Leseprobe
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The p-adic Simpson Correspondence
Takeshi Tsuji, Michel Gros, Ahmed Abbes, et al.
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
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Commutative diagram, Ring homomorphism, Diagram (category theory), Exact sequence, Morphism, Tensor product, Torsor (algebraic geometry), Higgs bundle, Topology, Morphism of schemes, Valuation ring, Direct limit, Summation, Theorem, Rational number, Sheaf (mathematics), Commutative property, Irreducible component, Formal scheme, Discrete valuation ring, Direct sum, Divisibility rule, Galois cohomology, Exact functor, Logarithm, Equivalence of categories, Fundamental group, Identity element, Algebraic closure, Base change, Vector bundle, Zariski topology, Endomorphism, Galois group, Coefficient, Inverse system, Functor, Subgroup, Connected component (graph theory), Cohomology, Spectral sequence, Computation, Covering space, Monoid, Presheaf (category theory), Embedding, Adjoint functors, Generic point, Initial and terminal objects, Field of fractions, Existential quantification, Derived category, Residue field, Automorphism, Inverse limit, Affine transformation, Homomorphism, Inverse image functor, Integral domain, Integral element, Corollary, Commutative ring, Integer, Closed immersion, Hodge theory, Profinite group, Maximal ideal, Cokernel, Mathematical induction, P-adic number