2014/12/10 by Ali S. Janfada, Janfada, Ali S., Sajad Salami +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1412.3258
11 pages, accepted to publish in Kodai Mathematical Journal
openalex publication_date 2014/12/10 · arxiv created 2014/12/12 · arxiv updated 2014/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbb K=\mathbb Q(√(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 (\mathbb K,θ)-congruent number if there is a triangle, called the (\mathbb K,θ, n)-triangles, with sides in \mathbb K having θ as an angle and nαθ as area, where αθ=√(r2-s2). Consider the (\mathbb K,θ)-congruent number elliptic curve En,θ: y2=x(x+(r+s)n)(x-(r-s)n) defined over \mathbb K. Denote the squarefree part of positive integer t by \rm sqf(t). In this work, it is proved that if m≠ \rm sqf(2r(r-s)) and mn≠ 2, 3, 6, then n is a (\mathbb K,θ)-congruent number if and only if the Mordell-Weil group En,θ(\mathbb K) has positive rank, and all of the (\mathbb K,θ, n)-triangles are classified in four types.