Relations de d\'ependance et intersections exceptionnelles (Dependence relations and exceptional intersections)

TL;DR

This paper discusses a key finiteness theorem for algebraic curves with multiplicatively independent functions and explores its conjectural generalizations to higher-dimensional semiabelian varieties, connecting to major Diophantine conjectures.

Contribution

It analyzes the finiteness of points satisfying multiple multiplicative relations on algebraic curves and discusses conjectural extensions to higher-dimensional varieties and related Diophantine problems.

Findings

Finiteness of points with multiple multiplicative relations on curves.

Connections between these finiteness results and Mordell-Lang or Manin-Mumford theorems.

Recent progress in arithmetic cases by Habegger, Rémond, and Viada.

Abstract

This text is devoted to the following result, stemming out works of Bombieri, Masser, Zannier, and Maurin: Let $X$ be an complex algebraic (projective, connected) curve and let us consider $n$ rational functions $f_1,...,f_n$ on $X$ which are multiplicatively independent. The points $x$ of $X$ where their values $f_1(x),...,f_n(x)$ satisfy at least two independent multiplicative dependence relations form a finite set. We discuss the conjectural generalizations of this theorem (Bombieri, Masser, Zannier; Zilber; Pink) concerning the finiteness of points of a $d$-dimensional subvariety $X$ of a semiabelian variety $G$ which belong to an algebraic subgroup of codimension $>d$ of $G$, their relations with theorems of Mordell-Lang or Manin-Mumford type, and, in the arithmetic case, recent results in this direction (Habegger; R\'emond; Viada). ----- Ce texte est consacr\'e au r\'esultat suivant, issus des travaux de Bombieri, Masser, Zannier et Maurin: Soit $X$ une courbe alg\'ebrique (projective, connexe) complexe et consid\'erons $n$ fonctions rationnelles $f_1,...,f_n$ multiplicativement ind\'ependantes sur $X$. Les points $x$ de $X$ o\`u leurs valeurs $f_1(x),...,f_n(x)$ v\'erifient au moins deux relations de d\'ependance multiplicative ind\'ependantes forment un ensemble fini. Nous discutons les g\'en\'eralisations conjecturales de ce th\'eor\`eme (Bombieri, Masser, Zannier; Zilber; Pink) concernant la finitude des points d'une sous-vari\'et\'e $X$ de dimension $d$ d'une vari\'et\'e semi-ab\'elienne $G$ qui appartiennent \`a un sous-groupe alg\'ebrique de codimension $>d$ dans $G$, leurs relations avec les th\'eor\`emes de type Mordell-Lang ou Manin-Mumford et, dans le cas arithm\'etique, les r\'esultats r\'ecents dans cette direction (Habegger; R\'emond; Viada).