Galois representations with open image
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Galois representations with open image Ralph Greenberg1 For Glenn Stevens on the occasion of his 60th birthday.
Received: 31 December 2014 / Accepted: 15 April 2015 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract We describe an approach to constructing Galois extensions of Q with Galois group isomorphic to an open subgroup of G L n (Z p ) for various values of n and primes p. The approach involves studying a certain topological generating set for a Sylow pro- p subgroup of S L n (Z p ). It also involves finding algebraic number fields which admit a Galois extension with Galois group isomorphic to a free pro- p on n generators. Keywords
Galois representations · Iwasawa theory · pro- p groups
Résumé Cet article décrit une méthode pour construire des extensions de Q de groupe de Galois isomorphe à un sous-groupe ouvert de G L n (Z p ), pour plusieurs valeurs différentes de n et du nombre premier p. Notre démarche consiste à étudier un certain système de générateurs topologiques d’un pro- p sous-groupe de Sylow de S L n (Z p ). Elle repose aussi sur la mise en évidence de certaines extensions de corps de nombres de groupe de Galois un pro- p-groupe libre sur n générateurs. Mathematics Subject Classification
11R23 · 11R32 · 22E20
1 Introduction Suppose that p is a prime and that n ≥ 1. Let G Q = Gal(Q/Q) be the absolute Galois group of Q. Our objective in this paper is to construct continuous representations ρ : G Q −→ G L n (Z p )
R. Greenberg’s research supported in part by National Science Foundation grant DMS-0200785.
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Ralph Greenberg [email protected] Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA
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whose image is open. Continuous n-dimensional representations ρ arise naturally in algebraic geometry for every value of n, but it seems difficult to find such examples where the image is open when n ≥ 3. The construction described in this paper is not at all geometric in nature. It depends on the structure of certain Galois groups and of certain subgroups of G L n (Z p ). We assume always that p is an odd prime. One typical result is the following. Proposition 1.1 Suppose that p is a regular prime and that p ≥ 4[ n2 ] + 1. Let K = Q(μ p ) and let M denote the maximal pro- p extension of K which is unramified outside of p. Then there exist continuous representations ρ : Gal(M/Q) → G L n (Z p ) with an open image. A theorem of Shafarevich shows that if p is a regular prime, then Gal(M/K ) is a free pro- p group on p+1 2 generators. On the other hand, it turns out that a Sylow pro- p subgroup S0 of S L n (Z p ) requires only n generators topologically. One can then define a surjective homomorphism σ0 from Gal(M/K ) to S0 if p ≥ 2n − 1. There are many choices. However, one must make the definition carefully enough so that σ0 can be extended to Gal(M/Q), giving a homomorphism ρ0 : Gal(M/Q) → G L n (Z p ). If n is even, we will need the slightly stronger inequality p ≥ 2n +1 to show that this is possible. The image of
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