A Topological Perspective on the Birch and Swinnerton Dyer Conjectures

TL;DR

This paper introduces a topological approach to the Birch and Swinnerton Dyer conjecture, linking elliptic curve ranks to topological features via four-dimensional embeddings and new computational methods.

Contribution

It proposes a novel topological framework connecting elliptic curve ranks with topological loops, supported by computational functions and examples, offering fresh insights into the conjecture.

Findings

Patterns in rank curves support the topological-rank correspondence

New computational functions F and F_(m,s) facilitate analysis

Connections with established number theory frameworks are identified

Abstract

This paper presents a topological framework for investigating the Birch and Swinnerton Dyer conjecture through four dimensional embeddings of elliptic curves. We propose a correspondence between the algebraic rank of an elliptic curve and the number of topologically independent loops in its embedding, which appears to be related to the order of vanishing of its L function at s=1. Our computational function F new and its generalization F_(m,s) provide methods for examining this relationship through asymptotic analysis. Examples with rank curves from 0 to 8 show patterns supporting this correspondence. The approach connects with established frameworks, including the Kolyvagin Flach machinery and the Gross Zagier formula, potentially offering new perspectives on this significant open problem in number theory.