Table Of Content

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 Chapter1 i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ChapterI:LocalTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §1. Weilrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §2. RepresentationsofGL(2,F)inthenon archimedeancase . . . . . . . . . . . 12 §3. Theprincipalseriesfornon archimedeanfields . . . . . . . . . . . . . . . . 46 §4. Examplesofabsolutelycuspidalrepresentations . . . . . . . . . . . . . . . . 62 §5. RepresentationsofGL(2,R) . . . . . . . . . . . . . . . . . . . . . . . . 77 §6. RepresentationofGL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . 111 §7. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 §8. Oddsandends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 ChapterII:GlobalTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 §9. TheglobalHeckealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 152 §10. Automorphicforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 §11. Hecketheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 §12. Someextraordinary representations . . . . . . . . . . . . . . . . . . . . . 203 ChapterIII:QuaternionAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . 216 §13. Zeta functionsforM(2,F) . . . . . . . . . . . . . . . . . . . . . . . . . 216 §14. Automorphicformsandquaternionalgebras . . . . . . . . . . . . . . . . . 239 §15. Someorthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . 247 §16. AnapplicationoftheSelbergtraceformula . . . . . . . . . . . . . . . . . . 260 Chapter1 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 theL 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 to one dimensional representations? Although wecannot answer thequestion definitively oneof the principalpurposesofthesenotesistoprovidesomeevidencethattheanswerisaffirmative. The evidence is presented in §12. It come from reexamining, along lines suggested by a recent paperofWeil,theoriginalworkofHecke. Anythingnovelinourreexaminationcomesfromourpoint ofviewwhichisthetheoryofgrouprepresentations. Unfortunately thefactswhichweneedfromthe representation theory ofGL(2)donot seemto bein theliterature so wehaveto review, in Chapter I, therepresentation theoryofGL(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 who really wants tounderstand L functions should takeat leasttheresultsseriously forthey areverysuggestive. §9and§10arepreparatorytotheHecketheorywhichisfinallytakenupin§11. Wewouldliketo stress,sinceitmaynotbeapparent,thatourmethodisthatofHecke. Inparticulartheprincipaltoolis theMellintransform. ThesuccessofthismethodforGL(2)isrelatedtotheequalityofthedimensions ofaCartansubgroupandtheunipotentradicalofaBorelsubgroupofPGL(2). Theimplicationisthat our methods do not generalize. The results, with theexception of the conversetheorem in the 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 suggest a relation between the characters of representations of GL(2)and thecharactersofrepresentationsofthemultiplicativegroupofaquaternionalgebrawhichisverified, usingtheresultsof§13,in§15. Thisrelationwaswell knownforarchimedeanfieldsbutitssignificance hadnotbeenstressed. Although ourproofleavessomethingtobedesiredtheresultitselfseemstous 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 on semi simplegroups that wecould not resistadding it. Although weare very dissatisfied with 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. Chapter1 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. Beforebeginningthestudyofautomorphicformswemustreviewtherepre sentationtheoryofthegenerallineargroupintwovariablesoveralocalfield. Inparticularwehaveto prove the existence of various series of representations. One of the quickest methods 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 involution sending 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. Chapter1 2 LetS(K)bethe spaceof Schwartz Bruhat functions onK. There is aunique Haar measuredx onK suchthatifΦbelongstoS(K)and Φ0(x)= Φ(y)ψ (xy)dy K ZK then Φ(0)= Φ0(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)Φ0(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| = 1andthatifìssufficientlylargeγ differsfrom ψ (xxι)dx F Zp−K` Chapter1 3 byapositivefactor. Thisequals ψ (xxι)|x| d×x = ψ (xxι)|xxι| d×x F K F F Zp−K` Zp−K` ifd×xisasuitablemultiplicativeHaarmeasure. Sincethekernelofthehomomorphismν iscompact theintegralontherightisapositivemultipleof ψ (x)|x| d×x. F F Zν(p−K`) Setk = 2ìfK/F isunramified and setk = ìfK/F isramified. Thenν(p−`) = 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 Foràndthereforeksufficientlylargethefirstintegralis0. 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 is0ifk−j > nbut equalsqk−j ifk−j ≤ n, whereq is the number of elementsin the 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. Inthenotation of[19]thethirdandfourthassertionscouldbeformulatedasanequality γ = λ(K/F,ψ ). F Itisprobablybestatthemomenttotakethisasthedefinitionofλ(K/F,ψ ). F IfK isnotaseparablequadraticextensionofF wetakeω tobethetrivialcharacter. Chapter1 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)= γΦ0(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) preservesall relations betweenthe generators. For all relations except(a), (b),and (c)this canbeseenbyinspection. (a)translatesintoaneasilyverifiablepropertyoftheFouriertransform. (b) translatesintotheequalityγ2 = ω(−1)whichfollowsreadilyfromLemma1.2. Ifa =1therelation(c)becomes Φ0(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 Φ0(−yι) and taking the inverseFourier transform ofthe right side. 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 may be extended to a unitary 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 differentiableinsomeneighborhood oftheidentity. 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) Chapter1 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 dependson thechoiceofψ . IfabelongstoF× andψ0 (x) = ψ (ax)let F F F r0 bethecorrespondingrepresentation. Theconstantγ0 = ω(a)γ. Lemma1.4 (i) The representation r0 is given by a 0 a−1 0 r0(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 r0(g)λ(b−1)=λ(b−1)r(g) and r0(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. LetK0bethecompactsubgroupofK×consistingofallxwithν(x)=xxι =1andletG bethe + subgroupofGL(2,F)consistingofallgwithdeterminantinν(K×). G hasindex2or1inGL(2,F). + Usingthelemmaweshalldecomposer withrespecttoK0 andextendrtoarepresentation ofG . + LetΩbeafinite dimensionalirreduciblerepresentationofK×inavectorspaceU overC. Taking thetensorproductofr withthetrivialrepresentation ofSL(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 K0 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×). Chapter1 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). + Thefirstpartoftheproposition isaconsequenceofthepreviouslemma. LetH bethegroup of 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 r0 be the representation associated ψ0 (x) = ψ (ax). By the first part of the F F previouslemmathisrelationreducesto r0 (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 clear until we begin to discuss 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.