Complete non-Archimedean Field Research Articles

The universal vector extension of an abeloid variety

Let $A$ be an abelian variety over a complete non-Archimedean field $K$. The universal cover of the Berkovich space attached to $A$ reflects the reduction behaviour of $A$. In this paper the universal cover of the universal vector extension $E(A)$ of $A$ is described. In a forthcoming paper ( arXiv:2007.04659), this will be one of the crucial tools to show that rigid analytic functions on $E(A)$ are all constant.

UNIFORM LOCAL CONSTANCY OF ÉTALE COHOMOLOGY OF RIGID ANALYTIC VARIETIES

AbstractWe prove some $\ell $ -independence results on local constancy of étale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighbourhood in the analytic topology such that, for every prime number $\ell $ different from the residue characteristic, the closed subscheme and the open neighbourhood have the same étale cohomology with ${\mathbb Z}/\ell {\mathbb Z}$ -coefficients. The existence of such an open neighbourhood for each $\ell $ was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.

Non-linearizability of power series over complete non-Archimedean fields of positive characteristic

In [7], Herman and Yoccoz prove that for any given locally analytic (at z=0) power series f(z)=z(λ+∑i=1∞aizi) over a complete non-Archimedean field of characteristic 0 if |λ|=1 and λ is not a root of unity, then f is locally linearizable at z=0. They ask the same question for power series over fields of positive characteristic.In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable. More precisely, given a complete non-Archimedean field K of positive characteristic and a power series f(z)=z(λ+∑i=1∞aizi)∈K〚z〛 with λ not a root of unity and |1−λ|<1, we prove a sufficient condition (Criterion ★) for f to be non-linearizable. This phenomenon of prevalence for power series over fields of positive characteristic being non-linearizable was initially conjectured in [6, p 147] by Herman, and formulated into a concrete question by Lindahl as [9, Conjecture 2.2].As applications of Criterion ★, we prove the non-linearizability of three families of polynomials.

Tate algebras and Frobenius non-splitting of excellent regular rings

An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent F -pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field k of characteristic p with no nonzero continuous k -linear maps k^{1/p} \to k . An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of F -singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of F -purity. The excellent Euclidean domains we construct also admit no nonzero R -linear maps R^{1/p} \rightarrow R . These are the first examples that illustrate that F -purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero p^{-1} -linear maps. The latter is particularly interesting from the perspective of the theory of test ideals.

An Intrinsic Characterization of Bruhat–Tits Buildings Inside Analytic Groups

Given a reductive group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat–Tits building into the analytic space associated with the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

On Schneider\u2019s Continued Fraction Map on a Complete Non-Archimedean Field

Let {mathcal {M}} denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote k^{times }, and a uniformizer we denote pi . In this paper, we consider the map T_{v}: {mathcal {M}} rightarrow {mathcal {M}} defined by Tv(x)=πv(x)x-b(x),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} T_v(x) = \\frac{\\pi ^{v(x)}}{x} - b(x), \\end{aligned}$$\\end{document}where b(x) denotes the equivalence class to which frac{pi ^{v(x)}}{x} belongs in k^{times }. We show that T_v preserves Haar measure mu on the compact abelian topological group {mathcal {M}}. Let {mathcal {B}} denote the Haar sigma -algebra on {mathcal {M}}. We show the natural extension of the dynamical system ({mathcal {M}}, {mathcal {B}}, mu , T_v) is Bernoulli and has entropy frac{#( k)}{#( k^{times })}log (#( k)). The first of these two properties is used to study the average behaviour of the convergents arising from T_v. Here for a finite set A its cardinality has been denoted by # (A). In the case K = {mathbb {Q}}_p, i.e. the field of p-adic numbers, the map T_v reduces to the well-studied continued fraction map due to Schneider.

Dual finite frames for vector spaces over an arbitrary field with applications

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

TOWARDS A NON-ARCHIMEDEAN ANALYTIC ANALOG OF THE BASS–QUILLEN CONJECTURE

We suggest an analog of the Bass–Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles $K_{0}$.

Non-archimedean closed graph theorems

We consider linear maps T: X ? Y, where X and Y are polar local convex spaces over a complete non-archimedean field K. Recall that X is called polarly barrelled, if each weakly* bounded subset in the dual X0 is equicontinuous. If in this definition weakly* bounded subset is replaced by weakly* bounded sequence or sequence weakly* converging to zero, then X is said to be l?-barrelled or c0-barrelled, respectively. For each of these classes of locally convex spaces (as well as the class of spaces with weakly* sequentially complete dual) as domain class, the maximum class of range spaces for a closed graph theorem to hold is characterized. As consequences of these results, we obtain non-archimedean versions of some classical closed graph theorems. The final section deals with the necessity of the above-named barrelledness-like properties in closed graph theorems. Among others, counterparts of the classical theorems of Mahowald and Kalton are proved.

Hensel's lemma and the intermediate value theorem over a non-Archimedean field

This paper proves that all power series over a maximal ordered Cauchy complete non-Archimedean field satisfy the intermediate value theorem on every closed interval. Hensel's lemma for restricted power series is the main tool of the proof.

Tropical Skeletons

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of X as the subset of the associated Berkovich space X an which collects all Shilov boundary points in the fibers of the Kajiwara–Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When X is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm–Katz.

Maximality of hyperspecial compact subgroups avoiding Bruhat–Tits theory

Let $k$ be a complete non-archimedean field (non trivially valued). Given a reductive $k$-group $G$, we prove that hyperspecial subgroups of $G(k)$ (i.e. those arising from reductive models of $G$) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of $\textrm{GL}_n$ and avoids all considerations on the Bruhat-Tits building of $G$.

Effective motives with and without transfers in characteristic p

We prove the equivalence between the category RigDMéteff(K,Q) of effective motives of rigid analytic varieties over a perfect complete non-archimedean field K and the category RigDAFrobéteff(K,Q) which is obtained by localizing the category of motives without transfers RigDAéteff(K,Q) over purely inseparable maps. In particular, we obtain an equivalence between RigDMéteff(K,Q) and RigDAéteff(K,Q) in the characteristic 0 case and an equivalence between DMéteff(K,Q) and DAFrobéteff(K,Q) of motives of algebraic varieties over a perfect field K. We also show a relative and a stable version of the main statement.

Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

Abstract Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger ‘skeletonized’ versions of some foundational results of Liu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of Λ-metrized complexes of k-curves lifts to a finite morphism of K-curves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Saïdi and Wewers. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Néron models, providing a negative answer to a question of Ribet motivated by number theory. This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves.

Double series over a non-Archimedean field

In the present paper we investigate the convergence of a double series over a complete non-Archimedean field and prove that, while the proofs are somewhat different, the Archimedean results hold true.

Forms of an affinoid disc and ramification

Let k be a complete nonarchimedean field and let X be an affinoid closed disc over k. We classify the tamely ramified twisted forms of X. Generalizing classical work of P. Russell on inseparable forms of the affine line we construct explicit families of wildly ramified forms of X. We finally compute the class group and the Grothendieck group of forms of X in certain cases.

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces X and Y, let \(\mathcal{M}_\flat \left( X \right)\), \(\mathfrak{M}\left( {V \to W} \right)\) and \(\mathfrak{D}_\flat \left( {X,Y} \right)\) be the ballean of X (the family of the balls in X), the space of mappings from X to Y, and the space of mappings from the ballean of X to Y, respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., \(\hat \rho _u\), \(\hat \beta _{X,Y}^\lambda\), \(\hat \beta _{X,Y}^{ * \lambda }\)) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable λ, including some normed algebra structure. To some extent, the class \(\hat \beta _{X,Y}^\lambda\) is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when X is compact and Y = K is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of K-valued measures on X.

Dynamics of Transcendental Entire Maps on Berkovich Affine Line

Dynamics of transcendental entire maps with coefficients in an algebraically closed and complete non-Archimedean field \(K\) is studied. It is shown, among other things, that periodic repelling points are dense in Berkovich Julia set, the forward union of any open set which intersects the Julia set is the whole Berkovich affine line, and a multi-connected Fatou component is wandering, in which all points go to infinity under iteration.

A criterion for weak convergence on Berkovich projective space

We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space \({{\bf P}^{N}_K}\) over a complete non-archimedean field K. As an application, we give a sufficient condition for a certain type of equidistribution on \({{\bf P}^{N}_K}\) in terms of a weak Zariski-density property on the scheme-theoretic projective space \({{\mathbb P}^N_{\tilde{K{}_{\vphantom{0}}}}}\) over the residue field \({\tilde{K}}\) . As a second application, in the case of residue characteristic zero we give an ergodic-theoretic equidistribution result for the powers of a point a in the N-dimensional unit torus \({{\mathbb T}^N_K}\) over K. This is a non-archimedean analogue of a well-known result of Weyl over \({\mathbb C}\) , and its proof makes essential use of a theorem of Mordell-Lang type for \({{\mathbb G}_m^N}\) due to Laurent.

Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

We investigate Bruhat-Tits buildings and their compactific ations by means of Berkovich analytic ge- ometry over complete non-Archimedean fields. For every redu ctive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G, k) to the Berkovich analytic space G an asscociated with G. Composing this map with the projection of G an to its flag varieties, we define a family of compactifi- cations of B(G, k). This generalizes results by Berkovich in the case of split g roups. Moreover, we show that the boundary strata of the compactifie d buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them.