Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

TL;DR

This paper presents an algorithm that, with computational help, proves the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of low rank, successfully verifying the formula for most such curves under a certain conductor threshold.

Contribution

The authors develop an algorithm to verify the BSD conjecture for elliptic curves of rank zero or one, achieving extensive computational verification for curves with conductor less than 5000.

Findings

Proved BSD conjecture for 16714 out of 16725 curves with conductor < 5000.

Developed an effective algorithm for verifying BSD for low-rank elliptic curves.

Demonstrated the feasibility of computational proof for a large class of elliptic curves.

Abstract

We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we have proved the formula for 16714 of the 16725 such curves of conductor less than 5000.