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

Let K be a complete valued field extension of Qp with perfect residue field. We consider p-adic representations of a finite product GK,∆ = G∆K of the absolute Galois group GK 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 BdR and BHT, and multivariable Sen theory. In particular, we associate to any p-adic representation V of GK,∆ an integrable p-adic differential system in several variables Ddif(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].