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

TL;DR

This paper introduces a new method to analyze the distribution of $C_2 times H$ extensions over number fields, providing explicit main terms and power-saving error bounds, thus extending previous results with fewer restrictions.

Contribution

It offers an alternative approach to Malle's conjecture for wreath product groups, achieving explicit main terms and error estimates with relaxed assumptions on $H$.

Findings

Explicit main term for $C_2 times H$ extensions

Power-saving error bounds established

Method applicable to arbitrary base fields

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 J\"urgen Kl\"uners 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 $H.$