. Let g∞ be the complexification of the Lie algebra of G∞ . The universal enveloping algebra of the complexification of a real Lie algebra l is denoted by U(l) and the center of U(l) is denoted by Z(l).  Let g∞ and gι denote the Lie  algebras of G∞ and Gι , respectively. Then, U(g∞ ) ∼ = ι∈Σ∞ U(gι ) and Z(g∞ ) ∼ = ι∈Σ∞ Z(gι ). For any function ϕ on GA , the right translate of ϕ by an element g ∈ GA is denoted by R(g)ϕ, i.e., [R(g)ϕ](x) = ϕ(xg). For a smooth function ϕ on GA , its rightderivation by a differential operator D ∈ U(g∞ ) is also denoted by R(D)ϕ. 2.3.2. Let v ∈ Σfin . The Hecke algebra for the pair (Gv , Kv ) is denoted by H(Gv ; Kv ). Recall that it consists of all the C-valued compactly supported functions on Gv constant on each Kv -double coset; it has the unit given by 1Kv = vol(Kv ; dgv )−1 χKv . We define an element Tv ∈ H(Gv , Kv ) by Tv =

1 v χ . 1 ]K v vol(Kv ; dgv ) Kv [

2.4. L-functions for idele-class characters We refer to [49], [42].

2.4. L-FUNCTIONS FOR IDELE-CLASS CHARACTERS

11

2.4.1. Let v ∈ Σfin . For a unitary character χv of Fv× , let f (χv ) ∈ N be the minimal integer f ∈ N such that χv |Uv (f ) is trivial, where Uv (f ) = 1 + pfv if f > 0 f (χv ) and Uv (0) = o× is the conductor of χv . The root number of χv is v . Thus pv defined by  W (χv ) = qv−f (χv )/2 χ¯v (ξv−dv −f (χv ) ) ψF,v (ξv−dv −f (χv ) ), ξ∈o× v /Uv (f (χv ))

which is a complex number of modulus 1 independent of the choice of the uni −d −f (χv ) −d −f (χv ) formizer v . The integral o× χv (uv v ) ψF,v (uv v ) d× u is called v the Gauss sum associated with χv and is denoted by G(χv ). We have (2.2)

¯v ) G(χv ) = qvf (χv )/2 vol(Uv (f (χv )); d× xv ) W (χ

¯v )/|G(χv )|. The local L-factor of χv is defined by and W (χv ) = G(χ  (1 − χv (v ) qv−s )−1 , (f (χv ) = 0), L(s, χv ) = 1, (f (χv ) > 0). When χv is unramified then it has of the form χv (x) = |x|av v with

av ∈ [0, 2πi(log qv )−1 ); in this case, we set a(χv ) = av , calling it the exponent of χv . Let v ∈ Σ∞ and χv a unitary character of Fv× ∼ = R× . If we write χv (x) = av

v |x|v sgn(x) , x ∈ R with av ∈ iR and v ∈ {0, 1}, then the local L-factor and the root number of χv is defined as L(s, χv ) = ΓR (s + av + v ),

W (χv ) = i v ,

respectively. We set a(χv ) = av (resp. (χv ) = v ), calling it the exponent (resp. the sign) of χv . 2.4.2. Let χ be a unitary idele-class character of F × ; in this article throughout, whenever we consider such a character, it is assumed that it has a trivial restriction f (χv ) for all to R× + . The conductor of χ is the oF -ideal fχ such that fχ ov = pv v ∈ Σfin . For latter purpose, following [21, Chapter 5], it is convenient to introduce the analytic conductor q(χ) of χ by setting  q(χ) = N(fχ ) (3 + |a(χv )|) v∈Σ∞

with (a(χv ))v∈Σ∞ the exponents of χ at archimedean places. Note that W (χv ) = G(χv ) = 1 for almost all v ∈ Σfin . Set   W (χ) = W (χv ), G(χ) = G(χv ). v∈Σfin ∪Σ∞

Lemma 2.1. Set  =



v∈Σ∞ v .

v∈Σfin

Then,

¯ W (χ) = i DF N(fχ )−1/2 {(oF /fχ )× } G(χ). 1/2

−d /2

−d /2−f

(1−qv−1 )−1 Proof. The local volume vol(Uv (f ); d× t) equals qv v or qv v  f (χ ) according to f = 0 or f > 0, respectively. Since (oF /fχ )× = f (χv )>0 qv v (1 − −1  qv ), we have the result from (2.2).

12

2. PRELIMINARIES

The L-function L(s, χ) of χ, initially defined on Re(s) > 1 by the convergent Euler product of local L-factors L(s, χv ) over all places v, is continued to a meromorphic function on C satisfying the functional equation (2.3)

1/2−s

L(s, χ) = W (χ)DF

N(fχ )1/2−s L(1 − s, χ). ¯

The function L(s, χ) is holomorphic except possible simple poles at s = 0, 1 with (2.4) Ress=1 L(s, χ) = δ(χ = 1) vol(F × \A1 ),

Ress=0 L(s, χ) = −δ(χ = 1) DF vol(F × \A1 ). 1/2

ordv a For any oF -ideal a, let xa = (xa,v ) ∈ A× for fin be an idele such that xa,v = v ordv (a) all v ∈ Σfin , where ordv (a) ∈ Z is defined by aov = pv . We designate the value χ(xa ) by χ(a). ˜ If χ is the trivial character 1, then L(s, 1) coincides with the Dedekind zeta function multiplied with the gamma factor, which is denoted by ζF (s) in this article. The Euler product of L(s, χv ) over v ∈ Σfin is denoted by Lfin (s, χ).

2.4.3. Let Ξ0 be the set of all the unramified unitary idele-class character of F × . There exists a constant C1 such that Lfin (s, χ) has no zero in a region of the form Re(s)  1 − C1 / log{q(χ)(2 + |Ims|)}, |Ims| = 0 ([21, Theorem 5.10]). Then, from the proof of [50, Theorem 3.11], we obtain the estimations (2.5) |Lfin (1 + it, χ)−1 | = O(log q(χ| |it A )),



L (1 + it, χ)/Lfin (1 + it, χ) = O(log q(χ| |it )), fin A

t∈R

with the implied constants independent of χ ∈ Ξ0 . It is known that there exists θ ∈ R such that the L-series Lfin (s, η) for an arbitrary idele-class character η of F × admits a bound on the critical line Re(s) = 1/2 of the form (2.6)

1/4+θ |Lfin (1/2 + it, η)|  q(η| |it , A)

t∈R

with the implied constant independent of η. Actually, the bound with θ = 0 (the convexity bound) is known for a while (see [33, 3.1]). Any bound (2.6) with θ < 0 is called a subconvexity bound, which is established by Michel and Venkatesh in this generality ([32]). 2.4.4. Let UF+ be the set of totally positive unit of oF , viewed as a subset of F∞ by the natural embeding. Then, the additive group log UF+ = {(lv )v∈Σ∞ | lv ∈ R, (exp(lv ))v ∈ UF+ } is a lattice in the dF − 1 dimensional real vector space V =  {(xv )v∈Σ∞ | xv ∈ R, v xv = 0 }. Let L0 be the dual lattice of log UF+ in V , i.e.,  L0 = {(bv )v∈Σ∞ ∈ V | bv lv ∈ Z (∀(lv ) ∈ log UF+ ) }. v

Let χ be a unitary idele-class character of F × . The exponents a(χv ) (v ∈ Σ∞ ) of χ at the archimedean places are purely imaginary numbers with the constraint  a(χv ) = 0, v∈Σ∞ × = 1. If we set bv (χ) = Im{a(χv )} for which comes from our convention χ|R+ v ∈ Σ∞ , then the vector b(χ) = (bv (χ)) belongs to the lattice L0 . The mapping χ → b(χ) is a surjection from Ξ0 onto L0 whose kernel Ξ00 coincides with the set of finite order characters in Ξ0 . Note that Ξ00 is a finite abelian group with order hF , the class number of F .

2.5. REPRESENTATIONS OF LOCAL GROUPS AND THEIR L-FACTORS

13

2.5. Representations of local groups and their L-factors Let v be a place of F and πv an irreducible unitarizable smooth representation of Gv . We always assume that πv is infinite dimensional, or equivalently generic; let Vπv = W(πv , ψF,v ) be the ψF,v -Whittaker realization of πv . 2.5.1. Let v ∈ Σfin . Then, the maximal compact subgroup Kv of Gv admits a decreasing filtration by open compact subgroups   K0 (pnv ) = { ac db ∈ Kv | c ≡ 0 (mod pnv ) }, n ∈ N. K (pn )

From [6], there exists a unique integer c(πv ) ∈ N such that dimC Vπv 0 v = sup(n− c(πv ) + 1, 0) for any n ∈ N. For any quasi-character ηv of Fv× , the integral  (2.7) φv,0 ([ 0t 01 ]) ηv (t) |t|s−1/2 d× t, φv ∈ Vπ v Zv (s, ηv ; φv ) = v Fv×

is called the local zeta integral. By the theory of local new forms (cf. [44, §1]), c(πv ) K (pv )

there exists a unique element φ0,v ∈ Vπv 0

such that

(2.8) × −dv dv (s−1/2) Z(s, ηv ; φ0,v ) = vol(o× qv L(s, πv ⊗ ηv ), v ; d t) ηv (v )

Re(s)  0

for any unitary unramified character ηv of Fv× . Let us recall the construction of unramified principal series. For any unramified quasi-character χv of Fv× , let Iv (χv ) be the space of all the smooth functions fv :      1/2 Gv → C satisfying fv t01 tx2 g = χv (t1 /t2 ) |t1 /t2 |v fv (g) for any t01 tx2 ∈ Bv . We let the group Gv act on Iv (χv ) by right translation. The space Iv (χv ) contains (χ ) (χ ) a unique Kv -invariant function f0,vv such that f0,vv (e) = 1. For our purpose, we need information on the structure of the K0 (pv )-fixed part of Iv (χv ). To describe it, set (2.9) (χ )

f1,vv = πv



v−1 0 0 1



(χ )

(χ )

f0,vv − Q(πv ) f0,vv

with Q(πv ) =

χv (v ) + χv (v )−1 1/2

qv

−1/2

.

+ qv

We define a Gv -intertwining operator Iv (χv ) → W(Iv (χv ), ψF,v ) by mapping f ∈ Iv (χv ) to the function φ ∈ W(Iv (χv ), ψF,v ) such that   1x   −dv [ 0 1 ] gv ψF,v (−xv ) dxv , (2.10) φ(gv ) = χv (v ) f 01 −1 gv ∈ Gv , 0 Fv

where the integral is interpreted as in [5, p.498, (6.9)]. Let φi,v (i = 0, 1) be (χ ) the image of (1 − χv (v )2 qv−1 )−1 fi,vv in W(Iv (χv ), ψF,v ) under the intertwining operator. Then, from [5, Proposition 4.6.8], we have (2.11) ν (m+dv +1) −ν (m+dv +1)  m  qv v − qv v φ0,v 0v 01 = δ(m + dv  0) qv−(m+dv )/2 , qvνv − qv−νv

m∈Z

with qvνv = χv (v ). We remark that a small modification is necessary because dv is not necessarily 0 in our case.

14

2. PRELIMINARIES

Lemma 2.2. Suppose Iv (χv ) is irreducible and unitarizable. Then, the number (χ ) (χ ) Q(πv ) is real. The vectors f0,vv and f1,vv yield a basis of the two dimensional K (pv )

space Vπv 0

such that

(2.12)

(f1,vv |f0,vv ) = 0,

(χ )

(χ )

(χ )

(χ )

f1,vv 2 = {1 − Q(πv )2 }f0,vv 2 ,

for any Gv -invariant hermitian inner product ( | ) on Iv (χv ). Let ηv be a unitary unramified character of Fv× . Then, for sufficiently large Re(s), we have (2.13) (2.14)

× −dv dv (s−1/2) Zv (s, η; φ0,v ) = vol(o× qv L(s, Iv (χv )), v ; d t) ηv (v )

Zv (s, ηv ; φ1,v ) = (ηv (v )qv1/2−s − Q(πv )) Zv (s, η; φ0,v ). (χ )

Proof. In this proof, we abbreviate fj,vv to fj,v . Let Tv be as in 2.3.2. From [5, Proposition 4.6.6], the operator πv (Tv ) acts on the space VπKv v by the scalar 1/2 qv (χv (v )+χv (v )−1 ). By the unitarity, we have the equation (πv (Tv )f0,v |f0,v ) = (f0,v |πv (Tv )f0,v ), from which Q(πv ) ∈ R is inferred. By the unitarity together with the Kv -invariance of f0,v , we have  −1    −1  (f0,v |πv 0v 01 f0,v ) = vol Kv 0v 01 Kv ; dgv (πv (Tv )f0,v |f0,v ) (2.15) 1/2 qv (χv (v ) + χv (v )−1 ) = f0,v 2 . 1 + qv     Note that vol Kv 0v 01 Kv ; dgv = (1 + qv ) vol(Kv ; dgv ) from [5, p.494, Eq.(6.4)]. Now, by (2.15), the formula (2.12) is proved In particular, f0,v and easily.  f1,v are  −1

−1

linearly independent. Using the relation πv 0v 01 f0,v ([ 0t 10 ]) = f0,v t0v 01 , the formula (2.14) is proved by a change of variable. When dv = 0, the formula (2.13) is inferred from [5, Proposition 3.5.3] together with [5, p.358, Eq.(7.33)]; the  argument is easily extended to the case dv > 0 with a minor modification. For our purpose, we need only representations πv with c(πv ) = 0 or 1: ∼ Iv (| |νv ) with • If c(πv ) = 0, i.e., πv is Kv -spherical, then, πv = v

(qv−νv , qvνv ),

(νv ∈ C/2πi(log qv )

−1

Z),

the Satake parameter of πv . We have L(s, πv ⊗ ηv ) = (1 − ηv (v )qv−(s+νv ) )−1 (1 − ηv (v )q −(s−νv ) )−1 for any unramified character ηv of Fv× . The local new form φ0,v corre−(1+2νv ) −1 (χv ) sponds to the vector (1 − qv ) f0,v in Iv (χv ) with χv = | |νvv . • If c(πv ) = 1, then there exists v ∈ {0, 1} such that πv is isomorphic to the twist of the Steinberg representation Stv ⊗ sgn vv , or equivalently, 1/2 the unique proper subrepresentation of Iv (sgn vv | |v ), where sgnv is the unique unramified character of Fv× such that sgnv (v ) = −1. We have L(s, πv ⊗ ηv ) = (1 − (−1) v ηv (v ) qv−(s+1/2) )−1 , (1/2, πv ⊗ ηv , ψF,v ) = −(−1) v ηv (v ) for any unramified character ηv of Fv× . By the intertwining operator Iv (χv ) → W(Iv (χv ), ψF,v ) defined by (2.10), the local new form φ0,v of πv

2.6. CUSPIDAL AUTOMORPHIC REPRESENTATIONS AND THEIR L-FUNCTIONS

corresponds to the vector (1 − qv−2 )−1 {f0,vv − (−1) v πv (χ )



v−1 0 0 1



15 (χ )

f0,vv }

1/2

in Iv (χv ) with χv = sgn vv | |v . 2.5.2. Let v ∈ Σ∞ and πv a Kv -spherical unitarizable (gv , Kv )-module. Then, πv is isomorphic to the principal series Iv (| |νv ) constructed in the same way as (ν) (ν) the pv -adic case: it contains a unique Kv -fixed vector f0,v such that f0,v (e) = 1. ν Then, the isomorphism πv ∼ = Iv (| |v ) can be fixed in such a way that the local new vector φ0,v is the element of the Whittaker model Vπv which corresponds to (ν) ΓR (1 + 2ν) f0,v . We identify the Poincar´e upper half plane H with the homogeneous   space SL2 (R)/SO(2) by assigning the point (ai + b)/(ci + d) of H to a matrix ac db in SL2 (R). Let Ωv be the Casimir element of Gv ∼ = GL(2, R) which corresponds to (−2) times the hyperbolic laplacian −y 2 (∂ 2 /∂x2 + ∂ 2 /∂y 2 ) on H. Then, πv (Ωv )φ0,v =

(2ν)2 − 1 φ0,v , 2

and L(s, πv ⊗ ηv ) = ΓR (s − ν + a + ) ΓR (s + ν + a + ) for any character ηv (x) = |x|a sgn(x) of R× with a ∈ iR,  ∈ {0, 1}. 2.6. Cuspidal automorphic representations and their L-functions Let (π, Vπ ) be an irreducible cuspidal automorphic representation of GA with trivial central character, where the representation space Vπ is contained in the subspace of K-finite functions in the L2 -space L2 (GF ZA \GA ; dg). Let η be a unitary idele-class character of F × . The global zeta integral of π ⊗ η is defined by  s−1/2 × (2.16) ϕ ([ 0t 01 ]) η(t) |t|A d t, ϕ ∈ Vπ , Z(s, η; ϕ) = F × \A×

which converges absolutely for any s ∈ C giving an entire function on C. The c(π ) conductor of π is, by definition, the oF -ideal fπ such that fπ ov = pv v for all v ∈ Σfin . Fix a family {πv }v of irreducible smooth representations of Gv for all places v such that π ∼ = v πv . 2.6.1. Suppose fπ is square free and π is K∞ -spherical; thus πv is one of representations recalled in 2.5.1 and 2.5.2. Let Vπv = W(πv , ψF,v ) be the ψF,v Whittaker realization and φv,0 ∈ Vπv the local new vector. At almost all v, the Whittaker function φ0,v is Kv -invariant and φv,0 (e) = 1 ([5, Proposition 3.5.3]). 2 We fix a Gv -invariant hermitian inner product ( | )v on Vπv such that φ 0,v  = 1, ∼ once and for all. Furthermore, we assume that the identification  Vπ = v Vπv is made so that if ϕ ∈ Vπ corresponds to the decomposable tensor v φv then the globalWhittaker function Wϕ (g) defined by the integral [5, Eq.(5.19)] coincides with v φv (gv ) for any g ∈ GA . Lemma 2.3. Let π be an irreducible cuspidal automorphic representation with trivial central character such that fπ is square free and π is K∞ -spherical. Let η be an idele-class character of F × such that η 2 = 1 and such that fη is relatively prime to fπ . Then, L(1/2, π) L(1/2, π ⊗ η) = 0

unless

η˜(fπ ) = 1.

16

2. PRELIMINARIES

Proof. From assumptions, we have

⎧ ⎪ ⎨ηv (−1), (1/2, πv , ψF,v ) v (1/2, πv ⊗ ηv , ψF,v ) = ηv (v ), ⎪ ⎩ 1,

(v ∈ Σ∞ ∪ S(fη )), (v ∈ S(fπ )), (v ∈ Σfin − S(fπ fη ).

Thus, (1/2, π) (1/2, π ⊗ η) = η˜(fπ ) = ±1. Since η 2 = 1, the central character of π ⊗ η is also trivial. Hence the claim is a consequence of the functional equation relating the values at s and 1 − s of L(s, π) L(s, π ⊗ η).  2.6.2. Fix an oF -ideal n such that n is square free. Then, the product K0 (n) = K0 (n)K∞ = {0} v∈Σfin K0 (nov ), is an open compact subgroup of Kfin . Suppose Vπ



K (no )

from now on. Then, Vπv 0 v = {0} for all v ∈ Σfin . This implies that fπ divides n. For any ideal c dividing nf−1 π , let ϕπ,c be the function in Vπ corresponding to the decomposable tensor   { φ1,v } ⊗ { φ0,v } v∈S(c)

v∈S(c)

 under the identification Vπ ∼ = v Vπv . The function ϕπ,oF is called the newform of π and is denoted by ϕnew π . Lemma 2.4. The functions ϕπ,c (nf−1 ⊂ c) form an orthogonal basis of the π K0 (n)K∞ , and invariant part Vπ  ϕπ,c 2 (2.17) = {1 − Q(πv )2 }. new 2 ϕπ  v∈S(c)

Proof. Since π is irreducible, there existsa positive constant C such that the 2 -inner product on Vπ coincides with C · { v ( | )v } by the isomorphism Vπ ∼ L =  V . Since c is relatively prime to f , π v πv  2 ϕπ,c  = C {1 − Q(πv )2 } v∈S(c)

by Lemma 2.2. Taking c = oF , we obtain C = ϕπ,oF 2 . This shows (2.17).



Let η be a unitary idele-class character of F × such that (2.18)

η 2 = 1,

(2.19)

ηv (−1) = 1 for any v ∈ Σ∞ ,

(2.20)

fη is relatively prime to n and η˜(n) = 1. K0 (n)K∞

Instead of the zeta integral (2.16), which vanishes identically for ϕ ∈ Vπ unless η = 1, we consider     ϕ ∈ VπK0 (n)K∞ , Z ∗ (s, η; ϕ) = η(x∗η ) Z s, η; π 10 x1η ϕ ,

−f (η )

where xη (resp. x∗η ) is the adele (resp. idele) whose v-component is v v or 0 (resp. 1) according to v ∈ Σfin or v ∈ Σ∞ , respectively. The factor η(xη ) is included to make the expression independent of the choices of uniformizers v . Lemma 2.5. Let η be as above. Then, for any c dividing nf−1 π , we have    s−1/2 ∗ 1/2−s ηv (v )qv G(η) { − Q(πv ) } L(s, π ⊗ η). Z (s, η; ϕπ,c ) = DF v∈S(c)

2.6. CUSPIDAL AUTOMORPHIC REPRESENTATIONS AND THEIR L-FUNCTIONS

17

Proof. By a standard argument found in [5, pp.335–336], the integral Z ∗ (s, η; ϕπ,c ) is decomposed to a product of local ones

    φv Zv∗ (s, ηv ; φv ) = ηv (x∗η,v ) Zv s, ηv ; πv 10 xη,v 1

over all v, where φv is φ1,v or φ0,v according as v belongs to S(c) or not. If ηv is unramified, then Zv∗ (s, ηv ; φ0,v ) = Zv (s, ηv ; φ0,v ) is given by (2.8). If πv and ηv are both unramified, then Zv∗ (s, ηv ; φ1,v ) is evaluated in Lemma 2.2. It re∗ mains to calculate  the integral   Zv (s, ηv ; φ0,v ) when πv is unramified and f (ηv ) > 0. Since φ0,v [ 0t 01 ] we have

Zv∗ (s, ηv ; φ0,v ) =

1 v−f (ηv ) 0 1

 Fv×

qv−(m+dv )/2

ν (m+dv +1)

qv v

qvνv

m−dv

 ×{

) φ0,v ([ 0t 10 ]), by substituting (2.11),

φ0,v ([ 0t 01 ]) ψF,v (tv−f (ηv ) ) ηv (tv−f (ηv ) )|t|s−1/2 d× t v



=

−f (ηv )

= ψF,v (tv

o× v

−ν (m+dv +1)

− qv v − qv−νv

ηv (v )m qv−(s−1/2)m

ψF,v (uvm−f (ηv ) ) ηv (uv−f (ηv ) ) d× u}.

The integral in the last line is evaluated as δ(m = −dv ) ηv (vdv ) G(ηv ). Thus, only the term for m = −dv survives in the last summation, yielding the identity d (s−1/2) Zv∗ (s, ηv ; φ0,v ) = qv v G(ηv ). Since πv is unramified and ηv is ramified, the L-factor L(s, πv ⊗ ηv ) = 1. This completes the proof.  Lemma 2.6. Let π be an irreducible cuspidal automorphic representation with K (n)K∞ trivial central character such that Vπ 0 = {0}. Let η be an idele-class character × of F satisfying the conditions (2.18), (2.19) and (2.20). Then, the number  Z ∗ (1/2, 1; ϕ) Z ∗ (1/2, η; ϕ) Pη (π; K0 (n)) = ϕ∈B K0 (n)K∞

is independent of the choice of an orthonormal basis B of Vπ (2.21) −1/2

Pη (π; K0 (n)) = DF

G(η) {



v∈S(nf−1 π )

. We have

1 + ηv (v ) L(1/2, π) L(1/2, π ⊗ η) . } 2 1 + Q(πv ) ϕnew π 

The number G(η)−1 Pη (π; K0 (n)) is non negative. Proof. The first assertion is obvious. To show the second statement, we take B = {ϕπ,c /ϕπ,c  | nf−1 π ⊂ c }. Then, by Lemmas 2.4 and 2.5 , 

Pη (π; K0 (n)) =

ϕπ,c −2 Z ∗ (1/2, 1; ϕπ,c ) Z ∗ (1/2, η; ϕπ,c )

nf−1 π ⊂c

={





v∈S(c) nf−1 π ⊂c

(ηv (v ) − Q(πv ))(1 − Q(πv )) L(1/2, π) L(1/2, π ⊗ η) } G(1)G(η) . 2 1 − Q(πv )2 ϕnew π 

In the last line, the sum in the bracket is evaluated as     1 + ηv (v ) ηv (v ) − Q(πv ) . 1+ = 1 + Q(πv ) 1 + Q(πv ) −1 −1 v∈S(nfπ )

v∈S(nfπ )

18

2. PRELIMINARIES −1/2

Since G(1) = DF , this completes the proof of the second statement. From Lemma 2.2 and (2.17), Q(πv ) ∈ (−1, 1). Combining this with the nonnegativity of L(1/2, π) L(1/2, π ⊗ η) proved in [17], we have the second statement.  Corollary 2.7. Let π be as in Lemma 2.6. Let η be a unitary idele-class character of F × satisfying the following condition together with (2.18), (2.19) and (2.20). ηv (v ) = −1 for any v ∈ S(n).

(2.22)

Then, Pη (π; K0 (n)) is zero unless fπ = n, in which case it equals −1/2

DF

G(η)

L(1/2, π) L(1/2, π ⊗ η) . 2 ϕnew π 

2.7. Eisenstein series and their L-functions ν/2

For any unitary idele-class character χ and for any ν ∈ C, let I(χ| |A ) be the space of all the smooth complex valued functions f on GA such that      (ν+1)/2 f t01 tx2 g = χ(t1 /t2 ) |t1 /t2 |A f (g) for all t01 tx2 ∈ BA . ν/2

The group GA acts on I(χ| |A ) by the right translation. Then, the hermitian pairing  ν/2 −¯ ν /2 f |f  = (2.23) f (k) f¯ (k) dk, f ∈ I(χ| |A ), f  ∈ I(χ| |A ) K

ν/2

is GA -invariant. For each place v, we have defined a local counterpart Iv (χv | |v ) in 2.5.1. In the context of Eisenstein series, we always consider the Gv -hermitian ν/2 −¯ ν /2 paring between Iv (χv | |v ) and Iv (χv | |v ) similarly defined as (2.23) by integraν/2 tion on Kv , with respect to which Iv (χv | |v ) is unitary if ν ∈ iR. The global space ν/2 ν/2 I(χ| |A ) is identified with the restricted tensor product of Iv (χv | |v ) with respect (χ | |ν/2 ) v

to the family of Kv -invariant unit vectors f0,vv

(ν)

abbreviated to f0,χv for any (χv | |ν/2 ) v

(ν)

v ∈ Σ∞ ∪ S(fχ ). We use similar abbreviation f1,χv for f1,v (2.24)

and set

(ν) (ν) f˜1,χv = χv (v )qv−ν/2 (1 + qv−1 ) L(ν + 1, χ2v ) f1,χv .

(ν) (ν) Lemma 2.8. Let v ∈ Σ∞ ∪ S(fχ ). If ν ∈ iR, the functions f0,χv and f˜1,χv ν/2

comprise an orthonormal basis of the 2 dimensional space Iv (χv | |v )K0 (pv ) . The ν/2 −ν/2 local intertwining operator Mv (ν) : Iv (χv | |v ) −→ Iv (χ−1 ), v | |v  [Mv (ν)fv(ν) ] (gv ) = fv(ν) (w0 [ 10 x1 ] gv ) dx Fv ν/2

(ν)

(ν)

with sufficiently large Re(ν) is computed on Iv (χv | |v )K0 (pv ) by the basis {f0,χv , f˜1,χv } as Lv (ν, χ2v ) (ν) (−ν) f −1 , (2.25) Mv (ν)f0,χv = qv−dv /2 L(ν + 1, χ2v ) 0,χv Lv (ν, χ2v ) ˜(−ν) (ν) f1,χ−1 . (2.26) Mv (ν)f˜1,χv = χv (v )2 qv−ν−dv /2 v L(1 − ν, χ−2 v )

2.7. EISENSTEIN SERIES AND THEIR L-FUNCTIONS

19

ν/2

Proof. If ν ∈ iR, the hermitian inner product on Iv (χv | |v ) is Gv -invariant. Using the easily confirmed relation

−2

2 −ν/2 1 − Q(Iv (χv | |ν/2 (1 + qv−1 ) L(ν + 1, χ2v ) , ν ∈ iR, v )) = χv (v )qv we infer the first statement from Lemma 2.2. The formula (2.25) is proved in [5, Proposition 4.6.7] when dv = 0; the general case is similar. By (2.9), we see (ν) (−ν) that Mv (ν) sends the vector f1,χv to the vector f1,χ−1 multiplied by the same v constant occurring in (2.25). Then, taking the modification factor in (2.24) into account, we obtain (2.26).  ν/2

2.7.1. Fix χ. A family of smooth functions f (ν) ∈ I(χ| |A ) with varying ν ∈ C is said to be a flat section if f (ν) |K is independent of ν. A flat section ν/2 f (ν) ∈ I(χ| |A ) is K-finite if R(K)f (0) spans a finite dimensional space in I(χ). For such f (ν) , the Eisenstein series is initially defined by the absolutely convergent series  f (ν) (γg), g ∈ GA , Re(ν) > 1. E(f (ν) ; g) = γ∈BF \GF

For each g ∈ GA , the holomorphic function ν → E(f (ν) ; g) on Re(ν) > 1 is continued meromorphically to the whole complex plane, holomorphic on the imaginary axis. The unique singularity of E(f (ν) ; g) on the half-plane Re(ν) > 0 is a possible pole at ν = 1, which occurs only when χ2 = 1. ν/2

2.7.2. Let n be a square free oF -ideal. Then, the invariant part I(χ| |A )K0 (n)K∞ is non zero only when fχ = oF ; we assume this from now on. For each oF -ideal c di(ν) ν/2 viding n, let fχ,c ∈ I(χ| |A ) be the function coming from the decomposable tensor    (ν) (ν) ν/2 ν/2 { v∈S(c) f˜1,χv } ⊗ { v∈S(c) f0,χv } by the isomorphism I(χ| |A ) ∼ = v Iv (χv | |v ). (ν)

ν/2

Lemma 2.9. The family of functions fχ,c ∈ I(χ| |A ) (ν ∈ C) is flat. When (ν) ν ∈ iR, the set of functions {fχ,c | n ⊂ c } is an orthonormal basis of the invariant ν/2 K0 (n)K∞ part I(χ| |A ) . (ν)

Proof. For any place v, the restriction f0,χv |Kv is identically 1. If v ∈ Σfin , (ν) 1/2 −1/2 according to k ∈ K0 (pv ) or k ∈ it turns out that f˜1,χv (k) equals qv or −qv Kv − K0 (pv ), respectively. From these observations, the first assertion is evident. Since the inner product (2.23) is a tensor product of inner products on local ν/2 (ν) spaces Iv (χv | |v ) which contains a unit vector f0,v at all places v, the second assertion follows from the first statement of Lemma 2.8.  (ν)

From now on, we abbreviate E(fχ,c ; g) to Eχ,c (ν; g). Let  ◦ Eχ,c (ν; g) = Eχ,c (ν; [ 10 x1 ] g) dx F \A

be the constant term of Eχ,c (ν; g). Then, Lemma 2.10. −1/2

◦ Eχ,c (ν; g) = fχ,c (g) + DF (ν)

Aχ,c (ν)

L(ν, χ2 ) (−ν) f −1 (g), L(ν + 1, χ2 ) χ ,c

g ∈ GA ,

20

2. PRELIMINARIES

where Aχ,c (ν) = χ ˜2 (c)N(c)−ν



L(1 + ν, χ2v ) −2 . L(1 − ν, χ v ) v∈S(c)

Proof. By a well-known procedure found in the proof of [5, Theorem 3.7.1], the proof is reduced to a computation of local intertwining operators on the vectors (ν) (ν) f0,χv and f˜1,χv . Thus, we are done by Lemma 2.8.  ◦ Though ϕ = Eχ,c (ν; −) − Eχ,c (ν; −) is not left GF -invariant, it is still BF ∗ invariant; thus the integral Z (s, η; ϕ) makes sense.

Lemma 2.11. Let η be a unitary idele-class character satisfying (2.18), (2.19) ◦ and (2.20). Then, Z ∗ (s, η; Eχ,c (ν) − Eχ,c (ν)) (Re(s)  0) equals η ˜ F/Q c) G(η) Bχ,c (s, ν) (DF N(c))−ν/2 χ(d

L(s + ν/2, χη) L(s − ν/2, χ−1 η) L(ν + 1, χ2 )

with (2.27) s−1/2

η Bχ,c (s, ν) = DF



  {(qv−1 + 1) L(1 + ν, χ2v ) ηv (v )qv1/2−s − Q(Iv (χv | |ν/2 v )) }.

v∈S(c)



Proof. Similar to Lemma 2.5.

Lemma 2.12. Let ν ∈ iR and η as in Lemma 2.11. The meromorphic function     1−ν −1   L −z + ν+1 η η 1 2 , χη L −z + 2 , χ Bχ,c −z + 2 , ν L(ν + 1, χ2 ) in z ∈ C is holomorphic except possible simple poles at z = (ν ± 1)/2, (−ν ± 1)/2. The relevant residues satisfy the relations Resz=(ν+1)/2 = −(−1)S(c) N(c) Resz=(−ν−1)/2 = δχ,η vol(F × \A1 ) χ(c) ˜ (DF N(c))(ν+1)/2 , Resz=(1−ν)/2 = −(−1)S(c) N(c) Resz=(−1+ν)/2 −1/2

= δχ,η vol(F × \A1 ) χ(c) ˜ (DF N(c))(ν+1)/2 DF

Aχ,c (ν)

ζF (ν) . ζF (ν + 1)

Proof. The first statement is evident. The formulas of residues are obtained by a direct computation with the aid of the easily confirmed relations  −ν  −1 η (ν+1)/2 Bη,c η˜(c), 2 , ν = (DF N(c))   η 1 − ν2 , ν = (DF−1 N(c))(ν−1)/2 η˜(c) (−1)S(c) Aη,c (ν), Bη,c ν  η (ν−1)/2 Bη,c N(c) η˜(c) Aη,c (ν), 2 , ν = (DF N(c))   η (ν+1)/2 ν Bη,c 1 + 2 , ν = (DF N(c)) (−1)S(c) N(c)−1 η˜(c), combined with the functional equation of L(ν, 1) = ζF (ν).



2.7. EISENSTEIN SERIES AND THEIR L-FUNCTIONS

21

2.7.3. The only singularity of Eχ,c (ν; −) lying on Re(ν) > 0 is a possible simple pole at ν = 1. Indeed, Lemma 2.13. The residue at ν = 1 of Eχ,c (ν; g) is eχ,c,−1 (g) = δ(χ2 = 1, c = oF )

2 vol(F × \A1 ) −1 χ (det g), vol(ZA GF \GA )

g ∈ GA .

We have −1/2

D RF 2 vol(F × \A1 ) = F . vol(ZA GF \GA ) ζF (2) with RF the residue of ζF (s) at s = 1. Proof. (cf. [20, Proposition 6.13]) Let us examine the function m(ν) = −1/2 DF Aχ,c (ν) L(ν, χ2 )/L(ν + 1, χ2 ), which controls the singularity of the constant ◦ term Eχ,c (ν; −) at ν = 1 by Lemma 2.10. The L-function L(ν, χ2 ) has a possible simple pole at ν = 1 which occurs if and only if χ2 = 1. When χ2 = 1, the factor Aχ,c (ν) has a zero at ν = 1 unless S(c) = ∅. Thus, the possible simple pole of Eχ,c (ν; −) at ν = 1 occurs only when χ2 = 1 and c = oF . The Maass-Selberg relation applied to our Eisenstein series takes the form (cf. [12, Formula (5.13)]) : (2.28)  ∧T Eχ,c (σ)2 vol(F × \A1 )−1 =

Tσ T −σ (σ) 2 (σ) 2  −  fχ,c M (σ)fχ,c σ σ (σ) (σ) (σ) (σ) |M (σ)fχ,c  log T − fχ,c |M  (σ)fχ,c }, + 2 δ(χ2 = 1) {fχ,c −σ/2

where ∧T is the truncation operator and M (σ) : I(χ| |A ) → I(χ−1 | |A ) is the global intertwining operator and M  (σ) its derivative at a point σ ∈ R − {1}. On the left-hand side, the norm is the L2 -norm of L2 (ZA GF \GA ), while, on the right-hand side, the norm (or the hermitian pairing) is considered for elements of σ/2 −σ/2 I(χ| |A ) ⊕ I(χ−1 | |A ) by the formula (2.23). Suppose χ2 = 1, c = oF and write (ν) (−ν) R m(ν) = ν−1 + O(ν − 1). Then, by M (ν)fχ,c = m(ν) fχ−1 ,c , the formula (2.28) allows us to compute σ/2

lim (σ − 1)2  ∧T Eχ,c (σ)2 vol(F × \A1 )−1 = 2R + O(T −1 )

σ→1

(ν)

on the one hand. On the other hand, from Lemma 2.10, Eχ,c (ν; g) = fχ,c (g) + (−ν) (−1) m(ν) fχ−1 ,c (g), which yields eχ,c,−1 (g) = R fχ−1 ,c (g) = R χ−1 (det g). By this, the same limit is computed as R2 vol(ZA GF \GA ). Thus, R = 2 vol(F × \A1 )/vol(ZA GF \GA ). This completes the first assertion. Since R = Resν=1 m(ν) −1/2

= δ(χ2 = 1) DF 2

A1,c (1)RF /ζF (2) −1/2

= δ(χ = 1, c = oF ) DF we also have the second statement.

RF /ζ(2), 

22

2. PRELIMINARIES

2.8. Adjoint square L-functions For π as in 2.6.1, its adjoint square L-function L(s, π; Ad) is defined to be the Euler product  L(s, π; Ad) = Lv (s, πv ; Ad) v∈Σfin ∪Σ∞

convergent on a right half-plane, where the local factors are given as follows. If πv is Kv -spherical, then  (1 − qv−s+2νv )−1 (1 − qv−s−2νv )−1 (1 − qv−s )−1 , v ∈ Σfin , Lv (s, πv ; Ad) = v ∈ Σ∞ ΓR (s + 2νv ) ΓR (s − 2νv ) ΓR (s), ∼ Iv (| |νv ). If v ∈ Σfin with c(πv ) = 1, then Lv (s, πv ; Ad) = with νv ∈ C such that πv = v −(s+1) −1 (1 − qv ) . It is known that L(s, π; Ad) is continued to an entire function on C ([11]). The ¯ ), which is product ζF (s) L(s, π; Ad) is the convolution L-function for the pair (π, π studied by the Rankin-Selberg integral involving the Eisenstein series ([22]); in our case, it is explicated as follows. ∈ Vπ be the newform of π. Then, for Re(s)  0, Lemma 2.14. Let ϕnew π  new (2.29) E1,oF (2s − 1; g) ϕnew π (g) ϕπ (g) dg GF ZA \GA

= [Kfin : K0 (fπ )]−1 N(fπ )s DF

s−3/2

ζF (2s)−1 ζF (s) L(s, π; Ad).

Proof. By the standard unfolding procedure, the integral is decomposed to the product of the local integrals   dk φ0,v ([ 0t 01 ] k) φ0,v ([ 0t 10 ] k) |t|s−1 d× t. Zv (s) = v Kv

Fv×

If v ∈ Σfin with c(πv ) = 0, then we obtain dv (s−1) Zv (s) = vol(o× ζF,v (2s)−1 ζF,v (s) Lv (s, πv ; Ad) v ) qv

(2.30)

using (2.11) as in [5, Proposition 3.8.1]. If v ∈ Σ∞ , then by φ0,v ([ 0t 01 ]) = 2 t1/2 Kνv (2πt),

t > 0,

we have Zv (s) = ΓR (2s)−1 ΓR (s + 2ν) ΓR (s − 2ν) ΓR (s)2 ,

Re(s) > 2|Re(νv )|

using the formula in [36, p.101]. Finally, letv ∈ Σfin with c(πv ) = 1. According to the disjoint dicomposition Kv = K0 (pv ) ∪ { ξ∈ov /pv 10 1ξ w0 K0 (pv )}, the integral Zv (s) breaks up to the sum of the following integrals.  (1) φ0,v ([ 0t 01 ]) φ0,v ([ 0t 10 ]) |t|s−1 d× t, Zv (s) = vol(K0 (pv )) v Fv×          (2) Zv (ξ; s) = vol(K0 (pv )) φ0,v [ 0t 01 ] 10 1ξ w0 φ0,v [ 0t 01 ] 10 1ξ w0 |t|s−1 d× t v Fv×

1/2

with ξ ∈ ov /pv . If ϕ0,v denotes the Kv -spherical Whittaker function of Iv (sgn vv | |v ), 1/2 which is given by (2.11) with q νv = (−1) v qv ,  −1  φ0,v ([ 0t 01 ]) = ϕ0,v ([ 0t 01 ]) − (−1) v ϕ0,v t0v 01 .

2.8. ADJOINT SQUARE L-FUNCTIONS

23

Using this, we calculate the integrals to obtain dv (s−1) Zv(1) (s) = vol(K0 (pv )) vol(o× (1 − qv−s+2νv )−1 (1 − qv−s−2νv )−1 (1 − qv−s )−1 v ) qv

× {(1 + qv−s )(1 + qv−s+1 ) − 2qv−s+1/2 (qv1/2 + qv−1/2 )}, dv (s−1) Zv(2) (ξ; s) = vol(K0 (pv )) vol(o× (1 − qv−s+2νv )−1 (1 − qv−s−2νv )−1 (1 − qv−s )−1 v ) qv

× {(1 + qvs−1 )(1 + qv−s ) − 2qv−1/2 (qv1/2 + qv−1/2 )}.

Then, a further computation reveals that Zv (s) is given by the formula (2.30) multiplied with vol(K0 (pv )) qvs .  Corollary 2.15. 2 −1 L(1, π; Ad). ϕnew π  = 2 N(fπ ) [Kfin : K0 (fπ )]

Proof. This is proved by taking the residue of (2.29) at s = 1. Use Lemma 2.13  to compute the residue of E1,oF (2s − 1; g). Corollary 2.16. Let n be an square free ideal of oF . Let π be an irreducible cuspidal automorphic representation with trivial central character such that K (n)K∞ Vπ 0 = {0}. Let η be an idele-class character of F × satisfying the conditions (2.18), (2.19) and (2.20). Then, −1/2

Pη (π; K0 (n)) = DF

G(η)

[Kfin : K0 (fπ )] { 2N(fπ )

 v∈S(nf−1 π )

1 + ηv (v ) L(1/2, π) L(1/2, π ⊗ η) } . 1 + Q(πv ) L(1, π; Ad)

Proof. This follows from (2.21) and Corollary 2.15.



CHAPTER 3

Preliminary analysis 3.1. A decomposition of GL(2, R) Consider the following subgroups of SL2 (R):   0   sinhr | t > 0 }, A = {ar = coshr H 1 = {h(t) = 0t t−1 sinhr coshr | r ∈ R },   For any g = ac db ∈ SL2 (R), define t(g) > 0 and r(g) ∈ R by the relations t(g) = (a2 + b2 )1/4 (c2 + d2 )−1/4 , sinh 2r(g) = ac + bd. The subgroups H 1 , A and K 1 = SO(2) provides us with a system of coordinates on SL2 (R). Lemma 3.1. The map φ : H 1 × A × K 1 −→ SL2 (R) (h(t), ar , k) ∈ H 1 × A × K 1   is a diffeomorphism. For any g = ac db ∈ SL2 (R), φ−1 (g) = (h(t), ar , k) with r = r(g), t = t(g) and k ∈ K 1 uniquely determined by the relation g = h(t)ar k. φ(h(t), ar , k) = h(t) ar k,

Lemma 3.2. For any compact set V ⊂ SL2 (R), we have (3.1)

t(g)  t(gy),

g ∈ G, y ∈ V,

(3.2)

r(g)  r(gy),

g ∈ G, y ∈ V.

Proof. The elements t(y) with varying y ∈ V stay in a compact subset of (0, +∞). Thus,  2 1/4  2 1/4     a t(y)2 + b2 t(y)−2 a + b2 t( ac db y) = t( ac db h(t(y))) =  = t(g). c2 t(y)2 + d2 t(y)−2 c2 + d2 This shows (3.1). Another one (3.2) is proved in a similar way.



Consider the open subgroup GL+ (2, R) = {g ∈ GL(2, R)| det g > 0 } of GL(2, R). Then, GL+ (2, R) = SL2 (R) {z 12 | z ∈ R× + } is a direct product and  0  + . GL(2, R) = GL (2, R) ∪ GL+ (2, R) 10 −1 The following integral formula for the measure dg on GL(2, R) fixed in 2.3 will be used later : (3.3) 

 ϕ(g) dg = 2

GL(2,R)

(R× )2

 

 ϕ

R

O2 (R)

t1 0 0 t2



 ar k

(r) d× t1 d× t2 dr dk,

ϕ ∈ L1 (GL(2, R))

with (r) = cosh2r. Let ι ∈ Σ∞ . The subgroup of Gι corresponding to A by the isomorphism GL(2, R) ∼ = Gι is denoted by Aι . Let A∞ be the direct product of Aι (ι ∈ Σ∞ ), 25

26

3. PRELIMINARY ANALYSIS

regarded as a subgroup of G∞ . Then, from Lemma 3.1, we have a decomposition G∞ = H∞ A∞ K∞ . 3.2. Convergence lemmas   Set BA1 = { a0 db | a, d ∈ A1 , b ∈ A } and fix a relatively compact subset ω ⊂ BA1 such that BF ω = BA1 . Then, any subset of the form y 0  S(t) = ω { 01 y | y1 , y2 > 0, y1 /y2 > t } K, t > 0, 2

is called a Siegel set of GA . We fix a Siegel set S = S(t0 ) once and for all so that GA = GF S ([27, §10], [14], [28, §7]). Note that S = A (G1A ∩ S). By the Iwasawa decomposition GA = BA K, let us define a function y : GA → R+ by    a b y a0 db k = |a/d|A , 0 d ∈ BA , k ∈ K. g12 For each place v of F and an element g = [ gg11 21 g22 ] ∈ Gv , set  2 2 2 2 1/2 2−1 (1 + | det g|−1 , v )(g11 + g12 + g21 + g22 ) gv = −1 sup{| det g|v |gij |v , |gij |v | 1  i, j  2 },

v ∈ Σ∞ , v ∈ Σfin .

We remark that kv = 1 if k ∈ Kv . Then, for an adele point g ∈ GA , its height gA is defined to be the product of gv v over all places v: gA = v gv v . The following properties of height will be used frequently. (3.4)

gA = g −1 A ,

(3.5)

g1 g2 A  g1 A g2 A ,

(3.6)

gA  y(g)1/2 ,

g ∈ GA , g1 , g2 ∈ GA .

g ∈ G1A ∩ S.

Lemma 3.3. Let V 1 ⊂ G1A be a compact subset of G1A . (1) (3.7)



χV 1 (h−1 γg)  y(h) χGF hV 1 (g),

h ∈ G1A ∩ S, g ∈ G1A .

γ∈GF

(2) We have the estimate (3.8)

−1/2 γgg1 −1 , A  y(g)

(γ, g, g1 ) ∈ GF × G1A × V 1 .

Proof. These are essentially [29, p.59–60] and [55, Proposition 5.9], respectively. We recall the proofs in our framework for completeness. (1) Since KV 1 is compact, there exist compact sets ωN ⊂ NA and ωH ⊂ HA such that KV 1 ⊂ ωN ωH K. Fix t0 such that inf a∈ωH y(a)1/2 > t0 t−1 0 , where t0 is the number used to define our S = S(t0 ). Then, consider another Siegel set S = S(t0 ). Let h ∈ S ∩ G1A , g ∈ G1A , γ  ∈ GF such that γ  g ∈ hV 1 . Observe that this forces that g ∈ GF hV 1 . Then, denoting by [h] any element of HA such that y([h]) = y(h), we have γ  g ∈ hV 1 ⊂ ω [h] KV 1 ⊂ ω [h] ωN ωH K = {ω · [h] ωN [h]−1 } · [h]ωH · K.

3.2. CONVERGENCE LEMMAS

27

From this, there exist γ ∈ BF \GF and δ ∈ BF such that γ  = δγ and such that (3.9)

γg ∈ S ,

(3.10)

δω ∩ ω · [h]ωN [h]−1 = ∅.

If some γ0 satisfying (3.9) is found, then the number of BF -coset γ satisfying (3.9) is bounded by {γ ∈ BF \GF | S ∩ γγ0−1 S = ∅ }; the latter number is finite by [28, Proposition 7.8]. From (3.10), the element [h]−1 δ[h] is contained in ([h]−1 ω [h]) · ωN · ([h]−1 ω −1 [h]) · ([h]−1 ω −1 [h]), which in turn stays in a compact subset ωB ⊂ BA1 independent of h ∈ S. Thus, the number of δ satisfying (3.10) is bounded by (BF ∩ [h]ωB [h]−1 ), which is majorized by y(h) for varying h ∈ S. This completes the proof. (2) The height of a vector ξ = [ xy ] ∈ A2 , which is primitive in the sense that xv ov + yv ov = ov for almost all v ∈ Σfin , is defined by   ξA = { sup(|xv |v , |yv |v )} { (x2ι + yι2 )1/2 }. v∈Σfin

ι∈Σ∞ 

Then, it is easy to establish the estimation g ξA  g  A ξA for any g  ∈ GA and for any primitive vector ξ ∈ A2 . Combining this with (3.4), we have yet another estimate :  −1 ξA . g  −1 A ξA  (g )

(3.11)

We also need the fact that g  ξA  ξA as long as g  ∈ GA varies in a compact subset of GA . Let ξ0 = [ 10 ] ∈ F 2 . Having these properties of height, for any (γ, g, g1 ) ∈ GF × G1A × V 1 , we have y(g)−1/2 = g −1 ξ0 A

(by Iwasawa decomposition) −1

= g1 (γgg1 ) γξ0 A    (γgg1 )−1 γξ0 

(since g1 stays in the compact set V 1 )

 γgg1 −1 A γξ0 A

(by (3.11))

A

 γgg1 −1 A . Here, in the last step, we use γ ξ0 A  1, γ ∈ GF , which is inferred from Artin’s product formula.  Lemma 3.4. Let ϕ : GF ZA \GA → C be a continuous function and b ∈ R a constant such that |ϕ(g)|  y(g)b , g ∈ G1A ∩ S. If b < 1, then ϕ is integrable on GF ZA \GA . Proof. By (2.1), we have the bound   |ϕ(g)| dg  S∩G1A

+∞

1

whose majorant is convergent by b < 1. Thus,

y b−1 d× y,

 GF ZA \GA

|ϕ(g)| dg < +∞ as desired. 

28

3. PRELIMINARY ANALYSIS

Combining the decompositions G∞ = H∞ A∞ K∞ and Gfin = Hfin Nfin Kfin , we have GA = HA A∞ Nfin K. For a finite subset S of Σfin and for positive numbers p and q, define a function Ξp,q,S : GA → R+ by setting       k = inf{|t1 /t2 |pA , |t1 /t2 |−p Ξp,q,S t01 t02 a∞ 10 xfin (cosh2rι )−q A } 1 ×



ι∈Σ∞

sup(1, |xv |v )

−q



δ(xv ∈ ov )

v∈Σfin −S

v∈S

    for t01 t02 ∈ HA , 10 x1fin ∈ Nfin , a∞ = (arι ) ∈ A∞ and k ∈ K. Then, Ξp,q,S is a left ZA HF -invariant and right K-invariant continuous function on GA . Thus the sum  Ξp,q,S (γg), g ∈ GA , Ξp,q,S (g) = γ∈HF \GF

is well defined. Lemma 3.5. Let p > 0 and q > 1. (1) The series Ξp,q,S (g) converges uniformly on any compact subset of GA , defining a left GF ZA -invariant continuous function on GA . (2) Suppose 1 + 2p < q. Then, we have the estimation Ξp,q,S (g)  y(g)1−p ,

(3.12)

g ∈ G1A ∩ S.

The function Ξp,q,S belongs to Ll (ZA GF \GA )K for any l > 0 such that l(1 − p) < 1. Proof. Let V be a relatively compact open neighborhood of the identity in GA of the form V = V∞ Vfin with V∞ and Vfin being open in G∞ and in Kfin , respectively. Then, Ξp,q,S (xyfin ) = Ξp,q,S (x) for any yfin ∈ Vfin . At an archimedean place ι, we apply Lemma 3.2 to control the behavior of Ξp,q,S (xy∞ ) with varying y∞ ∈ V∞ . From these, we see that the estimate C0 Ξp,q,S (x)  Ξp,q,S (xy),

(3.13)

x ∈ GA , y ∈ V

holds with a positive constant C0 . Let ωG ⊂ G1A be a compact subset. For each γ ∈ HF \GF , fix a point gγ ∈ ωG such that supg∈ωG Ξp,q,S (γg) = Ξp,q,S (γgγ ). Let χV 1 be the characteristic function of V 1 = V ∩G1A in G1A . By (3.13), C0 Ξp,q,S (γg)  Ξp,q,S (γgγ y) for any (g, y, γ) ∈ ωG × V × (HF \GF ), from which by taking integral in y ∈ V 1 and then taking summation over γ ∈ HF \GF , we obtain (3.14) C0 vol(V 1 )

 γ∈HF \GF





Ξp,q,S (γg) 

Ξp,q,S (γgγ y) dy

γ∈HF \GF



=



γ∈HF \GF



=

γ∈HF \GF

 =



Ξp,q,S (x) d1 x

γ∈HF \GF

=

V1

{

γgγ V 1



G1 A

χV 1 (gγ−1 γ −1 x) Ξp,q,S (x) d1 x



HF \G1 A



HF \G1 A γ∈GF

{



χV 1 (gγ−1 γ −1 δx)} Ξp,q,S (x) d1 x

δ∈HF

χV 1 (gγ−1 γx)} Ξp,q,S (x) d1 x

3.2. CONVERGENCE LEMMAS

29

for any g ∈ ωG . Combining this with the easily confirmed inequality  χV 1 (gγ γ −1 x)  (GF ∩ ωG V 1 (V 1 )−1 ωG ) < +∞, γ∈GF

we have 

 Ξp,q,S (γg) 

γ∈HF \GF

HF \G1A

Ξp,q,S (g  ) d1 g  ,

g ∈ ωG .

By decomposing the measure d1 g  along the decomposition G1A = (HA ∩ G1A ) A∞ Nfin K, the integral in the majorant is bounded by  +∞   { inf(t2p , t−2p ) d× t} { (cosh2rι )−q+1 drι }{ 0

ι

R

v∈S

sup(1, |xv |v )−q dxv },

Fv

which is finite by p > 0 and q > 1. Thus, the series Ξp,q,S (g) converges uniformly in g ∈ ωG . This completes the proof of (1). (2) Invoking the estimate (3.7), from (3.14) applied with ωG = {g} , we obtain   (3.15) Ξp,q,S (γg)  y(g) Ξp,q,S (g  ) d1 g  , g ∈ G1A ∩ S. HF \(GF gV 1 )

γ∈HF \GF

Combined with the decomposition GA = HA A∞ Nfin K and (3.5), Lemma 3.3 (2) gives us a constant C1 > 0 such that Ξp,q,S (g  ) = 0, g  ∈ GF g V 1 implies g  is of the form     tu 0 k g  = 01 t−1 u2 a∞ 10 xfin 1 with t > 0, u1 , u2 ∈ F × \A1 , a∞ = (arι ) ∈ A∞ , xfin ∈ Afin and k ∈ K satisfying xv ∈ ov , for any v ∈ Σfin − S,   sup(1, |xv |v )} y(g)−1/2  C1 inf(t, t−1 ). { e|rι | } { ι

v∈S

Thus, the integral in the majorant of (3.15) is bounded by   sup(1, |xv |v )−q dxv } { cosh(2rι )−q+1 drι } { v∈S

×

Fv



ι

R

+∞ 

C1 {

 y(g)

−p



ι

{

e−|rι | } {



v∈S

Fv

v∈S

t−2p d× t sup(1,|xv |v )−1 } y(g)1/2

sup(1, |xv |v )

2p−q

 dxv } { cosh(2rι )p−q+1 drι }. ι

R

The integrals in the majorant are convergent if q > 2p + 1. Thus, we obtain (3.12), from which the second statement is inferred by Lemma 3.4. 

30

3. PRELIMINARY ANALYSIS

3.3. A space of test functions ∞ Let C+ (AGF \GA ) be the space of all the C-valued smooth functions ϕ on AGF \GA such that −(1+ )

pD, (ϕ) = sup {|R(D)ϕ(g)| gA

(3.16)

g∈G1A

} < +∞

for any D ∈ U(g∞ ) and for any  ∈ (0, 1) (cf. [34, p.675]); this space becomes a GA -module by the right regular action of GA . For any open compact subgroup ∞ of K ⊂ Kfin , the functions pD, : C+ (AGF \GA )K → R+ are semi-norms on ∞ K C+ (AGF \GA ) , which define a Frechet space structure on the space. ∞ Lemma 3.6. If 1 < p < 2, then C+ (AGF \GA )K ⊂ Lp (AGF \GA )K for any open compact subgroup K ⊂ Kfin .

Proof. There exists  ∈ (0, 1) such that p(1 + ) < 2. Then, we have the estip(1+ )  y(g)p(1+ )/2 , g ∈ S ∩ G1A , from which the conclusion mation |ϕ(g)|p  gA follows by Lemma 3.4.  Let {ξδ }δ>0 be an approximate identity in G1A ([3, 2.4], [4, 1.6]), i.e., ξδ is a smooth function on G1A with compact support such that (a) ξδ  0, (b) supp(ξ δ ) → {e} (δ → 0+),  (c) G1 ξδ (g) d1 g = 1. A

For any measurable function ϕ on GA , the convolution product  ϕ(gx−1 ) ξδ (x) d1 x, g ∈ GA ϕ ∗ ξδ (g) = G1A

is a smooth function on GA . Lemma 3.7. Let K be an open compact subgroup of Kfin . For any semi-norm ∞ (AGF \GA )K , we have pD, on C+ lim pD, (ϕ ∗ ξδ − ϕ) = 0,

δ→+0

∞ ϕ ∈ C+ (AGF \GA )K .

Proof. Fix a semi-norm pD, and choose m ∈ N such that D ∈ Um , where U (m ∈ N) is the degree filtration of U(g∞ ). Note that Um is of finite dimension m and is preserved by the adjoint action of G∞ . Fix a basis  {Dj } of U . Then, there ∞ exist C -functions cj on G∞ such that Ad(g)D = j cj (g) Dj for any g ∈ G∞ . Fix an R-basis {Xi } of the Lie algebra g∞  = Lie(G∞ ). Let η > 0. Choose a small ball N in g∞ centered at 0 such that X = i ai (X) Xi ∈ N implies m

sup |ai (X)| < η,

sup |cj (exp(X)) − cj (12 )|  1

i

j

and such that U∞ = exp(N ) is an open neighborhood of the identity in G∞ . Then, by (b), there exists δ0 > 0 such that  (3.17) supp(ξδ ) ⊂ (U∞ K) ∩ G1A . δδ0

For any x ∈ GA , denote the function g → ϕ(g x) − ϕ(g) on GA by ϕx . Then, from  1 [R(X)ϕ](g exp(τ X)) dτ, X ∈ N , ϕ(g exp(X)) − ϕ(g) = 0

3.3. A SPACE OF TEST FUNCTIONS

we have



31

1

[R(D)ϕexp(X) ](g) =

[R(D)R(exp(τ X))R(X)ϕ](g) dτ 0



1

[R(Ad(exp(−τ X)D) R(X)ϕ](g exp(τ X)) dτ

= 0

=

 j

 ai (X)

1

cj (exp(−τ X)) [R(Dj Xi )ϕ](g exp(τ X)) dτ. 0

i

Thus, by setting CD = supj (1 + |cj (12 )|), we obtain  1 | [R(D)ϕexp(X) ](g) |  CD η |[R(Dj Xi )ϕ](g exp(τ X))| dτ j

 CD η

i

 j

0

 pDj Xi , (ϕ)

i

0

1

g exp(τ X)1+

A dτ.

Thus, by g exp(X)A  gA  exp(X)A , (X ∈ N ), X ∈ N , g ∈ GA | [R(D)ϕexp(X) ](g) |  η CD, g1+

A ,   with CD, = CD j i pDj Xi , (ϕ) {supX∈N  exp(X)1+

A }. From this, combined with (a) and (c), we get  (3.18) | [R(D)ϕx−1 ](g) | ξδ (x) d1 x  η CD, g1+

g ∈ G A , δ  δ0 . A , ∞ (U∞ K)∩G1A

By (3.17) and (3.18), for any g ∈ GA and for any δ  δ0 ,

      | [R(D)(ϕ ∗ ξδ )](g) − [R(D)ϕ](g) | =  [R(D)ϕx−1 ](g) ξδ (x) d1 x  G1   A  | [R(D) ϕx−1 ](g) | ξδ (x) d1 x  η CD, g1+ A . ∞ (U∞ K)∩G1 A

Thus, δ  δ0 =⇒ pD, (ϕ ∗ ξδ − ϕ)  η CD, . Since η > 0 is arbitrary, this completes the proof.



Lemma 3.8. Let K ⊂ Kfin be an open compact subgroup. Then, we have the inclusion ∞ (AGF \GA )K . Cc∞ (AGF \GA )K ⊂ C+ ∞ (AGF \GA )K . The space Cc∞ (AGF \GA )K is dense in C+

Proof. Let ϕ ∈ Cc∞ (AGF \GA ). Then, ϕ regarded as a function on the Siegel domain S is of compact support modulo A. Hence, there exists y0 > 0 such that ϕ(g) = 0 for any g ∈ S with y(g) > y0 . In particular, for any D, the function R(D)ϕ on S is bounded by any power of y(g). ∞ (AGF \GA )K . By Lemma 3.7, it To show the second statement, let ϕ ∈ C+ suffices to show that, for δ > 0, η > 0 and a seminorm pD, , there exists a function ϕ0 ∈ Cc∞ (AGF \GA )K such that pD, (ϕ0 −ϕ∗ξδ ) < η. Set κ = 1− and choose p and ∞ (AGF \GA )K , q such that 1 < p < 2/(1 + κ + /2), q −1 + p−1 = 1. Since ϕ ∈ C+ (1+κ+ /2)p κ p 1 we have the estimate {xA |ϕ(x)|}  xA on x ∈ GA ∩ S. Thus, if we fix a right K-invariant positive measurable function μ on AGF \GA such that

32

3. PRELIMINARY ANALYSIS

1 p μ(x) = xκp A on GA ∩ S, then ϕ ∈ L (AGF \GA , μ(x)dx). By Lemma 3.3 (1), if U is a relatively compact open set of G1A such that supp ξδ ⊂ U , there exists a constant C1 > 0 such that  ˇ δ ](x−1 γg)|  C1 y(g) χG gU (x), g ∈ S ∩ G1A , x ∈ G1A . (3.19) |[R(D)ξ F

γ∈GF

We easily confirm that, from Lemma 3.3 (2), there exists a constant C2 > 0 such that  q −qκ/2 (3.20) χGF gU (x) x−qκ , g ∈ S ∩ G1A . A dx  C2 y(g) G1A ∩S

Cc0 (AGF \GA )K

is dense in Lp (AGF \GA , μ(x)dx)K and since the GF -averaging Since map f → f˜ is surjective from Cc0 (A\GA ) onto Cc0 (AGF \GA ), there exists f0 ∈ Cc0 (A\GA ) such that  { (3.21) |ϕ(x) − f˜0 (x)|p μ(x) dx}1/p  (C1 C2 )−1 η. GF \G1A

For any g ∈ S ∩ G1A , we have



[R(D)(ϕ ∗ ξδ )](g) − [R(D)(f˜0 ∗ ξδ )](g) = 

G1 A

= 

G1 A

ˇ δ ](x) d1 x {ϕ(gx−1 ) − f˜0 (gx−1 )} [R(D)ξ ˇ δ ](x−1 g) d1 x {ϕ(x) − f˜0 (x)} [R(D)ξ {ϕ(x) − f˜0 (x)}

= GF \G1 A

From this, combined with (3.19), we obtain



ˇ δ ](x−1 γg) d1 x. [R(D)ξ

γ∈GF



| [R(D)(ϕ ∗ ξδ )](g) − [R(D)(f˜0 ∗ ξδ )](g) |  C1 y(g)

GF \(GF gU )

|ϕ(x) − f˜0 (x)| dx,

g ∈ S ∩ G1A .

Using (3.21) and (3.20), by H¨older’s inequality, we obtain  |ϕ(x) − f˜0 (x)| d1 x GF \(GF gU)   |ϕ(x) − f˜0 (x)| χGF gU (x) d1 x G1A ∩S

 

G1A ∩S

1/p

|ϕ(x) − f˜0 (x)|

 C1−1 η y(g)−κ/2 ,

p

xκp A



1

d x G1A ∩S

1/q 1 χGF gU (x) x−qκ A d x

g ∈ S ∩ G1A .

Hence, | [R(D)(ϕ∗ξδ )](g)−[R(D)(f˜0 ∗ξδ )](g) |  η y(g)1−κ/2 = η y(g)(1+ )/2 ,

g ∈ S∩G1A .

By [3, Proposition 5.10], this in turn yields pD, (ϕ ∗ ξδ − ϕ0 )  η with ϕ0 = f˜0 ∗ ξδ , completing the proof.



CHAPTER 4

Green’s functions on GL(2, R) We continue to use the notation introduced in 3.1. Lemma 4.1. Let Ψ : SL2 (R) − H 1 K 1 → C be a smooth function satisfying (4.1)

(h(t), g, k) ∈ H 1 × (SL2 (R) − H 1 K 1 ) × K 1

Ψ(h(t)gk) = t2z Ψ(g),

with z ∈ C. Then, for any r ∈ R − {0}, ΩΨ(ar ) = D(z) r Ψ(ar )

with

D(z) r =

2z 2 1 d2 sinh2r d + + . 2 2 dr cosh2r dr cosh2 2r

Proof. cf. [54, Lemma 10], [19, Proposition 4.3].



For (s, z) ∈ C2 such that Re(s) > 2|Re(z)| and for r ∈ R, set  + −  2 −(s+1)/2 Ψ(z) (4.2) 2 F1 sz , sz ; s/2 + 1; 1/cosh 2r , s (r) = cz (s) (cosh2r) −1    −  Γ sz Γ(s/2 + 1)−1 , cz (s) = √ Γ s+ z 8 π where s± z = (s ± 2z + 1)/4. Then, for a fixed (s, z) such that Re(s) > |2Re(z)|, the (z) mapping r → Ψs (r) is a continuous function on R, which is smooth on R − {0} and satisfies the differential equation s2 − 1 (z) Ψs (r), r ∈ R − {0}. 2 Moreover, its value and derivative at r = 0 satisfy the relations (cf. [54, Lemma 12]): (4.3)

(z) D(z) r Ψs (r) =

(4.4)

Ψ(z) s (0) =

(4.5)

− Γ(s+ −1 z ) Γ(sz ) , − 8 Γ(s+ z + 1/2) Γ(sz + 1/2) d (z) d (z) Ψs (r) − lim Ψs (r) = 1. lim r→+0 dr r→−0 dr

Lemma 3.1 shows that the relation (4.6)

(h(t), ar , k) ∈ H 1 × A × K 1

Ψ(z) (s; h(t) ar k) = t2z Ψ(z) s (r),

defines a continuous function Ψ(z) (s; −) : SL2 (R) → C which is smooth on the open dense subset SL2 (R) − H 1 K 1 . Lemma 4.2. We have the estimate (4.7)

(z)

Ψ (s; g)  t(g)2Re(z) {cosh2r(g)}−(Re(s)+1)/2 , 33

g ∈ SL2 (R), Re(s) > |2Re(z)|.

4. GREEN’S FUNCTIONS ON GL(2, R)

34

Proof. By [54, Lemma 13], we have

√

(z) −(Re(s)+1)/2 π

Ψs (r)  (cosh2r)

Γ(s+ Γ(Re(s− z) z ))

+ −

Γ(sz + 1/2) Γ(Re(sz ) + 1/2)

for r ∈ R and for (s, z) ∈ C2 such that Re(s) + 1 > |2Re(z)|. Combining this with the estimation |Γ(α) Γ(α + 1/2)−1 |  1,

Re(α)  1/4

inferred from [54, Lemma 14], we obtain

(z)

Ψ (s; h(t)ar k)  t2Re(z) (cosh2r)−(Re(s)+1)/2 , 1 (t, r, k) ∈ R× + × R × K , Re(s) > 2|Re(z)|



by (4.6). This completes the proof. Let us extend the domain of Ψ(z) (s; −) to GL(2, R) by setting  Ψ(z) (s; g (det g)−1/2 ), g ∈ GL+ (2, R), (z)  1 0   Ψ (s; g) = (4.8) (z) −1/2 s; g 0 −1 | det g| , g ∈ GL+ (2, R), Ψ or explicitly Ψ(z) (s; g) = | det g|(s+1)/2 cz (s) (a2 + b2 )−(s−2z+1)/4 (c2 + d2 )−(s+2z+1)/4   − 2 2 2 −1 2 (c + d2 )−1 × 2 F1 s+ z , sz ; s/2 + 1; (det g) (a + b )   for any g = ac db ∈ GL(2, R). Then we can easily confirm the relation   Ψ(z) (s; t01 t02 g k) = |t1 /t2 |z Ψ(z) (s; g), g ∈ GL(2, R) (4.9)

for any t1 , t2 ∈ R× and for any k ∈ O(2; R). For each ι ∈ Σ∞ , let Ψι (s; −) be the function on Gι corresponding to Ψ(z) (s; −) by the isomorphism Gι ∼ = GL(2, R). (z)

f

 4.3. Let ι−z∈ Σ∞ . Let f :t1G0ι → C be a smooth function such that  Lemma t1 0 f (g) for any 0 t2 ∈ Hι and for any k ∈ Kι . Suppose 0 t2 gk = |t1 /t2 |

2 m 

d

|Rez|

,

dr m f (ar )  (cosh2r) m=0

r ∈ R.

If Re(s) > 2|Rez| + 1, then the equality  2 (4.10) Ψ(z) ι (s; g) [R(Ωι − (s − 1)/2)f ](g) dg = f (e) Hι \Gι

holds with the integral being convergent absolutely. Proof. By the integral formula (3.3) and by Lemma 4.1, the left-hand side equals  2 2 Ψ(z) (s; ar ) (D(z) r − (s − 1)/2)[f (ar )] (r) dr, R

which in turn becomes f (e) by Lemma 4.4 below.



4. GREEN’S FUNCTIONS ON GL(2, R)

35

 Lemma 4.4. Let f ∈ C ∞ (R) be such that 2m=0 |f (m) (r)|  (cosh2r)a on R with some a ∈ R. Let φ(r) be a continuous function on R smooth on R − {0} such 2 that m=0 |φ(m) (r)|  (cosh2r)b on R with some b ∈ R and such that (4.11)

lim φ (r) − lim φ (r) = 1.

r→0+

r→0−

If b + a + 1 < 0, then   f (0) φ(r) (D(z) f )(r) (r) dr = f (r) (D(z) r r φ)(r) (r) dr + 2 R R with the integrals being absolutely convergent. Proof. This is proved by applying integration by parts twice. (cf. [54, Lemma 27]). 

CHAPTER 5

Green’s functions on GL(2, Fv ) with v a non archimedean place In this section, we fix a place v ∈ Σfin . The following lemma is a refined form of the Iwasawa decomposition Gv = Hv Nv Kv . Lemma 5.1. The group Gv is decomposed to a disjoint union of double (Hv , Kv )cosets:   ! −m Hv nm Kv with nm = 10 v1 . Gv = m∈N

For any m ∈ N,

 t1 Hv ∩ nm Kv n−1 m = { 0

0 t2



| t1 , t2 ∈ o× v , t1 ≡ t2

(mod pm v ) }.

Proof. This is shown by examining the condition for two (Hv , Kv )-double cosets represented by elements of Nv to coincide. The argument is straightforward.  Let z ∈ C. From Lemma 5.1, it turns out that there exists a unique function : Gv → C such that   t1 0  (z)  (5.1) Φ0,v ( t01 t02 nm k) = |t1 /t2 |zv δ(m = 0), 0 t2 ∈ Hv , m ∈ N, k ∈ Kv . (z) Φ0,v

Let Tv be as in 2.3.2. Given z ∈ C and s ∈ C/4πi(log qv )−1 Z, we consider the following inhomogeneous equation (5.2)

(z)

R(Tv )Ψ − (qv(1−s)/2 + qv(1+s)/2 ) Ψ = Φ0,v

with the unknown function Ψ : Gv → C possessing the (Hv , Kv )-equivariance:     t1 0  Ψ t01 t02 gk = |t1 /t2 |zv Ψ(g), (5.3) 0 t2 ∈ Hv , k ∈ Kv . Lemma 5.2. Suppose Re(s) > |2Re(z)−1|. Then, there exists a unique bounded (z) function Ψv (s; −) : Gv → C satisfying (5.2) and (5.3), whose values on Nv are given by (5.4)

−(s+1)/2 1 x (1 − qv−(s−2z+1)/2 )−1 (1 − qv−(s+2z+1)/2 )−1 Ψ(z) v (s; [ 0 1 ]) = −qv

sup(1, |x|v )−(s−2z+1)/2 ,

x ∈ Fv

Proof. From Lemma 5.1, the relation Ψ(nm ) = a(m) yields a linear bijection between the C-vector space of functions Ψ on Gv with the equivariance (5.3) and the C-vector space of sequences of complex numbers {a(m)}m∈N . Let a(m) = Ψ(nm ), m ∈ N. Then,   Ψ (nm gv ) dgv = Ψ (nm γ) . R(Tv )Ψ (nm ) =     γ∈Kv v 0 Kv /Kv 0 1

Kv v 0 Kv 0 1

37

38

5. GREEN’S FUNCTIONS ON GL(2, Fv ) WITH v A NON ARCHIMEDEAN PLACE

    As a set of representatives of Kv 0v 01 Kv /Kv , we take a point γ ∗ = 10 0v together with qv points γx = [ 0v x1 ] with x ∈ ov /pv . By a direct computation, using (5.3), we have ⎧ ⎪ (m > 0) ⎨a(m + 1), Ψ(nm γx ) = qv−z a(1), (m = 0, x = −1), ⎪ ⎩ a(0), (m = 0, x = −1),  a(m − 1), (m > 0), Ψ(mm γ ∗ ) = qvz a(0), (m = 0). Thus,  R(Tv )Ψ (nm ) =

qv1−z a(m + 1) + qvz a(m − 1), (qv − 1)qv−z a(1) + (qv−z + qvz ) a(0),

(m > 0), (m = 0).

(z)

Since Φ0,v (nm ) = δm,0 , the equation (5.3) is translated to the following recurrence relation among complex numbers a(m). (5.5) qv1−z a(m + 1) + qvz a(m − 1) = (qv(1−s)/2 + qv(1+s)/2 ) a(m),

(m > 0),

(5.6) (qv − 1)qv−z a(1) + (qv−z + qvz ) a(0) = (qv(1−s)/2 + qv(1+s)/2 ) a(0) + 1,

(m = 0).

Since (5.5) is a three term recurrence relation, its fundamental solutions are found (1−s)/2 (1+s)/2 + qv ) X + qvz = 0. by solving the characteristic equation: qv1−z X 2 − (qv (2z±s+1)/2 This is solved as X = qv . Thus, it is inferred that {a(m)}m∈N is a linear m(2z+s−1)/2 m(2z−s−1)/2 combination of {qv }m∈N and {qv }m∈N , among which only the latter one is bounded when Re(s) > |2Re(z) − 1|. Consequently, the sequence {a(m)} corresponding to a bounded function Ψ on Gv satisfying (5.2) and (5.3) should be of the form a(m) = C qvm(2z−s−1)/2 ,

m∈N

with a constant C. By (5.6), the constant is determined as C = −qv−(s+1)/2 (1 − qv−(s−2z+1)/2 )−1 (1 − qv−(s+2z+1)/2 )−1 . This completes the proof.



Lemma 5.3. We have the estimation −(Re(s)+1)/2+Re(z) 1 x |Ψ(z) v (s; [ 0 1 ]) |  sup(1, |x|v )

for (z, s; g) ∈ C × (C/4πi(log qv )−1 Z) × GA such that Re(s)/2 > |Re(z)| with an absolute implied constant. Proof. We obtain this from Lemma 5.2 by observing that the inequality −1/2 −2  (1−qv ) holds if Re(s)− 2|Re(z)| > 0.  −(s+1)/2 −(s−2z+1)/2 −1 −(s+2z+1)/2 −1 |qv (1−qv ) (1−qv ) |

5. GREEN’S FUNCTIONS ON GL(2, Fv ) WITH v A NON ARCHIMEDEAN PLACE

39

   Lemma 5.4. Let f : Gv → C be a smooth function such that f t01 t02 gk = × |t1 /t2 |−z v f (g) for any t1 , t2 ∈ Fv and for any k ∈ Kv . Then, the equality (5.7)  Hv \Gv

(1+s)/2 Ψ(z) + qv(1−s)/2 ) 1Kv )f ](g) dg = vol(Hv \Hv Kv ) f (e) v (s; g) [R(Tv − (qv

holds as long as the integral on the left-hand side converges absolutely. Proof. By a change of variables and by the equation (5.2), the left-hand side of (5.7) equals   (z) [R(Tv − (qv(1+s)/2 + qv(1−s)/2 ))Ψ(z) (s)](g) f (g) dg = Φ0,v (g) f (g) dg v Hv \Gv

Hv \Gv

= vol(Hv \Hv Kv ) f (e).  The value of lemma.

(z) Ψv (s; −)

at a lower unipotent matrix is known by the following

Lemma 5.5. When x ∈ ov , then [ x1 01 ] = an Iwasawa decomposition of [ x1 01 ].



x−1 0 0 x



[ 10 x1 ]

0

−1 1 x−1



∈ Hv Nv Kv is 

Proof. This is proved by a direct computation. (z) Φ0,v

(z) Ψv (s; −),

(z) Φ1,v

Other than and we need yet another function : Gv → C, which is defined as a unique function supported in Hv K0 (pv ) and satisfying the relation    t1 0  (z)  Φ1,v t01 t02 k = |t1 /t2 |zA , 0 t2 ∈ Hv , k ∈ K0 (pv ).

CHAPTER 6

Kernel functions 6.1. Test functions For a positive number C > 0, let B(C) be the space of all the entire functions β(z) on the region |Re(z)| < C such that β(−z) = β(z) and satisfying the estimation |β(σ + it)|  (1 + |t|)−l ,

(6.1)

σ ∈ [a, b]

for any [a, b] ⊂ (−C, C) and for any l > 0. Let B denote the space of all the entire functions β(z) on C such that β(z) restricted to |Re(z)| < C belongs to B(C) for all C > 0. 6.1.1. Let S a finite set of places of F . Set S∞ = S ∩ Σ∞ and Sfin = S ∩ Σfin ; thus S = S∞ ∪ Sfin is a disjoint union. Define a complex manifold XS to be the product   Cι × (C/4πi(log qv )−1 Z), ι∈S∞

v∈Sfin

where Cι denotes a copy of the complex plane C. For any point s = (sv ) ∈ XS , its real part Re(s) is defined to be the vector (Re(sv )) ∈ RS . 6.1.2. Let AS be the space of holomorphic functions α(s) on the complex manifold XS such that for any point s0 ∈ XS and for any v ∈ S, the one variable function s → α(s0 (v; s)) belongs to the space B. Here, s0 (v; s) denotes the point of XS having the same coordinates as s0 except the v-th component being replaced with s. For ι ∈ Σ∞ , set Aι = B; for v ∈ Sfin , let Av be the space of all the entire functions β(s) on C/4πi(log qv )−1 Z such that β(−s) = β(s). An element α ∈ AS is called decomposable if α(s) = v∈S αv (sv ) with αv ∈ Av , v ∈ S. 6.1.3. For a point c ∈ RS , let LS (c) be the submanifold {s ∈ XS | Re(s) = c } of XS . A multidimensional contour integral of a holomorphic function f (s) on XS along LS (c) is defined inductively as follows. If S = S  ∪ {v} is a disjoint union  and c = (c , cv ) with c ∈ RS , then "      f (s) dμS (s) = f (s , sv ) dμS  (s ) dμv (sv ), LS (c)

where (6.2)

Lv (cv )

 dμv (s) =

LS  (c )

s ds, (1+s)/2 (1−s)/2 2−1 log qv (qv − qv ) ds,

v ∈ Σ∞ , v ∈ Σfin ,

and Lv (cv ) denotes the contour (directed with increasing imaginary part), which, as a point set, is the line cv + iR in C or the circle cv + i(R/4π(log qv )−1 Z) in 41

42

6. KERNEL FUNCTIONS

C/4πi(log qv )−1 Z according to v ∈ Σ∞ or v ∈ Σfin respectively. We always assume that the integrand f (s) decays rapidly enough on LS (c) so that the order of integration does not matter. For s = (sv ) ∈ XS , the infinimum of numbers (Re(sv ) + 1)/4 with v ∈ S frequently occurs in this article; we denote it by q(s), i.e., q(s) = inf{(Re(sv ) + 1)/4| v ∈ S }. 6.2. Green’s functions on adele groups 6.2.1. From now on, we assume Σ∞ ⊂ S. Thus, S∞ = Σ∞ . Let n be a square free oF -ideal such that S ∩ S(n) = ∅. For s = (sv )v∈S ∈ XS , z ∈ C and g ∈ GA , set (6.3) Ψ(z) (n|s; g) =

 ι∈Σ∞

Ψ(z) ι (sι ; gι )



Ψ(z) v (sv ; gv )

v∈Sfin



(z)

Φ1,v (gv )

v∈S(n)



(z)

Φ0,v (gv ).

v∈S∪S(n)

(z) Φ0,v (gv )

= 1 at almost all places v ∈ S ∪ S(n), the product on the right-hand Since side makes sense. We compare this function with the function Ξp,q,Sfin studied in 3.2. Lemma 6.1. (1) The function g → Ψ(z)  g) is right K∞ K0 (n)-invariant  (n|s; continuous function on GA . For any t01 t02 ∈ HA , we have     Ψ(z) n|s; t01 t02 g = |t1 /t2 |zA Ψ(z) (n|s; g). (2) Let σ > 0. We have the estimation



(z)

Ψ (n|s; g)  Ξσ,q(s),Sfin (g) for (z, s; g) ∈ C × XS × (A∞ Nfin K) satisfying q(s) > |Re(z)| + 1 with an absolute implied constant. Proof. Thestatement  (1) is obvious from definition. To show (2), it suffices to estimate Ψ(z) n|s, a∞ 10 xfin k with varying (a∞ , xfin , k) ∈ A∞ × Afin × K, 1   which is a product of Ψ(z) (sι ; arι ) over ι ∈ Σ∞ and the values at the point 10 x1v (z) (v) of one of the functions Ψ(z) (sv ; −), Φ0,v and Φ1,v over non archimedean places v. At ι ∈ Σ∞ , we use Lemma 4.2 to have the majorant (cosh2rι )−(Re(sι )+1)/2 , which is less than (cosh2rι )−q(s) since q(s) > 0. At v ∈ S, by Lemma 5.3, sup(1, |xv |)−(Re(sv )+1)/2+Re(z) is a majorant, which is less than sup(1, |xv |v )−q(s) since Re(z) < q(s)  (Re(sv ) + 1)/4. We remark that q(s) > |Re(z)| + 1 implies Re(sv )/2 > |Re(z)| for any v ∈ S. The remaining factors are easily bounded by  δ(xv ∈ ov ). 6.2.2. For any z ∈ C, define a quasi-character χz : HF \HA → C× by setting   χz t01 t02 = |t1 /t2 |zA , t1 , t2 ∈ A × . Lemma 6.2. Let ϕ ∈ Cc∞ (AGF \GA ). Then, the integral  H,(z) (g) = ϕ(hg) χz (h) dh, z ∈ C, g ∈ GA ϕ AHF \HA

converges absolutely. We have ϕH,(z) (hg) = χz (h)−1 ϕH,(z) (g) for any h ∈ HA . For any  > 0, the estimation 2|Rez|+

|ϕH,(z) (g)|  gA

,

g ∈ G1A

6.3. SMOOTHED KERNELS

43

holds. Proof. Let ωϕ ⊂ G1A be a compact set such # that supp(ϕ)$ ⊂ A GF ωϕ . Suppose  t∞ 0   t∞ 1/2 0 1 g =  0 with t > 0, g ∈ G . Then, g = γh with γ ∈ GF ϕ ∞ A 0 1 0 t −1/2 ∞

and h ∈ ωϕ . Then, as in the proof of Lemma 3.3, # $−1 1/2 0 −1 1 −1 t∞ g [ ]  = g [ 10 ] A = h−1 γ −1 [ 10 ] A  1 t−1/2 A ∞ 0 0 t −1/2 ∞

 0   1/2 g = Thus, t∞  g −1 [ 10 ] A  gA . By the left A GF -invariance of ϕ, ϕ t∞ 0 1  −1   −1/2 t∞ 0 w0 g = 0, hence the bound t∞ 0 yields ϕ  gA similarly. Thus, 0 1 with some constants C1 , C2 > 0, we have   ∞ 

 t∞ x 0   Rez ×

ϕ g t∞ d t∞ |ϕ(hg) χz (h)| dh = vol(F × \A1 ) dx 0 1 AHF \HA A1 /F × 0  2|Rez|+

×  tRez . ∞ d t∞  gA 2 [C1 g−2 A ,C2 gA ]

 For s ∈ XS , consider the element       1 − s2ι ΩS (s) = { Ωv + Tv − (qv(1−sv )/2 + qv(1+sv )/2 ) 1Kv } }⊗{ 2 ι∈Σ∞ v∈Sfin  of the Hecke algebra Z(g∞ ) ⊗ { v∈Sfin H(Gv ; Kv )}, which acts on the space of smooth functions on GA by right regular action R. Lemma 6.3. Let q(s) > 2|Rez| + 1. Let ϕ ∈ Cc∞ (AGF \GA )K∞ K0 (n) . Then, Ψ (n|s; g)ϕH,(z) (g) is integrable on HA \GA for any z ∈ C. Moreover, the following equation holds.  (6.4) Ψ(z) (n|s; g) [R(ΩS (s)) ϕH,(z) ](g) dg = vol(Hfin \Hfin K0 (n)) ϕH,(z) (e). (z)

HA \GA

(z) H,(z) (g) with g = Proof.  1 x  From Lemmas 6.1 and 6.2, the function Ψ (n|s; g)ϕ k is bounded by (arι ) 0 fin 1

{

  (cosh2rι )−q(s)+|Rez|+/2 } { sup(1, |xv |v )−q(s)+2|Re(z)|+ } { ι

v∈Sfin



δ(xv ∈ ov )}.

v∈Σfin −S

If q(s) > 2|Rez| + 1, the majorant is integrable on HA \GA from the proof of Lemma 3.5. This proves the first statement. Since R(ΩS (s))ϕ also belongs to Cc∞ (AGF \GA ), the integral on the left-hand side of (6.4) converges absolutely. Then, (6.4) is inferred from Lemma 4.3 and 5.4.  6.3. Smoothed kernels For z ∈ C, α(s) ∈ AS and g ∈ GA , the contour integral (6.5) ˆ (z) (n|α; g) = Ψ



1 2πi

S  LS (c)

Ψ(z) (n|s; g) α(s) dμS (s),

(q(c) > |Re(z)| + 1),

44

6. KERNEL FUNCTIONS

converges absolutely and is independent of the choice of c by Lemma 6.1 (2) and by Cauchy’s theorem (cf. [54, Lemma 28]). In particular, for a fixed (α, g), the ˆ (z) (n|α; g) is entire on C. function z → Ψ Lemma 6.4. Let σ > 0, q > 0 and [σ1 , σ2 ] be any compact interval of R. Then, we have the estimate (6.6)

ˆ (z) (n|α; g)|  Ξσ,q,S (g), |Ψ fin

(z, g) ∈ ([σ1 , σ2 ] + iR) × (A∞ Nfin K).

Proof. This follows from Lemma 6.1 (2) immediately if we note that each cv can be taken arbitrarily large in (6.5).  6.4. Regularization Let β ∈ BC with C > 0. By the estimation (6.1), the contour integral (β(z)/z) dz converges absolutely for any σ ∈ (−C, C) − {0}. Lσ



Lemma 6.5. Let β ∈ BC . Then,  1 β(0) β(z) dz = sgn(σ) , 2πi Lσ z 2

σ ∈ (−C, C) − {0}.

Proof. For R > 0, let QR be the open rectangle with vertexes ±σ ± Ri. We consider the contour integral of β(z)/z along the boundary ∂QR oriented counterclockwisely. The only pole of β(z)/z lying in the rectangle QR is z = 0 with residue β(0). Thus, by the residue theorem,  σ−iR  −σ−iR β(z) β(z) I(R) = dz − dz − 2πi β(0) z z σ+iR −σ+iR is expressed as a sum of contour integrals of β(z)/z along horizontal segments [−σ, σ] ± iR, which vanish in the limit R → +∞ because of the estimate (6.1). By β(−z) = β(z), the first term of I(R) has the opposite sign to the second one. This completes the proof.  6.4.1. Renormalized Green function. Let (β, λ) ∈ B × C; for g ∈ GA and s ∈ XS such that q(s) > 0, we examine the contour integral  β(z) 1 {Ψ(z) (n|s; g) + Ψ(−z) (n|s; g)} dz (6.7) Ψβ,λ (n|s; g) = 2πi Lσ λ + z with σ ∈ R such that − inf{q(s) − 1, Re(λ)} < σ < q(s) − 1. Let DS = {(λ, s) ∈ C × XS | q(s) > 1, q(s) + Re(λ) > 1 }. Lemma 6.6. The integral (6.7) converges absolutely and locally uniformly in (λ, s; g) ∈ DS × GA , and is independent of the choice of a contour Lσ satisfying − inf{q(s) − 1, Re(λ)} < σ < q(s) − 1. Proof. If (λ, s) ∈ DS , we can choose σ ∈ R such that − inf{q(s) − 1, Re(λ)} < σ < q(s) − 1. Then, the first assertion of the lemma follows from the estimates (4.7) and (6.1). The function z → (β(z)/(z + λ)) {Ψ(z) (n|s; g) + Ψ(−z) (n|s; g)} is holomorphic on the region − inf{q(s) − 1, Re(λ)} < Re(z) < q(s) − 1; from this, the independence of σ is proved by Cauchy’s theorem. 

6.4. REGULARIZATION

45

Lemma 6.7. Let c ∈ RS be such that q(c) > 1 and b > 0. Then, for any σ ∈ R such that 0 < σ < inf(b, q(c) − 1), we have the estimate |Ψβ,λ (n|s; g)|  Ξσ,q(c),Sfin (g), (λ, s; g) ∈ Lb × LS (c) × GA .   Proof. Let g = t01 t02 a∞ nfin k with t1 , t2 ∈ A× , a∞ ∈ A∞ , nfin ∈ Nfin and k ∈ K. We first consider the case when |t1 /t2 |A  1. Let 0 < σ < inf(b, q(c) − 1). Then, the vertical stripe between Lσ and L−σ is contained in the region − inf(b, q(c) − 1) < Re(z) < q(c) − 1. Thus, from (6.7), by Cauchy’s theorem, we have  β(z) 1 (−z) |t1 /t2 |−z (n|s; a∞ nfin k) dz Ψβ,λ (n|s; g) = A Ψ 2πi Lσ λ + z   β(z) |t1 /t2 |zA Ψ(z) (n|s; a∞ nfin k) dz . + L−σ λ + z Now, we estimate two integrals on the right-hand side separately using Lemma 6.1(2) and (6.1) to show that each one is majorized by |t1 /t2 |−σ A Ξσ,q(s),Sfin (a∞ nfin k) = Ξσ,q(s),Sfin (g). The case |t1 /t2 |A  1 is settled in a similar way.  6.4.2. Renormalized smoothed kernel. Similarly, given (β, λ) ∈ B × C, we introduce the renormalized smoothed kernel by the contour integral (6.8)  β(z) ˆ (z) ˆ β,λ (n|α; g) = 1 ˆ (−z) (n|α; g)} dz, (σ > −Re(λ)) Ψ {Ψ (n|α; g) + Ψ 2πi Lσ λ + z for α ∈ AS and for g ∈ GA . Lemma 6.8. (1) The integral (6.8) converges absolutely and locally uniformly in (λ, g) ∈ C × GA , and is independent of the choice of a contour Lσ with σ > −Re(λ). (2) Given a compact interval [σ1 , σ2 ] ⊂ (0, +∞) and a positive number q, the estimation ˆ β,λ (n|α; g)|  Ξσ,q,S (g), (λ, g) ∈ ([σ1 , σ2 ] + iR) × GA |Ψ fin

holds for any σ such that 0 < σ < σ1 . Proof. The assertion (1) follows from (6.6) and (6.1). The estimate in (2) is deduced from (6.6) by an argument similar to Lemma 6.7.  ˆ β,λ (n|α; g) is Lemma 6.9. Let α ∈ AS and g ∈ GA . Then the function λ → Ψ entire on C and ˆ (0) (n|α; g). ˆ β,0 (n|α; g) = β(0) Ψ Ψ Proof. The first assertion is inferred from Lemma 6.8 (1). By Lemma 6.4, ˆ (z) (n|α; g)} belongs to the space B. Then, ˆ (z) (n|α; g) + Ψ the function z → β(z) {Ψ we apply Lemma 6.5 to obtain the second assertion, noting that σ > 0 in (6.8). 

CHAPTER 7

Regularized periods Throughout this section, we fix a unitary idele-class character η such that η 2 = 1. 7.1. Definition Let β ∈ B. For any t > 0 and λ ∈ C, set  β(z) z 1 ˆ βλ (t) = t dz, (7.1) 2πi Lσ λ + z

(σ > −Re(λ)).

By the estimation (6.1), the integral converges absolutely and is independent of the contour Lσ involved. Lemma 7.1. For any σ > −Re(λ), the estimation |βˆλ (t)|  inf{tσ , t−Re(λ) }, t > 0 holds. Proof. By (7.1) and (6.1), the estimation |βˆλ (t)|  tσ is immediate. Let σ > Re(λ). Then, by the residue theorem combined with the estimation (6.1), we have  β(z) z 1 ˆ t dz + β(−λ) t−λ , βλ (t) = 2πi L−σ λ + z 

after moving the contour from Lσ to L−σ . By this expression, we obtain |βˆλ (t)|  t−Re(λ) for t  1. This completes the proof.  ∞ Lemma 7.2. Let K ⊂ Kfin be an open subgroup and ϕ ∈ C+ (AGF \GA )K . Then, for any (β, λ) ∈ B × C such that Re(λ) > 1/2, the integral   t 0  1 xη  η(tx∗η ) d× t (7.2) {βˆλ (|t|A ) + βˆλ (|t|−1 Pηβ,λ (ϕ) = A )} ϕ [ 0 1 ] 0 1 F × \A×

converges absolutely. The linear form ϕ → Pηβ,λ (ϕ) belongs to the continuous dual ∞ (AGF \GA )K . of the Frechet space C+   Proof. Take  > 0 such that Re(λ) > (1 + )/2. Since |ϕ([ 0t 10 ] 10 x1η )|  −(1+ )/2

p1, (ϕ) {|t|A + |t|A } for any t ∈ A× , we have  ∞



η × 1 (ϕ)  p (ϕ) vol(F \A ) (t(1+ )/2 + t−(1+ )/2 ) {ˆ|βˆλ (t)| + |βˆλ (t)|} d× t.

Pβ,λ 1,

(1+ )/2

0

From this, by Lemma 7.1, we have a constant C(λ) > 0 such that

η

∞ ϕ ∈ C+ (AGF \GA )K .

Pβ,λ (ϕ)  C(λ) p1, (ϕ),  47

48

7. REGULARIZED PERIODS

Suppose λ → Pηβ,λ (ϕ) has a meromorphic extension to a neighborhood of λ = 0 for any β ∈ B. If the constant term CTλ=0 Pηβ,λ (ϕ) at λ = 0, regarded as a linear functional of β, is proportional to β → β(0), we call the proportionality constant the regularized η-period of ϕ and denote it by Pηreg (ϕ), i.e., CTλ=0 Pηβ,λ (ϕ) = β(0) Pηreg (ϕ) for any β ∈ B. This terminology is justified by the following lemma. Lemma 7.3. If ϕ : A GF \GA → C is rapidly decreasing on the Siegel domain G1A ∩ S, then Pηβ,λ (ϕ) converges for any (β, λ) ∈ B × C. We have Pηreg (ϕ) = Z ∗ (1/2, η; ϕ). m Proof. For any m > 0, |ϕ([ 0t 01 ])|  inf(|t|−m A , |t|A ). By this, the first part of z −z the lemma follows. Since z → β(z)(t + t ) belongs to B, the identity is proved by Lemma 6.5. 

7.2. Regularized periods of automorphic forms In this subsection, we compute the regularized periods for various automorphic forms constructed in 2.6.2 and 2.7.2. Let n be a square free oF -ideal. Let η be a unitary idele-class character of F × satisfying the conditions (2.18), (2.19) and (2.20). 7.2.1. Cusp forms. Let π be an irreducible cuspidal automorphic representation of GA with trivial central character whose conductor fπ divides n. Then, from Lemmas 7.3 and 2.5, we immediately obtain the following. η Lemma 7.4. For any c dividing nf−1 π , the function λ → Pβ,λ (ϕπ,c ) is entire on C and  Pηreg (ϕπ,c ) = G(η){ (ηv (v ) − Q(πv ))} L(1/2, π ⊗ η). v∈S(c)

7.2.2. Eisenstein series. We compute the regularized period for the Eisenstein series Eχ,c (ν; −) with ν ∈ iR constructed in 2.7.2. Lemma 7.5. Let c be an ideal dividing n and χ an unramified unitary idele-class ∞ (AGF \GA ). For any β ∈ B, character of F × . Let ν ∈ iR. Then, Eχ,c (ν; −) ∈ C+ η the function λ → Pβ,λ (Eχ,c (ν; −)) has a meromorphic continuation to the whole λ-plane regular at λ = 0. We have (7.3)

Pηreg (Eχ,c (ν; −)) = (DF N(c))−ν/2 χ(d ˜ F/Q c) G(η) η × Bχ,c (1/2, ν)

L((1 + ν)/2, χη) L((1 − ν)/2, χ−1 η) . L(ν + 1, χ2 )

◦ Proof. By [38, I.2.10], the difference Eχ,c (ν) − Eχ,c (ν) is rapidly decreasing 1 on the Siegel set GA ∩ S. By Lemma 2.10, it is evident that the constant term ◦ (ν; g) with ν ∈ iR is bounded by gA on G1A ∩ S. From these observations, we Eχ,c obtain the first statement. Set    t 0  1 xη   ◦ E  (ν; t) = Eχ,c ν; [ 0t 01 ] 10 x1η − Eχ,c ν; [ 0 1 ] 0 1

7.2. REGULARIZED PERIODS OF AUTOMORPHIC FORMS

49

for Re(λ) > 1. If we have the absolute convergence of integrals    (ν)  × Pχ (λ, ν) = fχ,c [ 0t 01 ] 10 x1η η(tx∗η ){βˆλ (|t|A ) + βˆλ (|t|−1 A )} d t, F × \A×   × Q± (η; λ, ν) = η(tx∗η )βˆλ (|t|± χ,c A ) E (ν; t) d t, F × \A×

then, from Lemma 2.10, L(ν, χ2 ) P −1 (λ, −ν) L(ν + 1, χ2 ) χ − + Q+ χ,c (η; λ, ν) + Qχ,c (η; λ, ν). −1/2

(7.4)

Pηβ,λ (Eχ,c (ν; −)) = Pχ (λ, ν) + DF

Aχ,c (ν)

  (ν)  (ν+1)/2 We have fχ,c [ 0t 01 ] 10 x1η = χ(t)|t|A N(c)1/2 from definition (see 2.7.2). Since Re(λ) > 1 and Re(ν) = 0, by Lemma 7.6, Pχ (λ, ν) is absolutely convergent and   β((−ν − 1)/2) β((ν + 1)/2) + (7.5) Pχ (λ, ν) = δχ,η vol(F × \A1 ) N(c)1/2 . λ − (ν + 1)/2 λ + (ν + 1)/2 By exchanging the order of integrals formally, we have     1 β(z) z + |t| dz E  (ν, t) η(tx∗η ) d× t Qχ,c (η; λ, ν) = (7.6) 2πi Lσ λ + z A F × \A×  "  β(z) 1  z ∗ × E (ν, t) |t|A η(txη ) d t dz. = 2πi Lσ λ + z F × \A× ◦ (ν; ·)) Note that the integral inside the bracket is just Z ∗ (z + 1/2, η; Eχ,c (ν; ·) − Eχ,c computed in Lemma 2.11. Thus,

(7.7) Q+ χ,c (η; λ, ν)

(DF N(c))−ν/2 χ(d ˜ F/Q c)G(η) = 2πi

with (7.8)

η fχ,c (z, ν)

=

η Bχ,c

   L z+ 1 z + 2, ν

 η fχ,c (z, ν) Lσ

β(z) dz, λ+z

(σ > 1/2)

  −1 L z + 1−ν η 2 ,χ . L(ν + 1, χ2 )

1+ν 2 , χη



By this, the integration exchange in (7.6) is justified easily. The right-hand side of (7.7) with varying σ > 1/2 is holomorphic on Re(λ) > −σ, providing a holomorphic continuation to the whole λ-plane. By the same computation, if we choose σ such that Re(λ) > σ > 1/2, Q− χ,c (η; λ, ν) =

(DF N(c))−ν/2 χ(d ˜ F/Q c)G(η) 2πi

 η fχ,c (−z, ν) L−σ

β(z) dz. λ+z

The right-hand side is holomorphic on Re(λ) > σ. To continue this to the region Re(λ) > −σ, we apply the residue theorem shifting the contour from L−σ to Lσ . The possible poles of the integrand swept by the moving contour are located at z = (−1 ± ν)/2 and (1 ± ν)/2. Thus, by using Lemma 2.12 to compute the residues

50

7. REGULARIZED PERIODS −1/2

and then by the relation G(χ) = χ(d ˜ F/Q ) DF (7.9) −

Qχ,c (η; λ, ν) = (DF N(c))

−ν/2

 χ(d ˜ F /Q c)G(η)

1 2πi

, we obtain

 η

Lσ

fχ,c (−z, ν)

β(z) dz λ+z

β((−ν + 1)/2) β((ν + 1)/2) η η Resz=(ν+1)/2 fχ,c (−z, ν) − Resz=(−ν+1)/2 fχ,c (−z, ν) λ + (ν + 1)/2 λ + (−ν + 1)/2  β((−ν − 1)/2) β((ν − 1)/2) η η Resz=(ν−1)/2 fχ,c (−z, ν) − Resz=(−ν−1)/2 fχ,c (−z, ν) − λ + (ν − 1)/2 λ + (−ν − 1)/2  ˜ F /Q c)G(η) (DF N(c))−ν/2 χ(d β(z) η fχ,c (−z, ν) dz = 2πi λ+z Lσ   β((ν + 1)/2) × 1 1/2 S(c) −1 β((ν + 1)/2) N(c) + (−1) − + δχ,η vol(F \A ) N(c) λ + (ν + 1)/2 λ − (ν + 1)/2   β((ν − 1)/2) × 1 1/2 S(c) −1 β((ν − 1)/2) + δχ,η vol(F \A ) N(c) N(c) + (−1) − λ − (ν − 1)/2 λ + (ν − 1)/2 −

−1/2

× DF

Aχ,c (ν)

ζF (ν) . ζF (ν + 1)

From (7.4), (7.5), (7.7) and (7.9), we get Pηβ,λ (Eχ,c (ν; −)) =Q0χ,c (η; λ, ν) + δχ,η vol(F × \A1 ) {N(c)1/2 + (−1)S(c) N(c)−1/2 }   β((ν + 1)/2) β((ν − 1)/2) −1/2 ζF (ν) + D Aχ,c (ν) × (7.10) λ − (ν + 1)/2 λ + (ν − 1)/2 F ζF (ν + 1) on Re(λ) > σ, where Q0χ,c (η; λ, ν) = (7.11)

(DF N(c))−ν/2 χ(d ˜ F/Q c)G(η) 2πi  ×

η η {fχ,c (z, ν) + fχ,c (−z, ν)} Lσ

β(z) dz, λ+z

(Re(λ) > −σ).

This gives us a meromorphic continuation of Pηβ,λ (Eχ,c (ν; −)) to the half plane Re(λ) > −σ. Obviously, the right-hand side of (7.10) is holomorphic at λ = 0 and allows us to compute the value CTλ=0 Pηβ,λ (Eχ,c (ν; −)) as (7.12)

˜ F/Q c)G(η) (DF N(c))−ν/2 χ(d 2πi ×

 η η {fχ,c (z, ν) + fχ,c (−z, ν)} Lσ

β(z) dz z

+ δχ,η vol(F \A ) {N(c) + (−1)S(c) N(c)−1/2 }   β((ν + 1)/2) β((ν − 1)/2) −1/2 ζF (ν) × + DF Aχ,c (ν) . −(ν + 1)/2 (ν − 1)/2 ζF (ν + 1) 1

1/2

It remains to compute the first term of (7.12). By the residue theorem, the contour integral along Lσ divided by 2πi is evaluated as the sum of residues of η η (z, ν) + fχ,c (−z, ν)} β(z) at poles lying between L−σ and Lσ , which z → 2−1 {fχ,c z are z = 0, (1 ± ν)/2 and (−1 ± ν)/2. The residues at z = (1 ± ν)/2, (−1 ± ν)/2 cancel with the second term of (7.12). The residue at z = 0 is easily computed as η (0, ν). Thus, CTλ=0 Pηβ,λ (Eχ,c (ν; −)) is equal to β(0) fχ,c η ˜ F/Q c)G(η) β(0) fχ,c (0, ν). (DF N(c))−ν/2 χ(d

This completes the proof.



7.2. REGULARIZED PERIODS OF AUTOMORPHIC FORMS

51

In the course of the proof, we have the following formula, which will be used later. ˜ (c) (7.13) Q0χ,c (η; 0, ν) =δχ,η vol(F × \A1 ) N   β((1 + ν)/2) ζF (ν) β((1 − ν)/2) −1/2 + DF Aχ,c (ν) × (1 + ν)/2 ζF (1 + ν) (−ν + 1)/2 η ˜ F/Q c) fχ,c (0, ν), + β(0) (DF N(c))−ν/2 G(η) χ(d

˜ (c) = N(c)1/2 + (−1)S(c) N(c)−1/2 . where N Lemma 7.6. Let ξ be an unitary idele-class characters of F × . Let w, λ ∈ C be such that Re(w) < Re(λ). Then, for  ∈ {0, 1},  × × 1 βˆλ (|t|A ) ξ(t) (log |t|A ) |t|w A d t = δξ,1 vol(F \A ) F × \A×



×

β(−w) δ ,1 β  (−w) −

+1 (λ − w) λ−w

 ,

where the integral on the left-hand side converges absolutely. 1 Proof. By A× = R× + A , our integral is decomposed as  +∞  1 ξ(u) d u βˆλ (t) (log t) tw d× t. F × \A1

0

The first factor is easily seen to be equal to δξ,1 vol(F × \A1 ). The last assertion follows from Lemma 7.1 obviously. By Re(w) < Re(λ), we can fix σ, σ  ∈ R such that −Re(λ) < σ  < −Re(w) < σ. Then, Lσ (β(z)/(λ + z)) (log t) tz dz =  (β(z)/(λ + z)) (log t) tz dz by Cauchy’s theorem. Moreover, Lσ  1 (−1)

(log t) tz+w d× t = , (z ∈ Lσ ), (z + w)1+

0  +∞  (−1) +1 (log t) tz +w d× t =  , (z  ∈ Lσ ).

+1 (z + w) 1 Using these, we have 

+∞

βˆλ (t) (log t) tw d× t    +∞  β(z) 1 (log t) tz dz tw d× t = 2πi Lσ λ + z 0      1  +∞  1 1 β(z) β(z) (log t) tz dz tw d× t + (log t) tz dz tw d× t = 2πi Lσ λ + z 2πi Lσ λ + z 0 1     1  +∞  β(z) β(z) 1 1  z w ×  z (log t) t dz t d t + (log t) t dz tw d× t = 2πi λ + z 2πi λ + z 0 1 Lσ Lσ   1  +∞     1 1 β(z) β(z)  z+w ×  z+w × = (log t) t d t dz + (log t) t d t dz 2πi Lσ λ + z 2πi Lσ λ + z 0 1   1 1 β(z) β(z) −(−1) (−1) dz + dz = +1 2πi Lσ λ + z (w + z) 2πi Lσ λ + z (w + z)+1

 β(−w) δ,1 β  (−w) (−1) β(z) = = Resz=−w − . +1 +1 λ + z (w + z) (λ − w) λ−w 0

52

7. REGULARIZED PERIODS

 7.2.3. Residual forms. Recall that eχ,c,−1 (g) denote the residue of Eχ,c (ν; −) at ν = 1. Lemma 7.7. For Re(λ) > 0, we have Pηβ,λ (eχ,c,−1 ) = δ(χ = η, c = oF )

4 vol(F × \A1 )2 β(0) . vol(ZA GF \GA ) λ

Moreover, Pηreg (eχ,c,−1 ) = 0. 

Proof. This follows from Lemmas 2.13 and 7.6 immediately.

Let eχ,c,0 (g) be the constant term of the Laurent expansion of Eχ,c (ν; g) at ν = 1. Thus, eχ,c,−1 (g) Eχ,c (ν; g) = (7.14) + eχ,c,0 (g) + O(ν − 1). ν−1 For an idele-class character ξ, let R(ξ), C0 (ξ) and C1 (ξ) be such that R(ξ) + C0 (ξ) + C1 (ξ) (s − 1) + O((s − 1)2 ) (s → 1). s−1 × \A1 ), CF = C0 (1) and CF = For ζF (s) = L(s, 1), we set RF = R(1) = vol(F √ C1 (1). We remark that CQ = CEuler /2 − log(2 π) with CEuler the Euler constant. L(s, ξ) =

(7.15)

Lemma 7.8. The integral Pηβ,λ (eχ,c,0 ) converges absolutely in Re(λ) > 1. There exists an entire function f (λ) on C such that Pηβ,λ (eχ,c,0 ) =δχ,η vol(F × \A1 ) {N(c)1/2 + (−1)S(c) N(c)−1/2 }

β(1) λ−1

(DF−1 N(c))1/2 + 2δχ,η vol(F × \A1 ) ζF (2)      β(0) ζ (2) Aη,c (1) + Aη,c (1) + CF Aη,c (1) × RF − F ζF (2) λ 2 RF ˜ (DF N(c))−1/2 + δχ,η χ(c) ζF (2)   β(0) β(0) η  η ˜η,c ˜η,c × (B −B ) (0) (0) 2 + f (λ) λ λ η η ˜χ,c for Re(λ) > 1, where B (z) = (N(fη ) DF )z Bχ,c (1/2 − z, 1). We have (7.16)  −1/2 χ(c ˜ f−1 N(cf−1 δη,χ 2 η ) η ) G(η) CTλ=0 Pηβ,λ (eχ,c,0 ) = − R δ(c = oF ) β  (0) W (η) L(2, χ2 ) 2 F    δχ,η 2 ˜ η  η ˜χ,c + C0 (χη)2 B RF (Bη,c ) (0) β(0) . (0) − 2δχ,η RF CF δ(c = oF ) + 2 Proof. From Lemma 2.10, we have the expression    (1) (2) eχ,c,0 [ 0t 01 ] 10 x1η = χ(t)|t|A N(c)1/2 + e0 (t) + e0 (t) (2)

with e0 (t) = E  (1; t) defined in the proof of Lemma 7.5 and  (1) −1/2 (−s+1)/2 −1 1/2 e0 (t) = χ(t) N(c) DF CTs=1 |t|A Aχ,c (s)

L(s, χ2 ) L(s + 1, χ2 )

 .

7.2. REGULARIZED PERIODS OF AUTOMORPHIC FORMS

By Lemma 7.6, 

F × \A×

∗ × × 1 χ(t) |t|A {βˆλ (|t|A ) + βˆλ (|t|−1 A )} η(txη ) d t = δχ,η vol(F \A )



β(−1) β(1) + λ+1 λ−1

53



if Re(λ) > 1. By (7.15), we have    −1/2 N(c)1/2 D L (2, χ2 ) (1) 2  A ) − (1) + A (1) + C0 (χ2 ) Aχ,c (1) e0 (t) = F R(χ χ,c χ,c L(2, χ2 ) L(2, χ2 )  1 − R(χ2 ) Aχ,c (1) log |t|A χ(t)−1 . 2 Thus, by Lemma 7.6,  (1) (7.17) e0 (t) {βˆλ (t) + βˆλ (t−1 )} η(tx∗η ) d× t F × \A×

−1/2

N(c)1/2 D = 2 δχ,η vol(F × \A1 ) F ζF (2)      ζF (2)  Aχ,c (1) + Aχ,c (1) + CF Aχ,c (1) λ−1 β(0) × RF − ζF (2) if Re(λ) > 0. The same computation as in the proof of Lemma 7.5 is applied to obtain the identity  (2) e0 (t) {βˆλ (t) + βˆλ (t−1 )} η(tx∗η ) d× t F × \A×

  β(z) η ˜ F/Q c)G(η) {Resz=−1 + Resz=0 + Resz=1 } fχ,c (−z, 1) = −(DF N(c))−1/2 χ(d λ+z + Q0χ,c (η; λ, 1) (Re(λ) > 1), where the integral Q0χ,c (η; λ, 1) defined by (7.11) with the movable contour Lσ η ˜η,c (σ > 1) is holomorphic on Re(λ) > −σ. It is easy to see B (1) = η˜(c) N(c) and η ˜η,c B (−1) = (−1)S(c) η˜(c); by these, we have Resz=1 = δχ,η vol(F × \A1 ) η˜(c) N(c) DF

β(1) , λ+1

β(−1) Resz=−1 = −δχ,η vol(F × \A1 ) η˜(c) (−1)S(c) DF , λ−1   2 β(0) β(0) 1/2 RF η  η ˜ ˜ Resz=0 = −δχ,η DF − Bχ,c (0) 2 . (Bχ,c ) (0) ζF (2) λ λ Thus, we have the first assertion by setting f (λ) = Q0χ,c (η; λ, 1). To compute the constant term, we note (7.18) CTλ=0 Pηβ,λ (eχ,c,0 ) = − δχ,η vol(F × \A1 ) × {N(c)1/2 + (−1)S(c) N(c)−1/2 } β(1) + Q0χ,c (η; 0, 1). By the same way as in the last part of the proof of Lemma 7.5, the integral Q0χ,c (η; 0, 1) equals the half of {Resz=−1 + Resz=1 + Resz=0 }   β(z) −1/2 η η × (DF N(c)) χ(d ˜ F/Q c) G(η) {fχ,c (z, 1) + fχ,c (−z, 1)} . z

54

7. REGULARIZED PERIODS

The sum of the first two residues cancel with the first term of (7.18). By computing η ˜η,c (0) = δ(c = oF ), we obtain (7.16).  the residue at z = 0 and by B η ˜η,c For later purpose, we explicate the special values of Aη,c (ν) and B (z) occurring in Lemma 7.8.

Lemma 7.9. When η is unramified, we have η ˜ Bη,c (0) = δ(c = oF ), Aη,c (1) = δ(c = oF ), N(c) log N(c) , N(c) − 1 N(c) log N(c) , Aη,c (1) = −δ(S(c) = 1) N(c)2 − 1  qv log qv N(c) (log N(c))2 η  ˜η,c (B ) (0) = δ(S(c) = 1) η˜(c) + 2δ(S(c) = 2) η˜(c) . N(c) − 1 qv − 1 η  ˜η,c ) (0) = δ(S(c) = 1) η˜(c) (B

v∈S(c)

Proof. A direct computation.



CHAPTER 8

Automorphic Green’s functions Let S be a finite set of places of F containing Σ∞ and n a square free oF -ideal such that S(n)∩S = ∅. For (β, λ, s) ∈ B×C×XS such that Re(λ) > 0, q(s) > 1 (see 6.1.3), we consider the average of the function g → Ψβ,λ (n|s; g) over the GF -orbits:  Ψβ,λ (n|s; g) = (8.1) Ψβ,λ (n|s; γg), g ∈ GA . γ∈HF \GF

Proposition 8.1. (1) The series (8.1) converges absolutely and locally uniformly in (λ; s, g) ∈ {Re(λ) > 0} × {q(s) > 1} × GA . For a fixed (λ, s) in this region, (8.1) defines a continuous function Ψβ,λ (n|s; −) on GA , which is left GF -invariant and right K0 (n)K∞ -invariant. (2) Let (λ, s) ∈ C × XS be such that 2Re(λ) > 1, q(s) > 2Re(λ) + 1. Then, for any σ ∈ (1/2, Re(λ)), the estimation |Ψβ,λ (n|s; g)|  y(g)1−σ ,

g ∈ G1A ∩ S

holds. Proof. This is a direct consequence of Lemmas 6.7 and 3.5.



Let D∗S = {(λ, s) ∈ C × XS | 2Re(λ) > 1, q(s) > 2Re(λ) + 1 } and K ⊂ Kfin an open compact subgroup. By Proposition 8.1 (1), for any fixed (λ, s) ∈ D∗S , the function Ψβ,λ (n|s; −) defines a distribution on AGF \GA by  (8.2) Ψβ,λ (n|s), ϕ = Ψβ,λ (n|s; g) ϕ(g) dg, ϕ ∈ Cc∞ (AGF \GA )K , AGF \GA

∞ (AGF \GA )K as follows. extended continuously to the Frechet space C+

Lemma 8.2. Let (λ, s) ∈ D∗S . Then, the integral (8.2) converges absolutely ∞ and defines a continuous functional on the space C+ (AGF \GA )K . For any ϕ ∈ ∞ C+ (AGF \GA )ZA K0 (n)K∞ , we have  −1/2 (8.3) Ψβ,λ (n|s), R(ΩS (s))ϕ = DF { (1 + qv )−1 } vol(F × \A1 ) P1β,λ (ϕ). v∈S(n)

Proof. Fix σ ∈ (1/2, Re(λ)) and  ∈ (0, 2σ − 1). Then, we obtain the estimation |Ψβ,λ (n|s; g) ϕ(g)|  p1, (ϕ) y(g)(3−2σ+ )/2 ,

∞ g ∈ G1A ∩ S, ϕ ∈ C+ (AGF \GA )K

by (3.16) and Proposition 8.1 (2). Since (3 − 2σ + )/2 < 1, the first statement is inferred from this by Lemma 3.4. Moreover, there exists a constant C > 0 such that ∞ (AGF \GA )K , | Ψβ,λ (n|s), ϕ |  C p1, (ϕ), ϕ ∈ C+ 55

56

8. AUTOMORPHIC GREEN’S FUNCTIONS

which shows the second statement. It remains to prove (8.3). By Lemma 3.8 and by the continuity of the both sides of (8.3) as linear functionals on ∞ C+ (AGF \GA )ZA K0 (n)K∞ ,

it suffices to establish the identity for ϕ ∈ Cc∞ (AGF \GA )ZA K0 (n)K∞ . For such ϕ, using Lemma 6.1 (1), we have Ψβ,λ (n|s), ϕ   = AGF \GA γ∈H \G F F

Ψβ,λ (n|s; γg) ϕ(γg) dg



Ψβ,λ (n|s; g) ϕ(g) dg

= AHF \GA





= HA \GA



AHF \HA



= HA \GA

AHF \HA

Ψβ,λ (n|s; hg) ϕ(hg) dh dg 

1 2πi

 Lσ

β(z) {Ψ(z) (n|s; g) χz (h) + Ψ(−z) (n|s; g) χ−z (h)} dz z+λ



× ϕ(hg) dh dg    1 β(z) = {Ψ(z) (n|s; g) ϕH,(z) (g) + Ψ(−z) (n|s; g) ϕH,(−z) (g)} dg dz. 2πi Lσ z + λ HA \GA To show the last equality, we should note that AZF \ZA ∼ = F × \A1 in the obvious way. By the estimation of the support of h → ϕ(hg) in the proof of Lemma 6.2, combined with Lemma 6.1 (2), it is easy to confirm the convergence of the triple integral    |β(z)| |dz| dg |Ψ(±z) (n|s; g) ϕ(hg) χ±z (h)| dh, |z + λ| Lσ HA \GA AHF \HA which justifies the exchange of order of integrals in the above computation. Apply the above computation for R(ΩS (s))ϕ in place of ϕ. Then, since (R(ΩS (s))ϕ)H,(z) (g) = R(ΩS (s))[ϕH,(z) ](g), by Lemma 6.3, we proceed with  β(z) 1 vol(Hfin \Hfin K0 (n)) {ϕH,(z) (e) + ϕH,(−z) (e)} dz 2πi Lσ z + λ  β(z) 1 = vol(Hfin \Hfin K0 (n)) vol(F × \A1 ) 2πi Lσ z + λ  ϕ(h) {χz (h) + χ−z (h)} dh dz × ZA HF \HA  × = vol(Hfin \Hfin K0 (n)) vol(F × \A1 ) ϕ ([ 0t 01 ]) {βˆλ (|t|A ) + βˆλ (|t|−1 A )} d t. F × \A×

P1β,λ (ϕ),

The last integral equals and the first volume factor is computed by the following lemma. Thus, we are done.  Lemma 8.3. −1/2

vol(Hfin \Hfin K0 (n)) = DF

−1/2

[Kfin : K0 (n)]−1 = DF

 v∈S(n)

(1 + qv )−1 .

8. AUTOMORPHIC GREEN’S FUNCTIONS

57

Proof.   our normalization of Haar measures (see 2.3),  Let v ∈ Σfin . From we have Hv \Gv f (gv ) dgv = Fv Kv f ([ 10 x1 ] k) dx dk for any integrable function f on Hv \Gv . Apply this to the characteristic function of Hv \Hv K0 (pn ). Then, vol(Hv \Hv K0 (pnv )) = vol(ov ) vol(K0 (pnv ); dkv ). Since vol(ov ) = qv−dv /2 and we are done.

vol(K0 (pv ); dkv ) = (1 + qv )−1 vol(Kv ; dkv ) = (1 + qv )−1 , 

CHAPTER 9

Automorphic smoothed kernels Let S and n be as in the previous section. For (β, λ) ∈ B × C and α ∈ AS such ˆ β,λ (n|α; g) translated that Re(λ) > 0, consider the average of the smoothed kernel Ψ over the discrete orbit space HF \GF :  ˆ β,λ (n|α; γg), g ∈ GA . ˆ β,λ (n|α; g) = Ψ Ψ (9.1) γ∈HF \GF

Proposition 9.1. (1) The series (9.1) converges absolutely and locally uniformly in (λ, g) ∈ {Re(λ) > 0} × GA . For a fixed g, the function ˆ β,λ (n|α; g) is entire on the half plane Re(λ) > 0. For a fixed λ → Ψ ˆ β,λ (n|α; g) on GA is continuous, left GF Re(λ) > 0, the function g → Ψ invariant and right K0 (n)K∞ -invariant. ˆ β,λ (n|α; −) ∈ Ll (AGF \GA ) for any l > 0 (2) Let Re(λ) > 0. Then, we have Ψ such that l(1 − Re(λ)) < 1. Proof. This is a direct consequence of Lemmas 6.8 and 3.5.



Lemma 9.2. Let Re(λ) > 1 and q(c) > sup(Re(λ) + 1, 2). Then, for ϕ ∈ ∞ (AGF \GA ), C+  (9.2)

ˆ β,λ (n|α), ϕ = Ψ

1 2πi

S  LS (c)

Ψβ,λ (n|s), ϕ α(s) dμS (s).

Proof. By the same way as in the proof of Lemma 8.2, ˆ β,λ (n|α), ϕ Ψ "  S   1 Ψβ,λ (n|s; g) α(s) dμS (s) ϕ(g) dg = 2πi AHF \GA LS (c) "   S  1 = Ψβ,λ (n|s; g) ϕ(g) dg α(s) dμS (s) 2πi LS (c) AHF \GA S   1 = Ψβ,λ (n|s), ϕ α(s) dμS (s). 2πi LS (c) To justify the change of order of integrals, we have to show the convergence of the integral   (9.3) dg|Ψβ,λ (n|s; g)| |ϕ(g)| |α(s)| |dμS (s)|. LS (c)

AHF \GA

59

60

9. AUTOMORPHIC SMOOTHED KERNELS

By Lemma 6.7 and by |ϕ(g)|  g1+

A , we majorize this integral by   { |α(s)| |dμS (s)|} { ΞRe(λ),q(c),Sfin (g) g1+

A dg} LS (c)

AHF \GA

with some  > 0 specified below. The first integral is convergent by the rapid decay of α(s) along LS (c) (see 6.1.2). Let ωA1 ⊂ A1 be a compact set such  that A1 = F × ωA1 . Then, by GA = HA A∞ Nfin K and AHF \HA ∼ = { yu0 1 u02 | y > 0, u1 , u2 ∈ ωA1 }, we bound the second integral by  ∞   1+

y −Re(λ)+1 d× y} { (cosh2rι )−q(c)+ 2 (rι ) drι } { 1

×{

 v∈S

ι∈Σ∞

R

sup(1, |xv |v )−q(c)+1+ dxv },

Fv

which is finite if  > 0 is small enough so that q(c) > 2 + .



ˆ β,λ (n|α; g) on GA is Lemma 9.3. Let Re(λ) > 1. Then, the function g → Ψ smooth. For any ι ∈ Σ∞ , ˆ β,λ (n|α; g) = Ψ ˆ β,λ (n|αm,ι ; g) R(Ωι )m Ψ with αm,ι (s) = ((s2ι − 1)/2)m α(s). 

Proof. cf. [54, Proposition 42]. 9.1. Spectral expansion of automorphic smoothed kernels

9.1.1. Let L2cus be the space of cuspidal functions in L2 (GF ZA \GA ). For a square free oF -ideal n, we denote by Lp (n) and L2cus (n) the K0 (n)K∞ -fixed subspace of Lp (GF ZA \GA ) and that of L2cus , respectively. Moreover, let Πcus (n) denote the set of all those irreducible cuspidal automorphic representations π with K (n)K∞ = {0}. For each π ∈ Πcus (n), fix trivial central characters such that Vπ 0 K (n)K∞ an orthonormal basis B(n)π of the finite dimensional space Vπ 0 and set  Bcus (n) = π∈Πcus (n) B(n)π . Then, Bcus (n) provides us with an orthonormal basis of L2cus (n). One way to obtain B(n)π is given in 2.6.2. 9.1.2. The orthogonal complement of L2cus in the total L2 -space is well-understood by the theory of Eisenstein series ([12]), which enables us to have the spectral expansion of an arbitrary L2 -function not necessarily cuspidal. Let us recall the result concisely. Recall the set Ξ0 (see 2.4.3). For χ ∈ Ξ0 such that χ2 = 1, set ϕχ (g) = χ(det g) vol(ZA GF \GA )−1/2 for any g ∈ GA . Let Bres (n) be the orthonormal system consisting of all functions ϕχ with χ ∈ Ξ0 such that χ2 = 1. Set B(n) = Bcus (n) ∪ Bres (n). If Ψ ∈ L2+ (n) for some  > 0, then Ψ is expanded to a weakly convergent series and integrals:    vol(F × \A1 )−1  Ψ(g) = Ψ|ϕ ϕ(g) + αΨ (χ, c; ν) Eχ,c (ν; g) dν 8πi iR ϕ∈B(n)

χ∈Ξ0 c|n

with

 αΨ (χ, c; ν) =

ZA GF \GA

Ψ(g) Eχ,c (ν; g) dg.

9.1. SPECTRAL EXPANSION OF AUTOMORPHIC SMOOTHED KERNELS

61

∞ 9.1.3. Let ϕ ∈ C+ (AGF \GA )K0 (n)K∞ be such that

(9.4) (9.5)

2 νϕ,ι −1 ϕ, νϕ,ι ∈ iR+ ∪ [0, 1) for any ι ∈ Σ∞ , and 2 R(Tv ) ϕ = (qv(1−νϕ,v )/2 + qv(1+νϕ,v )/2 ) ϕ for any v ∈ Sfin

R(Ωι ) ϕ =

with some νϕ,S = (νϕ,v )v∈S ∈ XS . In this case, the element νϕ,S will be called the spectral parameters at S of ϕ. For example, if ϕ ∈ B(n)π with π ∈ Πcus (n), then ν /2 νϕ,S is determined by the condition πv ∼ = Iv (| |vϕ,v ) for any v ∈ S (see 2.5). If ϕ = Eχ,c (ν; −) with ν ∈ iR and χ ∈ Ξ0 , then νϕ,S coincides with ν χ defined as ν χ = (ν + 2 a(χv ))v∈S with a(χv ) defined in 2.4.1. If ϕ = ϕχ with χ ∈ Ξ0 , χ2 = 1, then νϕ,S = (1 + 2 a(χv ))v∈S by a direct computation. Lemma 9.4. Let Re(λ) > 1. Let νϕ,S ∈ XS be the spectral parameters at S of ∞ (AGF \GA )K0 (n)K∞ ZA . Then, ϕ ∈ C+  ˆ β,λ (n|α)|ϕ = (−1)S D−1/2 { Ψ (1 + qv )−1 } α(νϕ,S ) P1β,λ (ϕ). F v∈S(n)

Proof. This follows from Lemmas 8.2 and 9.2 with the aid of the following Lemma 9.5. Note that the L2 -inner product on L2 (ZA GF \GA ) differs from the  pairing (8.2) by the factor vol(ZF A\ZA ) = vol(F × \A1 ). Lemma 9.5. Let αv ∈ Av . Then, for any ν ∈ C and c ∈ R such that c > |Re(ν)|,  c+i∞ 2 αv (s) 1 dμι (s) = −αv (ν) if v ∈ Σ∞ , 2πi c−i∞ ν 2 − s2  c+2iπ(log qv )−1 1 αv (s) dμv (s) 2πi c−2iπ(log qv )−1 qv(1−ν)/2 + qv(1+ν)/2 − qv(1−s)/2 − qv(1+s)/2 = −αv (ν),

if v ∈ Σfin ,

where dμv (s) is given by (6.2). Proof. This is easily confirmed by the residue theorem.



For α ∈ AS and χ ∈ Ξ0 , we define a one variable function α ˜ χ (ν) by α ˜ χ (ν) = α(ν χ ),

ν ∈ C.

Lemma 9.6. Let Re(λ) > 1. For any g ∈ GA , we have (9.6) ˆ β,λ (n|α; g) = (−1)S D−1/2 { Ψ F  ×



(1 + qv )−1 }

v∈S(n)



ϕ∈Bcus (n)

α(νϕ,S ) P1β,λ (ϕ) ¯ ϕ(g) +



α(νϕ,S ) P1β,λ (ϕ) ¯ ϕ(g)

ϕ∈Bres (n)

   vol(F × \A1 )−1  1 + α ˜ χ (ν) Pβ,λ (Eχ,c (ν; −)) Eχ,c (ν; g) dν . 8πi iR χ∈Ξ0 c|n

The series and integrals converges absolutely and uniformly on arbitrary compact subset of ZA GF \GA .

62

9. AUTOMORPHIC SMOOTHED KERNELS

Proof. By Lemmas 9.1 and 9.3, there exists some  > 0 such that ˆ β,λ (n|α; −) ∈ L2+ (n) R(Ωι )m Ψ for any ι ∈ Σ∞ and for any m ∈ N. Thus, we have the conclusion by Lemma 9.4 combined with an obvious variant of [34, Theorem 4.7].  9.2. Regularized automorphic smoothed kernels For α ∈ AS and g ∈ GA , set  C(α; g) =

|α(νϕ,S )| (|ϕ(g)| + |ϕ(gw0 )|)2 ,

ϕ∈Bcus (n)

with νϕ,S the spectral parameters at S of ϕ. Lemma 9.7. The series C(α; g) is convergent. For any m > 0, we have C(α; g)  y(g)−m ,

g ∈ G1A ∩ S.

Proof. This follows from Proposition 15.2 by noting that the series   |α(νϕ,S )| { (|νϕ,ι |2 + 1)m } ϕ∈Bcus (n)

ι∈Σ∞

is convergent for any m ∈ N by Weyl’s law.



ˆ β,λ (n|α; g) on Lemma 9.8. Let g ∈ GA . Then, the holomorphic function λ → Ψ Re(λ) > 1 has a holomorphic continuation to the whole λ-plane. Its value at λ = 0 equals    −1/2 { (1 + qv )−1 } α(νϕ,S ) Z ∗ (1/2, 1; ϕ) ϕ(g) (−1)S DF v∈S(n)

ϕ∈Bcus (n)

  vol(F × \A1 )−1 −1/2 + χ(d ˜ F/Q c)−1 DF 8πi χ∈Ξ0 c|n  × α ˜ χ (ν) (DF N(c))ν/2 Bχ1−1 ,c (1/2, −ν) iR

L((1 + ν)/2, χ) L((1 − ν)/2, χ−1 ) Eχ,c (ν; g) dν × L(1 − ν, χ−2 )    N(c)1/2 + (−1)S(c) N(c)−1/2 + c|n

× (˜ α1 (1)e1,c,0 (g) +



α ˜ 1 (1) e1,c,−1 (g))

β(0).

Proof. Let Ψcus (λ) = Ψcus (λ, α, g), Ψres (λ) = Ψres (λ, α, g) and Ψct (λ) = Ψct (λ, α, g) be the first series, the second series and the integral, respectively, inside the bracket of (9.6). Let us examine Ψres (λ). As we already remarked, νϕ1 = 1. By Lemma 7.7, (9.7)

Ψres (λ) = 2α1 (1) vol(ZA GF \GA )−1 vol(F × \A1 ) β(0) λ−1 ,

which is evidently meromorphic on C. We construct meromorphic continuations of the remaining terms Ψcus (λ) and Ψct (λ) separately. Recall that P1β,λ (ϕ) (ϕ ∈ Bcus (n)) is entire on C by Lemma 7.3. Thus, to continue Ψcus (λ), it suffices to show

9.2. REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

63

that the same series defining Ψcus (λ) on Re(λ) > 1 converges absolutely and locally uniformly on the whole λ-plane. By the Cauchy-Schwarz inequality,

 ϕ∈Bcus (n)

⎧ ⎨

α(νϕ,S ) P1β,λ (ϕ) ϕ(g)  ⎩ ×

 ϕ∈Bcus (n)

⎧ ⎨ ⎩

⎫1/2

1

2 ⎬ |α(νϕ,S )| Pβ,λ (ϕ) ⎭



|α(νϕ,S )| |ϕ(g)|2

ϕ∈Bcus (n)

⎫1/2 ⎬ ⎭

,



and P1β,λ (ϕ)  is bounded by  1

+∞

{|βˆλ (t)| + |βˆλ (t−1 )|}





+∞

1

F × \A1

ϕ1

 tu 0  0 1

| d1 u d× t  1/2

F × \A1





+∞

{|βˆλ (t)| + |βˆλ (t−1 )|}2 t−σ d× t d1 u



×

F × \A1

1

|ϕ1

 tu 0  0 1

1/2 ×

| t d td u 2 σ

1

for any σ > 0, where ϕ1 (g) = |ϕ(g)| + |ϕ(gw0 )|. Thus, (9.8)



2−1 vol(F × \A1 )−1 



α(νϕ,S ) P1β,λ (ϕ) ϕ(g) 

ϕ∈Bcus (n) +∞

1

{|βˆλ (t)| + |βˆλ (t−1 )|}2 t−σ d× t



+∞

× 1

 F × \A1

1/2

  0  σ × 1 t d td u C α; tu 0 1

1/2

C(α; g)1/2 .

By Lemma 7.1, the first factor on the right-hand side converges locally uniformly in Re(λ) > −σ/2. For any σ > 0, the second and the third factors are also finite by Lemma 9.7. By letting σ > 0 vary, the consideration yields the desired convergence of Ψcus (λ) on C. As a byproduct, for any m > 0, we have the estimate (9.9)

|Ψcus (λ, α, g)|  y(g)−m ,

g ∈ G1A ∩ S

with the implied constant locally uniform in λ. η Let us examine Ψct (λ). Let fχ,c (0, ν) and Q0χ,c (η; λ, ν) be as in the proof of Lemma 7.5. By the expression (7.10) with η = 1, on the region Re(λ) > 1, Ψct (λ)

64

9. AUTOMORPHIC SMOOTHED KERNELS

is written as a sum of the three terms:  1  ˜ β((−ν + 1)/2) N (c) E1,c (ν; g) dν, (9.10) Φ1 (λ) = α ˜ 1 (ν) 8πi λ − (−ν + 1)/2 iR c|n

(9.11)

1  ˜ −1/2 N (c) DF Φ2 (λ) = 8πi c|n  ζF (−ν) β((ν + 1)/2) E1,c (ν; g) dν, × α ˜ 1 (s) A1,c (−ν) ζ (1 − ν) λ − (ν + 1)/2 F iR

Φ3 (λ) =

 vol(F × \A1 )−1   α ˜ χ (ν) Q0χ−1 ,c (1; λ, −ν) Eχ,c (ν; g) dν, 8πi iR c|n χ∈Ξ0

˜ (c) = N(c)1/2 +(−1)S(c) N(c)−1/2 . By the functional equation of Eisenstein where N series ζF (−ν) −1/2 E1,c (ν; g) = E1,c (−ν; g), DF A1,c (−ν) ζF (1 − ν) we easily have the identity Φ1 (λ) = Φ2 (λ) for Re(λ) > 1. Thus, it suffices to continue Φ1 (λ) and Φ3 (λ) separately. Applying the residue theorem, we shift the contour in Φ1 (λ) from iR to Lc with c > 1 to obtain 1  ˜ N (c) Φ1 (λ) = 8πi c|n   β((1 − ν)/2) α ˜ 1 (1) β(0) × E1,c (ν; g) dν − 2πi e1,c,−1 (g) , α ˜ 1 (ν) λ − (1 − ν)/2 λ Lc which defines a holomorphic continuation of Φ1 (λ) to Re(λ) > (−c + 1)/2. By this,  1  ˜ β((1 − ν)/2) CTλ=0 Φ1 (λ) = N (c) α ˜ 1 (ν) E1,c (ν; g) dν. 8πi (ν − 1)/2 Lc c|n

By (7.11), the integral Φ3 (λ) is written as an absolutely convergent double integral, which yields the holomorphicity of Φ3 (λ) on the whole λ-plane. We have (9.12)

 vol(F × \A1 )−1   CTλ=0 Φ3 (λ) = α ˜ χ (ν) Q0χ−1 ,c (1; 0, −ν) Eχ,c (ν; g) dν. 8πi iR c|n χ∈Ξ0

Substituting (7.13) to (9.12) and using the functional equation of the Eisenstein series, we get CTλ=0 Φ3 (λ) =  1  ˜ β((1 − ν)/2) E1,c (ν; g) dν N (c) α ˜ 1 (ν) 4πi (1 − ν)/2 iR c|n

  β(0) χ(d ˜ F/Q c)−1 vol(F × \A1 )−1 8πi c|n χ∈Ξ0  (−1+ν)/2 × DF N(c)ν/2 α ˜ χ (ν) fχ1−1 ,c (0, −ν) Eχ,c (ν; g) dν.

+

iR

9.2. REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

65

By the residue theorem, the first term added by 2CTλ=0 Φ1 (λ) equals    β((1 − ν)/2) ˜ N (c) Resν=1 α E1,c (ν; g) , (9.13) ˜ 1 (ν) ν −1 c|n

which is evaluated as 

˜ (c){α N ˜ 1 (1) e1,c,−1 (g) + α ˜ 1 (1) e1,c,0 (g)} β(0)

c|n

by virtue of (7.14) combined with α ˜ 1 (ν) β((ν − 1)/2) = α ˜ 1 (1)β(0) + α ˜ 1 (1)β(0) (ν − 1)+O(ν −1). From the considerations so far, we infer that Ψct (λ) = 2Φ1 (λ)+Φ3 (λ) has a meromorphic continuation to the whole λ-plane with a simple pole at λ = 0 such that ˜ 1 (1) β(0) 1 ˜ α N (c) e1,c,−1 (g) Ψct (λ) = − 2 λ c|n

  β(0) vol(F × \A1 )−1 + χ(d ˜ F/Q c)−1 8πi c|n χ∈Ξ0  (−1+ν)/2 × DF N(c)ν/2 α ˜ χ (ν) fχ1−1 ,c (0, −ν) Eχ,c (ν; g) dν iR  ˜ (c){α N ˜ 1 (1) e1,c,−1 (g) + α + ˜ 1 (1) e1,c,0 (g)} β(0) + O(λ). c|n

From Lemma 2.13, the singular part at λ = 0 cancels with Ψres (λ). This completes the proof.  Lemma 9.8 allows us to define the regularized automorphic smoothed kernel ˆ reg (n|α; g) by the relation Ψ ˆ β,λ (n|α; g) = Ψ ˆ reg (n|α; g) β(0), CTλ=0 Ψ

β ∈ B,

providing us with its spectral expression (9.14) ˆ reg (n|α; g) Ψ = (−1)

S

−1/2 DF

{



v∈S(n)

(1 + qv )

−1

 }



α(νϕ,S ) Z ∗ (1/2, 1; ϕ) ϕ(g)

ϕ∈Bcus (n)

  vol(F × \A1 )−1 −1/2 χ(d ˜ F/Q c)−1 DF + 8πi χ∈Ξ0 c|n  × α ˜ χ (ν) (DF N(c))ν/2 Bχ1−1 ,c (1/2, −ν) iR

L((1 + ν)/2, χ) L((1 − ν)/2, χ−1 ) Eχ,c (ν; g) dν × L(1 − ν, χ−2 )    N(c)1/2 + (−1)S(c) N(c)−1/2 (˜ α1 (1)e1,c,0 (g) + α + ˜ 1 (1) e1,c,−1 (g)) . c|n

66

9. AUTOMORPHIC SMOOTHED KERNELS

Note that from the proof of Lemma 9.8 the series and the integral in (9.14) are convergent absolutely. To analyze this further, we need to know its behavior as a function in g ∈ GA . Lemma 9.9. Let m be an arbitrary positive integer and N the exponent occuring in Proposition 15.1. (1) On the Siegel domain G1A ∩ S, we have the following bounds.



(9.15)

α(νϕ,S ) Z ∗ (1/2, 1; ϕ) ϕ(g)  y(g)−m , ϕ∈Bcus (n)

(9.16)

 

χ∈Ξ0



|α ˜ χ (ν)| Bχ1−1 ,c (1/2, −ν)

iR



L((1 + ν)/2, χ) L((1 − it)/2, χ−1 )



L(1 − ν, χ−2 )

× |Eχ,c (ν; g)| dν  y(g)N , (9.17)

|e1,c,0 (g)| + |e1,c,−1 (g)|  y(g).

ˆ reg (n|α; g) on ZA GF \GA is (2) Set N1 = sup(N, 1). Then the function g → Ψ continuous and satisfies the estimate



ˆ

Ψreg (n|α; g)  y(g)N1 , g ∈ S ∩ G1A . Proof. The bound (9.15) is obtained from (9.9). The estimate (9.17) is proved by Lemma 2.13 and by the expression of e1,c,0 given in the proof of Lemma 7.8. Let us show (9.16). Invoking the convexity bound (2.6) and the estimates (2.5), together with Stirling’s formula, we obtain the bound (9.18)

|L((1 − it)/2, χ−1 ) L((1 + it)/2, χ) L(1 − it, χ−2 )  (1 + |t + a(χι )|) , t ∈ R, χ ∈ Ξ0

−1

|

ι∈Σ∞

for any  > 0. Combining this with Proposition 15.1, we infer that the integral in (9.16) with varying g ∈ S is majorized by y(g)N . The assertion (2) is obtained from (1) by (9.14). 

CHAPTER 10

Periods of regularized automorphic smoothed kernels: the spectral side Let n and S be as in the previous section and fix α ∈ AS . To simplify formulas, we set  −1/2 (10.1) { (1 + qv )−1 }. C(n, S) = (−1)S DF v∈S(n)

Let η be a unitary idele-class character of F × satisfying (2.18), (2.19) and (2.20). Let N be an eligible exponent of y(g) in Proposition 15.1 and set N1 = sup(N, 1) from now on. By Lemmas 9.9 and 7.1, the integral    1 xη  η t 0 ˆ ˆ η(tx∗η ) d× t, {βˆλ (|t|A )+ βˆλ (|t|−1 Pβ,λ (Ψreg (n|α)) = A )} Ψreg n|α; [ 0 1 ] 0 1 F × \A×

converges absolutely on the region Re(λ) > N1 , defining a holomorphic function. We continue this meromorphically to a neighborhood of λ = 0. In this section and the next, we carry out the necessary analysis and calculate the value at λ = 0 in two different ways. Lemma 10.1. For any Re(λ) > N1 , ˆ reg (n|α)) = C(n, S) {Pη (β, λ; α) + Pη (β, λ; α) + Pη (β, λ; α)} (10.2) Pηβ,λ (Ψ cus res eis with Pηcus (β, λ; α) =



α(νϕ,S ) Z ∗ (1/2, 1; ϕ) Pηβ,λ (ϕ),

ϕ∈Bcus (n)

Pηeis (β, λ; α)

   vol(F × \A1 )−1 −1/2 −1 χ(d ˜ F/Q c) DF = α ˜ χ (ν) (DF N(c))ν/2 8πi iR χ∈Ξ0 c|n

L((1 + ν)/2, χ) L((1 − ν)/2, χ−1 ) L(1 − ν, χ−2 ) η × Pβ,λ (Eχ,c (ν; −)) dν,   N(c)1/2 + (−1)S(c) N(c)−1/2 Pηres (β, λ; α) = × Bχ1−1 ,c (1/2, −ν)

c|n

  × α ˜ 1 (1) Pηβ,λ (e1,c,0 ) + α ˜ 1 (1) Pηβ,λ (e1,c,−1 ) ,

where the infinite series Pηcus (β, λ; α) and the integral Pηeis (β, λ; α) converge absolutely and locally uniformly on the whole λ-plane and on the half-plane Re(λ) > N , respectively. Proof. This follows from (9.14); we apply Fubini’s theorem to justify the termwise integral, invoking the estimations (9.15), (9.16) together with Lemma 7.1.  67

68

10. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

Lemma 10.2. The holomorphic function λ → Pηres (β, λ; α) (Re(λ) > N1 ) has a meromorphic continuation to the whole λ-plane with the singular part  α1 (1) β(0) (10.3) δη,1 vol(F × \A1 ) (1 + δ(n = oF )) (1 + N(n)−1 ) { (1 + qv )} λ−1 v∈S(n)

+ δη,1

−1/2 2 4DF RF

ζF (2)

     −ζF (2) CF β(0) + + A˜1 (n) α ˜ 1 (1) α ˜ 1 (1) + ζF (2) RF λ

−1/2

− δη,1

2 2DF RF β(0) α ˜ 1 (1) 2 , ζF (2) λ

where

1  1 − qv A˜1 (n) = log qv . 4 1 + qv

(10.4)

v∈S(n)

We have CTλ=0 Pηres (β, λ; α)

with (10.5) ˜ η (n) = B 0

 2 2 N(fη )1/2 G(η) RF δη,1  β (0) = − W (η) ζF (2) 2    C0 (η)2 ˜ η CF ˜1 (n) β(0) α B B + (n) − 2δ + 2δ ˜ 1 (1) η,1 η,1 0 2 RF RF

 1 − ηv (v ) q −1 v , −1 1 − q v v∈S(n)

(10.6)

 qv + 1 1 ˜1 (n) = 1 B (log qv )2 + 8 qv − 1 8 v∈S(n)

 {v1 ,v2 }⊂S(n) v1 =v2

qv1 qv2 + 1 log qv1 log qv2 . (qv1 − 1)(qv2 − 1)

Proof. This follows from Lemmas 7.7 and 7.8. To explicate the singular part, we use Lemma 7.9.  Lemma 10.3. Suppose η is unramified. As ν → −1, we have ν/2 L((1

DF

η η D−2 D−1 + ν)/2, η) L((1 − ν)/2, η) = + D0η + O(ν + 1), + ζF (1 − ν) (ν + 1)2 ν+1

with 4R(η)2 , ζF (2)   ζF (2) R(η)2 C0 (η) R(η) η +4 D−1 = η˜(dF/Q ) −4 , ζF (2) ζF (2)2  ζ  (2) C0 (η)2 + 2C1 (η)R(η) D0η = η˜(dF/Q ) −4C0 (η) R(η) F 2 + ζF (2) ζF (2)     2 ζ (2) ζ (2) . + 4R(η)2 − F 2 + 2 F 3 ζF (2) ζF (2) η D−2 = η˜(dF/Q )

Proof. A direct computation.



10. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

69

Lemma 10.4. The function λ → Pηeis (β, λ; α) (Re(λ) > N ) has a meromorphic continuation to C with the singular part     −1/2 2  4DF RF −ζF (2) CF β(0) + + A˜1 (n) α ˜ (1) − δη,1 α ˜  (1) + ζF (2) ζF (2) RF λ −1/2

+ δη,1

2 2DF RF β(0) α(1) ˜ . ζF (2) λ2

We have CTλ=0 Pηeis (β, λ; α) + CTλ=0 Pηres (β, λ; α) =   β(0)  −1/2 (10.7) G(η) DF vol(F × \A1 )−1 α ˜ χ (ν) L1χ,c (ν) Lηχ−1 ,c (−ν) dν 8πi iR +

{X0η (n) α ˜ 1 (1)

+

χ∈Ξ0 c|n η Y0 (n) α ˜ η (1)

+ Y1η (n) α ˜ η (1) + Y2η α ˜ η (1)} β(0),

where (10.8) Lηχ,c (ν) = (DF N(c))ν/2 Bχη −1 ,c (1/2, −ν)

L((1 + ν)/2, χη) L((1 − ν)/2, χ−1 η) L(1 − ν, χ−2 )

and −1/2

X0η (n) =

2DF ˜ η (n), C0 (η)2 δ(fη = oF ) B 0 ζF (2)

Y0η (n) =

2N(fη )1/2 G(η) ˜ η (n) C0 (η)2 B 0 W (η) ζF (2) −1/2

2 D RF + 4δη,1 F ζF (2)     CF ζF (2) ζF (2)2 ζF (2) ˜ 2CF ζF (2) ˜ −2 +4 A1 (n) + B2 (n) , +2 − × − RF ζF (2) ζF (2) ζF (2)2 RF ζF (2)  −1/2 2   D RF ζF (2) CF Y1η (n) = −8δη,1 F − A˜1 (n) , − ζF (2) ζF (2) RF

Y2η = 4δη,1

−1/2

2 DF RF . ζF (2)

˜2 (n) is defined by Here A1 (n) is given by (10.4) and B   1 − qv 2 1  1 + qv ˜2 (n) = 1 B (log qv )2 − (log qv )2 8 1 + qv 8 1 − qv v∈S(n)

+

1 4

v∈S(n)

 {v1 ,v2 }⊂S(n) v1 =v2

2

(qv1 qv2 + 1) . (qv21 − 1)(qv22 − 1)

η Proof. Let fχ,c (0, ν) and Q0χ,c (η; λ, ν) be as in the proof of Lemma 7.5.

˜η (c) = δ(fη = oF ) η˜(dF/Q c) {N(c)1/2 + (−1)S(c) N(c)−1/2 }. N From (7.10), on the region Re(λ) > N1 ,  −1/2 − Pηeis (β, λ; α) = vol(F × \A1 )−1 DF {Φ+ 1 (c; λ) + Φ1 (c; λ) + Φ2 (c; λ)} c|n

70

10. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

with

˜η (c)  vol(F × \A1 ) N β((ν + 1)/2) dν, = α ˜ η (ν) L1η,c (ν) 8πi λ − (ν + 1)/2 iR ˜η (c)  vol(F × \A1 ) N ζF (ν) −1/2 Φ− (c; λ) = α ˜ η (ν) L1η,c (ν) DF Aη,c (ν) 1 8πi ζF (ν + 1) iR β((ν − 1)/2) dν, × λ + (ν − 1)/2   χ(d ˜ F/Q c)−1 Φ2 (c; λ) = α ˜ χ (ν) L1χ,c (ν) Q0χ (η, λ, ν) dν. 8πi iR Φ+ 1 (c; λ)

χ∈Ξ0

From (2.27),

1 (1/2, ν) Bη,c

1 = N(c)ν Aη,c (ν) Bη,c (1/2, −ν) follows immediately. Comν−1/2

bining this with the functional equation ζF (1 − ν) = DF −1/2

L1η,c (ν) DF

(10.9)

Aη,c (ν)

ζF (ν), we obtain

ζF (ν) = L1η,c (−ν). ζF (ν + 1)

+ Thus, Φ− 1 (c; λ) = Φ1 (c; λ) with both sides being holomorphic on Re(λ) > 1/2. To + continue Φ1 (c; λ) beyond the region Re(λ) > 1/2, we shift the contour iR to L−σ with σ > 1/2 applying the residue theorem. Since Re(λ) > 1/2, the possible poles of the integrand swept by the moving contour is located only at ν = −1 and it comes from L((1 ± ν)/2, η). If we set 1 N(c)ν/2 Bη,c (1/2, −ν) = pη0 (c) + pη1 (c) (ν + 1) + pη2 (c) (ν + 1)2 + O((ν + 1)3 ),

β((ν + 1)/2) α ˜ η (ν) = q0 (λ) + q1 (λ)(ν + 1) + O((ν + 1)2 ), λ − (ν + 1)/2

ν → −1,

then a straightforward computation yields (10.10)

p10 (c) = δ(c = oF ),

(10.11)

q0 (λ) =

α ˜ η (1) β(0) , λ

log N(c) N(c)1/2 , p11 (c) = δ(S(c) = 1) 2 N(c) + 1    α ˜ η (1) α ˜ η (1) q1 (λ) = + β(0). λ 2λ2

and the relevand residue is calculated as   β((ν + 1)/2) 1 η Resν=−1 α Lη,c (ν) = p10 (c) q1 (λ) D−2 ˜ η (ν) λ − (ν + 1)/2 η η + p10 (c) q0 (λ) D−1 + p11 (c) q0 (λ) D−2 .

Thus, (10.12) Φ+ 1 (c; λ) =

˜η (c)  vol(F × \A1 ) N β((ν + 1)/2) dν α(ν) ˜ L1η,c (ν) 8πi λ − (ν + 1)/2 L−σ   1 η η η  1 1 + 2πi p0 (c) q1 (λ) D−2 + p0 (c) q0 (λ) D−1 + p1 (c) q0 (λ) D−2 , (Re(λ) > 1/2),

whose right-hand side is meromorphic on Re(λ) > (−σ + 1)/2. By substituting (7.7), the term Φ2 (c; λ) is expressed as an absolutely convergent double integral on the region Re(λ) > −σ. This gives us a holomorphic continuation of Φ2 (c; λ) to C.

10. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

71

As a consequence, we obtain a meromorphic continuation of Pηeis (β, λ; α) with the Laurent expansion at λ = 0 of the form 

−1/2

vol(F × \A1 )−1 DF

c|n



˜η (c)  vol(F × \A1 ) N β((ν + 1)/2) dν × α ˜ η (ν) L1η,c (ν) 4πi −(ν + 1)/2 L−σ    χ(d ˜ F/Q c)−1 + α ˜ χ (ν) L1χ,c (ν) Q0χ (η; 0, ν) dν 8πi iR χ∈Ξ0

+

1  −1/2 ˜ η η η Nη (c){p10 (c) q1 (λ) D−2 DF + p10 (c) q0 (λ) D−1 + p11 (c) q0 (λ) D−2 } 2 c|n

+ O(λ). The sum of the two contour integrals occurring above is the constant term of Pηeis (β, λ; α). To compute this further, we shift the contour L−σ in the first integral back to the imaginary axis iR accounting for the residue at ν = −1. In the second integral, we substitute the expression (7.13) for Q0χ (η, 0, ν). Thus, CTλ=0 Pηeis (β, λ; α) = 

−1/2

DF

c|n

 ˜  Nη (c) 1 ˜ β((ν + 1)/2) dν − N α ˜ η (ν) L1η,c (ν) η (c) Resν=−1 4πi iR −(ν + 1)/2 2   β((ν + 1)/2) × α ˜ η (ν) L1η,c (ν) −(ν + 1)/2  1 ˜η (c) + α ˜ η (ν) L1η,c (ν) N 8πi iR   β((1 + ν)/2) ζF (ν) β((1 − ν)/2) −1/2 + DF × Aη,c (ν) dν (1 + ν)/2 ζF (1 + ν) (ν − 1)/2    β(0) × 1 −1 1 −ν/2 η α ˜ χ (ν) Lη,c (ν) (DF N(c)) G(η) fχ,c (0, ν) dν . + vol(F \A ) 8πi iR ×

χ∈Ξ0

By (10.9), the first integral cancels with the second integral. The residue is evaluated as η η η Resν=−1 = −pη0 (c) D−2 α ˜ η (1) β(0) − 2 (pη0 (c) D−1 + pη1 (c) D−2 )α ˜ η (1) β(0)     −1 η  η η + pη0 (c) D−2 β (0) − 2D0η β(0) − 2D−2 pη2 (c) β(0) − 2 pη1 (c) D−1 β(0) 4 ×α ˜ η (1).

72

10. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

Thus, CTλ=0 (Pηeis (β, λ; α) + Pηres (β, λ; α)) is expressed as (10.7) but with 2  −1/2 ˜η (c) η˜(dF/Q ) C0 (η) pη (c), N X0η (n) = DF ζF (2) 0 c|n  −1/2 ˜η (c) N Y0η (n) = DF c|n

    C0 (η)2 η η × pη0 (c) D0η − η˜(dF/Q ) + pη2 (c) D−2 + pη1 (c) D−1 ζF (2)   −1 −1/2 2N(cfη ) G(η)  2 ˜ ˜ + −2δ1,η RF CF + 2δ1,η B1 (n) + C0 (η) B0 (n) , W (η) ζF (2)  −1/2 ˜η (n) (pη (c) Dη + pη (c) Dη ), N Y1η (n) = DF 0 −1 1 −2 c|n

Y2η =

−1/2 DF

2



˜η (n) pη (c) Dη . N 0 −2

c|n

We should remark that β  (0)-term in CTλ=0 Pηres (β, λ; α) cancel with the β  (0)term in Resν=−1 given above. By computing these quantities further with the aid of (10.10) and Lemma 10.3, we have the desired formulas of Y0η (n), Y1η (n) and Y2η .  Now, we have the main result of this section. Theorem 10.5. Let S be a finite set of places containing Σ∞ and n a square free oF -ideal such that S ∩ S(n) = ∅. Let η be a unitary idele-class character ˜ χ (ν) = α(ν χ ) satisfying (2.18), (2.19) and (2.20). Let α ∈ AS (see 6.1.2) and set α for ν ∈ C, where ν χ ∈ XS is the element defined in 9.1.3. Then, the function ˆ reg (n|α)) (Re(λ) > N1 ) has a meromorphic continuation to the whole λ → Pηβ,λ (Ψ λ-plane with the singular part −1/2

δη,1 (−1)S DF

vol(F × \A1 ) (1 + δ(n = oF )) (1 + N(n)−1 )

α(1) ˜ β(0) . λ−1

We have ˆ reg (n|α)) = C(n, S) {Iηcus (n|α) + Iη (n|α) + Dη (n|α)}, Pηreg (Ψ eis with (10.13)

Iηcus (n|α) =

(10.14)

Iηeis (n|α)



α(νϕ,S ) Z ∗ (1/2, 1; ϕ) Z ∗ (1/2, η; ϕ),

ϕ∈Bcus (n) −1/2

= G(η) DF  × iR

(10.15)

vol(F × \A1 )−1

  1 8πi

χ∈Ξ0 c|n

α ˜ χ (ν) L1χ,c (ν) Lηχ−1 ,c (−ν) dν,

Dη (n|α) = X0η (n) α ˜ 1 (1) + Y0η (n) α ˜ η (1) + Y1η (n) α ˜ η (1) + Y2η α ˜ η (1).

Here, C(n, S) and Lηχ,c (ν) are defined by (10.1) and by (10.8), respectively; νϕ,S ∈ XS denotes the spectral parameters at S of ϕ ∈ Bcus (n) (see 9.1.3); X0η (n), Y0η (n), Y1η (n) and Y2η are constants defined in Lemma 10.4.

10. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

73

Proof. This follows from Lemmas 10.1, 10.2 and 10.4. Note that, when summed up, the possible singularity at λ = 0 of Pηeis (β; λ, α) and that of Pηres (β; λ, α) cancel with each other, leaving a possible simple pole at λ = 1 which occurs only −1/2 when η = 1. We also note G(η) = η˜(dF/Q ) DF when fη = oF . 

CHAPTER 11

A geometric expression of automorphic smoothed kernels Let S, n be as in §8. Let η be a unitary idele-class character of F × satisfying (2.18), (2.19), (2.20) and S(fη ) ∩ S = ∅.

(11.1) Fix β ∈ B and α ∈ AS

11.1. Classification of double cosets For δ ∈ GF , set St(δ) = HF ∩ δ −1 HF δ. Then, St(δ)\HF ∼ = HF \HF δHF . Lemma 11.1. The following elements of GF form a complete set of representatives of the double coset space HF \GF /HF .   e = [ 10 01 ] , w0 = 01 −1 , 0   0 −1   1 1 u = [0 1], u ¯ = [ 11 01 ] , u w0 = 11 −1 , ¯ w0 = 1 −1 , u 0   −1 1+b 1 δb = , (b ∈ F − {0, −1}). 1 1 We have St(e) = St(w0 ) = HF and St(δ) = ZF for δ ∈ {u, u ¯ , u w0 , u ¯ w0 , δb }. 

Proof. This is straightforward. (cf. [41, Lemma 1]). Let t ∈ A× . By HF \GF =

!

HF \(HF δHF ) ∼ =

δ∈HF \GF /HF

!

(St(δ)\HF ),

δ∈HF \GF /HF

   ˆ β,λ n|α; [ t 0 ] 1 xη (Re(λ) > 0) as a we write the absolutely convergent series Ψ 01 0 1 sum of sub-series     ˆ β,λ n|α; δ γ [ t 0 ] 1 xη Jδ (β, λ, α; t) = Ψ 01

0 1

γ∈St(δ)\HF

with δ varying in the set HF \GF /HF , whose complete set of representatives is given by Lemma 11.1. We examine these terms separately as functions in the variable λ. 11.2. Terms associated to e and w0 Lemma 11.2. The functions λ → Je (β, λ, α; t) and λ → Jw0 (β, λ, α; t) are entire on C whose values at λ = 0 are β(0) Jid (α; t) and β(0) Jid (α; t) δ(n = oF ), respectively, where  S  1 (11.2) Υ1S (s) α(s) dμS (s) Jid (α; t) = δ(fη = oF ) 2πi LS (c) 75

76

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

with Υ1S (s) = {

 −1 Γ((sι + 1)/4)2  }{ (1 − qv−(sv +1)/2 )−1 (1 − qv(sv +1)/2 )−1 }. 2 8 Γ((sι + 3)/4)

ι∈Σ∞

v∈Sfin

Proof. Since St(e) = St(w0 ) = HF and w0 ∈ Kfin − K0 (pv ) for any v ∈ S(n), we have    ˆ β,λ n|α; [ t 0 ] 1 xη , Je (β, λ, α; t) = Ψ 01 0 1   1 0  ˆ β,λ n|α; [ 1 0 ] −x , Jw0 (β, λ, α; t) = δ(n = oF ) Ψ η 1 0 t which are entire on C by Lemma 6.9. We have    ˆ (0) n|α; [ t 0 ] 1 xη CTλ=0 Je (β, λ, α; t) = β(0)Ψ 01 0 1  S     1 Ψ(0) n|s; [ 0t 01 ] 10 x1η α(s) dμS (s). = β(0) 2πi LS (c) By Lemma 6.1 (1), we proceed with  S     1 = β(0) (11.3) Ψ(0) n|s; 10 x1η α(s) dμS (s). 2πi LS (c) The value of Ψ(0) occurring above is computed from (6.3) by (4.4) and Lemma 5.2. Since f (ηv ) = 0 for any v ∈ Sfin ∪ S(n) from (2.20) and (11.1), we have the formula (11.2). In a similar way,  S 1 CTλ=0 Jw0 (β, λ, α; t) = β(0) δ(n = oF ) 2πi    1 0  w0 α(s) dμS (s), × Ψ(0) n|s; −x η 1 LS (c)

which turns out to coincides with (11.3) by Lemma 5.5.



¯ w0 11.3. Terms associated to u, u ¯, uw0 and u Set Ju (β, λ, α; t) = Ju (β, λ, α; t) + Ju¯w0 (β, λ, α; t), Ju¯ (β, λ, α; t) = Juw0 (β, λ, α; t) + Ju¯ (β, λ, α; t). Lemma 11.3. For ∗ = u or u ¯ , the function λ → J∗ (β, λ, α; t) is continued holomorphically to C, and its value at λ = 0 is given by β(0) J∗ (α; t) with   S       1 xη  1 −1 (11.4) Ju (α; t) = Ψ(0) n|s; 10 at1 0 1 2πi a∈F × LS (c)    1 0 1 0  (0) n|s; at−1 1 −xη 1 w0 α(s) dμS (s), +Ψ  (11.5)

Ju¯ (α; t) =

1 2πi

S   a∈F ×

LS (c)



   1 0 ] 1 xη Ψ(0) n|s; [ at 1 0 1

   1 0  + Ψ(0) n|s; [ 10 at w α(s) dμS (s). ] 0 −xη 1 1 These series-integrals are absolutely convergent.

11.3. TERMS ASSOCIATED TO u, u ¯, uw0 AND u ¯w0

77

  −1   Proof. Let a ∈ F × and t ∈ A× . Then, by the relations u [ 0t 10 ] a 0 01 10 x1η =  −1     1 xη    0  1 0 1 0  −1   a t0 1 at−1 and u ¯w0 [ 0t 01 ] a 0 01 10 x1η = 10 a−1 −xη 1 w0 , 0 1 0 1 t at−1 1 0 1 from Lemma 6.1 (1), we have (11.6)

(11.7)

    ˆ β,λ n|α; u [ t 0 ] a−1 0 1 xη Ψ 01 0 1 0 1    S       1 xη  1 1 β(z) −1 = |t|zA Ψ(z) n|s; 10 at1 0 1 2πi LS (c) 2πi Lσ z + λ "     1 xη  −1 −z (−z) 1 at + |t|A Ψ n|s; 0 1 dz α(s) dμS (s), 0 1    −1   ˆ β,λ n|α; u ¯w0 [ 0t 01 ] a 0 01 10 x1η Ψ    S      1 0  1 β(z) 1 (z) |t|−z n|s; at1−1 10 −x w0 = η 1 A Ψ 2πi 2πi z + λ LS (c) Lσ "   1 0 1 0  z (−z) + |t|A Ψ n|s; at−1 1 −xη 1 w0 dz α(s) dμS (s).

From Lemma 5.2 and the formula (4.9), there exists a constant C0 > 0 and an oF -ideal a, depending on t, σ and c, such that

   1 xη 

(±z)  −1 n|s; 10 at1

 f (a),

Ψ 0 1

a ∈ F × , (s, z) ∈ LS (c) × Lσ ,

where f (a) = C0 {



−q sup(1, |at−1 }{ v |v )

v∈Sfin

×{





2 −q/2 (1 + |at−1 } ι |ι )

ι∈Σ∞

δ(a ∈ aov )},

a ∈ A×

v∈Σfin −S

with q = 2q(c) − σ. Thus,

 

 −1  

ˆ

Ψβ,λ n|α; u [ 0t 01 ] a 0 01 10 x1η  (|t|σA + |t|−σ A ) f (a)

 

β(z)

×

z + λ α(s) |dz| |dμS (s)| LS (c) Lσ holds for a ∈ F × . By the rapid decay of β and α, the double integral in the right-hand side is convergent.  The first assertion of our lemma follows from the convergence of the series a∈F × f (a), which in turn is proved by comparing it  with the integral A f (a) da. Since c can be chosen so that q(c) is arbitrarily large, the last integral is convergent. The identity CTλ=0 Ju (β, λ, α; t) = β(0) Ju (α; t) is then inferred by Lemma 6.5. The integral Ju¯ (β, λ, α; t) is analyzed in a similar way. 

78

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

11.4. Terms associated to δb ’s In this subsection, we analyze the integral (11.8)



Jhyp (β, λ, α; t) =

b∈F −{0,−1}



=

Jδb (β, λ, α; t) 

     ˆ β,λ n|α; b−1 +1 1 [ at 0 ] 1 xη , Ψ 0 1 0 1 1 1

t ∈ A×

b∈F −{0,−1} a∈F ×

in detail. Lemma 11.4. Let v ∈ Σfin and set f = f (ηv ). Given b, t ∈ Fv× , b = −1, write     −1    −f b +1 1 [ t 0 ] 1 v = u01 u02 10 y1 k (11.9) 0 1 1 1 0 1 with u1 , u2 ∈ Fv× , y ∈ Fv and k ∈ Kv . • Suppose f > 0. Then,  −f −f (−t ∈ vf Uv (f )), |t|−1 v |1 + tv |v |b + tv (b + 1)|v , |y|v = (−t ∈ vf Uv (f )), |t|v |b + 1|v  −f −2 (−t ∈ vf Uv (f )), |t|v |b|−1 v |1 + tv |v |u1 /u2 |v = −1 (−t ∈ vf Uv (f )). |t|−1 v |b|v • Suppose f = 0. Then,  |t|−1 (|t|v  1), v |b|v |y|v = |t|v |b + 1|v (|t|v > 1),

 |u1 /u2 |v =

|t|v |b|−1 v −1 |t|−1 |b| v v

(|t|v  1), (|t|v > 1).

If the relation (11.9) holds with k ∈ K0 (pv ), then t ∈ pv .      u−1   0 a12 1 b−1 +1 1 [ t 0 ] 1 v−f and examine Proof. Set [ aa11 ] = 10 −y −1 21 a22 0 1 1 1 1 0 1 0 u2 the condition for this matrix to belong to Kv , i.e., (11.10)

a11 ov + a12 ov = ov ,

(11.11)

a21 ov + a22 ov = ov ,

(11.12)

−1 −1 a11 a22 − a12 a21 = u−1 t ∈ o× v . 1 u2 b

By a direct computation, −1 a11 = u−1 ) − yu−1 1 t(1 + b 2 t,

−1 −1 −1 −f a12 = {u−1 ) − yu−1 1 t(1 + b 2 t}v + u1 − yu2 ,

a21 = u−1 2 t,

−1 −f a22 = u−1 2 tv + u2 .

From (11.11), we have (11.13)

|u2 |v = sup(|t|v , |1 + tv−f |v ).

Let us consider the case f = f (ηv ) > 0, separating cases according to the valuation of t. (i) Let t ∈ pfv +1 . Then, |u2 |v = 1 from (11.13), and thus |u1 /u2 |v = |b|−1 v |t|v from (11.12). Set (11.14)

−f −1 )}(1 + tv−f )−1 y = u2 u−1 1 {1 + tv (1 + b

11.4. TERMS ASSOCIATED TO δb ’S

79

so that a12 = 0. Then, since 1 + tv−f ∈ Uv (1) ⊂ o× v , we have −f −1 )|v . |y|v = |b|v |t|−1 v |1 + tv (1 + b −1 (1 + tv−f )−1 ; thus a11 ∈ o× A computation shows the equality a11 = u−1 v and 1 tb the condition (11.10) is satisfied. −1 (ii) Let t ∈ pfv . Then, |u2 |v = |tv−f |v from (11.13), and thus |u1 |v = |u2 |−1 t|v v |b −1 −1 2f −1 −f −2 from (11.12). Hence, |u1 /u2 |v = |b|v |t|v |v |v = |b|v |t|v |1 + tv | as desired. If we define y by (11.14), then |y|v = |v−f |v |b + tv−f (b + 1)|v , and −1 (1 + t−1 vf )−1 ; thus a11 ∈ o× a11 = vf u−1 v and the condition (11.10) is sat1 b isfied. (iii) Let t ∈ pfv − pfv +1 and write t = −vf u with u ∈ o× v . Assume u ∈ Uv (f ). from (11.12). Thus, Then, |u2 |v = |vf |v = |t|v from (11.13) and |u1 |v = |b|−1 v −1 −1 as desired. Let y = u2 u−1 ) so that a11 = 0. Then, |u1 /u2 |v = |b|−1 v |t|v 1 (1 + b −1 ∈ o× |y|v = |t|v |1 + b|v and a12 = −u−1 v is easily confirmed. Assume u ∈ Uv (f ). 1 b −f −f −1 Then, |u2 |v = |1 + tv |v and |u1 |v = |b|−1 from (11.13) and v |t|v |1 + tv |v −1 −f −2 (11.12). Thus, |u1 /u2 |v = |b|v |t|v |1 + tv |v as desired. Let us define y by −1 −f −f (11.14). Then, a12 = 0, a11 ∈ o× v and |y|v = |t|v |1 + tv |v |b + tv (b + 1)|v is confirmed by a computation. This complete the consideration for f (ηv ) > 0. The unramified case f (ηv ) = 0 is treated similarly. The condition k ∈ K0 (pv ) is equivalent to a21 ∈ pv ; by examining this, we have the last statement. 

Lemma 11.5. Let v ∈ Σfin . Let t ∈ Fv× and b ∈ Fv× − {−1}. Set f = f (ηv ). Then     −f (z)  −1 =0 Φ0,v b 1+1 11 [ 0t 01 ] 10 1v unless b ∈ p−f and v |t|v = qv−f

or

qv−f |b|v  |t|v  qv−f |b + 1|−1 v .

Proof. Let u1 , u2 and y be as in (11.9). From (5.1), we have that      −f (z)  −1 (z)  = |u1 /u2 |zv Φ0,v 10 y1 (11.15) Φ0,v b 1+1 11 [ 0t 01 ] 10 1v is not zero if and only if y ∈ ov , which we assume in the remaining part of this proof. Suppose f = 0; then |b|v  |t|v  1 or 1 < |t|v  |b + 1|−1 v by Lemma 11.4. Thus, b ∈ ov follows. If |t|v = 1, then |b|v < 1, |b + 1|v = 1 or 1 < |b + 1|−1 v , |b|v = 1. This shows the claim in the case when f = 0. Suppose f > 0 from now on. −f • Let t ∈ pfv +1 . Then, |t|−1 v |b + tv (b + 1)|v  1 from Lemma 11.4. Hence, (11.16)

|b + tv−f (b + 1)|v  |t|v  |vf +1 |v . We separate cases according to which one is larger, |b|v or |tv−f (b + 1)|v : Consider the case |b|v < |tv−f (b + 1)|v . Then the first inequality of (11.16) yields |b + 1|v  |vf |v < 1; thus b ∈ o× v . This, combined with |b|v < |tv−f (b + 1)|v and the second inequality of (11.16), leads to the inequality |v−1 |v < |b+1|v , which is contradictory to b ∈ ov . Consider the case |b|v = |tv−f (b + 1)|v . Then the second inequality in (11.16) gives us f |b|v < |v |v |b+1|v , which forces b+1 ∈ o× v . Thus, |t|v = |v b|v . Consider −f the case |b|v > |tv (b+1)|v . Then (11.16) becomes |b|v  |t|v < |vf +1 |v , −f which implies 1 + b ∈ o× v . Thus, |t|v  |b|v > |tv |v , leading to the

80

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

impossible inequality 1 > |v−f |v . Summing up the argument, we obtained the implication: (11.17)

t ∈ pfv +1

=⇒

b ∈ ov , |t|v = |vf b|v .

• Let t ∈ pfv . Then, from Lemma 11.4, the condition y ∈ ov becomes |v−f |v |b + tv−f (b + 1)|v  1, or equivalently (11.18)

|tv−f |v |1 + b(1 + t−1 vf )|v  |vf |v . Since |tv−f |v > 1, we have |1 + b(1 + t−1 vf )|v  |vf |v . Note that −1 f 1 + t−1 vf ∈ o× v )|v  |vf |v , v . If b ∈ ov , then 1 < |b|v = |1 + b(1 + t a contradiction. Thus, we have b ∈ ov . There are four cases to be considered: (i) |b|v = 1, |b + 1|v < 1, |b|v < |tv−f (1 + b)|v , (ii) |b|v = 1, |b + 1|v < 1, |b|v = |tv−f (1 + b)|v , (iii) |b|v = 1, |b + 1|v < 1, |b|v > |tv−f (1 + b)|v , (iv) |b|v < 1, |b + 1|v = 1. The condition (i) combined with (11.18) gives us two inequalities f −1 |t|v  |v2f |v |b + 1|−1 v and |v |v |b + 1|v < |t|v , leading to the impossible inequality |v2f |v > |vf |v . The condition (ii) gives us |t|v = |vf |v |b+1|−1 v . The condition (iii) combined with (11.18) yields the impossible inequality |v−f |v  1. The condition (iv) combined with (11.18) yields |t|v  |v2f |v ; this contradicts with t ∈ pfv . Summing up the consideration, we obtain the following implication:

(11.19)

t ∈ pfv

=⇒

b ∈ ov , |t|v = |vf |v |b + 1|−1 v .

• Let t ∈ pfv − pfv +1 , i.e., |t|v = qv−f . Consider the case −t ∈ vf Uv (f ). Then, by Lemma 11.4, the condition y ∈ ov becomes |t|v  |b + 1|−1 v , or −f ⊂ p . Consider the equivalently |b + 1|v  qvf . Hence, b ∈ −1 + p−f v v case −t ∈ vf Uv (f ); thus |1 + tv−f |v > |vf |v . From Lemma 11.4, the condition y ∈ ov becomes (11.20)

−f −f |1 + tv−f |v |t|−1 v |tv + b(1 + tv )|v  1. f −1 If |b(1 + tv−f )|v  |tv−f |v , then |b|v  |1 + tv−f |−1 v < |v |v . Thus, −f −f −f b ∈ pv . If |b(1 + tv )|v > |tv |v . Then, from (11.20), −f |1 + tv−f |v | |t|−1 v |b(1 + tv )|v  1.

Combined with |1 + tv−f |v > |vf |v , this gives us |v2f |v |t|−1 v |b|v  1, or . Summing up equivalently |b|v  |t|v |v−2f |v = |v−f |v . Thus, b ∈ p−f v the consideration, (11.21)

t ∈ pfv − pfv +1

=⇒

−f b ∈ p−f v , |t|v = qv .

By (11.21), (11.19) and (11.17), we are done.



11.4.1. In this paragraph, let v ∈ Σfin − S. For any t ∈ Fv× , b ∈ Fv× − {−1} and for any σ ∈ R, set σ −f (ηv ) , qv−2f (ηv ) |b|v  |t|v  |b + 1|−1 fv(σ) (t, b) = inf(1, |t|−2 v ) δ(b ∈ pv v ).

11.4. TERMS ASSOCIATED TO δb ’S

81

Corollary 11.6. Let v ∈ Σfin − (Sfin ∪ S(n)). For given σ ∈ R, there exists a constant Cv (σ) > 0 such that for any t ∈ Fv× , b ∈ Fv× − {−1} and for any z ∈ C with Re(z) = σ, the inequality

  

(z)  b−1 +1 1  t 0 1 v−f (ηv ) σ (σ) [ ]

Φ0,v

 Cv (σ) |b|−σ v |t|v fv (t, b) 0 1 1 1 0 1 holds. If f (ηv ) = 0, then we can take Cv (σ) = 1. Proof. This follows from Lemmas 11.4 and 11.5 combined with (11.15). Fv× ,



Fv×

Corollary 11.7. Let v ∈ S(n). Then, for any t ∈ b∈ − {−1} and z ∈ C, the inequality

  

(z)  b−1 +1 1  t 0 1 v−f (ηv ) σ (σ) [ ]

Φ1,v

 |b|−σ v |t|v δ(t ∈ pv ) fv (t, b) 0 1 1 1 0 1 holds with σ = Re(z). Proof. The condition v ∈ S(n) implies f (ηv ) = 0 by (2.20). Then, this is proved by the same reasoning as Corollary 11.6, except that the extra factor  δ(t ∈ pv ) arises from the last statement of Lemma 11.4. For ρ ∈ R and t ∈ Fv× , set  |t|ρv , vρ (t) = sup(0, − logqv |t|v ),

(ρ = 0), (ρ = 0).

Lemma 11.8. Let ρ ∈ R. If ρ = 0, then, for any integers A  B,  q −Aρ − qv−ρ qv−Bρ |t|ρv d× t = qv−dv /2 v . 1 − qv−ρ qv−B |t|v qv−A For any constant C0 , C1 > 0,  |t|ρv d× t  δ(C0 |b|v  C1 ) (1 + vρ (b)), C0 |b|v |t|v C1

If ρ < 0, then

 C0 |b|v |t|v

|t|ρv d× t  vρ (b),

b ∈ Fv× .

b ∈ Fv× . 

Proof. This is confirmed by a direct computation. Lemma 11.9.

(1) Let σ, ρ ∈ R. Then,

(11.22)  (ηv ) fv(σ) (t, b) |t|ρv d× t  δ(b ∈ p−f ) (1 + vρ (b)) sup(1, v2σ−ρ (b + 1)), v Fv×

b ∈ Fv× ,

(2) Suppose v ∈ Σfin − S(fη n). Then, if ρ = 0 and |b|v = |b + 1|v = 1, then  fv(σ) (t, b) |t|ρv d× t = qv−dv /2 . Fv×



If ρ = 0, 2σ, there exists a constant C(σ, ρ)  1, independent of v, such that

Fv×

fv(σ) (t, b) |t|ρv d× t  C(σ, ρ) qv−dv /2 sup(1, vρ (b)) sup(1, v2σ−ρ (b + 1))

for any b ∈ ov , b(b + 1) ∈ pv .

82

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

(3) Suppose v ∈ S(n). Then, If ρ = 0, there exists a constant C(ρ)  1, indepdent of v, such that

 Fv×



δ(t ∈ pv ) fv(σ) (t, b) |t|ρv d× t  C(ρ) qv−dv /2 δ(b ∈ pv ) sup(1, vρ (b)),

b ∈ ov − {0, −1}.

Proof. Set f = f (ηv ). We estimate the integral on pv as  ρ × fv(σ) (t, b) |t|ρv d× t = δ(b ∈ p−f ) δ(qv−2f |b|v  |t|v  |b + 1|−1 v v ) |t|v d t pv |t|v 2|σ|. Then, /small

(z)   b−1 +1 1  t 0  1 xη  σ (σ) [ ] t, b ∈ Fv× , (z, s) ∈ Lσ × Lc .

Ψv s;

 |b|−σ v |t|v fv (c; t, b), 0 1 0 1 1 1 Proof. This is inferred from Lemma 11.4 and (5.4).



11.4. TERMS ASSOCIATED TO δb ’S

83

Lemma 11.11. Let σ, ρ, c ∈ R be such that (c + 1)/2 > |ρ|. Then,

 Fv×

fv(σ) (c; t, b) |t|ρ+σ d× t (1 + vρ+σ (b)) sup(1, vσ−ρ (b + 1)) v × sup(1, |b + 1|v )−(c+1)/4+σ/2−|σ−ρ| Mv (σ, ρ, c; b),

Proof. We have



fv(σ) (c; t, b) |t|ρ+σ v

×



d t= |b|v |t|v 1

ov

|t|ρ+σ d× t + |b|−(c+1)/2+σ v v



×

b ∈ Fv× .

|t|v inf(1,|b|v )

|t|ρ+(c+1)/2 d× t v

 |b|ρ+σ v  δ(|b|v  1) (1 + vρ+σ (b)) + −(c+1)/2+σ |b|v

(|b|v  1), (|b|v > 1)

 (1 + vρ+σ (b)) sup(1, |b + 1|v )−(c+1)/2+σ , and  Fv× −ov

fv(σ) (c; t, b) |t|ρ+σ d× t v



=

1 3σ/2 + ρ + (σ + ρ)− + |σ − ρ|, σ + ρ > −1. 

Proof. This is proved by (11.23) immediately.

11.4.3. In this paragraph, let ι ∈ Σ∞ and identify Fι = R, Gι = GL(2, R). For any σ, c ∈ R and for any t, b ∈ R× , set fι(σ) (c; t, b) = {b2 + t2 (b + 1)2 }−(c+1)/4+σ/2 (1 + t−2 )−(c+1)/4−σ/2 |t|−2σ ,  (11.24) Mι (σ, ρ, c; b) = |b + 1|−(σ−ρ)− |b|(c+1)/4−σ/2 fι(σ) (c; t, b) |t|σ+ρ d× t. R×

Here, for q ∈ R, we put q− = inf(0, q). Lemma 11.13. Let ι ∈ Σ∞ . Let σ, c ∈ R be such that c > 2|σ|. Then,

(z)   b−1 +1 1  t 0  −σ σ (σ) s; [ ] |t| fι (c; t, b), t, b ∈ R× , (z, s) ∈ Lσ × Lc .

Ψ 0 1  |b| 1 1 Proof. This follows from Lemma 4.2 and (4.8) immediately.



Lemma 11.14. Let ι ∈ Σ∞ . Let σ, c, ρ ∈ R be such that (c + 1)/4 > |ρ| − σ/2 and (c + 1)/4 > σ/2, Then, (11.25)

Mι (σ, ρ, c; b)  |b + 1|−(c+1)/4+σ/2−(σ−ρ)− ,

b ∈ R× − {−1}.

84

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

Moreover, if  > 0 is such that ||ρ| − σ| + (σ − ρ)− < /3 < 1 and if (c + 1)/4 > σ/2 − (σ − ρ)− + 1, then the function b → |b(b + 1)| Mι (σ, ρ, c; b) is locally bounded on R. Proof. By the inequality b2 + t2 (b + 1)2  2|b| |b + 1| |t|, the integral in (11.24) is majorized by 2−(c+1)/4+σ/2+1 (|b| |b + 1|)−(c+1)/4+σ/2



+∞

t(c+1)/4+3σ/2+ρ−σ (1 + t2 )−(c+1)/4−σ/2 d× t,

0

which is convergent if (c + 1)/4 > |ρ| − σ/2. This shows (11.25). Let q = /3 − σ + |ρ|, which is positive by assumption. Then, the equality (1 + t2 )−σ tσ+ρ  (t + t−1 )q for any t > 0 is easily confirmed. Since {t−1 b2 + t(b + 1)2 }q  (2|b(b + 1)|)q (t > 0), we have that the integral in (11.24) is  ∞ [{t−1 b2 + t(b + 1)2 }(t + t−1 )]−(c+1)/4+σ/2 (t2 + 1)−σ tσ+ρ d× t 2 0

 2(2|b(b + 1)|)−q m(l; n), where m(l; n) is the function defined in 15.1 with l = c + 1 − 2σ − 4q and n = b(b + 1). From this, |b(b + 1)| Mι (σ, ρ, c; b) |b + 1|σ−|ρ|−(σ−ρ)− +/3 |b|(c+1)/4+σ/2−|ρ|+/3 |b(b + 1)|/3 m(l; b(b + 1)). Since |b(b + 1)|/3 m(l; b(b + 1)) is locally bounded on R from Lemmas 15.3 and 15.5 in the appendix, the majorant is locally bounded on R if σ − |ρ| − (σ − ρ)− + /3  0,

(c + 1)/4 + σ/2 − |ρ| + /3  0.

Since |ρ| − σ + (σ − ρ)− < /3 < 1 and (c + 1)/4 > σ/2 − (σ − ρ)− + 1, these conditions are satisfied. 

11.4.4. In this paragraph, putting estimations over various places obtained in the previous paragraphs together, we construct a majorant of the adelic Green function to discuss convergence of the series (11.8). For σ, ρ ∈ R, c ∈ RS , t ∈ A× and b ∈ F × − {−1}, set   fv(σ) (tv , b)}{ δ(tv ∈ pv ) fv(σ) (tv , b)} N(n|σ, c; t, b) = |t|σA { v∈Σfin −S∪S(n)

×{



v∈S(n)

fv(σ) (cv ; tv , b)},

v∈S

M(n|σ, ρ, c; b) = {



v∈Σfin −S

×{



δ(b ∈ f−1 η nov )} {



sup(1, vσ+ρ (b))}

v∈Σfin v +1)/4+σ/2 |b|−(c Mv (σ, ρ; cv , b)}. v

v∈S

Lemma 11.15. Let c ∈ RS and σ ∈ R be such that q(c) > |σ| + 1. Then,

 −1  



(z)  n|s; b 1+1 11 [ 0t 01 ] 10 x1η  N(n|σ, c; t, b), b ∈ F × − {−1}, t ∈ A× ,

Ψ (z, s) ∈ Lσ × LS (c). Proof. This follows from Lemmas 11.7, 11.10 and 11.13 by taking product. in majorants disapear in the product by We should note that the factors |b|−σ v Artin’s product formula |b|A = 1. 

11.4. TERMS ASSOCIATED TO δb ’S

85

Lemma 11.16. Let σ, ρ ∈ R and c ∈ RS be such that q(c) > σ/2 + |ρ| + 1 and σ = ±ρ. Then, for any  > 0,  A×

N(n|σ, c; t, b) |t|ρA d× t  M(n|σ, ρ, c; b) |N(b(b + 1))| N(n) ,

b ∈ F × − {−1},

with the implied constant independent of n. Proof. We remark that three sets S, S(n) and S(fη ) are mutually disjoint. We apply Lemmas 11.9 and 11.11 to bound our integral by N(b(b + 1)) N(n) times (11.26)  {

δ(b ∈ f−1 η nov )} {

v∈Σfin −S



×{



sup(1, vσ+ρ (b))} {

sup(1, |b + 1|vσ−ρ )}

v∈Σfin −S(n)

v∈Σfin

sup(1, |b + 1|v )



−(cv +1)/4+σ/2−|σ−ρ|

Mv (σ, ρ, cv ; b)}

v∈Sfin

×{



− |b|ι−(cι +1)/4+σ/2 |b + 1|(σ−ρ) Mι (σ, ρ, cι ; b)}. ι

ι∈Σ∞

Here the factor N(b(b + 1)) N(n) arises due to the estimations C(ρ, σ)#S(b(b+1))  N(b(b + 1)) and C(ρ)#S(n)  N(n) . If v ∈ S(n), then δ(b ∈ nov ) sup(1, |b + ) = 1. Thus, in the third factor in (11.26), the range of v is enlarged from 1|σ−ρ v Σfin − S(n) to Σfin without spoiling the majoration. We have {



δ(b ∈ f−1 η nov )}

v∈Σfin −S



={

sup(1, |b +





×{ ={



1|σ−ρ )} { δ(b v v∈Σfin −S

σ−ρ δ(b ∈ f−1 )} η nov ) sup(1, |b + 1|v

v∈Σfin −S

sup(1, |b + 1|v )|σ−ρ| } {

v∈Sfin

×{

σ−ρ ∈ f−1 )} η nov ) sup(1, |b + 1|v

− sup(1, |b + 1|v )|σ−ρ| |b + 1|(σ−ρ) } v

v∈Sfin



sup(1, |b + 1|σ−ρ ) v

v∈Σfin

v∈Sfin

{





− |b + 1|−(σ−ρ) } ι

ι∈Σ∞



δ(b ∈

f−1 η nov )

sup(1, |b + 1|v )|σ−ρ| }

ι∈Σfin −S

by using the easily confirmed relation {



− |b + 1|(σ−ρ) }{ v

v∈Sfin

×{



ι∈Σ∞



v∈Σfin −S

|b +

sup(1, |b + 1|σ−ρ )} = v 

− 1|−(σ−ρ) }{ sup(1, |b ι v∈Σfin −S

+ 1|v )|σ−ρ| .

86

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

  |σ−ρ| Note that ι∈Σfin −S δ(b ∈ f−1  v∈Σfin −S δ(b ∈ f−1 η nov ) sup(1, |b + 1|v ) η nov ). Thus, our integral is bounded by   { δ(b ∈ f−1 sup(1, vσ+ρ (b))} η nov )} { v∈Σfin −S

×{



v∈Σfin

sup(1, |b + 1|v )−(cv +1)/4+σ/2 Mv (σ, ρ, cv ; b)}

v∈Sfin

×{



|b|ι−(cι +1)/4+σ/2 Mι (σ, ρ, cι ; b)}.

ι∈Σ∞

From sup(1, |b + 1|v )  |b|v and (cv + 1)/4 > σ/2 + |ρ| + 1  σ/2, we have v +1)/4+σ/2 sup(1, |b + 1|v )−(cv +1)/4+σ/2  |b|−(c . v

By these, we find that M(n|σ, ρ, c; b) is a majorant of our integral. This completes the proof.  Corollary 11.17. Let σ, ρ ∈ R and c ∈ RS be such that q(c) > σ/2 + |ρ| + 1, σ = ±ρ. Let U be a compact subset of A× . Then, for any  > 0,  N(n|σ, c; at, b)  M(n|σ, ρ, c; b) |N(b(b + 1))| N(n) , b ∈ F × − {−1}, t ∈ U a∈F ×

with the implied constant independent of n.   Proof. Let V = v∈Σfin o× v ι∈Σ∞ {x ∈ Fι | |x − 1| < 1/2}. Then, V is a relatively compact neighborhood of 1 in A× . From definition of N(n|σ, c; t, b), there exists a constant C0 > 0 (depending only on σ, c) such that t ∈ A× , y ∈ V, b ∈ F × − {−1}.

C0 N(n|σ, c; t, b) |t|ρA  N(n|σ, c; ty, b) |ty|ρA ,

By the same argument as Lemma 3.5, we obtain the estimation   N(n|σ, c; at, b)|at|ρA  N(n|σ, c; a, b) |a|ρA d× a, t ∈ U, b ∈ F × − {−1}. A×

a∈F ×

From this, combined with Lemma 11.16, we have the conclusion. Let oF,(S) be the ring of S-integers in F , i.e.,   oF,(S) = F ∩ {F∞ ov Fv }. v∈Σfin −S

v∈Sfin

Then, oF,(S) is a Dedekind domain containing the maximal order oF of F . Lemma 11.18. Let σ, ρ ∈ R,  > 0 and c ∈ RS be such that q(c) > sup(3σ/2 + ρ + |σ − ρ| + 1, σ/2 − (σ − ρ)− + 1 + 2, |σ|/2 + |ρ|) and σ + ρ > −1, σ = ±ρ. If ||ρ| − σ| + (σ − ρ)− < /3 < 1, then the series    (11.27) { sup(1, vσ+ρ (b))} { Mv (σ, ρ, cv ; b)}N(b(b + 1))

b∈oF,(S) −{−1} v∈Sfin

is convergent.

v∈S



11.4. TERMS ASSOCIATED TO δb ’S

87

Proof. Let v ∈ S. From Lemma 11.14, we have the estimation |b(b + 1)| ι Mι (σ, ρ, cι ; b)  (1 + |b|ι )−(cι +1)/4+σ/2−(σ−ρ)− +2

 on R. Let Φ(b) be the function on FS obtained from v∈S Mv (σ, ρ, cv ; bv ) by replacing its ι-factor with Φι (bι ) = (1 + |bι |ι )−(cι +1)/4+σ/2−(σ−ρ)− +2 for each ι ∈ Σ∞ . Then, there exists a neighborhood of 0 in FS such that Φ(b +  x)  Φ(b), x ∈ V , b ∈ FS . Then, since oF,(S) is a discrete subgroup of FS = v∈S Fv , the series (11.27) is majorized by the integral     sup(1, vσ+ρ (b)) Mv (σ, ρ, cv ; b) dbv } { Φι (bι ) dbι }, { v∈Sfin

Fv

ι∈Σ∞

R



which is convergent by the condition on q(c).

Lemma 11.19. Let σ, ρ ∈ R and c ∈ R be such that (c + 1)/4 > 5|σ|/2 + 2|ρ| + 1 and σ+ρ > −1, σ = ±ρ. Let c ∈ RS be the diagonal image of c in RS . Then, for any  > 0 such that ||ρ|−σ|+(σ −ρ)− < /3 < 1 and (c+1)/4 > σ/2−(σ −ρ)− +1+2,  M(n|σ, ρ, c; b)N(b(b + 1))  N(n)−(c+1)/4+σ/2+|σ+ρ| b∈F −{0,−1}

with the implied constant independent of the ideal n. × Proof. By dividing f−1 η noF,(S) to oF,(S) -orbits, we have



M(n|σ, ρ, c; b) N(b(b + 1))

b∈F −{0,−1}



=

{



sup(1, vσ+ρ (b))} {

v∈Σfin b∈f−1 η noF,(S) −{0,−1}



=

{



|b|−(c+1)/4+σ/2 Mv (σ, ρ; c, b)} N(b(b + 1)) v

v∈S

sup(1, vσ+ρ (b))}{

× v∈Σfin −S b∈f−1 η noF,(S) /oF,(S)

⎛





|b|−(c+1)/4+σ/2 } v

v∈S

b =0,−1

⎞

  ⎜  ⎟ ×⎝ { sup(1, vσ+ρ (ub))}{ Mv (σ, ρ; c, ub) N(ub(ub + 1)) }⎠ . v∈Sfin u∈o× F,(S)

v∈S

From Lemma 11.18, the last series over u ∈ o× F,(S) converges and is bounded as a function of b. Thus, our series is majorized by the Euler product 

{



sup(1, vσ+ρ (b)) |b|(c+1)/4−σ/2 } v

× v∈Σfin −S b∈(f−1 η noF,(S) −{0})/oF,(S)



mv (σ, ρ − δ, c)

v∈Σfin −S

for any sufficiently small δ > 0 with  mv (σ, ρ, c) =

sup(1, qv−e(σ+ρ) ) qve(−(c+1)/4+σ/2) ,

e−f (ηv )+ordv (n)

which equals (1 − qv−(c+1)/4+σ/2−(σ+ρ)− )−1 qvev (−(c+1)/4+σ/2−(σ+ρ)− )  with ev = ordv (n) for v ∈ Σfin − (S ∪ S(fη )). Thus, v∈Σfin −S mv (σ, ρ − δ, c) is bounded from above by a constant times of  (1 − qv−(c+1)/4+σ/2−(σ+ρ−δ)− )−1 , N(n)−(c+1)/4+σ/2−(σ+ρ−δ)− v∈Σfin −S∪S(fη )

88

11. A GEOMETRIC EXPRESSION OF AUTOMORPHIC SMOOTHED KERNELS

which is convergent if (c + 1)/4 − σ/2 + (σ + ρ − δ)− > 1.



Lemma 11.20. Let U a compact subset of A× . Let σ, c ∈ R be such that c+1 9|σ| (11.28) > + 1, σ > −1. 4 2 Then, the series 

   1 xη 

0

Ψ(z) n|s; δb [ at 0 1] 0 1 b∈F −{0,−1} a∈F ×

converges uniformly in (t; z, s) ∈ U × Lσ × LS (c) and the integral   



(z)  n|s; δb [ a0 01 ] 10 x1η |a|ρA d× a

Ψ b∈F −{0,−1}

A×

converges uniformly in (z, s) ∈ Lσ × LS (c) for any ρ such that 0 < ||ρ| − σ| < , σ + ρ > −1 with a sufficiently small  > 0. Proof. By (11.28), there exists ρ ∈ R (with |ρ| being sufficiently close to σ) such that (c + 1)/4 > 5|σ|/2 + |ρ| + 1, ρ + σ > −1 and σ = ±ρ. Then, the desired convergence of series and integrals is inferred from Lemmas 11.15, 11.16, Corollary 11.17 and Lemma 11.19.  Now, we turn around to the series (11.8). Lemma 11.21. The holomorphic function Jhyp (β, λ, α; t) (Re(λ) > 1) is continued to an entire function on the whole λ-plane, whose value at λ = 0 is given by β(0) Jhyp (α; t) with     1 xη  0 Jhyp (α; t) = (11.29) . Ψ(0) n|α; δb [ at 0 1] 0 1 b∈F −{0,−1} a∈F ×

The series converges absolutely and locally uniformly in t ∈ A× .

Proof. From Lemma 11.20, if Re(λ) > 1, we have that Jhyp (β, λ, α; t) equals 1 2πi

S 

LS (c)



1 2πi



Lσ

β(z) z+λ

 

(z)

(Ψ

(−z)

+Ψ

   1 xη  at 0 ) n|s; δb [ 0 1 ] 0 1 dz

b∈F × a∈F × b =−1

× α(s) dμS (s).

Here, c, σ ∈ R are chosen so that q(c) > 9|σ|/2 + 1, σ > −Re(λ). By Lemma 11.20, we can move the contours LS (c) and Lσ obeying the condition (11.28) to obtain a holomorphic continuation of Jhyp (β, λ, α; t). This shows the first statement. Then, the second statement is proved by Lemma 6.9.  11.5. Conclusion From Lemmas 11.2, 11.3 and 11.21, we obtain an expression of the regularized automorphic smoothed kernel, which is a starting point to the next stage: Proposition 11.22. Let n be a square free oF -ideal and S a finite set of places such that Σ∞ ⊂ S, S ∩ S(n) = ∅. Let η be a unitary idele-class character of F × satisfying (2.18), (2.19), (2.20) and (11.1). Then, for any α ∈ AS , we have    ˆ reg n|α; [ t 0 ] 1 xη = (1 + δ(n = oF )) Jid (α; t) Ψ 0 1

0 1

+ Ju (α; t) + Ju¯ (α; t) + Jhyp (α; t),

t ∈ A× ,

11.5. CONCLUSION

89

where terms on the right-hand side are defined by (11.2), (11.4), (11.5) and (11.29).

CHAPTER 12

Periods of regularized automorphic smoothed kernels: the geometric side Let S, n, η be as in the previous section; we keep assuming all the conditions imposed on these objects. For β ∈ B, α ∈ AS and λ ∈ C, set  Jη∗ (β, λ; α)

= F × \A×

∗ × J∗ (α; t) {βˆλ (|t|A ) + βˆλ (|t|−1 A )} η(txη ) d t

with ∗ denoting id, u, u ¯ or hyp. Set (12.1) ΥηS (s) = {

 −1 Γ((sι + 1)/4)2  }{ (1 − qv(sv +1)/2 )−1 (1 − ηv (v ) qv−(sv +1)/2 )−1 }, 2 8 Γ((s ι + 3)/4) ι∈Σ v∈S ∞

fin

and (12.2)

  dF log qv (CEuler + 2 log 2 − log π) + CηS,∗ (n|s) = C0 (η) + R(η) log DF + (sv +1)/2 2 v∈Sfin 1 − qv 





sι + 3 sι + 1 1  +ψ ψ + 2 ι∈Σ 4 4 ∞  δ(n = oF ), (∗ = u) R(η) log N(n) × + 2 ¯) 2 − δ(n = oF ), (∗ = u

for s ∈ XS . Here, ψ(z) = Γ (z)/Γ(z) is the digamma function, CEuler the Euler constant, R(η) = Ress=1 L(s, η) = δη,1 vol(F × \A1 ) and C0 (η) = CTs=1 L(s, η). Note that CηS,∗ (s) (∗ = u, u ¯ ) reduces to the constant C0 (η) = L(1, η) when η = 1. We also consider the series-integral Kη (n|s) =

 b∈F −{0,−1}

 A×

   Ψ(0) n|s; δb [ 0t 10 ] 10 x1η η(tx∗η ) d× t,

which is absolutely convergent by Lemma 11.21 if Re(s) = c with c ∈ R such that (c + 1)/4 > 1. Theorem 12.1. (1) The integral Jη∗ (β, λ; α) converges absolutely and locally uniformly in Re(λ) > 1. The holomorphic function λ → Jη∗ (β, λ; α) has a meromorphic continuation to the half plane Re(λ) > −1. 91

92

12. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

(2) If we set Jηid (n|α) = 0, Jη∗ (n|α)

(12.3)

(12.4)

1 2πi

S

= (1 + δ(n =  × ΥηS (s) CηS,∗ (n|s) α(s) dμS (s), 

Jηhyp (n|α)

 1/2 oF )) DF G(η)

=

(∗ = u, u ¯),

LS (c)

1 2πi

S  LS (c)

Kη (n|s)α(s) dμS (s),

then the constant term CTλ=0 Jη∗ (β, λ; α) is equal to β(0) Jη∗ (n|α). We have the identity ˆ reg (n|α)) = Jηu (n|α) + Jηu¯ (n|α) + Jη (n|α), Pηreg (Ψ hyp

(12.5)

α ∈ AS .

The proof of this theorem is separated to several lemmas. 12.1. The integral Jηid Lemma 12.2. If Re(λ) > 0, then the integral Jηid (β, λ; α) converges absolutely and



Jηid (β, λ; α)



×

δη,1 vol(F \A )

=

1

1 2πi

"

S  LS (c)

Υ1S (s) α(s) dμS (s)

2β(0) . λ

Proof. Since Jid (α; t) is independent of t (Lemma 11.2), Lemma 7.6 shows the claim.  12.2. The integral Jηu Let q(Re(s)) > Re(λ) > σ > 1 and consider the integrals (12.6) ± U0,η (λ; s) =

1 2πi

(12.7) ± (λ; s) = U1,η

1 2πi

 L∓σ

 L∓σ

β(z) z+λ β(z) z+λ

 A×

 A×

   −1   1 xη  ∗ ±z × Ψ(0) n|s; 10 t 1 )|t| d t dz, η(tx η A 0 1    1 0 1 0  ∗ ±z × Ψ(0) n|s; t−1 ) |t| d t dz. w η(tx 0 −x 1 η A η 1

Set ΥηS (z; s)

−1/2

{(oF /fη )× }−1  −π −z/2 Γ((sι + 1)/4) Γ((sι + 2z + 1)/4) 1 − z  √ Γ } ×{ 2 8 π Γ((sι + 3)/4) Γ((sι − 2z + 3)/4) ι∈Σ∞  ×{ (1 − ηv (v ) qv−(z+(sv +1)/2) )−1 (1 − qv(sv +1)/2 )−1 }, z ∈ C, s ∈ XS .

= DF

v∈Sfin

± Lemma 12.3. The double integrals Uj,η (λ; s) (j = 0, 1) converge absolutely and

(12.8) ± Uj,η (λ; s) =

1 2πi

 L∓σ

β(z) N(fη )∓z N(n)±jz η˜(n)j δ(nj = oF ) L(∓z, η) ΥηS (±z; s) dz. z+λ

12.2. THE INTEGRAL Jη u

93

+ Proof. Let us examine U0,η (λ; s) first. From (6.3), the inner integral over A× is decomposed to the product





Jv (Ψ(0) v (sv ; −); z)

v∈S



(0)

Jv (Φ1,v ; z)

(0)

Jv (Φ0,v ; z),

v∈S∪S(n)

v∈S(n)

where, for any place v and for any function Φ on Gv , we set  Jv (Φ; z) =

Fv×

Φ



1 t−1 0 1

  1 xη,v  0

1

ηv (tx∗η,v )|t|zv d× t,

where xη,v = x∗η,v = v−f with f = f (ηv ) if v ∈ Σfin and xη,v = 0, x∗η,v = 1 if v ∈ Σ∞ .    1 −f  −1 (0)  v • Let v ∈ S ∪ S(n). If f > 0, then, from Φ0,v 10 t 1 = δ(−t ∈ 0 1 vf Uv (f )), we have Jv (Φ0,v ; z) = qv−f z (0)

 −vf (0)

If f (ηv ) = 0, then, from Φ0,v  (0) Jv (Φ0,v ; z)

ηv (tv−f ) d× t = ηv (−1) qv−f z vol(Uv (f ); d× t). Uv (f )



1 t−1 0 1

  1 −f  v

0

1

= δ(t−1 ∈ ov ), we have

ηv (t−1 ) |t−1 |zv d× t

= ov

=



× ηv (v )−l qvlz vol(o× v ; d t)

l0 × −1 z −1 qv ) = vol(o× v ; d t) (1 − ηv (v )

if Re(z) < 0.

• Let v ∈ S(n). From (2.20), f (ηv ) = 0. Thus, × −1 z −1 Jv (Φ1,v ; z) = vol(o× qv ) , v ; d t) (1 − ηv (v ) (0)

Re(z) < 0

is obtained in the same way as the last case. • Let v ∈ Sfin . From (11.1), f (ηv ) = 0. Thus, using (5.4), we compute −(s +1)/2

Jv (Ψv (sv ; −); z) = (1 − qv v )−1 (1 − qv v )−1 ⎧ ⎫ ⎨  ⎬  −l lz × l −lz −l(sv +1)/2 × × ηv (v ) qv d t + ηv (v ) qv qv d t −l × ⎩ ⎭  l o× v ov v (0)

l0

= (1 −  ×

(s +1)/2

l1

v

−(s +1)/2 −1 qv v ) (1

−

(s +1)/2 −1 qv v ) −(s +2z+1)/2

1 ηv (v ) qv v + −1 z −(s +2z+1)/2 1 − ηv (v ) qv 1 − ηv (v ) qv v



−(sv +2z+1)/2 −1

= (1 − ηv (v )−1 qvz )−1 (1 − ηv (v ) qv × (1 −

(s +1)/2 −1 qv v )

if Re(z) < 0, Re(sv + 2z + 1) > 0.

× vol(o× v ; d t),

)

× vol(o× v ; d t)

94

12. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

• Let v ∈ Σ∞ . From (2.18) and (2.19), the v-component ηv is trivial. By (4.9), we have 

(0)

Jv (Ψv (sv ; −); z) = 2 c0 (sv ) 0 1

 = c0 (sv )

+∞

(1 + t2 )−(sv +1)/4 2 F1



sv +1 sv +1 sv +2 , 4 ; 2 ; (1 4

x(sv +1)/4+z/2−1 (1 − x)−z/2−1 2 F1

0



+ t2 )−1



t−z d× t



sv +1 sv +1 sv +2 , 4 ; 2 ;x 4

dx

Γ((sv + 2)/2) Γ((sv + 2z + 1)/4) Γ(−z/2) Γ((1 − z)/2) Γ((sv + 1)/4) Γ((sv + 3)/4) Γ((sv − 2z + 3)/4) −1 Γ((sv + 1)/4) Γ((sv + 2z + 1)/4) Γ(−z/2) Γ((1 − z)/2) , = √ 8 π Γ((sv + 3)/4) Γ((sv − 2z + 3)/4) = c0 (sv )

using [16, Formula 7.512.3] to evaluate the integral when Re(z) < 0, Re(sv + 2z + 1) > 0. Putting the computations above together, we obtain that the inner integral of (12.18) is absolutely convergent if Re(z) < −1 and q(Re(s)) + Re(z)/2 > 0 and is equal to N(fη )−z L(−z, η)  ηv (−1) vol(Uv (f (ηv )); d× t)} ×{ v∈Σfin

×{



(1 − ηv (v ) qv−(sv +2z+1)/2 )−1 (1 − qv(sv +1)/2 )−1 }

v∈Sfin

 −π −z/2 Γ((sv + 1)/4) Γ((sv + 2z + 1)/4) Γ((1 − z)/2) √ }. 8 π Γ((sv + 3)/4) Γ((sv − 2z + 3)/4) v∈Σ∞   From (2.19), v∈Σfin ηv (−1) = ι∈Σ∞ ηι (−1) = 1. Hence,   ηv (−1) vol(Uv (f (ηv )); d× t) = qv−f (ηv )−dv /2 (1 − δ(f (ηv ) > 0) qv−1 )−1 ×{

v∈Σfin

v∈Σfin −1/2

= DF

{(oF /fη )× }−1 .

+ Thus, we obtain (12.8) for U0,η (λ; s). The other integrals are settled in the same way. This completes the proof. 

Since σ > 1, the integrand of (12.8) has poles at z = 0 and z = ±1 inside the vertical stripe between L−σ and Lσ . By applying the residue theorem to move the contour L−σ of (12.8) upto Lσ , we have (12.9)

 β(z) 1 N(fη )−z N(n)jz η˜(n)j δ(nj = oF ) L(−z, η) ΥηS (z; s) dz 2πi Lσ z + λ − Rj,−1 (λ; s) − Rj,0 (λ; s) − Rj,1 (λ; s)

+ Uj,η (λ; s) =

with Rj,q (s) denoting the residue at z = q. From (2.4), (12.10) Rj,−1 (λ; s) = −δη,1 N(n)−j η˜(n)j δ(nj = oF ) vol(F × \A1 ) ΥηS (−1; s) (12.11)

Rj,0 (λ; s) = δη,1 DF vol(F × \A1 ) ΥηS (0; s) δ(nj = oF ) 1/2

Lemma 12.4. The function Rj,1 (λ; s) is of the form  Rj,1 (λ; s) = φi (λ; s), i∈I

β(0) . λ

β(1) , λ−1

12.2. THE INTEGRAL Jη u

95

where {φi (λ; s)}i∈I is a finite family of functions with the following properties. • φi (λ; s) is a meromorphic function on XS holomorphic on the region q(Re(s)) > 0 . • For each i ∈ I, there exists vi ∈ Σ∞ such that φi (λ; s) is independent of svi , the vi -th coordinate of s. Proof. For each v ∈ Σ∞ , set Tv (z, sv ) =

−π −z/2 Γ((sv + 1)/4) Γ((sv + 2z + 1)/4) Γ((1 − z)/2) √ . Γ((sv + 3)/4) Γ((sv − 2z + 3)/4) 8 π

Then the Laurent expansion of Tv (z, sv ) at z = 1 is of the form ∞  1 1 + Tv (z, sv ) = av,m (sv ) (z − 1)m 4π z − 1 m=0

with av,m (sv ) holomorphic on Re(sv ) > −1. Thus, (12.12)



Tv (z, sv ) =

v∈Σ∞

dF ∞   bj (s∞ ) + am (s∞ ) (z − 1)m , j (z − 1) m=0 j=1

where bj (s∞ ) is a function in s∞ = {sv }v∈Σ∞ which is holomorphic on the region inf v∈Σ∞ Re(sv ) > −1 and is independent of the variable svj for some vj ∈ Σ∞ . The function (12.13)

β(z) −1/2 N(n)jz N(fη )−z L(−z, η) DF {(oF /fη )× }−1 z+λ  (1 − ηv (v ) qv−(z+(sv +1)/2) )−1 (1 − qv(sv +1)/2 )−1 , × v∈Sfin

∞ m being holomorphic at z = 1, is expanded as m=0 cm (λ; sfin ) (z − 1) , where cm (λ; sfin ) is a holomorphic function of the variable sfin = {sv }v∈Sfin on the region inf v∈Sfin Re(sv ) > −1. Since the integrand of (12.8) is the product of (12.12) dF and (12.13), we obtain Rj,1 (λ; s) = j=1 bj (s∞ ) cj−1 (λ; sfin ), from which the conclusion follows.  Lemma 12.5. Let c ∈ RS be such that q(c) > Re(λ). Then, for j = 0, 1, (12.14)  S  1 Rj,1 (λ; s) α(s) dμS (s) = 0, 2πi LS (c) (12.15)  S  1 1 Rj,0 (λ; s) α(s) dμS (s) = δ(nj = oF ) Jηid (β, λ; α), 2πi 2 LS (c) (12.16)  S  1 Rj,−1 (λ; s) α(s) dμS (s) = −δη,1 N(n)−j η˜(n)j δ(nj = oF ) 2πi LS (c) −1/2

× DF

vol(F × \A1 ) (−1)S

β(1) α ˜ (1) . λ−1

96

12. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

Proof. To prove (12.14), from Lemma 12.4, it suffices to show  φi (λ; s) α(s) dμS (s) = 0. LS (c)

Let vi ∈ Σ∞ be such that φi (λ; s) is independent of svi . Then, since α(s) is an even holomorphic function in Re(svi ) > −1, we have  φi (λ; s) α(s) svi dsvi = 0 (c > 1) Lc

by Cauchy’s theorem. The identity (12.15) follows from (12.11) and Lemma 12.2 immediately. To show (12.16), for simplicity, we assume that α(s) is decomposable, i.e.,  α(s) = v∈S αv (sv ) with functions αv ∈ Av for v ∈ S. Then, from (12.10), we have  S  1 Rj,−1 (λ; s) α(s) dμS (s) 2πi LS (c) −1/2 β(1) = −δη,1 N(n)−j η˜(n)j δ(nj = oF ) vol(F × \A1 ) DF λ−1  1  c+i∞ −2 αv (sv ) dμv (sv )} ×{ 2πi c−i∞ s2v − 1 v∈Σ∞  1  cv +2πi(log qv )−1 ×{ (1 − qv(1−sv )/2 )−1 (1 − qv(1+sv )/2 )−1 αv (sv ) dμv (sv )} 2πi cv −2πi(log qv )−1 v∈Sfin

−1/2

= −δη,1 N(n)−j η˜(n)j δ(nj = oF ) vol(F × \A1 ) DF

β(1)  (−αv (1)) λ−1 v∈S

−1/2

= −δη,1 N(n)−j η˜(n)j δ(nj = oF ) DF

vol(F × \A1 ) (−1)S

α ˜ (1) β(1) , λ−1

applying Lemma 9.5 to calculate the contour integrals.



Lemma 12.6. The integral Ju (β, λ; α) converges absolutely on the region Re(λ) > 1 and is equal to (12.17) −

β(1) α(1) ˜ 1 + δ(n = oF ) η −1/2 Jid (β, λ; α) + δη,1 DF {1 + N(n)−1 δ(n = oF )} vol(F × \A1 ) (−1)S 2 λ−1      β(z)  1 1 S N(fη )−z (1 + N(n)z η˜(n)δ(n = oF )) L(−z, η) ΥηS (z; s) + 2πi 2πi z +λ LS (c) Lσ   + N(fη )z (1 + N(n)−z η˜(n)δ(n = oF )) L(z, η) ΥηS (−z; s) dz α(s) dμS (s), (q(c) > σ > 1).

The function λ → Jηu (β, λ; α) (Re(λ) > 1) has a meromorphic extension to C holomorphic everywhere except a possible simple pole at λ = 0, 1. We have CTλ=0 Jηu (β, λ; α) = Jηu (n|α) β(0) with Jηu (n|α) defined by (12.3). Proof. Let Re(λ) > 1. By exchanging order of integrals, we obtain  Jηu (β, λ; α) =

1 2πi

S  LS (c)

+ − + − {U0,η (λ; s) + U0,η (λ; s) + U1,η (λ; s) + U1,η (λ; s)} α(s) dμS (s),

η 12.3. THE INTEGRAL Ju ¯

97

where c is taken so that q(c) is sufficiently large. Then, from Lemma 12.3, (12.9) and Lemma 12.5, we obtain the expression (12.17) with varying contours LS (c) and Lσ , which yields a meromorphic continuation of Jηu (β, λ; α) to the whole λ-plane. From (12.17), we have CTλ=0 Jηu (β, λ; α) = (12.18)

−1/2

− δη,1 DF {1 + N(n)−1 δ(n = oF )} vol(F × \A1 ) (−1)S β(1) α ˜ (1)    S   1 1 β(z) + {f (z) + f (−z)} dz α(s) dμS (s) 2πi 2πi z LS (c) Lσ

with f (z) = N(fη )−z {1 + N(n)z η˜(n)δ(n = oF )} L(−z, η) ΥηS (z; s). By a similar  β(z) 1 argument as in the proof of Lemma 6.5, the contour integral 2πi {f (z) + Lσ z f (−z)} dz is computed as the half of {Resz=−1 +Resz=1 +Resz=0 }{(β(z)/z) (f (z)+ f (−z))}. The half of Resz=−1 +Resz=1 cancels with the first term of the right-hand side of (12.18). By (2.3) and by the formula ψ(1/2) = −CEuler − 2 log 2 ([36, p.15]), a computation shows the identity 2−1 Resz=0 =β(0) (1 + δ(n = oF ) η˜(n)) DF G(η) ΥηS (s) CηS (s). 1/2



Note that η˜(n) = 1 from (2.20). Combining these, we have the result. 12.3. The integral Jηu¯ Let q(Re(s)) > Re(λ) > σ > 1 and consider the integrals (12.19) ˜ ± (λ; s) = 1 U 1,η 2πi (12.20) ˜ ± (λ; s) = 1 U 0,η 2πi

 L±σ

 L±σ

β(z) z+λ β(z) z+λ



   × n|s; [ 1t 01 ] 10 x1η η(tx∗η )|t|±z A d t

(0)

  1 0  × n|s; [ 10 1t ] −x w0 η(tx∗η ) |t|±z η 1 A d t

Ψ A×

 Ψ A×



(0)

dz,  dz.

˜ ± (λ; s) converge absolutely as double integrals. Lemma 12.7. The integrals U j,η We have (12.21) 1 ± ˜j,η (λ; s) = U 2πi

 L±σ

β(z) N(fη )∓z N(n)∓jz η˜(n)j δ(n1−j = oF ) L(±z, η) ΥηS (∓z; s) dz. z+λ

˜ + (λ; s). The inner integral over A× is decomposed to Proof. We examine U 0,η the product    (0) (0) J˜v (Ψ(0) J˜v (Φ1,v ; z) J˜v (Φ0,v ; z), v (sv ; −); z) v∈S

v∈S(n)

v∈S∪S(n)

where, for any place v and any smooth function Φ on Gv , we set      J˜v (Φ; z) = Φ [ 10 1t ] −x1η,v 10 w0 ηv (tx∗η,v ) |t|zv d× t, z ∈ C, Fv×

x∗η,v

where xη,v and are the same as in the proof of Lemma 12.3. We compute these integrals separately.  Let v ∈ S ∪ S(n). If f = f (ηv ) > 0, then, by the Iwasawa decomposition of −x1η,v 10 (Lemma 5.5), we have the formula   (0)  Φ0,v [ 10 1t ] −x1η,v 10 = δ(t ∈ vf Uv (f )).

98

12. PERIODS OF REGULARIZED AUTOMORPHIC SMOOTHED KERNELS

By this, (0) J˜v (Φ0,v ; z) = qv−f z vol(Uv (f ); d× t)    coincides with the is easily obtained. If f (ηv ) = 0, then Φ0,v [ 10 1t ] −x1η,v 10 (0) × × characteristic function of ov . Thus, J˜v (Φ0,v ; z) = vol(ov ; d t) (1 − ηv (v ) qv−z )−1 is proved in the same way as in Lemma 12.3. If v ∈ S(n), then     (0)  (0) (0)  −1 Φ1,v [ 10 1t ] −x1η,v 10 w0 = Φ1,v ([ 10 1t ] w0 ) = Φ1,v −t =0 1 0 (0) for all t ∈ Fv× . Thus, J˜v (Φ1,v ; z) = 0 identically. In the remaining case, i.e., v ∈ S (0)

and Φ = Ψv (sv ; −), by a similar computation made in the proof of Lemma 12.3, we easily have the same formula for J˜v (Φ; z) as Jv (Φ; −z).  The argument after Lemma 12.3 go through verbatim with an obvious modification. Consequently, we obtain the following lemma. Lemma 12.8. The integral Ju¯ (β, λ; α) converges absolutely on the region Re(λ) > 1 and is equal to (12.22) −

1 + δ(n = oF ) η β(1) α(1) ˜ −1/2 {δ(n = oF ) + N(n)−1 } vol(F × \A1 ) (−1)S Jid (β, λ; α) + δη,1 DF 2 λ−1      1 1 S β(z)  + N(fη )z (δ(n = oF ) + N(n)z η˜(n)) L(−z, η) ΥηS (z; s) 2πi 2πi z +λ LS (c) Lσ   + N(fη )−z (δ(n = oF ) + N(n)−z η˜(n)) L(z, η) ΥηS (−z; s) dz α(s) dμS (s), (q(c) > σ > 1).

The function λ → Jηu¯ (β, λ; α) (Re(λ) > 1) has a meromorphic extension to C holomorphic everywhere except a possible simple pole at λ = 0, 1. We have CTλ=0 Jηu¯ (β, λ; α) = Jηu¯ (n|α) β(0) with Jηu¯ (n|α) defined by (12.3). 12.4. The integral Jηhyp Lemma 12.9. The integral Jηhyp (β, λ; α) converges absolutely and defines a holomorphic function on the half plane Re(λ) > −1. We have CTλ=0 Jηhyp (β, λ; α) = β(0) Jhyp (n|α) with the absolutely convergent integral (12.4). Proof. Let c ∈ R, σ ∈ R be such that (c + 1)/4 > 9|σ|/2 + 1, 1 > σ > −1. Then, from Lemmas 11.15 and 11.16, we have 

     |β(z)|  (0)  × n|s; δb [ 0t 01 ] 10 x1η  |t|±σ { Ψ A d t}|dz| LS (c) Lσ |z + λ| b∈F −{0,−1} A×    |β(z)| |dz|} { |α(s)| |dμS (s)|} { M(n|0, ±σ, c; b)|N(b(b + 1))|δ }. { LS (c) Lσ |z + λ| 

|α(s)| |dμS (s)|

b∈F −{0,−1}

with some δ > 0. In the majorant, the first two factors are convergent by the rapid decay of α and β. The last series is convergent by Lemma 11.19. From this, the

ˆ 12.5. A REMARK ON THE SINGULAR PART OF Pη β,λ (Ψreg (n|α))

holomorphicity of

      β(z) 1 Ψ(0) n|s; δb [ 0t 01 ] 10 x1η 2πi z + λ × LS (c) Lσ b∈F −{0,−1} A   × (Re(λ) > −σ) × (|t|zA + |t|−z A ) d t dz α(s) dμS (s),

Jηhyp (β, λ; α) =

1 2πi

S 

99



on Re(λ) > −1 is inferred. This expression allows us to obtain the value at λ = 0 by Lemma 6.5.  ˆ reg (n|α)) 12.5. A remark on the singular part of Pηβ,λ (Ψ ˆ reg (n|α)) From Lemmas 12.2, 12.6, 12.8 and 12.9, the function λ → Pηβ,λ (Ψ (Re(λ) > 1) has a meromorphic continuation to the half-plane Re(λ) > −1 and its sigular part is α(1) ˜ β(0) −1/2 . δη,1 (−1)S DF vol(F × \A1 ) (1 + δ(n = oF )) (1 + N(n)−1 ) λ−1 This is consistent with the spectral side (cf. Theorem 10.5).

CHAPTER 13

Asymptotic formulas Let S be a finite set of places of F containing Σ∞ and η a unitary idele-class character of F × satisfying the conditions (2.18), (2.19) and (11.1). Let IS,η be the set of all the square free oF -ideals n satisfying the condition (2.20). By Dirichlet’s theorem on arithmetic progression extended to the case of number fields, the set IS,η contains infinitely many prime ideals of degree 1. Let n ∈ IS,η . From Theorem 10.5 and Theorem 12.1, we obtain the identity (13.1) C(n, S) {Iηcus (n|α) + Iηeis (n|α) + Dη (n|α)} = Jηu (n|α) + Jηu¯ (n|α) + Jηhyp (n|α),

α ∈ AS ,

which is regarded as an identity between linear functionals on the space AS and was ˆ reg (n|α)) in two different ways. This is our version obtained by computing Pηreg (Ψ of the Relative Trace Formula. In this section, we deduce some asymptotic results on spectral average of central L-values from the relative trace formula (13.1) by pluging suitable test functions α into it. 13.1. Weyl’s law In this section, we take S = Σ∞ and set Iη = IΣ∞ ,η . Let n ∈ Iη . Recall that Πcus (n) denotes the set of all the irreducible cuspidal representations of GA occurring in L2cus and having a non zero K∞ K0 (n)-fixed vector (see 9.1.1.). For π ∈ Πcus (n), let νπ,∞ = (νπ,ι ) ∈ XΣ∞ denotes the spectral parameters νϕ,Σ∞ at Σ∞ of any non  zero vector ϕ ∈ Vπ (see 9.1.3.), and νπ,∞  its Euclidean norm, i.e., νπ,∞ 2 = ι∈Σ∞ |νπ,ι |2 . Having these notation, for any n ∈ Iη and x > 0, we consider the counting function  −1/2 Nηcus (n|x) = DF [Kfin : K0 (n)]−1 G(η)−1 Pη (π; K0 (n)), π∈Πcus (n) νπ,∞ 2 x

where Pη (π; K0 (n)) is the number defined in Lemma 2.6 (see also Corollary 2.16). The following bound of Nηcus (n|x), which is rather weak but uniform in the ideal n, will be used later in the proof of the main theorem. Theorem 13.1. For any  > 0, we have Nηcus (n|x)  (1 + x)dF /2+ ,

x>0

with the implied constant independent of n ∈ Iη . For an individual counting function, we have a more precise asymptotic formula. We exclude the case η = 1, for which an asymptotic formula with a remainder term is given in the next section. 101

102

13. ASYMPTOTIC FORMULAS

Theorem 13.2. Let n ∈ Iη and η = 1. Then, 1/2

Nηcus (n|x) ∼

2(1 + δ(n = oF )) DF L(1, η) dF /2 √ x , (4 π)dF Γ(dF /2 + 1)

x → +∞.

13.1.1. A class of test functions. For any T > 0, set  2  s −1 βT (s) = exp T , s ∈ C. 2 Then, βT ∈ B is evident. Given T = (Tι ) ∈ (0, +∞)Σ∞ and a family of polynomials {Qι (X)}ι∈Σ∞ , let α(T, {Qι }) be a function on XΣ∞ defined by  βTι (sι ) Qι (s2ι ), s ∈ XΣ∞ . α(T, {Qι }; s) = ι∈Σ∞

When we fix {Qι } and let T vary, we write αT in place of α(T, {Qι }). It is clear to see that αT belongs to AΣ∞ . We apply the formula (13.1) to αT and analyze the behavior of each terms other than Iηcus (n|αT ) as Tι getting small. 13.1.2. The term Jηhyp (n|α). Lemma 13.3. Let ι ∈ Σ∞ . Let Q(s) be an even polynomial with m = deg Q. Then, for any q > 0 and for any T0 > 0,    

c+i∞ c−i∞

  t 0  Ψ(0) (s ; δ [ ]) β (s ) Q(s ) dμ (s ) ι b 0 1 T ι ι ι ι  ι

T −m/2−1



1+

log(|b|ι + |b + 1|ι ) √ 4 T

−q

fι(0)



 1 √ ; t, b , 2 T0

b ∈ R× , t ∈ R× , T ∈ (0, T0 ).

(σ)

Here, fι (c; t, b) is defined in 11.4.3. Proof. Set u = (1 + t−2 )(b2 + t2 (b + 1)2 ). Then, it is easy to confirm the inequality u  (|b| + |b + 1|)2  1. Set   s + 1 s + 1 s + 2 −1 . B(s; u) = s Q(s) c0 (s) 2 F1 , ; ;u 4 4 2 Then, the integral to be estimated is

 c+i∞   2

s −1 (13.2)

T ds

B(s; u) u−(s+1)/4 exp 2 c−i∞

  

(c + iy)2 −1/4 −T /2 =u T dy

e

B(c + iy; u) exp −(c + iy) L(u) + 2 R with L(u) = 14 log u. By Cauchy’s theorem, the integrals above does not depend on the choice of c > 0. Now, take c = L(u) T −1 . From the proof of Lemma 4.2, we have the estimation B(s; u)  (1 + |s|2 )(m+1)/2 ,

u ∈ [1, +∞), Re(s) > 0.

13.1. WEYL’S LAW

103

Using this, we bound (13.2) by

  

  −L(u)2

−1 2

u−1/4 e−T /2 exp B L(u)T + iy; u exp(−y T /2) dy

2T R

  

(m+1)/2

−L(u)2

 2

L(u)T −1 + iy 2

 u−1/4 exp 1 + exp(−y T /2) dy

2T R

  

−L(u)2 m+1 u−1/4 exp (1 + L(u)2 T −2 )(m+1)/2

(1 + |y|) exp(−y 2 T /2) dy

2T R   −L(u)2 −1/4 2 −2 (m+1)/2 −m/2−1 u exp T (1 + L(u) T ) 2T with absolute implied constants. Since −x2 < −x + 1 for√any x ∈ R, the last exponential factor, which equals exp(−x2 ) with x = L(u)( 2T )−1 , is majorized by       −L(u) L(u)2 L(u) −q √ exp − exp  1+ √ 2T 2 T 2 T   L(u) −q −1/(8√T ) u . = 1+ √ 2 T

(by e−x  (1 + 2−1/2 x)−q for x > 0)

Combining all the estimations obtained above together, we have







c+i∞

(0)

Ψι c−i∞

 

   L(u) −q+m+1 −1/(8√T )−1/4 sι ; δb 0t 10 βT (sι ) Q(sι ) sι dsι

 T −m/2−1 1 + √ u 2 T

with the implied constant depending only on q > 0. From u  (|b| + |b + 1|)2  1, the majorant can be replaced with  −q+m+1 √ log(|b| + |b + 1|) √ T −m/2−1 1 + u−1/(8 T0 )−1/4 4 T √ √ (0) as long as T ∈ (0, T0 ). Since fι (1/(2 T0 ); t, b) = u−1/(8 T0 )−1/4 , we are done.  For b ∈ F × and ι ∈ Σ∞ , set [b]ι = |b|ι + |b + 1|ι . For any δ > 1, consider the set X(δ) = {b ∈ F × | sup [b]ι > δ }. ι∈Σ∞

Lemma 13.4. Let q > 0, T0 > 0 and δ > 1. Then, −q  log[b]ι  T q/2 , T ∈ (0, T0 )Σ∞ , b ∈ X(δ). 1+ √ 4 T ι∈Σι Proof. Let b ∈ X(δ); then supι∈Σ∞ [b]ι = [b]w with some w ∈ Σ∞ . We have  −q  −q  −q log[b]w log δ log[b]w √ √   , 1+ √ 4 T 4 T 4 T  −q log[b]ι  1 (ι = w), 1+ √ 4 T from which −q   log[b]ι  T q/2 (4 log δ)−q 1+ √ 4 T ι∈Σ∞ is obtained.



104

13. ASYMPTOTIC FORMULAS

Lemma 13.5. Let {Qι }ι∈Σ∞ be a family of polynomials and consider αT = −1/2 α(T, {Qι }). Let q > 0, δ > 1 and T0 > 0. Set c = 2−1 T0 . Then,

  

ˆ (0) n|αT ; δb t 0 1

Ψ

xη 0 1

0 1





 |T|q N (n|0, c; t, b) ,

|T| = sup Tι ∈ (0, T0 ), t ∈ A× , b ∈ X(δ) ι

with the implied constant independent of n. ˆ (0) (n|αT ; g) equals Proof. Since Ψ {

 ι∈Σ∞

1 2πi



c+i∞ c−i∞



(0)

Ψι (sι ; g) βTι (sι ) Qι (s2ι ) dμι (sι )} {



(0)

Φ1,v (gv )} {

(0)

Φ0,v (gv )}

v∈Σfin −S∪S(n)

v∈S(n)

for any g ∈ GA , from Corollaries 11.6 and 11.7 and Lemma 13.3, the estimation  −q

  1 xη 

log[b]ι

ˆ (0)  t 0 n|αT ; δb [ 0 1 ] 0 1  { } N(n|0, c; t, b), 1+ (

Ψ 4 |T| ι∈Σ∞ |T| ∈ (0, T0 ), t ∈ A× , b ∈ F × 

is inferred. Applying Lemma 13.4, we have the conclusion.

Lemma 13.6. There exists a constant δ0 > 1, depending only on F , such that sup (|b|ι + |b + 1|ι )  δ0

ι∈Σ∞

for any b ∈ oF − {0, −1}. Proof. For B > 0, consider the punctured cube Q∗ (B) = {x ∈ F∞ − {0}| |xι |  B} in F∞ . Then Q∗ (B) ∩ oF is compact and discrete in F∞ , thus is a finite set. Hence, to prove the claim, it suffices to show that Q∗ (1) ∩ oF = {±1}. Since Q∗ (1) ∩ oF is a finite set stable under multiplication, it consists of roots of  unity in F . Since F is totally real, we certainly have Q∗ (1) ∩ oF = {±1}. Lemma 13.7. Let a be an oF -ideal. Then, sup [b]ι  N(a)1/dF ,

ι∈Σ∞

b ∈ aoF .

Proof. Let xa be an idele such that aov = xa,v ov for any v ∈ Σfin . Let b ∈ a; then b ∈ xa,v ov for any v ∈ Σfin . Thus,   |b|v  |xa,v |v = N(a)−1 . v∈Σfin



By the product formula, obtained. Since [b]ι  |b|ι ,

v∈Σfin

v∈Σfin

|b|v =

( sup [b]ι )dF  ι∈Σ∞

 ι∈Σ∞



|b|−1 ι . Thus,

 ι∈Σ∞

|b|ι  N(a) is

|b|ι  N(a).

ι∈Σ∞

 Lemma 13.8. Let αT = α(T, {Qι }) be as in Lemma 13.5. Let q > 0, T0 ∈ (0, 1) and δ > 1. Then, √ T0 )−1/4

|Jηhyp (n|αT )|  |T|q N(n)−1/(8

,

|T| = sup Tι ∈ (0, T0 ), n ∈ Iη . ι

13.1. WEYL’S LAW

105

Proof. From the proof of Lemma 12.9, Jηhyp (n|αT ) is expressed as the absolutely convergent series-integral      ˆ (0) n|αT ; δb [ t 0 ] 1 xη η(tx∗η ) d× t. Ψ 01 0 1 A× b∈F −{0,−1}

Then, the conclusion follows from Lemmas 13.5, 11.16, 11.19 together with Lemmas 13.6 and 13.7.  13.1.3. The terms Iηeis (n|α) and Dη (n|α). To control the behavior of the Eisenstein term (10.14), we need a bound of the function Bχ,c (1/2, ν) defined by (2.27). Lemma 13.9. For any  > 0, we have |Bχ,c (1/2, ν)|  N(c) ,

ν ∈ iR, χ ∈ Ξ0 , c ⊂ oF .

Proof. Since |L(1 + ν, χ2v )|  (1 − qv−1 )−1 for ν ∈ iR, from (2.27), we have   1 + qv  qv1/2 + 1 1 2 1 + = |Bχ,c (1/2, ν)|  . 1/2 −1/2 1/2 qv 1 − qv−1 qv + qv −1 v∈S(c) v∈S(c) qv The desired estimation follows from this, since limx→∞

x+1 x (x−1)

= 0.



Lemma 13.10. For any  > 0, |C(n, Σ∞ ) Iηeis (n|αT )|  T−dF /2− N(n)−1+ ,

T ∈ (0, +∞)Σ∞ , n ∈ Iη ,

where we set T = inf ι Tι . We have limT →0 T dF /2 C(n, Σ∞ ) Iηeis (n|αT ) = 0 uniformly in n ∈ IS,η . Proof. For any χ ∈ Ξ0 , we have to estimate Lηχ,c (ν) = (DF N(c))ν/2 Bχη −1 ,c (1/2, −ν) φ(ν, χ) (see (10.8)) explicating the dependence on ν ∈ iR, χ ∈ Ξ0 and c, where φ(ν, χ) = L((1 + ν)/2, χη) L((1 − ν)/2; χ−1 η) L(1 − ν, χ−2 )−1 . Let  > 0. From Lemma 13.9, the first two factors of Lηχ,c (ν) with varying ν ∈ iR, χ  and c is O(N(c) /2 ). From (9.18), |φ(ν, χ)| is bounded by ι∈Σ∞ (1 + |ν + a(χι )|) .  Combining these, we obtain Lηχ,c (ν) = O( ι (1 + |ν + a(χι )|) N(c) /2 ) with the implied constant independent of (ν, χ, c). Let b(χ) = (bι (χ))ι∈Σ∞ be the vector of the lattice L0 defined in  2.4.4. From what we recalled in 2.4.4, noting that |C(n, Σ∞ )|  N(n)−1 and c|n 1  log(1 + N(n)), we have |C(n, Σ∞ ) Iηeis (n|αT )| 

 

N(n)−1

χ∈Ξ0 c|n



×

R

η |L1 χ,c (iy)Lχ−1 ,c (−iy)| exp(−

 N(n)−1 (

 c|n

 N(n)−1+



1)

  b∈L0

RΣ ∞

R



Tι ((y + 2bι (χ))2 + 1)/2) dy

ι

(1 + |y + 2bι |)2 exp(−T((y + 2bι )2 + 1)/2) dy

(1 + |yι |)2 exp(−T((yι2 + 1)/2)



dyι

ι

which is O(T−dF /2− N(n)−1+ ). We argue as in Lemma 9.9 using a subconvex1/4+θ ity bound L(1/2 + it, χ)  q(χ| |it (t ∈ R, χ ∈ Ξ0 ) with θ < 0 ([32]), by A)

106

13. ASYMPTOTIC FORMULAS

the same way as above, |C(n, Σ∞ ) Iηeis (n|αT )| is bounded by the integral J(T ) =    −δ −T (yι2 +1)/2 −1 e log(1 + N(n)) with some ι∈Σ∞ (1 + |yι |) ι dyι times N(n) RΣ ∞ δ > 0. By the dominated convergence theorem, we have limT →+0 T dF /2 J(T ) = 0 easily.  From (10.15), the estimation |C(n, Σ∞ ) Dη (n|αT )|  N(n)−1+ ,

(13.3)

T ∈ (0, +∞)Σ∞ , n ∈ Iη .

is evident. 13.1.4. The term Jηu (n|αT ). Lemma 13.11. For  ∈ {0, 1}, we have the asymptotic expansion of the form     

 2 1+s 3+s −1 Γ((s + 1)/4)2 1 eT (s −1)/2 sds ψ + ψ 2 2πi iR 8 Γ((s + 3)/4) 4 4  ∞ ∞   −1 am T m/2 + log T bm T m ∼ √ (4 log T ) T −1/2 1 + 4 π m=1 m=1 as T → +0. Proof. This by the asymptotic zΓ((1 + z)/4)2 /Γ(3 + ∞ is proved ∞expansions 2 −j −j z)/4) = 4 + j=1 pj z and ψ(z) = log z + j=1 qj z as |z| → +∞, |argz| < π ([36, p.18], [51]) with the aid of [54, Lemma 57].  Lemma 13.12. We have

1/2

Jηu (n|αT ) = Jηu¯ (n|αT ) ∼ (1 + δ(n = oF )) (−1)dF DF G(η) L(1, η)

1 √

dF 

4 π

Tι−1/2

ι

as |T| → 0+. Proof. Note that Jηu = Jηu¯ from η = 1. By definition (12.3), this follows from Lemma 13.11.  13.1.5. Proof of Theorems 13.1. Take Qι (X) to be the constant 1 for any ι. Let T0 > 0 and  > 0. For any T ∈ (0, +∞)Σ∞ with |T| ∈ (0, T0 ), consider the function αT ∈ AΣ∞ . By (13.1) applied to αT , combined with Lemmas 13.8, 13.10 and 13.12, we have   (1 + qv )−1 Iηcus (n|αT ) = O( Tι−1/2− ), |T| ∈ (0, T0 ) ι

v∈S(n)

uniformly for n ∈ Iη . Now, Theorem 13.1 follows from this combined with the following lemma. Lemma 13.13. For x > 1, set Tx = (2(1 + x)−1 )ι . Then,  (1 + qv )−1 Iηcus (n|αTx ), x > 0. Nηcus (n|x)  e2dF v∈S(n)

Proof. For any π ∈ Πcus (n), its spectral parameter νπ,∞ satisfies νπ,ι ∈ iR ∪ [−1, +1] for any ι ∈ Σ∞ . Let νπ,∞ 2  x. If νπ,ι ∈ iR, then 2 2 (νπ,ι − 1)/(1 + x)  (νπ,ι − 1)/x > (−νπ,∞ 2 − 1)/x  −1 − 1/x > −2

13.2. CONVERGENCE OF SPECTRAL MEASURES

107

2 since νπ,ι is a negative real number. If νπ,ι ∈ [−1, 1], then 2 − 1)/(1 + x)  −1/(1 + x) > −1. (νπ,ι 2 − 1)/(1 + x))  e−2 for any x > 1 and for any π ∈ Πcus (n) such that Thus, exp((νπ,ι 2 νπ,∞   x. From this, combined with the non-negativity of G(η)−1 Pη (π; K0 (n)) (Lemma 2.6) as well as Lemma 8.3, we have the conclusion . 

13.1.6. Proof of Theorems 13.2. For T > 0, set ˆ η (n|T ) = D−1/2 [Kfin : K0 (n)]−1 G(η)−1 P F





Pη (π; K0 (n)) exp T

π∈Πcus (n)

2  νπ,ι −1 2 ι∈Σ

 .

∞

ˆ η (n|T ) = (−1)dF C(n, Σ∞ ) Iη (n|αT ). From Lemmas 13.8, 13.10 and 13.12 Then, P cus and also from (13.3), we have the asymptotic 1/2

oF )) DF L(1, η) −dF /2 ˆ η (n|T ) ∼ 2(1 + δ(n = √ P T , (4 π)dF

T → +0,

from which Theorem 13.2 is inferred by the Tauberian theorem [57, Thereom 4.3 (p.192)]. We should remark that a subconvexity bound of L(1/2 + it, χ) in the taspect uniform with respect to χ ∈ Ξ0 is necessary in the proof of Lemma 13.10.  13.2. Convergence of spectral measures s/2

13.2.1. Let v ∈ S. The unramified principal series Iv (| |v ) is irreducible and unitarizable if and only if either s or −s belongs to  i [0, +∞) ∪ (0, 1), v ∈ Σ∞ , 0+ Xv = −1 −1 v ∈ Sfin . i [0, 2π(log qv ) ] ∪ (0, 1) ∪ (2πi(log qv ) + (0, 1)), 0 0+ ηv Let X0v denote the tempered part of X0+ v , i.e., Xv = Xv ∩ iR. The measure dλv 0+ on Xv defined by (1.2) is also given by

dληvv (iy) =

iy/2 iy/2 L(1/2, Iv (| |v )) L(1/2, Iv (| |v )

L(1, ηv )

 −2 ⎧   ⎪ ⎨ 1 Γ iy  dy, ⊗ ηv )   4π 2 × ⎪ log q v ⎩ |1 − qv−iy |2 dy, 4π

v ∈ Σ∞ , v ∈ Sfin .

Note that the measure dληvv is non-negative. Remark : The meausre dληvv really depends only on the sign ηv (v ) = ±1. When ηv (v ) = +1, it turns out that dλv (iy) is the Plancherel measure describing the decomposition of the Kv -invariant part of L2 (Hv \Gv ). While dληvv (iy) with ηv (v ) = −1 coincides with dPv (y), the spherical Plancherel measure of PGL(2, Fv ). 0+ Define X0+ over all v ∈ S and consider the product S to be the product of Xv measure  3/2 dληvv dληS = 4DF L(1, η) v∈S

X0+ S .

S(X0+ S ) ∞

on the space Let such that f is of class C

be the space of continuous functions f : X0+ S → C on the purely imaginary part X0S and satisfies

sup |Df (s)|(1 + s∞ 2 )m < +∞

s∈X0S

108

13. ASYMPTOTIC FORMULAS

for any constant coefficient differential operator D on X0S and for any m ∈ N. Here, s∞ denotes the projection of s ∈ XS to XΣ∞ and s∞  the hermitian norm of s∞ . It is easy to confirm that ληS is tempered, i.e., as a linear functional, it has a continuous extension to S(X0+ S ). Indeed, Lemma 13.14. There exist constants C1 > 0 and m1 ∈ N such that | ληS , f |  C1 sup |f (s)|(1 + s∞ 2 )m1 , s∈X0S

f ∈ S(X0+ S ).

Proof. This is immediate form the explicit nature of our ληS .



Lemma 13.15. Let α ∈ AS and set f = α|X0+ S . Then, if n = oF , 4Jηu (n|α) = (−1)S DF−1 G(η) ληS , f . Proof. It suffices to confirm the identity for functions α of the form α(s) = 1/2 η v∈S αv (sv ) with αv ∈ Av . Then, Ju (α) is equal to DF G(η) L(1, η) times the product of the following integrals.  1 Υηv (s) αv (s) dμv (s), v ∈ S, Jv (αv ) = 2πi Lv (0) v



where Υηv (s) is the v-th factor of ΥηS (s) and Lv (0) denotes the imaginary axis or the interval [−2πi(log qv )−1 , 2πi(log qv )−1 ] on the imaginary axis according to v ∈ Σ∞ or v ∈ Sfin , respectively. If v ∈ Sfin , then, from (6.2), Jv (αv ) =

1 2π



2π(log qv )−1

{Υηv (iy) − Υηv (−iy)} 2−1 qv1/2 log qv (qviy/2 − qv−iy/2 ) αv (iy) dy.

0

−(s+1)/2 −1

Since Υηv (s) = (1 − qv )−1 (1 − ηv (v )qv η η that Υv (iy) − Υv (−iy) equals (s+1)/2

)

, a direct computation shows

qv−1/2 (qviy/2 − qv−iy/2 ) (1 − ηv (v )qv−1 ) × (1 − qv−(1+iy)/2 )−1 (1 − qv−(1−iy)/2 )−1 (1 − ηv (v )qv−(1+iy)/2 )−1 (1 − ηv (v )qv−(1−iy)/2 )−1 .

Thus, Jv (αv ) = − ληvv , αv |X0+ v . In the same way, we can show that this formula is also true for v ∈ Σ∞ .  13.2.2. For any n ∈ IS,η , the linear functional ληS (n), f 



= 2 DF G(η)−1 [Kfin : K0 (n)]−1 1/2

Pη (π; K0 (n)) f (νπ,S ),

f ∈ Cc0 (X0+ S )

π∈Πcus (n)

is a positive Radon measure. As shown by the following lemma, it has a continuous extension to S(X0+ S ). Lemma 13.16. There exists a constant C0 > 0 and m0 ∈ N independent of n such that | ληS (n), f |  C0 sup |f (s)|(1 + s∞ 2 )m0 , s∈X0+ S

Proof. For m ∈ N and n ∈ IS,η , set −1/2

Q(m) (n) = DF

[Kfin : K0 (n)]−1 G(η)−1

 π∈Πcus (n)

f ∈ S(X0+ S ), n ∈ IS,η .

Pη (π; K0 (n)) (1 + νπ,∞ 2 )−m .

13.2. CONVERGENCE OF SPECTRAL MEASURES

109

Since | ληS (n), f | Q(m0 ) (n) { sup |f (s) (1 + s∞ 2 )m0 }, s∈X0+ S

it suffices to show that there exists m0 ∈ N such that supn∈IS,η Q(m0 ) (n) < +∞. We start with the expression of Q(m0 ) (n) as a Stieltjes integral:  +∞ Q(m0 ) (n) = (1 + x)−m0 dNηcus (n|x). 0 −m0

We have limx→+∞ (1 + x) parts, and by Theorem 13.1,

Nηcus (n|x)

= 0 if m0 > dF /2. Then, by integration by

+ Q(m0 ) (n) = [(1 + x)−m0 Nηcus (n|x)]+∞ 0



+∞

Nηcus (n|x) 0



+∞

= m0

Nηcus (n|x) (1 + x)−(m0 +1) dx 

0



m0 dx (1 + x)m0 +1

+∞

(1 + x)dF /2+ −m0 −1 dx

0

with the implied constant independent of n. The last integral is convergent if  m0 > dF /2 + . This completes the proof. 13.2.3. The following is the main result of this article, which asserts that the globally defined measure ληS (n) with growing level n converges weakly to the locally defined measure ληS in the limit. Theorem 13.17. Let S be a finite set of places of F containing Σ∞ and η a unitary idele-class character of order 2 such that ηv (−1) = 1 for any v ∈ Σ∞ . Let IS,η denote the set of all the square free oF -ideals n such that S(n) is disjoint from S(fη ) ∪ S and η˜(n) = 1. Then, as N(n) → +∞ with n ∈ IS,η , the measure ληS (n) converges to ληS vaguely on the space S(X0+ S ), i.e., ληS (n), f → ληS , f

for any

f ∈ S(X0+ S ).

Proof : (cf. [8, Theorem 9.1]) Let C2 = max(C0 , C1 ) and m2 = max(m0 , m1 ), where C0 , m0 , C1 and m1 are constants in Lemmas 13.14 and 13.16. Let f ∈ S(X0+ S ) and take an arbitrary  > 0. Then, by [8, Lemma 9.3] and by the Weierstrass approximation theorem, we can find T = (Tι ) ∈ (0, +∞)Σ∞ and a family of polynomials {Qv }v∈S such that sup |f (s) − α(s)|(1 + s∞ 2 )m2
0, ληS (n), f0 = ληS , f0 + O(N(n)−1+ ),

n ∈ IS,η , n = oF .

110

13. ASYMPTOTIC FORMULAS

Proof. An analogue of Lemma 13.9 and the estimate (13.3) for Iηeis (n|α) and D (n|α) with S larger than Σ∞ is true obviously. For varying n ∈ IS,η , the estimation Jηhyp (n|α) = O(N(n)−q ) for any q > 0 is an easy corollary of the proof of Lemma 12.9. Thus, by the formula (13.1), η

C(n, S) Iηcus (n|α) − 2Jηu (n|α) = O(N(n)−1+ ), From Lemma 13.15, the left-hand side is (−1) This completes the proof.

S

DF−1

n ∈ IS,η .

G(η) { ληS (n), f0 − ληS , f0 }. 

By this lemma, there exists δ > 0 such that | ληS (n) − ληS , f0 |  /3 for any n ∈ IS,η such that N(n) > δ. From Lemma 13.16, | ληS (n), f − f0 |  C2 sup |f (s) − α(s)|(1 + s∞ 2 )m2  C2 s∈X0+ S

 = /3 3C2

for any n ∈ IS,η (1). Similarly, from Lemma 13.14, we have | ληS , f − f0 |  /3. Therefore, | ληS (n) − ληS , f |  | ληS (n), f − f0 | + | ληS (n) − ληS , f0 | + | ληS , f − f0 |  /3 + /3 + /3 =  for any n ∈ IS,η such that N(n) > δ. This completes the proof.



Corollary 13.19. Let J be any Borel subset of X0+ S whose boundary has measure zero with respect to ληS . Then, as N(n) → +∞ with n ∈ IS,η , we have   dληS (n) −→ dληS . J

J

Proof. This follows from Theorem 13.17 by [2, Proposition 22, Chap.IV, §5, n◦ 12].  Proof of Theorem 1.1 : Theorem 1.1 is obtained from Theorem 13.17. Indeed, the set I∗S,η defined in §1 is contained in IS,η . If n ∈ I∗S,η , then Pη (π; K0 (n)) is explicated by Corollary 2.7, from which it is confirmed that ληS (n) in Theorem 13.17 is identical  to the one defined in §1 when n ∈ I∗S,η . Proof of Corollary 1.2 : We may suppose Jv are bounded. Let J be the set of −ν /2 −ν /2 ν ∈ X0S such that (1 − νι2 )/4 ∈ Jι for any ι ∈ Σ∞ and qv v + qv v ∈ Jv η for any v ∈ Sfin . Then, vol(J; dλS ) = 0 is evident. Let δ > 0. Then, from Corollary 13.19, there exists an n ∈ I∗S,η such that N(n) > δ and |vol(J; dληS (n)) − vol(J; dληS )| < 2−1 vol(J; dληS ), in particular vol(J; dληS (n)) > 0. Therefore, there exists some π ∈ Πcus (n) such that fπ = n and L(1/2, π) L(1/2, π ⊗ η) = 0, This completes the proof. 

CHAPTER 14

An error term estimate in the Weyl type asymptotic law The Weyl group E = {±1}Σ∞ of (G∞ , H∞ ) acts on the space XΣ∞ by the sign-change of coordinates, i.e., if ε = (ει ) ∈ E and s = (sι ) ∈ XΣ∞ , then ε(s) is defined to be (ει sι ).For any function f (s) on XΣ∞ , its E-symmetrization f˜(s) is defined to be 2−dF ε∈E f (ε(s)). With this notation, we state our main result of this section as follows. Theorem 14.1. Let n be a square free oF -ideal and η an idele class charactern F × satisfying (2.18), (2.19) and (2.20). Let D be a compact subset of i−1 X0Σ∞ with smooth boundary which is regular (i.e., νι = 0 for all ι ∈ Σ∞ ) and symmetric (i.e., ε D = D for any ε ∈ E). For t > 0, set D(t) = t D. Then, for any  > 0, (14.1) 



    f˜0 (K) dK + O tdF −1 (log t)3 + O tdF (1+4θ)+2

Pη (π; K0 (n)) = 2 D(t)

π∈Πcus (n) νπ,Σ∞ ∈i D(t)

as t → +∞. Here, Pη (π; K0 (n)) is given by (2.21), θ is the constant in (2.6), dK the Euclidean measure of i−1 XΣ∞ and (14.2)



f0 (x) = ×

dF

 i xι Γ((i xι + 1)/4)2 4 Γ((i xι + 3)/4)2 ι∈Σ∞     " i xι + 1 ixι + 3 R(η)  η C (F, n) + ψ +ψ , 2 4 4

1 4π 

DF G(η) [Kfin : K0 (n)] (1 + δn,oF )

ι∈Σ∞

where dF R(η) (CEuler + 2 log 2 − log π) 2 with R(η) = Ress=1 L(s, η) and C(η) = CTs=1 L(s, η).  Cη (F, n) = C(η) + R(η) log(DF N(n)1/2 ) +

The proof of this theorem is given in the last part of this section. We fix an enumeration ιj (1  j  dF ) of the set Σ∞ once and for all to identify XΣ∞ with CdF ; thus i−1 X0Σ∞ is identified with the Euclidean space RdF . Our argument is a hybrid of [37, p.272–273] and [9, §6–§8]. Following [37], we consider a test function (m) αK,Δ ∈ AΣ∞ by setting   s − K 2 (m) −1 (14.3) eK,Δ (s) = exp − s) and αK,Δ (s) = (1 − s 2 )m e K,Δ (i Δ2 111

112

14. AN ERROR TERM ESTIMATE IN THE WEYL TYPE ASYMPTOTIC LAW

for m ∈ N, K = (Kj ) ∈ RdF and Δ (> 1). Here s 2 =

 j

s2j for any point (m)

s ∈ XΣ∞ = CdF . We analyze the formula (13.1) with S = Σ∞ , η = 1 and α = αK,Δ (m)

to estimate Iηcus (n|αK,Δ ) for large K and Δ such that Δ  log(2 + K). The (m) growth order of the unipotent term Jη (n|α ) is ΔdF f˜0 (K) (Lemma 14.9). It u

K,Δ

(m)

(m)

turns out that the order of magnitude of the terms Jηhyp (n|αK,Δ ) and Dη (n|αK,Δ ) can be made smaller than arbitrary negative power (1 + K2 )−δ (Lemma 14.10); thus, they are negligible compared with the unipotent term. By the bound (2.6), (m) the Eisenstein term Iηeis (n|αK,Δ ) can be estimated as O(Δ (1 + K2 )2θ1 +m+ ) if m (m)

is large enough (Lemma 14.11). From the asymptotic formula of Iηcus (n|αK,Δ ) with large m, together with a bound of the non-tempered contribution (Lemma 14.12), “short interval estimation” for the counting function of Pη (π; K0 (n)) for tempered cuspidal π’s is inferred (Lemma 14.14). With the aid of this, we can lower m down to m = 0 by a simple argument (Lemma 14.15). Then, theorem 14.1 is inferred (0) from the asymptotic formula of the tempered part of Iηcus (n|αK,Δ ) in a way similar to [9, Theorem 8.8]. 14.1. Preliminary Set d = dF . For any compact set D in Rd , let Bρ (D) denote the ρ-neighborhood of D: Bρ (D) = {x ∈ Rd | d(x, D)   }. Let μ be a Radon measure on Rd . We suppose that there exists some q > 0 such that |μ|(Bρ (0))  (1 + ρ)q , ρ > 0. The aim of this subsection is to study a growth of the integral  (m) eK,Δ (x) dμm (x), Iμ (K, Δ) = Rd

defined for m ∈ N, K ∈ R and Δ > 1. Here, dμm (x) = (1 + x2 )m dμ(x). d

Lemma 14.2. For any  > 0, |Iμ(m) (K, Δ)|

 (1 + Δ)

2m+q

 (q+1)/2+m+

K2 , 1+ Δ2

K ∈ Rd , Δ > 1.

Proof. Set p = (q + 1)/2 + m + . We have

−p x − K2 d|μm |(x) Δ2 Rd p 

−p

x2 K2 1 + 1+ d|μm |(x). Δ2 Δ2 Rd 

|Iμ(m) (K, Δ)|



1+

By dividing Rd to the shells BlΔ (0) − B(l−1)Δ (0) with l ∈ N∗ , the last integral is estimated as   −p 1 + l2 |μm |(BΔl (0))  (1 + l2 )−p · (1 + Δl)2m+q l

l

 (1 + Δ)2m+q



(1 + l)−2p+2m+q .

l

Since −2p + 2m + q < −1, the last series is convergent.



14.1. PRELIMINARY

113

Lemma 14.3. Given δ > 0, there exists B > 0 such that (1 + K2 )−m Iμ

(m)

  (0) (K, Δ) − Iμ (K, Δ) = O Δ (1 + K2 )−1/2 log(2 + K) μ(QB (K, Δ)) + O((1 + K2 )−δ )

as long as 1 < Δ  log(2 + K), where QB (K, Δ) = {x ∈ Rd | x − K  Δ(B log(2 + K)1/2 }.

(14.4)

Proof. By Taylor’s theorem, 

1 + x2 1 + K2

m

  − 1 = O Δ (1 + K2 )−1/2 log(2 + K)

if x ∈ QB (K, Δ). From this, (1 + K2 )−m



 eK,Δ (x) dμm (x) − QB (K,Δ)

eK,Δ (x) dμ(x) QB (K,Δ)

  = O Δ (1 + K2 )−1/2 log(2 + K) μ(QB (K, Δ)) .

(14.5)

  2  (2 + K)−3B/4 . This, combined Let x ∈ Rd − QB (K, Δ). Then, exp − 3x−K 4Δ2 with the factorization exp(−x) = exp(−3x/4) × exp(−x/4), yields     x − K2 x − K2 2 −3B/8 exp − exp −  (1 + K ) , Δ2 4Δ2 K ∈ Rd , Δ > 1, x ∈ Rd − QB (K, Δ). Taking integral and using Lemma 14.2, we obtain  Rd −QB (K,Δ)

eK,Δ (x) d|μm |(x)  (1 + K2 )−3B/8 × (1 + 2Δ)

2m+q

 (q+1)/2+m+1 K2 . 1+ 4Δ2

Since Δ  log(2 + K), this has a smaller magnitude of growth than (1 + K2 )−δ if B is large enough.  Lemma 14.4. Let D ⊂ Rd be a compact set. For any  > 0, 

 eK,Δ (x) dK  Δd

D



eK,2Δ (x) dK, B (∂D)



eK,Δ (x) dK  Δ

d

Rd −D

Δ > 1, x ∈ Rd − D,

eK,2Δ (x) dK, B (∂D)

Δ > 1, x ∈ D.

114

14. AN ERROR TERM ESTIMATE IN THE WEYL TYPE ASYMPTOTIC LAW

Proof. (cf. [9, Lemma 8.6]) Since K − x  d(x, ∂D) for any (K, x) ∈ D × (Rd − D),   eK,Δ (x) dK  eK,Δ (x) dK D K−xd(x,∂D)  = exp(−K2 /Δ2 ) dK 

Kd(x,∂D) +∞



exp(−ρ2 /Δ2 ) ρd−1 dρ d(x,∂D)

  d(x, ∂D)2  exp − exp(−ρ2 /(2Δ2 )) ρd−1 dρ 2Δ2 R   d(x, ∂D)2 d  Δ exp − . 2Δ2 For x ∈ Rd − D, choose x0 ∈ ∂D such that d(x, ∂D) = x − x0 ; then B (x0 ) ⊂ B (∂D). Thus,  eK,2Δ (x) dK B (∂D)   exp(−x − K2 /(2Δ)2 ) dK B (x0 )

  x − x0 + ω2 exp − dω (2Δ)2 ω∈B (0)       ω2 x − x0 2 x − x0 2 exp −  exp − dω  exp − . 2Δ2 2Δ2 2Δ2 B (0) 



Thus, we obtain the first estimate. The second estimate is similar.



Lemma 14.5. Let D be a compact subset of Rd and ∂D its boundary. Then, for any  > 0,    (0) (0) d/2 d Iμ (K, Δ) dK = π Δ μ(D) + O Δd I|μ| (K, 2Δ) dK D

B (∂D)

with the implied constant independent of Δ > 1. Proof. We closely follow the argument of [9, Theorem 8.5]. Set D = Rd − D. Then,         Iμ(0) (K, Δ) dK = − + eK,Δ (x) dμ(x) dK. K∈Rd

D

x∈D

K∈D 

x∈D

K∈D

x∈D 

The first integral is evaluated to be π d/2 Δd μ(D). With the aid of Lemma 14.4, the second integral is bounded by    (0) { eK,2Δ (x) dK} d|μ|(x)  Δd I|μ| (K, 2Δ) dK. x∈D

B (∂D)

The third integral is estimated in the same way.

B (∂D)



14.1. PRELIMINARY

115

14.1.1. Let f : Rd −→ R be a C 1 -function satisfying the estimations on Rd |f (x)|  (1 + x2 )m log(2 + x),

∂

2 m

∂xj f (x)  (1 + x ) h(x) log(2 + x),

(14.6) (14.7)

(1  j  d)

with some m ∈ N. Here, h(x) =

d 

(1 + |xj |)−1 ,

x = (xj ) ∈ Rd .

j=1 (0)

(0)

Set dμ(x) = f (x) dx and consider Iμ (K, Δ), which is denoted by If (K, Δ). Since μ(Bρ (0))  (1 + ρ)2m+d+1 , Lemma 14.2 gives us a trivial estimate : (0)

|If (K, Δ)|  (1 + Δ)2m+d+1 (1 + K2 /Δ2 )d+m+1 .

(14.8)

This is improved as follows. Lemma 14.6. If Δ = O(log(2 + K)) and 1 + K  minj |Kj |, then (0)

If (K, Δ) = π d/2 Δd f (K) + O(Δd+1 (1 + K2 )m h(K) log(2 + K)3/2 ). Proof. Let B > 0 be a large constant, whose magnitude should be specified later. For any K ∈ Rd , set QB (K, Δ) = {x ∈ Rd | x − K  Δ(B log(2 + K))1/2 }. Let I ∗ (K, Δ) and R(K, Δ) be the integrals of f (x) exp −x − K2 /Δ2 over QB (K, Δ) and over Rd − QB (K, Δ), respectively. Thus, If (K, Δ) is a sum of I ∗ (K, Δ) and R(K, Δ), which we estimate separately. Since exp(−3x − K2 /(4Δ2 ))  (1 + K)−3B/4 for x ∈ Rd − QB (K, Δ), we estimate      x − K2 3x − K2 |f (x)| exp − exp − dx |R(K, Δ)| = 4Δ2 4Δ2 Rd −QB (K,Δ)    x − K2 −3B/4  (1 + K) |f (x)| exp − dx 4Δ2 Rd  (1 + K)−3B/4 Δ2m+d+1 (1 + K2 /Δ2 )d+m+1 by (14.8). To estimate the integral I ∗ (K, Δ), we write it as a sum of the following three integrals.    x − K2 exp − I0∗ (K, Δ) = f (K) dx, Δ2 Rd    x − K2 exp − J1 (K, Δ) = −f (K) dx, Δ2 Rd −QB (K,Δ)    x − K2 J2 (K, Δ) = {f (x) − f (K)} exp − dx. Δ2 QB (K,Δ) The first integral is easily evaluated as π d/2 Δd f (K). The second one is estimated in the same way as for R(K, Δ):    x − K2 −3B/4 exp − dx |J1 (K, Δ)|  |f (K)| (1 + K) 4Δ2 Rd  Δd (1 + K)2m+ −3B/4 .

116

14. AN ERROR TERM ESTIMATE IN THE WEYL TYPE ASYMPTOTIC LAW

By the mean value theorem, the integral J2 (K, Δ) is written as    x − K2 exp − {f (L) − f (K)} dx Δ2 QB (K,Δ) with some L ∈ QB (K, Δ). By the mean value theorem and by (14.7),

) *

∂f

|x ∈ Q |f (L) − f (K)|  sup ∂x (x) (K, Δ), 1  j  d L − K

B j  (1 + K2 )m h(K) log(2 + K) · Δ(B log(2 + K))1/2 Thus, J2 (K, Δ) = O(Δd+1 (1 + K2 )m h(K) log(2 + K)3/2 ). Since Δ = O(log(2 + K)), by choosing the constant B greater than 2/3, the growing order of terms R(K, Δ) and J1 (K, Δ) can be made smaller than that of J2 (K, Δ). This completes the proof.  14.2. A key lemma We first note that, from (10.1), (12.1) and (12.2), it turns out the function f0 (x) (defined by (14.2)) coincides with  d −1 F 1/2 C(n, Σ∞ )−1 DF G(η) (1 + δ(n = oF )) 2πi ⎛ ⎞ dF  ×⎝ xj ⎠ ΥηΣ∞ (ix) 2−1 {CηΣ∞ ,u (n|ix) + CηΣ∞ ,¯u (n|ix)}. j=1

Now we state a key lemma. Lemma 14.7. Let n be a square-free oF -ideal. Let m ∈ N be such that m  2θ, where θ is the constant in (2.6). Then, there exist B1 , B2 > 0 such that, for any given  > 0,   (m) (1 + K2 )−m Iηcus (n|αK,Δ ) = 2π dF /2 ΔdF f˜0 (K) + O ΔdF +1 h(K) log3/2 (2 + K)   + O ΔdF (1 + K2 )2θdF +

holds for K ∈ RdF and Δ > 1 with B1 < Δ−1 log(2 + K) < B2 . The proof of this lemma is divided to several steps. First, from the formula (m) (13.1) applied to αK,Δ , (14.9)

Iηcus (n|αK,Δ ) = C(n, Σ∞ )−1 { Jηu (n|αK,Δ ) + Jηu¯ (n|αK,Δ ) + Jηhyp (n|αK,Δ )} (m)

(m)

(m)

(m)

(m)

(m)

− Iηeis (n|αK,Δ ) − Dη (n|αK,Δ ). The terms on the right-hand side should be estimated in the succeeding paragraphs. (m)

(m)

14.2.1. The term Jηu (n|αK,Δ ) + Jηu¯ (n|αK,Δ ). From (12.3), a simple manipulation shows the equality (m)

(m)

(0)

Jηu (n|αK,Δ ) + Jηu¯ (n|αK,Δ ) = 2 Ifm (K, Δ) with fm (x) = (1 + x2 )m f˜0 (x).

14.2. A KEY LEMMA

117

Lemma 14.8. We have a bound



∂ f˜

0

sup (x)  h(x) log(1 + x)

1idF ∂xi

|f˜0 (x)|  log(2 + x),

on the whole space RdF . If K stays in a cone contained in Rd+F , then f˜0 (K) equals  dF 1 DF G(η)[Kfin : K0 (n)](1 + δn,oF ) (Cη (F, n) + dF R(η) log K) 4π   + O (1 + K)−1 log(2 + K) . Proof. This follows from the asymptotic expansions recalled in the proof of Lemma 13.11.  Lemma 14.9. For K ∈ RdF , Δ > 1 with Δ = O(log(2 + K)), (m)

(m)

Jηu (n|αK,Δ ) + Jηu¯ (n|αK,Δ ) = 2 π dF /2 ΔdF (1 + K2 )m × {f˜0 (K) + O(Δ h(K) log(2 + K)3/2 )}. 

Proof. This follows from Lemma 14.8 and 14.6. (m)

14.2.2. The term Jηhyp (n|αK,Δ ). Lemma 14.10. For any B > 0, set YB = {(K, Δ) ∈ RdF × R+ | Δ  B log(2 + K) }. Then, if B > 0 is large enough, the following bound holds with C = 8−1 dF B log δ0 , where δ0 > 1 is an absolute constant. (m)

|Jηhyp (n|αK,Δ )|  (1 + K2 )m+dF /2−C Δ4m+3dF ,

(K, Δ) ∈ YB .

× Proof. For each t = (tj ), b = (bj ) ∈ F∞ , we set u = (uj ) with uj = (1 + 2 2 + tj (b + 1) ). Then, set   d d   sj + 1 sj + 1 sj + 2 −1 2 m , ; ; uj sj } sj c0 (sj ) 2 F1 Bm (s; u) = {1 − . 4 4 2 j=1 j=1

2 t−2 j )(bj

We proceed the same way as in the proof of Lemma 13.3. First, the estimation |Bm (s; u)|  {1 +

d 

|sj |2 }m+1/2 ,

u ∈ [1, +∞)d , Re(s) ∈ (0, +∞)d

j=1

is obtained. For simplicity, we write L(c) in stead of LΣ∞ (c). Then, from this by choosing the contour L(c) appropriately,

⎞ ⎛



d d d 2   

(sj − iKj ) ⎠ −(sj +1)/4

⎝ (14.10) B (s; u) u exp ds m j j

2 Δ

L(c)

j=1 j=1 j=1 is majorized by (14.11)

⎞m+d/2 ⎞ ⎛ d d 2   Δ −Δ −1/4 ⎝ { uj } 1+ L(uj )2 ⎠ exp ⎝ L(uj )2 ⎠ 4 4 j=1 j=1 j=1    2 y − K ×{ (1 + y2 )m+d/2 exp − dy}. Δ2 Rd d 

⎛

4

118

14. AN ERROR TERM ESTIMATE IN THE WEYL TYPE ASYMPTOTIC LAW

Here, L(uj ) = 4−1 log uj . The integral in (14.11) is estimated by (1 + K2 )m+d/2 Δ2m+2d easily. This, conbined with the inequality

⎞ ⎛ d d 2  Δ4  Δ 1+ L(uj )2  (1 + Δ2 ) ⎝1 + L(uj )2 ⎠ , 4 j=1 4 j=1

(14.11) is further majorized by {



d 

−1/4 uj }

j=1

d Δ2  1+ L(uj )2 4 j=1



m+d/2 exp

d −Δ2  L(uj )2 4 j=1

 Δ4m+3d (1 + K2 )m+d/2 .

√ Using (1 + x)m+d/2 exp(−x)  exp(− x/2) (x > 0), this is bounded by ⎞ ⎛ d d   −Δ −1/4 { { uj } exp ⎝ L(uj )2 }1/2 ⎠ Δ4m+3d (1 + K2 )m+d/2 . 4 j=1 j=1 If Δ > Δ0 , then the exponential factor is majorized by ⎛ ⎞   d d   −Δ Δ −Δ /16 exp ⎝ L(uj )⎠  uj 0 exp − L(uj ) 2 j=1 4 j=1 

d 

−Δ /16 uj 0

j=1 −1

since L(uj )  2 bounded by 4m+3d

Δ

  Δ exp − log(|bj | + |bj + 1|) , 8

log(|bj | + |bj + 1|). Thus, under the condition Δ > Δ0 , (14.10) is

(1 + K )

2 m+d/2

{

d 

−1/4−Δ0 /16 uj

j=1



 Δ exp − log(|bj | + |bj + 1|) }. 8

If b = (bj ) comes from a point of the ideal n, then |bj | + |bj + 1|  δ0 > 1 from Lemma 13.6. Combining this with the inequality Δ  B log(2 + K), the last exponential factor is bounded by   Δ log δ0 exp −  (1 + K)−(B log δ0 )/8 . 8 −1/4−Δ0 /16

(0)

We also note that fιj (Δ0 /4; tj , bj ) = uj have shown that (14.10) is majorized by 4m+3d

Δ

(1 + K )

2 m+d/2−C

d  j=1

. Summing up the argument, we 

fι(0) j

Δ0 ; tj , bj 4



× F∞

∩ n. Here, C = (dB log δ0 )/8. Then, by the same as long as Δ > Δ0 and b ∈ way as in Lemma 13.5, the estimation       (m) (0) t 0 Ψ (n|s; δb [ 0 1 ]) αK,Δ (s) dμΣ∞ (s) Δ4m+3d (1 + K2 )m+d/2−C N(n|0, c; t, b),   L(c)  t ∈ A× , b ∈ n − {0, −1}, Δ0 < Δ

14.3. CONTRIBUTION FROM THE NON-TEMPERD SPECTRUM

119

with c = Δ0 /8 and C = (dB log δ0 )/8 is obtained. We complete the proof by the same way as in Lemma 13.8.  (m)

14.2.3. The term Iηeis (n|αK,Δ ). Lemma 14.11. Let θ be the constant in (2.6). If m > −2θ, then for any sufficiently small  > 0 d 

(m)

|Iηeis (n|αK,Δ )|  Δd (1 + K2 )m

(1 + |Kj |2 )2θ+ ,

K ∈ RdF

j=1

as long as Δ  log(2 + K). Proof. Let c be a divisor of n. From the proof of Lemma 9.9, the estimation (2.6) yields |Lηχ,c (ν)| + |L1χ,c (ν)| 

d 

(1 + |ν + a(χι )|2 )θ+ ,

ν ∈ iR, χ ∈ Ξ0

j=1

for any  > 0. For χ ∈ Ξ0 , let b(χ) = (bj (χ)) be the vector defined in 2.4.4; it belongs to the lattice L0 in a dF − 1 dimensional vector space V ⊂ Rd consisting (m) of trace zero elements. Thus, |Iηeis (n|αK,Δ )| is majorized by   d   dF F      |y + bj (χ) − Kj |2   1 η 2 m Lχ,c (iy) Lχ−1 ,c (−iy) (1 + (y + bj (χ)) ) exp − dy   2   Δ R j=1

χ∈Ξ0



dF  

b∈L0

R j=1





(1 + RdF

(1 + |y + bj |2 )2θ+ (1 +

dF  j=1

j=1

dF 

|y + bj |2 )

j=1

yj2 )m

dF  m

exp −

j=1

|y + bj − Kj |2 Δ2

 dy



|yj − Kj |2 dy1 · · · dydF (1 + |yj |2 )2θ+ exp − Δ2 j=1 dF 

From Lemma 14.6, the last integral is O(ΔdF (1 + K2 )m  > 0 is small enough so that m + 2θ +  > 0.

dF

j=1 (1

+ |Kj |2 )2θ+ ) if 

14.2.4. Proof of Lemma 14.7. We start with (14.9). The “peripheral term” (m) Dη (n|αK,Δ ) is easily bounded by O(exp(−K2 /Δ2 ). The remaining terms on the right-side are estimated by Lemmas 14.10, 14.9 and 14.11. We should remark that (m) (m) (m) the term Jηhyp (n|αK,Δ ) is absorbed to the “error term” of Jηu (n|αK,Δ ) + Jηu¯ (n|αK,Δ ) by choosing the constant B large enough; this is possible, for δ0 > 1. 14.3. Contribution from the non-temperd spectrum For any subset J ⊂ Σ∞ = {1, . . . , dF }, set J  = Σ∞ − J and  X0J = {ν ∈ X0+ Σ∞ | νπ,ι ∈ iR for all ι ∈ J, and νπ,ι ∈ (0, 1) for all ι ∈ J },

Πcus (n; J) = {π ∈ Πcus (n) |νπ,Σ∞ ∈ X0J }. 0 Then, X0+ Σ∞ is a disjoint union of all XJ ’s. Note that Πcus (n; Σ∞ ) is the tempered dF part of Πcus (n). For each x ∈ C , let xJ be the natural projection of x to CJ .

120

14. AN ERROR TERM ESTIMATE IN THE WEYL TYPE ASYMPTOTIC LAW

Since the number of π ∈ Πcus (n; J) with νπ,J lying in a compact set is finite, we have a measure μ ˆJ on RJ such that  μ ˆJ (D) = Pη (π; K0 (n)) π∈Πcus (n;J) νπ,J ∈iD

for any compact subset D ⊂ RJ . From Theorem 13.1, |ˆ μJ |(RJ ∩ Bρ (0))  Nηcus (n|ρ2 )  (1 + ρ)dF +1 , Lemma 14.12. Set



(m)

Icus (αK,Δ ; J) =

ρ > 0.

(m)

Pη (π; K0 (n)) αK,Δ (νπ,Σ∞ ).

π∈Πcus (n;J)

Then, |Icus (αK,Δ ; J)|  exp(−KJ  2 /Δ2 ) Δ2m+dF +1 (1 + Δ−2 KJ 2 )(dF +2)/2+m+1 , (m)

K ∈ RdF , Δ > 1. Proof. Let ν ∈ X0J . Then, the first factor on the right-hand side of the equality   νJ  2 − KJ  2 |eK,Δ (i−1 ν)| = exp eKJ ,Δ (i−1 νJ ) Δ2 is bounded from below and above by exp(−KJ  2 /Δ2 ) since |νπ,ι |  1 for all ι ∈ J  . From this, we easily have

(m)

(m) |Icus (αK,Δ ; J)|  exp(−KJ  2 /Δ2 ) IμˆJ (KJ , Δ) . The required estimation follows from this combined with the estimation for (m) IμˆJ (KJ , Δ) provided by Lemma 14.2.  Corollary 14.13. There exist B1 , B2 > 0 such that, for any given  > 0 and q > 0,   (m) (1 + K2 )−m IμˆΣ (K, Δ) = 2π dF /2 ΔdF f˜0 (K) + O ΔdF +1 h(K) log3/2 (2 + K) ∞   + O ΔdF (1 + K2 )2dF θ+

⎛ ⎞  + O ⎝Δ2m+dF +1 h(KJ  )q (1 + KJ )(dF +3)/2 ⎠ J=Σ∞

holds for K ∈ RdF and Δ > 1 with B1 < Δ−1 log(2 + K) < B2 . (m)

Proof. From definition IμˆΣ (K, Δ) equals ∞  (m) (m) η (14.12) Icus (αK,Δ ; J). Icus (n|αK,Δ ) − J=Σ∞

The first term is given by Lemma 14.7. The remaining terms amount at most to ⎞ ⎛  O ⎝Δ2m+dF +1 h(KJ  )q (1 + KJ )(dF +1)/2+m+1 ⎠ J=Σ∞

by Lemma 14.12.



14.4. CONCLUSION

121

14.4. Conclusion We follow [37, p.272–273], where an asymptotic formula similar to Theorem 14.1 is deduced from Kuznetsov’s formula when dF = 1. First, we have a “short interval estimate” : Lemma 14.14. For any B > 0,

  μ ˆΣ∞ (QB (K, Δ)) = O ΔdF log(2 + K) .

as long as Δ  log(2 + K) and 1 + K  minj |Kj |. Here, QB (K, Δ) is defined by (14.4). Proof. Fix m ∈ N large enough. From the obvious bound χQB (K,Δ) (x)  (1 + K2 )−m (1 + x2 )m eK,Δ (x), by the non-negaitivity of G(η)−1 Pη (π; K0 (n)) (m) (Lemma 2.6), we obtain |ˆ μΣ∞ (QB (K, Δ))|  (1 + K2 )−m |IμˆΣ (K, Δ)| by in∞ tegration, whose majorant is equals to (14.12) multiplied by (1 + K2 )−m . The first term is estimated by O ΔdF log(2 + K) by Lemma 14.7. Compared with this, the remaining terms are negligible by Lemma 14.12. Here the constraint  1 + K  minj |Kj | on K is necessary to have h(KJ )  (1 + K)−1 . Lemma 14.15. Let n be a square free oF -ideal. Then, for any  > 0 and for any q > 0,   (0) IμˆΣ (K, Δ) = 2 π dF /2 ΔdF f˜0 (K) + O ΔdF +1 (1 + K2 )−1/2 log2 (2 + K) ∞   + O ΔdF +1 h(K) log3/2 (2 + K)   + O ΔdF (1 + K2 )2dF θ+

  + O ΔdF +1 h(K)q holds for K ∈ RdF and Δ > 1 with Δ  log(2 + K) and 1 + K  minj |Kj |. Proof. This follows from Lemmas 14.7 and 14.3 with the aid of Lemma 14.14.  Proof of Theorem 14.1 : Set D0 = D − 2−1 D. On the set D0 (t), we have 1 + K  minj |Kj |  (1+t). Since ∂D is smooth, we have vol(B (∂D0 (t))) = O(td−1 ). From (0) Lemmas 14.8 and 14.15, IμˆΣ (K, Δ) = O(ΔdF log(2 + K)) if Δ  log(2 + K). ∞ Hence,  (0) IμˆΣ (K, Δ) dK = O(ΔdF tdF −1 log(2 + t)) if Δ  log(2 + t). B (∂D0 (t))

∞

Combining this with Lemmas 14.15 and 14.5, we obtain ˆΣ∞ (D(t) − D(t/2)) (14.13) μ     2 f˜0 (K) + O Δ (1 + K2 )−1/2 log2 (2 + K) = D(t)−D(t/2)

  + O Δ h(K) log3/2 (2 + K) 

+ O (1 + K )

2 2dF θ+



 + O (Δ h(K) ) dK + O(tdF −1 log(2 + t)) q

122

14. AN ERROR TERM ESTIMATE IN THE WEYL TYPE ASYMPTOTIC LAW

as long as Δ  log(2 + t). Now, divide D(t) − D to a union of D(t/2j ) − D(t/2j+1 ) with 0  j  [log t/ log 2]). Since log(t/2j ) = log t − j log 2  log t, the condition Δ  log(2 + t) implies Δ  log(2 + t/2j ). Applying (14.13) to each piece D(t/2j ) − we conclude that μ ˆΣ∞ (D(t)−D) equals D(t/2j+1 ) and summing up the estimations,   the following expression up to O( tdF −1 log(2 + t) .     2 f˜0 (K) + O Δ (1 + K2 )−1/2 log2 (2 + K) D(t)−D

  + O Δ h(K) log3/2 (2 + K) 



 + O (Δ h(K) ) dK

+ O (1 + K )      =2 f˜0 (K) dK + O Δ (1 + t)dF −1 log2 (2 + t) + O (1 + t)dF (1+4θ)+2 . 2 2dF θ+

q

D(t)−D

Since Δ  log(2 + t), we are done.



Proof of Theorem 1.3 : For J ⊂ X0Σ∞ as in Theorem 1.3, we apply Theorem 14.1  with D = ε∈E ε(J). Then the integral occurring in (14.1) is evaluated by Lemma 14.8 as  dF 1 DF G(η) [Kfin : K0 (n)] (1 + δn,oF ) vol(D) tdF (dF R(η) log t + Cη (F, n)) 4π + O(tdF −1 log t). We note that vol(D) = 2dF vol(J). Theorem 1.3 follows from these, combined with Corollary 2.16.  Proof of Corollary 1.4 : Let J ⊂ X0Σ∞ be a closed cone, which is positive. Let Πcus (n)J = {π ∈ Πcus (n)| νπ,Σ∞ ∈ J }. Then, from Lemma 14.15,   P1 (π; K0 (n)) = O (1 + νπ,Σ∞ )2 sup(2dF θ,−1/2)+ , π ∈ Πcus (π)J for any  > 0. We use Corollary 2.15 to relate P1 (π; K0 (n)) and |L(1/2, π)|2 /L(1, π; Ad),  (2/(1 + Q(πv )) is bounded from below by 1. noting that the extra factor v∈S(nf−1 π ) From Stirling’s formula, |L(1/2, π)|2 |Lfin (1/2, π)|2  (1 + νπ,Σ∞ )−dF , L(1, π; Ad) Lfin (1, π; Ad)

π ∈ Πcus (n).

To conclude the proof, we invoke the bound Lfin (1, π; Ad) = O((1 + νπ,Σ∞ ) ) ([35, Theorem 1]). 

CHAPTER 15

Appendix In this section, we collect some bound of automorphic forms which is necessary in the proof of Lemmas 9.7 and 9.9, and provide an estimation of a certain integral used in the proof of Lemma 11.14. 15.1 We need bounds for Eisenstein series and for cusp forms. Proposition 15.1. Let χ be a unitary idele class character of F × . Let fχ be ν/2 a K∞ K0 (n)-invariant flat section for I(χ| |A ). Then, there exists a constant N such that (ν)

ν/2

|E(fχ(ν) ; g)|  {q(χ| |A )}N y(g)N ,

ν ∈ iR, g ∈ S ∩ G1A

holds with the implied constant independent of χ. Proof. This is proved by the same argument as [13, Corollary 2] with a minor modificatoin. The point is that, instead of a rather crude bound of automorphic L-functions, a precise bound (2.5) is available for us to explicate the dependence on the analytic conductor of χ.  Recall that Bcus (n) denotes an orthogonal basis of L2cus (n) (see 9.1.1). Proposition 15.2. For any m ∈ N, there exists Nm ∈ N such that the estimation |ϕ(g)|  {



(1 + |νι,ϕ |)Nm } y(g)−m ,

ϕ ∈ Bcus (n), g ∈ S ∩ G1A

ι∈Σ∞

holds. Proof. This is proved in the same way as [13, Corollary 2]. For completeness we reproduce the argument. From [38, Lemma I.2.4], there exists r ∈ R such that for any continuous function f on G1A of compact support, the estimation  (15.1) |f (x−1 γg)|  grA , x ∈ GA , g ∈ S ∩ G1A γ∈GF

holds. Having this r and the given m, we apply [38, Lemma I.2.16] to obtain a finite family of elemensts {Xj }1jq in U (g∞ ) such that, for any ϕ ∈ Bcus (n), (15.2) | ∧ ϕ(g)|  T

g−m A

q 

 sup{| [R(Xj )ϕ] (g  )| g  −r A | g ∈ GA },

g ∈ S ∩ G1A .

j=1

Let n be the largest degree of the elements Xj (1  j  q). As in the proof of [13, Corollary 2], we take functions f1 and f2 on G1A ∼ = A\GA with the properties: 123

124

15. APPENDIX

(a) f1 is smooth; (b) f2 is fixed by an open compact subgroup of Kfin and, for any gfin ∈ Gfin , f2 (gfin g∞ ) is n-times continuously differentiable in g∞ ∈ G∞ ; (c) the Dirac distributuion δ on G1A at the identity is expressed as δ = f1 + f2 ∗ P (Ω) with some polynonial of the Casimir elements Ωv (v ∈ Σ∞ ). Let ϕ ∈ Bcus (n). Since it is cuspidal, for any T > 1, we have ϕ = ∧T ϕ. Using (15.1), we estimate   |ϕ(x)| { |[R(Xj )fˇi ](x−1 γg)|} dx |[R(Xj ) R(fi )ϕ](g)|  AGF \GA

γ∈GF



 grA

GF \G1A

|ϕ(x)| dx  grA ϕ2 = grA

for g ∈ S ∩ G1A . From this, denoting by λϕ,P (Ω) the eigenvalue of P (Ω) on ϕ, we have |R(Xj )ϕ(g)|  |[R(Xj ) R(f1 )ϕ] (g)| + |λϕ,P (Ω) | |[R(Xj ) R(f2 )ϕ] (g)|  (1 + |λP (Ω) |) grA for g ∈ S ∩ G1A . Combining this estimate with (15.2), we obtain |ϕ(g)| = | ∧T ϕ(g)|  (1 + |λϕ,P (Ω) |) gr−m , A

g ∈ S ∩ G1A

with the implied constant independent of ϕ ∈ Bcus (n). Since |λϕ,P (Ω) | is majorized  by ι∈Σ∞ (1 + |νι,ϕ |)N with some N , we are done.  15.2 The aim of this paragraph is to provide several lemmas needed in 11.4.3. For any l > 0 and n ∈ R − {0}, define  (1 + 4n cosh2 x)−l/4 dx, (n > 0), m(l; n) = R 2 −l/4 (1 − 4n sinh x) dx, (n < 0). R Lemma 15.3. The function n → m(l; n) is positive, is smooth on R − {0}, and satisfies the relation  ∞ [(1 + t−2 ){b2 + t2 (b + 1)2 }]−l/4 d× t = m(l; b(b + 1)), b ∈ R − {0, −1}. 0

Proof. The first two asserions are evident. A computation shows the equality  1 + 4b(b + 1)cosh2 x, (b(b + 1) > 0), (1 + t−2 ){b2 + t2 (b + 1)2 } = 2 (b(b + 1) < 0) 1 − 4b(b + 1)sinh x, with ex = (|b|/|b + 1|)1/2 t−1 . By this, making the variable change from t to x, we are done.  Lemma 15.4. If n > 0, then  √ 1 π Γ(λ)2 Γ(l/4 − λ) m(l; n) = |4n|−λ dλ (15.3) 2πi (c) Γ(l/4) Γ(λ + 1/2) where (c) denotes the vertical line contour from c − i∞ to c + i∞ with 0 < c < l/4. If n < 0, then  1 1 Γ(λ)2 Γ(1/2 − λ) Γ(l/4 − λ) √ m(l; n) = |4n|−λ dλ, (15.4) 2πi (c) π Γ(l/4)

15.2

125

where (c) denotes the vertical line contour as above with 0 < c < inf(1/2, l/4). Proof. Let us compute the Mellin transform of ϕ(n) = m(l; n/4) (n > 0). Let λ ∈ C. By proceeding formally,  ∞ m(l; n/4) nλ d× n ϕ(λ) ˜ = 0   ∞  2 −l/4 = (1 + n cosh x) dx nλ d× n R 0   ∞ = dx (1 + n cosh2 x)−l/4 nλ d× n R 0  ∞  dx (1 + n)−l/4 (n(coshx)−2 )λ d× n = R 0  ∞   ∞  −2λ −l/4 λ−1 =2 (coshx) dx (n + 1) n dn . 0

0

To justify the above computation, we need to show the absolute convergence of the two integrals occuring in By the variable change coshx = z −1 ,  the last line.−1/2 1 1 λ−1 (1 − z) dz, a beta-integral which converges the first integral becomes 2 0 z 1 Γ(λ) Γ(1/2) absolutely if Re(λ) > 0 and evaluated as 2 Γ(λ+1/2) . By a variable change, the 1 second integral is also expressed as the beta-integral 0 z l/4−λ−1 (1−z)λ−1 dz, which Γ(λ) for such converges absolutely when 0 < Re(λ) < l/4 and is evaluated as Γ(l/4−λ) Γ(l/4) λ. Thus, we have shown the equality ϕ(λ) ˜ =

√

π

Γ(λ)2 Γ(l/4 − λ) , Γ(l/4) Γ(λ + 1/2)

(0 < Re(λ) < l/4).

Then, (15.3) is infered from this by the Melling inversion formula. The proof of (15.4) when b(b + 1) < 0 is similar.  The following lemma gives us the asymptotic behavior of m(l; n) as |n| getting small. Lemma 15.5. Let  ∈ (0, 1). Then, for any n ∈ R such that |n| ∈ (0, 1/2) and for any l > 0, we have

     

m(l; n) + ψ l + ψ 1 + log |4n| + 2CEuler  |n| l

4 2 with the implied constants independent of n and l. Here, ψ(z) = Γ (z)/Γ(z) is the digamma function. Proof. First assume n > 0. We shift the contour (c) in (15.3) to (−); by the residue theorem m(l; n) is expressed as a sum of a contour integral and the residue √  π Γ(λ)2 Γ(l/4 − λ) −λ |4n| , Resλ=0 Γ(l/4) Γ(λ + 1/2) which is calculated as −ψ(l/4) − ψ(1/2) − log |4n| − 2CEuler . Thus, m(l; n) =

1 2πi



(−)

√

πΓ(λ)2 Γ(l/4 − λ) |4n|−λ dλ + (−ψ(l/4) − ψ(1/2) − log |4n| − 2CEuler ) . Γ(l/4) Γ(λ + 1/2)

126

15. APPENDIX

From this,



       |Γ(− + it)2 | |Γ(l/4 +  − it)|



m(l; n) + ψ l + ψ 1 + log |4n| + 2CEuler  |4n| dt. √



4 2 2 π R Γ(l/4) |Γ(− + it + 1/2)|

Since |Γ(l/4 +  − it)|/|Γ(l/4)|  |Γ(l/4)

−1

 | 0

= |Γ(l/4)|−1



0

∞

∞

e−y |y l/4+ −it−1 | dy e−y y l/4+ −1 dy =

Γ(l/4 + )  l , Γ(l/4)

noting that |Γ(− + it)2 |/|Γ(− + it + 1/2)| is of exponential decay as |t| → ∞, we have the first estimation as desired when n > 0. From (15.4), by the same procedure, we obtain the same estimate in the case when n < 0 also. 

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SELECTED PUBLISHED TITLES IN THIS SERIES

1105 Tai-Ping Liu and Yanni Zeng, Shock Waves in Conservation Laws with Physical Viscosity, 2014 1104 Gerhard Hiss, William J. Husen, and Kay Magaard, Imprimitive Irreducible Modules for Finite Quasisimple Groups, 2014 1103 J.-M. Delort, Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres, 2014 1102 Jianyong Qiao, Julia Sets and Complex Singularities of Free Energies, 2014 1101 Jochen Denzler, Herbert Koch, and Robert J. McCann, Higher-Order Time Asymptotics of Fast Diffusion in Euclidean Space: A Dynamical Systems Approach, 2014 1100 Joel Friedman, Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture, 2014 1099 Anthony H. Dooley and Guohua Zhang, Local Entropy Theory of a Random Dynamical System, 2014 1098 Peter Keevash and Richard Mycroft, A Geometric Theory for Hypergraph Matching, 2014 1097 Xiaoye Fu and Jean-Pierre Gabardo, Self-Affine Scaling Sets in R2 , 2015 1096 Rapha¨ el Cerf, Critical Population and Error Threshold on the Sharp Peak Landscape for a Moran Model, 2015 1095 Detlef M¨ uller, Marco M. Peloso, and Fulvio Ricci, Analysis of the Hodge Laplacian on the Heisenberg Group, 2015 1094 Ian F. Putnam, A Homology Theory for Smale Spaces, 2014 1093 Ron Blei, The Grothendieck Inequality Revisited, 2014 1092 Yun Long, Asaf Nachmias, Weiyang Ning, and Yuval Peres, A Power Law of Order 1/4 for Critical Mean Field Swendsen-Wang Dynamics, 2014 1091 Vilmos Totik, Polynomial Approximation on Polytopes, 2014 1090 Ameya Pitale, Abhishek Saha, and Ralf Schmidt, Transfer of Siegel Cusp Forms of Degree 2, 2014 ˇ 1089 Peter Semrl, The Optimal Version of Hua’s Fundamental Theorem of Geometry of Rectangular Matrices, 2014 1088 Mark Green, Phillip Griffiths, and Matt Kerr, Special Values of Automorphic Cohomology Classes, 2014 1087 Colin J. Bushnell and Guy Henniart, To an Effective Local Langlands Correspondence, 2014 1086 Stefan Ivanov, Ivan Minchev, and Dimiter Vassilev, Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem, 2014 1085 A. L. Carey, V. Gayral, A. Rennie, and F. A. Sukochev, Index Theory for Locally Compact Noncommutative Geometries, 2014 1084 Michael S. Weiss and Bruce E. Williams, Automorphisms of Manifolds and Algebraic K-Theory: Part III, 2014 1083 Jakob Wachsmuth and Stefan Teufel, Effective Hamiltonians for Constrained Quantum Systems, 2014 1082 Fabian Ziltener, A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane, 2014 1081 Sy-David Friedman, Tapani Hyttinen, and Vadim Kulikov, Generalized Descriptive Set Theory and Classification Theory, 2014 1080 Vin de Silva, Joel W. Robbin, and Dietmar A. Salamon, Combinatorial Floer Homology, 2014 1079 Pascal Lambrechts and Ismar Voli´ c, Formality of the Little N -disks Operad, 2013

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/memoseries/.

Memoirs of the American Mathematical Society

9 781470 410193

MEMO/235/1110

Number 1110 • May 2015

ISBN 978-1-4704-1019-3