Upper Bounds for the Number of Number Fields with Alternating Galois Group

TL;DR

This paper improves upper bounds on the number of degree n number fields with Galois group A_n and bounded discriminant, using Pila's work on counting integral points, for a large range of n.

Contribution

It introduces a new upper bound for N(n, A_n, X) that is tighter than previous bounds for 5 < n < 84394, leveraging Pila's counting techniques.

Findings

New upper bound: N(n, A_n, X)  X^{(n^2 - 2)/4(n - 1)+}

Improvement over previous bounds by approximately 1/4 for 5 < n < 84394

Utilizes Pila's work on counting integral points on curves

Abstract

We study the number $N(n, A_n, X)$ of number fields of degree $n$ whose Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(n, A_n, X) \sim C_n X^{1/2} (\log X)^{b_n}$, for constants $b_n$ and $C_n$. For $5 < n < 84394$, the best known upper bound is $N(n, A_n, X) \ll X^{\frac{n + 2}{4}}$; this bound follows from Schmidt's Theorem, which implies there are $\ll X^{\frac{n + 2}{4}}$ number fields of degree $n$. (For $n > 84393$, there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that $N(n, A_n, X) \ll X^{\frac{n^2 - 2}{4(n - 1)}+\epsilon}$, thereby improving the best previous exponent by approximately 1/4 for $5 < n < 84394$.