On representing coordinates of points on elliptic curves by quadratic forms

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Given an elliptic quartic of type Y²=f(X) representing an elliptic curve of positive rank over, we investigate the question of when the Y-coordinate can be represented by a quadratic form of type ap²+bq². In particular, we give examples of equations of surfaces of type c₀+c₁x+c₂x²+c₃x³+c₄x⁴=(ap²+bq²)², a,b,c ∈ where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type Y²=f(a p²+b q²) where the polynomial f is of degree 3.