Lower bounds for $\text{GL}_2(\mathbb{F}_\ell)$ number fields

TL;DR

This paper establishes lower bounds on the number of degree $n$ number fields with Galois group $ ext{GL}_2(_ell)$ or $ ext{PGL}_2(_ell)$, advancing understanding of their distribution and providing computational methods.

Contribution

It provides the first explicit lower bounds for such fields with these Galois groups and introduces a method to compute bounds for any permutation representation.

Findings

Lower bounds for $ ext{GL}_2(_ell)$ and $ ext{PGL}_2(_ell)$ number fields established.

Method for computing bounds for any permutation representation of these groups.

Results applicable for primes $ell geq 13$.

Abstract

Let $\mathcal{F}_n(X;G)$ denote the set of number fields of degree $n$ with absolute discriminant no larger than $X$ and Galois group $G$. This set is known to be finite for any finite permutation group $G$ and $X \geq 1$. In this paper, we give a lower bound for the cases $G=\text{GL}_2(\mathbb{F}_\ell), \; \text{PGL}_2(\mathbb{F}_\ell)$ for primes $\ell \geq 13$. We also provide a method to compute lower bounds for any permutation representations of these groups.