This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. The Theorem is stated in various forms, related to each other and to their applications to number theory.

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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 Brandeis University. Jerry Shurman received his Ph.D from Princeton University in 1988 under the direction of Goro Shimura and now teaches at Reed College.

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