Rational configuration problems and a family of curves

Abstract

Given

, we consider the number of rational points on the genus one curve$H_{\eta }:y^2={\left(a\right(1-x^2)+b(2x\left)\right)}^2+{\left(c\right(1-x^2)+d(2x\left)\right)}^2.$ We prove that the set of η for which $H_{\eta }\left(Q\right)\ne \varnothing$ has density zero, and that if a rational point $(x_0,y_0)\in H_{\eta }\left(Q\right)$ exists, then $H_{\eta }\left(Q\right)$ is infinite unless a certain explicit polynomial in $a,b,c,d,x_0,y_0$ 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.