Hurwitz trees in non-Archimedean analytic geometry 

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Hurwitz trees and the Berkovich pointed unit disc

Hurwitz trees are mathematical objects parametrizing local actions in characteristic zero. The properties of order p automorphisms of the p-adic disc studied by Green-Matignon ([28]) and Raynaud ([52]) has been encoded combinatorially in an article by Henrio ([31]) by associating to such an automorphism 2 AutRR[[T]] a rooted metric tree H with additional information, encoding the position of the ramication points and the reduction of the action of . The tree is classically called Hurwitz tree, and the information, Hurwitz data. The existence and mutual compatibility of Hurwitz data is a necessary and sucient condition to the existence of an automorphism giving rise to that Hurwitz tree.
Brewis and Wewers ([14]) extended the denition of Hurwitz tree in order to describe local actions of any nite group G. In their denition, Hurwitz data are dened as characters arising from representations of G, describing the ramication theory of the local action in characteristic zero. They are called the depth character and the Artin character. Not all Hurwitz data are generalized in this denition, and the question if a Hurwitz tree in the sense of [31] can be dened for any nite group remains open.
In chapter 5, we characterize the Hurwitz tree as a non-Archimedean analytic object, in the sense of Berkovich. We start by proving the following result Theorem 1.0.11. Let T be the Hurwitz tree associated to the local action of a nite group in characteristic zero : G ,! AutR(R[[T]]). Then there is a metric embedding T ,!M(KfTg) of the Hurwitz tree in the Berkovich closed unit disc D(0; 1) such that the image is contained in the set of points xed by the action on M(KfTg).
This theorem permits the identication of a vertex v 2 V (T) with a closed disc D(v), and of an edge e 2 E(T) with an open disc D(e). We exploit this identication to translate Hurwitz data in analytic terms. We rst prove that the depth character v of [14] coincides with the evaluation of the analytic function (T) 􀀀 T on the point corresponding to v in the embedded Hurwitz tree. We then give a similar formulation for the Artin character ae. The identication permits to characterize the groups Gv as the stabilizers of closed discs D(v), and helps proving the following theorem, that gives an analytic description of good deformation data Theorem 1.0.12. Let : Z=pZ ,! Aut(R[[T]]) be a local action in characteristic zero, having L as set of xed points. Then there exists a metrized line bundle on D(0; 1) n L, such that the good deformation datum associated to a vertex v 2 V (T) is a section of the (Temkin) reduction ] ; v at v.

Weil representation and metaplectic group

In the paper [18] we dene and describe explicitly the Weil representation over any integral domain. This representation has been introduced together with the metaplectic group by Andre Weil in his Acta paper [62] in order to shed light on the results of Siegel on theta functions and to formulate them in an adelic setting. This led to various developments in number theory, for example the work of Shimura on modular forms of half-integral weight and the one of Jacquet and Langlands on automorphic representations of adele groups.
The construction of Weil is as follows: he considers a local eld F, a nite dimensional F-vector space X and the symplectic group Sp(W) over W = X X. He lets T be the multiplicative group of complex numbers of unitary absolute value and : F ! T be a non-trivial continuous character. Using he shows the existence of an action of Sp(W) over the Heisenberg group, dened up to multiplication by an element of T . Therefore he constructs a cover Mp(W) of Sp(W) such that ‘ lifts to a complex innite representation of Mp(W), the now-so-called Weil representation. Finally he shows that Mp(W) contains properly a double cover of Sp(W) on which the Weil representation can be restricted.
In chapter 6, we suppose that F is non-Archimedean of characteristic 6= 2 and residue eld of cardinality q = pe. We replace T by an integral domain R such that p 2 R, R contains a square root of q and pn-th roots of unity for every n, to ensure the existence of a nontrivial smooth character : F ! R. In this generality we are able to reproduce the results of Weil showing the existence of the reduced metaplectic group, dened in the following way. Firstly we construct the metaplectic group Mp(W) in such a way to have a non-split short exact sequence 1 􀀀! R 􀀀! Mp(W)􀀀!Sp(W) 􀀀! 1.

General k-analytic spaces, G-topology and regular functions

The anoid spaces are the local bricks that form general k-analytic spaces in the sense of Berkovich. The process of glueing is explained in [5], section 1.2. In the same paper, in section 1.3, the G-topology is dened over a k-analytic space, as well as a structure sheaf for this Grothendieck topology. We brie y review in this section the results that are necessary for our constructions. Since anoid spaces are compact, one can not perform glueing in the same way as scheme theoretically, namely by glueing locally ringed spaces. Berkovich construction takes into account this concern and relies on the notion of quasi-net and anoid atlas. Denition 2.1.8. A quasi-net on a locally separated topological space X is a collection of compact separated subsets of X such that each x 2 X has a neighborhood of the form [Vi for nitely many Vi 2 , with the property that x 2 \Vi. The notion of quasi-net is dened in such a way to provide a substitute of the notion of open covering in this setting.

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Functions on analytic curves

One of the features that is peculiar of Berkovich spaces with respect to other non-Archimedean analytic theories, is the natural presence of combinatorial structures. This is strictly related to the theory of regular functions and, more in general, sheaves of modules. We introduce in this section the notion of skeleton and the one of vector bundle on a k-analytic curve, pointing out the relations between these two objects.

The skeleton of a curve

Let C be a smooth, proper, connected k-analytic curve. If C is any ko-model with semi-stable reduction for C, there is an associated subset S = SC of C called the skeleton of C. Berkovich proved that the skeleton of C is homeomorphic to the dual graph 􀀀(Cs). It can therefore inherit a metric structure: we may think at S as a nite graph in which each edge is identied to anEuclidean line segment of some length, possibly innite. By results of the previous section on dual graphs, S is a subset of C. Berkovich proved also that SC is a deformation retract of C. Denition 2.4.1. Let X be a k-anoid space. We say that it is potentially isomorphic to the unit disc if there exists a nite separable extension k0 of k such that the k0-anoid space k k0 it is isomorphic to a nite sum of copies of the closed unit disc over k0. Lemma 2.4.2. A connected k anoid domain V A1;an k is potentially isomorphic to the unit disc if and only if its Shilov boundary is a singleton, @SV = fg, such that V coincides with the complement of the unique unbounded connected component of A1;an k n fg.
Denition 2.4.3. Let C be a ko-curve with simple semi-stable reduction, and let O(C) be the subset of C 􀀀 @S(C) whose elements are points having an anoid neighborhood potentially isomorphic to the unit disc. The skeleton of C is dened to be the complement SC = O(C)c. Remark 2.4.4. The denition of the skeleton by Berkovich is given for smooth k-analytic curves admitting distinguished formal coverings with semi-stable reduction in section 4.3 of [4], and generalized for nondegenerate pluri-stable formal schemes in chapter 4 of [6]. The characterization above for models with simple semi-stable reduction is due to Thuillier ([59]) and ts better to our purposes.

The Hurwitz tree for Z=pZ

We describe in this section the role of the Hurwitz tree as an object parametrizing local actions in characteristic zero in such a way to keep track of their reduction properties. Notations: basics on trees In what follows, the word tree denotes a nite oriented rooted tree T . This means that the set V (T ) of its vertices is endowed with a partial order relation, denoted by v ! v0, in such a way that the following conditions are satised:
there exists a unique v0 2 V (T ) with v0 ! v for every v 2 V (T ) .
for every couple v; v0 2 V (T ) such that v ! v0 and such that there is no v00 with v ! v00 ! v0 there is an unique edge connecting v and v0 .
conversely, for each edge of T joining two vertices v; v0 2 V (T ) these are such that v ! v0 and such that there is no v00 with v ! v00 ! v0. The vertex v0 is called the root of the tree. A vertex v such that there is no v0 2 V (T ) with v ! v0 is called a leaf. The couple associated to an edge e of T is noted vs e ! vte and they are respectively called starting vertex and ending vertex of e. Finally, we set e􀀀v for the unique edge having v as ending vertex and E+(v) = fe 2 E(T) : vs e = vg.

Table of contents :

1 Introduction 
2 Non-Archimedean analytic geometry 
2.1 Berkovich spaces
2.1.1 Construction and algebraic description
2.1.2 General k-analytic spaces, G-topology and regular functions
2.2 Reduction techniques
2.2.1 The Berkovich generic ber of a formal scheme
2.2.2 Boundaries
2.2.3 Reduction a la Temkin
2.3 Curves
2.3.1 Types of points
2.3.2 Semi-stable reduction
2.4 Functions on analytic curves
2.4.1 The skeleton of a curve
2.4.2 Vector bundles
3 Local actions on curves 
3.1 Local and global actions on curves
3.2 Lifting local actions
3.2.1 Geometric local actions
3.2.2 Local actions on boundaries
3.3 The Hurwitz tree for Z=pZ
3.3.1 The Hurwitz tree associated to an automorphism of order p
3.3.2 Good deformation data
3.4 Hurwitz trees of general type
3.4.1 Denition of Hurwitz tree
3.4.2 Comparaison with the Hurwitz tree for Z=pZ
3.5 The elementary abelian case
4 Explicit calculations 
4.1 Combinatorial rigidity of Hurwitz tree
4.1.1 Lifting actions to the closed unit disc
4.2 Lifting actions of elementary abelian p-groups
4.2.1 Lifting intermediate Z=pZ-extensions
4.2.2 Fp-vector spaces of multiplicative good deformation data
4.2.3 Actions of Z=3Z Z=3Z
4.2.4 The case \(4,1) »
5 Hurwitz trees in non-Archimedean analytic geometry 
5.1 Automorphisms of open and closed analytic unit discs
5.2 The Berkovich-Hurwitz tree
5.2.1 The embedding
5.2.2 Translation of Hurwitz data
5.3 Analytic good deformation data
5.3.1 Good deformation data and the analytic sheaf of deformations
5.4 Characterizations of the Hurwitz tree
5.4.1 Dynamical properties
5.4.2 The Berkovich-Hurwitz tree as tropicalization
5.4.3 Covers of Berkovich curves and metric structure of the Hurwitz tree
6 Weil representation and metaplectic groups over an integral domain 
6.1 Notation and denitions
6.2 The metaplectic group
6.2.1 The group B0(W)
6.2.2 The group B0(W)
6.2.3 The metaplectic group
6.3 The Weil factor
6.3.1 The Weil factor
6.3.2 Metaplectic realizations of forms
6.4 Fundamental properties of the Weil factor
6.4.1 The quaternion division algebra over F
6.4.2 The Witt group
6.4.3 The image of the Weil factor
6.5 The reduced metaplectic group


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