  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.
