Softcover ISBN:  9780821833681 
Product Code:  CBMS/102 
216 pp 
List Price:  $52.00 
Individual Price:  $41.60 
Electronic ISBN:  9781470417574 
Product Code:  CBMS/102.E 
216 pp 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 102; 2004MSC: Primary 11; 05;
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.ReadershipAdvanced graduate students and research mathematicians interested in number theory.

Table of Contents

Chapters

Chapter 1. Basic facts

Chapter 2. Integer weight modular forms

Chapter 3. Halfintegral weight modular forms

Chapter 4. Product expansions of modular forms on $\mathrm {SL}_2(\mathbb {Z})$

Chapter 5. Partitions

Chapter 6. Weierstrass points on modular curves

Chapter 7. Traces of singular moduli and class equations

Chapter 8. Class numbers of quadratic fields

Chapter 9. Central values of modular $L$functions and applications

Chapter 10. Basic hypergeometric generating functions for $L$values

Chapter 11. Gaussian hypergeometric functions


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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, 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.
Advanced graduate students and research mathematicians interested in number theory.

Chapters

Chapter 1. Basic facts

Chapter 2. Integer weight modular forms

Chapter 3. Halfintegral weight modular forms

Chapter 4. Product expansions of modular forms on $\mathrm {SL}_2(\mathbb {Z})$

Chapter 5. Partitions

Chapter 6. Weierstrass points on modular curves

Chapter 7. Traces of singular moduli and class equations

Chapter 8. Class numbers of quadratic fields

Chapter 9. Central values of modular $L$functions and applications

Chapter 10. Basic hypergeometric generating functions for $L$values

Chapter 11. Gaussian hypergeometric functions