Reduction Theory and Arithmetic Groups

374 Pages·2022·2.892 MB·English

by  Joachim Schwermer

ReductionTheoryandArithmeticGroups
Arithmeticgroupsaregeneralisations, tothesettingof algebraicgroupsover aglobal field,of the
subgroupsoffiniteindexinthegenerallineargroupwithentriesintheringofintegersofanalgebraic
numberfield.Theyarerich,diversestructures,andtheyariseinmanyareasofstudy.
Thistextenablesyoutobuildasolid,rigorousfoundationinthesubject.Itfirstdevelopsessential
geometricandnumberoftheoreticalcomponentstotheinvestigationsofarithmeticgroupsandthen
examinesanumberofdifferentthemes,includingreductiontheory,(semi)-stablelattices,arithmetic
groups in forms of the special linear group, unipotent groups and tori, and reduction theory for
adeliccoset spaces. Alsoincludedisathorough treatment of theconstruction ofgeometric cycles
in arithmetically defined locally symmetric spaces and some associated cohomological questions.
Writtenbyarenowned expert, thisbook willbeavaluable reference for researchers andgraduate
studentsalike.
JOACHIM SCHWERMERisEmeritusProfessorofMathematicsattheUniversityofVienna,and
recentlyGuestResearcherattheMax-Planck-InstituteforMathematics,Bonn.HewasDirectorofthe
Erwin-Schrödinger-InstituteforMathematicsandPhysics,Vienna,from2011to2016.Hisresearch
focuses on questions arising in the arithmetic of algebraic groups and the theory of automorphic
forms.
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Reduction Theory and Arithmetic Groups
JOACHIM SCHWERMER
UniversityofVienna
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Informationonthistitle:www.cambridge.org/9781108832038
DOI:10.1017/9781108937610
© JoachimSchwermer2023
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Firstpublished2023
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Names:Schwermer,Joachim,author.
Title:Reductiontheoryandarithmeticgroups/JoachimSchwermer.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2023.|
Series:Newmathematicalmonographs|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2022022858|ISBN9781108832038(hardback)|
ISBN9781108937610(ebook)
Subjects:LCSH:Arithmeticgroups.|Numbertheory.|
BISAC:MATHEMATICS/NumberTheory
Classification:LCCQA174.2.S3492023|DDC512.7/4–dc23/eng20220902
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Contents
Preface pageix
PARTI ARITHMETICGROUPSINTHEGENERAL
LINEARGROUP 1
1 Modules,Lattices,andOrders 3
1.1 Modules 4
1.2 Projective 𝑅-modules 7
1.3 Modulesoffractionsandlocalisation 10
1.4 Lattices 14
1.5 Integrality properties 18
1.6 Discretevaluation rings,Dedekinddomains,
andoverrings 20
1.7 ModulesoverDedekinddomains 26
1.8 Centralsimplealgebras 28
1.9 Ordersinalgebras 37
1.10 Non-abelian Galoiscohomology 43
2 TheGeneralLinearGroupoverRings 46
2.1 Elementarymatrices 47
2.2 ThestablestructureofGL𝑛 53
2.3 ThestablerangeofaDedekinddomain 59
2.4 Thestablerangeofordersindivision algebras 61
2.5 Anapplication: Mennicke symbolsandtheirproperties 63
3 AMenagerieofExamples:AHistorical Perspective 70
3.1 Reductiontheoryofquadratic forms 71
3.2 LatticesinEuclidean space 78
3.3 Unitgroupsofnumberfields 82
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vi Contents
3.4 Unitgroupsindivision algebras: thetheoremofKäteHey 83
3.5 Themodulargroup 85
4 ArithmeticGroups 90
4.1 RingsofS-integers:generalconcept andresults 91
4.2 Globalfields 93
4.3 RingsofS-integersinglobalfields 97
4.4 ArithmeticandS-arithmeticgroups 102
4.5 Arithmeticgroups:theirambientLiegroups 104
4.6 S-arithmetic groups:theirambientLiegroups 105
4.7 Thegenerallineargroupovertheringofadeles 108
4.8 Strongapproximation property andconsequences 113
4.9 ElementsoffiniteorderinGL𝑛(Z) 118
5 ArithmeticallyDefinedKleinianGroupsandHyperbolic3-Space 122
5.1 Kleiniangroupsactingonhyperbolic 3-space 122
5.2 Bianchigroups 124
5.3 ReductiontheoryforBianchigroups 126
5.4 Arithmetic groups originating from orders in quaternion
division algebras 135
PARTII ARITHMETICGROUPSOVERGLOBALFIELDS 139
6 Lattices:ReductionTheoryforGL𝒏 141
6.1 Thebasiccasesofaglobalfield:thenumberfieldQ
andarational function field 𝐹𝑞(𝑡) 142
6.2 Minkowskiinequalities orsuccessive minima 143
6.3 Mahler’scompactness criterion 150
7 ReductionTheoryand(Semi)-StableLattices 152
7.1 Euclidean Z-lattices 153
7.2 Arithmetic O𝑘-lattices 159
7.3 Canonical filtration,(semi)-stable lattices 164
7.4 Reductiontheoryandthecanonical filtrationforZ-lattices 167
7.5 Comparison 171
8 ArithmeticGroupsinAlgebraic 𝒌-Groups 174
8.1 Arithmeticgroups 175
8.2 ChevalleygroupschemesoverZ 178
8.3 Integral structuresforinner 𝑘-formsofSL𝑛 186
8.4 Integral structuresforarbitrary 𝑘-formsofSL𝑛 191
8.5 Divisionalgebras withprescribed localbehaviour 199
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Contents vii
9 ArithmeticGroups,AmbientLieGroups,andRelated
GeometricObjects 202
9.1 Homogeneous spaces, locallysymmetricspaces 203
9.2 S-arithmetic groupsandaffinebuildings 205
9.3 Arithmeticgroupsinunipotent groups 206
9.4 Arithmeticgroupsinalgebraic 𝑘-tori 211
9.5 Godement’s compactness criterion 219
9.6 Constructions ofcompactornon-compact arithmeticquotients 232
10 GeometricCycles 237
10.1 Construction ofgeometriccycles 238
10.2 Orientability 241
10.3 Intersection numbers, excessbundles, andEulernumbers 245
11 GeometricCyclesviaRationalAutomorphisms 249
11.1 Prelude 250
11.2 Fixedpointsandnon-abelian Galoiscohomology 252
11.3 Intersection numbersofspecial geometriccycles 256
11.4 TheEulernumberoftheexcessbundle 258
11.5 Non-vanishing of the intersection number of two geometric
cycles 268
11.6 Construction ofcohomology classes:anoutlook 272
12 ReductionTheoryforAdelicCosetSpaces 274
12.1 Preliminaries: adeliccosetspaces 275
12.2 Theadelegroups𝐺(A𝑘) and𝐺(A𝑘)1 276
12.3 Adelicheightsandtheirproperties 279
12.4 ReductiontheoryforGL𝑛:Minkowskirevisited 285
12.5 Compactness criterionandSiegeldomains 289
12.6 Thecaseofconnected reductive 𝑘-splitgroups:asketch 291
PARTIII APPENDICES 297
AppendixA LinearAlgebraicGroups:AReview 299
A.1 Affine 𝑘-groupschemeswith 𝑘 aring 299
A.2 Hopfalgebrasandaffine 𝑘-group schemes 304
A.3 Smoothness 307
A.4 Operations andrepresentations 308
A.5 Restrictionandinduction of𝐺-modules 311
A.6 Cohomology of𝐺-modules 313
A.7 Weilrestriction orrestrictionofscalars 314
A.8 Unipotent groups 317
A.9 Diagonalisable andmultiplicative groups 319
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viii Contents
A.10 Algebraic 𝑘-tori 321
A.11 Reductivegroups 322
A.12 Formsofalgebraicgroups 324
AppendixB GlobalFields 326
B.1 Absolutevaluesandlocalfields 326
B.2 Globalfields 328
B.3 Restrictedproducts oftopological spaces 330
B.4 Theringofadeles 331
B.5 Theidelegroup 333
B.6 S-integers andS-units 334
AppendixC TopologicalGroups,HomogeneousSpaces,
andProperActions 335
C.1 Topological groups 335
C.2 Topological transformation groups 339
C.3 Locallycompacttransformation groups 341
C.4 Propermaps 341
C.5 Properactions oftopological groups 343
C.6 Characterisation ofproperactions 346
References 350
Index 357
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Preface
Arithmetic groups are generalisations, to the setting of algebraic groups defined
over a global field, of the subgroups of finite index in the general linear group
GL𝑛(Λ) with entries in the ring of integers Λ of an algebraic number field 𝑘 or,
moregenerally, anorderΛinafinite-dimensional divisionalgebraover 𝑘.Histori-
cally suchgroups arosenaturally inthestudyofarithmetic properties ofquadratic
forms.Thestudyofreduction ofsuchforms,asdeveloped byGauss,Hermite,and
Minkowski, amongothers, gaveapowerfulwaytoselect, fromtheinfinitelymany
forms, that are integrally equivalent to a given form, one that is intrinsically char-
acterised by suitable conditions on its entries. Minkowski, following a suggestion
made by Gauss in 1831, created a new version of reduction theory by working
with lattices as geometric objects. His works, especially his geometric point of
view, served as substantial stimuli for Siegel’s studies of quadratic, symplectic, or
Hermitianformsandtheirassociateddiscontinuous groups.Sincethedevelopment
of the general theory of linear algebraic groups over fields, it has been natural to
view arithmetic groups as a rich integral extension of that algebro-geometric the-
ory.Thus,thankstotheworkofChevalley,Borel,Serre,Harder,andRaghunathan,
the study ofarithmetic groups today can start from the theory of algebraic groups
definedoverafield𝑘,whichiseitheranalgebraicnumberfieldorafiniteseparable
extension of 𝐹𝑞(𝑡),where 𝐹𝑞 isafinitefieldand𝑡 istranscendental over 𝐹𝑞,i.e. 𝑘
isaglobalfield.
An arithmetic group Γ acts on a homogeneous space which is defined by the
ambientalgebraicgroup.Thisactionandthestudyoftheorbitspaceareofinterest,
bothintrinsicallyandfortheinsightintothestructureofΓ.Thus,thereisanessential
geometric component to the investigations of these groups. Further, arithmetic
groups arise in a wide variety of mathematical contexts, ranging from differential
geometry,inparticular,thetheoryoflocallysymmetricspaces,topology,geometric
group theory, to number theory and arithmetic algebraic geometry, the theory of
automorphic formsoverglobalfields,andevenlatelyquantum computing.
https://doi.org/10.1017/9781108937610.001 Published online by Cambridge University Press
x Preface
Therefore, on theone hand, ‘Arithmetic Groups’ donot present themselves asa
coherent limitedtheory,sothatabookdealingwiththisareafacesachallenge. On
theother hand, there aresomeoverarching results andconstructions, inparticular,
regarding reduction theory, whichare fundamental tomany oftheareas discussed
above and which are likely to be used in different contexts. Thus, as a resolution,
inspired by the approach in Carl E.Schorske’s Fin-de-Siècle Vienna, each chapter
ofthismonographis‘issuedfromaseparateforayintotheterrain,varyinginscale
andfocusaccording tothenatureoftheproblem’.
Itisintendedtolayrigorousandsolidgroundwork indealingwithS-arithmetic
groupsinalgebraicgroupsdefinedoveraglobalfield 𝑘.Attheoutset,weexamine
the fundamental case ofthe general linear group. Additionally, wesurvey someof
thehistoricalsources,whichmightbehelpfulforunderstanding thegenesisofthis
mathematical area.
Webeginwithananalysisofthenormalsubgroupstructureofthegenerallinear
group GL𝑛 overa(non-commutative) ringwithidentity andanintroduction ofthe
basicconceptsregarding S-arithmeticgroupsinGL𝑛.Bydefinition,theringO𝑘,S
ofS-integersinaglobalfield 𝑘,associated withafinitesetS ofplacesof 𝑘 which
includes thearchimedean onesinthecaseofanalgebraic number field,isthering
ofelementsof𝑘integralateachplaceoutsideofS.If𝑘isanalgebraicnumberfield
andSconsistsonlyofthearchimedeanplacesof𝑘,theringofS-integersin𝑘isthe
usualringofintegersO𝑘 in𝑘.AnS-arithmeticsubgroupof𝐺𝐿𝑛(𝑘)isdefinedtobe
asubgroupwhichiscommensurablewith𝐺𝐿𝑛(O𝑘,S).Anyideal𝔮inO𝑘,S givesrise
totheprincipalS-congruencesubgroupGL𝑛(O𝑘,S,𝔮)oflevel𝔮.Itisdefinedasthe
kernel ofthegrouphomomorphism GL𝑛(O𝑘,S) −→ GL𝑛(O𝑘,S/𝔮),thusanormal
subgroup of finite index in the S-arithmetic group GL𝑛(O𝑘,S). Any S-arithmetic
subgroup that contains a principal S-congruence subgroup for some ideal 𝔮 is
called a congruence subgroup. We indicate how S-arithmetic groups, depending
on the form of S, can be naturally viewed as discrete subgroups in a reductive
(cid:2)
Lie group, real, 𝑝-adic or product of such groups, to be denoted 𝑣∈S𝐺𝑣 =:
𝐺S.Foreachplace𝑣 ∈ S,thereisacorresponding homogeneousspace 𝑋𝑣,andan
S-arithmetic sub(cid:2)group, viewedasadiscretesubgroup of𝐺S,naturally actsonthe
product 𝑋S := 𝑣∈S 𝑋𝑣. The resulting orbit spaces are the objects that concern
us. Having these essentials in place, we follow different thematic branches. As a
conclusiontothefirstpartofthebook,pointingtowardsthegeometricperspective,
wediscuss reduction theory inthe case ofarithmetically defined subgroups of the
groupoforientationpreservingisometriesofhyperbolic3-spaceandstudytheorbit
spaces.
The second part begins with the uniform construction of Siegel sets in the case
GL𝑛 overthebasic cases ofaglobal field,namely, thefield Qofrational numbers
or the field 𝐹𝑞(𝑡) of rational functions in the variable 𝑡 and having coefficients in
https://doi.org/10.1017/9781108937610.001 Published online by Cambridge University Press