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Hermitian Jacobi Forms and Congruences

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Hermitian Jacobi Forms and Congruences HERMITIAN JACOBI FORMS AND CONGRUENCES Jayantha Senadheera Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS August 2014 APPROVED: Olav K. Richter, Major Professor William Cherry, Committee Member Charles H. Conley, Committee Member Su Gao, Chair of the Department of Mathematics Mark Wardell, Dean of the Toulouse Graduate School Senadheera, Jayantha. Hermitian Jacobi Forms and Congruences. Doctor of Philosophy (Mathematics), August 2014, 60 pp., 30 numbered references. In this thesis, we introduce a new space of Hermitian Jacobi forms, and we determine its structure. As an application, we study heat cycles of Hermitian Jacobi forms, and we establish a criterion for the existence of U(p) congruences of Hermitian Jacobi forms. We demonstrate that criterion with some explicit examples. Finally, in the appendix we give tables of Fourier series coefficients of several Hermitian Jacobi forms. Copyright 2014 by Jayantha Senadheera ii ACKNOWLEDGEMENTS First and foremost, I would like to thank my thesis advisor, Dr. Olav Richter, for introducing me to the wonderful area of automorphic forms. I highly appreciate his excellent guidance, understanding, and patience throughout my work on automorphic foms. I am extremely grateful to the faculty and staff of the mathematics department at the University of North Texas for arranging a great research environment to carry out my studies. I take this opportunity to express my special thanks to Dr. Martin Raum, for providing SAGE codes to compute Hermitian Jacobi forms, which were extremely helpful at the initial stages of this research. I greatly appreciate the support of my friends Dhanyu and Fazeen, who assisted whenever I needed help with computer related issues. I will forever be thankful to Dr. W. Ramasinghe for encouraging me to return to mathematics again, and for offering me an academic position, even 10 years after I had received a bachelor's degree. I will also be forever thankful to Dr. Sunil Gunarathne for encouraging me to maintain my interest in number theory. Both of them gave me enormous support for pursuing a Ph.D. in the USA. I am extremely grateful to my beloved wife Janakee and my beloved son Matheesha for their unconditional support, patience, and encouragement during my studies. Finally, I dedicate my dissertation work to my loving parents, the late Mrs. Wimala Mahathanthile and Mr. Adwin Senadheera with a special feeling of gratitude. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ............................................................................................... iii CHAPTER 1. INTRODUCTION .................................................................................... 1 CHAPTER 2. MODULAR FORMS ................................................................................ 3 2.1 Definition and Examples ............................................................................ 3 2.2 U(p) Congruences of Modular Forms ......................................................... 6 CHAPTER 3. CLASSICAL JACOBI FORMS ................................................................ 8 3.1 Definition of Jacobi Group and Jacobi Forms ............................................ 8 3.2 Taylor Development of Jacobi Forms and the Structure Theorem for Jk,1 .................................................................................................................. 10 3.3 The Theta Decomposition ......................................................................... 12 CHAPTER 4. HERMITIAN JACOBI FORMS ............................................................. 14 4.1 The Hermitian Jacobi Group and Hermitian Jacobi Forms ...................... 14 4.2 The Theta Decomposition ......................................................................... 17 4.3 The Structure of Hermitian Jacobi Forms of Index 1 ............................... 22 CHAPTER 5. U(p) CONGRUENCES OF HERMITIAN JACOBI FORMS ................. 29 5.1 Congruences and Filtrations ..................................................................... 29 5.2 Examples ................................................................................................... 37 APPENDIX. FOURIER COEFFICIENTS OF HERMITIAN JACOBI FORMS .......... 41 BIBLIOGRAPHY ........................................................................................................... 58 iv CHAPTER 1 INTRODUCTION Eichler and Zagier [7] systematically developed a theory of Jacobi forms, which are holomorphic functions of two complex variables that satisfy certain transformation laws under the action of the Jacobi group. Jacobi forms appear naturally in different areas of mathematics and physics. In particular, they occur as Fourier-Jacobi coefficients of Siegel modular forms of degree 2. This link played an important role in the solution of the Saito- Kurokawa conjecture (see [15, 16, 17, 2, 30]). Hermitian modular forms are generalizations of Siegel modular forms, and Hermitian Jacobi forms are holomorphic functions of three complex variables that occur as Fourier- Jacobi coefficients of Hermitian modular forms of degree 2. First Haverkamp [9, 10], and later Sasaki [25] and Das [4, 5] studied Hermitian Jacobi forms over the Gaussian number field Q(i), and they established several properties of such Jacobi forms. The usual heat operator is an important tool in the study of classical Jacobi forms, which for example allows one to explore congruences and filtrations of Jacobi forms (see [23, 24]). As in the case of classical Jacobi forms, there is also a heat operator in the context of Hermitian Jacobi forms (see (12)). However, the action of that heat operator on the Hermitian Jacobi forms in [9, 25, 4, 5] is not natural, i.e., it cannot be \corrected" as in the case of classical Jacobi forms. Thus, one needs a different notion of Hermitian Jacobi forms. In this thesis, I introduce a more general definition of Hermitian Jacobi forms (see Definition 4.3 and Definition 4.5), which permits the desired action of the heat operator. I determine the structure of such forms if the index is 1. More precisely, I have shown (see Theorem 4.14) that the ring of Hermitian Jacobi forms of index 1 over Q(i) is a free module of rank 4 over the ring of elliptic modular forms. As an application, I study \heat cycles" of Hermitian Jacobi forms of index 1, and I establish a criterion for the existence of U(p) congruences of Hermitian Jacobi forms (see Theorem 5.11). I present several explicit examples to illustrate Theorem 5.11. Recall that 1 U(p) congruences of elliptic modular forms have applications in the context of traces of singular moduli and class equations (see Ahlgren and Ono [1], Elkies, Ono, and Yang [8], and Chapter 7 of Ono [20]). It would be interesting to see if U(p) congruences for Hermitian Jacobi forms also find further applications. This thesis is organized as follows. In Chapter 2, I briefly recall the notion of a modular form. In Chapter 3, I review the classical Jacobi forms of [7], and I state some of their basic properties. In Chapter 4, I give a new definition of Hermitian Jacobi forms over Q(i), and I prove Theorem 4.14. In Chapter 5, I discuss U(p) congruences of Hermitian Jacobi forms over Q(i) of index 1, and I prove Theorem 5.11. Finally, in the Appendix , I give tables of Fourier series coefficients of several Hermitian Jacobi forms. 2 CHAPTER 2 MODULAR FORMS This brief chapter is on modular forms. In Section 2.1 I recall the definition and basic examples of modular forms, and in Section 2.2 I review the notion of U(p) congruences for modular forms. The content of Section 2.1 can by found in any text on modular forms, and Ono [20] is a good reference for both sections. Throughout this chapter, k is a nonnegative integer. 2.1. Definition and Examples Let H be the usual complex upper half plane, τ 2 H be a typical variable, q := e2πiτ , and n a b o Γ := SL2(Z) = ( c d ) a; b; c; d 2 Z; ad − bc = 1 be the modular group. It is well known that Γ acts on H by linear fractional transformations, and modular forms are equivariant with respect to this action. Definition 2.1. A function f : H ! C is a holomorphic modular form of weight k on Γ if it satisfies the following conditions: (i) f is holomorphic on the upper half plane H. aτ + b (ii) f = (cτ + d)kf(τ) 8 ( a b ) 2 Γ. cτ + d c d (iii) The Fourier series expansion of f is of the form 1 X f(τ) = c(n)e2πinτ : n=0 If c(0) = 0, then f is called a cusp form. Let Mk denote the space of holomorphic modular forms of weight k and let Sk denote its subspace of cusp forms of weight k. The definition of modular forms can easily be extended to other groups and also to the case where k is a half-integer (see for example Ono [20]). In this thesis, I mostly deal with integer weight modular forms for the full modular group, but in Chapter 3 I briefly 3 encounter modular forms of half-integral weight for the congruence subgroup n a b o Γ0(4) := ( c d ) 2 Γ j c ≡ 0 (mod 4) : Next I recall Eisenstein series, which are essential examples of modular forms. Example 2.2. Let k ≥ 2. The Eisenstein series of weight k are defined by 1 2k X E (τ) := 1 − σ (n)qn; k B k−1 k n=1 where X k−1 σk−1(n) := d djn is the usual divisor function, and where the Bernoulli numbers Bk are defined by 1 x X xk = B : ex − 1 k k! k=0 If k ≥ 4, then Ek 2 Mk. If k = 2, then E2 is a so-called quasimodular form, which satisfies aτ + b 6c (1) E = (cτ + d)2 E (τ) + (cτ + d) 8 ( a b ) 2 Γ : 2 cτ + d 2 πi c d Finally, any modular form on Γ can be expressed in terms of the Eisenstein series E4 and E6. For example, the famous Ramanujan-Delta function can be written as 1 ∆ := (E3 − E2) 2 S : 1728 4 6 12 Other important examples of modular forms are theta functions. I now recall the classical Jacobi theta functions. Definition 2.3. Let a; b 2 R. Then X πi(a+n)2τ+2πi(n+a)(z+b) (2) θa;b(τ; z) = e : n2Z Of particular interest are the following special cases: P πin2τ+2πinz (i) θ0;0(τ; z) = e P πin2τ+2πin(z+ 1 ) (ii) θ 1 (τ; z) = e 2 0; 2 P πi(n+ 1 )2τ+2πi(n+ 1 )z (iii) θ 1 (τ; z) = e 2 2 2 ;0 4 P πi(n+ 1 )2τ+2πi(n+ 1 )(z+ 1 ) (iv) θ 1 1 (τ; z) = e 2 2 2 .
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