Preface

Modular forms appear in many ways in number theory. They play a central

role in the theory of quadratic forms; in particular, they are generating functions

for the number of representations of integers by positive definite quadratic forms

(for example, see [Gro]). They are also key players in the recent spectacular proof

of Fermat’s Last Theorem (see for example, [Bos, CSS]). Modular forms are

presently at the center of an immense amount of research activity.

Here I describe other roles that modular forms and q-series play in number

theory. Sarnak’s elegant monograph [Sar] describes the implications of bounding

Fourier coeﬃcients of modular forms. Recent books such as [Bos, CSS, Hi2]

focus on questions involving Galois representations associated to modular forms

and questions in arithmetical algebraic geometry. I focus on complementary issues

involving the arithmetic and combinatorics of modular forms, and their further

roles in number theory.

The vast amount of research currently being done makes it impossible to provide

a comprehensive account in eleven chapters. Obviously, the subjects I choose to

emphasize are a matter of personal taste, and they reflect the projects which have

most actively engaged my efforts. Here we consider recent work on partitions, basic

hypergeometric functions, Gaussian hypergeometric functions, super-congruences,

Weierstrass points on modular curves, singular moduli, class numbers, L-values,

and elliptic curves. This monograph is an expanded version of the ten lectures I

presented at the NSF-CBMS Regional Conference at the University of Illinois at

Urbana-Champaign. The ten lectures were:

• Preview.

• Facts and tools.

• Infinite products expansions of modular forms.

• Congruences for partitions and singular moduli.

• Supersingular j-invariants.

• Class numbers of quadratic fields.

• Nonvanishing of L-functions.

• Basic hypergeometric series and L-values.

• Gaussian hypergeometric functions and modular forms.

• Open Problems.

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