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

TL;DR

This paper classifies symplectic Galois representations with large residual images, especially those containing transvections, and applies these results to advance the inverse Galois problem.

Contribution

It provides a classification of symplectic representations with transvections and establishes conditions for huge images, enhancing inverse Galois problem solutions.

Findings

Classified symplectic representations with transvections into three types.

Established simple criteria for symplectic Galois representations to have huge images.

Strengthened applications to the inverse Galois problem using these classifications.

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 (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.