Ramification theory of the p-adic open disc and the lifting problem [Elektronische Ressource] / vorgelegt von Louis Hugo Brewis

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• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2009
• Nombre de lectures: 13
• Langue: English

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UniversitätUlm
InstitütfürReineMathematik
Ramificationtheoryofthe p-adicopendiscandthe
liftingproblem
Dissertation
zurErlangungdesDoktorgrades
Dr. rer. nat.
derFakultätfürMathematikundWirtschaftswissenschaften
derUniversitätUlm
vorgelegtvon
LouisHugoBrewis
ausKapstadt
Ulm,2009Amtierender Dekan: Prof. Dr. Werner Kratz
1. Gutachter: Prof. Dr. Irene Bouw
2. Prof. Dr. Werner Lütkebohmert
3. Gutachter: Prof. Dr. Stefan Wewers
Tag der Promotion: 3. Juli 2009Contents
Introduction 7
1 Swan conductors I : Kato’s differential character 13
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Assumption and setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Kato’s Swan conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 The value group of Kato’s Swan conductor . . . . . . . . . . . . . . . . . . 15
1.3.2 Order functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.3 Kato’s Swan conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.4 Functorial properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Z=pZ-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Kato’s Hasse–Arf theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Vector space property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
21.7.1 Example : (Z=pZ) -extensions . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.2 : generalized quaternion extensions . . . . . . . . . . . . . . . . . 25
1.7.3 Example : depths at double points . . . . . . . . . . . . . . . . . . . . . . . 26
2 Swan conductors II: Ramification groups 29
2.1 Notation and setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Ramification groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
^2.3 The simplified ramification groups G . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
^ ^2.3.2 The quotients G =G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37n n+1
2.4 Artin and depth characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 Relation to Kato’s Swan conductor . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.2 The upper ramification jumps . . . . . . . . . . . . . . . . . . . . . . . . . 42
22.4.3 Example : G = (Z=pZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 The local vector-space theorem: trivial filtration case . . . . . . . . . . . . . . . . . 45
2.6 The local vector general case . . . . . . . . . . . . . . . . . . . . . 46
3 Group actions on the open disc 49
3.1 Notations and setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Hurwitz trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 The multiplicative character . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Metric trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
34 CONTENTS
3.2.3 Hurwitz trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Group actions on the disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.2 The depth character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.3 The Artin character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.4 Relation between the depth and Artin characters . . . . . . . . . . . . . . . 57
3.5 Definition of the Hurwitz tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Applications to the lifting problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6.1 A new obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6.2 Simple quaternion actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6.3 Example : G =Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
3.7 Theorem of Green–Matignon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Hurwitz-tree obstruction to cyclic actions 67
4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
n4.2 Artin characters of Z=p Z-Galois extensions . . . . . . . . . . . . . . . . . . . . . 68
4.3 Vanishing of obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.1 Underlying tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.2 Monodromy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.3 Thicknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.4 Depth characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Towards the lifting problem 79
5.1 Notation and setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Towards differential Hurwitz trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.1 Example : nonlogarithmic nonexact differential Swan conductor . . . . . . . 80
5.2.2 : special extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Classification problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.1 Definitions of the extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.2 Galois theory of E =F and E =F . . . . . . . . . . . . . . . . . . . . . . 851 2
5.3.3 Conjugation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3.4 The Swan conductors of the extensions . . . . . . . . . . . . . . . . . . . . 87
5.4 The role of the simplified ramification groups . . . . . . . . . . . . . . . . . . . . . 89
6 Dihedral actions 93
6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Some Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.1 General Galois theory of -extensions . . . . . . . . . . . . . . . . . . . . 94D4
6.2.2 Cohomological Galois theory of fields . . . . . . . . . . . . . . . . . . . . . 95
6.2.3 Artin–Schreier theory of power series fields in characteristic 2 . . . . . . . . 98
6.2.4 Supersimple -extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 99D4
6.2.5 Classifying supersimple -extensions . . . . . . . . . . . . . . . . . . . . 101D4
6.3 Good reduction of Galois closures . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Lifting supersimple -actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106D4
6.5 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110CONTENTS 5
Bibliography 113
Acknowledgements 117
Zusammenfassung (Deutsch) 1196 CONTENTSIntroduction
In this thesis we study Galois covers of algebraic curves. In characteristic 0 , the Galois theory
of curves is well understood. This is in strong contrast to the situation in characteristic p where
less is known. One technique for studying Galois covers in characteristic p is to relate them to the
Galois theory of curves in characteristic 0 . Reducing Galois covers of curves in characteristic 0 to
characteristic p is one way to prove existence of Galois covers in characteristic p with a given Galois
group. In the opposite direction, the lifting problem asks which covers of curves in characteristic p
lift to characteristic 0 . It is known that not all Galois covers in p lift to
0 . In our work we prove results regarding the liftability of Galois covers of algebraic curves. For some
Galois covers in characteristic 2 , we explicitly construct lifts to characteristic 0 . We also introduce a
new necessary condition for the liftability of Galois covers. We then use our new necessary condition
to show that certain Galois covers do not lift to characteristic 0 .
Background
It is known that the set of Galois covers of a punctured Riemann surface can be described in terms of
its topological fundamental group. Indeed, if S is a surface of genus g with n punctures,
then the quotients of its topological fundamental group correspond to the Galois covers of S .g;n
An explicit description of the group is known in terms of the genus g and theg;n
number of punctures n , namely it is the free profinite group on 2g +n 1 generators for n> 0 .
This theory can also be used to describe the Galois theory of curves defined over the complex num-
bers. This is accomplished by using algebraization techniques, which essentially state that there is an
equivalence between the category of compact Riemann surfaces and the category of smooth projec-
tive algebraic curves defined over the complex numbers. This correspondence respects finite branched
covers. Furthermore, one knows that inertia groups of a characteristic- 0 Galois cover D! C , i.e.
the stabilizers of the fixed points of D , are always cyclic.
The Galois theory of curves becomes more difficult in characteristic p . One problem is that th