Seminar on p-adic Hodge theory

Fontaine and Fontaine-Jannsen's conjectures(CdR , Ccris , Cst ) are a series of conjectures comparing the p-adic ́etale cohomology of varieties over p-adic local fields with their de Rham cohomology. These conjectures, which have proven to be extremely important and influential in the modern arithmetic geometry, are already proved in three different ways: by Faltings using almost ring theory, by Niziol using K-theory, and Fontaine, Messing, Hyodo, Kato, Tsuji using syntomic complex. Recently, Beilinson has get a new proof to define the p-adic period map and then prove the above conjectures(CdR , Ccris , Cst )(the essential idea is using the conjugate filtration to prove a poinca ́e lemma). And the aim of our seminar is to study Beilinson's new approach to these important comparison conjectures.