CBMS Regional Conference Series in Mathematics 2004; 216 pp; softcover Number: 102 ISBN10: 0821833685 ISBN13: 9780821833681 List Price: US$49 Member Price: US$39.20 All Individuals: US$39.20 Order Code: CBMS/102
 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, supercongruences, 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. Readership Advanced graduate students and research mathematicians interested in number theory. Table of Contents  Basic facts
 Integer weight modular forms
 Halfintegral weight modular forms
 Product expansions of modular forms on \(\mathrm{SL}_2(\mathbb{Z})\)
 Partitions
 Weierstrass points on modular curves
 Traces of singular moduli and class equations
 Class numbers of quadratic fields
 Central values of modular \(L\)functions and applications
 Basic hypergeometric generating functions for \(L\)values
 Gaussian hypergeometric functions
 Bibliography
 Index
