In recent years, several problems in arithmetic statistics that seem completely intractable over number fields, have been resolved in the case of function fields by geometric methods; more specifically, by casting the problem in terms of the homology of a suitable moduli space of algebraic curves, and understanding the homology asymptotically. Perhaps the most notable is the work of Ellenberg-Venkatesh-Westerland and Landesman-Levy on Cohen-Lenstra heuristics.

I will talk about joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams, where this paradigm is applied to another problem in analytic number theory. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith (CFKRS) which allows for precise predictions for the asymptotics of moments of many different families of L-functions. We consider the family of all L-functions attached to hyperelliptic curves over some fixed finite field, in which case we are able to prove the CFKRS predictions for all moments, for all sufficiently large (but fixed) q.
We do this by studying the homology of the moduli space of hyperelliptic curves, with symplectic coefficients: we compute the stable homology groups, together with their structure as Galois representations, and prove a novel homological stability theorem in this setting. The proofs use homotopical ideas developed in connection with the Mumford conjecture, Borel's work on stable real cohomology of arithmetic groups, logarithmic algebraic geometry, and cellular E_k-algebras.