The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-Series

by Ken Ono

Modular forms appear in many ways in number theory. They play a central role in the theory of quadratic forms, in particular, as 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 at the center of an immense amount of current research activity. Also detailed in this volume are other roles that modular forms and $q$-series play in number theory, such as 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. The first three chapters provide some basic facts and results on modular forms, which set the stage for the advanced areas that are treated in the remainder of the book. Ono gives ample motivation on topics where modular forms play a role. Rather than cataloging all of the known results, he highlights those that give their flavor. At the end of most chapters, he gives open problems and questions. The book is an excellent resource for advanced graduate students and researchers interested in number theory.

Genres: Mathematics

216 pages, Paperback

First published December 22, 2003

Reviews

[Wissam Raji · Rating 5 out of 5]

One of the main strengths of this book is the presentation of different open problems that are interesting. The book presents different topics in integer and half integer weight modular forms with the concentration on the number theoretic connection to the theory. Thumbs up.