Nonabelian Cohen-Lenstra Heuristics over Function Fields

TL;DR

This paper investigates the distribution of Galois groups of maximal unramified pro-p extensions over function fields, confirming heuristic predictions and establishing moments that suggest similar behavior in number fields.

Contribution

It computes the moments of the proposed distribution, proves their uniqueness, and confirms the heuristics in the function field setting, leading to new conjectures for number fields.

Findings

Moments of the distribution match the heuristics in the function field case.

Uniqueness of the distribution with given moments is established.

Results support the conjecture that number field Galois groups behave similarly.

Abstract

Boston, Bush, and Hajir have developed heuristics, extending the Cohen-Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-p extensions of imaginary quadratic number fields for p an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of F_q(t), the Galois groups of the maximal unramified pro-p extensions, as q goes to infinity, have the moments predicted by the Boston, Bush, and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.