DOCSLIB.ORG

Patching on Berkovich Spaces and the Local-Global Principle Vlere Mehmeti

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Patching on Berkovich Spaces and the Local-Global Principle Vlere Mehmeti Patching on Berkovich Spaces and the Local-Global Principle Vlere Mehmeti To cite this version: Vlere Mehmeti. Patching on Berkovich Spaces and the Local-Global Principle. Algebraic Geometry [math.AG]. Normandie Université, 2019. English. NNT : 2019NORMC240. tel-02499757 HAL Id: tel-02499757 https://tel.archives-ouvertes.fr/tel-02499757 Submitted on 5 Mar 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE Pour obtenir le diplôme de doctorat Spécialité MATHEMATIQUES Préparée au sein de l'Université de Caen Normandie Ρatching οn Βerkοvich Spaces and the Lοcal-Glοbal Ρrinciple Présentée et soutenue par Vlere MEHMETI Thèse soutenue publiquement le 04/12/2019 devant le jury composé de M. DAVID HARARI Professeur des universités, Université Paris 11 Paris-Sud Rapporteur du jury Mme ANNETTE WERNER Professeur des universités, Goethe-Universität Rapporteur du jury M. ANTOINE DUCROS Professeur des universités, universités Sorbonne regroupées Membre du jury M. PHILIPPE GILLE Directeur de recherche, Université Lyon 1 Claude Bernard Membre du jury M. DAVID HARBATER Professeur, UNIVERSITY OF PENNSYLVANIA Membre du jury M. JÉRÔME POINEAU Professeur des universités, Université Caen Normandie Directeur de thèse Thèse dirigée par JÉRÔME POINEAU, Laboratoire de Mathématiques 'Nicolas Oresme' (Caen) Patching on Berkovich Spaces and the Local-Global Principle Vler¨eMehmeti Acknowledgements I would like to start by a heartfelt thank you to J´erˆome Poineau, without whom this thesis would not have been possible. I am very grateful for all the time that he has dedicated into introducing me to the world of research and preparing me to be a mathematician. J´erˆome’s support and guidance have been most precious to me during all this time, and have ceaselessly encouraged me to continue with my work, even at moments of demoralization from my part. My profound gratitude goes to both David Harari and Annette Werner for having accepted to be the referees of this thesis, as well as part of its defense committee. In particular, many thanks to David for his support and recommendations about the next step after Caen. A very big thanks to Antoine Ducros for his support and for accepting to participate in the defense committee. His enthousiasm and curiosity for my work even in its earlier stages have been highly motivating and have made me feel welcome in the domain of Berkovich spaces. The quality of this manuscript has benefited greatly from Antoine’s remarks. My sincere gratitude also goes to Philippe Gille and David Harbater for accepting to be part of my thesis committee and for expressing interest in my work. I am especially grateful to David Harbater and Julia Hartmann for a research visit they invited me to at the University of Philadelphia, and the thought-provoking discussions we had on that occasion. A very special thanks to Q¨endrim Gashi, one of my first professors (fortunately for me) at the University of Prishtina. His unwavering support has played a big role in shaping both my ambitions and the path to a PhD in mathematics. Shum¨efaleminderit p¨er mb¨eshtetjen, Q¨endrim! Upon my first trip to France, Q¨endrim put me in contact with Martin Andler. I am very thankful to Martin for his support which started then and continues to this day. It is, in part, what influenced my decision to come to France for my master’s studies. Whenever I had somewhat non-specific mathematical questions, there was always the very curious and enthousiastic Philippe Satg´eI could address them to, and who would always take his time to try and give me an answer. Merci de ton soutien, Philippe ! I am very happy to have been part of the LMNO for the previous three years. There is a great work environement, and I have learnt a lot from the many workshops organized here during this time. Many thanks to all of the members of the LMNO. Upon my arrival to the LMNO, the great atmosphere among the PhD students and postdocs made my integration here easy and my days very enjoyable. Many thanks for that to: Loulou, Sarra, Pablo, C´esar, Georges, Frank and Arnaud. The moments we shared during “les pauses caf´es” and “les soir´ees” are among the highlights of my stay in Caen. I am also very grateful for the wonderful three years in the company of: Emilie,´ Daniele, Julien, Stavroula, Guillaume, Etienne,´ Nacer, Angelot, Rub´en, Coumba, Tiphaine, Thien, Dorian, Tin, Hung and Gilberto. I wish we had spent more time together! I am very thankful to Mariagiulia for injecting to my life some (much needed) quirki- ness! She made my transition from Preshev¨eto Palaiseau much easier. A very special thanks to my friend, Sudarshan Arthur Shinde, the amicus omnibus. Many of my years in France would have been simply unbearable without our “irish” moments, long talks, and little trips. Shum¨efaleminderit nan¨es e babit p¨er p¨erkrahjen e tyre t¨evazhdueshme gjat¨et¨er¨e jet¨es time! Kjo doktoratur¨enuk do t¨ekishte qen¨ee mundur pa juve. Faleminderit q¨ekeni ardhur prej Presheve ve¸can¨erisht p¨er mbrojtjen time t¨edoktoratur¨es. Shpresoj se do t¨e g¨ezojm¨eedhe plot momente t¨etjera t¨elumtura bashk¨e. A very heartfelt thanks to my parents for their support for over 27 years now and for coming to my PhD defense. I am beyond happy that all of the choices I have made brought me to Caen. I would like to thank my partner, L´eonard, for all of his support and the great moments we have spent together. I look forward to all the ones that are to come. Abstract. Field patching, introduced by Harbater and Hartmann in [33], and extended by the aforementioned authors and Krashen in [34], has recently seen numerous appli- cations. We present an extension of this technique to the setting of Berkovich analytic geometry and applications to the local-global principle. In particular, we show that this adaptation of patching can be applied to Berkovich analytic curves, and as a consequence obtain local-global principles over function fields of curves defined over complete ultrametric fields. Because of the connection between the points of a Berkovich analytic curve and the valuations that its function field can be endowed with, one of these local-global principles is given with respect to completions, thus evoking some similarity with more classical versions. As an application, we obtain local-global principles for quadratic forms and results on the u-invariant. These findings generalize those of [34]. As a starting point for higher-dimensional patching in the Berkovich setting, we show that this technique is applicable around certain fibers of a relative Berkovich analytic curve. As a consequence, we prove a local-global principle over the germs of meromorphic functions on said fibers. By showing that said germs of meromorphic functions are algebraic, we also obtain local-global principles over function fields of algebraic curves defined over a larger class of ultrametric fields. Resum´ e.´ Recollement sur les espaces de Berkovich et principe local-global. Le recollement sur les corps, introduit par Harbater et Hartmann dans [33], et ´etendu par ces auteurs et Krashen dans [34], a r´ecemment trouv´ede nombreuses applications. Nous pr´esentons ici une extension de cette technique au cadre de la g´eom´etrie analytique de Berkovich et des applications au principe local-global. Nous montrons que cette adaptation du recollement peut s’appliquer aux courbes analytiques de Berkovich, et par cons´equent obtenons des principes locaux-globaux sur les corps de fonctions de courbes d´efinies sur des corps ultram´etriques complets. Grˆace `ala connexion entre les points d’une courbe analytique de Berkovich et les valuations dont on peut munir son corps de fonctions, nous obtenons un principe local-global par rapport `ades compl´et´es du corps de fonctions consid´er´e, ce qui pr´esente une ressemblance avec des versions plus classiques. En application, nous ´etablissons des principes locaux- globaux dans le cas plus pr´ecis des formes quadratiques et en d´eduisons des bornes sur l’u-invariant de certains corps. Nos r´esultats g´en´eralisent ceux de [34]. Comme point de d´epart pour le recollement en dimension sup´erieure dans un cadre d’espaces de Berkovich, nous montrons que cette technique peut s’appliquer autour de certaines fibres d’une courbe analytique relative. Nous l’utilisons ensuite pour d´emontrer un principe local-global sur les germes des fonctions m´eromorphes sur ces fibres. En montrant que ces germes de fonctions m´eromorphes sont alg´ebriques, nous obtenons aussi des principes locaux-globaux sur les corps de fonctions des courbes alg´ebriques d´efinies sur une famille plus vaste de corps ultram´etriques. Contents Introduction i Presentation of the major directions i Organization of the manuscript iv Introduction (in French) xiii Pr´esentation des directions majeures xiii Organisation du manuscrit xvi Chapter 1. Introduction to Berkovich Spaces 1 1.1. Banach rings and the Berkovich spectrum 2 1.1.1. Valued Fields 2 1.1.2. Semi-normed rings 4 1.1.3. The spectral radius 5 1.1.4. Semi-normed modules 6 1.1.5.
Load more

Recommended publications
  • The Arakelov-Zhang Pairing and Julia Sets
    THE ARAKELOV-ZHANG PAIRING AND JULIA SETS ANDREW BRIDY AND MATT LARSON Abstract. The Arakelov-Zhang pairing h ; φi is a measure of the \dynamical distance" between two rational maps and φ defined over a number field K. It is defined in terms of local integrals on Berkovich space at each completion of K. We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height hφ and the standard Weil height h, and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction. 1. Introduction Let K be a number field with fixed algebraic closure K. For z 2 P1(K), we use h(z) to denote the standard logarithmic Weil height of z and hφ(z) to denote the Call-Silverman canonical height of z with respect to φ 2 K(x). We recall background on these heights in Section 2. Let ; φ : P1 ! P1 be rational maps defined over K (equivalently, rational functions in K(x)) each of degree ≥ 2. In [21] (see also [23]), Petsche, Szpiro, and Tucker introduced the Arakelov-Zhang pairing h ; φi, a symmetric, non-negative, real-valued pairing on the space of rational maps. In Section 2, we present their definition of the pairing as a sum of integrals over Berkovich space at each place of K.
    [Show full text]
  • Berkovich Analytification
    Berkovich analytification Wouter Zomervrucht, November 10, 2016 1. Berkovich spaces Throughout, k is a non-archimedean field. −1 −1 Let’s recall some definitions. An affinoid algebra A is a quotient of kfr1 T1,..., rn Tng for some r1,..., rn > 0. Its Berkovich spectrum is the topological space bounded multiplicative (equivalence classes of) bounded X = M(A) = = . seminorms on A maps from A to a valuation field A domain D ⊆ X is a closed subset that admits a bounded homomorphism A ! AD universal for the property that M(AD) !M(A) has image inside D. For such D one has D = M(AD). The map A ! AD is flat. Domains form systems of neighborhoods at each point of X. In other words, domains behave exactly like the standard opens D( f ) ⊆ Spec A do for schemes. A subset Z ⊆ X is special if it is a finite union of domains. If Z = D1 [ ... [ Dn is such a = presentation, then we write AZ ker ∏i ADi ⇒ ∏i,j ADi\Dj . Define OX(U) = lim AZ. Z ⊆ U special Now (X, OX) is a locally ringed space, satisfying OX(X) = A. Remark 1.1. The functor M : faffinoid algebrasg ! flocally ringed spacesg is faithful, but not full. This problem, together with the lack of localizations, warrants the somewhat involved definition below. A quasi-affinoid space is a locally ringed space U equipped with the data of an affinoid algebra A and an open immersion U ,!M(A).A morphism of quasi-affinoid spaces f : (U, A) ! (V, B) is a map of locally ringed spaces such that for all domains D ⊆ U, E ⊆ V with f (D) ⊆ E the restriction f : D ! E is affinoid, i.e.
    [Show full text]
  • Weight Functions on Non-Archimedean Analytic Spaces and the Kontsevich–Soibelman Skeleton
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Spiral - Imperial College Digital Repository Algebraic Geometry 2 (3) (2015) 365{404 doi:10.14231/AG-2015-016 Weight functions on non-Archimedean analytic spaces and the Kontsevich{Soibelman skeleton Mircea Mustat¸˘aand Johannes Nicaise Abstract We associate a weight function to pairs (X; !) consisting of a smooth and proper va- riety X over a complete discretely valued field and a pluricanonical form ! on X. This weight function is a real-valued function on the non-Archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich{Soibelman skeleton of (X; !), and we prove that this skeleton is connected when X has geometric genus one and ! is a canonical form on X. This result can be viewed as an analog of the Shokurov{Koll´ar connectedness theorem in birational geometry. 1. Introduction 1.1 Skeleta of non-Archimedean spaces An important property of Berkovich spaces over non-Archimedean fields is that, in many geo- metric situations, they can be contracted onto some subspace with piecewise affine structure, a so- called skeleton. These skeleta are usually constructed by choosing appropriate formal models for the space, associating a simplicial complex to the reduction of the formal model and embedding its geometric realization into the Berkovich space. For instance, if C is a smooth projective curve of genus at least two over an algebraically closed non-Archimedean field, then it has a canonical skeleton, which is homeomorphic to the dual graph of the special fiber of its stable model [Ber90, x 4].
    [Show full text]
  • Definable Sets of Berkovich Curves
    DEFINABLE SETS OF BERKOVICH CURVES PABLO CUBIDES KOVACSICS AND JER´ OME^ POINEAU Abstract. In this article, we functorially associate definable sets to k-analytic curves, and definable maps to analytic morphisms between them, for a large class of k-analytic curves. Given a k-analytic curve X, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of X and show that they satisfy a bijective relation with the radial subsets of X. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of k-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly k-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting. Contents 1. Introduction 1 2. Preliminaries 2 3. Local and global structure of curves 12 4. Definable analytic curves and morphisms 23 5. Definable subsets of B 30 6. Definable subsets of curves 35 7. Radiality and definability 39 8. Comparison with Hrushovski-Loeser's spaces 48 References 52 1. Introduction Let (k; j·j) be a complete rank 1 non-archimedean algebraically closed non-trivially valued field.
    [Show full text]
  • Analytification and Tropicalization Over Non-Archimedean Fields
    ANALYTIFICATION AND TROPICALIZATION OVER NON-ARCHIMEDEAN FIELDS ANNETTE WERNER Abstract: In this paper, we provide an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff [BPR11] [BPR13] in the case of curves, we explain a result from [CHW14] comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of most of the results in [GRW14], where a general higher-dimensional theory is developed. In particular, we explain the construction of generalized skeleta in [GRW14] which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the com- parison between polyhedral substructures of Berkovich spaces and tropicalizations. 2010 MSC: 14G22, 14T05 1. Introduction Tropical varieties are polyhedral images of varieties over non-archimedean fields. They are obtained by applying the valuation map to a set of toric coordinates. From the very beginning, analytic geometry was present in the systematic study of tropical varieties, e.g. in [EKL06] where rigid analytic varieties are used. The theory of Berkovich spaces which leads to spaces with nicer topological properties than rigid varieties is even better suited to study tropicalizations. A result of Payne [Pay09] states that the Berkovich space associated to an algebraic variety is homeomorphic to the inverse limit of all tropicalizations in toric varieties. However, individual tropicalizations may fail to capture topological features of an analytic space. arXiv:1506.04846v1 [math.AG] 16 Jun 2015 The present paper is an overview of recent results on the relationship between analytic spaces and tropicalizations.
    [Show full text]
  • Berkovich Spaces and Tubular Descent
    Available online at www.sciencedirect.com Advances in Mathematics 234 (2013) 217–238 www.elsevier.com/locate/aim Berkovich spaces and tubular descent Oren Ben-Bassata,∗, Michael Temkinb a Mathematical Institute, University of Oxford, United Kingdom b Department of Mathematics, Hebrew University, Israel Received 9 February 2012; accepted 1 October 2012 Communicated by Tony Pantev Abstract We consider an algebraic variety X together with the choice of a subvariety Z. We show that any coherent sheaf on X can be constructed out of a coherent sheaf on the formal neighborhood of Z, a coherent sheaf on the complement of Z, and an isomorphism between certain representative images of these two sheaves in the category of coherent sheaves on a Berkovich analytic space W which we define. ⃝c 2012 Elsevier Inc. All rights reserved. Keywords: Tubular descent; Berkovich analytic spaces 1. Introduction The aim of this paper is to study how a coherent sheaf F on a k-scheme of finite type X can be reconstructed from its restrictions to an open subvariety U and the complement Z D X − U, or informally speaking, how F can be glued from FU and FZ . Clearly, FU and FZ do not determine F, so one should suggest a richer descent datum which does determine F but uses as little as possible information beyond the pair .FU ; FZ /. Note that at the very least one should know the restrictions of F onto all closed subschemes whose reduction is Z, or equivalently one should know the formal completion FbZ of F along Z. Furthermore, even the knowledge of FbZ and FU is insufficient, so we will introduce an intermediate geometric space W that will play the role of the “intersection” of X D bX Z and ∗ Corresponding author.
    [Show full text]
  • Homotopy Theory for Rigid Analytic Varieties
    Helene Sigloch Homotopy Theory for Rigid Analytic Varieties Dissertation zur Erlangung des Doktorgrades der Fakultat¨ fur¨ Mathematik und Physik der Albert-Ludwigs-Universitat¨ Freiburg im Breisgau Marz¨ 2016 Dekan der Fakultat¨ fur¨ Mathematik und Physik: Prof. Dr. Dietmar Kr¨oner Erster Referent: Dr. Matthias Wendt Zweiter Referent: Prof. Dr. Joseph Ayoub Datum der Promotion: 25. Mai 2016 Contents Introduction 3 1 Rigid Analytic Varieties 11 1.1 Definitions . 12 1.2 Formal models and reduction . 20 1.3 Flatness, smoothness and ´etaleness . 23 2 Vector Bundles over Rigid Analytic Varieties 27 2.1 A word on topologies . 28 2.2 Serre{Swan for rigid analytic quasi-Stein varieties . 30 2.3 Line bundles . 37 2.4 Divisors . 38 3 Homotopy Invariance of Vector Bundles 41 3.1 Serre's problem and the Bass{Quillen conjecture . 41 3.2 Homotopy invariance . 42 3.3 Counterexamples . 44 3.4 Local Homotopy Invariance . 48 3.5 The case of line bundles . 56 3.5.1 B1-invariance of Pic . 57 1 3.5.2 Arig-invariance of Pic . 59 4 Homotopy Theory for Rigid Varieties 65 4.1 Model categories and homotopy theory . 65 4.2 Sites and completely decomposable structures . 70 4.3 Homotopy theories for rigid analytic varieties . 71 4.3.1 Relation to Ayoub's theory . 75 4.3.2 A trip to the zoo . 76 5 Classification of Vector Bundles 79 5.1 Classification of vector bundles: The classical results . 79 5.2 H-principles and homotopy sheaves . 82 5.3 The h-principle in A1-homotopy theory . 86 5.4 Classifying spaces .
    [Show full text]
  • PATCHING OVER BERKOVICH CURVES and QUADRATIC FORMS Vlerë Mehmeti
    PATCHING OVER BERKOVICH CURVES AND QUADRATIC FORMS Vlerë Mehmeti To cite this version: Vlerë Mehmeti. PATCHING OVER BERKOVICH CURVES AND QUADRATIC FORMS. Compo- sitio Mathematica, Foundation Compositio Mathematica, 2019. hal-02437192 HAL Id: hal-02437192 https://hal.archives-ouvertes.fr/hal-02437192 Submitted on 13 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PATCHING OVER BERKOVICH CURVES AND QUADRATIC FORMS VLERE¨ MEHMETI Abstract. We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local-global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the u-invariant. The patching method we adapt was introduced by Harbater and Hartmann in [18], and further developed by these two authors and Krashen in [19]. The results presented in this paper generalize those of [19] on the local-global principle and quadratic forms. Resum´ e.´ Recollement sur les courbes de Berkovich et formes quadratiques. Nous ´etendonsla technique de recollement sur les corps au cadre de la g´eom´etrieana- lytique de Berkovich pour d´emontrer un principe local-global sur les corps de fonctions de courbes analytiques par rapport `acertains de leurs compl´et´es.Dans le contexte des formes quadratiques, nous le combinons avec des conditions suffisantes d'isotropie locale sur une courbe de Berkovich pour obtenir des applications au u-invariant.
    [Show full text]
  • Topics in Algebraic Geometry I. Berkovich Spaces
    Math 731: Topics in Algebraic Geometry I Berkovich Spaces Mattias Jonsson∗ Fall 2016 Course Description Berkovich spaces are analogues of complex manifolds that appear when replacing complex numbers by the elements of a general valued field, e.g. p-adic numbers or formal Laurent series. They were introduced in the late 1980s by Vladimir Berkovich as a more honestly geometric alternative to the rigid spaces earlier conceived by Tate. In recent years, Berkovich spaces have seen a large and growing range of applications to complex analysis, tropical geometry, complex and arithmetic dynamics, the local Langlands program, Arakelov geometry, . The first part of the course will be devoted to the basic theory of Berkovich spaces (affinoids, gluing, analytifications). In the second part, we will discuss various applications or specialized topics, partly depending on the interest of the audience. Contents 1 September 6 5 1.1 Introduction..............................................5 1.2 Valued fields.............................................5 1.3 What is a space?...........................................7 1.4 The Berkovich affine space (as a topological space)........................7 1.4.1 The Berkovich affine line..................................8 2 September 8 9 2.1 Banach rings [Ber90, 1.1]......................................9 2.1.1 Seminormed groupsx .....................................9 2.1.2 Seminormed rings...................................... 10 2.1.3 Spectral radius and uniformization............................. 11 2.1.4 Seminormed modules..................................... 12 2.1.5 Complete tensor products.................................. 12 2.2 The spectrum [Ber90, 1.2]..................................... 12 x 3 September 13 14 3.1 The spectrum (continued) [Ber90, 1.2]............................... 14 3.2 Complex Banach algebras [Jon16, App.x G]............................. 15 3.3 The Gelfand transform [Ber90, pp.
    [Show full text]
  • Buildings and Berkovich Spaces
    Buildings and Berkovich Spaces Annette Werner Goethe-Universit¨at, Frankfurt am Main DMV-Jahrestagung Munchen¨ 2010 Emmy Noether Lecture 1 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 This talk reports on joint work with Amaury Thuillier and Bertrand R´emy(Lyon). Our results generalize results of Vladimir Berkovich who investigated the case of split groups. 2 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Non-archimedean fields K non-Archimedean field, i.e. K is complete with respect to a non-trivial absolute value j jK satisfying ja + bjK 5 maxfjajK ; jbjK g: ∗ ∗ K is called discrete if the value group jK j ⊂ R is discrete. Non-archimedean analysis has special charms: 1 X an converges if and only if an ! 0: n=1 3 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Non-archimedean fields Examples: K = k(T ) formal Laurent series over any ground field k with j P a T nj = e−n0 if a 6= 0 n≥n0 n n0 K = CffT gg Puiseux series −vp(x) K = Qp, the completion of Q with respect to jxj = p algebraic extensions of Qp K = Cp, the completion of the algebraic closure of Qp 4 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Semisimple groups G semisimple group over K, i.e. G ,! GLn;K closed algebraic subgroup such that rad(G)(= biggest connected solvable normal subgroup) = 1 Examples: SLn; PGLn; Sp2n; SOn over K SLn(D) D central division algebra over K 5 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Main result Goal: Embed the Bruhat-Tits building B(G; K) associated to G in the Berkovich analytic space G an associated to G.
    [Show full text]
  • Berkovich Spaces Embed in Euclidean Spaces (.Pdf)
    BERKOVICH SPACES EMBED IN EUCLIDEAN SPACES EHUD HRUSHOVSKI, FRANÇOIS LOESER, AND BJORN POONEN Abstract. Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V an of any d-dimensional quasi-projective scheme V over K embeds in R2d+1. If, moreover, the value group of K is dense in R>0 and V is a curve, then we describe the homeomorphism type of V an by using the theory of local dendrites. 1. Introduction In this article, valued field will mean a field K equipped with a nonarchimedean absolute value j j (or equivalently with a valuation taking values in an additive subgroup of R). Let K be a complete valued field. Let V be a quasi-projective K-scheme. The associated Berkovich space V an [Ber90, §3.4] is a topological space that serves as a nonarchimedean analogue of the complex analytic space associated to a complex variety. (Actually, V an carries more structure, but it is only the underlying topological space that concerns us here.) Although the set V (K) in its natural topology is totally disconnected, V an is arcwise connected if and only if V is connected; moreover, the topological dimension of V an equals the dimension of the scheme V [Ber90, Theorems 3.4.8(iii,iv) and 3.5.3(iii,iv)]. Also, V an is locally contractible: see [Ber99,Ber04] for the smooth case, and [HL12, Theorem 13.4.1] for the general case.
    [Show full text]
  • Non-Archimedean Analytic Spaces 1 Introduction
    Non-archimedean analytic spaces Annette Werner Abstract: This paper provides an elementary introduction to Vladimir Berkovich's theory of analytic spaces over non-archimedean fields, focussing on topological aspects. We also discuss realizations of Bruhat-Tits buildings in non-archimedean groups and flag varieties. MSC(2010):14G20, 11G25, 32P05, 20E42 Keywords: Non-archimedean analytic spaces, Berkovich spaces, Bruhat-Tits buildings 1 Introduction About two decades ago, Vladimir Berkovich introduced a new approach to analytic ge- ometry over non-archimedean fields. At this time, Tate's theory of rigid analytic spaces was quite well developed and established. Rigid analytic spaces, however, have rather poor topological properties. Berkovich's analytic spaces contain more points, which leads for example to better connectivity properties. Meanwhile, Berkovich's approach to non- archimedean analytic geometry has become an active area of research with important application to various branches of geometry. This text aims at providing a very gentle introduction to the nice topological properties of Berkovich spaces. We hope to convey some of the fascination of non-archimedean geometry to researchers from other areas. Due to the introductory nature of this text, we focus on topological aspects. We start with a discussion of fields with non-archimedean absolute values, pointing out features which are different from the probably better known archimedean situation. Then we discuss in detail the structure of the Berkovich unit disc. Whereas the rigid analytic unit disc is the set of maximal ideals in the Tate algebra, the Berkovich unit disc is a set of suitable seminorms on the Tate algebra. This includes seminorms whose kernels are not maximal ideals.
    [Show full text]