On some conjectures on the Mordell-Weil and the Tate-Shafarevich groups of an abelian variety

Details On some conjectures on the Mordell-Weil and the Tate-Shafarevich groups of an abelian variety On some conjectures on the Mordell-Weil and the Tate-Shafarevich groups of an abelian variety

2008-01-07

arxiv.org/abs/0801.1054v1

Mathematics

Number Theory

22 pages

Scientific paper

We consider an abelian variety defined over a number field. We give conditonal bounds for the order of its Tate-Shafarevich group, as well as 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 generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich 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.

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