arXiv subjects
Publications and source records attributed to Konstantin Ardakov.
We prove a general irreducibility result for geometrically induced coadmissible equivariant $\mathcal{D}$-modules on rigid analytic spaces. As an application, we geometrically reprove the irreducibility of certain locally analytic representations previously constructed by Orlik-Strauch.
We show that the coefficients of a power series occurring in $p$-adic Fourier theory for $\mathbf{Q}_{p^2}$ have valuations that are given by an intriguing formula.
Let $F$ be a finite extension of $\mathbb{Q}_p$, let $Ω_F$ be Drinfeld's upper half-plane over $F$ and let $G^0$ the subgroup of $GL_2(F)$ consisting of elements whose determinant has norm $1$. Let $\mathscr{L}$ be a torsion $G^0$-equivariant line bundle with connection on $Ω_F$. We show that the strong dual of $\mathscr{L}(Ω_F)$ is an admissible locally $F$-analytic representation of $G^0$ of topological length at most $2$. It is topologically irreducible if and only if the underlying connection on $\mathscr{L}$ is non-trivial. We give an explicit formula for the length of the strong dual of the space of globally-defined rigid analytic functions on a $G^0$-equivariant finite étale rigid analytic covering of $Ω_F$ with abelian Galois group as an admissible locally $F$-analytic representation of $G^0$.
Let $F$ be a finite extension of $\mathbb{Q}_p$, let $\Omega_F$ be Drinfeld's upper half-plane over $F$ and let $G^0$ the subgroup of $GL_2(F)$ consisting of elements whose determinant has norm $1$. By working locally on $\Omega_F$, we construct and classify the torsion $G^0$-equivariant line bundles with integrable connection on $\Omega$ in terms of the smooth linear characters of the units of the maximal order of the quaternion algebra over $F$.
Let $p \geq 5$ be a prime number and let $G = SL_2(\mathbb{Q}_p)$. Let $Ξ$ = Spec$(Z)$ denote the spectrum of the centre $Z$ of the pro-$p$ Iwahori Hecke algebra of $G$ with coefficients in a field $k$ of characteristic $p$. Let $\mathcal{R} \subset Ξ\times Ξ$ denote the support of the pro-$p$ Iwahori Ext-algebra of $G$, viewed as a $(Z,Z)$-bimodule. We show that the locally ringed space $Ξ/\mathcal{R}$ is a projective algebraic curve over Spec$(k)$ with two connected components, and that each connected component is a chain of projective lines. For each Zariski open subset $U$ of $Ξ/\mathcal{R}$, we construct a stable localising subcategory $\mathcal{L}_U$ of the category of smooth $k$-linear representations of $G$.
This paper is motivated by an open question in $p$-adic Fourier theory, that seems to be more difficult than it appears at first glance. Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $o_L$ and let $\mathbb{C}_p$ denote the completion of an algebraic closure of $\mathbb{Q}_p$. In their work on $p$-adic Fourier theory, Schneider and Teitelbaum defined and studied the character variety $\mathfrak{X}$. This character variety is a rigid analytic curve over $L$ that parameterizes the set of locally $L$-analytic characters $λ: (o_L,+) \to (\mathbb{C}_p^\times,\times)$. One of the main results of Schneider and Teitelbaum is that over $\mathbb{C}_p$, the curve $\mathfrak{X}$ becomes isomorphic to the open unit disk. Let $Λ_L(\mathfrak{X})$ denote the ring of bounded-by-one functions on $\mathfrak{X}$. If $μ\in o_L [\![o_L]\!]$ is a measure on $o_L$, then $λ\mapsto μ(λ)$ gives rise to an element of $Λ_L(\mathfrak{X})$. The resulting map $o_L [\![o_L]\!] \to Λ_L(\mathfrak{X})$ is injective. The question is: do we have $Λ_L(\mathfrak{X}) = o_L [\![o_L]\!]$? In this paper, we prove various results that were obtained while studying this question. In particular, we give several criteria for a positive answer to the above question. We also recall and prove the ``Katz isomorphism'' that describes the dual of a certain space of continuous functions on $o_L$. An important part of our paper is devoted to providing a proof of this theorem which was stated in 1977 by Katz. We then show how it applies to the question. Besides $p$-adic Fourier theory, the above question is related to the theory of formal groups, the theory of integer valued polynomials on $o_L$, $p$-adic Hodge theory, and Iwasawa theory.
The center $Z(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category $\mathcal{C}$ is said to be stable if it is stable under essential extensions. The set $\mathbf{L}^{st}(\mathcal{C})$ of stable localizing subcategories of $\mathcal{C}$ is partially ordered under reverse inclusion. We show $\mathcal{L} \mapsto Z(\mathcal{C}/\mathcal{L})$ defines a sheaf of commutative rings on $\mathbf{L}^{st}(\mathcal{C})$ with respect to finite coverings. When $\mathcal{C}$ is assumed to be locally noetherian, we also show that the sheaf condition holds for arbitrary coverings.
Let $G$ be a locally profinite group and let $k$ be a field of positive characteristic $p$. Let $Z(G)$ denote the center of $G$ and let $\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra of natural endomorphisms of the identity functor on the category of smooth $k$-linear representations of $G$. We show that if $G$ contains an open pro-$p$ subgroup but no proper open centralisers, then there is a natural isomorphism of $k$-algebras $\mathfrak{Z}(Z(G)) \xrightarrow{\cong} \mathfrak{Z}(G)$. We also describe $\mathfrak{Z}(Z(G))$ explicitly as a particular completion of the abstract group ring $k[Z(G)]$. Both conditions on $G$ are satisfied whenever $G$ is the group of points of any connected smooth algebraic group defined over a local field of residue characteristic $p$. In particular, when the algebraic group is semisimple, we show that $\mathfrak{Z}(G) = k[Z(G)]$.
We prove an Induction Equivalence and a Kashiwara Equivalence for coadmissible equivariant D-modules on rigid analytic spaces. This allows us to completely classify such objects with support in a single orbit of a classical point with co-compact stabiliser. As an application, we use the locally analytic Beilinson-Bernstein equivalence to construct new examples of large families of topologically irreducible locally analytic representations of certain compact semisimple p-adic Lie groups.
We develop a dimension theory for coadmissible D-cap-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic D-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinte-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $H^i_Z(\mathcal{M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal{M}$ on $X$.
We prove that the category of coadmissible D-cap-modules on a smooth rigid analytic space supported on a closed smooth subvariety is naturally equivalent to the category of coadmissible D-cap-modules on the subvariety, and use this result to construct a large family of pairwise non-isomorphic simple coadmissible D-cap-modules.
We define coadmissible equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces and relate them to admissible locally analytic representations of semisimple $p$-adic Lie groups.
Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the algebra $\mathcal{E}$ of bounded $K$-linear endomorphisms of the structure sheaf. In the case of complex manifolds, Ishimura proved that the analogous sheaves are isomorphic. In the rigid analytic situation, we prove that the natural map $\widehat{\mathcal{D}} \to \mathcal{E}$ is an isomorphism if and only if the ground field $K$ is algebraically closed and its residue field is uncountable.
We prove that the canonical dimension of an admissible Banach space or a locally analytic representation of an arbitrary semisimple p-adic Lie group is either zero or at least half the dimension of a non-zero coadjoint orbit. This extends the results of Ardakov-Wadsley and Schmidt in the split semisimple case.
We introduce a sheaf of infinite order differential operators D-cap on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Fréchet-Stein algebras, and use this to define co-admissible sheaves of D-cap-modules. We prove analogues of Cartan's Theorems A and B for co-admissible D-cap-modules.
We give an overview of the theory of $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces and its applications to admissible locally analytic representations of $p$-adic Lie groups.
We establish the faithfulness of Verma modules for rational Iwasawa algebras of split semisimple compact $L$-analytic groups. We also prove the algebraic independence of Arens-Michael envelopes over Iwasawa algebras and compute the centre of affinoid enveloping algebras of semisimple $p$-adic Lie algebras.
Let G be a nilpotent complete p-valued group of finite rank and let k be a field of characteristic p. We prove that every faithful prime ideal of the Iwasawa algebra kG is controlled by the centre of G, and use this to show that the prime spectrum of kG is a disjoint union of commutative strata. We also show that every prime ideal of kG is completely prime. The key ingredient in the proof is the construction of a non-commutative valuation on certain filtered simple Artinian rings.