Explicit isogeny descent on elliptic curves

TL;DR

This paper presents an explicit method for performing l-isogeny descent on elliptic curves over Q, expressing Selmer groups via simple maps and providing evidence for the Birch and Swinnerton-Dyer conjecture.

Contribution

It introduces a new explicit approach to l-isogeny descent, linking Selmer groups to kernels of maps between finite-dimensional vector spaces.

Findings

Expressed Selmer groups as kernels of explicit maps.

Provided examples supporting the BSD conjecture for specific curves.

Demonstrated the method's effectiveness on small conductor curves.

Abstract

In this note, we consider an l-isogeny descent on a pair of elliptic curves over Q. We assume that l > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finite- dimensional F_l-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the l-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor.