arXiv subjects

Ken Willyard

Publications and source records attributed to Ken Willyard.

2 recordsLinked to original sources

arXiv2026

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

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.

math.NT↗

arXiv2025

The imaginary case of the nonabelian Cohen--Lenstra heuristics

For a finite group $\Gamma$, we study the distribution of the Galois group $G_{\emptyset}^{\#}(K)$ of the maximal unramified extension of $K$ that is split completely at $\infty$ and has degree prime to $|\Gamma|$ and $\textit{Char}(K)$, as $K$ varies over imaginary $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, we compute the moments of the distribution of $G_{\emptyset}^{\#}(K)$ by counting points on Hurwitz stacks. In order to understand the probability of the distribution, we prove that $G_{\emptyset}^{\#}(K)$ admits presentations of a specific form, then use this presentation to build random groups to simulate the behavior of $G_{\emptyset}^{\#}(K)$, and make the conjecture to predict the distribution using the probability measures of these random groups. Our results provide the imaginary analog of the work of Wood, Zureick-Brown, and the first author on the nonabelian Cohen--Lenstra heuristics.

math.NT↗