Journal of the European Mathematical Society

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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