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

HA Helfgott; A Venkatesh

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

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

HA Helfgott, A Venkatesh

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

46 Citations (Scopus)

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 ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice ([Sil6], [GS], [He]). 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.

Translated title of the contribution	Integral points on elliptic curves and 3-torsion in class groups
Original language	English
Pages (from-to)	527 - 550
Number of pages	24
Journal	Journal of the American Mathematical Society
Volume	19 (3)
DOIs	https://doi.org/10.1090/S0894-0347-06-00515-7
Publication status	Published - Jul 2006

Bibliographical note

Publisher: American Mathematical Society
Other identifier: IDS number 042FR

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title = "Integral points on elliptic curves and 3-torsion in class groups",

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 ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice ([Sil6], [GS], [He]). 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.",

author = "HA Helfgott and A Venkatesh",

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T1 - Integral points on elliptic curves and 3-torsion in class groups

AU - Helfgott, HA

AU - Venkatesh, A

N1 - Publisher: American Mathematical Society Other identifier: IDS number 042FR

PY - 2006/7

Y1 - 2006/7

N2 - 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 ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice ([Sil6], [GS], [He]). 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.

AB - 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 ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice ([Sil6], [GS], [He]). 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.

U2 - 10.1090/S0894-0347-06-00515-7

DO - 10.1090/S0894-0347-06-00515-7

M3 - Article (Academic Journal)

VL - 19 (3)

SP - 527

EP - 550

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

ER -

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

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