Constructing congruent number elliptic curves using 2-descent

TL;DR

This paper explores the relationship between congruent numbers and elliptic curves, proving the converse of a classical result and demonstrating how to construct elliptic curves with higher ranks using 2-descent.

Contribution

It provides a new proof of the connection between congruent numbers and elliptic curves, and shows how to construct curves with rank at least 2 and 3 using 2-descent.

Findings

Proved the converse of the classical congruent number theorem.

Developed a method to construct elliptic curves with higher rank.

Demonstrated applications of 2-descent in constructing specific elliptic curves.

Abstract

A positive integer that is the area of some rational right triangle is called a congruent number. In an algebraic point of view, being a congruent number means satisfying a system of equations. As early as the 1800s, it is understood that if $n$ is a congruent number, then the equation $nm^2 = uv(u^2 - v^2)$ has a solution in $\mathbb{Z}$. Using the relation between congruent numbers and elliptic curves $E_n: y^2 = x^3 - n^2 x$ which was established in the 1900s, we will prove that the converse of this two century-old result holds as well. In addition to this, we present another proof of the converse using the method of 2-descent. Towards the end of this paper, we demonstrate how one can use our proof to construct subfamilies of $E_n$ with rank at least 2 and 3.