Includes bibliographical references and index.

Modular forms, elliptic curves, and modular curves -- Modular curves as Riemann surfaces -- Dimension formulas -- Eisenstein series -- Hecke operators -- Jacobians and abelian varieties -- Modular curves as algebraic curves -- The Eichler-Shimura relation and L-functions -- Galois representations -- Hints and answers to the exercises -- List of symbols -- Index -- References.

A first course in modular forms: This textbook introduces the theory of modular forms and explains their deep connections with elliptic curves, modular curves, and modern number theory. The authors develop the subject from fundamental concepts through modular curves as Riemann surfaces and algebraic curves, dimension formulas, Eisenstein series, Hecke operators, Hecke eigenforms, Jacobians and Abelian varieties, the Eichler-Shimura relation, L-functions, and Galois representations. A major objective is to explain the Modularity Theorem and its relationship to rational elliptic curves. Designed for advanced undergraduates and beginning graduate students, the book provides a largely self-contained treatment, includes exercises throughout, and does not assume prior knowledge of algebraic number theory or algebraic geometry.