Crystalline representations and Wach modules in the relative case

Abstract

We study the notion of Wach modules in relative setting, generalizing the arithmetic case. Over an unramified base, for a $p$-adic representation admitting such structure, we examine the relationship between its relative Wach module and filtered $(\ddot{I}†,\partial )$-module. Moreover, we show that such a representation is crystalline (in the sense of Brinon), and one can recover its filtered $(\ddot{I}†,\partial )$-module from the relative Wach module. Conversely, for low Hodge-Tate weights $[0,p-2]$, we construct relative Wach modules from free relative Fontaine-Laffaille modules (in the sense of Faltings).

Accepted for publication in Annales de l'Institut Fourier