Combinatorial Objects at the Interface of q-series and Modular Forms Open Access
Jameson, Marie Kathleen (2014)
Abstract
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
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'D\Ω0D(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
Bibliography...111
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