A heuristic for boundedness of ranks of elliptic curves

TL;DR

This paper proposes a heuristic indicating that the ranks of elliptic curves over rationals are bounded, suggesting only finitely many with rank exceeding 21, based on modeling ranks and Shafarevich-Tate groups.

Contribution

It introduces a novel heuristic model linking ranks and Shafarevich-Tate groups, providing evidence for boundedness of elliptic curve ranks over the rationals.

Findings

Ranks of elliptic curves are heuristically bounded by 21.

Finitely many elliptic curves have rank greater than 21.

The model extends to elliptic curves over other global fields.

Abstract

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.