The p-adic Simpson correspondence /
Ahmed Abbes, Michel Gros, Takeshi Tsuji.
- 1 online resource
- Annals of Mathematics Studies ; 193 .
- Annals of mathematics studies ; no. 193. .
Includes bibliographical references and index.
Representations of the fundamental group and the torsor of deformations. An overview -- Representations of the fundamental group and the torsor of deformations. Local study -- Representations of the fundamental group and the torsor of deformations. Global aspects -- Cohomology of Higgs isocrystals -- Almost �etale coverings -- Covanishing topos and generalizations -- Facsimile : A p-adic Simpson correspondence -- Frontmatter -- Contents -- Foreword -- Chapter I. Chapter II. Chapter III. Chapter IV. Chapter V. Chapter VI. Bibliography -- Indexes.
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 �etale 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.
In English.
9781400881239 1400881234
10.1515/9781400881239 doi
22573/ctt193cjds JSTOR 9452437 IEEE
Group theory. p-adic groups. Geometry, Algebraic. Th�eorie des groupes. Groupes p-adiques. G�eom�etrie alg�ebrique. MATHEMATICS--Algebra--Intermediate. MATHEMATICS--Algebra--General. Geometry, Algebraic. Group theory. p-adic groups.