Special Seminar on Homological Stability and Number Theory

The braid group B_{2g+1} has a description in terms of the hyperelliptic mapping class group of a curve X of genus g.  This equips it with an action on V = H_1(X), and we may produce a wealth of new representations S^{\lambda}(V) by applying Schur functors to V.  The goal of this talk is to describe the stable (in g) group homology of these representations.  

Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space.  We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces.

This is joint work with Bergström, Diaconu, and Petersen.  Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.