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

By Harald Andrés Helfgott, Akshay Venkatesh. 2004

101 citations recorded by the source corpus.

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 quasi-orthogonality 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/115656d31924336e61838e221d9707e5e80e981c

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Effective equidistribution of eigenvalues of Hecke operators (2009, background): “Lillian Pierce, in [20], has obtained a bound of the form O(N27/56) and Helfgott and Venkatesh in [12] got the further improvement O(N .”
Reflection Principles and Bounds for Class Group Torsion (2007, background): “D̃iv 0 K is a number-field analogue of the group of “divisor classes of degree 0.””
Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves (2017, methodology): “12 of [9], proved in the same way but using the trivial bound h2(K) ≤ h(K).”
The 3‐part of Class Numbers of Quadratic Fields (2005, methodology): “4 The work of Helfgott and Venkatesh In [36], Helfgott and Venkatesh present a method for bounding the number of integral points on elliptic curves that yields the bounds h3(D) = O(|D| )”