Presentations of Galois groups of unramified extensions of global fields and its predicted distribution

TL;DR

This paper generalizes heuristics for the distribution of Galois groups of unramified extensions over global fields, introducing a new random group model that accounts for arbitrary base fields and local conditions.

Contribution

It constructs a new random group model predicting the distribution of Galois groups for $ ext{Gamma}$-extensions over any global field, extending previous heuristics.

Findings

Proves certain Galois groups have specific presentations.

Develops a new random group model for distribution prediction.

Generalizes non-abelian Cohen-Lenstra-Martinet heuristics.

Abstract

Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ -- for $\Gamma$-extensions $K/Q$, and prove they have presentations of a particular form. This presentation leads us to the construction of a new random group model as in the work of Liu, Wood, and Zureick-Brown that predicts the distribution of $G_{\emptyset}^T(K)$ as we vary among $\Gamma$-extensions $K/Q$ with prescribed local conditions at places in $T$, giving a generalization of the non-abelian Cohen-Lenstra-Martinet Heuristics. The key generalization is that $Q$ can be an arbitrary global field, while this comes at a cost of introducing a prime-to-$|\text{Cl}_T(Q)|$ condition in addition to avoiding roots of unity, $|\Gamma|$, and the characteristic if $Q$ is a function field.