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

  • Chambert-Loir, Antoine
Publication Date
Jun 12, 2011
Submission Date
Jan 25, 2011
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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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