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- W187293611 abstract "In recent years, the study of the deformation theory of Galois representations has become of central importance in arithmetic algebraic geometry. The most fundamental question in this field is the explicit determination of universal deformation rings associated to given residual representations. It was observed by Mazur in his first paper [Maz90, Section 1.6] on the subject that the solution of this problem is immediate if a certain Galois cohomology group associated to the residual representation vanishes. The goal of this paper is to provide an overview of a theorem of Flach which yields the vanishing of this cohomology group for many mod l representations coming from rational elliptic curves. We do not seek to give a complete proof; we hope only to make clear the main ideas. In the process we will touch on many facets of arithmetic algebraic geometry, including Tate’s duality theorems in Galois cohomology, generalized Selmer groups, Kolyvagin’s theory of Euler systems and the geometry of modular curves. The work we will describe actually has another, more direct, application: it can be used in many cases to prove the Taylor-Wiles isomorphism between a certain universal deformation ring and a certain Hecke algebra. We will not touch on this aspect; for details, see [Maz94] or [Wes00]. We have tried to keep the prerequisites to a minimum. The main requirement is a good familiarity with Galois cohomology. The algebraic geometry we use is mostly at the level of [Sil86, Chapters 1 and 2], which the exception of Appendix B, which is significantly more advanced. Some familiarity with elliptic curves is helpful, although with the exception of Appendix A we will use little more than the Tate module and the existence of the Weil pairing. I would like to thank Brian Conrad, Matthew Emerton and Karl Rubin for teaching me much of the material presented here. I would also like to thank Fernando Gouvea for encouraging the writing of this paper. Above all, I would like to thank Barry Mazur for his constant help and insights; I can only hope that his point of view is visible in the mathematics below." @default.
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- W187293611 date "2003-01-01" @default.
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- W187293611 title "AN OVERVIEW OF A THEOREM OF FLACH" @default.
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