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TRANSPORT PARALLÈLE ET CORRESPONDANCE DE SIMPSON $p$-ADIQUE

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry Curves

Published online by Cambridge University Press:  19 June 2017

DAXIN XU
DAXIN XU*
Affiliation:
Laboratoire Alexander Grothendieck, ERL 9216 du CNRS, Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France; daxinxu@ihes.fr

Abstract

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Deninger et Werner ont développé un analogue pour les courbes $p$-adiques de la correspondance classique de Narasimhan et Seshadri entre les fibrés vectoriels stables de degré $0$ et les représentations unitaires du groupe fondamental topologique pour une courbe complexe propre et lisse. Par transport parallèle, ils ont associé fonctoriellement à chaque fibré vectoriel sur une courbe $p$-adique, dont la réduction est fortement semi-stable de degré $0$, une représentation $p$-adique du groupe fondamental de la courbe. Ils se sont posé quelques questions : leur foncteur est-il pleinement fidèle ? La cohomologie des systèmes locaux fournis par celui-ci admet-elle une filtration de Hodge-Tate ? Leur construction est-elle compatible avec la correspondance de Simpson $p$-adique développée par Faltings ? Nous répondons à ces questions dans cet article.

Deninger and Werner developed an analogue for $p$-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree $0$ and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a $p$-adic curve whose reduction is strongly semi-stable of degree $0$ a $p$-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this functor admits a Hodge–Tate filtration; and whether their construction is compatible with the $p$-adic Simpson correspondence developed by Faltings. We answer these questions in this article.

MSC classification

Primary: 14H60: Vector bundles on curves and their moduli
Secondary: 11G20: Curves over finite and local fields 14G20: Local ground fields
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e13
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2017

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