JournalsjemsVol. 21, No. 9pp. 2859–2903

A heuristic for boundedness of ranks of elliptic curves

  • Jennifer Park

    University of Michigan, Ann Arbor, USA

  • Bjorn Poonen

    Massachusetts Institute of Technology, Cambridge, USA

  • John Voight

    Dartmouth College, Hanover, USA

  • Melanie Matchett Wood

    University of Wisconsin, Madison, USA
A heuristic for boundedness of ranks of elliptic curves cover

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Abstract

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. 2859–2903

DOI 10.4171/JEMS/893