Dyson's theorem for curves

Abstract

Let K be a number field and X 1 and X 2 two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X 1 × X 2 . If X i = P 1 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, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.