A heuristic for boundedness of ranks of elliptic curves

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.