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 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.