Algèbres de distributions et $\mathcal{D}$-modules arithmétiques

Abstract

Let $p$ be a prime number, $V$ a complete discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over Spec $V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level $m$, generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra over a finite extension $K$ of ${\mathbf{Q}}_p$. We then show that these distribution algebras can be identified with invariant arithmetic differential operators over $G$, and prove a coherence result when the ramification index of $K$ is $<p-1$.