Period morphisms and syntomic cohomology

Abstract: In 2017, 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 $(\varphi,\Gamma)$-modules theory which allows to reduce the period isomorphism to a comparison theorem between Lie algebras. In this talk, I will first give the geometric version of this construction before explaining how to globalize it. 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 Česnavičius and Koshikawa, which generalized the proof of the crystalline conjecture by Bhatt, Morrow and Scholze. In the second part of the talk, I will explain how we can use the previous map to show that the period morphism of Tsuji and the one of Česnavičius-Koshikawa are the same.

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html