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 $\mathbb {F}_p$. A random subspace chosen with respect to this measure is discrete with probability $1$, 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 $p$-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 $\mathbb {F}_p$. 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 $2$-Selmer groups in certain families of quadratic twists, and the average size of $2$- and $3$-Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay’s heuristics for $p$-torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most $1/2$. Many of our results generalize to abelian varieties over global fields.