Multivariable de Rham representations, Sen theory and $p$-adic differential equations

Abstract

Let K be a complete valued field extension of Q p with perfect residue field. We consider p-adic representations of a finite product G K,∆ = G ∆ K of the absolute Galois group G K of K. This product appears as the fundamental group of a product of diamonds. We develop the corresponding p-adic Hodge theory by constructing analogues of the classical period rings B dR and B HT , and multivariable Sen theory. In particular, we associate to any p-adic representation V of G K,∆ an integrable p-adic differential system in several variables D dif (V ). We prove that this system is trivial if and only if the representation V is de Rham. Finally, we relate this differential system to the multivariable overconvergent (ϕ, Γ)-module of V constructed by Pal and Zábrádi in [20], along classical Berger's construction [5].