A heuristic for boundedness of ranks of elliptic curves
Jennifer M. Park, Bjorn Poonen, John Voight, Melanie Matchett Wood · Journal of the European Mathematical Society · 2019
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.