Special Seminar on Homological Stability and Number Theory

A basic question in arithmetic statistics is:  what does the Selmer group of a random abelian variety look like?  This question is governed by the Poonen-Rains heuristics, later generalized by Bhargava-Kane-Lenstra-Poonen-Rains, which predict, for instance, that the mod p Selmer group of an elliptic curve has size p+1 on average.  Results towards these heuristics have been very partial but have nonetheless enabled major progress in studying the distribution of ranks of abelian varieties.

We will describe new work, joint with Aaron Landesman, which establishes a version of these heuristics for the mod n Selmer group of a random quadratic twist of a fixed abelian variety over a global function field.  This allows us, for instance, to bound the probability that a random quadratic twist of an abelian variety A over a global function field has rank at least 2, and to bound the discrepancy between odd and even Selmer rank.  The method is very much in the spirit of earlier work with Venkatesh and Westerland which proved a version of the Cohen-Lenstra heuristics over function fields by means of homological stabilization for Hurwitz spaces.  So part of the goal of this talk is to give a general picture of how questions in arithmetic statistics over function fields can be reduced to topological questions about homological stabilization, and more speculatively, which questions can be so reduced, which I hope will also serve as a good lead-in to talks by Petersen and Westerland later this week.