# Homology of Hurwitz spaces and the Cohen--Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)

## TL;DR

This paper explores the homological stability of Hurwitz spaces and connects it to the Cohen--Lenstra heuristic for function fields, providing a topological approach to understanding class group distributions.

## Contribution

It establishes a homological stability theorem for Hurwitz spaces and links it to the distribution of class groups in function fields, advancing the understanding of the Cohen--Lenstra heuristic.

## Key findings

- Proved a homological stability theorem for Hurwitz spaces.
- Connected topological stability results to number-theoretic heuristics.
- Provided new insights into the distribution of class groups in function fields.

## Abstract

Ellenberg, Venkatesh, and Westerland have established a weak form of the function field analogue of the Cohen--Lenstra heuristic, on the distribution of imaginary number fields with $\ell$-parts of their class groups isomorphic to a fixed group. They first explain how this follows from an asymptotic point count for certain Hurwitz schemes, and then establish this asymptotic by using the Grothendieck--Lefschetz trace formula to translate it into a difficult homological stability problem in algebraic topology, which they nonetheless solve. These are the notes accompanying my talk at the S\'eminaire Bourbaki, which focus on the remarkable homological stability theorem for Hurwitz spaces.