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Special values of L-functions and false Tate curve 8 extensions II 0 0 2 ThanasisBouganis n a February 3,2008 J 5 2 Abstract ] Inthispaperweshowhowonecancombinethep-adicRankin-Selbergproduct T constructionofHidawithfreenessresultsofHeckemodulesofWilestoestablish N interestingcongruencesbetweenspecialvaluesofL-functions.Thesecongruences . isapartofsomedeepconjecturalcongruencesthatfollowfromtheworkofKato h onthenon-commutativeIwasawatheoryofthefalseTatecurveextension. t a m 1 Introduction [ 1 LetEbeanellipticcurvedefinedoverQandparationalprime.Intheclassicalsetting v ofcyclotomicIwasawatheoryforellipticcurvesoneisconcernedwiththestudyofthe 9 twistsoftheellipticcurvebyfiniteordercharacterthatfactorthroughthecyclotomic 3 9 ZpextensionQcyc n≥0Q(µpn),whereµpn isthegroupofthepn-throotsofunity. ⊂∪ 3 TheaimofthetheoryistoobtainalinkbetweentheanalyticallydefinedLfunctions . attached to E, and its twists, and the arithmetic properties of the elliptic curve over 1 0 the cyclotomic tower. The cyclotomic Main Conjecture for elliptic curves gives to 8 thisconjecturallinkaverypreciseform. Wenotethatmuchhasalreadybeenproven 0 towardsthisMainConjecturebyKato[19],andSkinnerandUrbanhaveannounceda : v completeprooffor semi-stable E, subject to provingcertain results aboutthe Galois i representationsattachedtoautomorphicforms. X One of the key ingredients of the above Main Conjecture are the p-adic L func- r a tions. Theseareusuallyrealizedasp-adicmeasuresoverGaloisgroups,which,when evaluated at finite order characters, interpolate canonically modified values of the L function. Theirconstructionusuallyinvolvestwosteps. Thefirstoneistofindproper transcendentalnumbers,usuallycalledperiods,suchthattheratiooftheLvaluesover theseperiodsgivesanalgebraicnumber.Thesecondstepistoprovethatthesevalues, oraslightmodificationofthem,havethedesiredinterpolationandintegralityproper- ties. Lately there has been great interest in extending the classical Iwasawa theory to a non abelian setting, that is to replace the Z extension by more generalp-adic Lie p extensionswhoseGaloisgroupisnon-abelian. Infactin[5]apreciseanalogueofthe MainConjectureinthisnonabeliansettingforalargefamilyofp-adicLiegroupshas beenstated. 1 1 INTRODUCTION 2 Oneoftheextensionsthatisofparticularinterestisthesocalled“falseTatecurve” extensions. Thatisextensionsoftheform,QFT := n≥0Q(µpn, p√n m)forsomep- ∪ powerfreeintegerm > 1. NotethattheGaloisgroupisthesemi-directproductZ ⋉ p Z×.Thereisaconjecturaltheoryforp-adicLfunctionsthatshouldexistinthissetting. p InaworkwithV.Dokchitser[4]wehaveaddressedthefirstoftheabovementionedtwo steps,thatisalgebraicityofthecriticalvaluesoftheLfunctionsinvolved. Inordertomakethingsmoreexplicitletusfixsomemorenotation.WewriteEfor anellipticcurvedefinedoverQandN foritsconductor.Aswealreadymentionedwe E considertheextensionsQFT,n := Q(µpn, p√n m)andQFT = n≥0QFT,n. Wewrite ∪ ρ foranArtin representationthatfactorsthroughQ andN foritsconductor. Let FT ρ usalsowriteL(E,ρ,s)fortheLfunctionattachedtoEtwistedbyρ. Weconsiderthe valueofL(E,ρ,s)atthecriticalpoints=1. ThefactthattheArtinrepresentationsρ factorthroughthefalseTatecurveallowedustoestablishtheanalyticityofL(E,ρ,s) ats = 1andthenourmainresultin[4]isconcernedwiththealgebraicpropertiesof thesevalues. LetuswriteΩ (E)fortheNe´ronperiodsattachedtotheellipticcurve ± E. Thenwehaveshownthat L(E,ρ,1) Q. Ω (E)dim(ρ+)Ω (E)dim(ρ−) ∈ + − forallArtinrepresentationsρthatfactorthroughQ . Actuallywedidmore.Namely, FT involvingalsotheperiodthatshouldcorrespondtothe“Artinmotive”M(ρ)attached toρweestablishedtheperiodconjectureofDelignethatgivesaprecisedescriptionof thenumberfieldwherethisvaluelies. Let us now move to the second step that we mentioned above, that is the p-adic properties of these values. From now on we will assume that the elliptic curve has good ordinary reduction at p. We start by stating a conjectural congruence between theseLvaluesfordifferentArtinrepresentations.WedefinethequantityR(ρ)as P (ρˆ,u−1) L (E,ρ,1) R(ρ):=e (ρ)u−vp(Nρ) p {p,q|m} p P (ρ,w−1) · Ω (E)dim(ρ+)Ω (E)dim(ρ−) p + − where e (ρ) is a local epsilon factor of ρ suitably normalized, P (ρ,X) is the usual p p characteristicpolynomialassociatedtoρatpandu,warep-adicnumbersdefinedby, 1 a X +pX2 =(1 uX)(1 wX), u Z× and p+1 a =#E (F ) − p − − ∈ p − p p p HereρîsthedualrepresentationbutinourfalseTatecurvesettingitiseasytoseethat ρˆ=ρ. Finallythesubscript p,q m meansthatwehaveremovedtheEulerfactorsat ∼ { | } theseprimes. Thenwestate, Conjecture: Foreachn 1,letχ beacharacterofGal(Q /Q(µ ))ofexact n FT,n pn ≥ orderpn. Writeρ fortheinducedrepresentationofχ toGal(Q /Q)andσ for n n FT,n n therepresentationinducedtoGal(Q /Q)fromthetrivialoneoverQ(µ ). Then, FT,n pn thevaluesR(ρ )andR(σ )arep-adicallyintegralandsatisfy n n R(ρ ) R(σ ) <1 n n p | − | 1 INTRODUCTION 3 ormoregenerally R(ρ ψ) R(σ ψ) <1 n n p | ⊗ − ⊗ | whereψisafiniteordercharacterofGal(Qcyc/Q)and normalizedas p =p−1. p p |·| | | Letuscommentalittlebitmoreonthisconjectureanditsconnectiontononcom- mutative Iwasawa theory. The definition of the quantity R(ρ) describes the interpo- lation propertiesthat the conjectural, as in [5], non-abelianp-adic L-functionshould satisfy. Indeedthe authorsin [5] haveconjecturedthe existenceof an elementin the K oftheIwasawaalgebraassociatedtothisextensionthatinterpolatessuitablymod- 1 ified, as above, values of L(E,ρ,1) and plays the role of the non-abelian p-adic L function in their theory. Note that the representationsρ and σ are defined over Q n n andarecongruentmodulopthatisifweconsidertheirreductionmodulopthentheir semi-simplificationsareisomorphic. Hencetheexistenceofthenon-abelianp-adicL functionwouldimplythatitsvaluesshouldbealsop-adicallyclose. There is almost nothing known concerning the construction of this object for a generalp-adicLieextension.Howeverinthesettingthatweareinterestedin,thefalse Tatecurveextension,Katoin[18]hasrelatedtheexistenceofthisnon-abelianobject withcongruencesbetweenclassicalabelianp-adicLfunctionsovervariousfieldsofthe extension.Wetakesometimetoexplainthisasitwillhelpusmotivatetheresultsthat appearin thispaper. LetGbetheGaloisgroupofthefalse Tatecurveextensionand Λ(G) = Z [[G]]theIwasawaalgebraofG. WesetU(n) := ker(Z× (Z/pnZ)×). p p → ThemainresultofKatoin[18]istheconstructionofaninjectivehomomorphism θ :K (Λ(G)) Z [[U(n)]]× G 1 p → n≥0 Y and the explicit description of the image. In order to make this last statement a little bit more precise we write, for n m 0, N : Z [[U(m)]] Z [[U(n)]] m,n p p ≥ ≥ → forthe canonicalnormmap, φ be the ring homomorphismZ [[Z×]] Z [[Z×]] in- p p → p p duced by the rising to the power p map on Z×. Then the result of Kato says that p θ (K (Λ(G)))=(a ) with G 1 n n≥0 N (c )pi 1 mod p2n i,n i ≡ 0<i≤n Y withc =b φ(b )−1andb =a N (a )−1. Theelementsa haveanarithmetic n n n−1 n n 0,n 0 n meaning, they are abelian p-adic L functions. More precisely if we write ρ for the n Artin representation of G induced from a character of pn order of the Galois group Gal(Q(µpn, p√n m)/Q(µpn)),thentheelementsanaretheabelianp-adicL-functions interpolatingthevaluesL(E ρ χ,1),forχDirichletcharactersofthecyclotomic n ⊗ ⊗ extensionofQ. Theconjecturalcongruencesthatwehavewrittenabovecorrespondtothecaseof n = 1 of Kato’s congruences after evaluating the abelian p-adic L functions at the characterψ. Thereiscomputationalsupportfortheseconjectures;initiallybyBalister [1] and much more vastly by the Dokchitser brothers [11]. In the first part of this work[3]wehaveshowedtheexistenceoftheabelianp-adicL-functionsa appeared n in Katos’s congruencesand provedthe aboveconjecturalcongruencesup to an issue 1 INTRODUCTION 4 of periods. Namely there we have used notthe motivic periodsthatare stated in the congruencesbut automorphicperiods, the so called Eichler-Shimura-Harderperiods, thatappearquitenaturalinthesocalledmodularsymbolconstruction.Therewecame acrossto a ratherdeepproblem, namelythe relationof these automorphicperiodsas one use the functorialpropertiesof the L-functionsand especially base-change. We say a little bit more on this at the last section of this paper. Finally we note that in [7] an inductiveargumentwas used to showhow these congruences(for n = 1)can providecongruencesforn > 1intheformconjecturedbyKatobutunfortunatelynot modulotherightppower. Ouraiminthispaperistotackletheconjecturalcongruencesinsistingongetting therightmotivicperiods.Weachievethatforthecasewherep=3butwealsodiscuss possibleextensionsforthecaseofp > 3. Weneedtoimposesomefurtherconditions onE,otheroftechnicalnaturewhichwebelievecanberemovedandotherthatseem important. Namely fromnow on we assume that (a) The curveE is semi-stable and if we consider the minimaldiscriminant∆E = q|NEqiq then p does not divide iq for all q. Note that the last condition means that the conductor of E is equal to the Q Artin conductor of the mod p representation obtained by E. (b) We assume that m thatappearinthefalseTateextensionispowerfreewith(m,N ) = (m,p) = 1and, E (c)a ratherimportantassumption,thatE hasno rationalsubgroupoforderp, thatis theassociatedmoduloprepresentationisirreducible. Finallywementionherethatas ouraimhereistoaddresstheissueofmotivicversusautomorphicperiodswefocuson provingtheaboveconjecturesforψ = 1. Howeverwelayallimportantconstructions sothateverythingcanbeextendtothecaseψbeingnottrivial. Our proof can be divided into two parts. Let us write f S (Γ (N );Q) for 2 0 E ∈ therationalnewformthatwecanassociatetoE. Inthefirstpartwerelyonthework of Hida of the construction of a p-adic Rankin-Selberg product initiated in [13] and generalizedin [14]. We can associate a newformg of weightone to the Artin repre- sentationρandanEisensteinseries ofweightonewithσ. Usingthem,weconstruct E p-adic measuresdµ and dµ over Z× that are congruentmodulop, in the sense f,g f,E p thattheirvaluesateveryfinitecharacterofZ×arecongruent.Thesemeasuresinterpo- p late,p-adically,twistsofthecriticalvaluesoftheRankin-SelbergproductsD(f,g,s) andD(f, ,s)byfiniteordercharacters.Evaluatingthesemeasuresatthetrivialchar- E acterwegetafirstformofcongruencesbetweenD(f,g,1)andD(f, ,1). Underthe E semi-stable assumptionwe can easily relate the Rankin-Selbergproductto the twists oftheellipticcurveE. Howeverwedonotyetgetthecongruencesstatedinthetheoremabove. Weneed to work further two things. First, in order to establish the congruencesbetween the measuresabove,wehadtoclearadenominatorc(f,m)thatdependssolelyonf and m. Hence we getcongruencesafter multiplyingwith this constantc(f,m). Second, theperiodsthatweusetogettherationalityoftheRankin-Selbergproductareclosely related to the Petersson inner product< f,f >. These periodsmay not be equalto ourperiodsΩ (E) and Ω (E) up to a p-adicunit. These two problemsare related. + − Thatis,thereasonthatthedenominatorc(f,m)appearsinourp-adicinterpolationis thefactthatthePeterssoninnerproductisnottheproperautomorphicperiodinorder togetp-adicallyintegralratiosoftheform L−values . aut.periods In the second part we show, under the assumptions of the theorem, that indeed 2 BASICNOTATIONS 5 this is the case. This part relies heavily on the work of Wiles. We make use of two of his importantresults in [28]. The first one is an extension of a theorem of Mazur [24] onthe freeness, overa completedHecke algebra, of the first cohomologygroup ofmodularcurvesafterlocalizingitata propermaximalideal. Thesecondoneisan extension of a theorem of Ihara on the study of maps between Jacobiansof modular curvesofdifferentlevels. Herewewouldliketomentionhowhelpfulwasforusthe paperofDarmon,DiamondandTaylor[6]reviewingtheworkofWiles. Letusjustmentionthatwetriedtoapplythesameideasforp > 3. Hereinorder to bring things to the previous setting we use the fact that the base-change property forautomorphicrepresentationsofGL(2)hasbeenprovedforcyclicextensions[23]. Using this, we can work the congruences over the totaly real field F := Q(µ )+. p Howeverwefacetwoproblems. Firstthefactthatweworkwithaprimethatramifies inF putsrestrictionsonthefreenessresultsthatweneed.Secondweneedtorelateour definedautomorhicperiodsoverF with the onesoverQ, and evenstrongerwe need therelationtobeuptop-adicunitsaproblemmuchofthesamenaturethatwefacein ourwork[3]. Wedonothaveananswertothesequestionsyet. Acknowledgements: TheauthorwouldliketothankProfessorJohnCoatesforsug- gestingto work on Kato’scongruencesandfor recommendingto considerthe use of theRankin-Selbergmethodanditsp-adicversion. 2 Basic Notations LetHbethecomplexupperhalfplane. IfwedenotebyGL+(R)thetwobytworeal 2 matriceswithpositivedeterminant,thenweconsidertheactionofthemonHbyliner a c fractionaltransformations,z α(z)= az+b,forα= GL+(R). Welet 7→ cz+d b d ∈ 2 (cid:18) (cid:19) k 1beanintegerandwedefineanactionofGL+(R)onfunctionsf :H Cby ≥ 2 → f (f [α])(z)=det(α)k/2(cz+d)−kf(α(z)) k 7→ | a c for α = GL+(R). We denote by SL (Z) the two by two matrices b d ∈ 2 2 (cid:18) (cid:19) withdeterminant1andintegralentries. ForapositiveintegerN wehavethestandard notationsforthesubgroupsofSL (Z), 2 1 0 Γ(N)= γ SL (Z) γ mod N { ∈ 2 | ≡ 0 1 } (cid:18) (cid:19) Γ0(N)={γ ∈SL2(Z) | γ ≡ ∗0 ∗ mod N} (cid:18) ∗ (cid:19) 1 Γ1(N)={γ ∈Γ0(N) | γ ≡ 0 1∗ mod N} (cid:18) (cid:19) We write M (Γ (N)) (resp. S (Γ (N))) for the space of modular forms (resp. k 1 k 1 cuspforms)ofweightkwithrespecttoΓ (N).WewriteM (Γ (N),χ)(respS (Γ (N),χ) 1 k 0 k 0 formodularforms(resp. cuspforms)withrespecttoΓ (N)andNebentypeχ. 0 3 P-ADICMODULARFORMSANDMEASURES 6 Letusconsideracuspformf S (Γ (N),χ)andamodularformg M (Γ (N),ψ), k 0 l 0 ∈ ∈ forsomeintegerskandlwherewemoreoverassumek >l. LetuswritetheirFourier expansionsat cuspasf(z) = ∞ a(n,f)qn andg(z) = ∞ a(n,g)qn with ∞ n=1 n=0 q = e2πız. We alsodefinefρ(z) = ∞ a(n,f)qn S (Γ (N),χ¯). We consider the quantities L(f,g,s) := ∞Pa(n,nf=)1a(n,g)n−s∈andkthei0rPRankin-Selbergcon- n=1 P volution, D(f,g,s) := L (χψ,2s+2 k l)L(f,g,s) where we have removed N P − − theEulerfactorsatN fromL(χψ,s). Ifweassumethatf andg areactuallynormal- izedeigenformsandifwewritetheirLfunctionsL(f,s)= (1 α(q,f)q−s)(1 q{ − − β(q,f)q−s) −1andL(g,s)= (1 α(q,g)q−s)(1 β(q,g)q−s) −1thenwehave } q{ − − Q } that Q D(f,g,s)= (1 α(q,f)α(q,g)q−s)(1 α(q,f)β(q,g)q−s) { − − × q Y (1 β(q,f)α(q,g)q−s)(1 β(q,f)β(q,g)q−s) −1 − − } 3 p-adic modular forms and measures Inthissectionweintroducetheneededbackgroundinordertoobtainthep-adicversion of the Rankin-Selbergconvolution. For all this backgroundwe follow Hida’s papers [13,14].WeletpbeaprimenumberandwefixanembeddingQ֒ Q ֒ C ,where → p → p C isthep-adiccompletionofQ underthenormalizedp-adicabsolutevalue with p p |·|p p =p−1. ForanysubringR QweconsidertheR-modules, p | | ⊆ M (Γ (N),ψ;R):= f M (Γ (N),ψ) f(z)= a(n,f)qn, a(n,f) R k 0 k 0 { ∈ | ∈ } n≥0 X M (Γ (N);R):= f M (Γ (N)) f(z)= a(n,f)qn, a(n,f) R k 1 k 1 { ∈ | ∈ } n≥0 X MoreoverwedefineS (Γ (N),ψ;R)=S (Γ (N),ψ) M (Γ (N),ψ;R)andsim- k 0 k 0 k 0 ∩ ilar for S (Γ (N);R). For a modular form f M (Γ (N);Q) it is known that k 1 k 1 ∈ one can define the p-adic norm of f, f := sup a(n,f) . Let now K be p n≥0 p 0 | | | | any finite extension of Q and write K for the closure of K in C . We define the 0 p space M (Γ (N),ψ;K) (resp. M (Γ (N);K)) to be the p-adic completion of the k 0 k 1 space M (Γ (N),ψ;K ) (resp. M (Γ (N);K ) with respect to the norm in- k 0 0 k 1 0 p |·| side K[[q]] where we consider q as indeterminant. Then it is known by the work of Deligneand Rapoport[8] that, M (Γ (N),ψ;K) = M (Γ (N),ψ;K ) K, k 0 k 0 0 ⊗K0 M (Γ (N);K) = M (Γ (N);K ) K. Moreoveritisknownthatthedefinition k 1 k 1 0 ⊗K0 of M (Γ (N);K) and M (Γ (N),ψ;K) is independent of the choice of the dense k 1 k 0 subfieldK . Letusnowwrite forthep-adicringofintegersofK.Thenwedefine 0 K O thep-adicintegralmodularformsas, M (Γ (N),ψ; ):= f M (Γ (N),ψ;K) f 1 =M (Γ (N),ψ;K) [[q]], k 0 K k 0 p k 0 K O { ∈ || | ≤ } ∩O M (Γ (N); ):= f M (Γ (N);K) f 1 =M (Γ (N);K) [[q]] k 1 K k 1 p k 1 K O { ∈ || | ≤ } ∩O 3 P-ADICMODULARFORMSANDMEASURES 7 Definition1 (p-adic modular forms). Let A be either K or . We consider the K O spaces, M (N;A):= ∞ M (Γ (Npn);A) and M (N,ψ;A):= ∞ M (Γ (Npn),ψ;A) k ∪n=0 k 1 k ∪n=0 k 0 Then we define the space of p-adic modular forms of Γ (N), resp. of Γ (N) and 1 0 characterψ,asthecompletionoftheabovespaceswithrespecttothenorm . We p |·| denotethembyM (N;A),resp. M (N,ψ;A). k k We notethatalltheabovediscussioncanbedoneconsideringcuspformsinstead ofmodularforms. Inparticularwecanconsideralsop-adiccuspformswhichwewill denotebyS (N,A)andS (N,ψ;A). k k Remark1 For ourlater use, we mention thatthe space M (N,A) is actuallyinde- k pendentofkfork 2,sowemayalsowritejustM(N;A),see[14]. ≥ Nowwearegoingtodefinep-adicHeckeoperatorthatextendtheusualoneswhen restricted to the space of classical modularforms. For any integer n prime to N we n−1 o consideramatrixσ Γ (N),suchthatσ mod N. Itfollowsby n ∈ 0 n ≡ 0 n (cid:18) (cid:19) theworkofDeligneandRapoport[8]thattheactionf f σ onM (Γ (N);K)is k n k 1 7→ | integral,thatisitpreservestheintegralspaceM (Γ (N); ). We“define”theHecke k 1 K O operatorsT(ℓ) and S(ℓ), for every prime ℓ, acting on M (Γ (N);K) by describing k 1 theiractionontheq-expansion, a(ℓn,f)+ℓk−1a(n,f σ ), ifℓisprimetoN; a(n,T(ℓ)f)= ℓ |k ℓ a(ℓn,f), otherwise. (cid:26) ℓk−2a(n,f σ ), ifℓisprimetoN; a(n,S(ℓ)(f))= |k ℓ 0, otherwise. (cid:26) Notethatthesedefinitionsareconsistentwiththeonesontheclassicalellipticmodular forms.WedefinetheHeckealgebraH (Γ (N),ψ;A),resp.H (Γ (N);A)),forAei- k 0 k 1 therKor astheA-subalgebraofEnd (M (Γ (N),ψ;A)),resp.End (M (Γ (N);A)), K A k 0 A k 1 O generatedbyT(ℓ)andS(ℓ)forallprimesℓ. Similarlywedefineh (Γ (N);ψ;A)and k 0 h (Γ (N);A)whenwerestricttheactiontothespaceofcuspforms.Actuallyonehas k 1 that H (Γ (N),ψ;A) = H (Γ (N),ψ;Z) A and similarly for the other spaces. k 0 k 0 Z ⊗ FinallywenotethatwhenpN theactionoftheHeckeoperatorsisp-adicallyintegral | i.e. Tf f foreveryT H (Γ (N); ). p p k 1 K | | ≤| | ∈ O Wenowdefinep-adicHeckealgebras. Noticethatwehavethe -surjectiveho- K O momorphismsinducedbyrestrictionoftheHeckeoperators, H (Γ (Npm),ψ; ) H (Γ (Npn),ψ; ) for m n 1 k 0 K k 0 K O → O ≥ ≥ H (Γ (Npm); ) H (Γ (Npn); ) for m n 1 k 1 K k 1 K O → O ≥ ≥ Definition2 Wedefinethespaceofp-adicHeckealgebrasH (N,ψ; )(resp.H (N; )) k K k K O O by the projective limit, lim H (Γ (N),ψ; )(resp. lim H (Γ (N); )). Simi- n k 0 K n k 1 K O O larlywedefinethespac←es−h (N,ψ; )andh (N; ←)−. k K k K O O 3 P-ADICMODULARFORMSANDMEASURES 8 By definitionthisoperatorsacton the spacesM (N;A) andM (N,ψ;A) for A k k equaltoKor .Howeverthefacttheyarep-adicallyintegralallowustoextendtheir K O actionto the space of p-adicmodularformsM (N;A) and M (N,ψ;A). Ournext k k stepistodefineHida’sordinaryidempotenteattachedtotheHeckeoperatorT(p).We startwithagenerallemma, Lemma1 For any commutative -algebra R of finite rank over and for any K K O O x Rthelimitlim xn! existsandgivesanidempotentofR. n→∞ ∈ Proof See[16](p.201) . Definition3 Wedefineanidempotente inH (Γ (Npn,ψ; )andinH (Γ (Npn; ) n k 0 K k 1 K O O bythelimite =lim T(p)m!. MoreoverwedefineanidempotentinH (N; ) n m→∞ k K O andinH (N,ψ; )bytakingtheprojectivelimite=lim e . k K n n O ←− WewillbeinterestedinthespaceeM (N,ψ; ),usuallycalledtheordinarypart k K ◦ O ofM (N,ψ; )anddenotedbyM (N,ψ; ).Actuallythisspaceisnotthatlarge k OK k OK asthefollowinglemmaindicates, Lemma2 (Hida)Let C(ψ) be theconductorofthe characterψ. Definepositive in- tegers N′ and C(ψ)′ by writing N = N′pr and C(ψ) = C(ψ)′pt with (N′,p) = (C(ψ)′,p)=1. Lets:=max(t,1). Then, eM (N,ψ; ) M (Γ (N′ps),ψ; ) k K k 0 K O ⊂ O Proof: See[13]. Definition4 Wesaythatanormalizedeigenformf S (Γ (N )ψ)isan(p-)ordi- 0 k 0 0 ∈ naryformif, 1. ThelevelN oftheformf isdivisiblebyp. 0 2. TheFouriercoefficienta(p,f )isap-adicunit. 0 Thefollowinglemmaisprovedin[13](p.168), Lemma3 Letf S (Γ (N),ψ)beanewformwithk 2and a(p,f) =1. Then, k 0 p ∈ ≥ | | there is a uniqueordinary form f of weightk andcharacterψ such thata(n,f) = 0 a(n,f )forallnnotdivisiblebyp. Moreover,f isgivenby, 0 0 f(z), ifpdividesN; f (z)= 0 f(z) wf(pz), otherwise. (cid:26) − wherewistheuniquerootofX2 a(p,f)X+ψ(p)pk−1 =0with w <1.Moreover p − | | inthesecondcasei.e.(p,N)=1wehavethatN =Npandthata(p,f )=uwhere 0 0 uisthep-adicunitrootoftheaboveequation. 3 P-ADICMODULARFORMSANDMEASURES 9 LetusnowconsiderasurjectiveK-linearhomomorphismΦ:h (Γ (N ),ψ;K) k 0 0 → K thatisinducedbyanordinaryformf bysendingT(n) a(n,f ). Letusmore- 0 0 7→ overassumethatthismapissplit(wewillshowlaterthatinthecaseofinterestthiswill betrue)andinducesanalgebradirectdecomposition,h (Γ (N ),ψ;K)=K Afor k 0 0 ∼ × some summandA and let us denote by 1 the idempotentcorrespondingto the first f0 summandisomorphictoK.Wenowconsiderthelinearformℓ :S (N ,ψ;K) K f0 k 0 → definedby,ℓ (g) := a(1,1 e g). Notethat,bylemma2,thelinearformiswellde- f0 f0 fined. Proposition1 (Hida’slinear operator)Assume thatK containsall the Fourier co- 0 efficients of the ordinary form f . Then, the linear form ℓ has values in K on 0 f0 0 S (Γ (N pn),ψ;K ) forevery n 0. Furthermore, for g S (Γ (N pn),ψ;K ) k 0 0 0 k 0 0 0 ≥ ∈ wehave <h ,g > ℓ (g)=a(p,f )−npn(k/2) n N0pn f0 0 <h,f > 0 N0 0 1 whereh=f0ρ|k N0 −0 ,hn(z)=h(pnz). (cid:18) (cid:19) Proof See[13]p.175. Wenotethatifweconsideraconstantc(f ) suchthatc(f )1 h (Γ (N ); ) 0 ∈OK 0 f0 ∈ k 1 0 OK thenwehaveanintegralvaluedlinearformc(f )ℓ : S (N ,ψ; ) asthe 0 f0 k 0 OK → OK Heckeoperatorsarep-adicallyintegral p-adicmodularformsvaluedmeasures: Nowwe aregoingto definep-adicmea- suresassociatedwithp-adicmodularformsM(N; )forsomeN relativeprimeto K O p. Notethatitfollowsfromremark1thatwedonotneedtospecifytheweight. We let X to be a p-adic space that consists of some copies of Z and of a finite p productoffinite groups. For ourapplicationslater X isgoingto be justZ× = (1+ p ∼ pZ ) (Z/pZ)×. Let us write C(X; ) for the space of continuous functions of p K × O X withvaluesin andLC(X; )forthespaceoflocallyconstantfunctionson K K O O X. A measure µ on X with values in the space M(N; ) is just an -linear K K O O homomorphismfromC(X; )toM(N; ). K K O O LetusconsiderthespaceZ :=Z× (Z/NZ)×andforanelementz Z letus N p × ∈ N writez fortheprojectionofz tothefirstcomponent. WecandefineanactionofZ p N onthespaceM (Γ (Npr); )byf f z := zkf σ with σ asdefinedabove. k 1 OK 7→ | p |k z z ThisactioncanbeextendedtoM(N, )(see[14]p. 10). K O Definition5 (see[14])We saythatap-adicmeasureµ : C(X;O ) M(N;O ) K K → isarithmeticifthefollowingthreeconditionsaresatisfied, 1. Thereexistspositiveintegerksuchthatforeveryφ LC(X; ), K ∈ O µ(φ) M (Np∞; ) k K ∈ O Wewillcallktheweightofµ. 3 P-ADICMODULARFORMSANDMEASURES 10 2. There are continuous action Z X X and a finite order character ξ : N Z × such that µ(φ)z = z×kξ(z)→µ(φ(z x)) for every φ C(X;O ), N → OK | p · ∈ K wherektheweightofµ. Wethensaythatthearithmeticmeasureisofcharacter ξ. WesaythatthemeasureiscuspidalifµactuallytakesvaluesinS(N; ). K O Weareinterestedinattachingarithmeticmeasurestoagivenmodularform.Given a modularformf M (Γ (N),χ; ) with q-expansionf(z) = a(n,f)qn ∈ k 0 OK n≥0 wecanassociateameasuredµ onX :=Z× by, f p P dµ (φ) φ(n)a(n,f)qn, φ C(X; ) f K 7→ ∈ O n≥1 X wherewedefinetheactionofZ onZ×byz x z2x.Fromthefollowinglemmadue N p · 7→ p toShimuraweconcludethatdµ isanarithmeticmeasureofweightk andcharacter f χ. Lemma4 Letg = ∞ b(n,g)qn M (Γ (N),ω)andφanarbitraryfunctionon n=0 ∈ k 0 a b Ym =Z/NpmZ.DefiPneg(φ):= ∞n=0φ(n)b(n,g)qn.Thenforanyγ = c d ∈ (cid:18) (cid:19) Γ0(N2p2m),wehavethefollowiPngtransformationformula, g(φ) γ =ω(d)g(φ ) k a | whereφ (y)=φ(a−2y)forally Y =Z/NpmZ. a m ∈ Proof See[13](p. 190) ByaresultofHidain[14](p. 24corollary2.3)itfollowsthatactuallythemeasure µ ,onZ×,iscuspidal. f p Eisenstein measure and convolution: Of particular importance for us is the exis- tence, which follows from [21], of the following arithmetic measure of weight one, dE :C(Z ; ) S¯(L; )definedby, L K K O → O ∞ 2 φ(z)dE =  sgn(d)φ(d)qn K[[q]] ∈O ZZL (nXn,p=)1=1(dX,dL|n)=1      WecallthistheEisenstein-Katzmeasure. Forageneralarithmeticmeasureµ ofZ× g p associatedtoamodularformofweightℓandcharacterψwecandefineaconvolution operation,see forexample[13, 26], ofµ and dE. We considerthe action ofZ on g L C(Z×; )by(z⋆φ)(x) := ψ(z)zℓφ(z2x)forz Z andφ C(Z×; ). Fora p OK p p ∈ L ∈ p OK givenintegerk ℓandafiniteordercharacterχ:Z C× wedefinethearithmetic L ≥ → measure(µ dE) :C(Z×; ) S(L; )as g ∗ χ,k p OK → OK φ(x)(µL dE) := χ(z)zk−1(z−1⋆φ)(x)dE(z)dµL(x) ZZ×p g ∗ χ,k ZZ×p ZZL p g