Homology of braid groups,the Burau representation,and points on superelliptic curves over finite fields

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. In this paper, we calculate H _*(B _n ; V _n ). 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 F _q is q.