Second Isogeny Descents and the Birch and Swinnerton-Dyer Conjectural Formula

Details Second Isogeny Descents and the Birch and Swinnerton-Dyer Conjectural Formula Second Isogeny Descents and the Birch and Swinnerton-Dyer Conjectural Formula

2011-05-20

arxiv.org/abs/1105.4018v1

Mathematics

Number Theory

31 pages

Scientific paper

Let h be a p-isogeny of elliptic curves. We describe how to perform h-descents on the nontrivial elements in the Shafarevich-Tate group which are killed by the dual isogeny h'. This makes computation of p-Selmer groups of elliptic curves admitting a p-isogeny over Q feasible for p = 5,7 in cases where an isogeny descent is insufficient and a full p-descent would be infeasible. As an application we complete the verification of the full Birch and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank zero or one and conductor less than 5000.

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