Ahmed Abbes Directeur de recherche au CNRS Laboratoire Alexander Grothendieck, ERL 9216 du CNRS, IHÉS

The p-adic Simpson Correspondence Ahmed Abbes, Michel Gros & Takeshi Tsuji Annals of Mathematics Studies, Vol. 193 Princeton University Press Paperback | 2016 | ISBN: 9780691170299 Hardcover | 2016 | ISBN: 9780691170282 616 pp. | 7 x 10 eBook | ISBN: 9781400881239

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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The p-adic Simpson Correspondence and Hodge-Tate Local Systems Ahmed Abbes and Michel Gros Lecture Notes in Mathematics, volume 2345 Springer Softcover ISBN 978-3-031-55913-6 eBook ISBN 978-3-031-55914-3 DOI: 10.1007/978-3-031-55914-3 Number of Pages X, 443

Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated with T. Tsuji, we develop in this book new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our original construction.

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Les suites spectrales de Hodge-Tate Ahmed Abbes and Michel Gros Astérisque, volume 448 Société Mathématique de France ISBN : 978-2-85629-988-3 ISSN : 0303-1179 (print), 2492-5926 (electronic) DOI : 10.24033/ast.1221 Nombre de pages: 482

Ce livre présente deux résultats importants en théorie de Hodge p-adique suivant l'approche initiée par Faltings, à savoir (i) son principal théorème de comparaison p-adique, et (ii) la suite spectrale de Hodge-Tate. Nous établissons pour chacun de ces résultats deux versions, une absolue et une relative. Si les énoncés absolus pouvaient raisonnablement être considérés comme bien compris, notamment après leur extension aux variétés rigides par Scholze, l'approche initiale de Faltings pour les variantes relatives restait beaucoup moins étudiée. Bien que nous suivions la même stratégie que celle utilis&eactute;e par Faltings pour établir son principal théorème de comparaison p-adique, une partie de nos preuves est basée sur de nouveaux résultats. La suite spectrale de Hodge-Tate relative est nouvelle dans cette approche.