Counting $C_2 \wr S_4$ fields with a power saving error term

TL;DR

This paper improves the error term in counting degree 8 fields with Galois group $C_2 times S_4$, providing a stronger power saving than previous results, and extends bounds to other related groups.

Contribution

It establishes a sharper power saving error term for counting $C_2 times S_4$ fields and extends bounds to additional permutation groups of degree 8.

Findings

Improved error term: $O(X^{3/4-1/30})$ for $N_8(C_2\wr S_4,X)$

Confirmed Malle's conjecture with stronger bounds for specific groups

Extended bounds to groups $S_4$, $C_2^3 \rtimes S_4$, $GL_2(\mathbb{F}_3)$, and $Q_8 \rtimes S_4$

Abstract

Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|\Delta_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^{\alpha}(\log X)^\beta$. Previously, Kl\"uners proved Malle's conjecture for $G=C_2 \wr S_4$. His proof gives a power savings of $O(X^{7/8})$. We improve Kl\"uners' result by establishing a stronger power saving error term for the count of such fields. Specifically, we show $N_8(C_2\wr S_4,X)=CX+O(X^{3/4-1/30})$. Additionally, we obtain new bounds on $N_8(G,X)$ for the groups $S_4$, $C_2^3 \rtimes S_4$, $GL_2 (\mathbb{F}_3)$, and $Q_8\rtimes S_4$ as permutation subgroups of $S_8$.