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This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura-Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction. In this new second edition, a detailed description of Barsotti-Tate groups (including formal Li

Curves, Elliptic. --- Forms, Modular. --- Modular forms --- Forms (Mathematics) --- Elliptic curves --- Curves, Algebraic

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Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics-the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep-and often very mystifying-mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.

Counting. --- Curves, Elliptic. --- Elliptic functions. --- Number theory. --- Elliptic functions --- Curves, Elliptic --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Fonctions elliptiques --- Courbes elliptiques --- Théorie des nombres --- Number study --- Numbers, Theory of --- Elliptic curves --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Algebra --- Curves, Algebraic --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic