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

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 g...

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Bibliographic Details
Published in:arXiv.org 2014-05
Main Authors: Arias-de-Reyna, Sara, Dieulefait, Luis, Wiese, Gabor
Format: Article
Language:English
Subjects:
Conjugation
Fields (mathematics)
Image classification
Representations
Online Access:Get full text
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Summary: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 (n,p)-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 strenghtened application to the inverse Galois problem.
ISSN:2331-8422
DOI:10.48550/arxiv.1203.6552