Fermat's Last Theorem: Basic Tools
American Mathematical Soc., 1 nov. 2013 - 200 pages
This book, together with the companion volume, Fermat's Last Theorem: The proof, presents in full detail the proof of Fermat's Last Theorem given by Wiles and Taylor. With these two books, the reader will be able to see the whole picture of the proof to appreciate one of the deepest achievements in the history of mathematics. Crucial arguments, including the so-called $3$-$5$ trick, $R=T$ theorem, etc., are explained in depth. The proof relies on basic background materials in number theory and arithmetic geometry, such as elliptic curves, modular forms, Galois representations, deformation rings, modular curves over the integer rings, Galois cohomology, etc. The first four topics are crucial for the proof of Fermat's Last Theorem; they are also very important as tools in studying various other problems in modern algebraic number theory. The remaining topics will be treated in the second book to be published in the same series in 2014. In order to facilitate understanding the intricate proof, an outline of the whole argument is described in the first preliminary chapter, and more details are summarized in later chapters.
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A-module ap(E ap(f bijective Chapter coefficients commutative group scheme complete local O-algebra conditions are equivalent Corollary curve over Q cyclotomic character deﬁne defined deﬁnition denoted discrete valuation ring étale commutative group exists EZ(p ﬁeld finite extension finite flat commutative ﬁnitely generated free flat commutative group following conditions form f form of level functor G-torsor Galois representation geometric fiber GLn(A Hecke algebra Hecke operators injective isomorphism class kernel l-adic representation Lemma let F lifting mod l representation modular curves modular forms modular of level morphism morphism of schemes odd prime open subscheme positive integer prime number primitive form proﬁnitely profinitely generated complete projective limit Proof of Proposition proof of Theorem q-expansion quotient reduction modulo residue field ring homomorphism ring of integers satisﬁes satisfies condition satisfies the following semistable elliptic curve subgroup sufﬁces suffices to show Suppose surjection of RTM-triples surjective Tate module Theorem 0.3 unramified