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Caterina Consani Matilde Marcolli (Eds.) Noncommutative Geometry and Number Theory Aspects of Mathematics Edited by Klas Diederich Vol. E 6: G. Faltings/G. Wüstholz et al.: Rational Points* Vol. E15: J .-P. Serre: Lectures on the Mordell-Weil Theorem Vol. E16: K. Iwasaki/H. Kimura/S. Shimemura/M. Yoshida: From Gauss to Painlevé Vol. E19: R. Racke: Lectures on Nonlinear Evolution Equations Vol. E21: H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm Vol. E22: D. V. Anosov/A. A. Bolibruch: The Riemann-Hilbert Problem Vol. E27: D. N. Akhiezer: Lie Group Actions in Complex Analysis Vol. E28: R. Gérard/H. Tahara: Singular Nonlinear Partial Differential Equations Vol. E34: I. Lieb/J. Michel: The Cauchy-Riemann Complex Vol. E36: C.Hertling/M. Marcolli (Eds.): Frobenius Manifolds* Vol. E37: C.Consani/M. Marcolli (Eds.): Noncommutative Geometry and Number Theory* *A Publication of the Max-Planck-lnstitute for Mathematics, Bonn Caterina Consani Matilde Marcolli (Eds.) Noncommutative Geometry and Number Theory Where Arithmetic meets Geometry and Physics A Publication of the Max-Planck-Institute for Mathematics, Bonn Bibliografische information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutschen Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. Prof. Dr. Caterina Consani Department of Mathematics The Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218, USA [email protected] Prof. Dr. Matilde Marcolli Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn [email protected] Prof. Dr. Klas Diederich(Series Editor) Fachbereich Mathematik Bergische Universität Wuppertal Gaußstraße 20 D-42119Wuppertal [email protected] Mathematics Subject Classification Primary: 58B34, 11X Secondary:11F23, 11S30, 11F37, 11F41, 11F70, 11F80, 11J71, 11B57, 11K36, 11F32, 11F75, 11G18, 14A22, 14F42, 14G05, 14G40, 14K10, 14G35, 18E30, 19D55, 20G05, 22E50, 32G05, 46L55, 58B34, 58E20, 70S15, 81T40. Firstedition, March2006 All rights reserved ©Friedr. Vieweg & Sohn Verlag | GWVFachverlage GmbH, Wiesbaden 2006 Editorial Office: Ulrike Schmickler-Hirzebruch / Petra Rußkamp Vieweg is a company in the specialist publishing group Springer Science+Business Media. www.vieweg.de Nopartof this publication maybe reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder. Cover design: Ulrike Weigel, www.CorporateDesignGroup.de Printing and binding: MercedesDruck, Berlin Printed on acid-free paper Printed in Germany ISBN 3-8348-0170-4 ISSN 0179-2156 Preface In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. Research across these fields has now reached an impor- tant turning point, as shows the increasing interest with which the mathematical community approaches these topics. This volume collects and presents up-to-date researchtopics in arithmetic and noncommutative geometry and ideas from physics that point to possible new con- nections between the fields of number theory, algebraic geometry and noncommu- tative geometry. Thecontributionstothisvolumepartlyreflectthetwoworkshops“Noncommu- tativeGeometryandNumberTheory”thattookplaceattheMax–Planck–Institut fu¨r Mathematik in Bonn, in August 2003 and June 2004. The two workshopswere the first activity entirely dedicated to the interplay between these two fields of mathematics. An important part of the activities, which is also reflected in this volume, came from the hindsight of physics which often provides new perspectives onnumbertheoreticproblemsthatmakeitpossibletoemploythetoolsofnoncom- mutative geometry, well designed to describe the quantum world. Somecontributionstothevolume(Aubert–Baum–Plymen,Meyer,Nistor)cen- ter on the theory of reductive p-adic groups and their Hecke algebras, a promising direction where noncommutative geometry provides valuable tools for the study of objects of number theoretic and arithmetic interest. A generalization of the clas- sical Burnside theorem using noncommutative geometry is discussed in the paper byFel’shtynandTroitsky. ThecontributionofLacaandvanFrankenhuijsenrepre- sentsanotherdirectioninwhichsubstantialprogresswasrecentlymadeinapplying toolsofnoncommutativegeometrytonumbertheory: the constructionofquantum statistical mechanical systems associated to number fields and the relation of their KMS equilibrium states to abelian class field theory. The theory of Shimura vari- etiesisconsideredfromthenumbertheoreticsideinthecontributionofBlasius,on the Weight-Monodromy conjecture for Shimura varieties associated to quaternion algebrasovertotally realfields and the Ramanujan conjecture for Hilbert modular forms. An approach via noncommutative geometry to the boundaries of Shimura varietiesisdiscussedinPaugam’spaper. Modularformscanbestudied usingtech- niques from noncommutative geometry, via the Hopf algebra symmetries of the modular Hecke algebras, as discussed in the paper of Connes and Moscovici. The generalunderlying theory of Hopf cyclic cohomologyin noncommutative geometry is presented in the paper of Khalkhali and Rangipour. Further results in arith- metic geometry include a reinterpretation of the archimedean cohomology of the fibersatarchimedeanprimesofarithmeticvarieties(Consani–Marcolli),andKim’s vi PREFACE paper on a noncommutative method of Chabauty. Arithmetic aspects of noncom- mutativetoriarealsodiscussed(Boca–ZaharescuandPolishchuk). Theinputfrom physics and its interactions with number theory and noncommutative geometry is represented by the contributions of Kreimer, Landi, Marcolli–Mathai, and Ponge. The workshops were generously funded by the Humboldt Foundation and the ZIP Programof the German Federal Government, through the Sofja Kovalevskaya Prize awarded to Marcolli in 2001. We are very grateful to Yuri Manin and to Alain Connes, who helped us with theorganizationoftheworkshops,andwhoseideasandresultscontributedcrucially to bring number theory in touch with noncommutative geometry. Caterina Consani and Matilde Marcolli Contents Preface v The Hecke algebra of a reductive p-adic group: a view from noncommutative geometry Anne-Marie Aubert, Paul Baum, Roger Plymen 1 Hilbert modular forms and the Ramanujan conjecture Don Blasius 35 Farey fractions and two-dimensional tori Florin P. Boca and Alexandru Zaharescu 57 Transgressionof the Godbillon-Vey class and Rademacher functions Alain Connes and Henri Moscovici 79 Archimedean cohomology revisited Caterina Consani and Matilde Marcolli 109 A twisted Burnside theorem for countable groups and Reidemeister numbers Alexander Fel’shtyn and Evgenij Troitsky 141 Introduction to Hopf cyclic cohomology Masoud Khalkhali and Bahran Rangipour 155 The non-abelian (or non-linear) method of Chabauty Minhyong Kim 179 The residues of quantum field theory – numbers we should know Dirk Kreimer 187 Phase transitions with spontaneous symmetry breaking on Hecke C∗-algebras from number fields Marcelo Laca and Machiel van Frankenhuijsen 205 On harmonic maps in noncommutative geometry Giovanni Landi 217 Towards the fractional quantum Hall effect: a noncommutative geometry perspective Matilde Marcolli and Varghese Mathai 235 Homological algebra for Schwartz algebras of reductive p-adic groups Ralph Meyer 263 viii CONTENTS A non-commutative geometry approach to the representation theory of reductive p-adic groups: Homology of Hecke algebras, a survey and some new results Victor Nistor 301 Three examples of non-commutative boundaries of Shimura varieties Frederic Paugam 323 Holomorphic bundles on 2-dimensional noncommutative toric orbifolds Alexander Polishchuk 341 A new short proof of the local index formula of Atiyah-Singer Raphael Ponge 361 The Hecke algebra of a reductive p-adic group: a geometric conjecture Anne-Marie Aubert, Paul Baum, and Roger Plymen Abstract. Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on theasymptoticHeckealgebra. WeproveourconjectureforSL(2)andGL(n). We also prove part (1) of the conjecture for the Iwahori ideals of the groups PGL(n)andSO(5). Theconjecture, iftrue,leadstoaparametrizationofthe smoothdualofGbythepointsinacomplexaffinelocallyalgebraicvariety. 1. Introduction The reciprocity laws in number theory have a long development, starting from conjectures of Euler, and including contributions of Legendre, Gauss, Dirichlet, Jacobi, Eisenstein, Takagi and Artin. For the details of this development, see [Le]. The local reciprocity law for a local field F, which concerns the finite Galois extensions E/F such that Gal(E/F) is commutative, is stated and proved in [N, p. 320]. This local reciprocity law was dramatically generalized by Langlands. ThelocalLanglandscorrespondenceforGL(n)is anoncommutativegeneralization of the reciprocity law of local class field theory. The local Langlands conjectures, and the global Langlands conjectures, all involve, inter alia, the representations of reductive p-adic groups, see [BG]. To each reductive p-adic group G there is associated the Hecke algebra H(G), whichwenowdefine. LetKbeacompactopensubgroupofG,anddefineH(G//K) astheconvolutionalgebraofallcomplex-valued,compactly-supportedfunctionson G such that f(k xk )=f(x) for all k , k in K. The Hecke algebra H(G) is then 1 2 1 2 defined as (cid:1) H(G):= H(G//K). K The smooth representations of G on a complex vector space V correspond bijectively to the nondegenerate representations of H(G) on V, see [B, p.2]. In this article, we consider H(G) from the point of view of noncommutative (algebraic) geometry. Werecallthatthecoordinateringsofaffinealgebraicvarietiesarepreciselythe commutative, unital, finitely generated, reduced C-algebras,see [EH, II.1.1]. 2 ANNE-MARIEAUBERT,PAULBAUM,ANDROGERPLYMEN TheHeckealgebraH(G)isanon-commutative,non-unital,non-finitely-generated, non-reduced C-algebra,and so cannot be the coordinate ring of an affine algebraic variety. The Hecke algebra H(G) is non-unital, but it admits local units, see [B, p.2]. The algebra H(G) admits a canonical decomposition into ideals, the Bernstein decomposition [B]: (cid:2) H(G)= Hs(G). s∈B(G) Each ideal Hs(G) is a non-commutative, non-unital, non-finitely-generated, non-reduced C-algebra,and so cannot be the coordinate ring of an affine algebraic variety. (cid:3) In section 2, we define the extended centre Z(G) of G. At a crucial point in the construction of the centre Z(G) of the category of smooth representations of G, certain quotients are made: we replace each ordinary quotient by the extended quotient to create the extended centre. In section 3 we define morita contexts, following [CDN]. In section 4 we prove that each ideal Hs(G) is Morita equivalent to a unital k-algebraoffinite type, wherek isthe coordinateringofacomplexaffinealgebraic variety. We think of the ideal Hs(G) as a noncommutative algebraic variety, and H(G) as a noncommutative scheme. In section 5we formulateourconjecture. We conjecture thateach idealHs(G) is geometrically equivalent (in a sense which we make precise) to the coordinate ring of a complex affine algebraic variety Xs: Hs(G)(cid:1)O(Xs)=Z(cid:4)s(G). The ring Z(cid:4)s(G) is the s-factor in the extended centre of G. The ideals Hs(G) therefore qualify as noncommutative algebraic varieties. We have stripped awaythe homologyand cohomologywhich play such a dom- inant role in [BN], [BP], leaving behind three crucial moves: Morita equivalence, morphisms which are spectrum-preserving with respect to filtrations, and deforma- tion of central character. These three moves generate the notion of geometric equivalence. In section 6 we prove the conjecture for all generic points s∈B(G). In section 7 we prove our conjecture for SL(2). Insection8wediscusssomegeneralfeaturesusedinprovingtheconjecturefor certain examples. In section 9 we review the asymptotic Hecke algebra of Lusztig. The asymptoticHeckealgebraJ playsavitalroleinourconjecture, aswenow proceed to explain. One of the Bernstein ideals in H(G) corresponds to the point i ∈ B(G), where i is the quotient variety Ψ(T)/W . Here, T is a maximal torus f in G, Ψ(T) is the complex torus of unramified quasicharactersof T, and W is the f finite Weyl group of G. Let I denote an Iwahori subgroup of G, and define e as follows: (cid:5) vol(I)−1 if x∈I, e(x)= 0 otherwise. Then the Iwahori ideal is the two-sided ideal generated by e: Hi(G):=H(G)eH(G).

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In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics
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