Abstract
We describe an approach to constructing Galois extensions of \({\mathbf{Q}}\) with Galois group isomorphic to an open subgroup of \(GL_n({\mathbf{Z}}_p)\) for various values of n and primes p. The approach involves studying a certain topological generating set for a Sylow pro-p subgroup of \(SL_n({\mathbf{Z}}_p)\). It also involves finding algebraic number fields which admit a Galois extension with Galois group isomorphic to a free pro-p on n generators.
Résumé
Cet article décrit une méthode pour construire des extensions de Q de groupe de Galois isomorphe à un sous-groupe ouvert de \({GL}_{n}(\mathbf{Z}_{p})\), 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 \({SL}_{n}(\mathbf{Z}_{p})\). 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.
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For Glenn Stevens on the occasion of his 60th birthday.
R. Greenberg’s research supported in part by National Science Foundation grant DMS-0200785.
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Greenberg, R. Galois representations with open image. Ann. Math. Québec 40, 83–119 (2016). https://doi.org/10.1007/s40316-015-0050-6
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DOI: https://doi.org/10.1007/s40316-015-0050-6