The Distribution of Number Fields with Wreath Products as Galois Groups

TL;DR

This paper investigates the distribution of number fields with Galois groups formed by wreath products, establishing their asymptotic growth and confirming a conjecture of Malle for these groups.

Contribution

It determines the asymptotic behavior of counting functions for number fields with wreath product Galois groups, extending Malle's conjecture to these cases.

Findings

Counting functions grow linearly with discriminant norm

Results align with Malle's conjecture for these groups

Wreath product groups exhibit similar asymptotics as symmetric groups

Abstract

Let $G$ be a wreath product of the form $C_2 \wr H$, where $C_2$ is the cyclic group of order 2. Under mild conditions for $H$ we determine the asymptotic behavior of the counting functions for number fields $K/k$ with Galois group $G$ and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.