We present a heuristic that suggests that ranks of elliptic curves over are bounded. In fact, it suggests that there are only finitely many 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.
Cite this article
Jennifer Park, Bjorn Poonen, John Voight, Melanie Matchett Wood, A heuristic for boundedness of ranks of elliptic curves. J. Eur. Math. Soc. 21 (2019), no. 9, pp. 2859–2903DOI 10.4171/JEMS/893