Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula

作者：Creutz Brendan^*; Miller Robert L

来源：Journal of Algebra, 2012, 372: 673-701.

DOI：10.1016/j.jalgebra.2012.09.029

• Elliptic curve
• Birch and Swinnerton-Dyer conjecture
• Descent
• Tate-Shafarevich group

Let phi : E -> E' be an isogeny of prime degree l between elliptic curves defined over a number field. We describe how to perform phi-descents on the nontrivial elements in the Shafarevich-Tate group of E' which are killed by the dual isogeny 'phi'. This makes computation of l-Selmer groups of elliptic curves admitting an l-isogeny over Q feasible for l = 5,7 in cases where a phi-descent on E is insufficient and a full l-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.

• 出版日期2012-12-15