A heuristic for boundedness of ranks of elliptic curves

Abstract

We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E 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. Keywords: Elliptic curve; rank; Shafarevich–Tate group