The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Journal of the European Mathematical Society

Full-Text PDF (342 KB) | Metadata | Table of Contents | JEMS summary
Volume 21, Issue 9, 2019, pp. 2859–2903
DOI: 10.4171/JEMS/893

Published online: 2019-05-21

A heuristic for boundedness of ranks of elliptic curves

Jennifer Park[1], Bjorn Poonen[2], John Voight[3] and Melanie Matchett Wood[4]

(1) University of Michigan, Ann Arbor, USA
(2) Massachusetts Institute of Technology, Cambridge, USA
(3) Dartmouth College, Hanover, USA
(4) University of Wisconsin, Madison, USA

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.

Keywords: Elliptic curve, rank, Shafarevich–Tate group

Park Jennifer, Poonen Bjorn, Voight John, Wood Melanie Matchett: A heuristic for boundedness of ranks of elliptic curves. J. Eur. Math. Soc. 21 (2019), 2859-2903. doi: 10.4171/JEMS/893