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Special values of anticyclotomic l-functions Hamieh, Alia 2013

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Special Values of Anticyclotomic L-functions by Alia Hamieh  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2013 c Alia Hamieh 2013  Abstract This thesis consists of four chapters and deals with two different problems which are both related to the broad topic of special values of anticyclotomic L-functions. In Chapter 3, we generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic Zp -extension. Let g be a cuspidal Hilbert modular form of parallel weight (2, ..., 2) and level N over a totally real field F , and let K/F be a totally imaginary quadratic extension of relative discriminant D. We study the l-adic valuation of the special values L(g, χ, 12 ) as χ varies over the ring class characters of K of Ppower conductor, for some fixed prime ideal P. We prove our results under the only assumption that the prime to P part of N is relatively prime to D. In Chapter 4, we compute a basis for the two-dimensional subspace S k (Γ0 (4N ), F ) of half-integral weight modular forms associated, via the 2 Shimura correspondence, to a newform F ∈ Sk−1 (Γ0 (N )), which satisfies L(F, 12 ) = 0. Here, we let k be a positive integer such that k ≡ 3 mod 4 and N be a positive square-free integer. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined. The squares of the Fourier coefficients of these forms are known to be essentially proportional to the central critical values of the L-function of F twisted by some quadratic characters.  ii  Preface A version of Chapter 4 has been published under the title Ternary Quadratic Forms and Half-Integral Weight Modular Forms in LMS Journal of Computation and Mathematics [10].  iii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  vi  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction and Organization . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 1 8  2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Hilbert Modular Forms . . . . . . . . . . . . . . . . . . . . . 2.1.1 Classical Hilbert Modular Forms . . . . . . . . . . . . 2.1.2 Adelic Hilbert Modular forms . . . . . . . . . . . . . 2.2 Automorphic Forms on Totally Definite Quaternion Algebras 2.2.1 Quaternion Algebras . . . . . . . . . . . . . . . . . . 2.2.2 Jacquet Langlands Correspondence . . . . . . . . . . 2.3 Rankin-Selberg L-functions . . . . . . . . . . . . . . . . . . .  10 10 10 13 18 18 23 25  3 Special Values of Anticyclotomic L-functions 3.1 Preliminaries and Notations . . . . . . . . . 3.2 CM Points and Galois Action . . . . . . . . 3.3 Uniform Distribution of CM Points . . . . . 3.4 Toward Computing ordλ (Lal (π, χ, 12 )) . . . .  . . . . .  28 28 29 31 32  4 Fourier Coefficients of Half-Integral Weight Modular Forms 4.1 The Shimura Correspondence . . . . . . . . . . . . . . . . . . 4.2 A Modular Form in S k (Γ0 (4N ), F ) . . . . . . . . . . . . . . 2 4.3 Theta Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46 46 50 53  modulo λ . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . .  iv  Table of Contents 4.4 4.5  On The Result Of Waldspurger . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  55 65  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  72  v  List of Tables 4.1 4.2 4.3  Ratios of Fourier Coefficients . . . . . . . . . . . . . . . . . . The Local Factors C1 and C2 . . . . . . . . . . . . . . . . . . The Global Factors AF (t) . . . . . . . . . . . . . . . . . . . .  63 67 68  vi  Acknowledgements It is a great pleasure for me to express my heartfelt gratitude to my PhD advisor, Professor Vinayak Vatsal. He patiently provided the vision, encouragement and guidance that made it possible for me to see this project through to the end. I will never be able to thank him enough for his support and understanding over the course of my program. Besides my advisor, I would like to thank the members of my dissertation committee, Professor Michael Bennett and Professor Greg Martin, for their invaluable time and professional insight. I also extend my gratitude to the faculty and staff of the department of Mathematics at UBC for their time, commitment and support. I am especially grateful to Mrs. Lee Yupitun for her timely help in all of my queries. I wish to thank my parents for their unconditional love and immeasurable sacrifice throughout the years. I owe them everything, and I hope this work would make them proud. Many thanks go out to my siblings who always cheer me up and help me get through rough patches. I am also grateful to my mother-in-law for her warm wishes and support. I will always remain indebted to my husband Khalil for his unflagging support and patience. I thank him for sharing the highs and lows of this process with me and keeping the faith even when I had none. Most of all, I would like to thank darling Tiyana for being such a wonderful daughter and making it possible for me to complete what I started. If this work is worth dedicating, I dedicate it to her. Finally, I gratefully acknowledge the financial support from the University of British Columbia.  vii  Chapter 1  Introduction and Organization 1.1  Introduction  Let π be an irreducible cuspidal automorphic representation of GL2 over a totally real field F , and let K/F be a totally imaginary quadratic extension. Attached to π and a Hecke character χ of K is the Rankin-Selberg L-function L(π, χ, s). If w is the central character of π and χ · w = 1 on A∗F ⊂ A∗K , then L(π, χ, s) is entire and satisfies the functional equation L(π, χ, s) = (π, χ, s)L(π, χ, 1 − s), where (π, χ, 21 ) = (π, χ) ∈ {±1}. If (π, χ) = −1, the special value L(π, χ, 12 ) vanishes trivially. Hence, we assume throughout that (π, χ) = 1 in which case we say that the pair (π, χ) is even. Obtaining a formula for the special value L(π, χ, 21 ) in the even case has been the subject of extensive research for over 25 years now. There are several results in the literature relating L(π, χ, 21 ) to a finite sum a(x, χ) which is essentially the height of a twisted CM point on the Shimura curve associated to some carefully chosen totally definite quaternion algebra B. In 1985, Waldspurger proved a fundamental theorem (Th´eor`eme 2 in [28]) which states that under very mild conditions on π and χ, L(π, χ, 12 ) = 0 if and only if a(x, χ) = 0. However, this result doesn’t give a precise formula for the special value L(π, χ, 12 ) in terms of a(x, χ). Most authors refer to such a formula as a Gross-Zagier formula, and it is expected (but not known yet) that it exists in full generality. The general scheme for obtaining a Gross-Zagier formula involves the construction of a discrete set of CM points and a function ψ on this set induced by some automorphic form θ in the representation space of π , the unique cuspidal automorphic representation on B associated to π via the JacquetLanglands correspondence. In this framework, the Gross-Zagier sum a(x, χ) is given by a(x, χ) =  χ(σ)ψ(σ.x), Galab K  1  1.1. Introduction where x is a CM point of conductor equal to that of χ, and Galab K is the Galois group of the maximal abelian extension of K acting continuously on the set of CM points. In fact, one can easily check that a(x, χ) is a finite sum. In 1987, Gross established the main identity which expresses the special value L(π, χ, 12 ) in terms of the height pairing on the CM points (see [9]). In Gross’ result, π corresponds to a weight 2 newform of prime conductor and trivial central character. Subsequently, Zhang obtained a generalization of this result to the case where π corresponds to a Hilbert newform g of parallel weight (2, ..., 2) and trivial central character. More specifically, Zhang proved in [32] that if the level N of g, the conductor C of χ, and the discriminant D of K/F are pairwise coprime, then 1 C L(π, χ, ) = √ g, g |a(x, χ)|2 , 2 D where g, g is the Petersson inner product, D is the absolute norm of C 2 D and C is a non-zero constant independent of χ. Results as such were the point of departure in the work of Vatsal ([24], [25]) and Cornut-Vatsal ([7]) on the non-vanishing of L(π, χ, 21 ) in anticyclotomic towers. We will now give a brief account of the main results in [7], [24] and [25] which will be discussed more elaborately in Chapter 3 of this dissertation. Let E be an elliptic curve over Q of conductor N , and let K/Q be an imaginary quadratic field extension of discriminant D such that N and D are relatively prime. Denote by K∞ the anticyclotomic Zp -extension of K where p is a given prime number with p N D. In 2002, Vatsal succeeded in settling a conjecture of Mazur pertaining to the size of the Mordell-Weil group E(K∞ ). In fact, Mazur’s conjecture predicts that the group E(K∞ ) is finitely generated, and Vatsal proved in [24] that this is true, at least when E is ordinary at p, or when the class number of K is prime to p. In more concrete terms, Vatsal considered the modular form g associated to E and the family of Rankin -Selberg L-functions L(g, χ, s) as χ varies over ring class characters of K of p-power conductor. Under certain conditions on g and χ, the result of Vatsal asserts that the special values L(g, χ, 21 ) are non-vanishing for all but finitely many χ, provided that p is an ordinary prime for g or p does not divide the class number of K. One consequence of this result is the non-triviality of certain Euler systems as formulated by Bertolini-Darmon in [3] which in its turn implies that the desired statement about the Mordell-Weil group is indeed true. 2  1.1. Introduction In 2004, Cornut and Vatsal generalized in [7] the above mentioned work of Vatsal to totally real fields. Numerous technical complications arise due to the fact that a more general number field F is considered. However, the basic arguments are ultimately the same, as the authors invoke deep theorems of Ratner [19] on uniform distribution of unipotent orbits on p-adic Lie groups to deduce the desired result. In 2003, Vatsal extended the results and methods of [24] to study the variation of the λ-adic absolute value of Lal (g, χ, 12 ) as a function of χ, where ¯ with residue characteristic l. In Chapter 3 of this λ is a fixed prime of Q dissertation, we generalize this work to totally real fields while removing most of the restrictions on N , p, D and l (Theorem 3.4.14). We use the improved formalism developed in [7] to achieve this purpose. We now give an overview of our work in Chapter 3. Let π be an irreducible automorphic representation of GL2 over a totally real field corresponding to a cuspidal Hilbert modular newform g of level N , trivial character and parallel weight (2, ..., 2). The Hecke eigenvalues of g are denoted by av (Tv g = av g). Let P be a prime ideal of F such that P lies over an odd rational prime p. Let l be a rational prime, and denote by El the l-adically complete discrete valuation ring containing the Hecke eigenvalues of g. To simplify the exposition of the introduction, we assume that l = p although we should mention that the case l = p does not give rise to significant complications. Let χ be a ring class character of K of conductor P n such that χ = 1 when restricted to A∗F ⊂ A∗K . In addition to some mild restrictions, we assume that the prime to P part of N is relatively prime to the discriminant D of K/F . We also impose sufficient conditions to make the sign in the functional equation of L(π, χ, s) equal +1 for all but finitely many characters χ of the type considered above. Let G(n) be the Galois group of the ring class field of conductor P n over K. We have the decomposition G(n) = G0 × H(n) where G0 is known as the tame subgroup of G(n) and H(n) as the wild subgroup of G(n). By class field theory, we can view χ as a character of G(n). Hence, χ can be written as the product of a tamely ramified character χ0 of G0 and a wild character χ1 of H(n). It can be shown that G(n) acts simply transitively on the set of CM points of conductor P n on the Shimura curve associated to some carefully chosen totally definite quaternion algebra B. Given a CM point x of conductor P n and a ring class character χ of the same conductor, we define the Gross-Zagier sum a(x, χ) =  1 |G(n)|  χ(σ)ψ(σ.x), σ∈G(n)  3  1.1. Introduction where ψ is the El -valued function on the set of Heegner points, associated to g via the Jacquet-Langlands correspondence. In the light of the existing Gross-Zagier formulae, our job is reduced to studying the l-adic valuation of this sum. More precisely, our goal is to prove an analogue of Proposition 4.1 and Corollary 4.2 in [25] for a Hilbert modular form g over a totally real field F , while removing the assumptions on l and the class number of K. Before we state the results we obtained in this direction, it is perhaps more enlightening to shed some light on Vatsal’s result in which π corresponds to a weight 2 newform g for Γ0 (N ) such that N , p, and D are pairwise relatively prime. Theorem 1.1.1. Let the tame character χ0 be given such that its order is prime to p. Then, under some restrictions on l and the Hecke field of g, we have ordλ (a(x, χ)) < µ (1.1) for all n 0 and χ = χ0 χ1 , where µ is the smallest integer such that aq ≡ 1 + q mod λµ for some q pN D. Note that the restriction on the Hecke field of g, which will be made clear in Section 3.4, was overlooked in [25]. More importantly, we remark that (1.1) is mistakenly given as an equality in [25]. The source of this mistake is an error made in the proof of Proposition 5.3 part (2). We mention here that in order to make the necessary corrections, we need to modify the definition of the constant µ from the one given in [25]. We say a bit more about this issue in Chapter 3. Recall that El is an l-adically complete discrete valuation ring containing the Hecke eigenvalues of g. We may assume without loss of generality that El contains the values of χ0 and the p-th roots of unity. We consider the trace of a(x, χ) taken from El (χ1 ) to El : Tr(a(x, χ)) =  σ(a(x, χ)). σ∈Gal(El (χ1 )/El )  This trace expression is different than the average expression b(x, χ0 ) =  a(x, χ0 χ1 ) χ1 ∈H(n)  considered in the work of Vatsal and Cornut-Vatsal. In particular, given any χ = χ0 χ1 , the non-vanishing of Tr(a(x, χ)) implies that of a(x, χ). After a series of reductions, we prove the following proposition in two level raising steps (Proposition 3.4.9 and Proposition 3.4.12). 4  1.1. Introduction Proposition 1.1.2. The trace expression simplifies to Tr(a(x, χ)) =  |G2 |[El (χ1 ) : El ] |G(n)|  χ0 (σ)ψm,D (σ.xm,D ), σ∈G0 /G1  where G1 and G2 are certain subgroups of G(n), ψm,D is a function of higher level induced by ψ, and xm,D is a CM point of higher level and conductor P n. Hence, the problem is reduced to studying the λ-adic valuation of χ0 (σ)ψm,D (σ.xm,D ). σ∈G0 /G1  Finally, we state the following theorem which is the main result in Chapter 3 (Theorem 3.4.14). Theorem 1.1.3. Let χ0 be any character of G0 . For any CM point x of conductor P n with n 0, there exists some y ∈ G(n).x such that   ordλ   χ0 (σ)ψm,D (σ.y) < µ,  σ∈G0 /G1  where µ is precisely given in Definition 3.4.8. Consider an irreducible cuspidal automorphic representation of GL2 over Q corresponding to a newform F . An interesting situation occurs if we let K vary over the imaginary quadratic extensions of Q and let χ be the trivial character of K. In this case we have the decomposition 1 1 1 L(π, χ, ) = L(π, )L(π ⊗ η, ), 2 2 2 where η is the quadratic character of Q associated with the extension K/F . By a striking result of Waldspurger [29], we know that the special values L(π ⊗ η, 12 ) as K vary over the imaginary quadratic extensions of Q are related to the Fourier coefficients of some half-integral weight modular form corresponding to F via the Shimura correspondence. Hence, the GrossZagier formula yields a very interesting interpretation of the height pairing a(x, χ) in terms of the Fourier coefficients of some half-integral weight modular form. This observation motivates to some extent our work in Chapter 4 of this dissertation. The contents of this chapter have been published in the LMS Journal of Computation and Mathematics [10]. 5  1.1. Introduction Let k be a positive integer such that k ≡ 3 mod 4, and let N be a positive square-free integer. In Chapter 4, we compute a basis for the twodimensional subspace S k (Γ0 (4N ), F ) of half-integral weight modular forms 2 associated, via the Shimura correspondence, to a newform F ∈ Sk−1 (Γ0 (N )), which satisfies L(F, 21 ) = 0. Waldspurger’s Theorem asserts that there exists a basis for S k (4N, F ) such that for every positive integer n the Fourier 2 coefficient an (fi ) of a basis element fi is the product of two factors: • a product of local terms ci (n, F ) each of which is completely determined by the local components of F according to explicit formulae given in [29]; • a global factor AF (n) whose square is the central critical value of the Lfunction of the newform F twisted by a quadratic character depending on n. In view of this remarkable result, our task is reduced to computing the global factors AF (n). Toward this end, we use a construction drawn from the articles [9] and [4] to compute a non-zero modular form g as a linear combination of ternary theta series g([Ii ]) generated by the norm form N of some carefully chosen quaternion algebra B. This modular form belongs to the Kohnen subspace of S 3 (Γ0 (4N ), F ). The determination of the theta 2 series g([Ii ]) up to a precision T amounts to computing the number of times N (x) represents 1, 2, 3, ..., T as x varies in some ternary lattice Ri . Therefore, 3 it takes time roughly proportional to T 2 . We use the Brandt module package in Sage to compute g as a linear combination of g([Ii ]). Now we show how we use the form g to compute yet another half-integral weight modular form h which also maps to F via the Shimura correspondence. These two forms make up a basis for the space S k (Γ0 (N ), F ). Let the 2  bn q n and let g =  Fourier expansion of the given newform be F = n≥1  an q n n≥1  be the Fourier expansion of the half-integral weight modular form obtained k above. For every prime number p not dividing N , we put λp = bp p1− 2 , αp + αp = λp , and αp αp = 1. For a positive square-free integer t, we denote by ∆−t the fundamental discriminant corresponding to −t. We obtain the following theorems which are the main results in Chapter 4 (Theorem 4.4.5 and Theorem 4.4.7). Theorem 1.1.4. For a positive square-free integer t satisfying  ∆−t p  =  6  1.1. Introduction 1  −p 2 λp for all p such that p|N and p ∆−t , we have νt  AF (t) = r2 2 |∆−t | νt  2−k 4  a|∆−t | ,  1  2− 2 , and r is a non-zero complex constant depending only  where 2 2 = p|N p ∆−t  on F . In what follows, the terms C1 (n) and C2 (n) refer to certain products of the local factors ci (n, F ) determined in Waldspurger’s paper. We assume that α2 = α2 to simplify exposition. new (Γ (N )) with odd bn q n be a newform in Sk−1 0  Theorem 1.1.5. Let F = n≥1  square-free level N such that k ≡ 3 mod 4 and L(F, 1) = 0. Let g = an q n ∈ S k (Γ0 (4N ), F ) be the form obtained above. Put 2  n≥1  an (f1 )q n  f1 =  and  n≥1 ∆−nsf p  where if  an (f2 )q n ,  f2 = n≥1  1  = −p 2 λp for all p such that p|N and p ∆−nsf , we have an (f1 ) = 2 an (f2 ) = 2  νn 2  νn 2  2−k 4  n  2−k 4  n  a|∆−nsf | |∆−nsf | a|∆−nsf | |∆−nsf |  k−2 4  k−2 4  C1 (n) C2 (n),  and otherwise we have an (f1 ) = an (f2 ) = 0. Then f1 and f2 form a basis for S k (Γ0 (4N ), F ) 2  In order to use Theorem 1.1.5 as an effective tool for computing a basis for S 3 (Γ0 (4N ), F ), we make the following important observation. Let 1  2  an (h)q n . It turns out that the Fourier  h = 2 2 (f1 + f2 ) and write h = n≥1  coefficients of h can be expressed in terms of the Fourier coefficients of g by means of simple formulae. Take, for example, a positive square-free integer 1 n such that np = −p 2 λp whenever p|N and p n (since, for square-free n, an (h) = 0 otherwise). We have an (h) =  an (g)(b2 − 2(2, n)2 ) if n ≡ 3 mod 4; 2a4n (g) otherwise. 7  1.2. Organization We obtain a similar expression of an (h) in terms of an (g) for when n is a positive integer n = nsf (y)2 , which is not divisible by any square prime to 4N . A recursive formula is then used to compute an (h) for an arbitrary positive integer n. We implemented these formulae in Sage. The result is a function that outputs the Fourier expansion of h up to a desired precision. Thus, we get a basis for S 3 (Γ0 (4N ), F ) consisting of the modular forms g and h. 2  1.2  Organization  The thesis is organized as follows. Chapter 1 is an introduction to the thesis in which we discuss the motivation and related research to our work, and we give a brief account of our main results. In Chapter 2, we introduce the basic definitions and all the preliminary material that will be used throughout the thesis. In the first section, we recall some arithmetic aspects of Hilbert modular forms. In Section 2.2 we give an overview of some basics from the theory of Quaternion Algebras. We then explore the connection between automorphic forms on totally definite quaternion algebras and adelic Hilbert modular forms; this connection is made possible by means of the Jacquet-Langlands correspondence. In Section 2.3, we recall the definition and some of the basic properties of Rankin-Selberg L-functions. In Chapter 3, we generalize the results of Vatsal on studying the ladic valuation of the special values of Rankin-Selberg L-functions in an anticyclotomic Zp -extension. First we introduce some notation and fix a set of hypotheses which we require throughout the chapter. In the second section, we recall the construction of CM points associated to a totally definite quaternion algebra. In Section 3.3, we recall the fundamental result of Cornut-Vatsal [7] on uniform distribution of CM points. In Section 3.4, we explore the Gross-Zagier sum a(x, χ) which is closely related to the special value L(π, χ, 21 ). Then we prove some propositions and lemmas related to this sum leading to the main result Theorem 3.4.14. In Chapter 4, we compute a basis for the subspace S k (Γ0 (4N ), F ) of half2 integral weight modular forms associated, via the Shimura correspondence, to a newform F of weight k − 1 and odd square-free level N . First, we recall the definition and some basic properties of the Shimura correspondence. In Section 4.2, we present a construction of a half-integral weight modular form g which belongs to the Kohnen subspace of S k (Γ0 (4N ), F ) for a 2 new (Γ (N )). The squares of the Fourier coefficients given newform F ∈ Sk−1 0 8  1.2. Organization of this form are essentially proportional to the central critical values of the L-function of F twisted with some quadratic characters. In Section 4.3, we digress briefly to discuss the representation-theoretic interpretation of the Shimura correspondence as a theta correspondence. In Section 4.4, we state the full result of Waldspurger. Next, we show that the space S k (Γ0 (4N ), F ) 2 is in fact two-dimensional and has a distinguished basis {f1 , f2 } such that either f1 − f2 or f2 belongs to the Kohnen subspace. We use this to express the global factors AF (n) in terms of the coefficients of g (Theorem 4.4.5). Finally, we explicitly determine the Fourier coefficients of the two modular forms that generate S k (Γ0 (4N ), F ), thus, arriving at our main contribution 2 given in Theorem 4.4.7. The last section contains examples to illustrate the calculations carried out in Section 4.4.  9  Chapter 2  Background 2.1  Hilbert Modular Forms  We begin by recalling some arithmetic aspects of Hilbert modular forms which are primary objects of interest for this dissertation. The exposition in this section follows that of Shimura in [21]. Let F be a totally real number field of degree d over Q, and let JF = {τ1 , ..., τd } be the set of all real embeddings F → R. For k = (k1 , ..., kd ) ∈ Zd and z = (z1 , ..., zd ) ∈ Cd , we write z k = di=1 ziki . An element a ∈ F is said to be totally positive (a 0) if τ (a) > 0 for all τ ∈ JF . We denote by OF the ring of integers of F and by D its different over Q. For each prime ideal P of F , we denote by FP and OF,P the completions of F and OF , respectively, at P. Moreover, given a fractional ideal I of F , we define the localization IP of I at P as a submodule of FP , and we put I = IP . We let AF = A⊗Q F be the ring of P  adeles of F and F = Af ⊗Q F be the ring of finite adeles. The archimedean + is the identity component in F ∗ . component of AF is F∞ = F ⊗ R, and F∞ ∞ If z ∈ AF , we let z∞ be its archimedean component. For t ∈ A∗F , we denote by tOF the fractional ideal in F associated to t unless otherwise specified. Here A∗F is the group of ideles.  2.1.1  Classical Hilbert Modular Forms d  We have an embedding of GL2 (F ) into  GL2 (R) given by i=1  a b c d  →  τ (a) τ (b) τ (c) τ (d)  . τ ∈JF  d  We often identify GL2 (F ) with its image in  GL2 (R) under i=1 GL+ 2 (R) to be the  this embed-  ding. For any subring R in F , we define  subgroup of 10  2.1. Hilbert Modular Forms elements with totally positive determinant. More precisely, we have d  GL+ 2 (R)  GL+ 2 (R)},  = {γ ∈ GL2 (R) : (τ1 (γ), ..., τd (γ)) ∈ i=1  where GL+ 2 (R) = {γ ∈ GL2 (R) : det(γ) > 0}. Let Hd be the d-fold Cartesian product of the Poincar´e upper-half plane. The well-known action of GL+ 2 (R) on H by fractional linear transformad d GL+ 2 (R) on H : For every z =  tions extends naturally to an action of i=1 d  d GL+ 2 (R) on H with γi =  (z1 , ..., zd ) ∈ Hd and γ = (γ1 , ..., γd ) ∈ i=1  ai bi ci di  we let γ act on z by γ.z = (γ1 .z1 , ..., γd .zd ), γi .zi =  ai (zi ) + bi . ci (zi ) + di d  Moreover, for a given k = (k1 , ..., kd ) ∈  Zd ,  GL+ 2 (R) acts on  the group i=1  all complex-valued functions on Hd as follows. Given f : Hd → C and d  GL+ 2 (R) as above, we put  γ∈ i=1  d  (ci zi + di )−ki  (f |k γ)(z) =  f (γ.z)  i=1  and  d  (f ||k γ)(z) :=  ki  det(γi ) 2  (f |k γ)(z).  i=1  A subgroup Γ of GL+ 2 (F ) is called a congruence subgroup if it contains the subgroup ΓN =  γ=  a b c d  ∈ SL2 (OF ) :  a−1 b c d−1  ∈ N.M2 (OF )  for some positive integer N , and if Γ/Γ∩F is commensurable with PSL2 (OK ). 11  ,  2.1. Hilbert Modular Forms Two examples of congruence subgroups that appear frequently in literature are Γ0 (I, N ) =  γ=  a b c d  OF IN  ∈  I −1 OF  : det(γ) ∈ OF∗+  (2.1)  and Γ1 (I, N ) =  a b c d  ∈ Γ0 (I, N ) : d ≡ 1  mod N  ,  (2.2)  where I is a fractional ideal of F and N is an ideal of OF . Definition 2.1.1. Let k be a tuple in Zd and let Γ be a congruence subgroup. A holomorphic function f : Hd → C is a classical Hilbert modular form of weight k and level Γ if 1. f ||k γ = f for all γ ∈ Γ 2. f is holomorphic at the cusps Γ\P1 (F ) The space of all classical Hilbert modular forms of weight k and level Γ is denoted by Mk (Γ). The condition of holomorphicity at the cusps is unnecessary for [F : Q] > 1 since it is readily met by any holomorphic function f : Hd → C which satisfies the transformation formula f ||k γ = f for all γ in some congruence subgroup Γ. This result is known as Koecher’s principle. Every classical Hilbert modular form f admits a Fourier expansion of the form f (z) = c(ξ)eF (ξz), ξ  where z = (z1 , ..., zd ) ∈ Hd , eF (ξz) = e2πiTr(ξz) and Tr(ξz) = di=1 τi (ξ)zi . Here ξ runs over 0 and all totally positive elements (ξ 0) of some lattice in F , and c(ξ) are complex numbers. Definition 2.1.2. A classical Hilbert modular form of weight k and level Γ is a cusp form if the constant term in the Fourier expansion of f |k γ is 0 for all γ ∈ GL+ 2 (F ). The space of all classical Hilbert cusp forms of weight k and level Γ is denoted by Sk (Γ). The spaces Mk (Γ) and Sk (Γ) are finite dimensional C-vector spaces. In what follows we list some basic properties of these spaces. The reader is referred to Shimura [21] for proofs and more elaborate discussions on such results. 12  2.1. Hilbert Modular Forms Proposition 2.1.3. If Mk (Γ) = 0, then either all ki are positive or k = (0, ..., 0) in which case M0 (Γ) = C and S0 (Γ) = (0). Proposition 2.1.4. We have Mk (Γ) = Sk (Γ) unless k1 = ... = kd . Finally we introduce an inner product on Mk (Γ). Let f and g be classical Hilbert modular forms in Mk (Γ) such that f g is a cusp form. We put f, g = µ(Γ\Hd )−1  f (z)g(z)y k dµ(z), Γ\Hd  where dµ(z) = dν=1 yν−2 dxν dyν with zν = xν + iyν and µ(Γ\Hd ) is the measure of a fundamental domain for Γ\Hd with respect to dµ(z).  2.1.2  Adelic Hilbert Modular forms d  d  R∗ SO2 (R) the stabilizer of j = (i, ..., i) in  + = We denote by K∞ ν=1  GL+ 2 (R). ν=1  Definition 2.1.5. Let k be a tuple in Zd and let K be a an open compact subgroup of GL2 (F ). A function f : GL2 (AF ) → C is an adelic Hilbert modular form of weight k with respect to K if it is left GL2 (F )-invariant, right K-invariant and for all g ∈ GL2 (F ) the function d  fg : γ = (γ1 , ..., γd ) ∈  G+ ∞  det(γν )−  →  kν 2  (cν i + dν )kν  f (gγ)  ν=1 + factors through a holomorphic function of G+ ∞ /K∞  Hd . Here G+ ∞ is  d  GL+ 2 (R). ν=1  An adelic Hilbert modular form of weight k with respect to K is called a cusp form if f F \AF  1 x 0 1  g dx = 0,  for all g ∈ GL2 (AF ).  For an integral ideal N of OF , we define the following open compact subgroup of GL2 (F ): K0 (N ) = K0 (N )P where the product is taken over all prime ideal P of F and K0 (N )P =  P  a b c d  ∈ GL2 (OF,P ) : c ∈ NP  .  (2.3) 13  2.1. Hilbert Modular Forms This is the standard Iwahori congruence subgroup of GL2 (F ) of level N . We also define K1 (N ) =  a b c d  ∈ K0 (N ) : d ≡ 1  mod N  .  (2.4)  An adelic Hilbert modular form of weight k with respect to K0 (N ) is said to have level N . The space of all such forms is denoted by Mk (N ). The space of all adelic Hilbert cusp forms of weight k and level N is denoted by Sk (N ). ∗ OF,P , Since the image of K0 (N ) in F ∗ under the determinant map is P  the strong approximation theorem for GL2 yields a bijection between GL2 (F )\ GL2 (AF )/G+ ∞ K0 (N ) ∗  + O , the narrow ideal class group of F . This allows us to and F ∗ \A∗F /F∞ F relate the adelic and the classical ideal theoretic formulations of Hilbert modular forms as follows. Choose elements g1 , ..., gh in GL2 (F ) such that + O ∗ . We {det(g1 ), ..., det(gh )} forms a set of representatives for F ∗ \A∗F /F∞ F put ti = det(hi ) ∈ F ∗ for all 1 ≤ i ≤ h, and we let Ii be the fractional ideal of F associated to ti so that {I1 , ..., Ih } forms a set of representatives for the narrow ideal class group of F . The map f → fg for g ∈ GL2 (F ), which was introduced in Definition 2.1.5 induces the following isomorphisms:  Mk (N )  Mk (Γgi )  (2.5)  Sk (Γgi )  (2.6)  1≤i≤h  Sk (N ) 1≤i≤h  f → (fg1 , ..., fgh ), GL+ 2 (F )  (2.7)  gi−1 K0 (N )gi G+ ∞.  where Γgi = ∩ With the notation of 2.1, we have Γgi = Γ0 (Ii , N ). In what follows, we define the Fourier coefficients of an adelic Hilbert modular form f of weight k and level N . Let (f1 , ..., fh ) be the h-tuple of classical Hilbert modular forms associated to f via the isomorphism 2.5 . For 1 ≤ i ≤ h, we know that fi ∈ Mk (Γ0 (Ii , N )) for some fractional ideal 1 b Ii of F . Since Γ0 (Ii , N ) contains all elements of the form with 0 1 14  2.1. Hilbert Modular Forms b ∈ Ii−1 , fi has a Fourier expansion ai (ξ)eF (ξz).  fi (z) = ai (0) + D−1  ξ∈Ii ξ 0 k  Notice that ai (ξ)ξ − 2 for ξ = 0 depends only on the ideal ξOF . This follows from the simple observation that Γ0 (Ii , N ) contains all elements of the form k 0 with ∈ OF∗ so that ai ( ξ) = 2 ai (ξ). 0 1 Every non-zero integral ideal M of F can be written as M = ξIi−1 D for a unique i ∈ {1, ...h} and a unique totally positive ξ ∈ Ii D−1 . Define : k  c(M, f ) =  ai (ξ)ξ − 2 0  if M is integral and M = ξIi−1 D otherwise  Then f has the following adelic Fourier expansion at infinity: for y ∈ A∗F , y∞ 0 and x ∈ AF , we have f  y x 0 1  k  =  c(ξyOF , f )(ξy∞ ) 2 eF (ξjy∞ )χF (ξx) 0  +  ξ∈F  c0 (yOF )(N(yOF )−1 |y∞ |) 0  k1 2  if k1 = ... = kd otherwise,  where χF is the standard additive character of AF /F such that χF (x∞ ) = eF (x∞ ) for x∞ ∈ F∞ , and c0 is a function on the narrow ideal class group of F defined by c0 (ηIi−1 ) = ai (0)N(Ii )−  k1 2  for 0  η ∈ F.  One of the problems that is encountered when studying classical Hilbert modular forms is the absence of an action of Hecke operators under which the space of all such forms (of given weight and level) is stable. However, this difficulty is easily overcome by working with the larger space of adelic Hilbert modular forms which unlike its classical counterpart, is invariant under the action of Hecke operators defined naturally as follows. The space Mk (N ) admits a left action of the Hecke algebra of K0 (N )biinvariant compactly supported functions on GL2 (F ). More precisely, for every g ∈ GL2 (F ), the Hecke operator corresponding to the characteristic function of K0 (N )gK0 (N ) is denoted by [K0 (N )gK0 (N )] and maps f to i f (·gi ), where K0 (N )gK0 (N ) = j gj K0 (N ) is a disjoint union. 15  2.1. Hilbert Modular Forms We now introduce the standard Hecke operators TP , UP and [P]k . For a prime ideal P of F , we let πP be a uniformizer of FP . We define TP by putting πP 0 TP = [K0 (N ) K0 (N )]. 0 1 We write UP for TP if P | N . We also define the diamond operator [P]k by putting f (πP x) if P N ([P]k f )(x)) = 0 otherwise, for all x ∈ GL2 (F ). This definition can be extended to integral ideals of F by multiplicativity. Notice that the action of the diamond operators on the space Mk (N ) is trivial in our case because we limit our discussion to Hilbert Modular forms with the trivial character modulo N . For a finite order Hecke character ψ of F whose conductor divides N , the space of adelic Hilbert modular forms of weight k, level N and character ψ is given by: Mk (N , ψ) = {f ∈ Mk (K1 (N )) : f (zg) = ψ(z)f (g) for all z ∈ A∗F }, where the open compact subgroup K1 (N ) is as defined in 2.4. The subalgebra generated by the standard Hecke operators TP , UP and [P]k for all prime ideals P is commutative. It is often referred to as the standard Hecke algebra. Let f and g be adelic Hilbert cusp forms in Sk (N ), and let (f1 , ..., fh ) and (g1 , ..., gh ) be the associated h-tuples respectively. The Petersson inner product h  (f, g) =  fi , gi i=1  endows Sk (N ) with a structure of a Hermitian space with respect to which the operators TP are normal. The standard Hecke operators satisfy various identities which are all elegantly accounted for in the following formal Euler product: TM N(M)−s = M  (1 − TP N(P)−s + [P]k N(P)1−2s )−1 , P  where M runs over all integral ideals of F . Following the notation of Shimura [21], we let k0 be the largest of k1 , ..., kd and put k0 −2 TM = N(M) 2 TM , 16  2.1. Hilbert Modular Forms C(A, f ) = N(A)  k0 2  c(A, f ).  Notice that if A = ξIi−1 D for some totally positive element ξ ∈ F , we get C(A, f ) = N(Ii D−1 )−  k0 2  ai (ξ)ξ  k0 −k 2  ,  where k0 here is understood to be the d-tuple (k0 , ..., k0 ). Then the Fourier coefficients of TM f satisfy the following identity: N(C)k0 −1 C(C −2 AM, f ).  C(A, TM f ) = A+M⊂C  We also have C(A, UP f ) = C(AP, f ), where P is a prime ideal dividing the level N . A nonzero Hilbert modular form f ∈ Mk (N ) that is an eigenform for all standard Hecke operators is called an eigenform. If TM f = λM f , then C(M, f ) = λM C(OF , f ). An eigenform f is said to be normalized if C(OF , f ) = 1. Proposition 2.1.6. [21] The eigenvalues of TM are algebraic numbers. Moreover, if k1 = ... = kd , the eigenvalues of TM are algebraic integers. A Hilbert modular form in Mk (N ) is said to be primitive if it is orthogonal with respect to the Petersson inner product to all forms coming from lower levels. The interested reader may consult Miyake [16] for more details on the construction of the subspace of primitive forms. A normalized primitive eigenform in Mk (N ) is called a newform. The Strong Multiplicity One Theorem for GL2 ([5] Theorem 3.3.6) implies that a primitive form that is an eigenform for all Hecke operators TP (P N ) is necessarily an eigenform for the Hecke operators UP (P|N ) as well. Proposition 2.1.7. [21] Suppose that f is a newform of level N and weight k such that k1 ≡ ... ≡ kd mod 2, and let Q(f ) be the field generated over Q by C(M, f ) for all M. Then Q(f ) is either totally real or a totally imaginary quadratic extension of a totally real algebraic number field. Another important result states that there is a bijection between the newforms f in Sk (N ) and cuspidal automorphic representations π of GL2 , of conductor N and trivial central character, and such that π∞ belongs to the holomorphic discrete series of weight k.  17  2.2. Automorphic Forms on Totally Definite Quaternion Algebras Finally we mention that one can attach to f ∈ Mk (N ) a Dirichlet series C(M, f )N(M)−s ,  L(f, s) = M  where M runs over all integral ideals of F . This converges for sufficiently large Re(s), and can be continued to a meromorphic function on the whole s-plane. It is entire if f is a cusp form.  2.2  Automorphic Forms on Totally Definite Quaternion Algebras  In this section we recall some basics from the theory of Quaternion Algebras which are used extensively in Chapters 3 and 4 of this dissertation. We then explore the connection between automorphic forms on totally definite quaternion algebra and adelic Hilbert modular forms; this connection is made possible by means of the Jacquet-Langlands correspondence.  2.2.1  Quaternion Algebras  The references for the material covered in this subsection are [1], [26] and [27]. Definition 2.2.1. Let F be any field. A quaternion algebra B over F is a central simple F -algebra that has dimension 4 over F . Over a field F with characteristic different from 2, every quaternion algebra B has an F -basis {1, i, j, k} satisfying i2 = a, j 2 = b, ij = −ji and ij = k for some a, b ∈ F ∗ . In this case we denote the quaternion algebra B by a,b a,b admits a standard involution F . Every quaternion F -algebra B = F of the first kind ω → ω called conjugation. If ω = x + yi + zj + tij, with x, y, z, t ∈ F , then w = x − yi − zj − tij. The reduced trace and reduced norm are defined by trd(ω) = ω + ω and nrd(ω) = ωω, respectively. A quaternion algebra over F is either a division F -algebra or a matrix F algebra. If F is algebraically closed, then B is necessarily a matrix algebra. If F is a local field (= C), there exists a unique division quaternion F -algebra up to isomorphism. Let F be a number field and for each place v of F , let Fv be the corresponding local field.  18  2.2. Automorphic Forms on Totally Definite Quaternion Algebras Definition 2.2.2. Let B be a quaternion algebra over F . For each place v of F , Bv = Fv ⊗ B is a quaternion algebra over Fv . If Bv is a division algebra, we say that Bv is ramified at v; otherwise, we say that B is split or unramified at v (Bv M2 (Fv )). A quaternion algebra that ramifies at all the infinite places of F is said to be totally definite. Note that if F is not a totally real number field, every quaternion algebra over F is not totally definite. Definition 2.2.3. The reduced discriminant DB of a quaternion B over F is the square-free product of the prime ideals of OF that ramify in B. It is well-known that two quaternion F -algebra are isomorphic if and only if they are ramified at the same places. The following theorem classifies quaternion algebras over F . Theorem 2.2.4. A quaternion algebra B over F is ramified at a finite even number of places. Moreover, given an even number of noncomplex places of F , there exists a quaternion algebra B over F that ramifies exactly at these places; B is unique up to isomorphism. Definition 2.2.5. A field K containing F is splitting field for B if K ⊗F B is split, i.e. K ⊗F B M2 (K). Given a quaternion algebra B over F , there always exists a field K that splits B. In fact, if F is an algebraic closure of F , then F splits H. The following standard result provides necessary and sufficient conditions under which a quadratic field extension of F splits B. Proposition 2.2.6. Let B be a quaternion algebra over F and K a quadratic field over F . Then the following conditions are equivalent: (a) K splits B. (b) There exists an F -embedding K → B. (c) Every place v in F that ramifies in B is not totally split in K. (d) K is F -isomorphic to a maximal subfield of B containing F . In what follows, we recall the theory of integral structures in quaternion algebras. Note that this discussion applies to quaternion algebras over any field F that arises as the field of fractions of a Dedekind ring. For our purposes F will always be a number field or the completion of a number field at a finite place, and B will always be a quaternion algebra over F . 19  2.2. Automorphic Forms on Totally Definite Quaternion Algebras Definition 2.2.7. An element ω ∈ B is said to be integral over F if its reduced characteristic polynomial x2 − trd(ω)x + nrd(ω) has coefficients in OF . This is consistent with the notion of integrality from commutative algebra. However, unlike the commutative case, the set of integral elements in a quaternion algebra need not form a ring. Definition 2.2.8. An OF -lattice Λ of B is a finitely generated OF -submodule contained in B. An OF -ideal I of B is an OF -lattice such that F ⊗OF I B. The reduced norm nrd(I) of an OF -ideal I is the fractional ideal in F generated by the reduced norms of its elements. Definition 2.2.9. An OF -order R ⊂ B is an OF -ideal of B which is also a subring of B. Corollary 2.2.10. If R ⊂ B is an OF -order, then every ω ∈ R is integral over F . To simplify notation, we refer to OF -ideals and OF -orders of B as ideals and orders of B. An important construction of orders comes from associating left and right orders with ideals. Namely, if I ⊂ B is an ideal, we define the left and right orders to be the sets Rl (I) = {ω ∈ B : ωI ⊂ I} and Rr (I) = {ω ∈ B : Iω ⊂ I}, respectively. These sets are orders but not necessarily equal. If Rl (I) = Rr (I) = R, we say that I is a two-sided or bilateral fractional R-ideal. In general, we say that I is a left fractional Rl (I)-ideal, and a right fractional Rr (I)-ideal. Moreover, the ideal I is integral if and only if it is contained in Rl (I) ∩ Rr (I). Equivalently, the ideal I is integral if and only if it is a left ideal of Rl (I) and a right ideal of Rr (I) in the usual sense. Definition 2.2.11. Let I be an ideal in B. We say I is left invertible if there exists a left fractional Rr (I)-ideal I −1 ⊂ B such that I −1 I = Rr (I). The notion of right invertibility is defined analogously. However, right and left invertibility are equivalent in the presence of a standard involution, and so we simply say I is invertible. Notice that if I is an invertible twosided fractional R-ideal, then its right inverse is equal to its left inverse is equal to {ω ∈ B : IωI ⊂ I}. Proposition 2.2.12. An ideal I ⊂ B is invertible if and only if I is locally principal. 20  2.2. Automorphic Forms on Totally Definite Quaternion Algebras Just as in the commutative setting, the discriminant of an order is realized as the reduced norm of the different ideal. More precisely, we have the following definition. Definition 2.2.13. The different D(R) of an order R ⊂ B is the bilateral integral ideal of R defined as D(R) = (R# )−1 , where R# is the dual of R with respect to the trace form: R# = {ω ∈ B : trd(ωR) ⊂ OF }. The reduced discriminant of R, denoted by ∆(R), is the reduced norm of D(R). We now discuss ideal classes associated to quaternion orders. Let R ⊂ B be an order, and let I and J be invertible right fractional R-ideals. We say that I and J are equivalent on the left if I = bJ for some b ∈ B ∗ . The set Cl(R) of left-ideal classes of R is defined as the set of invertible right fractional R-ideals modulo equivalence on the left. One can define the set of right-ideal classes of R in a similar fashion. However, this doesn’t yield anything new because conjugation I → I = {ω : ω ∈ I} induces a bijection between the sets of right and left fractional R-ideals. The following well-known result provides an adelic interpretation of ideal classes in a quaternion algebra, which is the non-commutative analogue of the class field theoretic realization of ideal classes in a number field. It follows almost immediately from Proposition 2.2.12 above. Proposition 2.2.14. The set Cl(R) is in bijection with B ∗ \B ∗ /R∗ , where B is the ring of finite adeles of B and R = R ⊗ Z. Let S be a finite set of places of F containing the archimedean places and let OF,S be the ring of S-integers in F . Then OF,S is a Dedekind domain with field of fractions F . Let R be an OF,S -order of B. In what follows we study the set Cl(R) by examining the effect of the reduced norm map on the double coset space B ∗ \B ∗ /R∗ . It is easy to verify that nrd(B ∗ ) = F ∗ . A less obvious result known as Eichler’s theorem on norms asserts that ∗ , where F ∗ nrd(B ∗ ) = F(+) (+) is the set of all x ∈ F such that v(x) > 0 for all ramified real archimedean places v of F . Hence, the reduced norm map induces a surjective map: ∗ c : B ∗ \B ∗ /R∗ → F(+) \F ∗ /nrd(R∗ ).  (2.8)  Theorem 2.2.15. If S contains a place v such that B is unramfied at v, then the map 2.8 is a bijection for all OF,S -orders R ⊂ B. 21  2.2. Automorphic Forms on Totally Definite Quaternion Algebras This rather remarkable result follows immediately from Eichler’s strong approximation theorem for quaternion algebras. ∗  Theorem 2.2.16. Let B1∗ and B1 be the kernels of the reduced norm maps ∗ nrd : B ∗ → F(+) and nrd : B ∗ → F ∗ respectively. If S contains a place v ∗  such that B is unramfied at v, then B1∗ is dense in B1 . For the rest of this subsection we study two types of quaternion orders which will be used extensively in the following chapters. Definition 2.2.17. Let B be quaternion algebra over a number field or the completion of a number field at a finite place. An order R ⊂ B is maximal if it is not properly contained in another order. An order R ⊂ B is Eichler if it can be written as the intersection of two maximal orders. Needless to say, the above definition is valid for any quaternion algebra over a field F which arises as the field of fractions of a Dedekind ring. Let B be a quaternion algebra over a number field F . First we study Eichler orders in the local setting. To this end, we fix a finite place v of F and consider the quaternion algebra Bv = Fv ⊗ B. Let πv be a uniformizer of OFv , the ring of integers of Fv . Recall that Bv is either a division algebra or a matrix algebra. It suffices to study orders in these cases since one can easily check that the property of being an Eichler order is preserved by isomorphisms. Proposition 2.2.18. If Bv is a division algebra, then Bv contains a unique maximal order Rv = {ω ∈ Bv : nrd(ω) ∈ OFv }. Hence, Rv is the unique Eichler order in Bv . Lemma 2.2.19. If Bv is the matrix algebra M2 (Fv ), then the maximal orders in Bv are the GL2 (Fv )-conjugate orders of M2 (OFv ). Proposition 2.2.20. Let Rv ⊂ M2 (Fv ) be an order. The following conditions are equivalent: (a) Rv is an Eichler order (b) There exists a unique pair {R1 , R2 } of maximal orders of M2 (Fv ) such that Rv = R1 ∩ R2 . (c) There exists a unique non-negative integer n such that the order Rv is OFv OFv conjugate to the order , which is known as the stanπvn OFv OFv dard Eichler order of level πvn OFv . 22  2.2. Automorphic Forms on Totally Definite Quaternion Algebras The ideal NRv = πvn OFv , determined in statement (c), is called the level of the local Eichler order Rv ⊂ M2 (Fv ). Definition 2.2.21. Let Rv ⊂ Bv be an Eichler order. The level of Rv is the ideal NRv =  OFv if Bv is a division algebra, Nφ(Rv ) where φ : Bv → M2 (Fv ) is an isomorphism.  Let us now study Eichler orders in the global setting. Recall that B is a quaternion algebra over a number field F with ring of integers OF . For an order R of B and a place v of F , put Rv = OFv ⊗ R. If v is a finite place, Rv ⊂ Bv is a local OFv -order; if v is an infinite place, consider OFv = Fv and Rv = Bv . One can easily check that R = B ∩ v Rv . Proposition 2.2.22. Let R be an order of B. Then, R is a maximal (resp. Eichler) order if and only if Rv is a maximal (resp. Eichler) order for every finite place v of F . Hence, we arrive at the following definition of the level of a global Eichler order. Definition 2.2.23. The level NR of a global Eichler order R is the unique integral ideal N ⊂ OF such that Nv is the level of each local order Rv at each finite place v of F . Thus, N = v NRv . Note that maximal orders are characterized by their discriminants. In fact, an order R ⊂ B is maximal if and only if ∆(R) = DB . This is not the case for Eichler orders. Nevertheless, we have the following result. Proposition 2.2.24. If R ⊂ B is an Eichler order, then NR is coprime to DB and ∆(R) = NR DB .  2.2.2  Jacquet Langlands Correspondence  We follow the exposition given in Cornut-Vatsal [7] Section 5. Let B be a totally definite quaternion algebra over a number field F , and let G = ResF/Q (B ∗ ) be the algebraic group over Q whose set of points on a commutative Q-algebra A is given by G(A) = (B ⊗ A)∗ . Notice that G is a reductive group with center Z = ResF/Q (F ∗ ), and the reduced norm map nrd : B → F induces a morphism nrd : G → Z. Definition 2.2.25. An automorphic form of weight 2 on G is a smooth (=locally constant) function θ : G(Q)\G(Af ) → C. 23  2.2. Automorphic Forms on Totally Definite Quaternion Algebras Adapting the notation from [7], we denote the space of all automorphic forms of weight 2 on G by S2 . We let G(Af ) act on S2 by right translation: (g.θ)(x) = θ(xg),  g ∈ G(Af ) and x ∈ G(Q)\G(Af ).  This admissible left action gives rise to a semi-simple representation of G(Af ) known as the right regular representation, and the space S2 is the algebraic direct sum of its irreducible subrepresentations. Definition 2.2.26. An irreducible representation π of G(Af ) is automorphic if it occurs in S2 . We denote by S2 (π ) the corresponding subspace of S2 , so that S2 ⊕π S2 (π ). An automorphic representation π of G(Af ) is either of dimension one or infinite dimension. If π is 1-dimesional, it corresponds to a smooth character χ of G(Af ) which is trivial on G(Q) and factors through nrd : G(Af ) → Z(Af ). A function θ ∈ S2 is said to be Eisenstein if it belongs to the subspace spanned by these finite dimensional subrepresentations of S2 . Equivalently, θ is Eisenstein if and only if it factors through the reduced norm. If π is infinite dimensional, we say that π is cuspidal. A function θ ∈ S2 is said to be a cusp form if it belongs to the G(Af )-invariant subspace of S2 spanned by its irreducible cuspidal subrepresentations. We say that a function θ ∈ S2 has level H if it is fixed by some open compact subgroup H of G(Af ). Hence, the subspace S2H of all such functions may be identified with the finite dimensional space of all complex-valued functions on the finite double quotient MH = G(Q)\G(Af )/H. For each place v of F at which H is maximal and B is unramified, there is a natural Hecke action on S2H given by: Tv θ(x) =  θ(xv,i ),  x = [g] ∈ MH and xv,i = [gηv,i ] ∈ MH .  i∈Iv  0 Hv = i∈Iv ηv,i Hv with v a local uniformizer in Fv . 0 1 Let f be a cuspidal Hilbert newform of parallel weight 2 and level N such that the automorphic representation of GL2 associated with f is squareintegrable (modulo the center) at the places of F where B is ramified. The Jacquet-Langlands correspondence which will be stated shortly implies that there exists a function θ ∈ S2H , for some open compact subgroup H ⊂ G(Af ), Here, Hv  v  24  2.3. Rankin-Selberg L-functions such that θ is an eigenfunction for all Hecke operators Tv with the same eigenvalues as f . Here we will only mention that the subgroup H corresponds to an order R ⊂ B with ∆(R) = N . Theorem 2.2.27. There is a natural bijection between the set of infinite dimensional automorphic representations π of G and the the set of cuspidal representations π of GL2 which satisfy πv is square-integrable (modulo the center) for all places v of F at which B is ramified.  2.3  Rankin-Selberg L-functions  In this section we recall some of the basic properties of Rankin-Selberg Lfunctions with emphasis on the special type of such L- functions which we study in Chapter 3. We follow the exposition of Cornut-Vatsal in [7] Section 1. Let F be a number field and AF its adele ring. Let π1 and π2 be automorphic cuspidal representations of GL2 (AF ) with trivial central characters and conductors N1 and N2 respectively. The Rankin-Selberg L-function attached to the pair {π1 , π2 } is first defined as a product of Euler factors over all places of F : L(π1 , π2 , s) = Lv (π1,v , π2,v , s), v  where the factors are of degree smaller than or equal to four. For every finite place v of F not dividing N1 N2 , let α1 (v) and α2 (v) be the Satake parameters of π1,v , and let β1,v and β2,v be the Satake parameters of π2,v . Then 2  Lv (π1,v , π2,v , s) =  2  (1 − αi,v βj,v qv−s )−1 , i=1 j=1  where qv is the cardinality of the residue field at v. There is a similar but more complicated definition of the Euler factors at the remaining places of F . The reader is referred to [12] and [11] for an elaborate discussion on this topic. Let π be an automorphic cuspidal representation of GL2 over a totally real number field F , and let K be a totally imaginary quadratic extension of F . Given a quasi-character χ of A∗K /K ∗ , we denote by L(π, χ, s) the Rankin-Selberg L-function attached to the pair {π, π(χ)}, where π(χ) is the automorphic representation of GL2 associated with χ. It is well-known that 25  2.3. Rankin-Selberg L-functions L(π, χ, s) has a meromorphic extension to C with functional equation L(π, χ, s) = (π, χ, s)L(˜ π , χ−1 , 1 − s), where π ˜ is the contragredient of π and (π, χ, s) is a certain -factor. If w is the central character of π and χ·w = 1 on A∗F ⊂ A∗K , then L(π, χ, s) is entire and L(π, χ, s) = L(˜ π , χ−1 , s). Hence, the functional equation becomes L(π, χ, s) = (π, χ, s)L(π, χ, 1 − s), and the parity of the order of vanishing of L(π, χ, s) at s = 12 is determined by the value of 1 (π, χ) = (π, χ, ) ∈ {±1}. 2 We say that the pair (π, χ) is even if (π, χ) = 1. Otherwise, we say that pair (π, χ) is odd. Definition 2.3.1. A character χ is said to be a ring class character if there exists some OF -ideal C such that χ factors through the finite group ∗ A∗K /(K ∗ K∞ OC∗ ),  where K∞ = K ⊗ R and OC = OF + COK . The conductor c(χ) of χ is the largest such ideal C. For our purposes, we view the automorphic cuspidal representation π as being fixed of parallel weight (2, 2, .., 2) and level N . We let χ vary through the collection of ring class characters of P-power conductor, where P is a fixed maximal ideal in OF . We also assume that the prime to P part N of N is relatively prime to the discriminant D of K/F . Under these conditions, Cornut and Vatsal obtained the following result in [7]: Proposition 2.3.2. For all but finitely many ring class characters of P power conductor, we have (π, χ) = (−1)|S| , where S is the union of all archimedean places of F together with the those finite places of F which do not divide P , are inert in K and divide N to an odd power. In other words, we have (π, χ) = (−1)[F :Q] η(N ), where η is the quadratic Hecke character of F attached to K/F . For the rest of the current section, we assume that (−1)[F :Q] η(N ) = 1 in which case we say that the tuple (π, K, P ) is definite. Let B be the quaternion algebra over F such that Ram(B) = S. Let G = ResF/Q (B ∗ ) be the algebraic group over Q associated to B ∗ , and Z = ResF/Q (F ∗ ) be 26  2.3. Rankin-Selberg L-functions its center. Since every place in F that ramifies in B is inert in K, there exists an F -embedding K → B. After fixing such an embedding, the group T = ResF/Q (K ∗ ) can be viewed as a maximal sub-torus of G defined over Q. Let π be the automorphic representation of G that corresponds to π = JL(π ) via the Jacquet-Langlands correspondence. In 1985, Waldspurger proved a fundamental theorem (Th´eor`eme 2 in [31]) yielding a criterion for the non-vanishing of L(π, χ, 12 ) in a very general setting. Consider the linear form lχ : S2 (π ) → C defined by the period integral: χ(t)φ(t) dt  lχ (φ) = Z(A)T (Q)\T (A)  Theorem 2.3.3. For a ring class character χ of P -power conductor that satisfies χ.w = 1 on A∗F , we have 1 L(π, χ, ) = 0 ⇔ ∃φ ∈ S2 (π ) : lχ (φ) = 0 2 However, the result of Waldspurger doesn’t give a precise formula for the special value L(π, χ, 21 ). Several authors have subsequently taken up the task of specifying a test vector φ on which to evaluate the linear functional lχ and finding a Gross-Zagier formula for L(π, χ, 12 ) in terms of lχ (φ). In [32], Zhang has obtained a formula of this type under the assumption that the central character of π is trivial and that N , P and D are pairwise co-prime. Another significant improvement in this direction has been made in [15] by Martin and Whitehouse who established a Gross-Zagier formula under the only assumption that P does not divide N .  27  Chapter 3  Special Values of Anticyclotomic L-functions modulo λ In this chapter we generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic Zp -extension. Let g be a cuspidal Hilbert modular form of parallel weight (2, ..., 2) and level N over a totally real field F , and let K/F be a totally imaginary quadratic extension of relative discriminant D. We study the l-adic valuation of the special values L(g, χ, 12 ) as χ varies over the ring class characters of K of Ppower conductor, for some fixed prime ideal P. We prove our results under the only assumption that the prime to P part of N is relatively prime to D.  3.1  Preliminaries and Notations  Let us first fix some notation. We write A (resp. Af ) for the ring of adeles (resp. finite adeles) of Q. Let F be a totally real number field, and let K be a totally imaginary quadratic extension of F the discriminant of which we denote by D. The ring of adeles of F is AF = A ⊗Q F , and the ring of finite adeles of F is F = Af ⊗Q F . Similarly, we write AK (resp. K) for the ring of adeles (resp. finite adeles) of K. We consider a cuspidal Hilbert modular newform g of level N , trivial character and parallel weight (2, ..., 2). We denote by av the Hecke eigenvalues of g (Tv g = av g) and by π the automorphic irreducible representation of GL2 over F corresponding to g. Let P be a prime ideal of F such that P lies over an odd rational prime p, and let πP be a uniformizer of FP . By abuse of notation, we also denote the maximal ideal in OF,P by P. Let χ be a finite-order Hecke character of K of conductor P n . The data of the previous paragraph is to remain fixed and the following hypotheses are assumed throughout this chapter: 1. The representations π and π ⊗ η are distinct, where η is the quadratic 28  3.2. CM Points and Galois Action character associated to the extension K/F . We say that the pair (π, K) is non-exceptional. 2. The prime to P part N of N is relatively prime to the discriminant D of K/F 3. The character χ is a ring class character of P-power conductor, and it is trivial when restricted to A∗F ⊂ A∗K . 4. Let S be the set of all the Archimedean places of F , together with those finite places of F which do not divide P, are inert in K, and divide N to an odd power. We require S to have an even cardinality. It follows from the last condition that the sign in the functional equation of L(π, χ, s) is +1 for all but finitely many characters χ that satisfy condition (3) (see Lemma 1.1 in [7]). Let l be any rational prime. Fix an embedding Q → Ql , and denote by E the subalgebra of Ql generated by the images of the Hecke eigenvalues of g. Write El for the integral closure of E in its field of fractions and λ for the maximal ideal in El .  3.2  CM Points and Galois Action  Let B be the totally definite quaternion algebra over F such that Ram(B) = S. Let G = ResF/Q (B ∗ ) be the algebraic group over Q associated to B ∗ . Thus, the center of G is Z = ResF/Q (F ∗ ). Since every place in F that ramifies in B is inert in K, there exists an F -embedding K → B. After fixing such an embedding, the group T = ResF/Q (K ∗ ) can be viewed as a maximal sub-torus of G defined over Q. In what follows, we sketch the construction of an OF -order R of reduced discriminant N in B following [7] and [33]. Let N be the prime to P part of N , and write N = P δ N . Let R0 be an Eichler order of level P δ in B. We choose R0 such that the OF -order O = OK ∩ R0 has a P-power conductor. For example, if P does not divide N , we require that R0 optimally contains OK . Denote by NB the discriminant of B/F , and let MK be an ideal in OK which has relative norm N /NB . We may find such an ideal MK as follows. For each prime P dividing N , let PK be a prime of OK dividing P. We put [ord (N )/2] ord (N ) MK = PK P . PK P . P|NB  N P| N  B  29  3.2. CM Points and Galois Action Finally, we obtain R by the following formula: R = O + (O ∩ MK ).R0 . In particular, RP = R0,P is an Eichler order of level P δ . Define an open compact subgroup H of G(Af ) by H = R∗ . The subgroup H is sometimes referred to as the level structure. This gives rise to the finite sets MH = G(Q)\G(Af )/H, and NH = Z(Q)+ \Z(Af )/nrd(H). It also gives rise to the set of CM points CMH = T (Q)\G(Af )/H. Notice that any function on MH induces a function on CMH via the obvious reduction map red : CMH → MH . The action of T (Af ) on CMH by left multiplication in G(Af ) factors through the reciprocity map recK : T (Af ) →Galab K . This induces an action on CM . Hence, for x = [g] ∈ CM and σ ∈Galab of Galab H H K , we have K σ.x = [βg] where β ∈ T (Af ) is such that recK (β) = σ. Moreover, the reduced norm map on G(Af ) induces the map c : MH → ab NH . Hence, the action of Galab F on NH induces an action of GalK on NH . For x = [z] ∈ NH and σ ∈Galab K , we have σ.x = [nrd(β)z] where β ∈ T (Af ) is such that recK (β) = σ. We now introduce the notion of a CM point with a P-power conductor. Definition 3.2.1. We say that x = [g] is a CM point of conductor P n and ∗ write x ∈ CMH (P n ) if T (Af ) ∩ gHg −1 = OP n , where OP n ⊂ OK is the OF -order of conductor P n . Choose αP ∈ OK,P such that {1, αP } is an OF,P -basis of OK,P . Since n α } is an O OP n ,P = OF,P + P n OK,P , the set {1, πP P F,P -basis of OP n ,P . We fix the embedding KP → M2 (FP ) defined by a + bαP →  a + bTrαP −b  bNαP a  ,  where Tr and N denote the trace and norm maps.  30  3.3. Uniform Distribution of CM Points Lemma 3.2.2. Consider gP ∈ BP gP =  M2 (FP ) specified as:  n Nα πP P 0  0 1  .  −1 Then, for n large enough, the order gP RP gP in BP optimally contains the n OF,P -order in KP of conductor P .  Proof. Let τ = a + bαP be any element in KP . We have −1 gP τ gP =  a + bTrαP n Nα −bπP P  −n bπP a  .  Recall that, by construction, RP is an Eichler order (of level P δ ) in BP M2 (FP ). Without loss of generality, we identify RP with the order a b c d  ∈ M2 (OF,P ) : c ≡ 0  δ mod πP  .  −1 τ gP ∈ RP if and only if τ ∈ OP n ,P . In Then, for all n 0, we have gP −1 other words, the order gP RP gP in BP optimally contains the OF,P -order in KP of conductor P n .  In the sequel, we shall fix a choice of CM point x = [g] ∈ CMH (P n ) such that its P-th component is as specified in the previous lemma. ∗ is isomorphic to the Iwahori subgroup For later reference, notice that RP U0 (OF,P , P δ ) =  3.3  a b c d  ∈ GL2 (OF,P ) : c ≡ 0  δ mod πP  .  Uniform Distribution of CM Points  The CM points are uniformly distributed on the components of the Shimura curve associated to B. This was the most crucial idea behind Vatsal’s proof of Mazur’s conjecture for weight two modular forms over Q. In this section, we recall a crucial result on the uniform distribution of CM points due to Cornut-Vatsal, which we use to prove our main theorem. To describe this result, we need to introduce some more notation. Let K[P n ] be the ring class field over K of conductor P n . In other words, K[P n ] is the abelian extension of K associated by class field theory ∗ O∗ ∗ to the subgroup K ∗ K∞ P n of AK . Let G(n) denote the Galois group of this extension. We have G(n) = Gal(K[P n ]/K)  ∗ ∗ A∗K /(K ∗ K∞ OP n)  31  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) via the reciprocity map of K. Set K[P ∞ ] = ∪n≥0 K[P n ], so that G(∞) = Gal(K[P ∞ ]/K). The torsion subgroup of G(∞) is denoted by G0 . It is finite and G(∞)/G0 is a free Zp -module of rank [FP : Qp ]. The reciprocity map of K maps A∗F ⊂ A∗K onto the subgroup G2 P ic(OF ) of G0 . Let G(∞) be the subgroup of G(∞) generated by the Frobeniuses of the primes of K which are not above P. Write G1 = G0 ∩ G(∞) . Let D be the square-free product of the primes Q = P of F which ramify in K. Then G1 /G2 is an F2 -vector space with basis {σQ  mod G2 : Q|D },  where σQ = FrobQ and Q is the prime of K above Q. Loosely speaking, the uniform distribution result in [24] states the following. Let p1 and p2 be arbitrary double cosets in MH , and let σ be an arbitrary nontrivial element of G0 with σ ∈ / G1 . Then there exists a CM point x ∈ CM (P n ) such that red(x) = p1 and red(σ.x) = p2 whenever n is sufficiently large. In what follows, we describe the result of Cornut and Vatsal which extends and refines Vatsal’s theorem alluded to in the previous paragraph. Let R be a set of representatives for G0 /G1 containing 1. We have the following maps: R RED : CMH (P ∞ ) → MH , R R C : MH → NH ,  x → (red(τ.x))τ ∈R  (aτ )τ ∈R → (c(aτ ))τ ∈R  and the composite map R C ◦ RED : CMH (P ∞ ) → NH ,  which is G(∞)-equivariant. The following is the key theorem of Cornut-Vatsal as stated in [7]. However, the reader is referred to [6] for a proof of this result. Theorem 3.3.1 (See [7]). For all but finitely many x ∈ CMH (P ∞ ), RED(G(∞).x) = C−1 (G(∞).C ◦ RED(x))  3.4  Toward Computing ordλ (Lal (π, χ, 12 ))  Let π be the unique cuspidal automorphic representation on B that is associated to π by the Jacquet-Langlands correspondence (π = JL(π )). We 32  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) associate to g a unique function θ ∈ S2 (π ), where S2 (π ) is the representation space of π . One can view θ as θ : MH → Eg , where Eg is the Hecke field of g. This yields the function ψ = θ ◦ red on CMH . The space of functions on MH is endowed by an action of Hecke operators Tv . This action agrees with the classical Hecke action on the space of Hilbert modular forms in the sense that θ has the same eigenvalues as g for all Tv , v N . In particular, we may view θ as taking values in El : θ : MH → El . Without loss of generality, we may also assume that θ([g]) is a λ-adic unit for some [g] ∈ MH . There are several results in the literature which relate the special value L(π, χ, 21 ) to |a(x, χ)|2 , for some CM point x ∈ CMH (P n ), with a(x, χ) =  1 |G(n)|  χ(σ)ψ(σ.x). σ∈G(n)  In 1985, Waldspurger proved a fundamental theorem (Th´eor`eme 2 in [31]) yielding a criterion for the non-vanishing of L(π, χ, 12 ) in a very general setting. Roughly speaking, the result of Waldspurger states that, under very mild conditions on π and χ, L(π, χ, 21 ) = 0 if and only if |a(x, χ)|2 = 0. However, this result doesn’t give a precise formula for the special value L(π, χ, 21 ) in terms of |a(x, χ)|2 . Most authors refer to such a formula as a Gross-Zagier formula, and it is expected (but not known yet) that it exists in full generality. Nevertheless, Zhang has proven a formula of this type in [32] under the assumption that the central character of π is trivial and that N , P and D are pairwise co-prime. Another significant improvement in this direction has been made in [15] by Martin and Whitehouse who established a Gross-Zagier formula under the only assumption that P does not divide N. Results as such were the point of departure in the work of Vatsal [24] and Cornut-Vatsal [7] on the non-vanishing of L(π, χ, 12 ). However, in order ¯ Vatsal employed in to study this special value modulo a given prime λ ∈ Q, [25] a construction of Hida to define a canonical period Ωcan such that g L(π, χ, 12 ) 1 Lal (π, χ, ) = 2 Ωcan g 33  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) is a λ-adic integer. Then Vatsal incorporated this period into the GrossZagier formula obtained by Zhang to deduce in one simple step an exact statement about the λ-adic valuation of L(π, χ, 12 ) from that of a(x, χ). Unfortunately, such a construction is not yet known in the degree of generality required in this work. Hence, we will only concern ourselves with studying ¯ as χ varies over the the value of a(x, χ) modulo a given prime ideal λ ⊂ Q ring class characters of K of P-power conductor. Adapting the notation from [7], we identify ring class characters of Ppower conductor with finite-order characters of G(∞). Hence, given a character χ0 of G0 such that χ0 = 1 on G2 , denote by P (n, χ0 ) the set of primitive characters of G(n) (do not factor through G(n − 1)) and induce χ0 on G0 . In [24] and [7], the authors proved that for each character χ0 of G0 and all but finitely many n, there exists a character χ ∈ P (n, χ0 ) such that L(π, χ, 21 ) = 0. Moreover, Vatsal showed in [24] that if χ0 has order prime to p and the Hecke field of g is linearly disjoint from the field generated over Q by the p-th roots of unity, then L(π, χ, 12 ) = 0 for all χ ∈ P (n, χ0 ) with n sufficiently large. We remark that this statement differs slightly from the statement given in [24] since the condition on the Hecke field of g was overlooked there. In [25], Vatsal extended the results and methods of [24] to study the algebraic part of the special value L(π, χ, 21 ) modulo a given ¯ of characteristic l. Vatsal proved that for all n prime λ ∈ Q 0, there exists χ ∈ P (n, χ0 ) such that ordλ  (L(π, χ, 21 ) Ωcan g Ccsp  < ordλ (C2Eis ),  (3.1)  where CEis is a constant that measures the congruence between g and the space of Eisenstein Series, and Ccsp is a constant that measures the congruence between g and some cusp forms of lower levels. Moreover, if χ0 has order prime to p, the Hecke field of g is linearly disjoint from the field generated over Q by the p-th roots of unity, and l satisfies certain conditions (see next paragraph), then (3.1) is true for all χ ∈ P (n, χ0 ) with n sufficiently large. We remark that (3.1) is mistakenly given as an equality in [25], the source of the mistake being an error made in the proof of Proposition 5.3 part (2). We provide a correct version of this result in Proposition 3.4.12 below. We mention here that establishing the correct statement involves modifying the choice of the constant CEis . In fact, the statement becomes ˜ µ where λ ˜ is a λ-adic uniformizer in El and µ is correct if we take CEis = λ the constant given in Definition 3.4.8 below. Given x = [g] ∈ CMH (P n ) and a ring class character χ of conductor P n , 34  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) we define the Gross-Zagier sum a(x, χ) =  1 |G(n)|  χ(σ)ψ(σ.x). σ∈G(n)  In order to prove that a family of values is non-vanishing, it is a standard technique to compute their average. Hence, in order to show that a(x, χ) = 0 is non-vanishing for some χ ∈ P (n, χ0 ), it suffices to show that a(x, χ) = 0.  b(x, χ0 ) = χ∈P (n,χ0 )  If the order of χ0 is prime to p, then all the characters in P (n, χ0 ) are in fact conjugates under the action of Aut(C). If, in addition, the Hecke field of g is linearly disjoint from the field generated over Q by the p-th roots of unity, it follows that the sums a(x, χ) are also conjugates for all χ ∈ P (n, χ0 ). Hence, the non-vanishing of a(x, χ) for some χ ∈ P (n, χ0 ) forces the non-vanishing of a(x, χ) for all χ ∈ P (n, χ0 ). Moreover, Vatsal noticed in [25] that if l splits completely in the field Q(χ0 ) generated by the values of χ0 , and if it is inert in the field Q(µp∞ ) generated by all p-power roots of unity, then all the characters in P (n, χ0 ) are conjugates under the action of a decomposition group Dλ . Thus, the sums a(x, χ) have the same λ-adic valuation for all χ ∈ P (n, χ0 ). Our goal is to prove an analogue of Theorem 1.2 in [25] (see also Proposition 4.1 and Corollary 4.2) for a Hilbert modular form g over a totally real field F , while removing the above mentioned assumptions on l and the order of χ0 . More precisely, given a character χ0 of G0 , we prove that ordλ (a(x, χ)) < µ for all χ ∈ P (n, χ0 ) with n 0, where µ is a constant to be specified at a later stage (see Definition 3.4.8). We know by Lemma 2.8 in [7] that we can identify G0 with its image G0 (n) in G(n) whenever n is sufficiently large. We denote the quotient group G(n)/G0 (n) by H(n). Suppose that χ0 is a fixed character of G0 and let χ ∈ P (n, χ0 ). One can express χ as χ = χ0 χ1 , where χ0 is some character of G(n) inducing χ0 on G0 (n) G0 , and χ1 is some character of H(n). Recall that El is an l-adically complete discrete valuation ring containing the Fourier coefficients of g. Enlarge El if necessary to contain the values of χ0 , and let El (χ1 ) be the field obtained by adjoining to El the values of χ1 . We assume without loss of generality that the order of χ0 is divisible by p. 35  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) Consider the trace of a(x, χ) taken from El (χ1 ) to El : σ(a(x, χ)).  Tr(a(x, χ)) = σ∈Gal(El (χ1 )/El )  This trace expression is different than the average expression a(x, χ)  b(x, χ) = χ∈P (n,χ)  considered in the work of Vatsal and Cornut-Vatsal. Nevertheless, the same approach is followed to study both expressions. Evidently, given any χ ∈ P (n, χ0 ), the non-vanishing of Tr(a(x, χ)) would then imply the nonvanishing of a(x, χ). Notice that Tr(a(x, χ)) =  1 Tr |G(n)| El (χ1 )/El  1 = |G(n)|  χ0 (σ)χ1 (τ )ψ(στ.x) σ∈G0 (n) τ ∈H(n)  χ0 (σ) σ∈G0  [El (χ1 ) : El ] = |G(n)|  ψ(στ.x)TrEl (χ1 )/El χ1 (τ ) τ ∈H(n)  χ0 (σ) σ∈G0  ψ(στ.x)χ1 (τ ). τ ∈H(n) χ1 (τ )∈El  Let m be the highest power of p dividing the order of χ0 , and put Z(n, m) = {τ ∈ H(n) : order(τ ) | pm }. Since Z(n, m) = {τ ∈ H(n) : χ1 (τ ) ∈ El }, we get Tr(a(x, χ)) =  [El (χ1 ) : El ] |G(n)|  Lemma 3.4.1. For n  χ1 (τ )ψ(στ.x).  χ0 (σ) σ∈G0  0, Z(n, m)  τ ∈Z(n,m)  OF /P m .  Proof. It follows from the definition of Z(n, m) that Z(n, m) = ker(H(n) H(n − m)). We also have an isomorphism between Z(n, m) and ker(G(n) G(n − m)) induced by the natural quotient map G(n) H(n). The reciprocity map induces an isomorphism between ker(G(n) G(n − m)) and ∗ ∗ ∗ K ∗ OP n−m /K OP n  ∗ ∗ ∗ OP n−m ,P /OP n−m OP n ,P .  36  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) ∗ ∗ ∗ Notice that OP n−m = OF is contained in OP n ,P for sufficiently large n, so that ∗ ∗ ker(G(n) G(n − m)) OP n−m ,P /OP n ,P .  On the other hand, n−m ∗ ∗ OP n−m ,P /OP n ,P = {1 + aαP πP  ∗ m mod OP n ,P : a ∈ OF,P /P },  where αP ∈ OK,P is such that {1, αP } is an OF,P -basis of OK,P . This yields the desired isomorphism. Define a function θm on G(Af ) by: θm (g) =  −m 1 aπP 0 1  χ1 (τa )θ(g.(1, 1, ..., a∈OF,P  /P m  , ..., 1, 1)),  P th place  where τa ∈ Z(n, m) is the element associated to a ∈ OF,P /P m . To simplify notation, we identify (1, 1, .., bP , .., 1, 1) ∈ G(Af ) with bP ∈ GL2 (FP ) BP∗ . For example, we write θm (g) =  −m 1 aπP 0 1  χ1 (τa )θ(g. a∈OF,P  /P m  ).  ∗  Lemma 3.4.2. The function θm has level Hm = Rm for some OF -order ∗ Rm which agrees with R outside P. At P, the subgroup Rm,P is isomorphic to the Iwahori subgroup a b c d  U1 (OF,P , P max(2m,δ) ) =  ∈ U0 (OF,P , P max(2m,δ) ) : a ≡ 1  max(2m,δ)  mod πP  where δ is such that N = P δ N and N is prime to P. Proof. Let γ be any element in U0 (OF,P , P max(2m,δ) ). The Iwahori factor1 0 d1 0 1 u ization of γ yields γ = . We view γ as an l 1 0 d2 0 1 element in G(Af ). For g ∈ G(Af ), we have θm (gγ) =  χ1 (τa )θ(gγ a∈OF,P  /P m  =  −m 1 aπP 0 1  χ1 (τad−1 d2 )θ(g  −m 1 aπP 0 1  χ1 (τad−1 d2 )θ(g  −m 1 aπP 0 1  1  a∈OF,P /P m  =  1  a∈OF,P /P m  ) −m −2m 1 − laπP −la2 πP −m l 1 + laπP  )  37  )  ,  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) If we further assume that γ ∈ U1 (OF,P , P max(2m,δ) ), then d1 ≡ d2 ≡ 1 max(2m,δ) mod πP . Hence, θm (gγ) = θm (g). We now make the following important observation. Since a(γ.x, χ) = χ−1 (γ)a(x, χ) for any γ ∈ G(n), and G(n) acts simply and transitively on CMH (P n ), it suffices to study the λ-adic valuation of a(y, χ) for any y ∈ CMH (P n ). Henceforth, we shall take for x the CM point obtained in Lemma 3.2.2 and fix the choice of representative g determined in the lemma as well. We denote by xm the class of g in CMHm . Lemma 3.4.3. Let ψm denote the function induced by θm on CMHm . We have χ1 (τ )ψ(τ.x) = ψm (xm ). τ ∈Z(n,m) n−m in Proof. By Lemma 3.4.1, we view τ ∈ Z(m, n) as the class of 1+aαP πP ∗ ∗ m OP n−m ,P /OP n ,P for some a ∈ OF,P /P . We then identify τ with its image in GL2 (OF,P ):  τ→  n−m NαP aπP 1  n−m TrαP 1 + aπP n−m −aπP  .  Write x = [g], where g ∈ G(Af ) is as defined in Lemma 3.2.2. We have χ1 (τ )ψ(τ.x) =  χ1 (τa )ψ(τa .x) a∈OF,P /P m  τ ∈Z(n,m)  =  χ1 (τa )θ( a∈OF,P  /P m  =  χ1 (τa )θ(g  n−m TrαP 1 + aπP 2n−m −aπP NαP  χ1 (τa )θ(g  −m 1 aπP 0 1  a∈OF,P /P m  = a∈OF,P  n−m TrαP 1 + aπP n−m −aπP  /P m  n−m NαP aπP 1 −m aπP 1  )  )  = ψm (xm ). The fourth line follows from the fact that n−m 1 + aπP TrαP 2n−m −aπP NαP  −m aπP 1  =  where d1 , d2 ∈ OF,P , and l ∈ P δ for n  −m 1 aπP 0 1  d1 0 0 d2  1 0 l 1  ,  0. 38  g)  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) Since χ0 = 1 on G2 , we have: Tr(a(x, χ)) =  |G2 |[El (χ1 ) : El ] |G(n)|  χ0 (σ)ψm (σ.xm ). σ∈G0 /G2  We can reduce the above sum into something even simpler by means of another level raising step. Proposition 3.4.4. There exists an OF -order Rm,D , a nonzero function ∗ θm,D of level Hm,D = Rm,D on G(Af ), and for each n ≥ 0, a Galois equivariant map x → xm,D from CMH (P n ) to CMHm,D (P n ) such that ψm,D (xm,D ) =  χ0 (τ )ψm (τ.xm ), τ ∈G1 /G2  where ψm,D = θm,D ◦ red. Proof. The reader is referred to the proof of Lemma 5.9 in [7] Hence, the trace expression simplifies to Tr(a(x, χ)) =  |G2 |[El (χ1 ) : El ] |G(n)|  χ0 (σ)ψm,D (σ.xm,D ). σ∈G0 /G1  We now study the λ-adic valuation of the sum χ0 (σ)ψm,D (σ.xm,D ), σ∈R  where R is a set of representatives for G0 /G1 containing 1. Definition 3.4.5. Let k be any ring. A k-valued function φ on MH is said to be Eisenstein if it factors through NH via the map c, where as φ is said to be exceptional if there exists z ∈ NH such that φ is constant on c−1 (σ.z) for all σ ∈ Galab K. ∗  Choose an ideal C in OF such that nrd(H) contains all elements of OF congruent to 1 modulo C. Such an integral ideal exists because nrd(H) is open in F ∗ . For a finite prime v of F , let qv be the cardinality of the residue class field at v. Denote by S the set of all finite places of F that do not divide N and correspond to a principal prime ideal aOF with a ≡ 1 mod C and a is totally positive. 39  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) Lemma 3.4.6. If φ is Eisenstein modulo λr for some positive integer r, then av ≡ qv + 1 mod λr for all v ∈ S. Proof. Let v be a finite place in F corresponding to a principal prime ideal Q = aOF with a ≡ 1 mod C and a is totally positive. Choose x ∈ MH such that φ(x) is a λ-adic unit. By definition, we know that Tv φ(x) =  φ(xηv,i ). i∈Iv  a 0 Hv = i∈Iv ηv,i Hv . Notice that c(xηv,i ) = c(xηv ), where 0 1 a 0 , 1, ..., 1). ηv = (1, .., 1, 0 1 Since φ is Eisenstein modulo λr , we get Here Hv  Tv φ(x) ≡ (1 + qv )φ(xηv )  mod λr .  Choose d ∈ G(Q) such that nrd(d) = a. Notice that nrd(ηv−1 d) = (a, ..., a, 1, a, ..., a) ≡ 1  mod C.  We thus obtain an element h ∈ H such that nrd(h) = nrd(ηv−1 d). Hence, φ(xηv ) ≡ φ(xηv d−1 ) ≡ φ(xh−1 ) ≡ φ(x)  mod λr .  On the other hand, we know that Tv φ(x) = av φ(x). Putting all of this together gives av φ(x) ≡ (1+qv )φ(x) mod λr , which implies that av ≡ 1+qv mod λr since φ(x) is a λ-adic unit. Lemma 3.4.7. If φ is exceptional but non-Eisenstein modulo λr for some positive integer r, then av is a λ-adic non-unit for all finite places v of F that are inert in K and do not divide N . Proof. The argument given here is drawn from [7]. Recall that we have an action of the group Galab K on NH , and one can show that there are at most ab two GalK -orbit in NH . If there were only one Galab K -orbit in NH , then any exceptional function on MH would also be Eisenstein. Since φ is exceptional and non-Eisenstein modulo λr , we know there must be exactly two Galab Korbits in NH , which we denote by X and Y with φ being constant modulo λr on c−1 (z) for all z ∈ X. Since φ is non-Eisenstein modulo λr , there exist y ∈ Y and some x1 , x2 ∈ c−1 (y) such that φ(x1 ) ≡ φ(x2 ) mod λr . 40  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) Let v be a finite place of F that is inert in K and does not divide N . For any x ∈ MH , we know that Tv φ(x) = av φ(x) =  φ(xηv,i ). i∈Iv  We also know that if x ∈ c−1 (y) then xηv,i ∈ c−1 (Frobv .y). Since v is inert in K, we get Frobv .y ∈ X, so that φ is constant modulo λr on c−1 (Frobv .y) with φ(v, y) being the common value. Hence, av φ(x1 ) ≡ (1 + qv )φ(v, y) ≡ av φ(x2 )  mod λr .  It follows that av is a λ-adic non-unit, since otherwise φ(x1 ) and φ(x2 ) would be congruent modulo λr . We shall assume henceforth that g satisfies the condition: av is a λ-adic unit for some v inert in K, v N . Definition 3.4.8. Let µ be the smallest integer such that av ≡ 1 + qv mod λµ for some v ∈ S, v D. It follows immediately from the definition of µ that the function θ is non-exceptional modulo λµ . We let f be the inertia degree of P over p. If λ lies above p (l = p), we let e be the corresponding ramification index. In this case, we denote by k the ring El /λemf +µ El . If λ does not lie above p (l = p), we denote by k the ring El /λµ El . We shall view θ, θm and θm,D as k-valued functions. Proposition 3.4.9. The function θm : MHm → k is a non-zero eigenfunction for all Hecke operators Tv (v PN D ) with Tv θm = av θm . Proof. It is clear that θm is an eigenfunction for all Hecke operators Tv (v PN D ) with Tv θm = av θm . If θm = 0 as a k-valued function, then 0= u∈(OF,P /P m )∗  u 0 0 1  =  .θm χ1 (τua )  u∈(OF,P  /P m )∗  = a∈OF,P /P m  a∈OF,P  /P m  −m 1 aπP 0 1  −m 1 aπP 0 1  .θ  .θ  χ1 (τua ) u∈(OF,P /P m )∗  41  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) Let q = pf denote the cardinality of the residue class field OF /P. By means of Lemma 3.4.11 below, we get 0 = q m−1 (q − 1)θ − q m−1 a∈(OF,P /P)∗ −1 1 aπP 0 1  = q m θ − q m−1 a∈OF,P /P   1 0  = q m−1 qθ − a∈OF,P /P  =q  −1 aπP  1  −1 1 aπP 0 1  .θ  .θ  .θ  m−1 +  θ .  This yields a contradiction since q m−1 θ+ is non-zero by Lemma 4.12 in [7]; the proof of this lemma uses the fact that θ is non-Eisenstein modulo λµ . The reader is referred to [7] for a description of the function θ+ and its properties (see, for example, Section 1.6, Theorem 5.10 and the Appendix). Corollary 3.4.10. θm is non-exceptional as a k-valued function. Lemma 3.4.11. For a ∈ OF /P m , we have  m−1 (q − 1) a ∈ Pm  q m−1 −q a ∈ P m−1 /P m and a ∈ / Pm χ1 (τua ) =  0 otherwise u∈(OF /P m )∗ Proof. The statement of the lemma follows trivially for a ≡ 0 mod P m . For the remaining cases, we write χ1 (τua ) −  χ1 (τua ) = u∈(OF  /P m )∗  u∈OF  /P m  χ1 (τua ) u∈P/P m  Notice that if a ∈ P m−1 /P m , we have χ1 (τua ) = q m−1 .  χ1 (τua ) = 0 and u∈OF /P m  u∈P/P m  Otherwise, we get χ1 (τua ) = u∈OF /P m  χ1 (τua ) = 0 u∈P/P m  . 42  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) Proposition 3.4.12. The function θm,D : MHm,D → k is a non-zero eigenfunction for all Hecke operators Tv away from PN D with Tv θm,D = av θm,D . Proof. By definition (see [7] p. 57), θm,D =  χ0 (σd )(αd .θm ), d|D  αQ , and αQ is any element in RQ ∼ M2 (OF,Q ). Notice that  where αd = Q|d  θm,D is left-invariant under Hm,D = Rm,D where Rm,D is the unique OF −1 order which agrees with Rm outside D and equals RQ ∩αQ RQ αQ at Q | D . If we fix a prime divisor Q of D , it is easy to see that θm,D can be rewritten as θm,D =  χ0 (σd )(αd .θm ) + χ0 (σQ )αQ . d| D Q  χ0 (σd )(αd .θm ). d| D Q  Let ϑ1 and ϑ2 be k-valued functions on MHm satisfying Tv ϑi = av ϑi for all v PN D . We claim that any nontrivial linear combination aϑ1 +bαQ .ϑ2 is nonzero in k. If aϑ1 + bαQ .ϑ2 = 0 for some scalars a and b, then aϑ1 = −bαq .ϑ2 ∗ and α R∗ α−1 which contains the is fixed under the group spanned by RQ Q Q Q kernel of the reduced norm map BP → FP . It follows from the strong approximation theorem ([26] p. 81) that ϑ1 factors through the norm map as a k-valued function, which is a contradiction to the fact that av ≡ qv + 1 mod λµ for some v ∈ S (Lemma 3.4.6). Hence, aϑ1 +bαQ .ϑ2 is non-zero. Not only this, but aϑ1 + bαQ .ϑ2 is also an eigenfunction for all Tv (v PN D ) with the same eigenvalues as ϑ1 and ϑ2 . In light of the above observation, we proceed by induction on the number of prime ideal divisors of D to prove that θm,D is non-zero and satisfies Tv θm,D = av θm,D for all v PN D . This reduces the problem to the case of θm which satisfies the required hypothesis by Proposition 3.4.9. Corollary 3.4.13. θm,D is non-exceptional as a k-valued function. Finally, we state and prove the main result in this chapter. This result gives an upper bound for the λ-adic valuation of the sum χ0 (τ )ψm,D (τ.xm,D ), τ ∈R  43  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) which we recall is related to the Gross-Zagier sum a(x, χ) by the formula Tr(a(x, χ)) =  |G2 |[El (χ1 ) : El ] |G(n)|  χ0 (τ )ψm,D (τ.xm,D ). τ ∈R  Theorem 3.4.14. Let χ0 be any character of G0 . For any x ∈ CMHm,D (P n ) with n 0, there exists some y ∈ G(∞).x such that χ0 (τ )ψm,D (τ.y) ≡ 0  (in k).  τ ∈R  Proof. We follow the proof of Corollary 5.7 in [7]. Since θm,D is nonexceptional as a k-valued function, there exists σ ∈ G(∞) such that θm,D is non-constant as a k-valued function on c−1 (c ◦ red(σ.x)). Choose p1 , p2 ∈ c−1 (c ◦ red(σ.x)) such that θm,D (p1 ) ≡ θm,D (p2 ) (in k). If n is sufficiently large, Theorem 3.3.1 guarantees the existence of y1 , y2 ∈ G(∞).x such that red(y1 ) = p1 ,  red(y2 ) = p2  and red(τ.y1 ) = red(τ.x) = red(τ.y2 )  for all τ = 1 in R.  We thus obtain χ0 (τ )ψm,D (τ.y1 ) − τ ∈R  χ0 (τ )ψm,D (τ.y2 ) = θm,D (p1 ) − θm,D (p2 ) τ ∈R  ≡ 0 (in k). Therefore, at least one of the sums χ0 (τ )ψm,D (τ.y1 ) τ ∈R  or  χ0 (τ )ψm,D (τ.y2 ) τ ∈R  is non-zero in k. Finally, we remark that one can easily obtain a lower bound on the ladic valuation of the Gross-Zagier sum a(x, χ). In fact, let ν be the largest  44  3.4. Toward Computing ordλ (Lal (π, χ, 21 )) integer such that θ is Eisenstein modulo λν . Then χ(σ)θ ◦ red(σ.x) =  χ(σ)θ(red(σ.x)) σ∈G(n)  σ∈G(n)  χ(σ)θ(c ◦ red(σ.x))  ≡ σ∈G(n)  χ(σ)θ(σ.c ◦ red(x))  ≡ σ∈G(n)  χ(σ)θ(nrd(β)c ◦ red(x))  ≡ σ∈G(n)  ≡0  mod λν ,  where the last line follows from the orthogonality property of group characters. Hence,   ordλ   χ(σ)ψ(σ.x) ≥ ν.  σ∈G(n)  Assume for simplicity that l = p, then it is obvious that this simple observation combined with Theorem 3.4.14 would give an exact value for the l-adic valuation of a(x, χ) if we have ν + 1 = µ. However, it is not clear to us whether the statement ν + 1 = µ is true or not. This is a very interesting question, but we choose not to discuss it in this work. We remark only that the answer seems to be connected to multiplicity-one-type results for the component group of a Shimura curve at Eisenstein primes.  45  Chapter 4  Fourier Coefficients of Half-Integral Weight Modular Forms Let k be a positive integer such that k ≡ 3 mod 4, and let N be a positive square-free integer. In this chapter, we compute a basis for the twodimensional subspace S k (Γ0 (4N ), F ) of half-integral weight modular forms 2 associated, via the Shimura correspondence, to a newform F ∈ Sk−1 (Γ0 (N )), which satisfies L(F, 12 ) = 0. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.1  4.1  The Shimura Correspondence  Let k be an odd positive integer, and M be a positive integer divisible by 4. A modular form of half-integral weight k2 for Γ0 (M ) is a holomorphic function f on the upper half-plane which is also holomorphic at the cusps and transforms like the k-th power of the theta series 2z  e2πin  θ(z) = 1 + 2 n≥1  under fractional linear transformations of Γ0 (M ). If f vanishes at all the cusps, we say that it is a cusp form and write f ∈ S k (Γ0 (M )). We also 2  denote by S k (M, χ), the space of cusp forms with central character χ 2  of conductor dividing M . The Kohnen subspace in S k (M, χ) consists of 2 1  The contents of this chapter have been published in the LMS Journal of Computation and Mathematics [10]  46  4.1. The Shimura Correspondence an e2πinz with the Fourier coefficients an satisfying  f (z) = n≥1  an = 0, when χ(2) (−1)(−1)  k−1 2  n ≡ 2, 3  mod 4,  where χ(2) is the 2-primary component of χ. One can define an action of Hecke operators on the space of half-integral weight modular forms. The different Hecke operators commute (Tm2 Tn2 = Tm2 n2 , gcd(m, n) = 1), and the operator Tp2v is a polynomial in Tp2 . However, one can have non-trivial operators Tm only for square m or for (m, M ) = 1. Hence, for a half-integral weight modular form which is an eigen-form for all Hecke operators, one can only relate its coefficients whose indices differ by a perfect square ([13], Proposition 14). For a more detailed exposition on half-integral weight modular forms the reader is referred to [13]. In 1973, Shimura proved a fundamental result which gives a correspondence between modular forms of half-integral weight and modular forms of even integral weight. Let S 3 (M, χ) denote the orthogonal complement 2 (with respect to the Petersson inner product) of the subspace of S 3 (M, χ) 2  spanned by the Shimura theta series ∞ 2 mz  ψ(n)ne2πin  θψ,m (z) = n=−∞  for all positive integers m and odd primitive Dirichlet characters ψ (see [23]). For an odd integer k ≥ 5, we put for notational convenience S k (M, χ) = 2 S k (M, χ). Adapting the notation from [29], we let 2  new Sk−1 (χ2 ) =  new Sk−1 (N, χ2 ), N >0  new (N, χ2 ) is the (finite) subset of newforms in S 2 where Sk−1 k−1 (N, χ ). If new 2 F ∈ Sk−1 (χ ) is such that Tp F = bp F for all p, one defines the Shimura lift of F to be the subspace  S k (M, χ, F ) = {f ∈ S k (M, χ) : Tp2 f = bp f for almost all p |M }. 2  2  Shimura showed that if f ∈ S k (M, χ) is an eigenform for almost all Hecke 2  new (χ2 ) such that f ∈ S (M, χ, F ). operators, then there exists a unique F ∈ Sk−1 k 2 This assignment is at the heart of the Shimura correspondence. The following is a simplified version of Shimura’s original theorem.  47  4.1. The Shimura Correspondence Theorem 4.1.1. (See [20]) Let f ∈ S k (M, χ) be a common eigenfunction 2 for all Tp2 with λp being the corresponding eigenvalue. Define the sequence of complex numbers {bn } by the formal identity ∞  bn n−s = n=1  p  1 − λp  p−s  1 + χ(p2 )pk−2−2s  ∞ 2πinz belongs to S 2 Then F (z) = k−1 (N , χ ) for some integer N n=1 bn e 2 which is divisible by the conductor of χ .  Thus, each eigenform of weight k2 is associated to a form of integral weight k − 1. However, it is not clear (and in general very far from true) that this correspondence is bijective on the level of eigenforms. Indeed, the principal point for us in this chapter is that the number of eigenforms of weight k2 of a given level and character associated to a fixed F is a rather subtle invariant. To clarify the situation, we introduce the following setup. new Let F ∈ Sk−1 (χ2 ) be a newform of even weight k − 1, level N and character 2 χ defined modulo N . Let M be a positive integer divisible by 4, and suppose that χ is defined modulo M . (Here N may or may not divide M .) Consider the space S k (M, χ, F ) defined in a previous paragraph. We are 2 led to the following questions: 1. What is the dimension of the space S k (M, χ, F )? 2  2. If it is non-zero, can we compute a basis for this space? It turns out that the answer to these questions is extremely delicate, and relies in a fundamental way on the representation theory of the metaplectic covers of SL2 and GL2 . The answer is rendered complicated by two main factors: firstly, that a certain global theta correspondence is trivial when a certain L-function vanishes, and secondly, that there are severe local complications in the representation theory of the metaplectic group. The first causes a natural construction of forms in S k (M, χ, F ) to vanish, while the 2 second shows that there is no theory of newforms on the half-integral side, and that the space one is trying to construct may have high dimension. The construction of forms in S k (M, χ, F ) if non-empty for some M and 2 χ was taken up by Shintani [22], and solved in general by Flicker [8]. It is not at all clear, a priori, which characters χ and which integers M one has to take in order to obtain a non-zero space. On the other hand, the questions of finding the dimension and a basis for S k (M, χ, F ) were taken up by Waldspurger in [29]. In fact, Waldspurger 2  48  4.1. The Shimura Correspondence proved that, under quite general conditions on F , N and χ, there exists a basis for S k (M, χ, F ) such that for every positive integer n the Fourier 2 coefficient an (fi ) of a basis element fi is the product of two factors: a product of local terms ci (n, F ) each of which is completely determined by the local components of F according to explicit formulae given in [29], and a global factor AF (n) whose square is the central critical value of the L-function of the newform F twisted by a quadratic character depending on n. A more detailed discussion of the representation-theoretic subtleties of this circle of questions is given in Section 3 below. The main point of our work is to answer these questions in the simplest case, when χ is trivial, and M = 4N , with N odd and square-free. More precisely, we will compute a basis for S k (Γ0 (4N ), F ) for an odd square-free integer N and k ≡ 3 2  mod 4 provided that L(F, 12 ) = 0 (the L-function is normalized so that the functional equation is with respect to s → 1 − s rather than s → k − 1 − s). In the light of Waldspurger’s work, our task is reduced to computing the global factors AF (n). The organization of the chapter is as follows. In Section 2, we present a construction of a half-integral weight modular form g which belongs to the new (Γ (N )). Kohnen subspace of S k (Γ0 (4N ), F ) for a given newform F ∈ Sk−1 0 2 The squares of the Fourier coefficients of this form are essentially proportional to the central critical values of the L-function of F twisted by some quadratic characters. In Section 3 we digress briefly to discuss the representation-theoretic interpretation of the Shimura correspondence as a theta correspondence. In Section 4, we state the full result of Waldspurger. Next, we show that the space S k (Γ0 (4N ), F ) is in fact two-dimensional and 2 has a distinguished basis {f1 , f2 } such that either f1 −f2 or f2 belongs to the Kohnen subspace. We use this to express the global factors AF (n) in terms of the coefficients of g (Theorem 4.4.5). Finally, we explicitly determine the Fourier coefficients of the two modular forms that generate S k (Γ0 (4N ), F ), 2 thus, arriving at our main contribution given in Theorem 4.4.7. The last section contains examples to illustrate the calculations carried out in Section 4. In conclusion, we remark that the problem of finding a basis for S k (M, χ, F ) 2 (in a more general setting than the one assumed in our present work) can undoubtedly be solved by representation-theoretic techniques, and by generalizing the framework sketched in Section 3 of the present chapter. We hope to take this up in a future work.  49  4.2. A Modular Form in S k (Γ0 (4N ), F ) 2  4.2  A Modular Form in S k (Γ0 (4N ), F ) 2  We first recall the definition of the L-function associated to a newform F ∈ new (Γ (N )). Following Waldspurger in [29], we set Sk−1 0 L(F, s) = 2(2π)1−s−k/2 N  s−1+k/2 2  ∞  bn (F )n1−s−k/2  Γ(s−1+k/2) n=1  3 Re s > . 2  Then L(F, s) extends to an entire function which satisfies a functional equation with respect to s → 1 − s. Moreover, since we assume at the outset that F is a newform with the trivial character, F is also an eigenform for the Atkin-Lehner Operator WN , so that WN F = wN F for some wN ∈ C. Hence, the functional equation for L(F, s) takes on the form: L(F, s) = ik−1 wN L(F, 1 − s). It is known that WN is an involution when k − 1 is even, so wN = ±1. In particular, if k ≡ 3 mod 4 and L(F, 12 ) = 0, then the root number in the functional equation is ik−1 wN = −wN = 1. Otherwise, L(F, 12 ) would vanish trivially. Let F be a newform in S2new (Γ0 (N )) with an odd square-free level N such that L(F, 21 ) = 0. For each prime divisor p of N we denote by wp the eigenvalue of the Atkin-Lehner involution Wp . Since wN = wp and wN = −1 by the above discussion, the set S = {p|N : wp = −1} has an odd cardinality. Notice that if we consider a newform of higher even weight k − 1, then |S| is odd if we also assume that k ≡ 3 mod 4. Consider the definite quaternion algebra B over Q ramified at S and ∞. Let O be an order in B such that Oq is maximal in Bq for all q ∈ S and is of index p in a maximal order for the remaining primes p dividing N . Such an order is called an Eichler order of square-free level N in B. The relation between orders of level N and modular forms on Γ0 (N ) will be clear in what follows. The Brandt module which we shall denote by X(R) is the free abelian group on the left ideal classes of a quaternion order of level N along with a natural Hecke action determined by the Brandt matrices. Pizer proved in [18] that there exists a Hecke algebra isomorphism between the Brandt module and a subspace of modular forms containing all the newforms of level N ([18], Corollary 2.29 and Remark 2.30). In the representation theoretic language of Jacquet-Langlands, Pizer’s result can be interpreted as giving a correspondence between automorphic forms on the adelization B(A) and automorphic forms on GL(2, A). 50  4.2. A Modular Form in S k (Γ0 (4N ), F ) 2  Recall that a left O-ideal I is a lattice in B such that Ip = Op ap (for some ap ∈ Bp∗ ) for every prime p. The set of all such ideals is denoted by C. Two left O-ideals I and J are equivalent if there exists a ∈ B ∗ such that I = Ja. This gives rise to the set C¯ = {[I1 ], [I2 ], ..., [IH ]} of left O-ideal classes which has a finite cardinality H. We also define the norm of an ideal I to be the positive rational number which generates the fractional ideal of Q generated by {N (x) : x ∈ I}. Here N (x) denotes the reduced norm of the quaternion element x. Define the right order of a left O-ideal I to be the set Or (I) = {a ∈ B : Ia ⊆ I}; this is also an order of level N in B. ¯ the set of left ODenote by X the free abelian group with basis C, ideal classes. We shall now illustrate the well-known construction of Hecke operators acting on the R-vector space X(R) = X ⊗Z R. We define a height pairing ( , ) on X with integer values by setting ([Ii ], [Ij ]) = 0 if [Ii ] = [Ij ] and ([Ii ], [Ii ]) = 21 #Or∗ (Ii ), then extending bi-additively. This height pairing induces an inner product on X(R). For each n ≥ 1, the Hecke operator tn : X(R) → X(R) is defined by : H  tn ([Ii ]) =  (B(n))ij [Ij ]. j=1  The entries of the Brandt matrix B(n) are calculated using (B(n))ij =  1 N (x) × #{x ∈ Ij −1 Ii : = n}, ej N (Ij −1 Ii )  where ej = #{x ∈ Or (Ij ) : N (x) = 1} = #Or∗ (Ij ). In other words, ej (B(n))ij is the nth coefficient in the Fourier expansion of the theta series N (x) −1 I ) i  q N (Ij  θij (z) = x∈Ij  −1 I  (q = exp(2πiz)).  i  In [18], Pizer described an explicit algorithm to compute these matrices. (x) The main procedure computes the number of times Q(x) = N N (I) represents 1, 2, 3, .., T for some given T as x varies over the lattice I in our quaternion algebra. The graph of Q(x) with x ∈ R4 is a 4-dimensional paraboloid which has a unique minimum point, and as we move away from this point in any  51  4.2. A Modular Form in S k (Γ0 (4N ), F ) 2  direction, the values given by Q(x) will always increase. That was the very simple idea behind Pizer’s method. The Hecke operators tn generate a commutative ring T of self adjoint operators ([18], Proposition 2.22). The spectral theorem implies that X(R) has an orthogonal basis of eigenvectors for T. As outlined in Section 1, we shall now construct a non-zero modular form g in the Kohnen subspace of S 3 (Γ0 (4N ), F ) as a linear combination 2 of theta series generated by the norm form of B evaluated on some ternary lattices. Needless to say, the ideas here are all well-known and largely drawn from articles [9] and [4]. Our main result is in Section 4 where we use the form g to compute yet another half-integral weight modular form h which also maps to F via the Shimura correspondence. These two forms make up a basis for the space S k (Γ0 (N ), F ). 2 For every left O-ideal Ii in C, let Oi be its right order, and let Ri be the subgroup of trace zero elements in the suborder Z + 2Oi . For every left O-ideal class [Ii ], we associate the ternary theta series g([Ii ]) =  1 2  q N (x) = x∈Ri  1 2  aD ([Ii ])q D ,  (4.1)  D≥0  then extend this association by linearity to X(R). Since, the ternary quadratic form N (x) for x ∈ Ri is a positive definite integral quadratic form with level 4N and a square discriminant, these modular forms have weight 23 , level 4N and the trivial character. The determination of the theta series g([Ii ]) up to a precision T amounts to computing the number of times N (x) represents 1, 2, 3, ..., T as x varies over all trace zero elements in Z + 2Oi . Therefore, it takes time roughly pro3 portional to T 2 . We compute these theta series by implementing a method similar to that of Pizer in [18] (see Section 5 for examples). The action of the (half-integral) Hecke operators Tp2 on the theta series g(I) for I ∈ X(R) is compatible with the action of the Hecke operators tn on I. More precisely, one can show that Tp2 (g(I)) = g(tp (I)) for all p 4N and all I ∈ X(R) (see Proposition 1.7 in [17]). Theorem 4.2.1. ([4] Theorem 3.2) If IF is a non-zero element in the F isotypical component of X(R), then g = g(IF ) is in the Kohnen subspace of S 3 (Γ0 (4N ), F ), and g(IF ) is non-zero if and only if L(F, 12 ) = 0. 2  We compute the eigenvector IF by using the Brandt module package in H  Sage. Hence, if IF =  ei [Ii ], then g is computed as the linear combination i=1  52  4.3. Theta Lifts H  ei g([Ii ]). i=1  The above procedure can be generalized to compute modular forms of higher weights by using what are known as generalized theta series. Given a newform F in Sk−1 (Γ0 (N )) with an odd square-free level N such that k ≡ 3 mod 4 and L(F, 21 ) = 0, the work of B¨ocherer and Schulze-Pillot guarantees the existence of a non-zero modular form g = gF in the Kohnen subspace of S k (Γ0 (4N ), F ). The form g is obtained as a linear combination 2 of generalized theta series attached to the ternary quadratic form N (x) and some homogeneous harmonic polynomials of degree k−3 2 in three variables. The reader is referred to [4] for a thorough discussion of this construction.  4.3  Theta Lifts  In the preceding treatment, we have only considered half-integral weight modular forms with the trivial character. A naturally arising problem is to compute a half-integral weight modular form g with a quadratic character χ, which belongs to the Kohnen subspace and maps to F via the Shimura correspondence. Unfortunately, the above classical construction does not generalize in any obvious or simple way to yield such a form. In theory, however, we are guaranteed the existence of g (under some restrictions on χ) by a beautiful result of Baruch and Mao (Theorem 10.1 in [2]) which, in parts, states the following. Consider a newform F of even weight k − 1, odd square-free level N and of trivial character. Let SN be the set of primes dividing N , and let S be a subset of SN . Write N = p∈S p, and let χ = p|2N χ(p) be any even Dirichlet character defined modulo 4N N such that χ(p) ≡ 1 when p|(N/N ) and χ(p) (−1) = −1 when p|N . Set χ = χ.χs−1 where χ−1 is the Dirichlet character modulo 4 defined by χ−1 (n) = −1 and s is the size n of S. In particular, χ is unramified at the prime 2. There exists a unique (up to scalar multiple) cusp form gS that is in the Kohnen subspace of S k (4N N , χ, Fχ ). The squares of the Fourier coefficients of gS are related 2 to the central values of the quadratic twists of the L-function of F . In [2], the construction of gS is of adelic nature because the authors worked in the setting of automorphic representations. A careful translation from the adelic language to the modular form language is needed to write down an explicit expression of gS as a linear combination of classical theta series. We hope to pursue this work in the future.  53  4.3. Theta Lifts Before we proceed into the next section, some remarks on the representation theoretic version of the Shimura correspondence seem to be in order. The theta correspondence for the pairs (SL2 , P B ∗ ) and (SL2 , PGL2 ) lies at the heart of the above classical constructions. Following Waldspurger’s treatment in [28] and [29], the half-integral weight modular form g can be realized as a theta lift from PGL2 (A) to SL2 (A), the two-fold metaplectic cover of SL2 (A). Moreover, one can obtain g as a theta lift from P B ∗ (A) to SL2 (A) (see [30]). To clarify things in the reader’s mind, let us discuss these ideas briefly. Let A˜ k (M, χ0 ) denote the space of cuspidal automorphic forms φ˜ = ⊗v φ˜v 2  k−1  on SQ \SL2 (A) with character χ0 = χ.χ−12 and level M . This space is determined by a set of local conditions satisfied by the vectors φ˜v (see [29] pp. 381–388 for notation and other details). It is known that there is a natural bijection between S k (M, χ) and A˜ k (M, χ0 ). Thus, half-integral 2  2  weight modular forms may be viewed as automorphic forms on SQ \SL2 (A) whenever convenient. In this framework, the Shimura correspondence has the following formulation. Suppose that g → F under the Shimura map. Let π ˜ be the automorphic representation of SQ \SL2 (A) associated to g, and let π be the automorphic representation of PGL2 (A) associated to F . Then, Sψ (˜ π ) = π ⊗ χ−1 0 , where ψ is the usual additive character on A/Q. The space Sψ (˜ π ) is defined in [28] as Sψ (˜ π ) = Θ(˜ π , ψ ν ) ⊗ χν for any choice of ν ∈ Q∗ such that Θ(˜ π , ψ ν ) = 0. Denote by π the automorphic representation of P B ∗ associated to π via the Jacquet-Langlands correspondence. In the previous section, the form g is obtained as a special vector in the representation space of Θ(π , ψ) using the theta correspondence for the pair (SL2 , P B ∗ ). Notice that g coincides (up to scalar multiple) with the form gS for S = φ. This follows directly from the theorem described in the second paragraph of the present section since N = 1 if χ is the trivial character. We shall now shed some light on the construction of the form gS for any S as carried out in [2]. Using the theta correspondence of the pair (SL2 , PGL2 ), the form gS is obtained as a special vector in the representation space Vπ˜ D of Θ(π ⊗ χD , ψ |D| ) = π ˜ D = ⊗v π ˜vD for some fundamental discriminant D determined by S. Let h be a non-zero modular form in S k (4N N , χ, Fχ ), and denote by 2  54  4.4. On The Result Of Waldspurger π ˜h the automorphic representation of SQ \SL2 (A) generated by h. One gets that π ˜h = π ˜ gS = π ˜ D . Moreover, the space Vπ˜ D = A˜ k (4N N , χ0 ) ∩ Vπ˜ D 2 is two-dimensional. In the next section, we verify this fact for S = φ. Roughly speaking, this owes to the fact that the space of vectors in π ˜2D that satisfy the 2-adic condition forced on a form φ˜ ∈ A˜ k (4N N , χ0 ) is two2 dimensional. This is the space Vπ˜ D ,2 in the factorization Vπ˜ D = ⊗v Vπ˜ D ,v . Moreover, if we let A˜+k (4N N , χ0 ) be the image of the Kohnen subspace 2  under the natural bijection S k (4N N , χ) → A˜ k (4N N , χ0 ), we get that 2 2 A˜+k (4N N , χ0 ) ∩ Vπ˜ D is one-dimensional, and gS is a non-zero vector in this 2  space. In fact, A˜+k (4N N , χ0 ) turns out to be the space of forms φ˜ = ⊗v φ˜v in 2  A˜ k (4N N , χ0 ) with φ˜2 being a carefully chosen vector in Vπ˜ D ,2 (see Section 2 9.4 in [2]). It is exactly this choice of φ˜2 that forces gS to lie in the Kohnen subspace.  4.4  On The Result Of Waldspurger  First, we state Waldspurger’s Theorem in its most general form. Let F ∈ new (χ2 ), and let π be the irreducible automorphic representation of GL (A) Sk−1 2 associated to F . The newform F is required to satisfy the following condition: when πp = π(µ1,p , µ2,p ) belongs to the principal series, then both characters µ1,p and µ2,p are even. Notice that µ1,p (−1) = µ2,p (−1) = 1 whenever πp is unramified principal series. In fact, Flicker proved that this condition is satisfied if and only if S k (L, χ, F ) = 0 for some positive integer 2 L. We also require that one of the following conditions is satisfied: 1. The level of F is divisible by 16 2. The conductor of the character χ0 is divisible by 16, where χ0 (n) = χ(n)  −1 n  k−1 2  3. π2 is not supercuspidal For every positive integer e and prime number p, Waldspurger used the local components of π and χ0 to define an integer np and a set Up (e, F ) of finitely many functions cp : Q∗p → C. According to these definitions, given an integer E ≥ 1, for all but finitely many p the set Up (vp (E), F ) is a singleton and np = 0. In fact, we are interested in the finite set U (E, F ) = Up (vp (E), F ). p prime  55  4.4. On The Result Of Waldspurger Let Nsf be the set of positive square-free integers, and let A : Nsf → C be any function. Denote square-free part of a positive integer n by nsf . Given an element cE = (cE,p ) ∈ U (E, F ), we define the function f (cE , A) on H by: an (cE , A)q n ,  f (cE , A) = n≥1  where an (cE , A) = A(nsf )n  k−2 4  cE,p (n). p  pnp is denoted by N (F ), and the complex vector space gen-  The integer p  erated by the set {f (cE , AF )}cE ∈U (E,F ) is denoted by U (E, F, AF ). Given a Dirichlet character ν, let L(ν, s) be the associated L-function. Recall that, for sufficiently large Re s, L(ν, s) is defined as L(ν, s) = π −  s+δ 2  Γ  s+δ 2  ∞  νˆ(n)n−s , n=1  where δ is such that ν(−1) = (−1)δ and νˆ is the primitive Dirichlet character associated to ν. It is known that L(ν, s) has a meromorphic continuation to all s. Define (ν, s) as the -factor that appears in the functional equation L(ν −1 , 1 − s) = (ν, s)L(ν, s). For any t ∈ Z∗ , denote √ by χt the quadratic Dirichlet character associated with the extension √Q( t)/Q. Notice that χt (n) = ( ∆nt ), where ∆t is the discriminant of Q( t). In particular, χt (n) = 1 for all n ∈ Z∗ if t is a perfect square. Theorem 4.4.1. (See [29]) Given a newform F satisfying the above hypotheses, there exists a function AF : Nsf → C, such that: −1 1 1 1. (AF (t))2 = L(F ⊗ χ−1 0 χt , 2 ) (χ0 χt , 2 )  2. S k (4N, F, χ) = U (E, F, AF ); the direct sum being taken over all 2 integers E such that N (F )|E|4N Our goal is thus reduced to computing the individual spaces U (E, F, AF ). This amounts to computing the half-integral weight modular forms f (cE , AF ) for all cE ∈ U (E, F ). Recall that given cE = (cE,p ) ∈ U (E, F ), the function f (cE , AF ) is written as an (cE , AF )q n , and the Fourier coefficients n≥1  56  4.4. On The Result Of Waldspurger an (cE , AF ) are defined by the formula an (cE , AF ) = AF (nsf )n  k−2 4  cE,p (n). p  cE,p (n) is completely determined by the local properties  The local factor p  of F and χ0 . In fact, the various local objects involved in this calculation are given explicitly in [29]. However, the determination of the global factor A(nsf ) requires more effort because we are only given its square value. One way to resolve this complication is to use the Fourier coefficients of the half-integral weight modular form g that was computed in Section 2. Let us now return to our usual setting. Recall that we are given a newform F = bn q n in Sk−1 (Γ0 (N )) with odd square-free level N such n≥1  an q n ∈ S k (Γ0 (4N ), F ) be  that k ≡ 3 mod 4 and L(F, 12 ) = 0. Let g =  2  n≥1  the form obtained in Section 2. k For every prime number p not dividing N , we put λp = bp p1− 2 , αp +αp = k  λp , and αp αp = 1. We also set λp = bp p1− 2 for every prime divisor p of N . We shall now apply the theorem of Waldspurger to compute a basis for the space S k (Γ0 (4N ), F ). In order to compute the local factors cp 2  p  of the Fourier coefficients of the desired basis elements, we use the explicit formulae given in [29] Section 8. Since the newform F has a square-free level and the trivial character modulo N , we get   2 if p = 2; np = 1 if p|N ;   0 otherwise. Hence, N (F ) =  pnp = 4N . We also get  S k (Γ0 (4N ), F ) =  U (E, F, AF ) = U (4N, F, AF ),  2  N (F )|E|4N  where U (E, F, AF ) is the vector space generated by the modular forms {f (cE , AF )}cE ∈U (E,F ) , and U (E, F ) is the finite set Up (vp (E), F ). Rep  call that for a given prime number p and a positive integer E, the set  57  4.4. On The Result Of Waldspurger Up (vp (E), F ) consists of finitely many local functions cp . Using the fact that E = 4N and N is odd and square-free, we get U2 (2, F ) =  {c2 [α2 ], c2 [α2 ]} if α2 = α2 ; {c2 [α2 ], c2 [α2 ]} otherwise,  where the local functions c2 and c2 are defined below. If p is an odd prime we get, Up (vp (4N ), F ) =  {c0p [λp ]} {csp [λp ]}  if p N ; otherwise.  Before we proceed any further, we need to recall the definition of the Hilbert symbol. Given two non-zero elements a and b in a local field K, the Hilbert symbol is defined by (a, b) =  1 −1  if z 2 = ax2 + by 2 has a non-zero solution in K 3 ; otherwise.  In particular, if a, b ∈ Z, write a = 2α s and b = 2β t such that s and t are odd integers. The Hilbert symbol over the 2-adics is (a, b)2 = (−1)  (s) (t)+αω(t)+βω(s)  ,  2  x −1 where (x) = x−1 2 and ω(x) = 8 . In order to evaluate the local functions cp at an integer n, we need to express n as up2h (vp (u) ∈ {0, 1}), then use the following formulae: 1  c2 [δ](n) =  δ h (δ − 2− 2 (2, u)2 ) δh,  if v2 (u) = 0 and (u, −1)2 = −1; otherwise.   n n  δ(c2 [δ]( 4 ) + c2 [δ]( 4 )) if v2 (n) ≥ 2 c2 [δ](n) = δ if v2 (n) = 0 and (n, −1)2 = −1;   0 otherwise.  h  if vp (u) = 1;  δ csp [δ](n) =  1  2 2 δh   0,  if vp (u) = 0 and  −u p  1  = −p 2 δ;  otherwise.  58  4.4. On The Result Of Waldspurger    1 c0p [δ](n) = bh − bh−1   b  if vp (u) = 0 and h = 0; −u p  p  − 12   bh =  1  2h  (4.2)  otherwise,  h  where  if vp (u) = 0 and h ≥ 1;  h 2   h + 1 h−2i 2 δ (δ − 4)i  . 2i + 1  i=0  If α2 = α2 , the set U (4N, F ) consists of csp [λp ]  c1 = c2 [α2 ]  c0p [λp ] and c2 = c2 [α2 ] p 2N  p|N  csp [λp ]  c0p [λp ]. p 2N  p|N  If α2 = α2 , the set U (4N, F ) consists of csp [λp ]  c1 = c2 [α2 ] p|N  c0p [λp ] and c2 = c2 [α2 ] p 2N  csp [λp ]  c0p [λp ]. p 2N  p|N  Theorem 4.4.1 gives a basis for S k (Γ0 (4N ), F ) that consists of the functions 2  an (c1 , AF )q n and f (c2 , AF ) =  f (c1 , AF ) = n≥1  an (c2 , AF )q n if α2 = α2 n≥1  and an (c1 , AF )q n and f (c2 , AF ) =  f (c1 , AF ) = n≥1  an (c2 , AF )q n if α2 = α2 . n≥1  Recall that an (c1 , AF ) = AF (nsf )n  k−2 4  csp [λp ](n)  c2 [α2 ](n)  p 2N  p|N  an (c2 , AF ) = AF (nsf )n  k−2 4  c0p [λp ](n)  csp [λp ](n)  c2 [α2 ](n)  c0p [λp ](n) p 2N  p|N  and an (c2 , AF ) = AF (nsf )n  k−2 4  csp [λp ](n)  c2 [α2 ](n) p|N  c0p [λp ](n). p 2N k−1  Lemma 4.4.2. For every positive integer n such that (−1) 2 n = −n ≡ 2, 3 mod 4, we have an (c1 , AF ) = an (c2 , AF ) and an (c2 , AF ) = 0. 59  4.4. On The Result Of Waldspurger Proof. Write n as u22h with v2 (u) ∈ {0, 1}. If n ≡ 1 mod 4, we get c2 [α2 ](n) = (α2 )0 = 1 = (α2 )0 = c2 [α2 ](n) and c2 [α2 ](n) = 0 since h = 0, v2 (u) = 0 and (u, −1)2 = 1. If n ≡ 2 mod 4, we also get c2 [α2 ](n) = (α2 )0 = 1 = (α2 )0 = c2 [α2 ](n) and c2 [α2 ](n) = 0 since h = 0 and v2 (u) = 1. Thus, an (c1 , AF ) = an (c2 , AF ) and an (c2 , AF ) = 0 for all n ≡ 1, 2 mod 4. Lemma 4.4.3. If α2 = α2 , the function f (c1 , AF ) − f (c2 , AF ) belongs to the Kohnen subspace of S k (Γ0 (4N ), F ). Otherwise, the function f (c2 , AF ) 2 belongs to the Kohnen subspace of S k (Γ0 (4N ), F ) 2  Proof. The coefficients an (c1 , AF ) − an (c2 , AF ) and an (c2 , AF ) are zero for k−1 all n such that (−1) 2 n ≡ 2, 3 mod 4. Proposition 4.4.4. If α2 = α2 , the half-integral weight modular form g is a non-zero scalar multiple of f (c1 , AF ) − f (c2 , AF ). Otherwise, g is a non-zero scalar multiple of f (c2 , AF ). Proof. We know by the multiplicity one result of Kohnen (Theorem 2 in [14]) that the Kohnen subspace of S k (Γ0 (4N ), F ) is one-dimensional. We 2 also know from Section 1 that g belongs to the Kohnen subspace. This forces g to be equal to a non-zero scalar multiple of f (c1 , AF ) − f (c2 , AF ) when α2 = α2 and a non-zero scalar multiple of f (c2 , AF ) when α2 = α2 . Let t be a positive square-free integer and denote by ∆−t the fundamental discriminant corresponding to −t. Recall that g = an q n . Let r1 and r2 n≥1  be the non-zero complex constants such that f (c1 , AF ) − f (c2 , AF ) = r1 g if α2 = α2 and f (c2 , AF ) = r2 g if α2 = α2 . We know by the previous proposition that a|∆−t | (c1 , AF ) − a|∆−t | (c2 , AF ) = r1 a|∆−t |  if α2 = α2  and a|∆−t | (c2 , AF ) = r2 a|∆−t |  if α2 = α2 .  We also know that r1 a|∆−t | = a|∆−t | (c1 , AF ) − a|∆−t | (c2 , AF ) = AF (t)|∆−t |  k−2 4  csp [λp ](|∆−t |)  c2 [α2 ](|∆−t |) − c2 [α2 ](|∆−t |) p|N  60  4.4. On The Result Of Waldspurger and r2 a|∆−t | = a|∆−t | (c2 , AF ) = AF (t)|∆−t |  k−2 4  csp [λp ](|∆−t |).  c2 [α2 ](|∆−t |) p|N  Notice that the local factors for p 2N do not appear in the above expressions. In fact, c0p [λp ](|∆−t |) = 1 for all such p since |∆−t | is not divisible by the square of any odd prime. Hence, if α2 = α2 , we get AF (t)|∆−t |  k−2 4  csp [λp ](|∆−t |) = r1 a|∆−t | .  c2 [α2 ](|∆−t |) − c2 [α2 ](|∆−t |) p|N  Otherwise, we get AF (t)|∆−t |  k−2 4  csp [λp ](|∆−t |) = r2 a|∆−t | .  c2 [α2 ](|∆−t |) p|N  Since ∆−t is a fundamental discriminant, it is not divisible by any square of any odd prime, and it satisfies ∆−t ≡ 1 mod 4 or ∆−t ≡ 8, 12 mod 16. Thus, c2 [α2 ](|∆−t |) − c2 [α2 ](|∆−t |) = α2 − α2 and c2 [α2 ](|∆−t |) = α2 . We still need to investigate the value of p|N csp [λp ](|∆−t |). Since |∆−t | is not divisible by the square of any odd prime csp [λp ](|∆−t |) = 1 for all p such that p|N and p|∆−t . If p|N and p ∆−t then 1  csp [λp ](|∆−t |)  =  ∆−t p  1  = −p 2 λp ;  22  if  0  otherwise.  Therefore,  csp [λp ](|∆−t |) = p|N       1  22  if  1  ∆−t p  = −p 2 λp ∀p, p|N and p ∆−t ;  p|N  p ∆−t    0  otherwise.  Putting this together gives AF (t)|∆−t |  k−2 4  1  (α2 − α2 )  2 2 = r1 a|∆−t |  if α2 = α2  p|N p ∆−t  61  4.4. On The Result Of Waldspurger and AF (t)|∆−t |  k−2 4  1  2 2 = r2 a|∆−t |  α2  if α2 = α2  p|N p ∆−t 1  whenever ∆p−t = −p 2 λp for all p such that p|N and p ∆−t . Hence we obtain the following theorem which is the same as Theorem 1.1.4 stated in Chapter 1. Theorem 4.4.5. For a positive square-free integer t satisfying  ∆−t p  =  1 2  −p λp for all p such that p|N and p ∆−t , we have νt  AF (t) = r2 2 |∆−t | νt  2−k 4  a|∆−t | ,  1  2− 2 , and r is a non-zero complex constant depending only  where 2 2 = p|N p ∆−t  on F . Remark 4.4.6. Let n be a positive integer such that its square-free part 1 ∆−nsf which we denote by nsf satisfies = p 2 λp for some prime p such p that p|N and p  ∆−nsf . For this particular p, write n = up2h (vp (u) ∈ −u p  {0, 1}). Since p ∆−nsf , we get vp (u) = 0. Moreover,  =  ∆−nsf p  =  1 2  p λp . This forces csp [λp ] to vanish at n. Therefore, an (c1 , AF ), an (c2 , AF ), an (c2 , AF ) vanish for all such n regardless of the value of AF (nsf ). Before stating our main theorem, we shall verify that f (c1 , AF ) and f (c2 , AF ) are not scalar multiples of each other even though they share the same Hecke eigenvalues at the primes p N . We assume that α2 = α2 purely for simplicity of exposition. Let n be a positive integer such that an (c1 , AF ) = 0. In particular, AF (nsf ) = 0 and csp [λp ](n) = 0. We set r = |∆−nsf | and w = 4|∆−nsf | to simplify notation. Hence, csp [λp ](r) = p|N  1  csp [λp ](w) = p|N  22 .  (4.3)  p|N pr  Moreover, for δ ∈ {α2 , α2 } we get 1  c2 [δ](r) =  δ − 2− 2 (2, r)2 δ  if r ≡ 3 mod 4; otherwise,  (4.4) 62  4.4. On The Result Of Waldspurger and  1  δ(δ − 2− 2 (2, r)2 ) δ2  c2 [δ](w) =  if r ≡ 3 mod 4; otherwise.  (4.5)  Notice that the coefficients of q r and q w in f (c1 , AF ) and f (c2 , AF ) are nonzero. The following table shows that these coefficients are not in a constant (c1 ,AF ) 1 ,AF ) ratio to each other. In fact, aaww (c = αα2 aarr (c (c2 ,AF ) . 2 ,AF ) 2  Table 4.1: Ratios of Fourier Coefficients r ≡ 3 mod 4 r ≡ 4, 8 mod 16 1  ar (c1 ,AF ) ar (c2 ,AF )  α2 −2− 2 (2,r)2  aw (c1 ,AF ) aw (c2 ,AF )  α2 (α2 −2− 2 (2,r)2 )  −1 2  α2 −2  α2 α2  (2,r)2 1  ( αα2 )2  1  α2 (α2 −2− 2 (2,r)2 )  2  In what follows, the terms K1 (n), K2 (n) and K2 (n) refer to the formulae K1 (n) = 2  νn 2  a|∆−nsf | |∆−nsf |  2−k 4  n  k−2 4  csp [λp ](n)  c2 [α2 ](n)  c0p [λp ](n) p 2N  p|N  (4.6) K2 (n) = 2  νn 2  a|∆−nsf | |∆  2−k 4 −nsf  |  n  k−2 4  csp [λp ](n)  c2 [α2 ](n)  c0p [λp ](n) p 2N  p|N  (4.7) and K2 (n) = 2  νn 2  a|∆−nsf | |∆−nsf |  2−k 4  n  k−2 4  csp [λp ](n)  c2 [α2 ](n) p|N  c0p [λp ](n). p 2N  (4.8) The following Theorem is the main result of the current chapter and is the same as Theorem 1.1.5 stated in Chapter 1. new (Γ (N )) with odd bn q n be a newform in Sk−1 0  Theorem 4.4.7. Let F = n≥1  square-free level N such that k ≡ 3 mod 4 and L(F, 21 ) = 0. Let g = an q n ∈ S k (Γ0 (4N ), F ) be the form obtained in Section 2. Put 2  n≥1  an (f1 )q n , f2 =  f1 = n≥1  an (f1 ) =  an (f2 )q n and f2 = n≥1  an (f2 )q n with n≥1  ∆−nsf p  K1 (n)  if  0  otherwise,  1  = −p 2 λp ∀p, p|N and p ∆−nsf ;  63  4.4. On The Result Of Waldspurger  an (f2 ) =  ∆−nsf p  K2 (n)  if  0  otherwise,  K2 (n)  if  0  otherwise.  1  = −p 2 λp ∀p, p|N and p ∆−nsf ;  and an (f2 ) =  ∆−nsf p  1  = −p 2 λp ∀p, p|N and p ∆−nsf ;  Then S k (Γ0 (4N ), F ) is generated by f1 and f2 if α2 = α2 , and it is generated 2 by f1 and f2 if α2 = α2 . In order to use Theorem 4.4.7 as an effective tool for computing a basis for S 3 (Γ0 (4N ), F ), we need to make the following observations. Once again, 2 we assume that α2 = α2 to simplify the exposition and avoid unrewarding details. 1 Let h = 2 2 (f1 + f2 ) and write h = an (h)q n . First, we compute n≥1  the Fourier coefficients an (h) for positive square-free integers n such that 1 n 2 p = −p λp whenever p|N and p n (since, for square-free n, an (h) = 0 otherwise). A straightforward calculation using Theorem 4.4.7, (4.2), (4.3), (4.4), (4.5), (4.6), (4.7), and (4.8) shows that an (h) =  an (g)(b2 − 2(2, n)2 ) if n ≡ 3 mod 4; 2a4n (g) otherwise.  Let us now consider a positive integer n, which is not divisible by any sf square prime to 4N , and write n = nsf y 2 . We also assume that np = 1  −p 2 λp whenever p|N and p nsf . To simplify notation, we set s = v2 (y). Another simple calculation shows that if nsf ≡ 3 mod 4, then an (h) = 2  s+1 2  1  v (y)  ansf (g)(α2s+1 + α2s+1 − 2− 2 (2, nsf )2 (α2s + α2s ))  bpp  .  p|N  Otherwise, we get s  v (y)  an (h) = 2 2 −1 ansf (h)(α2s + α2s )  bpp  .  p|N  Finally, we use Proposition 14 on page 210 in [13] to compute an (h) for an arbitrary positive integer n. Suppose that n0 is a positive integer, which 64  4.5. Examples is not divisible by the square of any prime p 4N . Then this proposition provides us with the following recursive formulae which hold for all p 4N and n1 prime to p: an0 n21 p2 (h) = an0 n21 (h) bp −  −n0 p  and an0 n2 p2(v+1) (h) = an0 n21 p2v (h) bp − 1  −n0 p  − pan0 n2 p2(v−1) (h) 1  v = 1, 2, ...  We implemented all of the above formulae in Sage. The result is a function that outputs the Fourier expansion of h up to a desired precision. Thus, we get a basis for S 3 (Γ0 (4N ), F ) consisting of the modular forms g 2 and h.  4.5  Examples  We are interested in computing a basis for the space S 3 (Γ0 (60), F ), where 2  F is the newform in S2new (Γ0 (15)) corresponding to the elliptic curve y 2 + xy + y = x3 + x2 − 10x − 10. We know that bn q n = q − q 2 − q 3 − q 4 + q 5 + q 6 + 3q 8 + q 9 − q 10 − 4q 11 + O(q 12 ).  F = n≥1  Since W3 (F ) = F and W5 (F ) = −F , we consider the definite quaternion algebra B ramified at 5. Let O be an order in B of level N = 15. O has two left ideal classes [I1 ] and [I2 ] which give rise to two orders O1 = Or (I1 ) and O2 = Or (I2 ). The subgroups R1 and R2 of trace zero elements in Z + 2O1 and Z + 2O2 have bases {i + 31 j + 43 k, 23 j + 83 k, 3k} and { 43 j − 32 k, i + 23 j − 1 3 k, j − 2k} respectively. The two theta series generated by the norm form of the quaternion algebra evaluated on R1 and R2 are: q N (x) =  g(I1 ) = x∈R1  1 + q 3 + q 12 + 3q 20 + 6q 23 + q 27 + 6q 32 + 6q 47 + q 48 2  + 3q 60 + 6q 63 + 6q 68 + 6q 72 + q 75 + 3q 80 + 6q 83 + 6q 87 + 6q 92 + 6q 95 + O(q 100 )  65  4.5. Examples and q N (x) =  g(I2 ) = x∈R2  1 + 2q 8 + q 12 + q 15 + q 20 + 4q 23 + 2q 27 + 4q 32 + 2q 35 2  + 8q 47 + 3q 48 + q 60 + 6q 63 + 6q 68 + 4q 72 + 5q 80 + 4q 83 + 4q 87 + 10q 92 + 6q 95 + O(q 100 ). The vector IF = I1 − I2 is in the F -isotypical component of X(R). Recall that IF is unique up to a scalar multiple and satisfies tp (IF ) = bp IF for all p N , where the action of the Hecke operators {tp }p N is determined by the Brandt matrices {Bp }p N . Hence, one can find the eigenvector IF using the Brandt module package in Sage by computing a few Brandt matrices and knowing a few Fourier coefficients of F . Hence, the modular form g = g(I1 ) − g(I2 ) = q 3 − 2q 8 − q 15 + 2q 20 + 2q 23 − q 27 + 2q 32 − 2q 35 − 2q 47 − 2q 48 + 2q 60 + 2q 72 + q 75 − 2q 80 + 2q 83 + 2q 87 − 4q 92 + O(q 100 ) lies in the space S 3 (Γ0 (60), F ). 2 We shall now apply the theorem of Waldspurger to compute a basis for this space. In order to determine the local objects necessary to calculate the local factor cp (n), we use the explicit formulae given in [29]. As a result of these calculations, we get N (F ) = 22 × 3 × 5 and S 3 (Γ0 (60), F ) = 2 U (60, F, AF ). We also get: 1  U3 (1, F ) = {cs3 [λ3 ]}  (λ3 = 3− 2 b3 ),  U5 (1, F ) = {cs5 [λ5 ]}  (λ5 = 5− 2 b5 ),  1  1  U2 (2, F ) = {c2 [α2 ], c2 [α2 ]} (α2 + α2 = 2− 2 b2 , α2 α2 = 1), Up (0, F ) = {c0p [λp ]}, for all p = 2, 3, 5  1  (λp = p− 2 bp ).  Since the set U (60, F ) constitutes of two elements c1 and c2 , the space U (60, F, AF ) is generated by the two functions f (c1 , AF ) and f (c2 , AF ) whose Fourier coefficients are calculated as follows: 1  an (c1 , AF ) = AF (nsf )n 4 cs3 [λ3 ](n)cs5 [λ5 ](n)c2 [α2 ](n)  c0p [λp ](n) p=2,3,5  C1 (n)  66  4.5. Examples 1  an (c2 , AF ) = AF (nsf )n 4 cs3 [λ3 ](n)cs5 [λ5 ](n)c2 [α2 ](n)  c0p [λp ](n) . p=2,3,5  C2 (n)  We carried out these computations and obtained the first few local factors C1 and C2 in Table 4.2 below.  n 2 3 4 5 6 7  Table 4.2: The Local Factors C1 and C2 C1 (n) C2 (n) n C1 (n) C2 (n) 2√ 2√ 13 0 0 1+i 7 1−i 7 14 0√ 0√ 2 2 −3+i 7 −3−i √ √ 7 0 0 15 2 2 2 2 √ √ 2 2 16 0 0 0 0 17 2 2 0 0 18 − √23 − √23  8  √ −1+i √ 7 2  √ −1−i √ 7 2  19  9 10 11 12  0 0 0 √ − 2  0 0 0 √ − 2  20 21 22 23  0  √ −1+i 7 2  0 0√  −3+i √ 7 2  0  √ −1−i 7 2  0 0√  −3−i √ 7 2  Therefore, 1 4  2  f (c1 , AF ) = AF (2)2 (2)q + AF (3)3 √  1 4  √ 1+i 7 2  1 4  1 4  8  − AF (2)2 (1 − i 7)q − 2AF (3)3 q 1 4  + 2AF (17)17 q − AF (23)23  1 4  17  1 4  − 2AF (2)2 q  √ 3−i 7 √ 2  18  12  1 √ q 3 + AF (5)5 4 ( 2)q 5  − AF (15)15  − AF (5)5  1 4  1 4  √ 3−i 7 √ 2 2  √ 1−i 7 √ 2  q 15  q 20  q 23 + O(q 24 ),  67  4.5. Examples and 1 4  2  f (c2 , AF ) = AF (2)2 (2)q + AF (3)3 √  1 4  √ 1−i 7 2  1 4  1 4  8  − AF (2)2 (1 + i 7)q − 2AF (3)3 q 1 4  + 2AF (17)17 q − AF (23)23  1 4  17  1 4  − 2AF (2)2 q  √ 3+i 7 √ 2  18  12  1 √ q 3 + AF (5)5 4 ( 2)q 5  − AF (15)15  − AF (5)5  1 4  √ 3+i 7 √ 2 2  √ 1+i 7 √ 2  1 4  q 15  q 20  q 23 + O(q 24 ).  Using Theorem 4.4.5 to calculate the global factors AF (t) gives the values shown in Table 4.3 below. Table 4.3: The Global Factors AF (t) t AF (t) 1 2 −r8− 4 1 1 3 r2− 2 3− 4 1 5 r5− 4 1 15 −r15− 4 17 0 1 23 r23− 4 Notice that the global factor AF (17) is zero because the coefficient c68 of g is zero. We apply the following linear combinations: √ 2 f = √ (f (c2 , AF ) − f (c1 , AF )) i 7 1  1  1  1  1  = 2 2 AF (3)3 4 q 3 + 2AF (2)8 4 q 8 + AF (15)15 4 q 15 + 2AF (5)5 4 q 20 1  + 2AF (23)23 4 q 23 + O(q 24 ) and h=  √  2(f (c1 , AF ) + f (c2 , AF )) 3  1  1  1  3  = 4AF (2)2 4 q 2 + AF (3)2 2 3 4 q 3 + 4AF (5)5 4 q 5 − 2AF (2)2 4 q 8 1  1  1  1  1  3  − 4AF (3)2 2 3 4 q 12 − 3AF (15)15 4 q 15 + 4AF (17)2 2 17 4 q 17 − 4AF (2)2 4 q 18 1  1  − 2AF (5)5 4 q 20 − 6AF (23)23 4 q 23 + O(q 24 ). 68  4.5. Examples After substituting the values of AF (t) obtained above, we get f = g = q 3 − 2q 8 − q 15 + 2q 20 + 2q 23 + O(q 24 ) and h = −4q 2 + q 3 + 4q 5 + 2q 8 − 4q 12 + 3q 15 + 4q 18 − 2q 20 − 6q 23 + O(q 24 ). Therefore, the modular forms g and h form a basis for the space S 3 (Γ0 (60), F ). 2 In what follows, we generate more examples of the modular forms g and h which form a basis for the space S 3 (Γ0 (4N ), F ), using the functions 2 we created in Sage to implement calculations of h (as described following Theorem 3.7) and g (as described in Section 1). These functions are available from the author upon request. The open source mathematical software package may be found at http : //www.sagemath.org/. Example 1 We compute a basis {g1, h1} for the space S 3 (Γ0 (44), F 1) 2 with F 1 being the newform in S2new (Γ0 (11)) corresponding to the elliptic curve y 2 + y = x3 − x2 − 10x − 20. sage: F1=Newforms(11)[0] sage: g1=shimura_lift_in_kohnen_subspace(F1,11,40) sage: g1 −q 3 + q 4 + q 11 + q 12 − q 15 − 2q 16 − q 20 + q 23 + q 27 + q 31 + O(q 40 ) sage: h1=shimura_lift_not_in_kohnen_subspace(F1,11,40) sage: h1 2q−2q 4 −2q 5 +2q 12 −4q 14 +4q 15 +2q 20 +4q 22 −4q 23 +4q 26 −4q 31 −2q 33 −4q 34 −2q 37 +4q 38 +O(q 40 ) Example 2 We compute a basis {g2, h2} for the space S 3 (Γ0 (84), F 2) 2 with F 2 being the newform in S2new (Γ0 (21)) corresponding to the elliptic curve y 2 + xy = x3 − 4x − 1. sage: F2=Newforms(21)[0] sage: g2=shimura_lift_in_kohnen_subspace(F2,21,40) 69  4.5. Examples sage: g2 q 3 − q 7 − 2q 19 − 2q 24 + q 27 + 2q 28 + 2q 31 + O(q 40 ) sage: h2=shimura_lift_not_in_kohnen_subspace(F2,21,40) sage: h2 q 3 −4q 6 +3q 7 +8q 10 −4q 12 −8q 13 −2q 19 +4q 21 +2q 24 +q 27 +2q 28 −6q 31 +8q 33 +O(q 40 ) Example 3 We compute a basis {g3, h3} for the space S 3 (Γ0 (132), F 3) 2 with F 3 being the newform in S2new (Γ0 (33)) corresponding to the elliptic curve y 2 + xy = x3 + x2 − 11x. sage: F3=Newforms(33)[0] sage: g3=shimura_lift_in_kohnen_subspace(F3,33,40) sage: g3 q 3 + q 11 + 2q 12 − 2q 15 − 4q 20 − 2q 23 − q 27 + O(q 40 ) sage: h3=shimura_lift_not_in_kohnen_subspace(F3,33,40) sage: h3 3q 3 −8q 5 +3q 11 −2q 12 +8q 14 +2q 15 −4q 20 +2q 23 −8q 26 −3q 27 +8q 33 +16q 38 +O(q 40 ) Example 4 We compute a basis {g4, h4} for the space S 3 (Γ0 (140), F 4) 2 with F 4 being the newform in S2new (Γ0 (35)) corresponding to the elliptic curve y 2 + y = x3 + x2 + 9x + 1. sage: F4=Newforms(35,names=’a’)[0] sage: g4=shimura_lift_in_kohnen_subspace(F4,35,40) sage: g4 −2q 4 + 2q 11 + 2q 15 − 2q 35 − 4q 36 + 2q 39 + O(q 40 ) sage: h4=shimura_lift_not_in_kohnen_subspace(F4,35,40) sage: h4 −4q−8q 9 +4q 11 +8q 14 −4q 15 +8q 16 −4q 21 +4q 25 +12q 29 +8q 30 −4q 35 −4q 39 +O(q 40 ) 70  4.5. Examples Example 5 We compute a basis {g5, h5} for the space S 3 (Γ0 (148), F 5) 2 with F 5 being the newform in S2new (Γ0 (37)) corresponding to the elliptic curve y 2 + y = x3 + x2 − 23x − 50. sage: F5=Newforms(37)[1] sage: g5=shimura_lift_in_kohnen_subspace(F5,37,40) sage: g5 q 8 − q 15 + q 19 − q 20 − q 24 − q 35 + q 39 + O(q 40 ) sage: h5=shimura_lift_not_in_kohnen_subspace(F5,37,40) sage: h5 2q 2 −2q 5 −2q 6 −2q 13 +2q 14 +2q 15 +2q 17 +2q 19 −2q 22 −4q 32 −2q 35 +2q 37 −2q 39 +O(q 40 ) Example 6 We compute a basis {g6, h6} for the space S 3 (Γ0 (924), F 6) 2 with F 6 being the newform in S2new (Γ0 (231)) corresponding to the elliptic curve y 2 + xy + y = x3 + x2 − 34x + 62. sage: F6=Newforms(231,names=’a’)[0] sage: g6=shimura_lift_in_kohnen_subspace(F6,231,40) sage: g6 4q 8 + 2q 11 − 4q 32 + 4q 35 + O(q 40 ) sage: h6=shimura_lift_not_in_kohnen_subspace(F6,231,40) sage: h6 8q 2 − 4q 8 + 2q 11 − 8q 18 − 8q 21 + 16q 29 − 16q 30 − 12q 32 + 4q 35 + O(q 40 )  71  Bibliography [1] M. Alsina and P. Bayer. Quaternion Orders, Quadratic Forms, and Shimura Curves, volume 22 of CRM Monograph Series. American Mathematical Society, 2004. [2] E. Baruch and Z. Mao. Central value of automorphic L-functions. GAFA, Geom. funct. anal., 17:333–384, 2007. [3] M. Bertolini and H. Darmon. A rigid analytic Gross-Zagier formula and arithmetic applications. Ann. of Math., 146 (1):111–147, 1997. [4] S. B¨ ocherer and R. Schulze-Pillot. Vector valued theta series and Waldspurger’s theorem. Abh. Math. Sem. Univ. Hamburg, 64:211–233, 1994. [5] D. Bump. Automorphic Forms and Representations, volume 55. Cambridge Studies in Advanced Mathematics, 1996. [6] C. Cornut and V Vatsal. CM points and quaternion algebras. Doc. Math., 10:263–309 (electronic), 2005. [7] C. Cornut and V. Vatsal. Nontriviality of Rankin-Selberg L-functions and CM points. L-functions and Galois Represenations, Ed. Burns, Buzzard and Nekovar, Cambridge University Press:121–186, 2007. [8] Y. Flicker. 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