Titlebook: Local Systems in Algebraic-Arithmetic Geometry; Hélène Esnault Book 2023 The Editor(s) (if applicable) and The Author(s), under exclusive

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Lecture 7: Companions, Integrality of Cohomologically Rigid Local Systems and of the Betti Moduli Se (.-adic version) of it (L. Lafforgue (Invent Math 147(1):1–241, 2002, Théorème VII.6) in dimension 1, Drinfeld (Moscow Math J 12(3):515–542, 2012, Theorem 1.1) in higher dimension in the smooth case), explain how we used Drinfeld’s theorem in the proof of Simpson’s integrality conjecture for cohom

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Lecture 8: Rigid Local Systems and ,-Isocrystals, of ., yield .-isocrystals. This is proved in Esnault and Groechenig (Acta Math 225(1):103–158, 2020, Theorem 1.6), using the theory of Higgs-de Rham flows on the mod . reduction of .. We give here a .-adic proof of this theorem, obtained with Johan de Jong, which relies on the fact that for ., the

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0075-8434 e presently out of reach.Proposes sub-conjectures that mightThe topological fundamental group of a smooth complex algebraic variety is poorly understood. One way to approach it is to consider its complex linear representations modulo conjugation, that is, its complex local systems. A fundamental pro

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Lecture 7: Companions, Integrality of Cohomologically Rigid Local Systems and of the Betti Moduli Sow we combined this idea together with de Jong’s conjecture in order to define and obtain an integrability property of the Betti moduli space (de Jong and Esnault, Integrality of the Betti moduli space, 18 pp. Trans. AMS, to appear, Theorem 1.1).

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