A First Course in Modular Forms (Google eBook)

Fred Diamond

Springer, Jan 19, 2005 - Mathematics - 436 pages

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.

Contents

• 1: 1
• Figure 11 A contour: 11
• 2: 45
• 3: 64
• 4: 109
• 5: 163
• 6: 211
• 7: 248
• 8: 309
• Igusas Theorem: 347
• 9: 365
• Hints and Answers to the Exercises: 400

Popular passages

Page 6 - A cusp form of weight k is a modular form of weight k whose Fourier expansion has leading coefficient OQ = 0, ie, n=l The set of cusp forms is denoted 5fe(SL2(Z)).‎

About the author

Fred Diamond teaches at Brandeis University.

Bibliographic information

• Title: A First Course in Modular Forms; Volume 228 of Graduate Texts in Mathematics, ISSN 0072-5285
• Author: Fred Diamond
• Editor: Jerry Shurman
• Edition: illustrated
• Publisher: Springer, 2005
• ISBN: 038723229X, 9780387232294
• Length: 436 pages
• Subjects: Mathematics / Geometry / Algebraic; Mathematics / Number Theory