On $\theta$-congruent numbers on real quadratic number fields

Abstract

Let $\mathbb{K}=\mathbb{Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0<\theta <\pi$ has rational cosine, say $\cos (\theta )=s/r$ with $0<|s|<r$ and $\gcd (r,s)=1$. A positive integer $n$ is called a $(\mathbb{K},\theta )$-congruent number if there is a triangle, called the $(\mathbb{K},\theta ,n)$-triangles, with sides in $\mathbb{K}$ having $\theta$ as an angle and $n{\alpha }_{\theta }$ as area, where ${\alpha }_{\theta }=\sqrt{r^2-s^2}$. Consider the $(\mathbb{K},\theta )$-congruent number elliptic curve $E_{n,\theta }:y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over $\mathbb{K}$. Denote the squarefree part of positive integer $t$ by $\mathrm{sqf}(t)$. In this work, it is proved that if $m\ne \mathrm{sqf}(2r(r-s))$ and $mn\ne 2,3,6$, then $n$ is a $(\mathbb{K},\theta )$-congruent number if and only if the Mordell-Weil group $E_{n,\theta }(\mathbb{K})$ has positive rank, and all of the $(\mathbb{K},\theta ,n)$-triangles are classified in four types.

11 pages, accepted to publish in Kodai Mathematical Journal