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arXiv:1602.01431math.NT

A heuristic for boundedness of ranks of elliptic curves

Jennifer Park, Bjorn Poonen, John Voight, Melanie Matchett Wood

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Abstract

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves 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.

arXiv comments
41 pages, typos fixed in torsion table in section 8
Journal reference
DOI
Subjects
math.NTNumber Theory

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  1. An elliptic curve of rank ≥ 30 1 comment

    • > The record before this morning was 29 and people suspected that was as high as it got. Exactly this. A fundamental question in the subject is, whether elliptic curve ranks are bounded. Contrast with e.g. prime numbers, of which are known to be infinitely many. If you set a new…

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