A First Course in Modular Forms

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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 curv...