Jordan Ellenberg

Abstract:   A Hurwitz space H_{G,n} is an algebraic variety parametrizing branched covers of the projective line with some fixed finite Galois group G. Hurwitz spaces have been much studied, especially in the classical case where G is a symmetric group and the monodromy around each branch point is a transposition in G. We will prove that, under some hypotheses on G, the rational i'th homology of the Hurwitz spaces stabilizes when the number of branch points is sufficiently large compared to i.

Our original motivation for this purely topological theorem comes from number theory. We will explain how two popular distributional conjectures in algebraic number theory -- the Cohen-Lenstra heuristics and Bhargava's conjectures on densities of discriminants - -- have function-field analogues related to counting points on Hurwitz spaces over finite fields. We explain how our theorem gives new results towards the Cohen-Lenstra heuristics over function fields, and describe a natural conjecture in topology which would imply, among other interesting facts, both the Cohen-Lenstra and Bhargava conjectures over function fields.

This is joint work with Akshay Venkatesh and Craig Westerland.