The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-Series (CBMS Regional Conference Series in Mathematics) (Cbms Regional Conference Series in Mathematics, 102)

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. 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. Preface Basic Facts Integer Weight Modular Forms Half-Integer Weight Modular Forms Product Expansions on Modular Forms on $SL_2(\mathbb(Z))$ Partitions Weierstrass Points on the Modular Curve Traces of Singular Moduli and Class Equations Class Numbers of Quadratic Fields Central Values of Modular $L$-functions and Applications Basic Hypergeometric Generating Functions of $L$-values Gaussian Hypergeometric Functions Bibliography Index 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."--Jacket