Andrea Surroca Ortiz
Published 2008-01-07, updated 2020-01-14Version 2
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.
Comments: 38 pages. Submitted version. This version improves substantially, in many ways, the unpublished previous version arXiv:0801.1054. In particular, Masser's lower bound for non-torsion points is replaced by Bosser-Gaudron's lower bound. The analytic estimates on the L-function, as well as the part relying the Tamagawa numbers and the Faltings' height were also substantially modified
The growth of the rank of Abelian varieties upon extensions
Categories of abelian varieties over finite fields I. Abelian varieties over $\mathbb{F}_p$
On reduction maps and support problem in K-theory and abelian varieties
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