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

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.MSC (2010): 11F80 (Galois representations); 20G14 (Linear algebraic groups over finite fields), 12F12 (Inverse Galois theory).

On the images of the Galois representations attached to generic automorphic representations of GSp(4)

“…Proof. We will closely follow the proof of Theorem 1.5 of [ADW16]. As in the previous section we will proceed by cases.…”

Lübeck's classification of representations of finite simple groups of Lie type and the inverse Galois problem for some orthogonal groups

“…Assume that for every ℓ > kn! + 1 a twist of ρ ℓ by some power of the cyclotomic character is regular with tame inertia weights at most k (in the sense of Section 3.2 of [ADW16]) and that for all ℓ = p, t, ρ ℓ is maximally induced of O-type at t of order p. Then, for all primes ℓ different from p and t, the image of ρ ℓ cannot be contained in a maximal subgroup of GO ± n (F ℓ r ) of geometric type. Moreover, for 12-dimensional compatible systems of orthogonal Galois representations as in the previous theorem, it can be proven that the image of ρ proj ℓ is a finite orthogonal group for almost all ℓ.…”

On the images of the Galois representations attached to certain RAESDC automorphic representations of $\mathrm{GL}_n (\mathbb{A}_\mathbb{Q})$

“…An immediate consequence of this results is that the previous groups occurs as a Galois group over Q for infinitely many primes ℓ and infinitely many positive integers s. To the best of our knowledge, the orthogonal groups mentioned above are not previously known to be Galois over Q, except for s = 1 which was studied in [Zyw]. The symplectic case was previously studied in [ADW17], [ADW16], [ADSW15] and [KLS08].…”