A First Course in Modular Forms

Graduate Texts in Mathematics #228

by Fred Diamond, Jerry Shurman

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers 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. 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. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Genres: Mathematics, Algebra

466 pages, Hardcover

First published April 30, 2007

Reviews

[Wissam Raji · Rating 5 out of 5]

One of the best references for the theory of modular forms. An excellent buildup to the several versions of Shimura-Taniyama conjecture with a very nice exposition of the theory of modular forms on subgroups of the full modular group. Highly recommended book on the subject that requires some background in advanced number theory, topology, analysis, and abstract algebra.

[Fan Wang · Rating 4 out of 5]

The book does a very good job in explaining the modularity Theorem and it contains friendly exercises which are closely related to the material. It is particularly good at sweeping all the dirty things under the carpet (before it gets a bit out of control in the last two chapters). You prepared popcorns, invited friends and were ready for a film, but in the end you realised you had been spending the whole day watching a trailer.

[Will · Rating 5 out of 5]

Excellent story arc. Slow build-up, but accelerates nicely to a fascinating, but predictable ending.

[Sophia · Rating 5 out of 5]

absolute banger and a thousand chef's kisses. instructive and a good pillow 4 the grad office. keeping the book on my desk to cry and enjoy over again.