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Armand Borel (1923–2003), Volume 51, Number 5 Armand Borel (1923–2003) James Arthur, Enrico Bombieri, Komaravolu Chandrasekharan, Friedrich Hirzebruch, Gopal Prasad, Jean-Pierre Serre, Tonny A. Springer, and Jacques Tits Citations within this article. Borel’s Collected Papers are [Œ], and his 17 books are referred to as [1] through [17]. These are all listed in a sidebar on page 501. An item like [Œ 23] is Borel paper number 23 in [Œ]. Ci- tations of work by people other than Borel are by letter combinations such as [Che], and the de- tails appear at the end of the article. their root systems. He spent the year 1949–50 in 1 Institute for Advanced Study, 1999. Paris, with a grant from the CNRS . A good choice Armand Borel (for us, as well as for him), Paris being the very spot where what Americans have called “French Topol- Jean-Pierre Serre ogy” was being created, with the courses from The Swiss mathematician Armand Borel died Leray at the Collège de France and the Cartan sem- August 11, 2003, in Princeton from a rapidly evolv- inar at the École Normale Supérieure. Borel was an ing cancer. Few foreign mathematicians had as active participant in the Cartan seminar while many connections with France. He was a student closely following Leray’s courses. He managed to of Leray, he took part in the Cartan seminar, and understand the famous “spectral sequence”, not an he published more than twenty papers in collabo- easy task, and he explained it to me so well that I ration with our colleagues Lichnerowicz and Tits, have not stopped using it since. He began to apply as well as with me. He was a member of Bourbaki it to Lie groups, and to the determination of their for more than twenty years, and he became a for- cohomology with integer coefficients. That work would make a thesis, defended at the Sorbonne eign member of the Académie des Sciences in 1981. (with Leray as president) in 1952, and published im- French mathematicians feel that it is one of their mediately in the Annals of Mathematics. Meanwhile own who has died. Borel returned to Switzerland. He did not stay long. He was born in La Chaux-de-Fonds in 1923 and He went for two years (1952–54) to the Institute for was an undergraduate at Eidgenössische Technis- Advanced Study in Princeton and spent the year che Hochschule of Zürich (the “Poly”). There he met 1954–55 in Chicago, where he benefited from the H. Hopf, who gave him a taste for topology, and presence of André Weil by learning algebraic geom- E. Stiefel, who introduced him to Lie groups and etry and number theory. He returned to Switzer- Jean-Pierre Serre is professor emeritus of the Collège de land, this time to Zürich, and in 1957 the Institute France. His email address is [email protected]. This part of for Advanced Study offered him a position as per- the article is a translation of the presentation made by him manent professor, a post he occupied until his to the Académie des Sciences in Paris on September 30, death (he became a professor emeritus in 1993). 2003 (© Académie des Sciences and printed with permis- sion). Anthony W. Knapp assisted with the translation. 1Centre National de la Recherche Scientifique. 498 NOTICES OF THE AMS VOLUME 51, NUMBER 5 again, Borel’s extensive knowledge of the literature James Arthur played a role. Inspired by a recent paper of V. P. My topic is Armand Borel and the theory of au- Platonov, he had written a note proving the analog tomorphic forms. Borel’s most important contri- for simple groups over algebraically closed fields butions to the area are undoubtedly those estab- of a classical result of V. V. Morozov: a maximal lished in collaboration with Harish-Chandra [Œ 54, proper subalgebra of a semisimple complex Lie al- 58]. They include the construction and properties gebra either is semisimple or contains a maximal of approximate fundamental domains, the proof of nilpotent subalgebra. Borel sent a copy of the man- finite volume of arithmetic quotients, and the char- uscript of this note to a mutual friend of ours, acterization in terms of algebraic groups of those who showed it to me. I observed that the argument arithmetic subgroups that give compact quotients. of Borel’s proof could be adapted to a considerably These results created the opportunity for working more general setting and provided results over ar- in the context of general algebraic groups. They laid bitrary fields, for instance the following fact, which the foundations of the modern theory of auto- morphic forms that has flourished for the past turned out to be of great importance in finite group forty years. theory: if G is a reductive k-group (k being any field) 25 The classical theory of modular forms concerns and U is a split unipotent subgroup, then there holomorphic functions on the upper half plane H exists a parabolic k-subgroup P of G whose unipo- that transform in a certain way under the action tent radical contains U such that every k-auto- of a discrete subgroup Γ of SL(2, R). The multi- morphism of G preserving U also preserves P; plicative group SL(2, R) consists of the 2 × 2 real then, in particular, P contains the normalizer of U. matrices of determinant 1, and each element (Much earlier, I had conjectured this fact and given γ = ab of SL(2, R) acts on H by the linear cd a case-by-case proof “in most cases”, but the new az+b fractional transformation z → . For example, approach, using Borel’s argument, gave a uniform cz+d one can take Γ to be the subgroup SL(2, Z) of and much simpler proof.) integral matrices or, more generally, the congruence I mention en passant the complements [Œ 94] subgroup to “Groupes réductifs”, containing results con- ∈ Z ≡ cerning, among other things, the closure of Bruhat Γ (N)= γ SL(2, ): γ I mod N cells in topological reductive groups and the fun- attached to a positive integer N. The theory began damental group of real algebraic simple groups. as a branch of complex analysis. However, with the The last joint paper by Borel and me is a Comptes work of E. Hecke, it acquired a distinctive number Rendus note [Œ 97] that in personal discussions we theoretic character. Hecke introduced a commut- nicknamed “Nonreductive groups”. Indeed, we had ing family of linear operators on any space of au- discovered that many of our results on reductive tomorphic forms for Γ (N), one for each prime not groups could, after suitable reformulation, be gen- dividing N, with interesting arithmetic properties. eralized to arbitrary connected algebraic groups. We now know that eigenvalues of the Hecke oper- ators govern how prime numbers p split in certain Examples are the conjugacy by elements rational nonabelian Galois extensions of the field Q of ra- over the ground field, of maximal split tori, of tional numbers [Sh], [D]. Results of this nature are maximal split unipotent subgroups, and of mini- known as reciprocity laws and are in some sense 26 mal pseudoparabolic subgroups, or the existence the ultimate goal of algebraic number theory. They of a BN-pair (hence of a Bruhat decomposition). can be interpreted as a classification for the num- Complete proofs of those results never got pub- ber fields in question. The Langlands program con- lished, but both Borel and I, independently, lectured cerns the generalization of the theory of modular about them (he at Yale University, and I at the Col- forms from the group of 2 × 2 matrices of deter- lège de France). minant 1 to an arbitrary reductive group G. It is believed to provide reciprocity laws for all finite al- gebraic extensions of Q. Let us use the results of Borel and Harish- Chandra as a pretext for making a very brief ex- cursion into the general theory of automorphic forms. In so doing, we can follow a path illuminated 25 A unipotent group over a field k is said to be split if it by Borel himself. The expository articles and mono- k has a composition series over , all quotients of which are graphs of Borel encouraged a whole generation of additive groups. 26 Pseudoparabolic subgroups are a substitute for para- James Arthur is professor of mathematics at the Univer- bolic subgroups that one must use when dealing with sity of Toronto and president elect of the AMS. His email nonreductive groups over nonperfect fields. address is [email protected]. 518 NOTICES OF THE AMS VOLUME 51, NUMBER 5 mathematicians to pursue the study of automor- organized jointly by Borel and G. D. Mostow. It phic forms for general algebraic groups. Together was a systematic attempt to make the emerging the- with the mathematical conferences he organized, ory of automorphic forms accessible to a wider au- they have had extraordinary influence. dience. Borel himself wrote four articles [Œ 73, 74, The general theory entails two reformulations 75, 76] for the proceedings, each elucidating a dif- of the classical theory of modular forms. The first ferent aspect of the theory. is in terms of the unitary representation theory of The second reformulation is in terms of adeles. the group SL(2, R). Though harder to justify at first, the language of The action of SL(2, R) on H√ is transitive. Since adeles ultimately streamlines many fundamental the stabilizer of the point i = −1 is the special or- operations on automorphic forms. The relevant thogonal group Boulder articles are [T] and [K].
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