Homology of braid groups, the Burau representation, and $\mathbb{F}_q$-points on superelliptic curves

TL;DR

This paper computes the homology of braid groups with coefficients in the Burau representation and links it to the expected number of points on superelliptic curves over finite fields, revealing a deep connection between topology and arithmetic geometry.

Contribution

It provides the first calculation of the homology of braid groups with coefficients in the Burau representation and connects this to counting points on superelliptic curves over finite fields.

Findings

Homology of braid groups with Burau coefficients is explicitly calculated.

Expected number of points on superelliptic curves over $ ext{F}_q$ is exactly $q$.

Topological methods yield arithmetic predictions with algebraic proofs.

Abstract

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the disc with $n$ punctures. The group homology $H_*(B_n;V_n)$ of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over $\mathbb{F}_q$ is exactly $q$.