Dyson Theorem for curves

ABSTRACT. Let K be a number field and X1 and X2 two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson Theorem for the product X1 ×X2. If Xi = P1 we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell–Weil theorems. 1 Introduction. After the proof of the Mordell conjecture by Faltings (the first proof is in [Fa], but [Fa2], [B2] and [Vo2] are nearer to the spirit of this paper), most of the qualitative results in the diophantine approximation of algebraic divisors by rational points over curves are solved