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Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations


Authors: Sara Arias-de-Reyna, Luis Dieulefait and Gabor Wiese
Journal: Trans. Amer. Math. Soc. 369 (2017), 887-908
MSC (2010): Primary 11F80, 20C25, 12F12
DOI: https://doi.org/10.1090/tran/6708
Published electronically: May 2, 2016
MathSciNet review: 3572258
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Abstract: This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem.

In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.


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Additional Information

Sara Arias-de-Reyna
Affiliation: Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg
Address at time of publication: Departamento de Álgebra, Universidad de Sevilla, Avda. Reina Mercedes s/n. Apdo. 1160, CP 41080, Sevilla, Spain
Email: sara_arias@us.es

Luis Dieulefait
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
Email: ldieulefait@ub.edu

Gabor Wiese
Affiliation: Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg
Email: gabor.wiese@uni.lu

DOI: https://doi.org/10.1090/tran/6708
Received by editor(s): September 23, 2013
Received by editor(s) in revised form: January 30, 2015
Published electronically: May 2, 2016
Additional Notes: The first author worked on this article as a fellow of the Alexander-von-Humboldt foundation. She thanks the Université du Luxembourg for its hospitality during a long term visit. She was also partially supported by projects MTM2013-46231-P and MTM2015-66716-P of the Ministerio de Economía y Competitividad of Spain
The second author was supported by projects MTM2012-33830 and MTM2015-66716-P of the Ministerio de Economía y Competitividad of Spain and by an ICREA Academia Research Prize
The third author was partially supported by the DFG collaborative research centre TRR 45, the DFG priority program 1489 and the Fonds National de la Recherche Luxembourg (INTER/DFG/12/10/COMFGREP)
Article copyright: © Copyright 2016 American Mathematical Society