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

Let \(K\) be a complete valued field extension of \(\mathbf{Q}_p\) with perfect residue field. We consider \(p\)-adic representations of a finite product \(G_{K,\Delta}=G_K^\Delta\) 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 \(\mathsf{B}_{\rm dR}\) and \(\mathsf{B}_{\rm HT}\), and multivariable Sen theory. In particular, we associate to any \(p\)-adic representation \(V\) of \(G_{K,\Delta}\) an integrable \(p\)-adic differential system in several variables \(\mathsf{D}_{\rm dif}(V)\). 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 \((\varphi,\Gamma)\)-module of \(V\) constructed by Pal and Zábrádi, along classical Berger's construction.