Roth's Theorem for Ruled Surfaces

摘要

This paper addresses conjectures of E. Bombierei and P. Vojta in the special case211u001eof ruled surfacs not birational to P2. Apart from this implicit restriction to P1 211u001ebundles S over an elliptic curve, the ultimate question of the arithmetic of 211u001epairs (S,D) for a divisor D requires further restrictions on D which turn the 211u001eproposed conjectures into the study of Roth's theorem on approximation of 211u001ealgebraic numbers alpha, but for alpha now parametrized by an elliptic curve. 211u001eWith these restrictions, best possible answers are obtained. The same study may 211u001ealso be carried out for holomrophic maps, and this is done simultaneously.