The distribution of 2-Selmer groups in quadratic twist families

The Poonen–Rains and Bhargava–Kane–Lenstra–Poonen–Rains give striking predictions for the distribution of Selmer groups in the family of all elliptic curves over Q. In particular, they predict that the average size of the p-Selmer group is 1+p, a result proved for p=2,3,5 by Bhargava and Shankar. 

However, these models do not accurately describe families of elliptic curves with isogenies, where the average p-Selmer size can even be infinite. In this talk, I will report on work in progress to determine the distribution of 2-Selmer groups in the family of quadratic twists of an elliptic curve with a 2-torsion point. I will present a theorem that shows that the distribution of the 2-Selmer groups coincides with a distribution arising from the kernels of random matrices. This work is joint with Harald Helfgott, Zev Klagsbrun, and Jennifer Park.