  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 $q$.
