Counting number fields whose Galois group is a wreath product of symmetric groups

TL;DR

This paper establishes asymptotic lower bounds for counting number fields with Galois groups that are iterated wreath products of symmetric groups, extending understanding related to Malle's conjecture and dynamical systems.

Contribution

It introduces new Galois theoretic techniques to estimate the number of such fields, particularly when the Galois group is a wreath product of symmetric groups.

Findings

Provides asymptotic lower bounds for counting fields with specified wreath product Galois groups.

Connects the problem to Malle's conjecture and dynamical systems.

Develops novel Galois theoretic methods for these counting problems.

Abstract

Let $K$ be a number field and $k\geq 2$ be an integer. Let $(n_1,n_2, \dots, n_k)$ be a vector with entries $n_i\in \mathbb{Z}_{\geq 2}$. Given a number field extension $L/K$, we denote by $\widetilde{L}$ the Galois closure of $L$ over $K$. We prove asymptotic lower bounds for the number of number field extensions $L/K$ with $[L:K]=\prod_{i=1}^k n_i$, such that $Gal(\widetilde{L}/K)$ is isomorphic to the iterated wreath product of symmetric groups $S_{n_1}\wr S_{n_2}\wr \dots \wr S_{n_k}$. Here, the number fields $L$ are ordered according to discriminant $|\Delta_L|:=|Norm_{K/\mathbb{Q}} (\Delta_{L/K})|$. The results in this paper are motivated by Malle's conjecture. When $n_1=n_2=\dots =n_k$, these wreath products arise naturally in the study of arboreal Galois representations associated to rational functions over $K$. We prove our results by developing Galois theoretic techniques that have their origins in the study of dynamical systems.