Crystalline representations and Wach modules in the relative case (Abhinandan, Université de Lille)Séminaire « Arithmétique »
M2 Kampé de Fériet
In this talk, we will introduce the notion of Wach modules in the relative setting, generalizing the arithmetic case. Over an unramified base, for a p-adic representation admitting such structure, we will examine the relationship between its relative Wach module and filtered $(\varphi, \partial)$-module. Further, we will show that such a representation is crystalline (in the sense of Brinon), and one can recover its filtered $(\varphi, \partial)$-module from the relative Wach module. Conversely, for low Hodge-Tate weights [0, p-2], we will construct relative Wach modules from free relative Fontaine-Laffaille modules (in the sense of Faltings).