Over an unramified extension `$F/\mathbb{Q}_p$`

, by the works of Fontaine, Wach, Colmez and Berger, it is well-known that a crystalline representation of the absolute Galois group of `$F$`

is of finite height. Moreover, in this case, to a crystalline representation one can functorially attach a lattice inside the associated etale `$(\varphi, \Gamma)$`

-module called the Wach module. Berger showed that the aforementioned functor induces an equivalence between the category of crystalline representations and Wach modules. Furthermore, this categorical equivalence admits an integral refinement. In this talk, our goal is to generalize the notion of Wach modules to relative `$p$`

-adic Hodge theory. For a “small” unramified base (in the sense of Faltings) and its etale fundamental group, we will generalize the result of Berger to an equivalence between crystalline representations and relative Wach modules as well as establish its integral refinement.

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Last updated: 24 Mar 2023