Hyperelliptic curves, the scanning map, and moments of families of quadratic L-functions

We compute the stable homology of the braid group with coefficients in any Schur functor applied to the integral reduced Burau representation. This may be considered as a hyperelliptic analogue of the Mumford conjecture (Madsen–Weiss theorem) with twisted coefficients. We relate the result to the function field case of conjectures of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of families of quadratic $L$-functions. Combined with a recent homological stability theorem of Miller-Patzt-Petersen-Randal-Williams, our homological calculations confirm the Conrey-Farmer-Keating-Rubinstein-Snaith predictions for all large enough prime powers $q$.