Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves

TL;DR

This paper proposes a probabilistic model for the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves, supported by theoretical proofs and connections to existing conjectures.

Contribution

It introduces a natural probability distribution on certain modules related to elliptic curves and conjectures its alignment with observed distributions, providing new theoretical evidence.

Findings

Established a probability distribution for Selmer and Sha groups

Connected the distribution to Delaunay's predictions for Sha

Proved new results on fppf cohomology of elliptic curves

Abstract

Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution of the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over Z_p, and we prove that the two probability distributions are compatible with each other and with Delaunay's predicted distribution for Sha. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.