On representing coordinates of points on elliptic curves by quadratic forms

TL;DR

This paper studies when the coordinates of points on elliptic curves can be expressed as quadratic forms, providing examples of surfaces with infinitely many rational points and analyzing specific polynomial cases.

Contribution

It introduces new conditions and examples for representing elliptic curve points via quadratic forms, expanding understanding of rational solutions on related surfaces.

Findings

Examples of surfaces with infinitely many rational points

Conditions for representing Y-coordinates as quadratic forms

Analysis of degree 3 polynomial cases

Abstract

Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we give examples of equations of surfaces of type $c_0+c_1x+c_2x^2+c_3x^3+c_4x^4=(ap^2+bq^2)^2$, $a,b,c \in \Q$ where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type $Y^2=f(a p^2+b q^2)$ where the polynomial $f$ is of degree $3$.