By Haruzo Hida
This booklet presents a accomplished account of the speculation of moduli areas of elliptic curves (over integer jewelry) and its software to modular varieties. the development of Galois representations, which play a primary position in Wiles' evidence of the Shimura - Taniyama conjecture, is given. furthermore, the ebook offers an summary of the evidence of various modularity result of two-dimensional Galois representations (including that of Wiles), in addition to a number 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 further to bankruptcy 1. As an software, a down-to-earth description of formal deformation conception 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, notwithstanding constrained to boring circumstances, newly integrated are Ribet's theorem of complete photograph of modular p-adic Galois illustration and its generalization to 'big' lambda-adic Galois representations lower than light assumptions (a new results of the author). notwithstanding a number of the notable advancements defined above is out of the scope of this introductory ebook, the writer supplies a flavor of modern examine within the sector of quantity idea on the very finish of the ebook (giving a superb account of modularity conception of abelian Q-varieties and Q-curves).
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