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.
This video is part of the Institute for Advanced Study‘s Special Seminar on Homological Stability and Number Theory.
