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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
