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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. Apart from this implicit restriction to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] 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 α, but for α 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.