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

Helfgott, H.; Venkatesh, A.

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

Integral points on elliptic curves and $3$-torsion in class groups
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by H. A. Helfgott and A. Venkatesh

J. Amer. Math. Soc. 19 (2006), 527-550

DOI: https://doi.org/10.1090/S0894-0347-06-00515-7

Published electronically: January 19, 2006

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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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