Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

TL;DR

This paper establishes a homological stability theorem for Hurwitz spaces, leading to new results on the distribution of class groups in quadratic extensions over function fields, connecting topology and number theory.

Contribution

It introduces a homological stabilization theorem for Hurwitz spaces and applies it to derive arithmetic results related to class groups over function fields.

Findings

Positive density of quadratic extensions with prescribed class group structure

Homological stability for Hurwitz spaces proven

Connection between topology and arithmetic over function fields

Abstract

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let l>2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t) have the l-part of their class group isomorphic to A.