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

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 $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n$ on $X$ which are multiplicatively independent. The points $x$ of $X$ where their values $f_1(x),\mathrm{.}\mathrm{.}\mathrm{.},f_n(x)$ 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 $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n$ multiplicativement ind'ependantes sur $X$. Les points $x$ de $X$ o`u leurs valeurs $f_1(x),\mathrm{.}\mathrm{.}\mathrm{.},f_n(x)$ 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).