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Integral points on elliptic curves and $ 3$-torsion in class groups


Authors: H. A. Helfgott and A. Venkatesh
Journal: J. Amer. Math. Soc. 19 (2006), 527-550
MSC (2000): Primary 11G05, 11R29; Secondary 14G05, 11R11
DOI: https://doi.org/10.1090/S0894-0347-06-00515-7
Published electronically: January 19, 2006
MathSciNet review: 2220098
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Abstract | References | Similar Articles | Additional Information

Abstract: 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
Email: helfgott@dms.umontreal.ca

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
Email: akshay@ias.edu

DOI: https://doi.org/10.1090/S0894-0347-06-00515-7
Keywords: Class groups, elliptic curves, integral points.
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.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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