Morphismes de p\'eriodes et cohomologie syntomique

TL;DR

This paper establishes a geometric comparison theorem linking p-adic nearby cycles and syntomic sheaves, leading to a proof of the semi-stable conjecture, using advanced cohomological and $(, )$-modules techniques.

Contribution

It provides a new geometric construction of the period isomorphism and proves the semi-stable conjecture, unifying previous approaches by Tsuji and Cesnavicius-Koshikawa.

Findings

Constructed a global isomorphism using 'more general coordinates' method.

Proved the semi-stable conjecture of Fontaine-Jannsen.

Showed the equivalence of Tsuji's and Cesnavicius-Koshikawa's period morphisms.

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.