Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday

655 Pages·2006·3.085 MB·English

by  Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

Progress in Mathematics
Volume253
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
Algebraic Geometry
and Number Theory
In Honor of Vladimir Drinfeld’s
50th Birthday
Victor Ginzburg
Editor
Birkha¨user
Boston • Basel • Berlin
VictorGinzburg
UniversityofChicago
DepartmentofMathematics
Chicago,IL60637
U.S.A.
[email protected]
MathematicsSubjectClassification(2000):03C60,11F67,11M41,11R42,11S20,11S80,
14C99,14D20,14G20,14H70,14N10,14N30,17B67,20G42,22E46(primary);
05E15,11F23,11G45,11G55,11R39,11R47,11R58,14F20,14F30,14H40,14H42,14K05,
14K30,14N35,22E67,37K20,53D17(secondary)
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VladimirDrinfeld
Preface
VladimirDrinfeld’smanyprofoundcontributionstomathematicsreflectbreadthand
greatoriginality. Thetenresearcharticlesinthisvolume,coveringadiversityoftopics
predominantlyinalgebraandnumbertheory,reflectDrinfeld’svisioninsignificant
areasofmathematics,andarededicatedtohimontheoccasionofhis50thbirthday.
ThepaperbyGoncharovandFockisdevotedtothestudyofclustervarietiesand
theirquantizations. ThissubjecthasitsoriginsintheworkofFominandZelevinsky
on cluster algebras and total positivity on the one hand, and, on the other hand, on
various attempts to understand Kashiwara’s theory of crystals and quantizations of
modulispacesofcurves.
StartingwithasplitsemisimplerealLiegroupGwithtrivialcenter,Goncharov
and Fock define a family of varieties with additional structures called cluster X-
varieties. These varieties have a natural Poisson structure. The authors define a
PoissonmapfromaclustervarietytothegroupGequippedwiththestandardPoisson–
LiestructureasdefinedbyV.Drinfeld. ThemapisbirationalandthusprovidesGwith
canonicalrationalcoordinates. Further,GoncharovandFockshowhowtoconstruct
complicatedclusterX-varietiesfrommoreelementaryonesusinganamalgamation
procedure. This is used, in particular, to produce canonical (Darboux) coordinates
forthePoissonstructureonaZariskiopensubsetofthegroupG.
SomeoftheclustervarietiesareverycloselyrelatedtothedoubleBruhatcells
studiedbyA.Berenshtein,S.Fomin,andA.Zelevinisky. Ontheotherhand,theresults
ofthepaperplayakeyroleindescribingtheclusterstructureofthemodulispaces
oflocalsystemsonsurfaces,asstudiedbyGoncharovandFockinanearlierwork.
TheimportantroleofDrinfeld’sideas—indeed,oneofthecentralthemesofhis
research—isevidentinthepaperbyFrenkelandGaitsgory,whichisdevotedtothe
(local) geometric Langlands correspondence from the point of view of D-modules
andtherepresentationtheoryofaffineKac–Moodyalgebras.
LetgbeasimplecomplexLiealgebraandGaconnectedalgebraicgroupwith
Lie algebra g. The affine Kac–Moody algebra(cid:1)g is the universal central extension
oftheformalloopagebrag((t)). Representationsof (cid:1)g haveaparameter,aninvar-
iant bilinear form on g, which is called the level. Representations corresponding
to the bilinear form that is equal to minus one-half of the Killing form are called
viii Preface
representations of critical level. Such representations can be realized in spaces of
globalsectionsoftwistedD-modulesonthequotientoftheloopgroupG((t))byits
“compact’’subgroupK equaltoG[[t]],ortotheIwahorisubgroupI.
ThisworkbyFrenkelandGaitsgoryisthefirstinaseriesofpapersdevotedto
thestudyofthecategoriesofrepresentationsoftheaffineKac–Moodyalgebra(cid:1)gof
thecriticallevelandD-modulesonG((t))/K fromthepointofviewofageometric
versionofthelocalLanglandscorrespondence. ThelocalLanglandscorrespondence
sets up a relation between two different types of data. Roughly speaking, the first
data consist of the equivalence classes of homomorphisms from the Galois group
ofalocalnon-archimedeanfieldKtoG(C)∨,theLanglandsdualgroupofG. The
seconddataconsistoftheisomorphismclassesofirreduciblesmoothrepresentations,
denotedbyπ,ofthelocallycompactgroupG(K).
Anaiveanalogueofthiscorrespondenceinthegeometricsituationseekstoassign
toaG(C)∨-localsystemontheformalpunctureddiscarepresentationoftheformal
loopgroupG((t)). However,theauthorsshowthatincontrasttotheclassicalsetting,
thisrepresentationofG((t))shouldbedefinednotonavectorspace,butonacategory,
asexplainedinthepaper.
InthecontributionbyIhara,andinthecloselyrelatedappendixbyTsfasman,the
authorsstudytheζ-functionζK(s)ofaglobalfieldK. Specifically,theyareinterested
intheso-calledEuler–KroneckerconstantγK, arealnumberattachedtothepower
seriesexpansionoftheζ-functionatthepoints =1. Inthespecialcaseofthefield
K=Qofrationalnumbers,thisconstantreducestotheEulerconstant
(cid:2) (cid:3)
1 1 1
γQ = lim 1+ + +···+ −logn .
n→∞ 2 3 n
TheconstantγKplaysanimportantroleinanalyticnumbertheory. Ontheother
hand,forK=F (X),thefieldofrationalfunctionsonacompletealgebraiccurveX
q
overafinitefield,thecorrespondingEuler–Kroneckerconstantiscloselyrelatedto
thenumberofF -rationalpointsofX.
q
IharaaddressesthequestionofhownegativetheconstantγKmaybe,depending
on the field K. In the number field case, this happens when K has many primes
withsmallnorm. Inthefunctionfieldcase, thereareknowntowersofcurveswith
manyFq-rationalpoints;theauthorstudiesthebehaviorofγKusingthegeneralized
Riemannhypothesis. Inthisway,heobtainsveryinterestingexplicitestimatesofγK.
Forinstance,inthecaseK=F (X)Iharaestablishesanupperbound
q
γK ≤2log((g−1)logq)+logq,
whereg denotesthegenusofthecurveX. Healsoobtainssimilarestimatesforthe
lowerbound.
HrushovskiandKazhdanintheirpaperlaythefoundationsofintegrationtheory
over, not necessarily locally compact, valued fields of residue characteristic zero.
A valued field is a field K equipped with a “ring of integers’’O ⊂ K, satisfying
the property that K = O∪(O\{0})−1. In particular, the authors obtain new and
base-fieldindependentfoundationsforintegrationoverlocalfieldsoflargeresidue
characteristic,extendingresultsofDenef,Loeser,andCluckers.
Preface ix
The work of Hrushovski and Kazhdan is on the border of logic and algebraic
geometry. Their methods involve an analysis of definable sets. Specifically, they
obtainaprecisedescriptionoftheGrothendiecksemigroupofdefinablesetsinterms
ofrelatedgroupsovertheresiduefieldandvaluegroup. Thisyieldsnewinvariantsof
alldefinablebijections,aswellasinvariantsofmeasure-preservingbijections. Their
results are intended to be applied to the construction of Hecke algebras associated
withreductivegroupsoveranotnecessarilylocallycompactvaluedfield. Inthecase
ofatwo-dimensionallocalfield,thecorrespondingHeckealgebraisexpectedtobe
closelyrelatedtothedoubleaffineHeckealgebraintroducedbyCherednik.
Kisin’spaperisdevotedtop-adicalgebraicgeometryandnumbertheory. This
subjectisrapidlydevelopingatthispointintime. FollowingtheideasofBergerand
Breuil, Kisin gives a new classification of crystalline representations. The objects
involvedmaybeviewedaslocal,characteristic0analoguesofthe“shtukas’’intro-
ducedbyDrinfeld. Kisinalsogivesaclassificationofp-divisiblegroupsandfinite
flatgroupschemes,conjecturedbyBreuil. Furthermore,heshowsthatacrystalline
representationwithHodge–Tateweights0,1arisesfromap-divisiblegroup—aresult
conjecturedbyFontaine.
Letkbeaperfectfieldofcharacteristicp >0,W =W(k)itsringofWittvectors,
K = W(k)[1], and K : K a finite totally ramified extension. Breuil proposed a
0 p 0
new classification of p-divisible groups and finite flat group schemes over the ring
of integers OK of K. For p-divisible groups and p > 2, this classification was
established in an earlier paper by Kisin, who also used a variant of Breuil’s theory
todescribeflatdeformationrings,andtherebyestablishamodularityliftingtheorem
forBarsotti–TateGaloisrepresentations.
Inthepresentpaper,theauthorgeneralizesBreuil’stheorytodescribecrystalline
representationsofhigherweightor,equivalently,theirassociatedweaklyadmissible
modules.
Krichever’spaperanalyzesdeepandimportantrelationsbetweenthetheoryofin-
tegrablesystemsandtheRiemann–Schottkyproblem. TheRiemann–Schottkyprob-
lemonthecharacterizationoftheJacobiansofcurvesamongabelianvarietiesismore
than120yearsold. QuiteafewgeometricalcharacterizationsoftheJacobianshave
beenfound. Noneofthem,however,providesanexplicitsystemofequationsforthe
imageoftheJacobianlocusintheprojectivespaceunderthelevel-2thetaimbedding.
Thelinkofthisproblemtointegrablesystemswasfirstdiscoveredinthe1980s.
Specifically,T.ShiotaestablishedthefirsteffectivesolutionoftheRiemann–Schottky
problem, known as Novikov’s conjecture. The conjecture says the following: An
indecomposable principally polarized abelian variety (X,θ) is the Jacobian of a
curveofagenusgifandonlyifthereexistg-dimensionalvectorsU (cid:7)=0,V,W such
thatthefunction
u(x,y,t)=−2℘2lnθ(Ux+Vy+Wt +Z)
x
isasolutionoftheKadomtsev–Petviashvilii(KP)equation
3u =(4u +6uu −u ) .
yy t x xxx x
x Preface
(Hereθ(Z)=θ(Z|B)istheRiemannthetafunction.)
Inthepresentpaper,Kricheverprovesthatanindecomposableprincipallypolar-
izedabelianvarietyXistheJacobianofacurveifandonlyifthereexistvectorsU (cid:7)=
0,V suchthattherootsx (y)ofthethetafunctionalequationθ(Ux+Vy+Z)=0
i
satisfy the equations of motion of the formal infinite-dimensional Calogero–Moser
system.
ThemaingoalofLaumon’spaperistoidentifythefibersoftheaffineSpringer
resolutionforthegroupGL withcoveringsofcompactifiedJacobiansofprojective
n
singular curves. This work is part of a more general project of obtaining a geo-
metric version of the “Fundamental Lemma’’that appears in Langlands’works on
automorphicforms.
LetF bealocalnon-archimedeanfieldofequalcharacteristic,letO beitsring
F
of integers, and let k be the residue field. Let E be a finite-dimensional F-vector
space. TheauthorconsiderstheaffineGrassmannianformedbyO -latticesM inE.
F
Given a regular semisimple and topologically nilpotent endomorphism γ of E,
one defines the affine Springer fiber, X , as the closed reduced subscheme of the
γ
affineGrassmannianformedbytheγ-stablelatticesM ⊂ E. KazhdanandLusztig
haveshownthatX isascheme,locallyoffinitetypeoverk. Moreover,thisscheme
γ
comesequippedwithanaturalfreeactionofanabelianalgebraicgroup(cid:6) suchthat
γ
thequotientZ =X /(cid:6) isaprojectivek-scheme.
γ γ γ
Inhispaper,theauthorattachestoγ aprojectivealgebraiccurveC overkwith
γ
asinglesingularpointsuchthatthecompletedlocalringatthispointisisomorphic
toO [γ]⊂F[γ]⊂Aut (E). Furthermore,theauthorrelatesthevarietiesX and
F F γ
Z withthecompactifiedJacobianofthecurveC . Thisallowshimtoreprovesome
γ γ
irreducibility results about compactified Jacobians due toAltman and Kleiman. In
addition,thetechniquesdevelopedinthepaperprovideanapproachtoanimportant
“purityconjecture’’concerningthecohomologyofcertainaffineSpringerfibers,due
toGoresky,Kottwitz,andMacPherson.
ThegoaloftheworkofManinpresentedinthisvolumeistostudypropertiesof
theiteratedintegralsofmodularformsintheupperhalf-plane. Thissettinggeneral-
izessimultaneouslythetheoryofmodularsymbolsandthatofmultiplezetavalues.
Multiplezetavaluesarethenumbersgivenbythek-multipleDirichletseries
(cid:4)
1
ζ(m ,...,m )= (0.1)
1 k nm1···nmk
0<n1<···<nk 1 k
or,equivalently,bythem-multipleiteratedintegralsm=m +···+m ,
1 k
(cid:5) (cid:5) (cid:5) (cid:5)
ζ(m ,...,m )= 1 dz1 z1 dz2 z2··· zmk−1 dzmk ··· . (0.2)
1 k z z 1−z
0 1 0 2 0 0 mk
Multiplezetavaluesareinterestingbecausetheyandtheirgeneralizationsappear
inmanydifferentcontextsinvolvingmixedTatemotives,deformationquantization
(Kontsevich),knotinvariants,etc.
Multiplezetavaluessatisfycertaincombinatorialrelations,calleddouble-shuffle
relations. The relations in question can be succinctly written in terms of formal
Preface xi
generating series for (regularized) iterated integrals (0.2). Such integrals appeared
more than 15 years ago in the celebrated work by Drinfeld on what is nowadays
knownastheDrinfeldassociator. However,thequestionaboutinterdependenceof
(double-)shuffleandassociatorrelationsdoesnotseemtobesettledatthemoment.
Inthepaper, theauthordefines1-formsofmodularandcuspmodulartypeand
studies iterated integrals and the total Mellin transform for families of such forms.
ThefunctionalequationforthetotalMellintransformisdeduced. Thisresultextends
theclassicalfunctionalequationforLseries. Theauthoralsointroducesaniterated
modular symbol as a certain noncommutative 1-cohomology class of the relevant
subgroupofthemodulargroup. Thepaperestablishessomeanaloguesoftheclassical
identity (0.1) = (0.2) but different from it in two essential respects. First, iterated
integrals are only linear combinations of certain multiple Dirichlet series. Second,
theidentitiesobtainedinthepaperinvolveintegralswhicharenotoftheusualtype,
(cid:4) a ···a
1,n1 n,nk;
nm1···nmk
0<n1<···<nk 1 k
infact,theircoefficientsdependonpairwisedifferencesn −n .
j i
In the paper by Eskin and Okounkov, the authors prove that natural generating
functionsforenumerationofbranchedcoveringsofthepillowcaseorbifoldarelevel-
2quasimodularforms. Thisgivesusawaytocomputethevolumesofthestrataof
themodulispaceofquadraticdifferentials.
ConsideracomplextorusT2 =C/L,whereL⊂Cisalattice. Itsquotient
P =T2/±1
by the automorphism z (cid:8)→ −z is a sphere with four (Z/2)-orbifold points, which
is sometimes called the pillowcase orbifold. The map T2 → P is essentially the
Weierstraß℘-function. Thequadraticdifferential(dz)2onT2descendstoaquadratic
differentialonP. ViewedasaquadraticdifferentialontheRiemannsphere, (dz)2
hassimplepolesatcornerpoints.
Let µ be a partition and ν a partition of an even number into odd parts. The
authorsareinterestedinenumerationofdegree2d maps
π :C →P (0.3)
withthefollowingramificationdata. Viewedasamaptothesphere, π hasprofile
(ν,2d−|ν|/2)over0∈P andprofile(2d)overtheotherthreecornersofP.Addition-
ally,π hastheprofile(µi,12d−µi)oversome(cid:8)(µ)givenpointsofP andunramified
elsewhere. Here(cid:8)(µ)isthenumberofpartsinµ.
ThepaperbySchechtmanmaybeviewedasacontinuationoftheworkbyGor-
bunov,Malikov,andSchechtmanonthechiraldeRhamcomplex. Specifically,the
paperinthevolumeintroducesacertainchiralanalogueofthethirdChern–Simons
classofavectorbundle.
·
LetXbeasmoothvarietyoverafieldkofcharacteristiczero,andwrite(cid:9) for
X
thedeRhamcomplexofX. AssociatedwithanyvectorbundleE onX,onehasthe
correspondingChernclasses