The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

TL;DR

This paper extends Fermat's method to generate new rational triangles associated with $ heta$-congruent numbers, linking geometric constructions to the elliptic curve group law, and provides new proofs and classifications of torsion points.

Contribution

It generalizes Fermat's algorithm for right triangles to arbitrary angles $ heta$, connecting it with the addition law on the associated elliptic curves and classifying torsion points.

Findings

Generalization of Fermat's algorithm to arbitrary $ heta$-triangles.

Method to produce new rational $ heta$-triangles from existing ones.

Complete classification of torsion points on the elliptic curves $E_N^ heta( ext{Q})$.

Abstract

A positive integer $N$ is called a $\theta$-congruent number if there is a $\ta$-triangle $(a,b,c)$ with rational sides for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $N \sqrt{r^2-s^2}$, where $\theta \in (0, \pi)$, $\cos(\theta)=s/r$, and $0 \leq |s|<r$ are coprime integers. It is attributed to Fujiwara \cite{fujw1} that $N$ is a $\ta$-congruent number if and only if the elliptic curve $E_N^\ta: y^2=x (x+(r+s)N)(x-(r-s)N)$ has a point of order greater than $2$ in its group of rational points. Moreover, a natural number $N\neq 1,2,3,6$ is a $\ta$-congruent number if and only if rank of $E_N^\ta(\Q)$ is greater than zero. In this paper, we answer positively to a question concerning the existence of methods to create new rational $\theta$-triangle for a $\theta$-congruent number $N$ from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle $\theta$ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in $E_N^\theta({\mathbb Q})$. Then, based on the addition of two distinct points in $E_N^\theta({\mathbb Q})$, we provide a way to find new rational $\ta$-triangles for the $\theta$-congruent number $N$ using given two distinct ones. Finally, we give an alternative proof for Fujiwara's theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $E_N^\theta({\mathbb Q})$ with corresponding rational $\theta$-triangles