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By Haruzo Hida

This booklet presents a entire account of the idea of moduli areas of elliptic curves (over integer jewelry) and its program to modular kinds. the development of Galois representations, which play a basic function in Wiles' facts of the Shimura - Taniyama conjecture, is given. additionally, the ebook provides an overview of the facts of numerous modularity result of two-dimensional Galois representations (including that of Wiles), in addition to a few of the author's new leads to 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 software, 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 evidence of regularity of the moduli of elliptic curve extra conceptual), and in bankruptcy four, even though constrained to bland circumstances, newly included are Ribet's theorem of complete photo of modular p-adic Galois illustration and its generalization to 'big' lambda-adic Galois representations below light assumptions (a new results of the author). although a number of the impressive advancements defined above is out of the scope of this introductory booklet, the writer offers a style of modern day examine within the quarter of quantity conception on the very finish of the ebook (giving an exceptional account of modularity concept of abelian Q-varieties and Q-curves).

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