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

Abstract

Motivated by the work of Liu, we study certain canonical quotients of $G_{\varnothing }^T(K)$ -- 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 $G_{\varnothing }^T(K)$ 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.

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