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We compute the stable homology of the braid group with coefficients in anySchur functor applied to the integral reduced Burau representation. This may beconsidered as a hyperelliptic analogue of the Mumford conjecture (Madsen-Weisstheorem) with twisted coefficients. We relate the result to the function fieldcase of conjectures of Conrey-Farmer-Keating-Rubinstein-Snaith on moments offamilies of quadratic L-functions. In particular, we formulate a purelytopological homological stability conjecture, which when combined with ourcalculations would imply a precise asymptotic formula for all moments in therational function field case.

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