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.
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