Sturm Bounds for Siegel Modular Forms OlavK.RichterandMartinWesterholt-Raum* Abstract 5 WeestablishSturmboundsfordegreeg Siegelmodularformsmoduloaprime p,whichare 1 vitalforexplicitcomputations.OurinductiveproofexploitsFourier-JacobiexpansionsofSiegel 0 modularformsandpropertiesofspecializationsofJacobiformstotorsionpoints.Inparticular, 2 our approachiscompletely differentfromthe proofsofthepreviously knowncases g =1,2, n whichdonotextendtothecaseofgeneralg. a MSC 2010: Primary 11F46; Secondary 11F33 J 0 3 LET p beaprime. AcelebratedtheoremofSturm[Stu87]impliesthatanellipticmodularform ] T withp-integralrationalFourierseriescoefficientsisdeterminedbyits“firstfew"Fourierseries N coefficientsmodulop.Sturm’stheoremisanimportanttoolinthetheoryofmodularforms(forex- . ample,see[Ono04;Ste07]forsomeofitsapplications). PoorandYuen[PY](andlater[CCK13]for h t p≥5)provedaSturmtheoremforSiegelmodularformsofdegree2. Theirworkhasbeenapplied a m indifferentcontexts,andforexample,itallowed[CCR11;DR10]toconfirmRamanujan-typecon- gruencesforspecificSiegelmodularformsofdegree2. In[RR],wegaveacharacterizationofU(p) [ congruencesofSiegelmodularformsofarbitrarydegree,but(lackingaSturmtheorem)wecould 1 onlydiscussoneexplicitexamplethatoccurredasaDuke-Imamog˘lu-Ikedalift.IfaSiegelmodular v 3 formdoesnotariseasalift,thenoneneedsaSturmtheoremtojustifyitsU(p)congruences. 3 In thispaper,we provide such aSturmtheoremforSiegel modularformsofdegree g ≥2. Our 7 proofis totallydifferentfromtheproofsof thecases g =1,2in [CCK13;PY;Stu87],which donot 7 0 havevisibleextensionstothecaseg >2.Moreprecisely,weperformaninductiononthedegreeg. . 1 Asin[BWR14],weemployFourier-JacobiexpansionsofSiegelmodularforms,andwestudyvanish- 0 ingordersofJacobiforms.However,incontrastto[BWR14]weconsiderrestrictionsofJacobiforms 5 totorsionpoints(insteadoftheirthetadecompositions),whichallowustorelatemodp diagonal 1 : vanishingorders(definedinSection1)ofJacobiformsandSiegelmodularforms. Wededucethe v i followingtheorem. X TheoremI. LetF beaSiegelmodularformofdegreeg ≥2,weightk,andwithp-integralrational r a Fourierseriescoefficientsc(T).Supposethat 4 g k c(T)≡0(modp) forallT =(t )with t ≤ . ij ii ³ ´ 3 16 Thenc(T)≡0(modp)forallT. IfaSiegelmodularformarisesasalift,thenonecansometimesinferthatithasintegralFourier seriescoefficients(see[PRY09]). ThesituationismorecomplicatedforSiegelmodularformsthat arenotlifts. However, ifthe“firstfewdiagonal"coefficientsofaSiegelmodularformareintegral (orp-integralrational),thenTheoremIimpliesthatallofitsFourierseriescoefficientsareintegral (orp-integralrational). ThefirstauthorwaspartiallysupportedbySimonsFoundationGrant#200765.ThesecondauthorthankstheMaxPlanck InstituteforMathematicsfortheirhospitality.Thepaperwaspartiallywritten,whilethesecondauthorwassupported bytheETHZurichPostdoctoralFellowshipProgramandbytheMarieCurieActionsforPeopleCOFUNDProgram. –1– SturmBounds—1 Preliminaries O.K.Richter,M.W.-Raum CorollaryII. Let F be a Siegel modular form of degree g ≥2, weightk, and withrationalFourier seriescoefficientsc(T).Supposethat 4 g k c(T)∈Z forallT =(t )with t ≤ . (0.1) ij ii ³ ´ 3 16 Thenc(T)∈ZforallT. Remarks. (1) TheoremIandCorollaryIIareeffectiveforexplicitcalculationswithSiegelmodular forms,sinceonlyfinitelymanyT satisfytheconditiont ≤(4)g k foralli. ii 3 16 (2) If p ≥5,thenTheorem3.2showsthatthebounds(4)g k inTheoremIandinCorollaryIIcan 3 16 bereplacedbytheslightlybetterbounds(4)g 9k . 3 160 (3) If(0.1)inCorollaryIIisreplacedbytheassumptionthatc(T)is p-integralrationalforallT = (t )witht ≤(4)g k ,thenconsideringthecaseq=pintheproofofCorollaryIIyieldsthatc(T)is ij ii 3 16 p-integralrationalforallT. (4) One can remove the assumption that c(T)∈Q in Corollary II. More precisely, if F is a Siegel modular form of degree g ≥2, weight k, and with Fourier series coefficients c(T)∈C such that (0.1)holds, thenresultsof[CF90]showthatF isalinearcombination ofSiegel modularformsof degreeg ≥2,weightk,andwithrationalFourierseriescoefficients,andapplyingCorollaryIIyields thatc(T)∈ZforallT. Thepaperisorganizedasfollows. InSection1,wegivesomebackgroundonJacobiformsand Siegel modular forms. In Section 2, we explore diagonal vanishing orders of Jacobi formsandof theirspecializationstotorsionpoints.InSection3,weinductivelyestablishdiagonalslopebounds forSiegelmodularformsofarbitrarydegree,andweproveTheoremIandCorollaryII. 1| Preliminaries Throughout,g,k,m≥1areintegers,andp isarationalprime. Weworkoverthemaximalunram- ifiedextension Qur of Q . Note thatQur contains all N-throotsof unityif N and p arerelatively p p p prime. WealwayswriteptodenoteaprimeidealinQupr,andOp standsforthelocalizationofQupr atp.Moreover,werefertotheelementsofthelocalringZ ∩Qasp-integralrationalnumbers. p Finally,letH betheSiegelupperhalfspaceofdegreeg,Sp (Z)bethesymplecticgroupofde- g g gree g overtheintegers,andρ bearepresentationofSp (Z)withrepresentationspaceV(ρ),and g suchthat kerρ:Sp (Z) <∞. £ g ¤ (g) §1.1 Siegelmodularforms. LetM (ρ)denotethevectorspaceofSiegel modularformsofde- k greeg,weightk,typeρ,andwithcoefficientsinOp(see[Shi78]).Ifρistrivial,thenwesimplywrite M(g).RecallthatanelementF ∈M(g)(ρ)isaholomorphicfunctionF :H →V(ρ)withtransforma- k k g tionlaw F (AZ+B)(CZ+D)−1 =ρ(M)det(CZ+D)kF(Z) ¡ ¢ forallM= A B ∈Sp (Z).Furthermore,F hasaFourierseriesexpansionoftheform ¡C D¢ g F(Z)= c(T)e2πitr(TZ), X T=tT≥0 wheretrdenotesthetrace, tT isthetransposeofT,andwherethesumisoversymmetric,positive semi-definite,andrationalg×g matricesT. IfF ∈M(g)(ρ)suchthatF 6≡0(modp),i.e.,ifthereexistsaFourierseriescoefficientc(T)ofF such k thatc(T)6≡0(modp),thenthemodpdiagonalvanishingorderofF isdefinedby ordpF :=max 0≤l ∈Z:∀T =(tij),tii ≤l forall1≤i ≤g :c(T)≡0(modp) . (1.1) © ª –2– SturmBounds—1 Preliminaries O.K.Richter,M.W.-Raum If F has p-integral rational coefficients such that F 6≡0(modp), then ord F is defined likewise. p Finally,themodpdiagonalslopeboundfordegreeg (scalar-valued)Siegelmodularformsisgiven by k ρ(g) :=inf inf , (1.2) diag,p k F∈M(g) ordpF k F6≡0(modp) (g) and the definition of the mod p diagonal slope bound ρ for degree g (scalar-valued) Siegel diag,p modularformswithp-integralrationalcoefficientsiscompletelyanalogous. §1.2 Jacobiforms. Ziegler [Zie89]introducedJacobi formsofhigherdegree (extending[EZ85]). (g) LetJ (ρ)denotetheringofJacobiformsofdegreeg,weightk,indexm,typeρ,andwithcoeffi- k,m cientsinOp. Ifρistrivial,thenwesuppressitfromthenotation. RecallthatJacobiformsoccuras Fourier-JacobicoefficientsofSiegelmodularforms: LetF ∈Mk(g+1)(ρ),andwriteZ =³τz τtz′´∈Hg+1, whereτ∈H ,z∈Cg isarowvector,andτ′∈H tofindtheFourier-Jacobiexpansion: g 1 F(Z)=F(τ,z,τ′)= φ (τ,z)e2πimτ′, X m 0≤m∈Z whereφ ∈J(g) (ρ).WenowbrieflyrecollectsomedefiningpropertiesofsuchJacobiforms. m k,m LetGJ:=Sp (R)⋉(R2gטR)betherealJacobigroupofdegreeg (see[Zie89])withgrouplaw g [M,(λ,µ),κ]·[M′,(λ′,µ′),κ′]:=[MM′,(λ˜+λ′,µ˜+µ′),κ+κ′+λ˜tµ′−µ˜tλ′], where (λ˜,µ˜):=(λ,µ)M′. For fixed k and m, define the following slash operator on functions φ: H ×Cg →V(ρ): g φ| A B ,(λ,µ),x (τ,z) := ρ−1 A B det(Cτ+D)−k (1.3) ³ k,m£¡C D¢ ¤´ ¡C D¢ ·exp 2πim −(Cτ+D)−1(z+λτ+µ)C t(z+λτ+µ) +λτtλ+2λtz+µtλ+x ³ ¡ ¢´ ·φ (Aτ+B)(Cτ+D)−1,(z+λτ+µ)(Cτ+D)−1 ¡ ¢ forall A B ,(λ,µ),x ∈GJ. AJacobi formof degree g, weight k, andindex m is invariant under £¡C D¢ ¤ (1.3)whenrestrictedto A B ∈Sp (Z), (λ,µ)∈Z2g, andκ=0. Moreover, everyφ∈J(g) (ρ)hasa ¡C D¢ g k,m Fourierseriesexpansionoftheform φ(τ,z)= c(T,R)e2πitr(Tτ+zR), X T,R where the sum is over symmetric, positive semi-definite, andrational g×g matrices T andover columnvectorsR∈Qg suchthat4mT −RtR ispositivesemi-definite. Finally,westatetheanalogof(1.1)forJacobiforms. Letφ∈J(g) (ρ)suchthatφ6≡0(modp),i.e., k,m thereexistsaFourierseriescoefficientc(T,R)ofφsuchthatc(T,R)6≡0(modp). Thenthemodp diagonalvanishingorderofφisdefinedby ordpφ:=max 0≤l ∈Z:∀R,T =(tij),tii ≤l forall1≤i ≤g :c(T,R)≡0(modp) , (1.4) © ª andifφhasp-integralrationalcoefficientssuchthatφ6≡0(modp),thenonedefinesord φinthe p sameway. –3– SturmBounds—2 VanishingordersofJacobiforms O.K.Richter,M.W.-Raum 2| Vanishing orders of Jacobi forms In this section, we discuss diagonal vanishing orders of Jacobi forms and of their evaluations at torsionpoints. Throughout,N isapositiveintegerthatisnotdivisiblebyp.ConsidertheCvectorspace V¡ρ[N]¢:=Ch¡N1Zg/NZg¢2i=span©eα,β :α,β∈ N1Zg/NZgª, (2.1) C andtherepresentationρ onV ρ ,whichisdefinedbytheactionofSp (Z)on(1Zg/NZg)2: [N] ¡ [N]¢ g N ρ[N]¡M−1¢eα,β:=eα′,β′, where ¡α′,β′¢:=¡α,β¢M forM∈Spg(Z). (2.2) Ifφ∈J(g) ,thenφ[N]isitsrestrictiontotorsionpointsofdenominatoratmostN,i.e., k,m φ[N]:H(g)−→V ρ [N] ¡ ¢ φ[N](τ):= φ| [I ,(α,β),0] (τ,0) , (2.3) ³¡ k,m g ¢ ´α,β∈1Zg/NZg N where I stands for the g×g identity matrix. It is easy to see thatφ[N]is a vector-valued Siegel g modularform(seealsoTheorem 1.3of[EZ85]andTheorem1.5of[Zie89]): Lemma2.1. Letφ∈J(g) .Thenφ[N]∈M(g)(ρ ). k,m k [N] Proof. Wefirstarguethatφ[N]iswell-defined:Ifa,b∈Zg,then φ| [I ,(α+Na,β+Nb),0]=φ| [I ,(Na,Nb),Nαtb−Nβta]| [I ,(α,β),0]. k,m g k,m g k,m g Note that κ:=Nαtb−Nβta ∈Z does not contribute to the action, andwe findthat the defining expressionforφ[N]isindependentofthechoiceofrepresentativesofα,β∈ 1Zg/NZg. N Nextweverifythebehaviorundermodulartransformationofφ[N].LetM∈Sp (Z).Then g [I ,(α,β),0]·[M,(0,0),0)]=[M,(0,0),0]·[I ,(α′,β′),0] g g with α′,β′ = α,β M,whichimpliesthat ¡ ¢ ¡ ¢ φ[N] | M= φ| [I ,(α,β),0] (·,0) | M= φ| [M,(0,0),0]·[I ,(α′,β′),0] (·,0) ¡ α,β¢k ¡ k,m g ¢ ¢k ¡ k,m g ¢ = φ[N]α′,β′ . ¡ ¢ Thenextlemmarelatesthemod pdiagonalvanishingordersofaJacobiformφanditsspecial- izationφ[N]. Lemma2.2. Letφ∈J(kg,m) .Thenordpφ[N]≥ordpφ−m4. Proof. Letφ(τ,z)= c(T,R)e2πi(tr(Tτ)+zR).Thenφ[N](τ)equals T,R P ¡φ|k[Ig,(α,β),0]¢(τ,0)=e2πim(ατtα+βtα) Xc(T,R)e2πi¡tr(Tτ)+(ατ+β)R¢ T,R (2.4) =e2πimβtα c(T,R)e2πiβRe2πitr³¡T−41mRtR+m1 t¡mα+21tR¢¡mα+12tR¢¢τ´. X T,R Observethatc(T,R)e2πiβ(tα+R)∈Op. Itsufficestoshowthatc(T,R)vanishesmod pifthediagonal entriesti′i ofT′:=T−41mRtR arelessthanordpφ−m4. ConsiderT,R suchthat ti′i ≤ordpφ−m4 forsome fixedi. Note thatc(T,R)remainsunchanged whenreplacingT 7→T+1(Rλ+tλtR)+mtλλandR7→R+2mtλ,whichcorrespondstotheinvariance 2 ofφunder| [I ,(λ,0),0]. HenceweonlyhavetoconsiderthecaseofR= t(r ,...,r )with−m≤ k,m g 1 g ri ≤m.Inthiscase,ti′i =tii−41mri2≤ordpφ−m4 impliesthattii ≤ordpφ,i.e.,c(T,R)≡0(modp). –4– SturmBounds—2 VanishingordersofJacobiforms O.K.Richter,M.W.-Raum Thefollowinglemmaassociatesthemod pdiagonalvanishingordersofscalar-valuedandvector- valuedSiegelmodularforms. Lemma2.3. Supposethatthereexistsamod pdiagonalslopebound̺(g) fordegreeg ≥1. Letρ diag,p bearepresentationofSpg(Z)definedoverOp,andassumethatitsdualρ∗isalsodefinedoverOp. If F ∈M(kg)(ρ)suchthatordpF >k±̺(dgia)g,p,thenF ≡0(modp). Proof. LetvbealinearformonV(ρ),i.e.,v∈V(ρ)∗(Op).Then〈F,v〉:=v◦F isascalar-valuedSiegel modularformofweightkforthegroupkerρ.Weobtainascalar-valuedSiegelmodularformforthe fullgroupSp (Z)viathestandardconstruction(seealsotheproofofProposition1.4of[BWR14]) g F := 〈F,v〉| M= 〈F,ρ∗(M)v〉∈M(g), v Y k Y dk M:kerρ\Sp (Z) M:kerρ\Sp (Z) g g whered :=£kerρ:Spg(Z)¤. Observethatρ∗(M)v∈V(ρ)∗(Op),andhencetheFourierseriescoeffi- cientsofFv dobelongtoOp. TheassumptionordpF >k±̺(dgia)g,p impliesthatordpFv >dk±̺(dgia)g,p, andsinceF isofweightdk,wefindthatF ≡0(modp)forallv. Hence〈F,v〉vanishesmod pfor v v everyv,whichprovesthatF ≡0(modp). The finalresult in thisSection on the mod pdiagonal vanishing ordersof scalar-valued Jacobi formsandSiegel modularformsisanimportantingredientintheproofofTheoremIinthenext Section. Proposition2.4. Supposethatthereexistsamod pdiagonalslopebound̺(g) fordegreeg ≥1.Let diag,p φ∈J(kg,m) suchthatordpφ>m4 +k±̺(dgia)g,p.Thenφ≡0(modp). Proof. Letφ(τ,z)= c(T,R)e2πi(tr(Tτ)+zR).Lemmata2.2and2.3implythatφ[N]≡0(modp)for T,R P all N thatare relatively primeto p. Weprove by induction on the diagonalentries(t )of T that ii c(T,R)≡0(modp). The constant Fourierseries coefficient of φ[1] equals c(0,0). Hence c(0,0)≡ 0(modp),i.e.,thebasecaseholds.Next,letT bepositivesemi-definiteandsupposethatc(T′,R)≡ 0(modp)forallT′=(t′ )witht′ <t foralli. IfR= t(r ,...,r )suchthat|r |>mforsomei,then ij ii ii 1 g i (asin theproof ofLemma 2.2)usethe modularinvariance ofφtorelatec(T,R)tosome c(T′,R′) with t′ <t . Thatis, itsuffices toshow thatc(T,R)≡0(modp)forR with −m ≤r ≤m foralli. ii ii i Now, fix a prime N 6= p such that 2m < N−2. If β= t(β ,...,β )∈ 1Zg, then φ[N]≡0(modp) 1 g N impliesthat(seealso(2.4)) c(T,R)e2πiβR≡ c(T,R)e2πiβR≡0(modp), X X R R |ri|≤N2−1 wherethefirstcongruencefollows fromtheinduction hypothesisandtheassumptionthat2m < N−2(seealsotheproofofLemma2.2). Notethate2πiβR areintegersintheN-thcyclotomicfield. Moreover,if A:=¡e2πiβR¢ R∈Zg,1−2N<ri≤N2−1 , β∈N1Zg,0≤Nβi≤N−2 then(observingthatN isprime)detA=(−1)N−1NN−2 isthediscriminantoftheN-thcyclotomic field.Inparticular,detA6≡0(modp),andweconcludethatc(T,R)≡0(modp). –5– SturmBounds—3 SlopeboundsforSiegelmodularforms O.K.Richter,M.W.-Raum 3| Slope bounds for Siegel modular forms (g) Weprovebyinductionthatthereexistsadiagonalslopebound̺ forSiegelmodularformsof diag,p degreeg ≥1,whichthenyieldsTheoremIandCorollaryII. Proposition3.1. If ̺(g−1) is a diagonal slope bound for degree g −1 Siegel modular forms, then diag,p ̺(g) := 3̺(g−1) isadiagonalslopeboundfordegreeg Siegelmodularforms. diag,p 4 diag,p Proof. Supposethatthereexistsan06≡F ∈M(g)whosediagonalslopemodulopislessthan̺(g) = k diag,p 3̺(g−1), i.e., the diagonal vanishing order of F is greater than k ̺(g) . Consider Fourier-Jacobi 4 diag,p ± diag,p coefficients06≡φ ∈J(g−1)ofF.Ifm≤k ̺(g) ,then m k,m ± diag,p k m 3 k m k ordpφm> ̺(g) ≥ 4 +4̺(g) = 4 +̺(g−1) , diag,p diag,p diag,p andProposition2.4impliesthatφ ≡0(modp). m If m >k ̺(g) , then an induction on m shows that φ ≡0(modp). More specifically, fix an ± diag,p m index m and suppose that φm′ ≡0(modp) for all m′ <m. Thus, the mod p diagonal vanishing orderofφ isatleastm,andwe applyagainProposition 2.4tofindthatφ ≡0(modp). Hence m m F ≡0(modp),whichyieldstheclaim. Proposition3.1holdsforanyprimeidealpinQur,andhencealsofortherationalprimep. Asa p consequencewediscoverexplicitslopebounds,whichimmediatelyimplyTheoremI. Theorem3.2. Letg ≥1.Thereexistadiagonalslopebound̺(g) suchthat diag,p 3 g ̺(g) ≥16· . diag,p ³4´ If,inaddition,g ≥2andp≥5,then ̺(g) ≥ 160· 3 g. diag,p 9 ³4´ Proof. WeapplyProposition3.1tothebasecase̺(1) =12(see[Stu87]),andifp≥5,tothebase diag,p case̺(2) =10(see[CCK13]). diag,p Example3.3. Ifp≥5,thenforg =3,4,5,6weobtain ̺(3) ≥7.5, ̺(4) ≥5.6, ̺(5) ≥4.2, ̺(6) ≥3.1. diag,p diag,p diag,p diag,p Finally,weproveCorollaryII. ProofofCorollaryII. LetF ∈M(g) with rationalFourierseriescoefficients c(T)such thatc(T)∈Z k for all T =(t ) with t ≤ 4 g k for all i. Note that F has bounded denominators (this follows ij ii ¡3¢ 16 from[CF90]),i.e.,thereexistsan0<l ∈ZsuchthatlF ∈M(g)hasintegralFourierseriescoefficients. k Letl beminimalwiththisproperty. Weneedtoshowthatl =1. Ifl 6=1,thenthereexistsaprime q such that q|l. Hence lc(T)≡0(modq)for all T with t ≤ 4 g k , andTheorem I asserts that ii ¡3¢ 16 lc(T)≡0(modq)forallT.Thiscontradictstheminimalityofl,andweconcludethatl =1. –6– SturmBounds—4 References O.K.Richter,M.W.-Raum 4| References [BWR14] J.BruinierandM.Westerholt-Raum.“Kudla’smodularityconjectureandformalFourier-Jacobi series”.Preprint.2014. [CCK13] D.Choi,Y.Choie,andT.Kikuta.“SturmtypetheoremforSiegelmodularformsofgenus2 modulop”.ActaArith.158.2(2013),pp.129–139. 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[Shi78] G.Shimura.“Oncertainreciprocity-lawsforthetafunctionsandmodularforms”.Acta.Math. 141.1-2(1978),pp.35–71. [Ste07] W.Stein.Modularforms:Acomputationalapproach.Vol.79.GraduateStudiesinMathematics. WithanappendixbyP.Gunnells.AmericanMathematicalSociety,Providence,RI,2007. [Stu87] J.Sturm.“Onthecongruenceofmodularforms”.NumberTheory(NewYork,1984–1985). Vol.1240.LectureNotesinMath.Springer,1987,pp.275–280. [Zie89] C.Ziegler.“Jacobiformsofhigherdegree”.Abh.Math.Sem.Univ.Hamburg59(1989), pp.191–224. DepartmentofMathematics,UniversityofNorthTexas,Denton,TX76203,USA E-mail:[email protected] Homepage:http://www.math.unt.edu/~richter/ MaxPlanckInstituteforMathematics,Vivatsgasse7,D-53111,Bonn,Germany E-mail:[email protected] Homepage:http://raum-brothers.eu/martin –7–