A Heuristic for Boundedness of Ranks of Elliptic Curves

@inproceedings{Park2016AHF,
  title={A Heuristic for Boundedness of Ranks of Elliptic Curves},
  author={Jennifer Park and Bjorn Poonen and John Voight and Melanie Matchett Wood},
  year={2016}
}
We present a heuristic that suggests that ranks of elliptic curves E over 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. 

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