Multivariable de Rham Representations, Sen Theory and p-adic Differential Equations

(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.  Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.