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

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

2012-03-29

arxiv.org/abs/1203.6552v1

Mathematics

Number Theory

25 pages

Scientific paper

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. We prove a classification result on those subgroups of a general symplectic group over a finite field that contain a nontrivial transvection. Translating this group theoretic result into the language 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.

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