$\theta$-Congruent Numbers, Tiling Numbers and the Selmer Rank of Related Elliptic Curves: odd n

Abstract

Several discrete geometry problems are closely related to the arithmetic theory of elliptic curves defined on the rational fields $\mathbb{Q}$. In this paper we consider the $\theta$-congruent number for $\theta=\frac{\pi}{3}$ and $\frac{2\pi}{3}$ and tiling number n. For the case that $n\geqslant 2$ is square-free odd integer, we determine all $n$ such that the Selmer rank of elliptic curve $E_{n,\frac{\pi}{3}}:\ y^2=x(x-n)(x+3n)$ or/and $E_{n,\frac{2\pi}{3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provide several series of non $\theta$-congruent numbers for $\theta=\frac{\pi}{3}$ and $\frac{2\pi}{3}$, and non tiling numbers n with arbitrary many of prime divisors.