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

2004/05/11 by H. A. Helfgott, Helfgott, H. A., Akshay Venkatesh +1 · 6 citations
Mathematics · #11G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0405180

openalex publication_date 2004/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• On the 2-torsion in class groups of number fields
• Counting rational points on elliptic and hyperelliptic curves over function fields
• Absolutely Abelian Hilbert Class Fields and ℓ-torsion conjecture
• Additive Rigidity for x-Coordinates of Rational Points on Elliptic Curves
• Number of integral points on quadratic twists of elliptic curves
• Counting points on a family of degree one del Pezzo surfaces

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• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
• Integral points on elliptic curves with j-invariant 0 over k(t)
• Local torsion primes and the class numbers associated to an elliptic curve over ℚ
• On the class numbers of the fields of the pⁿ-torsion points of certain elliptic curves over Q
• Uniform Bounds on S-Integral Torsion Points for \mathbbG_m and Elliptic Curves