Affinoid Space Research Articles

The existence of Zariski dense orbits for endomorphisms of projective surfaces

Let f f be a dominant endomorphism of a smooth projective surface X X over an algebraically closed field k \mathbf {k} of characteristic 0 0 . We prove that if there is no rational function H ∈ k ( X ) H \in \mathbf {k}(X) such that H ∘ f = H H \circ f = H , then there exists a point x ∈ X ( k ) x \in X(\mathbf {k}) such that the forward orbit of x x under f f is Zariski dense in X X . This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of smooth projective surfaces. We also define a new topology on varieties over algebraically closed fields with finite transcendence degree over Q \mathbb {Q} , which we call “the adelic topology”. This topology is stronger than the Zariski topology, and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic version of the Zariski dense orbit conjecture that is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of smooth projective surfaces, for endomorphisms of abelian varieties, and for split polynomial maps. This yields new proofs of the original conjecture in the latter two cases. In Appendix A, we study endomorphisms of k k -affinoid spaces. We show that for certain endomorphisms f f on a k k -affinoid space X X , the attractor Y Y of f f is a Zariski closed subset and that the dynamics of f f is semi-conjugate to its restriction to Y Y . A special case of this result is used in the proof of the main theorem. In Appendix B, written in collaboration with Thomas Tucker, we prove the Zariski dense orbit conjecture for endomorphisms of ( P 1 ) N (\mathbb {P}^1)^N .

Non-archimedean pinchings

We develop the theory of pinchings for non-archimedean analytic spaces. In particular, we show that although pinchings of affinoid spaces do not have to be affinoid, pinchings of Hausdorff analytic spaces always exist in the category of analytic spaces.

Vanishing and comparison theorems in rigid analytic geometry

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

On a dynamical Mordell–Lang conjecture for coherent sheaves

AbstractWe introduce a dynamical Mordell–Lang‐type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module‐theoretic variant of Strassman's theorem that we prove in the appendix.

Analytic functions on tubes of nonarchimedean analytic spaces

Let $k$ be a discretely valued non-Archimedean field. We give an explicit description of analytic functions whose norm is bounded by a given real number $r$ on tubes of reduced $k$-analytic spaces associated to special formal schemes (those include $k$-affinoid spaces as well as open polydiscs). As an application we study the connectedness of these tubes. This generalizes (in the discretely valued case) a result of Siegfried Bosch. We use as a main tool a result of A.J. de Jong relating formal and analytic functions on special formal schemes and a generalization of de Jong's result which is proved in the joint appendix with Christian Kappen.

Overconvergent subanalytic subsets in the framework of Berkovich spaces

We study the class of overconvergent subanalytic subsets of a k -affinoid space X when k is a non-archimedean field. These are the images along the projection X \times \mathbb B^n \to X of subsets defined with inequalities between functions of X \times \mathbb B^n which are overconvergent in the variables of \mathbb B^n . In particular, we study the local nature, with respect to X , of overconvergent subanalytic subsets. We show that they behave well with respect to the Berkovich topology, but not to the G -topology. This gives counter-examples to previous results on the subject, and a way to correct them. Moreover, we study the case dim (X)=2 , for which a simpler characterisation of overconvergent subanalytic subsets is proven.

Dagger geometry as Banach algebraic geometry

In this article, we look at analytic geometry from the perspective of relative algebraic geometry with respect to the categories of bornological and Ind-Banach spaces over valued fields (both Archimedean and non-Archimedean). We are able to recast the theory of Grosse-Klönne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers) and for the condition of a family of morphisms to be a cover. We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds coming together.

SUR LES COMPOSANTES CONNEXES D’UNE FAMILLE D’ESPACES ANALYTIQUES -ADIQUES

RésuméSoit $X=\mathcal{M}(\mathscr{A})$ un espace affinoïde et soient $f,g\in \mathscr{A}$. Nous étudions les ensembles de composantes connexes des espaces définis par une inégalité de la forme $|f|\le r\, |g|$, avec $r\ge 0$. Nous montrons qu’il existe une partition finie de $\mathbf{R}_{+}$ en intervalles sur lesquels ces ensembles sont canoniquement en bijection et que les bornes de ces intervalles appartiennent à $\sqrt{\rho (\mathscr{A})}$.

Slope filtrations in families

This paper concerns arithmetic families of -modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons are lower semicontinuous around any rigid point. We further prove that if the slope polygons are locally constant around a rigid point, then around this point, the family has a global slope filtration after base change to some extended Robba ring.

On families of filtered (φ,N)-modules

In this paper, as a generalization of Berger's construction, we give a functor from the category of families of filtered (ϕ, N )-modules (with certain condition) to the category of families of (ϕ, Γ)-modules. Combining this with Kedlaya and Liu's theorem we show that, when the base is a reduced affinoid space, every family of weakly admis- sible filtered (ϕ, N )-modules can locally be converted into a family of semistable Galois representations.

On families ofφ,Γ-modules

Berger and Colmez introduced a theory of families of overconvergent etale (φ,Γ)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. However, in contrast with the classical theory of (φ,Γ)-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) etale (φ,Γ)-modules can locally be converted into a family of p-adic representations in a unique manner, providing the “local” equivalence. There is a global mod p obstruction related to the moduli of residual representations.

A new proof of the Gerritzen-Grauert theorem

The Gerritzen-Grauert theorem ([GG], [BGR, 7.3.5/1]) is one of the most important foundational results of rigid analytic geometry. It describes so called locally closed immersions between affinoid varieties, and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains. In its turn, the latter fact allowed one to extend Tate’s theorem (see [Tate], [BGR, 8.2.1/1]) on acyclicity of the Cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains. The same fact also plays an important role in foundations of non-Archimedean analytic geometry developed by V. Berkovich in [Ber1] and [Ber2]. Recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry (the latter were called in [Ber1] strictly affinoid) and, besides, the valuation on the ground field is not assumed to be nontrivial. In the recent papers by A. Ducros [Duc, 2.4] and the author [Tem, 3.5], the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces, but its proof was based on the case of strictly affinoid ones (i.e., affinoid varieties). The original proof of the Gerritzen-Grauert theorem is not easy, and since then the only different proof was found by M. Raynaud in the framework of his approach to rigid analytic geometry (see [Ray], [BL]). Although that proof is more conceptual, it is based on a complicated algebraic technics. The purpose of this paper is to give a new proof of the Gerritzen-Grauert theorem which uses basic properties of affinoid algebras in a standard way. The only novelty is in using the whole spectrum M(A) of an affinoid algebra A, introduced in [Ber1], instead of the maximal spectrum Max(A), considered in rigid analytic geometry. The use of the whole spectrum allows one to apply additional but standard compactness arguments. In §§1-2, we work in the setting of rigid analytic geometry, i.e., the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered. In §1, we recall basic definitions of an affinoid algebra, an affinoid domain, all notions necessary for the formulation of the Gerritzen-Grauert theorem, and formulate it (Theorem 1.1). The only new fact is Proposition 1.2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a

Parties semi-alg�briques d'une vari�t� alg�brique p-adique

Let k be a field complete with respect to an ultrametric absolute value, let A be a finitely generated k-algebra and let X be its spectrum. We denote by X an the analytification a la Berkovich of the algebraic variety X. We say that a subset of X an is semi-algebraic if it can be defined by a boolean combination of inequalities |f|⋈λ|g| where f and g are in A, where ⋈ is a symbol belonging to { ,≤,≥} and where λ is a positive real number. In this text we show that the image of any semi-algebraic subset under an algebraic map between affine varieties is semi-algebraic, and that a semi-algebraic subset has only finitely many connected components, each of which is semi-algebraic. In fact our results concern not only algebraic varieties, but also analytic families of algebraic varieties which are parametrized by an affinoid space.

The class group of a one-dimensional affinoid space

A curve X over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space Y is trivial if and only if Y is a subspace of P 1 . As a consequence, X has locally a trivial class group if and only if the stable reduction of X has only rational components.

Orthonormalbasen in der nichtarchimedischen Funktionentheorie

In this paper one finds a new method to calculate problems concerning affinoid algebras. The method which uses orthonormal bases in normed vector spaces is developed in the first two paragraphs and is applied to affinoid algebras later on. In a simple way there are obtained nearly all results about affinoid algebras which are already known. Further this method gives new information about the functor F which associates to each affinoid space X an affine algebraic variety\(\tilde X\). In detail: F is compatible with extensions of the field k (if affinoid spaces\(X \subset k^{ \circ n} \) are considered, k algebraically closed, n arbitrary) and F is compatible with the cartesian product. These problems are treated in the language of affinoid algebras.