On θ-congruent numbers on real quadratic number fields

Abstract

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