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,