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An excursion into $$p$$-adic Hodge theory: from foundations to recent trends. (English, French) Zbl 1430.14001
Panoramas et Synthèses 54. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-913-5/pbk). xii, 268 p. (2019).
Publisher’s description: This volume offers a progressive and comprehensive introduction to $$p$$-adic Hodge theory. It starts with Tate’s works on $$p$$-adic divisible groups and the cohomology of $$p$$-adic varieties, which constitutes the main concrete motivations for the development of $$p$$-adic Hodge theory. It then moves smoothly to the construction of Fontaine’s p-adic period rings and their apparition in several comparison theorems between various $$p$$-adic cohomologies. Applications and generalizations of these theorems are subsequently discussed. Finally, Scholze’s modern vision on p-adic Hodge theory, based on the theory of perfectoids, is presented.
The articles of this volume will be reviewed individually.
Indexed articles:
Freixas i Montplet, Gerard, An introduction to Hodge-Tate decompositions, 1-17 [Zbl 1443.14023]
Caruso, Xavier, An introduction to $$p$$-adic period rings, 19-92 [Zbl 1443.13018]
Brinon, Olivier, Filtered $$(\varphi,N)$$-modules and semi-stable representations, 93-129 [Zbl 1443.14021]
Yamashita, Go, An introduction to $$p$$-adic Hodge theory for open varieties via syntomic cohomology, 131-157 [Zbl 1443.14024]
Hattori, Shin, Integral $$p$$-adic Hodge theory and ramification of crystalline representations, 159-203 [Zbl 07219334]
Brinon, Olivier; Andreatta, Fabrizio; Brasca, Riccardo; Chiarellotto, Bruno; Mazzari, Nicola; Panozzo, Simone; Seveso, Marco, An introduction to perfectoid spaces, 207-265 [Zbl 1443.14022]
Caruso, Xavier, Introduction, vii-xii [Zbl 1442.14005]
##### MSC:
 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14G45 Perfectoid spaces and mixed characteristic