Roth's theorem for ruled surfaces

This paper addresses conjectures of E. Bombieri and P Vojta in the special case of ruled surfaces not birational to P-2. Apart from this implicit restriction to P-1 bundles S over an elliptic curve, the ultimate question of the arithmetic of pairs (S, D) for a divisor D requires further restrictions on D which turn the proposed conjectures into the study of Roth's theorem on approximation of algebraic numbers alpha, but for alpha now parametrized by an elliptic curve. With these restrictions, best possible answers are obtained. The same study may also be carried out for holomorphic maps, and this is done simultaneously.