Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$

Abstract

Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolute discriminant bounded above by $X$ is asymptotically at least $\frac{X^{\frac{\ell}{12(\ell-1)\# GL_2(\mathbb{F}_\ell)}}}{\log X}$. We also obtain a similar result for the number of surjective homomorphisms $\rho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{F}_\ell)$ ordered by the prime to $\ell$ part of the Artin conductor of $\rho$.