(joint work with O. Brinon and N. Mazzari) Let K be a complete valued field extension of ${\mathbb Q}_p$ 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 ${\mathbb B}_{\rm dR}$ and ${\mathbb B}_{\rm HT}$, 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 ${\mathbb D}_{\rm dif} (V)$. 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 Zabradi along classical Berger's construction.
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Ahmed Abbes
