Dyson's theorem for curves

TL;DR

This paper extends Dyson's theorem to products of smooth projective curves over number fields, providing a new proof that simplifies existing methods and offers insights into effectiveness and integral points.

Contribution

It introduces a novel proof of Dyson's theorem for curves that avoids classical complex tools, applicable to hyperbolic curves and related to Siegel's theorem.

Findings

Proves Dyson's theorem for product of curves over number fields.

Provides a new, elementary proof avoiding Roth, Mordell-Weil, and Schmidt theorems.

Offers insights into effectiveness and integral points on hyperbolic curves.

Abstract

Let $\scriptstyle K$ be a number field and $\scriptstyle X_1$ and $\scriptstyle X_2$ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson Theorem for the product $\scriptstyle X_1 \times X_2$. If $\scriptstyle X_i = {\bb P}_1$ 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.