On representations attached to semistable vector bundles on Mumford curves [Elektronische Ressource] / vorgelegt von Gabriel Herz

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Gabriel HerzOn representations attached to semistable vectorbundles on Mumford curves{2005{Reine MathematikOn representations attached to semistable vector bundles onMumford curvesInaugural-Dissertation zur Erlangung des Doktorgrades derNaturwissenschaften im Fachbereich Mathematik und Informatik derMathematisch-Naturwissenschaftlichen Fakult at der Westf alischenWilhelms{Universit at Munstervorgelegt vonGabriel Herzaus Dusseldorf{2005{Dekan: Prof. Dr. Klaus HinrichsErster Gutachter: Prof. Dr. Christopher DeningerZweiter Gutachter: Prof. Dr. Annette Werner(Universit at Stuttgart)Tag der mundlic hen Prufung: 30. Juni 2005Tag der Promotion: 13. Juli 2005Deutschsprachige ZusammenfassungUber Darstellungen, die semistabilen Vektorbundeln aufMumfordkurven zugeordnet sindIn vorliegender Arbeit werden die beiden folgenden Konstruktionen vonDarstellungen miteinander verglichen.Sei X eine Mumfordkurve ub er einer endlichen K orpererweiterung von Q .pGerd Faltings konstruiert zu jedem semistabilen Vektorbundel vom Grade0 ub er X eine Darstellung der Schottkygruppe von X. Marius van der Putund Marc Reversat k onnen diese Konstruktion auf Mumfordkurven ub ernicht notwendigerweise diskret bewerteten nicht-Archimedischen K orpernverallgemeinern.Sei X eine glatte und projektive algebraische Kurve ub er einer endlichenErweiterung von Q .

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• Mathematics

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• Publié par: westfalische_wilhelms-universitat_munster
• Publié le: 01 janvier 2005
• Nombre de lectures: 36
• Langue: Deutsch

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On

Gabriel

Herz

representations attached to

bundles

on Mumford –2005–

semistable

curves

vector

Reine Mathematik

On representations attached to semistable vector bundles on Mumford curves

Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-NaturwissenschaftlichenFakult¨atderWestf¨alischen Wilhelms–Universit¨atM¨te uns r

vorgelegt von Gabriel Herz ausD¨usseldorf –2005–

Dekan: Erster Gutachter: Zweiter Gutachter:

Tagdermu¨ndlichen Tag der Promotion:

Pr¨ufung:

Prof. Dr. Klaus Hinrichs Prof. Dr. Christopher Deninger Prof. Dr. Annette Werner (Universita¨tStuttgart) 30. Juni 2005 13. Juli 2005

Deutschsprachige Zusammenfassung ¨ UberDarstellungen,diesemistabilenVektorbu¨ndelnauf Mumfordkurven zugeordnet sind In vorliegender Arbeit werden die beiden folgenden Konstruktionen von Darstellungen miteinander verglichen. SeiXerweiterunnKg¨voornperneldciehebernireurdk¨uveMuneormfieQ p . GerdFaltingskonstruiertzujedemsemistabilenVektorbu¨ndelvomGrade 0¨uberXeine Darstellung der Schottkygruppe vonX van der Put. Marius undMarcReversatk¨onnendieseKonstruktionaufMumfordkurven¨uber nichtnotwendigerweisediskretbewertetennicht-ArchimedischenK¨orpern verallgemeinern. SeiXenchlindreneeieralgeuettnieuKvr¨eburbiacsehtivealgendprojek Erweiterung vonQ p dtieinfienere-nAAnbrenenhceiehsrzlick¨uriner.Ine netteWernerundChristopherDeningereinen´etalenParalleltransportfu¨r einegewisseKlassevonVektorbu¨ndelnaufX C p ¨uhrionfersuketKo.nDsitt unmittelbar zu einem Funktor von dieser Klasse in die Kategorie der steti- genC p -Vektorraumdarstellungen der algebraischen Fundamentalgruppe von X C p .

Im Falle, daßXmuMenievrukdrofeierube¨lindrenehcnerEewtirenug vonQ p ,riwdßadegeinnannKtestonktruneioist,beweisenelelzipeesinre¨unf KlassevonsemistabilenVektorb¨undelnvomGrade0u¨berX C p isomorphe Darstellungen definieren. Wichtige Hilfsmittel zum Beweis dieser Aussage sind GAGA-Resultate zwis- chen rigider, formeller und algebraischer Geometrie sowie die Fundamental- ¨ gruppe der endlichen topologischen Uberlagerungen, welche in vorliegender Arbeit definiert und untersucht wird.

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Contents

Deutschsprachige Zusammenfassung Introduction Chapter 1. Preliminaries 1. Notations and conventions 2. Reduction and formal rigid spaces 3. GAGA-results 4. Fundamental groups of analytic manifolds 5. Mumford curves Chapter 2. Representations attached to vector bundles 1. The constructions of Faltings, van der Put–Reversat and Deninger–Werner 2. Comparison of the constructions Chapter 3. Applications 1. Fundamental groups of a Tate curve 2. Vector bundles on a Tate curve 3. Vector bundles on a Mumford curve of genus 2 Bibliography

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5 9 13 13 14 18 24 32 37 37 45 55 55 56 57 59

Introduction

AclassicalresultbyAndre´Weil,provedin1938,assertsthataholo- morphic vector bundle on a Riemann surface is given by a representation of the fundamental group if and only if each indecomposable component is of degree zero (cf. [Wei38]). Furthermore it is known that unitary rep- resentations of the fundamental group are in one-to-one correspondance to polystable vector bundles of degree 0 (cf. [NS65]). In 1983 Gerd Faltings introduced the notion ofφ-boundedrepresenta- tions and proved a corresponding result inp-adic analysis for vector bundles on a Mumford curve over a discrete non-Archimedean field. He proved an equivalence of categories between semistable vector bundles of degree zero on a Mumford curve andφ-bounded representations of its Schottky group. In his proof he used the theory of formal schemes and was therefore limited to discrete fields. In 1986 Marius van der Put and Marc Reversat generalised Faltings’ result to arbitrary non-Archimedean fields by using methods from rigid geometry. Unfortunately the functor they constructed does not com- mute with duals or tensor products. In the case of line bundles it is easy to see that this is inherently caused by the definiton ofφ the-boundedness. In sequel we will call their representationsPR-representations. LetXa projective, smooth and geometrically connected algebraic curvebe over a finite field extension ofQ p . In 2004 Annette Werner and Christopher Deningerconstructedan´etaleparalleltransportforvectorbundlesofacer- tain subcategory of all semistable vector bundles of degree zero onX C p (cf. [DWc]). Restricted to the algebraic fundamental group this is a functor that associates a continuousC p -vector space representation (in the follow- ing calledDW-representation) of the algebraic fundamental group ofX C p to every vector bundle of this class. This functor isC p -linear, exact and commutes with duals, tensor products, internal homs and exterior powers. LetXcurve over a finite field extension ofbe a Mumford Q p  this. In thesis we compare the DW-representations attached to a class of semistable vector bundles of degree zero onX C p to the PR-representations defined for this class of vector bundles. Performing this, obvious problems occur: 9

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INTRODUCTION (1) it is not known whether the DW-representation does exist for all semistable vector bundles of degree zero. (2) Different groups are represented. (3) The functors of Deninger–Werner and van der Put–Reversat have different properties. For example the DW-functor commutes with duals and tensor products, the PR-functor in general does not. The solution of all of these problems is (1) to consider only the subcategory of semistable vector bundles of degree 0 that have a vector bundle model on the extension of the minimal regular model ofXto the ring of integersoofC p , (2) to introduce the notion of finite topological coverings and the finite topological fundamental group ofX (3) and at last to prove that the DW-representation factorises through this fundamental group. Having done this, we can prove that the DW-representation and the pro- finitely completed PR-representation attached to vector bundles in the afore- said class are isomorphic. The PR-representations attached to vector bun- dles in this class are isomorphic to representations which have image in GL rk (o) if rk is the rank of the vector bundle considered. At least for line bundles the aforesaid class is the best possible on which both representations agree, since for line bundles whose PR-representation is not represented by numbers of norm equal to one, the PR-representation does not commute with tensor products and duals, but the DW-functor does. Introducing the fundamental group of finite topological coverings suggests itself, since we have to compare representations of the topological funda- mental group (that is the Schottky group) with representations of the finite ´etalefundamentalgroup.Thecoveringswhicharetopologicalandfinite e´taleareexactlythefinitetopologicalcoverings. We prove our result by extensive use of GAGA theorems between rigid, for - mal and algebraic geometry. Because we have some GAGA results only in the case of discrete valuation we reduce the case that the vector bundle is only defined after extension toC p to the case that it is already defined over a discrete valuation field by an argument from non-abelian cohomology. As an application we will have a closer look at Mumford curves of genus 1 and 2 and at vector bundles on them. This thesis is organised as follows. In the first chapter we remind the reader of some notions of rigid and analytic spaces which are important for us. We cite some GAGA results and prove slight extensions of them. Es- pecially quotients of schemes by finite groups are considered more closely.

INTRODUCTION 11 We state the basic notions of Galois theory, introduce the finite topological fundamental group and prove that it satisfies the six axioms of Grothendieck for a Galois theory. At the end of the first chapter we introduce Mumford curves and discuss their stable and minimal regular model. In the second chapter we describe the constructions of van der Put and Reversat, Faltings and Deninger–Werner. We characterise the semistable vector bundles of degree zero which have a vector bundle model on the minimal reguar model of the Mumford curve by their PR-representation and we compare the construction of Faltings with the one of van der Put– Reversat. In the next section we compare the DW-representation with the PR-representation. We deduce the case that the vector bundle is only de- fined after base change toC p from the case that it is already defined over a discrete valuation field and prove this case first. In the last chapter we give some applications. We study the various Galois groups of a Tate curve and investigate vector bundles on Tate curves and on Mumford curves of genus 2 in more detail. I would like to thank my supervisior Prof. Christopher Deninger and Prof. Annette Werner for introducing my to this interesting topic. The final version benefited from discussions with Jan Kohlhaase, Sylvain Maugeais, Roland Olbricht, Matthias Strauch and Stefan Wiech. I am grateful that Sylvain Maugeais and Stefan Wiech read a preliminary version. This thesis was partially supported by theDeutsche Forschungsgemeinschaft at the SFB 478Geometrische Strukturen in der Mathematik.