Rational configuration problems and a family of curves

TL;DR

This paper studies rational points on a family of genus one curves arising from configurations of points with rational distances, proving density zero results and conditions for infinitude of rational solutions.

Contribution

It introduces a new family of curves related to rational distance configurations and characterizes when these curves have rational points, including density and infinitude conditions.

Findings

Set of matrices with rational points has density zero.

Existence of rational points implies infinite solutions unless a polynomial vanishes.

Application to rational distances in geometric configurations.

Abstract

Given $\eta=\begin{pmatrix} a&b\\c&d \end{pmatrix}\in \text{GL}_2(\mathbb{Q})$, we consider the number of rational points on the genus one curve \[H_\eta:y^2=(a(1-x^2)+b(2x))^2+(c(1-x^2)+d(2x))^2.\] We prove that the set of $\eta$ for which $H_\eta(\mathbb{Q})\neq\emptyset$ has density zero, and that if a rational point $(x_0,y_0)\in H_\eta(\mathbb{Q})$ exists, then $H_\eta(\mathbb{Q})$ 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 $\mathbb{R}^n$ with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in $\mathbb{R}^2$ 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.