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

2008/01/07 by Andrea Surroca Ortiz, Ortiz, Andrea Surroca
Mathematics · #11G10 (Secondary) #11G40 (Primary) 11G50 #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.0801.1054

openalex publication_date 2008/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Néron-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.

Citations

• Good Reduction of Abelian Varieties
• Abelian varieties over cyclotomic fields with good reduction everywhere
• Modular Elliptic Curves and Fermat's Last Theorem

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