Jordan Ellenberg (Wisconsin)
Random matrices, random permutations, conjectures of
arithmetic distribution over function fields, topology of Hurwitz
spaces
A Hurwitz space HG,n
is an algebraic variety parametrizing
branched covers of the projective line with some fixed finite Galois
group G. We will prove that, under some hypotheses on G, the
rational ith homology of the Hurwitz spaces stabilizes when the
number of branch points is sufficiently large compared to i.
This purely topological theorem has some interesting number-theoretic
consequences. It implies, for instance, a weak form of the Cohen-
Lenstra conjectures over rational function fields, and some
quantitative inverse Galois results over function fields. For
instance, we show that the average size of the p-part of the class
number of a hyperelliptic genus-g curve over Fq is bounded
independently of g, when q is large enough relative to p; the key
point here is q can be held fixed while g grows.
I will try to give a general overview of the dictionary between
conjectures about topology of moduli spaces, on the one hand, and
arithmetic distribution conjectures (Cohen-Lenstra, Bhargava, Malle,
inverse Galois....) on the other. In particular, I will explain how
vanishing statements in cohomology imply that a natural Frobenius
action behaves like a random matrix, a random permutation, or a
random element of some other more exotic monodromy group.
(joint work with Akshay Venkatesh and Craig Westerland.)