  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.
