#### Vol. 281, No. 1, 2016

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Compatible systems of symplectic Galois representations and the inverse Galois problem II: Transvections and huge image

### Sara Arias-de-Reyna, Luis Dieulefait and Gabor Wiese

Vol. 281 (2016), No. 1, 1–16
##### Abstract

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem.

This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. A key ingredient is a classification of symplectic representations whose image contains a nontrivial transvection: these fall into three very simply describable classes, the reducible ones, the induced ones and those with huge image. Using the idea of an $\left(n,p\right)$-group of Khare, Larsen and Savin, we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strengthened application to the inverse Galois problem.

##### Keywords
compatible systems of symplectic Galois representations, inverse Galois problem
##### Mathematical Subject Classification 2010
Primary: 11F80, 20G40, 12F12
##### Milestones
Received: 16 July 2015
Accepted: 20 July 2015
Published: 9 February 2016
##### Authors
 Sara Arias-de-Reyna Faculté des Sciences, de la Technologie et de la Communication University of Luxembourg 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg Luxembourg Luis Dieulefait Departament d’Álgebra i Geometria, Facultat de Matematiques Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 Barcelona Spain Gabor Wiese Faculté des Sciences, de la Technologie et de la Communication University of Luxembourg 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg Luxembourg