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Morphismes de périodes et cohomologie syntomique

Abstract : Recently, Colmez and Nizioł proved a comparison theorem between arithmetic p-adic nearby cycles and syntomic cohomology sheaves. To prove it, they gave a local construction using (\phi,\Gamma)-modules which allows to reduce the period isomorphism to a comparison theorem between Lie algebras. In this thesis, we first give the geometric version of this construction before globalizing it to get a global period isomorphism. This period morphism can be used to describe the étale cohomology of rigid analytic spaces. In particular, we deduce the semi-stable conjecture of Fontaine-Jannsen, which relates the étale cohomology of the rigid analytic variety associated to a formal proper semi-stable scheme to its Hyodo-Kato cohomology. This result was also proved by (among others) Tsuji, via the Fontaine-Messing map, and by Cesnavicius and Koshikawa, which generalized the proof of the crystalline conjecture by Bhatt, Morrow and Scholze. In the second part of the thesis, we use the previous map to show that the period morphism of Tsuji and the one of Cesnavicius-Koshikawa are the same.
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Submitted on : Tuesday, October 13, 2020 - 2:51:30 PM
Last modification on : Wednesday, October 14, 2020 - 3:29:39 AM


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Sally Gilles. Morphismes de périodes et cohomologie syntomique. Théorie des nombres [math.NT]. Université de Lyon, 2020. Français. ⟨NNT : 2020LYSEN049⟩. ⟨tel-02965753⟩



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