2023/10/04 by Jonathan Love, Love, Jonathan R.
Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2310.02534
openalex publication_date 2023/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given η=\beginpmatrix a&b c&d \endpmatrix∈ GL2(ℚ), we consider the number of rational points on the genus one curve Hη:y2=(a(1-x2)+b(2x))2+(c(1-x2)+d(2x))2. We prove that the set of η for which Hη(ℚ)≠∅ has density zero, and that if a rational point (x0,y0)∈ Hη(ℚ) exists, then Hη(ℚ) is infinite unless a certain explicit polynomial in a,b,c,d,x0,y0 vanishes. Curves of the form Hη naturally occur in the study of configurations of points in ℝn with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in ℝ2 passes through a rational point on the unit circle, then it contains a dense set of points P such that the distances from P to each of the three points (0,0), (0,1), and (1,1) are all rational. We also prove some results regarding whether a rational number can be expressed as a sum or product of slopes of rational right triangles.