| Number theorists have long been interested in questions about distribution of arithmetic objects: what is the frequency that the class group of a random quadratic imaginary field contains (Z/3Z)^r? How many extensions K/Q with Galois group S_5 have discriminant less than N? How many elliptic curves over Q are there with conductor at most N? In general, little is known about such questions. It turns out that the analysis of such questions over function fields over finite fields reveals surprising connections with problems of current interest in topology and algebraic geometry. In particular, we explain how a purely topological theorem about stabilization of cohomology for families of congruence subgroups of mapping class groups (or, in the algebraic geometer's notation, familes of Hurwitz spaces over C) would imply function-field versions of several distributional conjectures in number theory (and several other statements which have not previously been conjectured), and we describe some partial progress towards the desired topological result. (joint work with Akshay Venkatesh and Craig Westerland) |