A closedness theorem and applications in geometry of rational points over Henselian valued fields
Krzysztof Jan Nowak
备注：
This paper has been published in Journal of Singularities 21 (2020), 233-254. arXiv admin note: substantial text overlap with arXiv:1704.01093, arXiv:1703.08203, arXiv:1702.07849
We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Hölder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.
We prove the closedness theorem over Henselian valued fields, which was established over rank one valued fields in one of our recent papers. In the proof, as before, we use the local behaviour of definable functions of one variable and the so-called fiber shrinking, which is a relaxed version of curve selection. Now our approach applies also relative quantifier elimination for ordered abelian groups due to Cluckers--Halupczok. Afterwards the closedness theorem will allow us to achieve i.a. the Łojasiewicz inequality, curve selection and extending hereditarily rational functions as well as to d…
The main purpose of the paper is to establish a closedness theorem over Henselian valued fields $K$ of equicharacteristic zero (not necessarily algebraically closed) with separated analytic structure. It says that every projection with a projective fiber is a definably closed map. This remains valid also for valued fields with analytic structure induced by a strictly convergent Weierstrass systems, including the classical, complete rank one valued fields with the Tate algebra of strictly convergent power series. As application, we prove two theorems on existence of the limit and on piecewise c…
We prove the existence of definable retractions onto arbitrary closed subsets of $K^{n}$ definable over Henselian valued fields $K$. Hence directly follows non-Archimedian analogues of the Tietze--Urysohn and Dugundji theorems on extending continuous definable functions. The main ingredients of the proof are a description of definable sets due to van den Dries, resolution of singularities and our closedness theorem.
This paper provides, over Henselian valued fields, some theorems on implicit function and of Artin--Mazur on algebraic power series. Also discussed are certain versions of the theorems of Abhyankar--Jung and Newton--Puiseux. The latter is used in analysis of functions of one variable, definable in the language of Denef--Pas, to obtain a theorem on existence of the limit, proven over rank one valued fields in one of our recent papers. This result along with the technique of fiber shrinking (developed there over rank one valued fields) were, in turn, two basic tools in the proof of the closednes…
We give an elementary proof of a version of the implicit function theorem over Henselian valued fields $K$. It yields a density property for such fields (introduced in a joint paper with J. Koll{á}r), which is indispensable for ensuring reasonable topological and geometric properties of algebraic subsets of $K^{n}$.
Abhyankar showed that for a finite tame extension $L_1/K$ and a finite extension $L_2/K$ of $\mathfrak{P}$-adic fields, the condition $[νL_1 : νK]$ divides $[νL_2 : νK]$ is sufficient to eliminate ramification, that is, $L_1 \cdot L_2 / L_2$ is unramified. In this paper, we show that the above condition is not sufficient in the case of an arbitrary henselian valued field. We construct a counterexample illustrating that fact. We also give a necessary and sufficient condition for the elimination of tame ramification of a henselian field after a finite extension of the base field.
Consider a Henselian rank one valued field $K$ of equicharacteristic zero with the three-sorted language $\mathcal{L}$ of Denef--Pas. Let $f: A \to K$ be a continuous $\mathcal{L}$-definable (with parameters) function on a closed bounded subset $A \subset K^{n}$. The main purpose is to prove that then $f$ is Hölder continuous with some exponent $s\geq 0$ and constant $c \geq 0$, a fortiori, $f$ is uniformly continuous. Further, if $f$ is locally Lipschitz continuous with a constant $c$, then $f$ is (globally) Lipschitz continuous with possibly some larger constant $d$. Also stated are some pro…
Consider a Henselian rank one valued field $K$ of equicharacteristic zero along with the language $\mathcal{L}^{P}$ of Denef--Pas. Let $f: A \to K$ be an $\mathcal{L}^{P}$-definable (with parameters) function on a subset $A$ of $K^{n}$. We prove that $f$ is piecewise continuous; more precisely, there is a finite partition of $A$ into $\mathcal{L}^{P}$-definable locally closed subsets $A_{1},\ldots,A_{s}$ of $K^{n}$ such that the restriction of $f$ to each $A_{i}$ is a continuous function.
Let $K$ be a Henselian, non-trivially valued field with separated analytic structure. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$. Hence directly follow definable non-Archimedean versions of the extension theorems by Tietze--Urysohn and Dugundji. This generalizes our previous paper dealing with complete non-Archimedean fields with separated power series and remains true for Henselian valued fields with strictly convergent analytic structure, because every such a structure can be extended in a definitional way to a separated analytic stru…
The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is a complete rank one valued fields with the Tate algebra of strictly convergent power series. The algebraic case was treated in our previous papers. Here we are going to carry over the research to the general analytic settings. Also considered are the local rings of analytic function germs induced by a given separated Weierstrass system, which turn out to be…
This paper develops algebraic geometry over Henselian real valued (i.e. of rank 1) fields $K$, being a sequel to our paper about that over Henselian discretely valued fields. Several results are given including: a certain concept of fiber shrinking (a relaxed version of curve selection) for definable sets, the canonical projection $K^{n} \times K\mathbb{P}^{m} \to K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, a version of the Lojasiewicz inequality for continuous rational functions and the theorem on extending continuo…
We develop geometry of affine algebraic varieties in $K^{n}$ over Henselian rank one valued fields $K$ of equicharacteristic zero. Several results are provided including: the projection $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ and blow-ups of the $K$-rational points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, curve selection for definable sets, a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the $K$-topology and extending continuous hereditarily rational functions, established for the real…
We are concerned with rigid analytic geometry in the general setting of Henselian fields $K$ with separated analytic structure, whose theory was developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore the usual global resolution of singularities from rigid analytic ge…
Nash structure of curves over a valued field and tame henselian rationality
Sorbonne Université, Université Paris-Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG（索邦大学，巴黎第七大学，法国国家科学研究中心，儒勒·让苏-巴黎左岸数学研究所）；Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS（索邦大学，儒勒·让苏-巴黎左岸数学研究所，法国国家科学研究中心）
We extend the formalism and results on motivic integration from ["Constructible motivic functions and motivic integration", Invent. Math., Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to prev…
This paper is concerned with algebraic geometry over complete discretely valued fields $K$ of equicharacteristic zero. Several results are given including: the canonical projection $K^{n} \times K\mathbb{P}^{m} \longrightarrow K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties are definably closed maps, a descent principle, a version of the Lojasiewicz inequality for continuous rational functions, curve selection for semialgebraic sets and the theorem on extending continuous hereditarily rational functions, established for the real field in our joint paper with J. Kollar. Our approa…
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A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tamely ramified extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2.
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Let $X$ be a proper variety over a henselian discretely valued field. An important obstruction to the existence of a rational point on $X$ is the index, the minimal positive degree of a zero cycle on $X$. This paper introduces a new invariant, the specialization index, which is a closer approximation of the existence of a rational point. We provide an explicit formula for the specialization index in terms of an $snc$-model, and we give examples of curves where the index equals one but the specialization index is different from one, and thus explains the absence of a rational point. Our main re…
Let $K$ be a complete non-Archimedean field $K$ with separated power series, treated in the analytic Denef--Pas language. We prove the existence of definable retractions onto an arbitrary closed definable subset of $K^{n}$, whereby definable non-Archimedean versions of the extension theorems by Tietze--Urysohn and Dugundji follow directly. We reduce the problem to the case of a simple normal crossing divisor, relying on our closedness theorem and desingularization of terms. The latter result is established by means of the following tools: elimination of valued field quantifiers (due to Clucker…
The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over $\FF_q((t))$ of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields $K$ with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over $K$. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper…
In this note, we establish a version of the local Cauchy-Crofton formula for definable sets in Henselian discretely valued fields of characteristic zero. It allows to compute the motivic local density of a set from the densities of its projections integrated over the Grassmannian.
If $G$ is a finite $\ell$-group acting on an affine space $\mathbb{A}^n$ over a finite field $K$ of cardinality prime to $\ell$, Serre has shown that there exists a rational fixed point. We generalize this to the case where $K$ is a henselian discretely valued field of characteristic zero with algebraically closed residue field and with residue characteristic different from $\ell$. We also treat the case where the residue field is finite of cardinality $q$ such that $\ell$ divides $q-1$. To this aim, we study group actions on weak Néron models.
Let (K,v) be a valued field, Y a K-variety, G an algebraic group over K (not necessarily smooth), and f: X->Y a G-torsor over Y. We consider the induced map X(K)-->Y(K), which is continuous for the topologies deduced from the valuation. Let I denote the image of this map. We prove that if (K,v) is henselian and its completion is a separable extension, then: - I is locally closed in Y(K); - the induced surjection X(K)-->I is a principal bundle with group G(K) (also topologized by the valuation).
This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.
We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.
Let $(K,v)$ be a valued field, and $μ$ an inductive valuation on $K[x]$ extending $v$. Let $G_μ$ be the graded algebra of $μ$ over $K[x]$, and $κ$ the maximal subfield of the subring of $G_μ$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $κ$ and the residual polynomial operator $R_μ: K[x]\toκ[y]$, where $y$ is another indeterminate, without any need to perform computations in the graded algebra. This leads to an OM algorithm to compute the factorization of separable defectless polynomials over henselian fields.
We define "t-stratifications", a strong notion of stratifications for Henselian valued fields $K$ of equi-characteristic 0, and prove that they exist. In contrast to classical stratifications in Archimedean fields, t-stratifications also contain non-local information about the stratified sets. For example, they do not only see the singularities in the valued field, but also those in the residue field. Like Whitney stratifications, t-stratifications exist for different classes of subsets of $K^n$, e.g. algebraic subvarieties or certain classes of analytic subsets. The general framework are defi…
Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I. We can in fact give a complete description of the finite extensions L/K such that C has an L-rational point. Applications are discussed to the corresponding problem over number fields. S. Sharif, in his 2006 Berkeley thesis, has independently obtained similar (but not identical) results. Our proof, however, is different: via deformation theory, we reduce to t…
We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on $p$-adic subanalytic sets, and we continue the study of non-archimedean fields with analytic structure of [LR3], [CLR1] and [CL1]. We show that the language $L_K$ consisting of the language of valued fields together with all strictly convergent power series over a complete, rank one valued field $K$ can be expanded, in a definitial way, to a larger language corresponding to an analytic structure (with separated power series) from [CL1], he…
