Collision of Orbits on an Elliptic Surface

Let C be a smooth projective curve defined over Q ― Q̄ double struck upper Q overbar , let π : E ⟶ C π :E C pi colon script upper E long right arrow upper C be an elliptic surface and let σ P 1 , σ P 2 , σ Q σ P₁,σ P₂,σ Q sigma Subscript upper P 1 Baseline comma sigma Subscript upper P 2 Baseline comma sigma Subscript upper Q Baseline be sections of π π pi (corresponding to points P 1 , P 2 , Q P₁,P₂, Q upper P 1 comma upper P 2 comma upper Q of the generic fibre E of E E script upper E ). We obtain a precise characterisation, expressed solely in terms of the dynamical relations between the points P 1 , P 2 ,