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G-torseurs en théorie de Hodge p-adique

Part of: Algebraic groups Algebraic number theory: local and $p$-adic fields Curves

Published online by Cambridge University Press:  24 November 2020

Laurent Fargues
Laurent Fargues*
Affiliation:
CNRS, Institut de Mathématiques de Jussieu, 4 place Jussieu, 75252 Paris, France laurent.fargues@imj-prg.fr
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Résumé

Étant donné un groupe réductif $G$ sur une extension de degré fini de $\mathbb {Q}_p$ on classifie les $G$-fibrés sur la courbe introduite dans Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodge $p$-adique, Astérisque 406 (2018)]. Le résultat est interprété en termes de l'ensemble $B(G)$ de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.

Abstract

Abstract

Given a reductive group $G$ over a finite extension of $\mathbb {Q}_p$ we classify the $G$-bundles over the curve introduced in Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodge $p$-adique, Astérisque 406 (2018)]. The result is interpreted in terms of Kottwitz set $B(G)$. We moreover compute the étale cohomology of the curve with torsion coefficients and relate the result to local class field theory.

Keywords

p-adic Hodge theoryreduction theoryprincipal G-bundles

MSC classification

Primary: 11S31: Class field theory; $p$-adic formal groups 14H99: None of the above, but in this section 14L05: Formal groups, $p$-divisible groups 14L24: Geometric invariant theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 10 , October 2020 , pp. 2076 - 2110
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Copyright
© The Author(s) 2020

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Footnotes

L'auteur a bénéficié du support du projet ANR-14-CE25 ‘PerCoLaTor’.

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