On representing coordinates of points on elliptic curves by quadratic forms

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 $3$.