Combinatorial Objects at the Interface of q-series and ModularForms Open Access

Jameson, Marie Kathleen (2014)

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In this work, the author proves various results related to q-series and modular forms by employing a broad range of tools from analytic number theory, combinatorics, the theory of modular forms, and algebraic number theory. More specifically, the circle method, the connection between modular forms and elliptic curves, continued fractions, period polynomials, and several other tools from the theory of modular forms are used here. These allow the author to prove a number of results related to q-series and partition functions, modular forms, period polynomials, and certain quadratic polynomials of a fixed discriminant. This includes a proof of the Alder-Andrews conjecture on certain restricted partition functions, and a resolution of a speculation of Don Zagier regarding the Eichler integrals of a distinguished class of modular forms.

Table of Contents

1 Introduction...1

1.1 Forward...1 1.2 Background on modular forms...3

1.2.1 Integer weight modular forms on SL2(Z)...3 1.2.2 Modular forms on congruence subgroups and modular forms of half-integral weight...6 1.2.3 Operators on spaces of modular forms...10 1.2.4 Congruences between modular forms and Sturm's Theorem...12 1.2.5 Partition functions and modular forms...12 1.2.6 Structure of this thesis...14

2 The Alder-Andrews Conjecture...17

2.1 The Alder-Andrews conjecture...17

2.1.1 Estimate of Qd(n) with explicit error bound...19 2.1.2 Estimate of qd(n) with explicit error bound...27 2.1.3 Proof of Alder's Conjecture...38

2.2 A conjecture of Andrews...39

2.2.1 Proof of Andrews's conjecture in the limit...41

3 Congruences for broken k-diamond partitions...44

3.1 Introduction and Statement of Results...44 3.2 Proof of Theorem 3.1...46 3.3 Proof of Theorem 3.2...47

3.3.1 Preliminaries...47 3.3.2 Proof of Theorem 3.2...49

4 Ramanujan congruences modulo powers of 7 and 11...51

4.1 Introduction and Statement of Results...51 4.2 The Work of Watson and Atkin...57 4.3 Properties of the Forms Fi and Gi...60 4.4 Proof of Theorem 4.1 and Corollaries 4.2 and 4.3...65

5 A problem of Zagier on quadratic polynomials and continued fractions...68

5.1 Introduction and Statement of Results...68 5.2 Nuts and Bolts...73

5.2.1 Background on Continued Fractions...73 5.2.2 Elementary Facts about AD(x) and ΩD(x)...74

5.2.3 Defining Ψ(a; b; c; n; X) and Ω0D(x)...76 5.2.4 A Useful Partition of Ω'D0D(x)...79

5.3 Proofs of Theorem 5.1 and Corollaries 5.2 and 5.3...87

5.3.1 Proof of Theorem 5.1...87 5.3.2 Proof of Corollary 5.2...88 5.3.3 Proof of Corollary 5.3...89

5.4 Examples...90 5.5 Proof of Theorem 5.4 and Corollary 5.5...92

6 Quadratic polynomials, period polynomials, and Hecke operators...97

6.1 Preliminaries...101

6.1.1 Background on Period Polynomials and Hecke Operators...101 6.1.2 Congruence Results from the Theory of Modular Forms...103 6.1.3 Zagier's Fk(D; x) and its connection to the theory of modular forms...105 6.1.4 Examples for small k...106

6.2 Proofs...108

6.2.1 Proof of Theorem 6.1...108 6.2.2 Congruences for period polynomials of modular forms...108 6.2.3 Proof of Theorem 6.2...109 6.2.4 Proof of Theorem 6.3...110


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