Variation of Selmer groups in quadratic twist families of abelian varieties over function fields

Abstract: 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. 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; in other words, the main argument is topological in nature. I will try to embed the talk in a general discussion of how one gets from topological results to consequences in arithmetic statistics, and what the prospects for further developments in this area look like.

number theory

Audience: researchers in the topic

Harvard number theory seminar