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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Integral points on elliptic curves and $3$-torsion in class groups
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by H. A. Helfgott and A. Venkatesh PDF
J. Amer. Math. Soc. 19 (2006), 527-550 Request permission


We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques and methods based on quasi-orthogonality in the Mordell-Weil lattice. We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the $3$-torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.
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Additional Information
  • H. A. Helfgott
  • Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
  • Address at time of publication: Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal QC H3C 3J7, Canada
  • MR Author ID: 644718
  • Email:
  • A. Venkatesh
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139–4307
  • Address at time of publication: Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540
  • MR Author ID: 693009
  • Email:
  • Received by editor(s): May 21, 2004
  • Published electronically: January 19, 2006
  • Additional Notes: The second author was supported in part by NSF grant DMS-0245606.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 19 (2006), 527-550
  • MSC (2000): Primary 11G05, 11R29; Secondary 14G05, 11R11
  • DOI:
  • MathSciNet review: 2220098