抄録
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 nαθ as area, where αθ = $\sqrt{r^2-s^2}$. Consider the (K,θ)-congruent number elliptic curve En,θ: y2 = 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(2r(r − s)) and mn ≠ 2, 3, 6, then n is a (K,θ)-congruent number if and only if the Mordell-Weil group En,θ(K) has positive rank, and all of the (K, θ, n)-triangles are classified in four types.
