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:
• 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.