Morphismes de p\'eriodes et cohomologie syntomique

Abstract

We give the geometric version of a construction of Colmez-Niziol which establishes a comparison theorem between arithmetic p-adic nearby cycles and syntomic sheaves. The local construction of the period isomorphism uses $(\phi,\Gamma)$-modules theory and is obtained by reducing the period isomorphism to a comparison theorem between cohomologies of Lie algebras. By applying the method of "more general coordinates" used by Bhatt-Morrow-Scholze, we construct a global isomorphism. In particular, we deduce the semi-stable conjecture of Fontaine-Jannsen. 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. We use the previous map to show that the period morphism of Tsuji and the one of Cesnavicius-Koshikawa are the same.