arXiv subjects
Publications and source records attributed to Christine Huyghe.
We establish a version of Kashiwara's theorem for twisted sheaves of Berthelot's arithmetic differential operators for a closed immersion between smooth p-adic formal schemes. As an application, we construct simple modules for crystalline distribution algebras of reductive groups using the twisted version of the crystalline localisation theorem.
Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.
Let ${\mathfrak o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and ${\mathfrak X}_0$ a smooth formal scheme over the formal spectrum of ${\mathfrak o}$. Given an admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ we introduce sheaves of differential operators ${\mathscr D}^\dagger_{{\mathfrak X},k}$ on ${\mathfrak X}$, for every integer $k \ge k_{\mathfrak X}$, where $k_{\mathfrak X}$ depends on the blow-up morphism ${\mathfrak X}\rightarrow {\mathfrak X}_0$. This generalizes Berthelot's construction of sheaves of arit hmetic differential operators on ${\mathfrak X}_0$. The coherence of these sheaves and several other basic properties are proven. In the second part we study the projective limit sheaf ${\mathscr D}_{{\mathfrak X},\infty} = \varprojlim_k {\mathscr D}^\dagger_{{\mathfrak X},k}$ and so-called coadmissible modules for ${\mathscr D}_{{\mathfrak X},\infty}$. The inductive limit of the sheaves ${\mathscr D}_{{\mathfrak X},\infty}$, over all admissible blow-ups ${\mathfrak X}$ of ${\mathfrak X}_0$, gives rise to a sheaf ${\mathscr D}_{\langle {\mathfrak X}_0 \rangle}$ on the Zariski-Riemann space of ${\mathfrak X}_0$. Analogues of Theorems A and B are shown to hold in each of these settings, i.e., for ${\mathscr D}^\dagger_{{\mathfrak X},k}$, ${\mathscr D}_{{\mathfrak X},\infty}$, and ${\mathscr D}_{\langle {\mathfrak X}_0\rangle}$.
Let G be the group of L-rational points of a connected split reductive group over a finite extension L of Q_p. We show that formal models of the algebraic flag variety X of G are D-affine for certain sheaves of arithmetic differential operators. We then introduce the category of coadmissible G-equivariant arithmetic D-modules on the system of formal models of X and prove that it is anti-equivalent to the category of admissible locally L-analytic G-representations with trivial infinitesimal character. We compute the equivariant arithmetic D-modules of certain classes of representations.
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 $p$).
In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the crystalline cohomology of the curve, which is a lattice in the rigid cohomology. Following a method of Wach, we first explain how to use this lattice to compute the $(φ,Γ)$-module associated to an hyperelliptic curve. We also explain an alternative way to get the $(φ,Γ)$-module mod $p$ that relies on the Deligne-Illusie morphism.