A power-saving error term in counting $C_2 \wr H$ extensions of an arbitrary base field parametrized by discriminants

TL;DR

The paper analyzes counting number field extensions with Galois group $C_2 \wr H$ over a number field, addressing Malle's conjecture. It develops an alternative method to extract an explicit main term and a power-saving error term for these wreath-product extensions and relaxes the hypotheses on $H$. The approach reduces the problem to sums over degree-$n$ $H$-extensions and quadratic extensions over those fields, leveraging recent bounds on $2$-torsion in class groups and density results for specific base-field cases (notably $S_3$ and $C_3$). The results include improved error terms for several cases, such as $\#\mathcal{F}_{12}(\mathbb{Q}, C_2 \wr S_3, X) = c X + O(X^{5/7+\varepsilon})$ and $\#\mathcal{F}_6(\mathbb{Q}, C_2 \wr C_3, X) = c X + O(X^{2(1+\delta)/5+\varepsilon})$ with $\delta\approx 0.2784$, along with conditional enhancements under the $2$-torsion conjecture. These contributions sharpen the understanding of how fast the counts converge to their asymptotic and provide practical tools for explicit computation of constants in these families.

Abstract

We study Malle's conjecture for the group $C_2 \wr H$ where $H$ is a permutation group. Malle's conjecture for this case was proved by Jürgen Klüners in \cite{arXiv:1108.5597} under mild conditions for $H$. In this article, we provide an alternative method to obtain the explicit main term and a power-saving error term for $C_2 \wr H$ extensions of an arbitrary number field. Furthermore, our method allows us to relax the assumptions for $H.$

Theorems & Definitions (24)

• Theorem 1.1: Klüners, 2012
• Theorem 1.3: Cohen, Diaz Y Diaz, Olivier; 2001
• Theorem 1.3: Cohen, Diaz Y Diaz, Olivier; 2001
• Theorem 1.4: McGown, Tucker; 2024
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Conjecture 1.10