Ari Shnidman (Temple)
Event Date
2025-10-01
Event Time
03:15 pm ~ 04:45 pm
Event Location
Wachman 413
The rational points on an elliptic curve ($y^2 = x^3 + Ax + B$) form a finitely generated abelian group. Heuristics imply that 50% of elliptic curves have rank 0 and 50% have rank 1. For example, this follows from Poonen-Rains' heuristics for the dimensions of the p-Selmer groups of elliptic curves, for all primes p. Bhargava-Shankar computed the first moment of these distributions for p = 2,3,5 and recently Bhargava-Shankar-Swaminathan gave an upper bound on the second moment when p = 2. I'll give a leisurely explanation of the definitions and the methods, and then describe work in progress with Bhargava-Ho-Swaminathan where we prove something close to the full Poonen-Rains distribution for p = 2 in certain large families of elliptic curves.