Crystalline representations and Wach modules in the relative case
Abhinandan
TL;DR
The paper extends Fontaine’s Wach module framework from the absolute/arithmetic Galois setting to a relative context over unramified bases, linking relative Wach modules to filtered (φ,∂)-modules and Brinon’s relative crystalline theory. It introduces positive finite q-height representations, defines relative Wach modules, and proves a central comparison: positive finite q-height representations V are crystalline and their relative D_cris(V) is recoverable from the relative Wach module N(T) via fat period rings, with compatibility across φ, ∂, Fil, and Γ_R. A parallel theory with relative Fontaine–Laffaille modules is developed, giving an explicit method to construct Wach modules from Fontaine–Laffaille data in low weights and establishing an equivalence of categories in the relative setting. Overall, the work provides a robust relative p-adic Hodge framework for studying crystalline lattices and representations, enabling explicit constructions and deepens connections between Wach modules, Brinon’s relative periods, and Fontaine–Laffaille theory. These results pave the way for computational approaches to relative Galois cohomology using syntomic-type complexes with relative D_cris coefficients and illuminate the structure of relative crystalline representations in families over formal tori.
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 $(\varphi, \partial)$-module. Moreover, we 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 construct relative Wach modules from free relative Fontaine-Laffaille modules (in the sense of Faltings).