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A First Course in Modular Forms

  • Fred Diamond
  • Jerry Shurman

Part of the Graduate Texts in Mathematics book series (GTM, volume 228)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Fred Diamond, Jerry Shurman
    Pages 1-44
  3. Fred Diamond, Jerry Shurman
    Pages 45-63
  4. Fred Diamond, Jerry Shurman
    Pages 65-108
  5. Fred Diamond, Jerry Shurman
    Pages 109-162
  6. Fred Diamond, Jerry Shurman
    Pages 163-214
  7. Fred Diamond, Jerry Shurman
    Pages 215-251
  8. Fred Diamond, Jerry Shurman
    Pages 253-312
  9. Fred Diamond, Jerry Shurman
    Pages 313-369
  10. Fred Diamond, Jerry Shurman
    Pages 371-411
  11. Back Matter
    Pages 413-453

About this book

Introduction

This book introduces the theory of modular forms with an eye toward the Modularity Theorem:
All rational elliptic curves arise from modular forms.

The topics covered include

• elliptic curves as complex tori and as algebraic curves,

• modular curves as Riemann surfaces and as algebraic curves,

• Hecke operators and Atkin–Lehner theory,

• Hecke eigenforms and their arithmetic properties,

• the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms,

• elliptic and modular curves modulo p and the Eichler–Shimura Relation,

• the Galois representations associated to elliptic curves and to Hecke eigenforms.

As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.
A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it contains exercises throughout.
Fred Diamond received his Ph.D from Princeton University in 1988 under the direction of Andrew Wiles and now teaches at King's College London. Jerry Shurman received his Ph.D from Princeton University in 1988 under the direction of Goro Shimura and now teaches at Reed College.
   

Keywords

Abelian variety Dimension Grad algebraic curve algebraic geometry number theory

Authors and affiliations

  • Fred Diamond
    • 1
  • Jerry Shurman
    • 2
  1. 1.Department of MathematicsKing’s College London StrandLondonUK
  2. 2.Department of MathematicsReed CollegePortlandUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-27226-9
  • Copyright Information Springer-Verlag New York 2005
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-2005-8
  • Online ISBN 978-0-387-27226-9
  • Series Print ISSN 0072-5285
  • Series Online ISSN 2197-5612
  • Buy this book on publisher's site
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