Arithmetic $\D$-modules and Representations

King Fai Lai (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic analogue of Kashiwara's theory of $D$-modules for real Lie groups i.e. it should give a realization of the $p$-adic representations of a $p$-adic Lie group as spaces of overconvergent solutions of arithmetic $D$-modules which will come equipped with an action of the Galois group. We shall discuss the case of Siegel modular varieties as a possible testing ground for the proposal.

Mots clés: arithmetic D modules; representations of p-adic group; p-adic modular forms

Domaines: Théorie des nombres [math.NT]; Théorie des représentations [math.RT]

King Fai Lai. Arithmetic $\D$-modules and Representations. 2007. ⟨hal-00256409⟩