Distributions of unramified extensions of global fields

TL;DR

This paper investigates the distribution of unramified extensions of global fields, linking function field results to conjectures about number fields, and introduces new invariants to refine the understanding of Galois group distributions.

Contribution

It establishes distribution results for Galois groups of unramified extensions over function fields and proposes a new conjecture on their distribution over number fields, incorporating homological invariants.

Findings

Distribution of Galois groups in function fields analyzed

New homological invariant introduced to refine group classification

Non-existence results support conjectures in number field case

Abstract

Given a finite group $\Gamma$, we prove results on the distribution of the prime-to-$q|\Gamma|$ part of fundamental groups of $\Gamma$-covers of the projective line $\mathbb P^1_{\mathbb F_q}$ over a finite field $\mathbb F_q$ as $q\to\infty$. Equivalently, this is a result on the distribution of the Galois groups of maximal unramified extensions of $\Gamma$-extensions of $\mathbb F_q(t)$, and thereby motivates a new conjecture on the distribution of Galois groups of maximal unramified extensions of $\Gamma$-extensions of a number field. In particular, this allows us to see and predict the effect of roots of unity in the base field on such distributions. We introduce the idea to study these groups along with the class in their 3rd homology group that arises from Artin-Verdier Duality. This invariant refines the lifting invariant that, in the function field setting, corresponds to stable components of Hurwitz space. One major input into our function field results is an application of our recently developed methods to determine a distribution of groups (or more general algebraic structures) from its moments. We prove non-existence results in the number field case that support our conjectures in the case where our conjectures predict certain kinds of groups occur with probability zero.