Version 1 has a mistake in the proof of Theorem 2.27.
From the 1980s to 1990s, J.-M. Fontaine constructed an equivalence of categories between the category of p-adic representations and the category of (φ,Γ)-modules. In this paper, we determine the (φ,Γ)-modules corresponding to crystalline representations. This result can be seen, in a sense, as a generalization of Wach modules in the ramified case.
I made a poster (in Japanese).
This paper is based on the great ideas of Laurent Berger and Takeshi Tsuji. We give an explicit description of the intersection of A_inf and the imperfect coefficient ring of (phi, Gamma)-modules, which was remarked by Nathalie Wach in 1996 without proof. This result indicates the difficulty of constructing the coefficient ring of Wach modules in the ramified case.
This is a survey of my papers and an expanded version of my talk in the conference "Algebraic Number Theory and Related Topics 2024" at Kyoto University.