Random maximal isotropic subspaces and Selmer groups

Abstract

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F p \mathbb {F}_p . A random subspace chosen with respect to this measure is discrete with probability 1 1 , and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable. We then prove that the p p -Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F p \mathbb {F}_p . By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2 2 -Selmer groups in certain families of quadratic twists, and the average size of 2 2 - and 3 3 -Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay’s heuristics for p p -torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1 / 2 1/2 . Many of our results generalize to abelian varieties over global fields.