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

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