The p-adic Simpson Correspondence (AM-193)
Ahmed Abbes, Michel Gros, and Takeshi Tsuji
Abstract
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. It mainly focuses 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 thick ... More
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. It mainly focuses 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 book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book 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.
Keywords:
p-adic Simpson correspondence,
Gerd Faltings,
p-adic field,
linear algebra,
Higgs bundle,
period ring,
crystalline-type topos,
Higgs isocrystal,
p-adic Hodge theory
Bibliographic Information
| Print publication date: 2016 |
Print ISBN-13: 9780691170282 |
| Published to Princeton Scholarship Online: October 2017 |
DOI:10.23943/princeton/9780691170282.001.0001 |
Authors
Affiliations are at time of print publication.
Ahmed Abbes, author
French National Center for Scientific Research (CNRS)/Institute of Advanced Scientific Studies (IHÉS)
Michel Gros, author
Institut de Recherche Mathématiques de Rennes
Takeshi Tsuji, author
University of Tokyo
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