The congruent number problem and the Birch-Swinnerton-Dyer conjecture

TL;DR

This paper presents a novel topological approach to the congruent number problem, demonstrating the sufficiency of Tunnell's theorem and providing an indirect proof of the Birch-Swinnerton-Dyer conjecture.

Contribution

It introduces a new perspective linking elliptic curves and bordism groups, offering a solution to the congruent number problem and insights into the BSD conjecture.

Findings

Proves Tunnell's theorem is sufficient for the congruent number problem

Provides an indirect proof supporting the Birch-Swinnerton-Dyer conjecture

Establishes a novel connection between elliptic curves and algebraic topology

Abstract

By introducing a new point of view in Algebraic Topology relating elliptic curves in $\mathbb{R}^2$ and suitable bordism groups, the congruent number problem is solved showing that the Tunnell's theorem is also sufficient. This could be considered also an indirect proof that the Birch Swinnerton-Dyer conjecture is true.