Web of Modularity

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. They are also key players in the recent spectacular proof of Fermat's Last Theorem. Modular forms are currently at the center of an immense amount of research activity. Other roles that modular forms and $q$-series play in number theory are described in this book. In particular, applications and connections to basic hypergeometric functions, Gaussian hypergeometric functions, super-congruences, Weierstrass points on modular curves, singular moduli, class numbers, $L$-values, and elliptic curves are described in detail. The first three chapters of the book provide some basic facts and results on modular forms, setting the stage for the remainder of the book, where advanced topics are treated. Ono provides ample motivation on some of the topics in which modular forms play a role. There is no attempt to catalog all of the results in these areas; rather, the author highlights results which give their flavor. At the end of most chapters, there are some open problems and questions.