$D$ -模块arithmétiques，distributions et localisation
$D$-modules arithmétiques, distributions et localisation
摘要：
设$p$是素数，$V$是具有不等特征的离散赋值环，$(0,p)$，$G$是$Spec \,V$上的光滑仿射代数群.利用Berthelot的部分分幂技巧，构造了具有水平$m$的算术分布代数，推广了经典分布代数的构造.我们还构造了经典分布代数的弱完备化。然后，我们证明了这些分布代数可以确定不变的算术微分算子在$G$。最后，我们将这些构造应用于约化群的情况，并获得了由$p$ -adic完备化获得的形式标志簇上算术微分算子层的局部化定理，推广了Ardakov-Wadsley之前的结果（对于级别0，并在$p$上具有某些条件）。
英文摘要：
Let $p$ be a prime number, $V$ a 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. We then show that these distribution algebras can be identified with invariant arithmetic differential operators over $G$. We finally apply these constructions in the case of a reductive group and obtain a localization theorem for the sheaf of arithmetic differential operators on the formal flag variety obtained by $p$-adic completion, generalizing a previous result of Ardakov-Wadsley (for the level 0 and with some condition on $p$).
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