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

Helfgott, H.^1, 2 ; Venkatesh, A.^3, 4

¹ Department of Mathematics, Yale University, New Haven, Connecticut 06520
² Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal QC H3C 3J7, Canada
³ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139–4307
⁴ Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540

Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 527-550

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-06-00515-7

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