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 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. RésuméSoient p un nombre premier, V un anneau de valuation discrète complet d'inégales caractéristiques (0, p), G un groupe algébrique affine lisse sur Spec V . En utilisant les techniques de puissances divisées de niveau m de Berthelot, nous construisons dans ce cadre des algèbres de distributions arithmétiques, avec des niveaux m, généralisant la construction classique. Nous construisons aussi la complétion faible de l'algèbre des distributions classique sur une extension finie K de Q p . Nous montrons alors que ces algèbres de distribution s'identifient aux opérateurs différentiels arithmétiques invariants sur G et donnons un théorème de cohérence si l'indice de ramification de K est < p − 1.

Kashiwara's theorem for twisted arithmetic differential operators

“…Since pD : T ,Q q T " D : pT q Q according to [11,Thm. 4.4.9.2], the ring D: X,Q is locally, on an open subset trivializing the torsor, isomorphic to D : X,Q b K D : pT q Q .…”

“…Let D : pT q be its crystalline distribution algebra, cf. [11,12]. Fix a character λ : D : pT q Q ÝÑ K.…”

Intermediate extensions and crystalline distribution algebras

“…Note, as for differential operators, that this dagger-algebra appears with coefficients tensored by Q. For more details on the basic theory of the algebra D : pGq we refer to [26,27].…”

“…In [26] we have introduced and studied the crystalline distribution algebra D : pGq associated to the p-adic completion G of G. It is a certain weak completion of the classical universal enveloping algebra Upgq. The interest in the algebra D : pGq comes at least from two sources.…”

Intermediate extensions and crystalline distribution algebras

“…In [39] we have introduced and studied the crystalline distribution algebra D : pGq associated to the p-adic completion G of G. It is a certain weak completion of the classical universal enveloping algebra U pgq. The interest in the algebra D : pGq comes at least from two sources.…”