Rendiconti del Seminario Matematico della Università di Padova

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Volume 139, 2018, pp. 1–76
DOI: 10.4171/RSMUP/139-1

Published online: 2018-06-06

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

Christine Huyghe[1] and Tobias Schmidt[2]

(1) Université de Strasbourg, France
(2) Université de Rennes I, France

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 $< p-1$.

Keywords: Arithmetic $\mathcal D$-modules, $p$-adic distribution algebras

Huyghe Christine, Schmidt Tobias: Algèbres de distributions et $\mathcal D$-modules arithmétiques. Rend. Sem. Mat. Univ. Padova 139 (2018), 1-76. doi: 10.4171/RSMUP/139-1