Conjectural estimates on the Mordell-Weil and Tate-Shafarevich groups of an abelian variety

TL;DR

This paper provides conditional bounds on the Tate-Shafarevich group and Mordell-Weil generators of an abelian variety over a number field, extending previous results under classical conjectures like BSD.

Contribution

It generalizes and improves bounds on Tate-Shafarevich groups and heights of Mordell-Weil generators, extending methods from elliptic curves to abelian varieties.

Findings

Conditional bounds on Tate-Shafarevich group size

Bounds on Néron-Tate heights of generators

Extension of Manin's algorithm to abelian varieties

Abstract

We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\'eron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.