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The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and \(q\)-series
Ken Ono, University of Wisconsin, Madison, WI
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
2004; 216 pp; softcover
Number: 102
ISBN-10: 0-8218-3368-5
ISBN-13: 978-0-8218-3368-1
List Price: US$49
Member Price: US$39.20
All Individuals: US$39.20
Order Code: CBMS/102
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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.

Readership

Advanced graduate students and research mathematicians interested in number theory.

Table of Contents

  • Basic facts
  • Integer weight modular forms
  • Half-integral 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
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