Enumerating Galois extensions of number fields

TL;DR

This paper derives asymptotic formulas for counting Galois extensions of number fields with bounded discriminant, introducing a new upper bound that improves previous estimates and decays with the size of the Galois group.

Contribution

It provides the first bound for general Galois groups with an exponent decreasing as the group size increases, advancing understanding of Galois extension enumeration.

Findings

Asymptotic formulas for Galois extensions with bounded discriminant

New upper bound on the number of extensions with fixed Galois group

Improved decay rate of bounds as group size increases

Abstract

Let $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with absolute discriminant bounded by some $X \geq 1$, as $X\to\infty$. We also provide an asymptotic formula for the closely related count of extensions $K/k$ whose normal closure has discriminant bounded by $X$. The key behind these results is a new upper bound on the number of Galois extensions of $k$ with a given Galois group $G$ and discriminant bounded by $X$; we show the number of such extensions is $O_{[k:\mathbb{Q}],G} (X^{ \frac{4}{\sqrt{|G|}}})$. This improves over the previous best bound $O_{k,G,\epsilon}(X^{\frac{3}{8}+\epsilon})$ due to Ellenberg and Venkatesh. In particular, ours is the first bound for general $G$ with an exponent that decays as $|G| \to \infty$.