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9 results on '"Amaury Thuillier"'

1. Wonderful compactifications of Bruhat-Tits buildings

Author

Bertrand Remy, Amaury Thuillier, and Annette Werner

Subjects
mathematics - group theory, mathematics - algebraic geometry, Mathematics, QA1-939
Abstract

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

Published
2017
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2. AN INTRINSIC CHARACTERIZATION OF BRUHAT-TITS BUILDINGS INSIDE ANALYTIC GROUPS

Author

Bertrand Rémy, Amaury Thuillier, Annette Werner, REMY, Bertrand, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), Institut fur Mathematik Goethe Universität Frankfurt-am-Main, Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Algèbre, géométrie, logique (AGL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Institut Camille Jordan [Villeurbanne] (ICJ), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
14G22, General Mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Berkovich analytic geometry, Group Theory (math.GR), [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 20E42, 51E24, 14L15, 14G22, Mathematics - Algebraic Geometry, 51E24, Bruhat-Tits building AMS classification (2000): 20E42, FOS: Mathematics, valued field, algebraic group, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics::Representation Theory, Algebraic Geometry (math.AG), Mathematics - Group Theory, 14L15, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]
Abstract

Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding., 12 pages, 1 figure. arXiv admin note: text overlap with arXiv:1110.1362

Published
2021

3. Wonderful compactifications of Bruhat-Tits buildings

Author

Annette Werner, Bertrand Rémy, and Amaury Thuillier

Subjects
Pure mathematics, Algebra and Number Theory, Analytic space, Mathematics::General Topology, Group Theory (math.GR), Ground field, Mathematics::Group Theory, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Algebraic group, Euclidean geometry, FOS: Mathematics, Embedding, Geometry and Topology, Compactification (mathematics), Mathematics::Representation Theory, Local field, Algebraic Geometry (math.AG), Mathematics - Group Theory, Mathematics
Abstract

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

Published
2017

4. Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings

Author

Annette Werner, Amaury Thuillier, and Bertrand Rémy

Subjects
Linear algebraic group, Pure mathematics, Absolutely irreducible, General Mathematics, 010102 general mathematics, General linear group, 01 natural sciences, 010101 applied mathematics, Analytic geometry, Algebraic group, Embedding, Equivariant map, Compactification (mathematics), 0101 mathematics, Mathematics
Abstract

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

Published
2011

5. Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Author

Amaury Thuillier, Bertrand Rémy, and Annette Werner

Subjects
Pure mathematics, Analytic space, Bruhat decomposition, General Mathematics, Algebraic group, Compactification (mathematics), Local field, Mathematics
Abstract

We investigate Bruhat-Tits buildings and their compactific ations by means of Berkovich analytic ge- ometry over complete non-Archimedean fields. For every redu ctive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G, k) to the Berkovich analytic space G an asscociated with G. Composing this map with the projection of G an to its flag varieties, we define a family of compactifi- cations of B(G, k). This generalizes results by Berkovich in the case of split g roups. Moreover, we show that the boundary strata of the compactifie d buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them.

Published
2010

6. Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels

Author

Amaury Thuillier

Subjects
Pure mathematics, Number theory, Analytic geometry, General Mathematics, Homotopy, Elementary proof, Perfect field, Algebraic geometry, Algebraic number, Invariant (mathematics), Mathematics
Abstract

V.G. Berkovich’s non-Archimedean analytic geometry provides a natural framework to understand the combinatorial aspects in the theory of toric varieties and toroidal embeddings. This point of view leads to a conceptual and elementary proof of the following results: if X is an algebraic scheme over a perfect field and if D is the exceptional normal crossing divisor of a resolution of the singularities of X, the homotopy type of the incidence complex of D is an invariant of X. This is a generalization of a theorem due to D. Stepanov.

Published
2007

7. Bruhat-Tits Buildings and Analytic Geometry

Author

Annette Werner, Bertrand Rémy, and Amaury Thuillier

Subjects
Algebra, Galois geometry, Analytic geometry, Algebraic group, Special linear group, Compactification (mathematics), Algebraic geometry and analytic geometry, Mathematics
Abstract

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group.

Published
2014

8. Automorphisms of Drinfeld half-spaces over a finite field

Author

Annette Werner, Bertrand Rémy, Amaury Thuillier, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut für Mathematik, and Goethe-Universität Frankfurt am Main

Subjects
Automorphism group, Pure mathematics, Algebra and Number Theory, Berkovich spaces, 010102 general mathematics, birational transformations, Base field, Automorphism, 01 natural sciences, 14G22, 14G15, 14E05, Mathematics - Algebraic Geometry, Finite field, Analytic geometry, 0103 physical sciences, Drinfeld upper half-space, FOS: Mathematics, Non-archimedean analytic geometry, 010307 mathematical physics, Projective linear group, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Valuation (algebra), Mathematics, Vector space
Abstract

We show that the automorphism group of Drinfeld's half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field., Comment: 15 pages

Published
2012

9. Mesures de Mahler et équidistribution logarithmique

Author

Antoine Chambert-Loir, Amaury Thuillier, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Chambert-Loir, Antoine, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics::Number Theory, 01 natural sciences, Mesure de Mahler, [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT], Mathematics - Algebraic Geometry, 14G40, 14G22, 32U05, Mathematics::Algebraic Geometry, 0103 physical sciences, systèmes dynamiques, 0101 mathematics, Mathematics, équidistribution, Algebra and Number Theory, Mathematics - Number Theory, 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], espaces analytiques de Berkovich, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], points de petite hauteur, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG], géométrie d'Arakelov, 010307 mathematical physics, Geometry and Topology, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Humanities, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Abstract

Let X be a projective integral normal scheme over a number field F; let L be a ample line bundle on X together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are 1) A formula which computes the height (relative to L) of a Cartier divisor D on X as generalized "Mahler measures", i.e. by integrating Green functions for D against measures attached to L. 2) a theorem of equidistribution of points of "small" height valid for functions with logarithmic singularities along a divisor D, provided the height of D is "minimal". In the context of algebraic dynamics, "small" means of height converging to 0, and "minimal" means height 0., Comment: 33 pages, in french

Published
2009

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