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 $a{p}^2+b{q}^2$. In particular, we give examples of equations of surfaces of type $c_0+c_{1}x+c_2{x}^2+c_3{x}^3+c_4{x}^4=(a{p}^2+b{q}^2{)}^2$, $a,b,c \in \Q$ 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 $3$.

20 pages; to appear in Acta Arithmetica