By Haruzo Hida

This publication presents a finished account of the speculation of moduli areas of elliptic curves (over integer earrings) and its software to modular types. the development of Galois representations, which play a primary function in Wiles' facts of the Shimura - Taniyama conjecture, is given. additionally, the publication offers an summary of the evidence of numerous modularity result of two-dimensional Galois representations (including that of Wiles), in addition to many of the author's new ends up in that path. during this new moment variation, a close description of Barsotti - Tate teams (including formal Lie teams) is extra to bankruptcy 1. As an program, a down-to-earth description of formal deformation concept of elliptic curves is integrated on the finish of bankruptcy 2 (in order to make the facts of regularity of the moduli of elliptic curve extra conceptual), and in bankruptcy four, although restricted to boring circumstances, newly integrated are Ribet's theorem of complete picture of modular p-adic Galois illustration and its generalization to 'big' lambda-adic Galois representations less than gentle assumptions (a new results of the author). notwithstanding many of the amazing advancements defined above is out of the scope of this introductory booklet, the writer supplies a flavor of state-of-the-art study within the quarter of quantity conception on the very finish of the e-book (giving a superb account of modularity conception of abelian Q-varieties and Q-curves).

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