**CBMS Regional Conference Series in Mathematics**

Volume: 102;
2004;
216 pp;
Softcover

MSC: Primary 11; 05;

Print ISBN: 978-0-8218-3368-1

Product Code: CBMS/102

List Price: $49.00

Individual Price: $39.20

**Electronic ISBN: 978-1-4704-1757-4
Product Code: CBMS/102.E**

List Price: $49.00

Individual Member Price: $39.20

#### Supplemental Materials

# The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series

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*Ken Ono*

A co-publication of the AMS and CBMS

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.

#### Table of Contents

# Table of Contents

## The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Basic facts 110 free
- Integer weight modular forms 2130
- Half-integral weight modular forms 4958
- Product expansions of modular forms on 𝑆𝐿₂(ℤ) 6978
- Partitions 8594
- Weierstrass points on modular curves 109118
- Traces of singular moduli and class equations 121130
- Class numbers of quadratic fields 133142
- Central values of modular 𝐿-functions and applications 149158
- Basic hypergeometric generating functions for 𝐿-values 165174
- Gaussian hypergeometric functions 183192
- Bibliography 207216
- Index 215224 free
- Back Cover Back Cover1226

#### Readership

Advanced graduate students and research mathematicians interested in number theory.