Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sambhabi Bose, Kevin J. McGown, Ishan Panpaliya, Natalie Welling, Laney Williams

TL;DR

The paper advances the arithmetic statistics of number fields by deriving a stronger power-saving error term for counting octic fields with Galois group $C_2 \wr S_4$, proving $N_8(C_2 \wr S_4,X)=CX+O(X^{3/4-1/30})$. The authors achieve this by decomposing octic fields into quadratic extensions over base quartic fields with Galois group $S_4$, leveraging Bhargava’s quartic-field counts and McGown–Tucker’s relative-quadratic-extension asymptotics, and employing explicit bounds on 2-torsion in class groups. In addition to sharpening the main result, they establish new upper bounds for $N_8(G,X)$ for four other permutation groups $G$ that occur as subgroups of $S_8$: $S_4$, $GL_2(\mathbb{F}_3)$, $C_2^3 \rtimes S_4$, and $Q_8\rtimes S_4$, with exponents $1/2$, $3/5$, $9/14$, and $3/5$ respectively. These improvements contribute to the broader program towards Malle’s conjecture by furnishing power-saving error terms for additional groups and illustrating a unified framework for octic-field counts via base-field quartic data and controlled ramification. The methods combine discriminant-splitting, Selmer-group bounds, and tail estimates for quartic fields, highlighting the role of 2-torsion and ramification geometry in refining asymptotics for high-degree number fields.

Abstract

Let $N_d(G,X)$ denote the number of degree $d$ extensions of $\mathbb{Q}$ with Galois closure $G$ and $|Δ_K|\leq X$. Malle's conjecture predicts an asymptotic of the form $N_d(G,X)\sim CX^α(\log X)^β$. Previously, Klüners proved Malle's conjecture for $G=C_2 \wr S_4$. His proof gives a power savings of $O(X^{7/8})$. We improve Klüners' 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 $S_8$.

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

TL;DR

The paper advances the arithmetic statistics of number fields by deriving a stronger power-saving error term for counting octic fields with Galois group , proving . The authors achieve this by decomposing octic fields into quadratic extensions over base quartic fields with Galois group , leveraging Bhargava’s quartic-field counts and McGown–Tucker’s relative-quadratic-extension asymptotics, and employing explicit bounds on 2-torsion in class groups. In addition to sharpening the main result, they establish new upper bounds for for four other permutation groups that occur as subgroups of : , , , and , with exponents , , , and respectively. These improvements contribute to the broader program towards Malle’s conjecture by furnishing power-saving error terms for additional groups and illustrating a unified framework for octic-field counts via base-field quartic data and controlled ramification. The methods combine discriminant-splitting, Selmer-group bounds, and tail estimates for quartic fields, highlighting the role of 2-torsion and ramification geometry in refining asymptotics for high-degree number fields.

Abstract

Let denote the number of degree extensions of with Galois closure and . Malle's conjecture predicts an asymptotic of the form . Previously, Klüners proved Malle's conjecture for . His proof gives a power savings of . We improve Klüners' result by establishing a stronger power saving error term for the count of such fields. Specifically, we show . Additionally, we obtain new bounds on for the groups , , , and as permutation subgroups of .
Paper Structure (8 sections, 10 theorems, 56 equations, 1 table)

This paper contains 8 sections, 10 theorems, 56 equations, 1 table.

Table of Contents

  1. Introduction
  2. Known results for groups in $\mathcal{G}$
  3. Counting octic extensions $L/K/\mathbb{Q}$ that are quadratic over $S_4$-quartic
  4. An upper bound for octic fields with Galois group $S_4$
  5. An upper bound for octic fields with Galois group $C_2^3\rtimes S_4$
  6. An upper bound for octic fields with Galois group $\mathop{\mathrm{GL}}\nolimits_2(\mathbb{F}_3)$
  7. An upper bound for octic fields with Galois group $Q_8\rtimes S_4$
  8. Acknowledgements

Key Result

Theorem 1.1

The number of octic fields with Galois group $C_2\wr S_4$ and discriminant bounded by $X$ equals where

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • proof : Proof of Theorem \ref{['T:2']} for $G=S_4$
  • Lemma 5.1
  • proof
  • proof : Proof of Theorem \ref{['T:2']} for $G=C_2^3\rtimes S_4$
  • ...and 14 more