A heuristic for boundedness of ranks of elliptic curves (Q2327704)

Summary: We present a heuristic that suggests that ranks of elliptic curves \(E\) over \(\mathbb{Q}\) 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.