Arithmetic $\D$-modules and Representations

TL;DR

This paper proposes a novel approach to the $p$-adic Langlands correspondence by utilizing arithmetic $D$-modules to realize $p$-adic group representations as solutions of these modules, linking Galois actions.

Contribution

It introduces a new framework connecting arithmetic $D$-modules with $p$-adic representations, aiming to realize the Langlands correspondence through overconvergent solutions.

Findings

Suggests arithmetic $D$-modules as a $p$-adic analogue of Kashiwara's theory.

Proposes a realization of $p$-adic representations via overconvergent solutions.

Discusses the application to Siegel modular varieties.

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.