Lower bounds for the number of number fields with Galois group $G{L}_2({\mathbb{F}}_{\ell })$

Abstract

Let $\ell \ge 5$ 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 $G{L}_2({\mathbb{F}}_{\ell })$ and absolute discriminant bounded above by $X$ is asymptotically at least $\frac{X^{\frac{\ell }{12(\ell -1)\#G{L}_2({\mathbb{F}}_{\ell })}}}{\log X}$. We also obtain a similar result for the number of surjective homomorphisms $\rho :Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to G{L}_2({\mathbb{F}}_{\ell })$ ordered by the prime to $\ell$ part of the Artin conductor of $\rho$.

Version 4: Minor corrections following referee report. Accepted for publication in Forum Mathematicum

References in corpus (1)

• Arithmetic statistics and diophantine stability for elliptic curves