# Arithmetic $\D$-modules and Representations

Abstract : 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.
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Cited literature [30 references]

https://hal.archives-ouvertes.fr/hal-00256409
Contributor : Marie-Annick Guillemer <>
Submitted on : Friday, February 15, 2008 - 2:13:35 PM
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### Identifiers

• HAL Id : hal-00256409, version 1
• ARXIV : 0802.2196

### Citation

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

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