Points entiers et mod�les des courbes alg�briques

Abstract

LetC: F(X, Y)=0 be an algebraic curve of genus ≥1, over a number fieldK. In this work we construct a modelG(Z,W)=0 of the curveC, over a fixed number fieldL with $$K \subseteq L$$ , having the following property: ifx, y are algebraic integers ofK withF(x, y)=0, thenz=Z(x, y), w=W(x, y) are algebraic integers ofL withG(z, w)=0. Also, the total degree and the height of the polynomialG are bounded. As an application of this result, we give a reduction of the problem to determine effectively the integer points on a curve of genus 2, over a number field, to the problem to determine effectively the integer solutions of an equation of degree 4, over a number field. Also we consider a family of curvesF(X, Y)=0, defined over a number fieldK, which are cyclic coverings ofP 1 and we calculate, using our previous results, an explicit upper bound for the height of the integer points ofF(X, Y)=0 overK.