Abbes / Gros / Tsuji The p-adic Simpson Correspondence

1. Auflage 2016
ISBN: 978-1-4008-8123-9
Verlag: De Gruyter
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 616 Seiten

Reihe: Annals of Mathematics Studies

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches, one by Ahmed Abbes and Michel Gros, the other by Takeshi Tsuji. The authors mainly focus on generalized representations of the fundamental group that are p-adically close to the trivial representation.

The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The authors show the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the volume contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored. The authors present a new approach based on a generalization of P. Deligne's covanishing topos.

Autoren/Hrsg.

Abbes, Ahmed

Gros, Michel

Tsuji, Takeshi

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis

Inhaltsverzeichnis

Frontmatter, pg. i

Contents, pg. vii
Foreword, pg. ix
Chapter I. Representations of the fundamental group and the torsor of deformations. An overview, pg. 1
Chapter II. Representations of the fundamental group and the torsor of deformations. Local study, pg. 27
Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects, pg. 179
Chapter IV. Cohomology of Higgs isocrystals, pg. 307
Chapter V. Almost étale coverings, pg. 449
Chapter VI. Covanishing topos and generalizations, pg. 485
Facsimile: A p-adic Simpson correspondence, pg. 577
Bibliography, pg. 595
Indexes, pg. 599

Über Autor(innen)

Ahmed Abbes is director of research at the French National Center for Scientific Research (CNRS) and the Institute of Advanced Scientific Studies (IHÉS), France. Michel Gros is a researcher at the CNRS. Takeshi Tsuji is a professor in the Graduate School of Mathematical Sciences at the University of Tokyo.