Special values of L-functions for orthogonal groups

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Number theory/Harmonic analysis

Special values of L-functions for orthogonal groups

Valeurs spéciales de fonctions L pour les groupes orthogonaux

Chandrasheel Bhagwat

¹

, A. Raghuram

IndianInstituteofScienceEducationandResearch,Dr.HomiBhabhaRoad,Pashan,Pune411008,India

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received18July2016

Acceptedafterrevision18January2017 Availableonline6February2017 PresentedbytheEditorialBoard

This is an announcement of certain rationality results for the critical values of the degree-2n L-functionsattachedto GL1×^SO(n,n) overQ^{for aneven positiveinteger n.}

Theprooffollowsfromstudyingtherank-oneEisensteincohomologyforSO(n+¹,n+¹).

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

r é s um é

Dans cette Note,nous présentonsdes résultatsde rationalité pour lesvaleurscritiques desfonctionsLdedegré2n,attachéesàGL1×^SO(n,n)surQ^{,oùnestunentierpositif.La} preuverésulted’uneétudedelacohomologied’Eisensteinderangun,pourSO(n+¹,n+¹).

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introductionandstatementofthemainresult

Tomotivatethemainresult,letusrecallawell-knowntheoremofShimura[13].

Theorem1.1(Shimura).Letf=

a_nqⁿ∈^Sk

(

N

, χ )

andg=

b_nqⁿ∈^Sl

(

N

, ψ)

beprimitivemodularformsofweightsk andl,with nebentypuscharacters

χ

and

ψ

for

0

(

N

)

.LetQ

(

f

,

g

)

bethenumberfieldobtainedbyadjoiningtheFouriercoefficients{^an}^and {^bn}^toQ^.Assumethatk

>

l.Let

DN

(

s

,

f

,

g

) :=

^LN

(

2s

+

²

−

^k

−

^l

, χ ψ )

∞

n=1

a_nb_n n^s

bethedegree4Rankin–SelbergL-functionattachedtothepair

(

f

,

g

)

.Then,foranyintegerm withl≤^m

<

k,wehave:

D_N

(

m

,

f

,

g

) ≈ (

2

π

ⁱ

)

^l+1−2m

g(ψ )

u⁺

(

f

)

u⁻

(

f

),

E-mailaddresses:[email protected](C. Bhagwat),[email protected](A. Raghuram).

1 C.B.ispartiallysupportedbyDST-INSPIREFacultyscheme,awardnumber[IFA-11MA-05].

http://dx.doi.org/10.1016/j.crma.2017.01.016

1631-073X/©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

where≈^{meansuptoanelementof}Q

(

f

,

g

)

,u^±

(

f

)

arethetwoperiodsattachedto f byShimura,andg(ψ)istheGausssumof

ψ

. Furthermore,theratiooftheL-valueinthelefthandsidebytherighthandsideisequivariantunderGal

(Q/Q)

^.

Theintegersl≤^m

<

kareallthecriticalpointsforDN

(

s

,

f

,

g

)

.(Therearenocriticalpoints ifl=^k.)Supposek≥^l+^2, andwe lookattwo successivecritical valuesthenthe onlychange intheright-hand side is

(

2πⁱ

)

⁻² whichmaybe seen to be exactly accounted forby the

-factors at infinity. Suppose L

(

s

,

f×g

)

denotes thecompleted degree-4 L-function attachedto

(

f

,

g

)

,normalizedinaclassicalwayasinthetheoremabove,thenwededuce:

L

(

l

,

f

×

g

) ≈

L

(

l

+

1

,

f

×

g

) ≈ · · · ≈

L

(

k

−

1

,

f

×

g

).

(1.2) Theaboveresultisastatement forL-functionsforGL2×^GL2 overQ^{.LaterShimura}generalizedthistoHilbertmodular forms[14],i.e.,forGL2×^GL2 overatotallyrealfield F.Notethat

(

GL2×^GL2

)/

GL1^GSO

(

2

,

2

)

,i.e.Shimura’sresultmay beconstruedasatheoremforL-functionsfororthogonalgroupsinfourvariables.

Themainaimofthisarticleistoannouncethatweexpectthefollowinggeneralizationoftheresultin(1.2)to L-functionsfor GL1×^SO

(

n

,

n

)

overatotallyrealfieldF ,andwhenn=^2r≥^{2.Forsimplicityof}exposition,wewillworkover F=Q^.

Theorem1.3.Letn=^2r≥^{2beanevenpositiveinteger.ConsiderSO}

(

n

,

n

)/

Q^{definedsothatthesubgroupofall}upper-triangular matricesisaBorelsubgroup.Let

μ

beadominantintegralweightwrittenas

μ

=

( μ

1≥

μ

2≥ · · · ≥

μ

n−1≥ |

μ

n|)^,with

μ

j∈Z^.Let

σ

beacuspidalautomorphicrepresentationofSO

(

n

,

n

)/

Q^.Assumethat:

(1)theArthurparameter

σiscuspidalonGL_2n

/Q

^; (2)

σ

isgloballygeneric;

(3)

σ

_{∞|SO(n,n)(R)}0isadiscreteseriesrepresentationwithHarish–Chandraparameter

μ

+

ρ

n.

Let^◦

χ

beafiniteordercharacterofQ^×\A^{×.Thenthecriticalsetforthe}degree-2n completedL-functionL

(

s

,

^◦

χ

_×

σ )

isthefiniteset ofcontiguousintegers

{

¹

− | μ

n

|,

²

− | μ

n

|, . . . , | μ

n

|}.

Assumealsothat|

μ

n|≥^{1,sothatthecriticalsetisnonempty;andinthiscasethereareatleasttwocriticalpoints.Wehave}

L

(

1

− | μ

n

|,

^◦

χ × σ ) ≈

^L

(

2

− | μ

n

|,

^◦

χ × σ ) ≈ · · · ≈

^L

(| μ

n

|,

^◦

χ × σ ),

where ≈ ^{meansuptoanelementofa numberfield} Q

(

^◦

χ , σ )

,andfurthermore,allthesuccessiveratiosareequivariantunder Gal

(Q/Q)

^.

2. Thecombinatoriallemmaandarestatementofthemaintheorem

The strategyofproof followstheparadigm inHarder–Raghuram[7,8].Inoursituation,thisinvolvesstudyingtherank- one EisensteincohomologyofG=^SO

(

n+¹

,

n+¹

)

,especiallythecontributioncoming froma parabolicsubgroup P with LeviquotientM_P=^GL1×^SO

(

n

,

n

)

.Asinloc.cit.,certainWeylgroupcombinatoricsplayanimportantrole—essentiallysaying thataparticularcontextinvolvingthecohomologyofarithmeticgroupsisviableexactlywhentheinterveningL-valuesare critical.

Lemma2.1.Let

μ

=

( μ

1≥

μ

2≥ · · · ≥

μ

n−1≥ |

μ

n|)^{beadominantintegralweight,and}

σ

beacuspidalautomorphicrepresentation forSO

(

n

,

n

)/Q

^asinTheorem 1.3.Letd∈Z^andput

χ

= ||^−d⊗^◦

χ

where^◦

χ

isafinite-ordercharacter.LetG=^SO

(

n+¹

,

n+¹

)

andP themaximalparabolicsubgroupobtainedbydeletingthe‘first’simpleroot,inwhichcasetheLevidecompositionP=^MPNP lookslike:MP=^GL1×^SO

(

n

,

n

)

anddim

(

NP

)

=^{2n.Thefollowingare}equivalent:

(1) −^{n and1}−^{n arecriticalforthecompleted}degree-2n L-functionL

(

s

, χ

_×

σ )

; (2) 1− |

μ

n|≤ ⁿ+^d ≤ |

μ

n|−^1;

(3)thereisauniquew∈^{WP(hereWPisthesetofKostant}representativesforP ;wehaveW_G=^WMPW^P)suchthatw⁻¹·(^d×

μ )

isdominantforG andl

(

w

)

=^dim

(

NP

)/

2.

Asdrunsthroughtherangeprescribedby(2),theratioofcriticalvalues L

( −

ⁿ

, χ _× σ )

L

(

1

−

n

, χ × σ )

(wherethecriticalityisassuredby(1))runsthroughthesetofallsuccessiveratiosofcriticalvalues

L

(

1

− | μ

n

|,

^◦

χ × σ )

L

(

2

− | μ

n

| ,

^◦

χ _× σ ) , . . . ,

^L

(| μ

n

| −

¹

,

^◦

χ × σ )

L

( | μ

n

| ,

^◦

χ _× σ )

.

This saysthat when the method ofEisenstein cohomologyis invokedfor rationality results,then we get a resultfor ratiosofallpossiblesuccessivecriticalvalues,nomoreandnoless!TowardsTheorem 1.3,weprovethefollowingtheorem.

Theorem2.2.Letthenotationson

χ

and

σ

beasinthelemmaabove,andassumethattheconditionsond aresatisfied.Thenthe quantity

c_∞

( χ

_∞

, σ

_∞

)

^Lf

(−

ⁿ

, χ × σ )

L_f

(

1

−

n

, χ × σ )

isalgebraicandisGal

(

Q

/

Q

)

-equivariant.(Herec_∞

( χ

_∞

, σ

_∞

)

isanonzerocomplexnumberthatdependsonlyonthedataatinfinity, andL_fisthefinitepartoftheL-function.PleaserefertoSection4formoredetails.)

3. Commentsontheconsequencesofvarioushypothesesofthemaintheorem 3.1. Adiscreteseriesrepresentationasthelocalrepresentationatinfinity

Thisis thesimplestkindofrepresentationwithnontrivialrelative Liealgebra cohomology;infact, ithasnonzero co- homology onlyin the middledegree.Furthermore, thisimpliesthat the finitepart

σ

f contributesto thecohomology of a locally symmetric space of SO

(

n

,

n

)

with coefficients in the local system attachedto

μ

. Using arguments asin Gan–

Raghuram[5],we show that

σ

_f isdefinedover a numberfield Q

( σ )

andthat thereis aGal

(

Q¯

/

Q

)

-actionon theset of cuspidalrepresentationsthatsatisfiesthehypotheses(1),(2)and(3).Intheproof,weneedtouseArthur’swork[1].Inthe statementofthetheoremabove,Q

(

^◦

χ , σ )

isthefieldgeneratedbythevaluesof^◦

χ

andQ

( σ )

.

3.2. Thetransfer

σiscuspidalonGL_2n

/Q

Thisisneededfortworeasons.(1)We donotwantthe L-function L

(

s

,

^◦

χ

_×

σ )

to breakupintosmaller L-functions;

although,evenifitdid,withaninductiveargument,atleastinthecasewhen

_σ istempered,wewouldverylikelystill havethemaintheorem.(2) Thesecondreasonisfarmoreseriousandverydelicate.Weneedtoprovea‘Manin–Drinfeld’

principle:thatthereisaHecke-projectionfromthetotalboundarycohomology(oftheBorel–Serreboundary)totheisotypic componentoftherepresentationofGinducedfrom

χ

⊗

σ

ofMP.SeeSection4below.Forthistowork,wehavetoexclude thepossibilityof

σ

being,forexample,aCAPrepresentation(whichalsogetsguaranteedbythenexthypothesis).

3.3.

σ

isgloballygeneric

This hypothesis plays several roles: itis used in proving the existence of the Galoisaction mentioned in Section 3.1 above.Shahidi’sresults[12]onlocalconstants(seeSection4below)needgenericityoftherepresentationatinfinity.

3.4. CompatibilitywithDeligne’sconjecture

TheabovetheoremiscompatiblewithDeligne’sconjecture[4]onthecriticalvaluesofmotivic L-function,becausewe havethefollowingperiodrelation:LetM beapureregularmotiveofrank-2n overQ^{withcoefficientsinanumberfield E.Suppose} M isoforthogonaltype(i.e.thereisamapSym²

(

M

)

→Q

(

−^w

)

wherewisthepurityweightofM),thenDeligne’speriodsc^±

(

M

)

are relatedas:

c⁺

(

M

) =

^c−

(

M

),

as elements of

(

E

⊗ C )

^×

/

E^×

.

ThiswasknownifM isatensorproductoftworank-twomotives;seeBlasius[3,2.3].

3.5. Langlandstransferandspecialvalues

ItisimportanttoprovethistheorematthelevelofL-functionsforGL1×^SO

(

n

,

n

)

,andnotasL-functionsforGL1×^GL2n aftertransferring. Wewouldseethissubtlepoint alreadyinthecontext ofShimura’stheorem,because (i)the Langlands transfer f^{g,whichisa cuspidal}representationofGL₄ doesnot seethePeterssonnorm^f

,

f^{ofonlyone ofthecon-} stituents;and(ii)foranL-functionL

(

s

,

π

)

withπ^cuspidalonGL4

/

Q^{,successiveL-valueswouldseec+}

(

π

)

andc⁻

(

π

)

,and intheautomorphicworld,itisnot(yet)knownthatifπ^{cameviatransferfromGL}2×^GL2 thenc⁺

(

π

)

≈^c−

(

π

)

.Inasimilar vein,onemayaskifthemainresultof[8]appliedtoGL1×^GL2n impliesthemainresultofthispaper;thiswouldbesoif wecouldprovethattherelativeperiods,denoted ε therein,fortherepresentation

σ ofGL_2n aretrivial—atthismoment, wehavenoideahowonemightprovesuchaperiodrelation—henceourinsistenceonworkingintrinsicallyinthecontext oforthogonalgroups.

3.6. Furthergeneralizations

AllthisshouldworkforL-functionsforGL1×^GSpin

(

2n

)

overatotallyrealfield F.Wesayshouldbecauseofthehypoth- esis“the Arthurparameter

σ beingcuspidal.”We mayappealtothe workofAsgari–Shahidi [2]andHundley–Sayag[9]

since weonlywantthecaseofgenerictransferfromGSpin

(

2n

)

toGL2n.However, aswe seebelow,thishypothesisisalso neededfortheManin–Drinfeldprincipleforboundarycohomology,andforthiswewillneedArthur’swork[1].

IfinsteadofGSpin

(

2n

)

,weconsidergeneralizingtoGSO

(

n

,

n

)

(orGO

(

n

,

n

)

),thenwecannothopetogetanynewresult, sincethestandarddegree-2n L-functionL

(

s

, σ )

foracuspidalrepresentation

σ

ofthegroupGSO

(

n

,

n

)

(orGO

(

n

,

n

)

)issame asthestandardL-functionforanyirreducibleconstituentoftherestrictionof

σ

toSO

(

n

,

n

)

.

4. AnadumbrationoftheproofofTheorem 2.2

Thebasicidea,following[7]and[8],istogiveacohomologicalinterpretationtotheconstanttermtheoremofLanglands, by studyingthe rank-one EisensteincohomologyofSO

(

n+¹

,

n+¹

)

.Letthe notationsbe asinthecombinatoriallemma above.A consequenceofthislemmaisthattherepresentationalgebraically(un-normalized)andparabolicallyinducedfrom

χ

f⊗

σ

f appearsinboundarycohomology:

aInd^G_P⁽_(A^Af)

f)

( χ

f

⊗ σ

f

)

^Kf

→

H^q0

(∂

PS_K^G_f

,

_M

_λ,E

),

where q₀=middle-dimension-of-symmetric-space-of-M_P+^dim

(

N_P

)/

2;

λ

=^w−1·

(

d+

μ )

; K_f is a deep-enough open- compactsubgroupofG

(

Af

)

;

∂

P denotesthepartcorrespondingto P oftheBorel–Serreboundaryofthelocallysymmetric spaceS_K^G_f ^{forG withlevelstructure K}f; Mλ,E isthe sheafcorrespondingtothefinite-dimensionalrepresentationMλ,E of thealgebraic group G×^{E.(Thereader isreferred to [7,Sect. 1] fora quickprimer on thesecohomologygroupsand} forthefundamentallong exactsequencethatcomesout oftheBorel–Serrecompactification.) Thefield E istakentobe a large enoughGaloisextension ofQ^{; forexample, E couldcontain} Q

( χ , σ )

.Torelateto thetheoryofautomorphicforms, we can passto C^{via an embedding}

ι

:^E→C^{.The induced} representationsandthe cohomologygroupsare allmodules foraHecke-algebraH^G_Kf^{,andinwhatfollowsbelow,werestrictourattentiontoa}commutativesub-algebraH^S_G ^ignoringa finitesetSofallramifiedplaces.Next,oneobservesthatthestandardintertwiningoperatorTst,atthepointofevaluation s= −^ngoesas:

T_st

:

^aIndG_P⁽_(A^Af_f⁾₎

( χ

f

⊗ σ

f

) −→

^aIndG_P⁽_(A^Af_f⁾₎

( χ

⁻_f¹

(

2n

) ⊗

^κ

σ

f

),

where

(

2n

)

denotesa Tate-twist,and

κ

is anelement ofO

(

n

,

n

)

butoutside SO

(

n

,

n

)

.Certaincombinatorial detailsabout Kostantrepresentatives allow usto observethatthe inducedrepresentationinthe targetalso appearsinboundarycoho- mologyas:

aInd^G_P⁽_(A^Af)

f)

( χ

⁻_f¹

(

2n

) ⊗

^κ

σ

f

)

^Kf

→

^Hq0

(∂

PS_K^G_f

,

_M

_λ,E

),

forthesamedegreeq0andthesameweight

λ

.Let

I^S_P

( χ

_f

, σ

_f

)

^Kf

:=

^aIndG_P^(A_(Af^f)₎

( χ

_f

_⊗ σ

_f

)

^Kf

⊕

^aIndG_P^(A_(A^f_f⁾₎

( χ

⁻_f¹

(

2n

) ⊗

^κ

σ

_f

)

^Kf

.

TheManin–DrinfeldprincipleamountstoshowingthatwegetaH^S_G-equivariantprojectionfromboundarycohomologyonto I^S_P

( χ

f

, σ

f

)

^Kf,andthetargetisisotypic,i.e.itdoesnotweakly intertwinewiththequotientofthe boundarycohomology by I^S_P

( χ

f

, σ

f

)

^Kf.Denotethisprojectionas:

R :

^Hq0

(∂

S^G_Kf

,

_M

_λ,E

) −→

^ISP

( χ

_f

, σ

_f

)

^Kf

.

If we denote the restriction map from global cohomology to the boundary cohomology as r^∗: ^Hq0

(

S^GK_f

,

Mλ,E

)

→ H^q0

(∂

S^G_Kf

,

Mλ,E

)

,thenthemaintechnicalresultonEisensteincohomologyinvolvestheimageofthecompositionR◦r^∗:

H^q0

(

S^G_Kf

,

_M

_λ,E

) −→

^{r∗ Hq0}

(∂

S_K^G_f

,

_M

_λ,E

) −→

^{R IS}_P

( χ

f

, σ

f

)

^Kf

.

For simplicityof explanation, let uspretend (and thiscould very well happen in some cases) that I^S_P

( χ

_f

, σ

_f

)

^Kf is a two-dimensional E-vector space. Ourmain result on Eisensteincohomology will then say that the image of R◦r^∗ is a one-dimensional subspace of this two-dimensional ambient space. We then look atthe slope of this line. Passing to a transcendentallevelviaan

ι

:^E→C^{,andusingtheconstanttermtheorem,oneprovesthattheslopeisinfact}

c_∞

( χ

_∞

, σ

_∞

)

^Lf

(−

n

, χ × σ )

L_f

(

1

−

ⁿ

, χ _× σ ) ,

where c_∞

( χ

_∞

, σ

_∞

)

isa nonzero complex numberdepending only onthe data atinfinity, and Lf

(

s

, χ

_×

σ )

is the finite partoftheL-function.Thisprovesthattheabovequantityliesin

ι (

E

)

.Studyingthebehaviorofthecohomologygroupson varying E thenprovesGaloisequivariance.

Alongtheway,weneedtoaddresscertainlocalproblems.Atthefiniteramifiedplaces,weprovethatthelocalnormal- izedintertwiningoperatorisnonzeroandpreservesrationalityusingtheresultsofKim[10],Mœglin–Waldspurger[11]and Waldspurger[16].AttheArchimedeanplace,yetanotherconsequenceofthecombinatoriallemmaisthattherepresentation

aInd^G_P^(R)_(R)

( χ

_∞

_⊗ σ

_∞

)

isirreducible;thisfollowsfromtheresultsofSpeh–Vogan[15].UsingShahidi’sresults[12]onlocalfactors,wethendeduce that the standard intertwining operator is an isomorphism and induces a nonzero isomorphism in relative Lie algebra cohomology.Butthesecohomologygroupsatinfinity areone-dimensional,andafterfixing baseson eithersidewe geta nonzeronumberc_∞

( χ

_∞

, σ

_∞

)

.We expectthat acarefulanalysis, asinHarder [6],oftherationality propertiesofrelative Liealgebracohomologygroups,shouldgiveusthatc_∞

( χ

_∞

, σ

_∞

)

isthesameasL

(

−ⁿ

, χ

_∞×

σ

_∞

)/

L

(

1−ⁿ

, χ

_∞×

σ

_∞

)

upto anonzerorationalnumber,justifyingourclaimaboutarationalityresultforcompletedL-valuesasinTheorem 1.3.

Acknowledgements

Itisourgreatpleasuretothanktherefereeforveryinsightfulandimportantcommentsandsuggestions.

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