Random maximal isotropic subspaces and Selmer groups

TL;DR

This paper constructs a probabilistic model for Selmer groups of elliptic curves using maximal isotropic subspaces in quadratic spaces, explaining known distribution phenomena and predicting average ranks.

Contribution

It introduces a novel measure on maximal isotropic subspaces to model Selmer groups, connecting geometric structures with arithmetic properties of elliptic curves.

Findings

Model explains distribution of Selmer ranks in various families.

Predicts average elliptic curve rank over number fields is at most 1/2.

Aligns with existing heuristics and results for 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. 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 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.