Sturm Bounds for Siegel Modular Forms

0.12 MB·English

by  Olav K. Richter

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
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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–