DOCSLIB.ORG

Deux Applications Arithmétiques Des Travaux D'arthur

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Deux Applications Arithmétiques Des Travaux D'arthur Deux applications arithmétiques des travaux d’Arthur Olivier Taïbi To cite this version: Olivier Taïbi. Deux applications arithmétiques des travaux d’Arthur. Théorie des nombres [math.NT]. Ecole Polytechnique X, 2014. Français. pastel-01066463 HAL Id: pastel-01066463 https://pastel.archives-ouvertes.fr/pastel-01066463 Submitted on 20 Sep 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. École Doctorale 447 Thèse de doctorat Discipline : Mathématiques présentée par Olivier Taïbi Deux applications arithmétiques des travaux d’Arthur dirigée par Gaëtan Chenevier Soutenue le 19 septembre 2014 devant le jury composé de : Gaëtan Chenevier École Polytechnique Directeur de thèse Laurent Clozel Université Paris XI Examinateur Guy Henniart Université Paris XI Examinateur Colette Mœglin Université Paris VI Rapporteuse David Renard École Polytechnique Examinateur Jean-Loup Waldspurger Université Paris VI Examinateur Rapporteurs absents lors de la soutenance : Toby Gee Imperial College London Rapporteur Benedict Gross Harvard University Rapporteur Remerciements Je tiens en premier lieu à exprimer ma reconnaissance à Gaëtan Chenevier pour avoir dirigé ma thèse. Travailler sous sa direction fut un réel plaisir. Au-delà de ses grandes connais- sances et de sa disponibilité, j’ai bénéficié de son enthousiasme et de ses encouragements. Sa hauteur de vue lui permet de motiver des objets sophistiqués par des applications arith- métiques simples et élégantes. J’ai été frappé par cette qualité, précieuse pour un thésard, lorsque je l’ai vu exposer ses travaux avec Laurent Clozel sur les corps de nombres peu ramifiés. Toby Gee, Benedict Gross et Colette Mœglin ont accepté de rapporter ma thèse. Il s’agit d’une lourde tâche et je les en remercie vivement. Laurent Clozel, Guy Henniart, David Renard et Jean-Loup Waldspurger me font l’honneur de faire partie du jury, je voudrais pour cela leur exprimer ma gratitude. Je souhaite à nouveau remercier Colette Mœglin et Jean-Loup Waldspurger. Alors que je cherchais une démonstration trop compliquée du lemme 2.6.1.1, ils m’ont gentiment expliqué qu’il découlait simplement d’un de leurs lemmes. La seconde partie de cette thèse a nécessité d’effectuer des calculs parallèles, mobilisant presque tous les ordinateurs du DMA. Je remercie donc le SPI pour sa bienveillance. J’ai eu la chance d’enseigner à l’ENS pendant ces trois dernières années. Cette expé- rience fut passionnante, et a permis un équilibre mathématique. Elle m’a également placé dans l’atmosphère conviviale et chaleureuse du DMA. Je remercie donc tous mes collègues et amis avec qui j’ai passé d’agréables moments, mathématiques ou non ; en particulier Arnaud, Henri et Quentin. J’ai la chance de confondre travail et passion. C’est l’une des nombreuses choses que je dois à mes parents. Je profite donc de cette occasion pour leur exprimer ma gratitude. Merci enfin à Chloé, grâce à qui cette thèse fut préparée dans le bonheur. 1 Table des matières 1 Introduction4 1.1 Formes modulaires................................5 1.2 Formes automorphes et représentations galoisiennes..............7 1.3 Résultats obtenus dans cette thèse.......................9 2 Eigenvarieties for classical groups and complex conjugations in Galois representations 10 2.1 Introduction.................................... 10 2.2 Assumptions on forthcoming results of Arthur................. 13 2.3 The eigenvariety for definite symplectic groups................ 14 2.3.1 The eigenvariety............................. 14 2.3.2 Unramified and “completely refinable” points............. 18 2.4 Galois representations associated with automorphic representations of sym- plectic groups................................... 23 2.4.1 A consequence of Arthur’s description of the discrete spectrum for classical groups.............................. 23 2.4.2 Galois representations stemming from symplectic forms are generi- cally almost irreducible......................... 26 2.5 Similar results for even orthogonal groups................... 32 2.6 The image of complex conjugation: relaxing hypotheses in Taylor’s theorem 33 2.6.1 Regular, L-algebraic, self-dual, cuspidal representations of GL2n(AF ) having Iwahori-invariants........................ 34 2.6.2 Regular, L-algebraic, self-dual, cuspidal representations of GL2n+1(AF ) having Iwahori-invariants........................ 37 2.6.3 Almost general case........................... 41 2.6.4 A supplementary, non-regular case................... 43 3 Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula 47 3.1 Introduction.................................... 47 3.2 Notations and definitions............................ 50 3.3 Computation of the geometric side of Arthur’s trace formula......... 51 3.3.1 Elliptic terms............................... 52 3.3.2 Computation of the elliptic terms in the trace formula........ 54 2 3.3.3 Computation of the parabolic terms using elliptic terms for groups of lower semisimple rank......................... 69 3.4 Endoscopic decomposition of the spectral side................. 76 3.4.1 The spectral side of the trace formula................. 76 3.4.2 Euler-Poincaré characteristic of cohomological archimedean Arthur packets.................................. 86 3.5 Application to vector-valued Siegel modular forms.............. 96 3.5.1 Bounded symmetric domains of symplectic type and holomorphic discrete series............................... 97 3.5.2 Siegel modular forms and automorphic forms............. 100 3.5.3 Example: genus 4 ............................. 104 3.5.4 Some dimensions in the scalar case................... 105 3.6 Reliability..................................... 106 3.7 Tables....................................... 108 3.7.1 Masses................................... 108 3.7.2 Some essentially self-dual, algebraic, level one, automorphic cuspidal representations of GLn for n ≤ 13 .................... 113 3 Partie 1 Introduction Dans cette introduction destinée aux mathématiciens non spécialistes, nous expliquons dans quel contexte s’inscrivent les résultats de cette thèse. Le titre annonce qu’il s’agit d’arithmétique, ce qui n’est peut-être pas évident à la lecture du texte. L’un des buts de cette branche des mathématiques est l’étude des équations diophantiennes, c’est-à-dire les équations “à coefficients entiers” dont on cherche les solutions entières. Un problème for- mulé aussi vaguement ne peut admettre de solution générale, et l’histoire de l’arithmétique est jalonnée par les découvertes d’outils permettant d’étudier seulement certaines classes d’équations diophantiennes. Citons deux outils fondamentaux pour l’étude des systèmes d’équations polynomiales, c’est-à-dire de la forme 8 P (x ; : : : ; x ) = 0 > 1 1 m < : : > :Pn(x1; : : : ; xm) = 0 où chaque Pi est un polynôme à coefficients entiers en les variables X1;:::;Xm. Afin de simplifier le problème, contentons-nous de travailler sur le corps Q des nombres rationnels plutôt que sur son sous-anneau Z des nombres entiers. Galois nous enseigne que les solutions rationnelles d’un tel système d’équations sont m les solutions (x1; : : : ; xm) 2 Q , où Q désigne une clôture algébrique de Q, qui sont fixées par Gal(Q=Q), le groupe de Galois absolu de Q qui agit sur Q. L’intérêt de ce point de vue est que, comme souvent en mathématiques, il est plus aisé d’étudier les propriétés de solutions dont l’existence est connue a priori que de s’attaquer directement au problème d’existence. Plus généralement, il est naturel de voir un objet défini sur Q comme un objet défini sur Q et muni d’une action de Gal(Q=Q). Notons toutefois que l’utilisation de nombres algébriques sur Q remonte à Gauss qui introduisit en 1832 (avant la publication des résultats de Galois) l’anneau des “entiers de Gauss” Z[i] = fa + ib j a; b 2 Zg dans le but de formuler la loi de réciprocité biquadratique. Une autre approche consiste à compléter le corps Q pour une de ses valuations v. D’après un théorème d’Ostrowski, à un exposant inessentiel près, v est soit la valuation archimédienne usuelle, soit la valuation p-adique pour un nombre premier p. Dans le corps complété Qv correspondant, on dispose d’outils analytiques (théorème des valeurs intermé- diaires, lemme de Hensel . .) simplifiant grandement la résolution du système d’équations. Afin de prendre en compte toutes ces valuations (également appelées “places”) à la fois, on 4 considère l’anneau des adèles Y0 Y 0 A = Qv = R × Qp v p premier qui est le produit restreint de ces corps et qui contient Q comme sous-anneau, plongé dia- gonalement. Ici on a noté Qp le complété Qv de Q pour la valuation p-adique v. L’existence d’une solution à coefficients rationnels implique donc l’existence d’une solution à coefficients adèliques. Toute la difficulté consiste à aller dans l’autre sens, c’est-à-dire à déterminer les obstructions “globales” contrôlant l’existence d’une solution rationnelle lorsque l’on suppose l’existence d’une solution adèlique. Par exemple le théorème de Hasse-Minkowski implique que dans le cas d’une seule équation
Load more

Recommended publications
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
    [Show full text]
  • Cohomology Theory of Lie Groups and Lie Algebras
    COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf.
    [Show full text]
  • Interview with Henri Cartan, Volume 46, Number 7
    fea-cartan.qxp 6/8/99 4:50 PM Page 782 Interview with Henri Cartan The following interview was conducted on March 19–20, 1999, in Paris by Notices senior writer and deputy editor Allyn Jackson. Biographical Sketch in 1975. Cartan is a member of the Académie des Henri Cartan is one of Sciences of Paris and of twelve other academies in the first-rank mathe- Europe, the United States, and Japan. He has re- maticians of the twen- ceived honorary doc- tieth century. He has torates from several had a lasting impact universities, and also through his research, the Wolf Prize in which spans a wide va- Mathematics in 1980. riety of areas, as well Early Years as through his teach- ing, his students, and Notices: Let’s start at the famed Séminaire the beginning of your Cartan. He is one of the life. What are your founding members of earliest memories of Bourbaki. His book Ho- mathematical inter- mological Algebra, writ- est? ten with Samuel Eilen- Cartan: I have al- berg and first published ways been interested Henri Cartan’s father, Élie Photograph by Sophie Caretta. in 1956, is still in print in mathematics. But Cartan. Henri Cartan, 1996. and remains a standard I don’t think it was reference. because my father The son of Élie Cartan, who is considered to be was a mathematician. I had no doubt that I could the founder of modern differential geometry, Henri become a mathematician. I had many teachers, Cartan was born on July 8, 1904, in Nancy, France. good or not so good.
    [Show full text]
  • Title: Algebraic Group Representations, and Related Topics a Lecture by Len Scott, Mcconnell/Bernard Professor of Mathemtics, the University of Virginia
    Title: Algebraic group representations, and related topics a lecture by Len Scott, McConnell/Bernard Professor of Mathemtics, The University of Virginia. Abstract: This lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development, and its relationship to the theory of finite group representations. Other topics of a Lie-theoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits. CHAPTER VI Linear Algebraic Groups in the 20th Century The interest in linear algebraic groups was revived in the 1940s by C. Chevalley and E. Kolchin. The most salient features of their contributions are outlined in Chapter VII and VIII. Even though they are put there to suit the broader context, I shall as a rule refer to those chapters, rather than repeat their contents. Some of it will be recalled, however, mainly to round out a narrative which will also take into account, more than there, the work of other authors. §1. Linear algebraic groups in characteristic zero. Replicas 1.1. As we saw in Chapter V, §4, Ludwig Maurer thoroughly analyzed the properties of the Lie algebra of a complex linear algebraic group. This was Cheval­ ey's starting point.
    [Show full text]
  • Theory of Lie Groups I
    854 NATURE December 20. 1947 Vol. t60 Theory of lie Groups, I deficiency is especially well treated, and the chapters By Claude Chevalley. (Princeton Mathematical on feeding experiments contain one of the best Series.) Pp. xii + 217. (Princeton, N.J.: Princeton reasoned discussions on the pros and cons of the University Press; London: Oxford University paired-feeding technique. In the author's view this Press, 1946.) 20s. net. method would appear to find its largest usefulness N recent years great advances have been made in in comparisons in which food consumption is not I our knowledge of the fundamental structures of markedly restricted by the conditions imposed, and analysis, particularly of algebra and topology, and in which the measure is in terms of the specific an exposition of Lie groups from the modern point effect of the nutrient under study, instead of the more of view is timely. This is given admirably in the general measure of increase in weight. It is considered book under review, which could well have been called that there are many problems concerning which the "Lie Groups in the Large". To make such a treatment use of separate experiments of both ad libitum and intelligible it is necessary to re-examine from the new controlled feeding will give much more information point of view such subjects as the classical linear than either procedure alone. groups, topological groups and their underlying The chapters on growth, reproduction, lactation topological spaces, and analytic manifolds. Excellent and work production are likewise excellent. In its accounts of these, with the study of integral manifolds clarity and brevity this work has few equals.
    [Show full text]
  • On Chevalley's Z-Form
    Indian J. Pure Appl. Math., 46(5): 695-700, October 2015 °c Indian National Science Academy DOI: 10.1007/s13226-015-0134-7 ON CHEVALLEY’S Z-FORM M. S. Raghunathan National Centre for Mathematics, Indian Institute of Technology, Mumbai 400 076, India e-mail: [email protected] (Received 28 October 2014; accepted 18 December 2014) We give a new proof of the existence of a Chevalley basis for a semisimple Lie algebra over C. Key words : Chevalley basis; semisimple Lie algebras. 1. INTRODUCTION Let g be a complex semisimple Lie algebra. In a path-breaking paper [2], Chevalley exhibitted a basis B of g (known as Chevalley basis since then) such that the structural constants of g with respect to the basis B are integers. The basis moreover consists of a basis of a Cartan subalgebra t (on which the roots take integral values) together with root vectors of g with respect to t. The structural constants were determined explicitly (up to signs) in terms of the structure of the root system of g. Tits [3] provided a more elegant approach to obtain Chevalley’s results which essentially exploited geometric properties of the root system. Casselman [1] used the methods of Tits to extend the Chevalley theorem to the Kac-Moody case. In this note we prove Chevalley’s theorem through an approach different from those of Chevalley and Tits. Let G be the simply connected algebraic group corresponding to g. Let T be a maximal torus (the Lie subalgebra t corresponding to it is a Cartan subalgebra of G).
    [Show full text]
  • Gerhard Hochschild (1915/2010) a Mathematician of the Xxth Century
    GERHARD HOCHSCHILD (1915/2010) A MATHEMATICIAN OF THE XXth CENTURY WALTER FERRER SANTOS Abstract. Gerhard Hochschild’s contribution to the development of mathematics in the XX century is succinctly surveyed. We start with a personal and mathematical biography, and then consider with certain detail his contributions to algebraic groups and Hopf algebras. 1. The life, times and mathematics of Gerhard Hochschild 1.1. Berlin and South Africa. Gerhard Paul Hochschild was born on April 29th, 1915 in Berlin of a middle class Jewish family and died on July 8th, 2010 in El Cerrito where he lived after he moved to take a position as professor at the University of California at Berkeley in 1958. His father Heiner was an engineer working as a patent attonery and in search of safety sent Gerhard and his older brother Ulrich, to Cape Town in May of 1933. The boys were some of the many germans escaping from the Nazis that were taking over their native country. The first 18 years of his life in Germany were not uneventful. In 1924 his mother Lilli was diagnosed with a lung ailment and sent –together with his younger son Gerhard that was then nine years old– to a sanatorium in the Alps, near Davos. Later in life he commented to the author that reading in Der Zauberberg (The Magic Mountain) by Thomas Mann, about Hans Castorp –Mann’s main character in the book– he evoked his own personal experiences. Castorp was transported away from his ordered and organized family life to pay a visit to his cousin interned also in a sanatorium in Davos.
    [Show full text]
  • Mathematical Communities
    II~'~lvi|,[~]i,~.~|[.-~-nl[,~.],,n,,,,.,nit[:;-1 Marjorie Senechal, Editor ] Nicolas Bourbaki, 1935-???? vided for self-renewal: from time to If you are a mathematician working to- time, younger mathematicians were in- The Continuing day, you have almost certainly been in- vited to join and older members re- fluenced by Bourbaki, at least in style signed, in accordance with mandatory and spirit, and perhaps to a greater ex- "retirement" at age fifty. Now Bourbaki Silence of tent than you realize. But if you are a itself is nearly twenty years older than student, you may never have heard of any of its members. The long-running it, him, them. What or who is, or was, Bourbaki seminar is still alive and well Bourbaki- Bourbaki? and living in Paris, but the voice of Check as many as apply. Bourbakl Bourbaki itself--as expressed through An Interview with is, or was, as the case may be: its books--has been silent for fifteen years. Will it speak again? Can it speak 9the discoverer (or inventor, if you again? prefer) of the notion of a mathemat- Pierre Cartier, Pierre Cartier was a member of ical structure; Bourbaki from 1955 to 1983. Born in 9one of the great abstractionist move- June 18, 1997 Sedan, France in 1932, he graduated ments of the twentieth century; from the l~cole Normale Supdrieure in 9 a small but enormously influential Marjorie Senechal Paris, where he studied under Henri community of mathematicians; Caftan. His thesis, defended in 1958, 9 acollective that hasn't published for was on algebraic geometry; since then fifteen years.
    [Show full text]
  • The University of Chicago Symmetry and Equivalence
    THE UNIVERSITY OF CHICAGO SYMMETRY AND EQUIVALENCE RELATIONS IN CLASSICAL AND GEOMETRIC COMPLEXITY THEORY A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF COMPUTER SCIENCE BY JOSHUA ABRAHAM GROCHOW CHICAGO, ILLINOIS JUNE 2012 To my parents, Jerrold Marvin Grochow and Louise Barnett Grochow ABSTRACT This thesis studies some of the ways in which symmetries and equivalence relations arise in classical and geometric complexity theory. The Geometric Complexity Theory program is aimed at resolving central questions in complexity such as P versus NP using techniques from algebraic geometry and representation theory. The equivalence relations we study are mostly algebraic in nature and we heavily use algebraic techniques to reason about the computational properties of these problems. We first provide a tutorial and survey on Geometric Complexity Theory to provide perspective and motivate the other problems we study. One equivalence relation we study is matrix isomorphism of matrix Lie algebras, which is a problem that arises naturally in Geometric Complexity Theory. For certain cases of matrix isomorphism of Lie algebras we provide polynomial-time algorithms, and for other cases we show that the problem is as hard as graph isomorphism. To our knowledge, this is the first time graph isomorphism has appeared in connection with any lower bounds program. Finally, we study algorithms for equivalence relations more generally (joint work with Lance Fortnow). Two techniques are often employed for algorithmically deciding equivalence relations: 1) finding a complete set of easily computable invariants, or 2) finding an algorithm which will compute a canonical form for each equivalence class.
    [Show full text]
  • Arxiv:2002.01446V3 [Math.GR] 12 Jun 2021
    TWISTED CONJUGACY CLASSES IN TWISTED CHEVALLEY GROUPS SUSHIL BHUNIA, PINKA DEY, AND AMIT ROY ABSTRACT. Let G be a group and ϕ be an automorphism of G. Two elements x, y ∈ G are said to be ϕ-twisted conjugate if y = gxϕ(g)−1 for some g ∈ G. We say that a group G has the R∞-property if the number of ϕ-twisted conjugacy classes is infinite for every automorphism ϕ of G. In this paper, we prove that twisted Chevalley groups over a field k of characteristic zero have the R∞-property as well as the S∞-property if k has finite transcendence degree over Q or Aut(k) is periodic. 1. INTRODUCTION Let G be a group and ϕ be an automorphism of G. Two elements x, y of G are −1 said to be twisted ϕ-conjugate, denoted by x ∼ϕ y, if y = gxϕ(g) for some g ∈ G. Clearly, ∼ϕ is an equivalence relation on G. The equivalence classes with respect to this relation are called the ϕ-twisted conjugacy classes or the Reidemeister classes of ϕ. If ϕ = Id, then the ϕ-twisted conjugacy classes are the usual conjugacy classes. The ϕ-twisted conjugacy class containing x ∈ G is denoted by [x]ϕ. The Reidemeister number of ϕ, denoted by R(ϕ), is the number of all ϕ-twisted conjugacy classes. A group G has the R∞-property if R(ϕ) is infinite for every automorphism ϕ of G. The Reidemeister number is closely related to the Nielsen number of a selfmap of a manifold, which is homotopy invariant.
    [Show full text]
  • Armand Borel 1923–2003
    Armand Borel 1923–2003 A Biographical Memoir by Mark Goresky ©2019 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. ARMAND BOREL May 21, 1923–August 11, 2003 Elected to the NAS, 1987 Armand Borel was a leading mathematician of the twen- tieth century. A native of Switzerland, he spent most of his professional life in Princeton, New Jersey, where he passed away after a short illness in 2003. Although he is primarily known as one of the chief archi- tects of the modern theory of linear algebraic groups and of arithmetic groups, Borel had an extraordinarily wide range of mathematical interests and influence. Perhaps more than any other mathematician in modern times, Borel elucidated and disseminated the work of others. family the Borel of Photo courtesy His books, conference proceedings, and journal publica- tions provide the document of record for many important mathematical developments during his lifetime. By Mark Goresky Mathematical objects and results bearing Borel’s name include Borel subgroups, Borel regulator, Borel construction, Borel equivariant cohomology, Borel-Serre compactification, Bailey- Borel compactification, Borel fixed point theorem, Borel-Moore homology, Borel-Weil theorem, Borel-de Siebenthal theorem, and Borel conjecture.1 Borel was awarded the Brouwer Medal (Dutch Mathematical Society), the Steele Prize (American Mathematical Society) and the Balzan Prize (Italian-Swiss International Balzan Foundation). He was a member of the National Academy of Sciences (USA), the American Academy of Arts and Sciences, the American Philosophical Society, the Finnish Academy of Sciences and Letters, the Academia Europa, and the French Academy of Sciences.1 1 Borel enjoyed pointing out, with some amusement, that he was not related to the famous Borel of “Borel sets." 2 ARMAND BOREL Switzerland and France Armand Borel was born in 1923 in the French-speaking city of La Chaux-de-Fonds in Switzerland.
    [Show full text]
  • The Cohomology of Left-Invariant Elliptic Involutive Structures On
    The cohomology of left-invariant elliptic involutive structures on compact Lie groups Max Reinhold Jahnke∗ [email protected] Department of Mathematics Federal University of São Carlos, SP, Brazil. December 2, 2019 Abstract Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on compact Lie groups, we prove that, under certain algebraic and topological conditions, the cohomology associated to left-invariant elliptic, and even hypocomplex, involutive structures on compact Lie groups can be computed by using only Lie algebras, thus reducing the analytical problem to a purely algebraic one. The main tool is the Leray spectral sequence that connects the result obtained Chevalley and Eilenberg to a result by Bott on the Dolbeault cohomology of a homogeneous manifold. 1 Introduction The initial motivation for this work was an article by Chevalley and Eilenberg [7], where it is proved that the de Rham cohomology of a compact Lie group can be studied by using only left-invariant forms, thus reducing the study of the cohomology to a purely algebraic problem. This led us to question whether something similar could be done to study the cohomology associated with general involutive structures defined on compact Lie groups. It is clear that the involutive structures must have algebraic properties from the Lie group. Therefore it is natural to restrict our attention to involutive structures that are invariant by the action of the group, these are called left-invariant involutive structures. Some particular cases of left-invariant involutive structures, as well as involutive structures that are invariant by the group actions in homogeneous spaces, have been studied previously.
    [Show full text]