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

TL;DR

This paper develops a method for performing second isogeny descents on elliptic curves to compute p-Selmer groups, enabling verification of the Birch and Swinnerton-Dyer conjecture for many curves with small conductors.

Contribution

It introduces a new descent technique for elliptic curves with p-isogenies, improving computational feasibility for verifying BSD conjecture.

Findings

Verified BSD conjecture for all elliptic curves over Q with rank 0 or 1 and conductor < 5000.

Developed practical methods for p-Selmer group computation for p=5,7.

Enhanced understanding of the relationship between isogeny descents and the BSD formula.

Abstract

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.