Noncommutative Geometry and Number Theory: Where Arithmetic meets Geometry and Physics

373 Pages·2006·6.38 MB·English

by  Anne-Marie Aubert

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).