Automofphic forms on GL(2)

105 Pages·1979·0.677 MB·English

by  Jacquet H., Langlands R.P.

Automorphic Forms on GL(2)
Herve´ Jacquet and Robert P. Langlands
Formerly appeared as volume #114 in the Springer Lecture Notes in Mathematics, 1970, pp. 1-548
Chapter 1 i
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ChapterI:LocalTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1. Weilrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§2. RepresentationsofGL(2,F)inthenon-archimedeancase . . . . . . . . . . . 15
§3. Theprincipalseriesfornon-archimedeanfields . . . . . . . . . . . . . . . . 58
§4. Examplesofabsolutelycuspidalrepresentations . . . . . . . . . . . . . . . . 77
§5. RepresentationsofGL(2,R) . . . . . . . . . . . . . . . . . . . . . . . . 96
§6. RepresentationofGL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . 138
§7. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
§8. Oddsandends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
ChapterII:GlobalTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
§9. TheglobalHeckealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 189
§10. Automorphicforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
§11. Hecketheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
§12. Someextraordinaryrepresentations . . . . . . . . . . . . . . . . . . . . . 251
ChapterIII:QuaternionAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 267
§13. Zeta-functionsforM(2,F) . . . . . . . . . . . . . . . . . . . . . . . . . 267
§14. Automorphicformsandquaternionalgebras . . . . . . . . . . . . . . . . . 294
§15. Someorthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . 304
§16. AnapplicationoftheSelbergtraceformula . . . . . . . . . . . . . . . . . . 320
Chapter 1 ii
Introduction
Two of the best known of Hecke’s achievements are his theory of L-functions with gro¨ssen-
charakter,whichareDirichletserieswhichcanberepresentedbyEulerproducts,andhistheoryofthe
Eulerproducts,associatedtoautomorphicformsonGL(2). Sinceagro¨ssencharakterisanautomorphic
formonGL(1)oneistemptedtoaskiftheEulerproductsassociatedtoautomorphicformsonGL(2)
play a role in the theory of numbers similar to that played by the L-functions with gro¨ssencharakter.
In particular do they bear the same relation to the Artin L-functions associated to two-dimensional
representations of a Galois group as the Hecke L-functions bear to the Artin L-functions associated
toone-dimensional representations? Although wecannot answerthequestiondefinitively oneof the
principalpurposesofthesenotesistoprovidesomeevidencethattheanswerisaffirmative.
The evidence is presented in §12. It come from reexamining, along lines suggestedby a recent
paperofWeil,theoriginalworkofHecke. Anythingnovelinourreexaminationcomesfromourpoint
ofviewwhichisthetheoryofgrouprepresentations. Unfortunatelythefactswhichweneedfromthe
representation theoryof GL(2)donotseemtobeintheliterature sowehavetoreview, inChapterI,
therepresentationtheoryofGL(2,F)whenF isalocalfield. §7isanexceptionalparagraph. Itisnot
usedintheHecketheorybutinthechapteronautomorphicformsandquaternionalgebras.
Chapter I is long and tedious but there is nothing hard in it. Nonetheless it is necessary and
anyone whoreally wants tounderstand L-functions shouldtakeatleasttheresultsseriouslyforthey
areverysuggestive.
§9and§10arepreparatorytotheHecketheorywhichisfinallytakenupin§11. Wewouldliketo
stress,sinceitmaynotbeapparent,thatourmethodisthatofHecke. Inparticulartheprincipaltoolis
theMellintransform. ThesuccessofthismethodforGL(2)isrelatedtotheequalityofthedimensions
ofaCartansubgroupandtheunipotentradicalofaBorelsubgroupofPGL(2). Theimplicationisthat
ourmethods donot generalize. The results, with theexceptionof the conversetheorem inthe Hecke
theory,may.
The right way to establish the functional equation for the Dirichlet series associated to the
automorphic formsisprobably thatofTate. In§13weverify,essentially, thatthismethodleadstothe
samelocalfactorsasthatofHeckeandin§14weusethemethodofTatetoprovethefunctionalequation
for the L-functions associated to automorphic forms on the multiplicative group of a quaternion
algebra. The results of §13 suggestarelation betweenthe characters of representations of GL(2)and
thecharactersofrepresentationsofthemultiplicativegroupofaquaternionalgebrawhichisverified,
usingtheresultsof§13,in§15. Thisrelationwaswell-knownforarchimedeanfieldsbutitssignificance
hadnotbeenstressed. Althoughourproofleavessomethingtobedesiredtheresultitselfseemstous
tobeoneofthemorestrikingfactsbroughtoutinthesenotes.
Both§15and§16areafterthoughts; wedidnotdiscovertheresultsinthemuntiltherestofthe
noteswerealmostcomplete. Theargumentsof§16areonlysketchedandweourselveshavenotverified
all the details. However the theorem of §16 is important and its proof is such a beautiful illustration
ofthepowerandultimatesimplicityoftheSelbergtraceformulaandthetheoryofharmonicanalysis
onsemi-simplegroups that wecouldnot resistadding it. Although weare verydissatisfiedwith the
methods of the first fifteen paragraphs we see no way to improve on those of §16. They are perhaps
themethodswithwhichtoattackthequestionleftunsettledin§12.
We hope to publish a sequel to these notes which will include, among other things, a detailed
proofofthetheoremof§16aswellasadiscussionofitsimplicationsfornumbertheory. Thetheorem
has,asthesethingsgo,afairlylonghistory. Asfarasweknowthefirstformsofitwereassertionsabout
therepresentabilityofautomorphicformsbythetaseriesassociatedtoquaternaryquadraticforms.
Chapter 1 iii
As we said before nothing in these notes is really new. We have, in the list of references at
the end of each chapter, tried to indicate our indebtedness to other authors. We could not however
acknowledgecompletelyourindebtednesstoR.Godementsincemanyofhisideaswerecommunicated
orallytooneofusasastudent. Wehopethathedoesnotobjecttothecompanytheyareforcedtokeep.
Thenotes∗ weretypedbythesecretariesofLeetOliverHall. Thebulkoftheworkwasdoneby
MissMaryEllenPetersandtoherwewouldliketoextendourspecialthanks. Onlytimecantellifthe
mathematicsjustifieshergreatefforts.
NewYork,N.Y. August,1969
NewHaven,Conn.
∗ thatappearedintheSLMvolume
ChapterI:LocalTheory
§1Weilrepresentations. Beforebeginning the studyof automorphic formswemustreviewtherepre-
sentationtheoryofthegenerallineargroupintwovariablesoveralocalfield. Inparticularwehaveto
prove the existence of various series of representations. One of the quickestmethods of doing this is
tomakeuseoftherepresentations constructedbyWeilin[1]. Webeginbyreviewinghisconstruction
adding,atappropriateplaces,someremarkswhichwillbeneededlater.
InthisparagraphF willbealocalfieldandK willbeanalgebraoverF ofoneofthefollowing
types:
(i) ThedirectsumF ⊕F.
(ii) AseparablequadraticextensionofF.
(iii) TheuniquequaternionalgebraoverF. K isthenadivisionalgebrawithcentreF.
(iv) ThealgebraM(2,F)of2×2matricesoverF.
InallcasesweidentifyF withthesubfieldofK consistingofscalarmultiplesoftheidentity. In
particularifK =F ⊕F weidentifyF withthesetofelementsoftheform(x,x). Wecanintroducean
involutionιofK,whichwillsendxtoxι,withthefollowingproperties:
(i) Itsatisfiestheidentities(x+y)ι =xι +yι and(xy)ι =yιxι.
(ii) IfxbelongstoF thenx= xι.
(iii) ForanyxinK bothτ(x) =x+xι andν(x)= xxι = xιxbelongtoF.
IfK =F ⊕F andx =(a,b)wesetxι =(b,a). IfK isaseparablequadraticextensionofF the
involutionιistheuniquenon-trivialautomorphismofK overF. Inthiscaseτ(x)isthetraceofxand
ν(x)isthenormofx. IfK isaquaternionalgebraauniqueιwiththerequiredpropertiesisknownto
exist. τ andν arethereducedtraceandreducednormrespectively. IfK isM(2,F)wetakeιtobethe
involutionsending
a b
x =
c d
(cid:18) (cid:19)
to
d −b
x=
−c a
(cid:18) (cid:19)
Thenτ(x)andν(x)arethetraceanddeterminantofx.
Ifψ =ψ isagivennon-trivialadditivecharacterofF thenψ =ψ ◦τ isanon-trivialadditive
F K F
characterofK. Bymeansofthepairing
hx,yi =ψ (xy)
K
wecanidentifyK withitsPontrjagindual. Thefunctionν isofcourseaquadraticformonK whichis
avectorspaceoverF andf = ψ ◦ν isacharacterofsecondorderinthesenseof[1]. Since
F
ν(x+y)−ν(x)−ν(y) =τ(xyι)
and
f(x+y)f−1(x)f−1(y)= hx,yιi
theisomorphismofK withitselfassociatedtof isjustι. Inparticularν andf arenondegenerate.
Chapter 1 2
LetS(K)bethe spaceof Schwartz-Bruhat functions onK. There is aunique Haar measuredx
onK suchthatifΦbelongstoS(K)and
Φ′(x) = Φ(y)ψ (xy)dy
K
ZK
then
Φ(0)= Φ′(x)dx.
ZK
Themeasuredx,whichisthemeasureonKthatweshalluse,issaidtobeself-dualwithrespecttoψ .
K
SincetheinvolutionιismeasurepreservingthecorollarytoWeil’sTheorem2caninthepresent
casebeformulatedasfollows.
Lemma1.1. There is a constant γ which depends on the ψ and K, such that for every function Φ
F
in S(K)
(Φ∗f)(y)ψ (yx)dy = γf−1(xι)Φ′(x)
K
ZK
Φ∗f istheconvolutionofΦandf. Thevaluesofγ arelistedinthenextlemma.
Lemma1.2 (i) If K =F ⊕F or M(2,F) then γ =1.
(ii) If K is the quaternion algebra over F then γ =−1.
(iii) If F = R, K =C, and
ψ (x)= e2πiax,
F
then
a
γ = i
|a|
(iv) If F is non-archimedean and K is a separable quadratic extension of F let ω be the quadratic
character of F∗ associated to K by local class-field theory. If U is the group of units of F∗
F
let m= m(ω) be the smallest non-negative integer such that ω is trivial on
Um ={a∈ U |α ≡1(modpm)}
F F F
and let n =n(ψ ) be the largest integer such that ψ is trivial on the ideal p−n. If a is any
F F F
generator on the ideal pm+n then
F
ω−1(α)ψ (αa−1)dα
γ = ω(a) UF F .
R ω−1(α)ψ (αa−1)dα
UF F
(cid:12) (cid:12)
(cid:12)R (cid:12)
ThefirsttwoassertionsareprovedbyW(cid:12) eil. Toobtainthethirdap(cid:12)plythepreviouslemmatothe
function
Φ(z)= e−2πzzι.
Weprovethelast. ItisshownbyWeilthat|γ| = 1andthatifℓissufficientlylargeγ differsfrom
ψ (xxι)dx
F
Zp−Kℓ
Chapter 1 3
byapositivefactor. Thisequals
ψ (xxι)|x| d×x = ψ (xxι)|xxι| d×x
F K F F
Zp−Kℓ Zp−Kℓ
ifd×xisasuitablemultiplicativeHaarmeasure. Sincethekernelofthehomomorphism ν iscompact
theintegralontherightisapositivemultipleof
ψ (x)|x| d×x.
F F
Zν(p−Kℓ)
Setk = 2ℓifK/F isunramifiedandsetk = ℓifK/F isramified. Thenν(p−ℓ) = p−k ∩ν(K).
K F
Since1+ω istwicethecharacteristicfunctionofν(K×)thefactorγ isthepositivemultipleof
ψ (x)dx+ ψ (x)ω(x)dx.
F F
Zp−Fk Zp−Fk
Forℓandthereforeksufficientlylargethefirstintegralis0. IfK/F isramifiedwell-knownproperties
ofGaussiansumsallowustoinferthatthesecondintegralisequalto
α α
ψ ω dα.
F
a a
ZUF (cid:16) (cid:17) (cid:16) (cid:17)
Sinceω =ω−1 weobtainthedesiredexpressionforγ bydividingthisintegralbyitsabsolutevalue. If
K/F isunramifiedwewritethesecondintegralas
∞
(−1)j−k ψ (x)dx− ψ (x)dx
F F
Xj=0 (ZpF−k+j ZpF−k+j+1 )
Inthiscasem= 0and
ψ (x)dx
F
ZpF−k+j
is 0if k −j > n butequals qk−j if k−j ≤ n, where q is the numberof elementsinthe residue class
field. Sinceω(a) = (−1)nthesumequals
∞
1
ω(a) qm + (−1)jqm−j 1−
 q 
 Xj=0 (cid:18) (cid:19)
A little algebra shows that this equals 2ω(a)qm+1 so that γ = ω(a), which upon careful inspection is
q+1
seentoequaltheexpressiongiveninthelemma.
Inthenotationof[19]thethirdandfourthassertionscouldbeformulatedasanequality
γ = λ(K/F,ψ ).
F
Itisprobablybestatthemomenttotakethisasthedefinitionofλ(K/F,ψ ).
F
IfK isnotaseparablequadraticextensionofF wetakeω tobethetrivialcharacter.
Chapter 1 4
Proposition1.3 There is a unique representation r of SL(2,F) on S(K) such that
α 0
1/2
(i) r Φ(x) =ω(α)|α| Φ(αx)
0 α−1 K
(cid:18)(cid:18) (cid:19)(cid:19)
1 z
(ii) r Φ(x)= ψ (zν(x))Φ(x)
0 1 F
(cid:18)(cid:18) (cid:19)(cid:19)
0 1
(iii) r Φ(x)= γΦ′(xι).
−1 0
If S(K)(cid:18)is(cid:18)given its(cid:19)u(cid:19)sual topology, r is continuous. It can be extended to a unitary representation
of SL(2,F) on L2(K), the space of square integrable functions on K. If F is archimedean and Φ
belongs to S(K) then the function r(g)Φ is an indefinitely differentiable function on SL(2,F) with
values in S K).
(
ThismaybededucedfromtheresultsofWeil. Wesketchaproof. SL(2,F)isthegroupgenerated
α 0 1 z 0 1
by the elements , , and w = with α in F× and z in F subject to the
0 α−1 0 1 −1 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
relations
α 0 α−1 0
(a) w = w
0 α−1 0 α
(cid:18) (cid:19) (cid:18) (cid:19)
−1 0
(b) w2 =
0 −1
(cid:18) (cid:19)
1 a −a−1 0 1 −a 1 −a−1
(c) w w = w
0 1 0 −a 0 1 0 1
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
α 0 1 z
together with the obvious relations among the elementsof the form and . Thus
0 α−1 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
the uniqueness of r is clear. To prove the existence one has to verify that the mapping specified by
(i), (ii), (iii) preservesallrelations betweenthe generators. Forallrelations except(a), (b),and(c)this
canbeseenbyinspection. (a)translatesintoaneasilyverifiablepropertyoftheFouriertransform. (b)
translatesintotheequalityγ2 = ω(−1)whichfollowsreadilyfromLemma1.2.
Ifa =1therelation(c)becomes
Φ′(yι)ψ (ν(y))hy,xιidy = γψ (−ν(x)) Φ(y)ψ (−ν(y))hy,−xιidy (1.3.1)
F F F
ZK ZK
which can be obtained from the formula of Lemma 1.1 by replacing Φ(y) by Φ′(−yι) and taking the
inverseFouriertransform ofthe rightside. Ifaisnot1the relation (c)canagain bereducedto(1.3.1)
providedψ isreplacedbythecharacterx→ ψ (ax)andγ anddxaremodifedaccordingly. Werefer
F F
to Weil’s paper for the proof that r is continuous and maybe extendedto aunitary representation of
SL(2,F)inL2(K).
Now take F archimedean. It is enough to show that all of the functions r(g)Φ are indefinitely
differentiableinsomeneighborhoodoftheidentity. Let
1 x
N = x ∈ F
F 0 1
(cid:26)(cid:18) (cid:19) (cid:12) (cid:27)
(cid:12)
(cid:12)
(cid:12)
Chapter 1 5
andlet
α 0
A = α ∈ F×
F 0 α−1
(cid:26)(cid:18) (cid:19) (cid:12) (cid:27)
(cid:12)
Then N wA N is a neighborhood of the identity whi(cid:12)ch is diffeomorphic to N ×A ×N . It is
F F F F F F
(cid:12)
enoughtoshowthat
φ(n,a,n ) =r(nwan)Φ
1
is infinitely differentiable as a function of n, as a function of a, and as a function of n and that
1
the derivations are continuous on the product space. For this it is enough to show that for all Φ all
derivativesofr(n)Φandr(a)ΦarecontinuousasfunctionsofnandΦoraandΦ. Thisiseasilydone.
Therepresentation r dependsonthechoiceofψ . IfabelongstoF× andψ′ (x) = ψ (ax)let
F F F
r′ bethecorrespondingrepresentation. Theconstantγ′ = ω(a)γ.
Lemma1.4 (i) The representation r′ is given by
a 0 a−1 0
r′(g) =r g
0 1 0 1
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(ii) If b belongs to K∗ let λ(b)Φ(x)=Φ(b−1x) and let ρ(b)Φ(x)= Φ(xb). If a= ν(b) then
r′(g)λ(b−1) =λ(b−1)r(g)
and
r′(g)ρ(b)=ρ(b)r(g).
In particular if ν(b) =1 both λ(b) and ρ(b) commute with r.
Weleavetheverificationofthislemmatothereader. TakeKtobeaseparablequadraticextension
ofF oraquaternionalgebraofcentreF. Inthefirstcaseν(K×)isofindex2inF×. Inthesecondcase
ν(K×)isF× ifF isnon-archimedeanandν(K×)hasindex2inF× ifF isR.
LetK′bethecompactsubgroupofK×consistingofallxwithν(x) =xxι =1andletG bethe
+
subgroupofGL(2,F)consistingofallgwithdeterminantinν(K×). G hasindex2or1inGL(2,F).
+
UsingthelemmaweshalldecomposerwithrespecttoK′ andextendrtoarepresentationofG .
+
LetΩbeafinite-dimensionalirreduciblerepresentationofK×inavectorspaceU overC. Taking
thetensorproductofrwiththetrivialrepresentationofSL(2,F)onU weobtainarepresentationon
S(K)⊗CU =S(K,U)
whichwestillcallrandwhichwillnowbethecentreofattention.
Proposition1.5 (i) If S(K,Ω) is the space of functions Φ in S(K,U) satisfying
Φ(xh)=Ω−1(h)Φ(x)
for all h in K′ then S(K,Ω) is invariant under r(g) for all g in SL(2,F).
(ii) The representation r of SL(2,F) on S(K,Ω) can be extended to a representation r of G
Ω +
satisfying
a 0
1/2
r Φ(x)=|h| Ω(h)Φ(xh)
Ω 0 1 K
(cid:18)(cid:18) (cid:19)(cid:19)
if a =ν(h) belongs to ν(K×).
Chapter 1 6
(iii) If η is the quasi-character of F× such that
Ω(a) =η(a)I
for a in F× then
a 0
r =ω(a)η(a)I
Ω 0 a
(cid:18)(cid:18) (cid:19)(cid:19)
(iv) The representation r is continuous and if F is archimedean all factors in S(K,Ω) are
Ω
infinitely differentiable.
(v) IfU isaHilbertspaceandΩ isunitaryletL2(K,U)bethespaceofsquareintegrablefunctions
from K to U with the norm
kΦk2 = kΦ(x)k2dx
Z
If L2(K,Ω) is the closure of S(K,Ω) in L2(K,U) then r can be extended to a unitary
Ω
representation of G in L2(K,Ω).
+
Thefirstpartofthepropositionisaconsequenceofthepreviouslemma. LetH bethegroupof
matricesoftheform
a 0
0 1
(cid:18) (cid:19)
with a in ν(K×). It is clear that the formula of part (ii) defines a continuous representation of H on
S(K,Ω). MoreoverG isthesemi-directofH andSL(2,F)sothattoprove(ii)wehaveonlytoshow
+
that
a 0 a−1 0 a 0 a−1 0
r g =r r (g)r
Ω 0 1 0 1 Ω 0 1 Ω Ω 0 1
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19)(cid:19)
Let a = ν(h) and let r′ be the representation associated ψ′ (x) = ψ (ax). By the first part of the
F F
previouslemmathisrelationreducesto
r′ (g) =ρ(h)r (g)ρ−1(h),
Ω Ω
whichisaconsequenceofthelastpartofthepreviouslemma.
Toprove(iii)observethat
a 0 a2 0 a−1 0
=
0 a 0 1 0 1
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
andthata2 = ν(a)belongstoν(K×). Thelasttwoassertionsareeasilyproved.
Wenow insert someremarks whose significance will not be clearuntil we begin todiscuss the
localfunctionalequations. WeassociatetoeveryΦinS(K,Ω)afunction
W (g)=r (g)Φ(1) (1.5.1)
Φ Ω
onG andafunction
+
a 0
ϕ (a)= W (1.5.2)
Φ Φ 0 1
(cid:18)(cid:18) (cid:19)(cid:19)
onν(K×). ThebothtakevaluesinU.