The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces

TL;DR

This paper calculates the average number of surjections from class groups of quadratic function fields to finite groups, confirming Cohen--Lenstra heuristics through topological methods involving Hurwitz spaces.

Contribution

It provides the first known moments of these class groups over function fields by linking algebraic number theory with the stable homology of Hurwitz spaces.

Findings

Computed stable rational homology groups of Hurwitz spaces.

Established moments of class groups matching Cohen--Lenstra heuristics.

Extended understanding of class group distributions in function fields.

Abstract

We compute the average number of surjections from class groups of quadratic function fields over $\mathbb F_q(t)$ onto finite odd order groups $H$, once $q$ is sufficiently large. These yield the first known moments of these class groups, as predicted by the Cohen--Lenstra heuristics, apart from the case $H = \mathbb Z/3\mathbb Z$. The key input to this result is a topological one, where we compute the stable rational homology groups of Hurwitz spaces associated to non-splitting conjugacy classes.