Crystalline representations and Wach modules in the relative case (Abhinandan, Université de Lille)

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).