Relations de dépendance et intersections exceptionnelles (Dependence relations and exceptional intersections)

Details Relations de dépendance et intersections exceptionnelles (Dependence relations and exceptional intersections) Relations de dépendance et intersections exceptionnelles (Dependence relations and exceptional intersections)

2011-01-25

arxiv.org/abs/1101.4738v2

Mathematics

Number Theory

S\'eminaire Bourbaki, 63e ann\'ee, 2010-11, Expos\'e n{\deg} 1032. In French

Scientific paper

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).

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