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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.

Group theory. --- p-adic groups. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Groups, p-adic --- Group theory --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Dolbeault generalized representation. --- Dolbeault module. --- Dolbeault representation. --- Faltings cohomology. --- Faltings extension. --- Faltings ringed topos. --- Faltings site. --- Faltings topos. --- Galois cohomology. --- Gerd Faltings. --- Higgs bundle. --- Higgs bundles. --- Higgs crystals. --- Higgs envelopes. --- Higgs isocrystal. --- HiggsДate algebra. --- HodgeДate representation. --- HodgeДate structure. --- HodgeДate theory. --- Hyodo's theory. --- Koszul complex. --- additive categories. --- adic module. --- almost faithfully flat descent. --- almost faithfully flat module. --- almost flat module. --- almost isomorphism. --- almost tale covering. --- almost tale extension. --- cohomology. --- covanishing topos. --- crystalline-type topos. --- deformation. --- discrete AЇ-module. --- finite tale site. --- fundamental group. --- generalized covanishing topos. --- generalized representation. --- inverse limit. --- linear algebra. --- locally irreducible scheme. --- morphism. --- overconvergence. --- p-adic Hodge theory. --- p-adic Simpson correspondence. --- p-adic field. --- period ring. --- ringed covanishing topos. --- ringed total topos. --- small generalized representation. --- small representation. --- solvable Higgs module. --- tale cohomology. --- tale fundamental group. --- torsor.