Second moments for 2-Selmer structures on elliptic curves, and applications

A key prediction of the Poonen–Rains heuristics is that every nonnegative integer $r$ occurs as a 2-Selmer rank for a positive proportion of elliptic curves over $\mathbb{Q}$, but this prediction was not previously known for any $r$. In this talk, we prove that a positive proportion of elliptic curves over $\mathbb{Q}$ have 2-Selmer rank $r$, for small values of $r$.