A First Course in Modular Forms

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,

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