some applications of classical modular forms to number theory

Copyright by Riad Mohamad Masri 2005 The Dissertation Committee for Riad Mohamad Masri Certifies that this is the approved version of the following dissertation: SOME APPLICATIONS OF CLASSICAL MODULAR FORMS TO NUMBER THEORY Committee: Fernando Rodriguez-Villegas, Supervisor David Boyd Sean Keel David J. Saltman John Tate SOME APPLICATIONS OF CLASSICAL MODULAR FORMS TO NUMBER THEORY by Riad Mohamad Masri, B.S.; M.S. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN August 2005 UMI Number: 3204214 3204214 2006 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. To Bonnie Acknowledgments To begin, I want to thank my advisor, Fernando Rodriguez-Villegas, for helpful dis- cussions and encouragement during the last three years. I owe much to my friend and collaborator Jim Kelliher for his patience while listening to me explain my ideas, and Misha Vishik for his constant encouragement. I benefited from the advice and sug- gestions of many mathematicians, including Bill Duke, Solomon Friedberg, Farshid Hajir, Gergely Harcos, Angel Kumchev, Jeff Lagarias, David Saltman, John Tate, and Jeff Vaaler. This list is by no means complete. I want to thank Haskell Rosen- thal, who taught me analysis and was instrumental in my coming to the University of Texas. Most importantly, I want to thank my wife, Bonnie Plott, for the love and fulfillment she has brought to my life over the past year. Part of this research was supported by a Joseph Patrick Brannen Fellowship in Mathematics. v SOME APPLICATIONS OF CLASSICAL MODULAR FORMS TO NUMBER THEORY Publication No. Riad Mohamad Masri, Ph.D. The University of Texas at Austin, 2005 Supervisor: Fernando Rodriguez-Villegas In this thesis we use classical modular forms to study several problems in number theory. In chapter 2 we use non-holomorphic Eisenstein series for the Hilbert modular group to obtain a formula for the relative class number of certain abelian extensions of CM number fields. In chapter 3 we compute the scattering determi- nant for the Hilbert modular group, and explain how this can be used to prove that the subspace of cuspidal, square integrable eigenfunctions for the Laplacian on products of rank one symmetric spaces is infinite dimensional. In chapter 4 we use zeta functions of quadratic forms over number fields to sharpen a certain constant appearing in C. L. Siegel’s lower bound for the residue of the Dedekind zeta function at s = 1. vi Table of Contents Acknowledgments iv Abstract v Chapter 1. Introduction 1 Chapter 2. Relative class numbers of abelian extensions of CM num- ber fields 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Fourier expansion of the non-holomorphic Eisenstein series . . . . . . . . 5 2.3 Taylor expansion of E(s, z; a, b) at s = 0 . . . . . . . . . . . . . . . . . . 13 2.4 Analytic and modular properties of log{Ψ(z)} . . . . . . . . . . . . . . . 15 2.5 CM-points on Hilbert modular varieties . . . . . . . . . . . . . . . . . . 18 2.6 The fundamental identity . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.7 Proof of Theorem 2.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.8 Proof of Theorem 2.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 3. The scattering determinant for the Hilbert modular group 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 The spectral decomposition of ∆ . . . . . . . . . . . . . . . . . . . 26 3.1.3 The dimension of the space of cusp forms . . . . . . . . . . . . . . 29 3.2 Eisenstein series associated to products of Q-rank one symmetric spaces 35 3.3 The scattering determinant for SL 2 (O K ) . . . . . . . . . . . . . . . . . . 41 3.4 The trace of Φ(s) at s = 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Zeta functions of quadratic forms . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Proof of Theorem 4.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Proof of Theorem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.8 The determinant of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 vii 3.9 Proof of Theorem 3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.10 Proof of Theorem 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter 4. A lower bound for the residue of the Dedekind zeta func- tion at s = 1 59 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.2 Zeta functions of quadratic forms . . . . . . . . . . . . . . . . . . 61 4.1.3 Functional equations and residues . . . . . . . . . . . . . . . . . . 62 4.1.4 A theorem of Siegel . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1.5 Convexity Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Proof of Theorem 4.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Proof of Theorem 4.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4 A Hecke type integral representation . . . . . . . . . . . . . . . . . . . . 69 4.5 Proof of Theorem 4.1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.1 Upper bound for h K R K . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.2 Lower bound for h K R K . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.7 Proof of Theorem 4.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.8 Proof of Theorem 4.1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Bibliography 82 Vita 84 viii Chapter 1 Introduction This thesis consists of three self-contained chapters. In chapter 2 we prove a formula for the relative class number of certain abelian extensions of CM number fields. Let H be the Hilbert class field of an imaginary quadratic extension K of a totally real field number field F over Q. We obtain a formula which expresses the relative class number h H /h K in terms of the determinant of a matrix whose entries are logarithms of ratios of a higher analog of the Dedekind eta function evaluated at CM-points on a Hilbert modular variety. This generalizes work of C. L. Siegel [Si2] in the case F = Q. The ratios obtained by Siegel are elliptic units in the Hilbert class field of K = Q( √ −D). The proof involves evaluating the leading term at s = 0 of the abelian L–function of a non-trivial character χ of Gal(H/K). This vanishes to order |F : Q| at s = 0. The formula we obtain is similar to that appearing in Stark’s conjecture [St2] in the case F = Q. In chapter 3 we prove that the scattering determinant for the Hilbert modular group SL 2 (O K ) over a number field K of degree r 1 + 2r 2 is (essentially) a ratio of Dedekind zeta functions of the Hilbert class field of K. This generalizes work of Efrat and Sarnak [ES] in the case K imaginary quadratic of discriminant D = 1, 3. Given the appropriate Weyl’s law, we explain how this formula can be used to prove that the subspace of L 2 ((H 2 ) r 1 ×(H 3 ) r 2 /SL 2 (O K )) consisting of cuspidal eigenfunctions for the Laplacian ∆ is infinite dimensional. In chapter 4 we study analytic properties of zeta functions of quadratic forms over numb er fields. We associate a zeta function Z K (Q, s) to each quadratic form Q in the symmetric space of positive n-forms over a number field K. We prove a functional equation for Z K (Q, s), and compute the residue at the simple pole at s = n 2 . We use the functional equation to obtain a Hecke type integral representation for Z K (Q, s), completed by the appropriate gamma factors. Using the integral representation, we adapt C. L. Siegel’s orgininal argument [Si1] to obtain a sharpening of his lower bound for the residue of the Dedekind zeta function of K 1 [...]... equation to obtain Phragmen-Lindel¨ff type o convexity bounds for ZK (Q, s) on vertical lines 2 Chapter 2 Relative class numbers of abelian extensions of CM number fields 2.1 Introduction Let H be the Hilbert class field of an imaginary quadratic extension K of a totally real number field F of degree n over Q In this chapter, we prove a Stark type formula for the leading term at s = 0 of the L–function of a... z; OF , OF ) satisfies the functional equation G(1 − s)E(s, z; OF , OF ) = G(2(1 − s))E(1 − s, z; OF , OF ), where G(s) is the gamma factor s/2 π −s/2 Γ G(s) = dF s 2 n To prove Theorem 2.2.2, we will need formulas for the Fourier coefficients of the function |N (z + a)|−2s , f (z) = Re(s) > 1 a∈a Let a∗ be the dual lattice of a, T be the trace, and vol(P ) be the volume of a fundamental parallelotope... provides a meromorphic continuation of E(s, z) to C in the s variable In section 2.3, we use the Fourier expansion to compute the Taylor expansion of E(s, z) at s = 0, E(s, z) = En−1 sn−1 + En (z)sn + O(sn+1 ) (2.2) The number En−1 is essentially the regulator of F , and the function En (z) is a multiple of Ψ(z), where Ψ : Hn → C is a modular function analogous to the modulus of the Dedekind eta function... a = b = OF Then by applying the functional equations G(s)ζF (s, C) = G(1 − s)ζF (1 − s, C) and K−v (z) = Kv (z) in (2.9), one can show that E(s, z; OF , OF ) satisfies the functional equation G(1 − s)E(s, z; OF , OF ) = G(2(1 − s))E(1 − s, z; OF , OF ) 2.3 Taylor expansion of E(s, z; a, b) at s = 0 We now use the Fourier expansion (2.9) to compute the first two terms in the Taylor expansion of E(s,... class group, hM the class number, wM the number of roots of unity, r(M ) the regulator, and dM the absolute value of the discriminant Given an integral ideal A in M , define the norm by NM/Q (A) = |OM : A| When A = (α), the norm is given by the product over the embeddings of M, |σ(α)| NM/Q ((α)) = σ Let χ be a non-trivial character of Gal(H/K) Definition 2.1.1 The L–function of χ is defined by L(H/K,... combination of the functions E(s, Φ(zC )) In section 2.7, we combine this expression with the Taylor expansion (2.2) to obtain the evaluation formula For a vector z ∈ Hn , let N (y(z)) denote the product of the imaginary parts of its components The evaluation formula is given in the following result Theorem 2.1.3 Let H be the Hilbert class field of an imaginary quadratic extension K of a totally real number. .. ideal class of K, and let A be an integral ideal in C Then NK/F (A) ∈ ai [˜] for some i ∈ I It follows from Lemma 2.5.1 that there is a decoma position A = ai ω1 + OF ω2 , where ω1 ∈ a−1 OK and ω2 ∈ OK Up to multiplication i by a unit in F , we may assume that the imaginary parts of the components of zC := ω2 /ω1 under a given choice of n real embeddings of F are positive Since K is a totally imaginary... primes p of K The L-function of χ can be expressed as L(H/K, χ, s) = χ(C)ζK (s, C), C∈cl(K) 3 (2.1) where ζK (s, C) is the Dedekind zeta function of the ideal class C of K, NK/Q (A)−s , ζK (s, C) = Re(s) > 1 A∈C We now outline our approach to evaluating the leading term of L(H/K, χ, s) at s = 0 In section 2.2, we compute the Fourier expansion of a non-holomorphic Eisenstein series E(s, z) associated to F... NK/F (A) lies in the ideal class of the form a[˜], a being an integral ideal in a −1 O F , if and only if there exist ω1 ∈ a K and ω2 ∈ OK such that A = aω1 + OF ω2 Choose a complete set of representatives {aj }j∈J of ideal classes of F Among the ideal classes {aj [˜]}j∈J of F , choose the sub-collection {ai [˜]}i∈I of ideal classes a a which contain the relative norm of an ideal in K We may assume... of a non-trivial character of Gal(H/K) We combine this result with the Frobenius determinant relation to obtain a formula for the relative class number of the extension H/K This extends work of C L Siegel [Si2] in the case n = 1 The following notation will remain fixed throughout this chapter For a + number field M , let OM denote the ring of integers, UM the units, UM the totally positive units, cl(M . Saltman John Tate SOME APPLICATIONS OF CLASSICAL MODULAR FORMS TO NUMBER THEORY by Riad Mohamad Masri, B.S.; M.S. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas. brought to my life over the past year. Part of this research was supported by a Joseph Patrick Brannen Fellowship in Mathematics. v SOME APPLICATIONS OF CLASSICAL MODULAR FORMS TO NUMBER THEORY Publication. Mohamad Masri Certifies that this is the approved version of the following dissertation: SOME APPLICATIONS OF CLASSICAL MODULAR FORMS TO NUMBER THEORY Committee: Fernando Rodriguez-Villegas, Supervisor David