Ranks of quadratic twists of elliptic curves

Mark Watkins[1]; Stephen Donnelly[1]; Noam D. Elkies[2]; Tom Fisher[3]; Andrew Granville[4]; Nicholas F. Rogers[5]

  • [1] School of Mathematics and Statistics, Carslaw Building (F07), University of Sydney, NSW 2006, AUSTRALIA
  • [2] Department of Mathematics, Harvard University, One Oxford Street, Cambridge MA 02138, UNITED STATES OF AMERICA
  • [3] Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UNITED KINGDOM
  • [4] Départment de mathématiques et de statistique, Pavilion André-Aisenstadt, 2920, chemin de la Tour, Montréal (Québec) H3T 1J4, CANADA
  • [5] UR Mathematics, 915 Hylan Building, University of Rochester, RC Box 270138, Rochester, NY 14627, UNITED STATES OF AMERICA

Publications mathématiques de Besançon (2014)

  • Issue: 2, page 63-98
  • ISSN: 1958-7236

Abstract

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We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.

How to cite

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Watkins, Mark, et al. "Ranks of quadratic twists of elliptic curves." Publications mathématiques de Besançon (2014): 63-98. <http://eudml.org/doc/275740>.

@article{Watkins2014,
abstract = {We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.},
affiliation = {School of Mathematics and Statistics, Carslaw Building (F07), University of Sydney, NSW 2006, AUSTRALIA; School of Mathematics and Statistics, Carslaw Building (F07), University of Sydney, NSW 2006, AUSTRALIA; Department of Mathematics, Harvard University, One Oxford Street, Cambridge MA 02138, UNITED STATES OF AMERICA; Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UNITED KINGDOM; Départment de mathématiques et de statistique, Pavilion André-Aisenstadt, 2920, chemin de la Tour, Montréal (Québec) H3T 1J4, CANADA; UR Mathematics, 915 Hylan Building, University of Rochester, RC Box 270138, Rochester, NY 14627, UNITED STATES OF AMERICA},
author = {Watkins, Mark, Donnelly, Stephen, Elkies, Noam D., Fisher, Tom, Granville, Andrew, Rogers, Nicholas F.},
journal = {Publications mathématiques de Besançon},
keywords = {elliptic curves; quadratic twists; Selmer groups; explicit formula; Birch–Swinnerton-Dyer conjecture; algebraic rank},
language = {eng},
number = {2},
pages = {63-98},
publisher = {Presses universitaires de Franche-Comté},
title = {Ranks of quadratic twists of elliptic curves},
url = {http://eudml.org/doc/275740},
year = {2014},
}

TY - JOUR
AU - Watkins, Mark
AU - Donnelly, Stephen
AU - Elkies, Noam D.
AU - Fisher, Tom
AU - Granville, Andrew
AU - Rogers, Nicholas F.
TI - Ranks of quadratic twists of elliptic curves
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
IS - 2
SP - 63
EP - 98
AB - We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family.
LA - eng
KW - elliptic curves; quadratic twists; Selmer groups; explicit formula; Birch–Swinnerton-Dyer conjecture; algebraic rank
UR - http://eudml.org/doc/275740
ER -

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