A First Course in Modular Forms (Graduate Texts in Mathematics, Vol. 228) by Fred Diamond

Graduate Texts in Mathematics #228

by Fred Diamond

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 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.

Genres: Mathematics, Algebra

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.