Galois Representations and Elliptic Curves
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چکیده
An elliptic curve over a field K is a projective nonsingular genus 1 curve E over K along with a chosen K-rational point O of E, which automatically becomes an algebraic group with identity O. If K has characteristic 0, the n-torsion of E, denoted E[n], is isomorphic to (Z/nZ) over K. The absolute Galois group GK acts on these points as a group automorphism, hence it acts on the inverse limit lim ←−n which is isomorphic to ∏ ` T`, where T` = lim ←− ] is the Tate module. The Tate module is isomorphic to Z` as an abelian group, and since the Galois group acts on finite quotients, it acts continuously, and we get a continuous homomorphism ρE,` : GK → GL2(Z`). This paper presents of a proof of Serre’s open image theorem, which states that the image is an open subgroup. The bulk of the proof involves facts about `-adic Galois representations constructed from global class field theory. The paper assumes standard results in algebraic number theory, algebra, and Lie group theory, as well as the basics of the arithmetic of elliptic curves. The paper quotes some results coming from p-adic Hodge theory of étale cohomology in the section on Hodge-Tate decompositions. Most of the material is from [Ser89], the original source of this theorem.
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تاریخ انتشار 2012