By Ahmed Abbes,Michel Gros,Takeshi Tsuji
The p-adic Simpson correspondence, lately initiated via Gerd Faltings, goals at describing all p-adic representations of the basic crew of a formal delicate kind over a p-adic box by way of linear algebra—namely Higgs bundles. This booklet undertakes a scientific improvement of the idea following new methods, one by means of Ahmed Abbes and Michel Gros, the opposite by means of Takeshi Tsuji. The authors often specialize in generalized representations of the elemental workforce which are p-adically as regards to the trivial representation.
The first method depends on a brand new family members of interval earrings outfitted from the torsor of deformations of the diversity over a common p-adic thickening outlined by means of J. M. Fontaine. the second one process introduces a crystalline-type topos and replaces the thought of Higgs bundles with that of Higgs isocrystals. The authors express the compatibility of the 2 buildings and the compatibility of the correspondence with the average cohomologies. The final a part of the quantity comprises result of wider curiosity in p-adic Hodge idea. The reader will discover a concise creation to Faltings' conception of virtually étale extensions and a bankruptcy dedicated to the Faltings topos. notwithstanding this topos is the final framework for Faltings' technique in p-adic Hodge idea, it is still really unexplored. The authors current a brand new strategy according to a generalization of P. Deligne's covanishing topos.