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Register a new userShafarevich-Tate groups of abelian varieties
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Authors
Igor Nikolaev
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The Shafarevich-Tate group $W (mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (mathscr{A})cong Cl~(Lambda)oplus Cl~(Lambda)$ or $W (mathscr{A})cong left(mathbf{Z}/2^kmathbf{Z}right) oplus Cl_{~mathbf{odd}}~(Lambda)oplus Cl_{~mathbf{odd}}~(Lambda)$, where $Cl~(Lambda)$ is the ideal class group of a ring $Lambda$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Lambda)$. The case of elliptic curves with complex multiplication is considered in detail.
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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 Neron-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.
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