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Alg`ebres simples centrales sur les corps de fonctions de deux variables

Jean-Louis Colliot-Th'el`ene
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Abstract

Let F be a function field in two variables over an algebraically closed field of characteristic zero. The main part of this Bourbaki talk describes A. J. de Jong's proof (Duke Math. J. 123 (2004) 71-94) that index and exponent coincide for central simple algebras over F. The text follows the simplified approach sketched at the end of de Jong's paper. Applications to linear algebraic groups are given. A last section surveys results on the comparison between index and exponent for central simple algebras over a function field in one variable over a p-adic field, with application to the isotropy of quadratic forms over such fields. ----- `A toute classe dans le groupe de Brauer d'un corps F sont associ'es deux entiers, l'indice (degr'e d'un corps gauche repr'esentant la classe) et l'exposant (ordre de la classe dans le groupe de Brauer). L'exposant divise l'indice, mais ne lui est pas n'ecessairement 'egal. Lorsque F est un corps de nombres, c'est un th'eor`eme des ann'ees 1930 qu'exposant et indice coincident. A. J. de Jong (Duke Math. J. 123 (2004) 71-94) a montr'e r'ecemment qu'ils coincident aussi lorsque F est un corps de fonctions de deux variables sur le corps des complexes. Apr`es des rappels sur le groupe de Brauer (Azumaya et Grothendieck), l'expos'e d'ecrit l'essentiel de la d'emonstration, dans la version simplifi'ee sugg'er'ee `a la fin de l'article de de Jong. On donne ensuite quelques cons'equences pour les groupes lin'eaires. Un dernier paragraphe passe en revue des r'esultats comparant exposant et indice pour les corps de fonctions d'une variable sur les corps p-adiques, avec application `a l'isotropie des formes quadratiques sur de tels corps.

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Jean-Louis Colliot-Th'el`ene. 2005-07-04. Alg`ebres simples centrales sur les corps de fonctions de deux variables. https://arxiv.org/abs/math/0507070

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