Rational configuration problems and a family of curves

Abstract

Given eta = (ab ) is an element of GL2(Q), we consider the number of c d rational pointson the genus one curve H eta : y 2 = (a(1 - x 2 ) + b (2 x )) 2 + (c(1 - x 2 ) + d (2 x )) 2 . We prove that the set of eta for which H eta (Q) not equal 0 has density zero, and that if a rational point ( x 0 , y 0 ) E H eta (Q) exists, then H eta (Q) is infinite unless a certain explicit polynomial in a, b, c, d, x0, y0 vanishes. Curves of the form H eta naturally occur in the study of configurations of points in R n with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in R 2 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org /licenses /by /4 .0/).