Generators for a module of vector-valued Siegel modular forms of degree 2

0.18 MB·English

by  C. H. van Dorp

GENERATORS FOR A MODULE OF VECTOR-VALUED
SIEGEL MODULAR FORMS OF DEGREE 2
∗
Christiaan van Dorp
January 15, 2013
3
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0
2
Abstract

n
a Inthispaperwewilldescribeallvector-valuedSiegelmodularformsofdegree2andweight
J Sym6(St)⊗detk(St)withkodd. Thesevector-valuedformsconstituteamoduleoverthering

4 ofclassicalSiegelmodularformsofdegree2andevenweightandthismoduleturnsoutto
1
befree. Inordertofindgenerators,wegeneralizecertainRankin-Cohendifferentialoper-

atorsontriplesofclassicalSiegelmodularformsthatwerefirstconsideredbyIbukiyama
]
G andwefindaRankin-Cohenbracketonvector-valuedSiegelmodularforms.
A
.
h 1 Introduction
t
a
m
In comparison to elliptic modular forms, Siegel modular forms and especially vector-valued
[
Siegelmodularformsaremuchlessunderstood. Althoughthedimensionsofthevectorspaces

1 of genus 2 modular forms are known due to Tsushima [18], explicit generators are unknown
v
in almost all cases. For some instances however, such generators are known. These ‘first few’
0
1 examplesareduetoSatoh[17]andIbukiyama[12,13].
9
Tsushima’sdimensionformulacangiveclues—apartfromthedimensions—aboutthestructure
2
. of the modules of forms. For instance, the dimension formula can predict relations and in
1
0 those cases the modules will not be free. When no relations are predicted, the modules could
3 be freely generated over the ring of classical modular forms of even weight. Ibukiyama has
1
conjecturedthatoneofthemoduleshestudiedisinfactfree,buthedidnotprovethis[13]. In
:
v
this paper we will prove his hypothesis and in order to do this, we develop some methods for
i
X constructingvector-valuedSiegelmodularformsofgenus2. Ourmethodallowsustocompute
r afeweigenvaluesfortheHeckeoperators. Ourresultsagreewithcalculationsdonebyvander
a
GeerbasedonhisjointworkwithFaber[6,7].
1.1 Siegel modular forms
We will first introduce some notions and notation. Let V be a finite dimensional (cid:67)-vector
space, g a positive integer and let ρ:GL(g,(cid:67))→GL(V) be a representation. A Siegel modular
form f of weight ρ and genus (degree) g≥2 is then a V-valued holomorphic function on the
∗
Korteweg- de Vries Institute for Mathematics, Universiteit van Amsterdam, Amsterdam. Current address:
TheoreticalBiologyandBioinformatics,UniversiteitUtrecht,Utrecht.
1
Siegel upper half-space H that satisfies for every element γ of the symplectic group Γ :=
g g
Sp(2g,(cid:90))thefunctionalequation
f(γ·τ)=ρ(j(γ,τ))f(τ), τ∈H .
g
Herewewrite j(γ,τ):=cτ+d forthefactorofautomorphyandγ·τ:=(aτ+b)(cτ+d)−1,where
(cid:179) (cid:180)
a,b,c,d∈gl(g,(cid:90))andγ= a b ∈Γ . ASiegelmodularform f hasaFourierseries
c d g
f(τ)=(cid:88)a(n)qn, qn:=e2πiσ(nτ),
n
where the sum is taken over the set Sg := (cid:169)n=(nij)(cid:175)(cid:175) nii∈(cid:90),2nij∈(cid:90),nij=nji(cid:170) of all half-
integer, symmetric g×g matrices n and σ(x) denotes the trace of a square matrix x. We will
(cid:48)
write x forthetransposeofamatrix xandwhen yisasquarematrixofappropriatesize,then
y[x]:=x(cid:48)yx.
TheFouriertransforma ofaSiegelmodularform f =(cid:80)a (n)qn definesafunctiona :S →V
f f f g
and if g>1, then a (n)(cid:54)=0 =⇒ n(cid:186)0 (the Koecher principle). Denote by S+ the subset of S
f g g
ofsemi-positivematrices. Aform f forwhich a (n)vanishesfornon-positive n∈S+ iscalleda
f g
cusp form. We will denote the space of modular forms of weight ρ by Mρ(Γg) and the space of
(cid:179) (cid:180)
cusp forms by Sρ(Γg). The general linear group GL(g,(cid:90))(cid:44)→Γg, embedded by u(cid:55)→ u0 u(cid:48)0−1 , acts
onS+ by u:n(cid:55)→unu(cid:48) andtheFouriertransform a of f behaveswellunderthisaction:
g f
a (unu(cid:48))=ρ(u)a (n). (1)
f f
From now on, we assume that g =2 unless otherwise specified. For convenience, we often
(cid:179) (cid:180)
write square matrices a b as (a,b;c,d) and when no confusion can be possible, we will write
c d
(n,r/2;r/2,m)∈S as(n,m,r). TheirreduciblerepresentationsρofGL(2,(cid:67))canbecharacterized
2
bytheirhighestweightvector(λ1≥λ2)andifλ2<0,thendim(cid:67)Mρ(Γ2)=0. Thereforeweonly
have to consider ‘polynomial’ representations. If we write j=λ −λ and k=λ , then ρ will
1 2 2
beisomorphictoSymj(St)⊗detk(St),whereStdenotesthestandardrepresentationofGL(2,(cid:67))
and Symj and detk denote the j-fold symmetric product and k-th power of the determinant
respectively. The representation Symj(St)⊗detk(St) is abbreviated to (j,k) and we write k to
denotetheweight(0,k).
The structure of the ring of classical Siegel modular forms M∗ :=(cid:76)kMk(Γ2) of genus 2 was
determinedbyIgusa[14]. Thesubring M∗0:=(cid:76)k≡0(2)Mk(Γ2)⊆M∗ isapolynomialring:
M0=(cid:67)[ϕ ,ϕ ,χ ,χ ],
∗ 4 6 10 12
where ϕ and ϕ are Eisenstein series of weight 4 and 6 (with Fourier series normalized
4 6
at(0,0,0))andχ andχ arecuspformsofweight10and12(withFourierseriesnormalized
10 12
at (1,1,1)). Satoh has determined the structure of the M∗0-module M(ij,∗) :=(cid:76)k≡i(2)M(j,k)(Γ2)
for(j,i)=(2,0)andIbukiyamadidthesamefor(j,i)=(2,1),(4,0),(4,1)and(6,0). Inthispaper
wewilldeterminethestructureof M1 . Ourmainresultcanbeformulatedasfollows.
(6,∗)
MainTheorem. TheM0-moduleM1 isfreelygeneratedbysevenelementsF ofweight(6,k)
∗ (6,∗) k
with k∈{11,13,15,17,19,21,23}.
We shall give the elements mentioned in the above theorem explicitly, but in order to do this
weneedtointroduceRankin-CohendifferentialoperatorsonSiegelmodularforms.
2
1.2 Rankin-Cohen operators
Rankin-Cohenoperators(RC-operators)sendt-tuplesofclassicalSiegelmodularformsto(pos-
sibly vector-valued) Siegel modular forms. They are therefore useful when we want to find
generators for modules of Siegel modular forms. RC-operators were studied in full general-
ity by (among others) Ibukiyama, Eholzer and Choie [11, 5, 3]. The general construction of
RC-operatorscanbequitecumbersomeandthereforeweonlyexplainthegenus2casehere.
WriteH =(cid:169)p∈(cid:67)[x,y]|∀λ∈(cid:67):p(λx,λy)=λjp(x,y)(cid:170)forthespaceofhomogeneouspolynomials
j
of degree j in two variables x and y. The representation ρ:=Symj(St)⊗det(cid:96)(St):GL(2,(cid:67))→
GL(H )isgivenexplicitlyby:
j
(ρ(G)·p)(x,y)=det(G)(cid:96)·p((x,y)G), G∈GL(2,(cid:67)), p∈H .
j
Let R =(cid:67)[rs |0<i≤ j≤2,0<s≤t]bethepolynomialringinthe3tvariablesrs andconsider
t ij ij
an element P of Hj⊗(cid:67)Rt. It is convenient to write P(r111,...,r2t2,x,y)=:P(r1,...,rt;v), where
rs =(cid:161)rs ,rs ;rs ,rs (cid:162) is a symmetric 2×2 matrix and v=(x,y)(cid:48). Using this notation, we can
11 12 12 22
givethefollowingdefinition.
Definition1.1. Anelement P of Hj⊗(cid:67)Rt iscalledρ-homogeneousif
P(Gr1G(cid:48),...,GrtG(cid:48);v)=det(G)(cid:96)·P(r1,...,rt;G(cid:48)v)
forallG∈GL(2,(cid:67)).
Ibukiyama refers to certain special elements P of the space Hj⊗(cid:67)Rt as ‘associated poly-
nomials’, since they are ‘associated’ to other polynomials P˜ called ‘pluri-harmonic’ polyno-
mials [11]. These polynomials P˜ can be constructed from P by first choosing an element
k=(k ,...,k )∈(cid:90)t , which we will refer to as a type, and then replacing the matrices rs by
1 t >0
ξsξs(cid:48) withξs=(ξs )a2×2k matrixofindeterminates:
ij s
P(cid:55)→P˜ : Hj⊗(cid:67)Rt→Hj⊗(cid:67)(cid:67)[ξsij|0<i≤2,0< j<2ks,0<s≤t],
P˜(ξ1 ,...,ξt ,x,y):=P(ξ1ξ1(cid:48),...,ξtξt(cid:48);v).
11 2,2kt
Note that the map P (cid:55)→P˜ depends on the choice of the type k and that for P ∈Hj⊗(cid:67)Rt, the
polynomial P˜ isnotnecessarilypluri-harmonic. Wecannowgivethefollowingdefinition.
Definition 1.2. An element P ∈Hj⊗(cid:67)Rt is called k-harmonic if P˜ is harmonic in the sense
that
∆P˜:= (cid:88) ∂2P˜ =0.
(∂ξs )2
i,j,s ij
Polynomials that are ρ-homogeneous and k-harmonic can be used to define RC-operators as
shown in the following theorem and therefore we will refer to these polynomials as RC-
polynomials. ThespaceofRC-polynomialsisdenotedbyHρ(k)⊂Hj⊗(cid:67)Rt. Writeτ=(τ1,z;z,τ2)
foranelementτ∈H andwrite|k|=(cid:80) k .
2 s s
Theorem 1.3 (Ibukiyama). Suppose that P ∈ Hρ(k) with ρ = (j,(cid:96)) and let f1,...,ft be
classical Siegel modular forms on Γ of weight k ,...,k respectively. Write d/dτs :=
2 1 t
(cid:161)∂/∂τs,1∂/∂zs;1∂/∂zs,∂/∂τs(cid:162). The H -valuedfunction
1 2 2 2 j
D[P](f1,...,ft)(τ):= (2πi1)j/2+(cid:96) P(d/dτ1,...,d/dτt;v)f1(τ1)···ft(τt)(cid:175)(cid:175)τ1=···=τt=τ, τ∈H2 (2)
3
isaSiegelmodularformofweightρ⊗det|k|(St)=(j,(cid:96)+|k|)andgenus2.
A more general version of Theorem 1.3 was proven by Ibukiyama [11]. Ibukiyama and oth-
ers also give explicit examples of RC-polynomials and all RC-polynomials for type of length 2
were determined explicitly by Miyawaki [15]. If the length t of the type k equals 2, then a
non-zero Siegel modular form that has been constructed using a RC-operator will only have
weight (j,(cid:96)) with (cid:96) odd if a classical Siegel modular form of odd weight has been used in this
construction [17, 12]. The cusp form χ ∈M (Γ ) is such a classical Siegel modular form of
35 35 2
oddweight,butifweweretouseitinaconstructionwithanRC-operator,theweightofthere-
sultingSiegelmodularformwillhave(cid:96)≥39. Thedimensionofe.g. M (Γ )equals1,hence
(2,21) 2
we need RC-operators with a higher type length in order to get forms of ‘low’ weight. This
‘trade-off’ causes further problems when we increase j and we will give a (partial) solution to
this problem below by introducing a Rankin-Cohen bracket on vector-valued Siegel modular
forms.
IbukiyamaconstructedRC-polynomialsofweight(2,1)and(4,1)andtypelength t=3inorder
tofindgeneratorsfor M1 and M1 . Wewillgeneralizethesepolynomialstofindgenerators
(2,∗) (4,∗)
for M1 , but our generalization can also be used to find other vector-valued Siegel modular
(6,∗)
formsofweight(j,(cid:96))with j≥2and(cid:96)≥15odd.
1.3 Acknowledgements
We would like to thank Gerard van der Geer, who stimulated the author to write this paper,
andFabienCléryfortheusefuldiscussionsandtheircorrectionsandcomments.
2 Results
TheHilbert-Poincaréseriesforthedimensionsof M(6,(cid:96))(Γ2)with(cid:96)oddisgivenby(cf. [18])
(cid:88) dim(cid:67)M(6,(cid:96))(Γ2)·X(cid:96)= X11(+1−XX134+)(1X−15X+6X)(117−+XX1019)(+1−XX211+2)X23. (3)
(cid:96)≡1(2)
This suggests that we should look for generators for M1 of weights (6,(cid:96)) with (cid:96) =
(6,∗)
11,13,...,23. We will first give two methods for constructing forms of weight (6,(cid:96)) with
(cid:96)=15,17,...,23and(cid:96)=11,13respectivelyandthenwewillgiveexplicitgenerators.
2.1 RC-polynomials of weight (j,1)
We start with an example by Ibukiyama and Eholzer [5]. The RC-polynomials for elliptic
modular forms were studied by Rankin and Cohen [4] and they can be used to construct RC-
polynomialsofweight(j,0). Suchagenus1RC-polynomial p ∈(cid:67)[r ,r ]canbewrittenas
j,k 1 2
pj,k(r1,r2)=(cid:88)j/2(−1)i(cid:179)j/i2(cid:180)(k1+j/2−1)i(k2+j/2−1)j/2−ir1j/2−ir2i , j≡0(2). (4)
i=0
Here we use the Pochhammer symbol (x) := x(x−1)(x−2)···(x−n+1). The corresponding
n
RC-polynomial P ∈H (k)isthengivenby
j,k (j,0)
P (r1,r2;v):=p (r1[v],r2[v]),
j,k j,k
4
anditiseasytoverifythat P isindeed(j,0)-homogeneousand k-harmonic. Givenapolyno-
j,k
mial pinthevariablesr ,...,r ,wethendenotethemapthatreplacesthevariablesr byrs[v]
1 t s
byΨ:
Ψ:p(r1,...,rt)(cid:55)→p(r1[v],...,rt[v]):(cid:67)[r1,...,rt]−→(cid:77)Hj⊗(cid:67)Rt.
j≥0
We will now give a construction that uses elliptic (i.e. genus g=1) RC-polynomials and pro-
ducesRC-polynomialsofweight(j,1)andtype k=(k ,k ,k ). Wefirstconsiderthecrossprod-
1 2 3
uct on the space of 2×2 symmetric matrices. Let J=(0,1;−1,0) and define for A=(a,b;b,c)
andB=(a(cid:48),b(cid:48);b(cid:48),c(cid:48))
(cid:181) 2ab(cid:48)−2ba(cid:48) ac(cid:48)−ca(cid:48) (cid:182)
A×B:=AJB−BJA= , (5)
ac(cid:48)−ca(cid:48) 2bc(cid:48)−2cb(cid:48)
then(GAG(cid:48))×(GBG(cid:48))=det(G)·G(A×B)G(cid:48) forallG∈GL(2,(cid:67)).
Definition2.1. TheoperatorMk:(cid:67)[r1,r2,r3]−→(cid:76)jHj⊗(cid:67)R3 isdefinedasfollows:
(cid:179) (cid:180) (cid:179) (cid:180) (cid:179) (cid:180)
M p=r1×r2[v]Ψ k p+r ∂ p −r1×r3[v]Ψ k p+r ∂ p +r2×r3[v]Ψ k p+r ∂ p .
k 3 3∂r 2 2∂r 1 1∂r
3 2 1
Proposition 2.2. Let p= pj,(k1+1,k2+1) be the elliptic RC-polynomial defined by equation (4),
thenM p isaRC-polynomialofweight(j+2,1)andtype k=(k ,k ,k ).
k 1 2 3
TheproofofProposition2.2iselementarybutverytedious. Wewillthereforeomitthedetails.
The operator M can be shown to ‘commute’ with the Laplacian ∆ at the cost of a shift in the
k
type (k ,k ,k )(cid:55)→(k +1,k +1,k +1). The above result then follows immediately. Note that
1 2 3 1 2 3
wecouldalsouseotherappropriatepolynomials p∈(cid:67)[r ,r ,r ].
1 2 3
We now have a recipe to construct vector-valued Siegel modular forms of weight (j,(cid:96)) with (cid:96)
odd. First take f ∈ M (Γ ) for s = 1,2,3, then choose an elliptic RC-polynomial p =
s ks 2
pj−2,(k1+1,k2+1) and define P =M(k1,k2,k3)p ∈H(j,1)(k1,k2,k3). Theorem 1.3 then tells us that
thefunctionD[P](f1,f2,f3)isanelementof M(j,|k|+1)(Γ2). Thiswouldbeofnouseiftheresult-
ingSiegelmodularformsvanishidentically. WecanshowthatD[P](f ,f ,f )isnon-vanishing
1 2 3
bycomputinganon-zeroFouriercoefficient. If f =(cid:80) a (n)qn andD[P](f ,f ,f )=(cid:80) b(n)qn,
s n s 1 2 3 n
then
(cid:88)
b(n)= P(n ,n ,n ;v)a (n )a (n )a (n ). (6)
1 2 3 1 1 2 2 3 3
(nn1,n+2n,n+3)n∈(=S+2n)3
1 2 3
Althoughthisisasimpleformula,thefactthatwetakeasumoveralltriples(n ,n ,n )∈(S+)3
1 2 3 2
such that n +n +n =n can result in a computationally difficult problem, since the number
1 2 3
ofpartitions n +n +n =ngrowsveryfastasthetraceσ(n)grows.
1 2 3
Example2.3. TheoperatorM definedinDefinition2.1generalisesthepolynomialsgivenby
k
Ibukiyama[12,13]. Thesepolynomialsaregivenby
M 1∈H (k) and M (cid:161)(k +1)r −(k +1)r (cid:162)∈H (k)
k (2,1) k 1 2 2 1 (4,1)
with k=(k ,k ,k ). Ibukiyama uses these polynomials to define Rankin–Cohen brackets on
1 2 3
triplesofclassicalSiegelmodularformsandshowsthattheresultingvector-valuedSiegelmod-
ularformsgeneratethemodules M1 and M1 . TheoperatorD[M 1]canalsobedescribed
(2,∗) (4,∗) k
5
in terms of the cross product (5) and Satoh’s RC-brackets [17]. Satoh uses the space of sym-
metric2×2complexmatricesasarepresentationspaceforSym2(St)(wheretheGL(2,(cid:67))action
isgivenbyG:A(cid:55)→GAG(cid:48) for A asymmetricmatrixandG∈GL(2,(cid:67)))andthendefines
[f1,f2]:=k1f1ddτf2−k2f2ddτf1∈M(2,k1+k2)(Γ2), fs∈Mks(Γ2).
Let f ∈M (Γ )for s=1,2,3. Then
s ks 2
F:=[f1,f2]×[f1,f3]∈M(2,2k1+k2+k3+1)(Γ2)
and unraveling the definitions easily shows that F is divisible by f1 in the M∗-module M(2,∗).
Ibukiyama’sRC-brackets[f ,f ,f ](in[12])arethengivenby[f ,f ,f ]:=cF/f forsomenon-
1 2 3 1 2 3 1
zeroconstant c.
2.2 A Rankin-Cohen bracket on vector-valued Siegel modular forms
OuraimistofindallSiegelmodularformsofweight(6,(cid:96))with(cid:96)odd. Formula(3)showsthat
dim(cid:67)M(6,11)(Γ2)=1andthelowest(cid:96)forwhichwecanusetheoperatorMk toconstructanon-
zeroformofweight(6,(cid:96))equals15. Notethatbytaking k=(4,4,4)and p=p ,wecanget
4,(5,5)
a form D[M p](ϕ ,ϕ ,ϕ )∈M (Γ ), but unfortunately this form vanishes identically. This
k 4 4 4 (6,13) 2
means that the above described construction with M is not sufficient for our purpose, unless
k
we would be able to divide by a classical Siegel modular form, but this appears to be quite
difficult without prior knowledge of the quotient. Ibukiyama encountered a similar problem
whenhedeterminedallmodularformsofweight(6,(cid:96))with(cid:96)evenandhesolvedthisbyusing
athetaseriesΘ ofweight(6,8)(cf.[13,9])andaKlingen-Eisensteinseries E ofweight(6,6)
8 6
(cf. [2]). These forms can not be constructed directly by means of RC-operators. However, as
wewillpointoutbelow, wecanuseRC-operatorstocomputetheirFouriercoefficients. Inthis
paperwehavescaledΘ suchthattheFouriercoefficientofΘ at(1,1,1)equals x4y2+2x3y3+
8 8
x2y4. Theform E isscaledsuchthatitsFouriercoefficientat(1,0,0)equals x6.
6
The forms Θ and E can be used to construct forms of weight (6,13) and (6,11). In order to
8 6
see how this can be done, we must first give a new interpretation of the operators D[M p],
k
where p is an elliptic RC-polynomial. Choose p= pj−2,(k1+1,k2+1) and let fs ∈ Mks(Γ2) for s=
1,2,3. Also let q= pj,(k1,k2) and define F=D[Ψq](f1,f2)∈M(j,k1+k2)(Γ2). We can re-write G:=
D[M p](f ,f ,f )insuchawaythat
k 1 2 3
G=c·(cid:169)F,f (cid:170)
3
tfohreseoxmacetbdielisnceriaprtifoonrmof(cid:169)(cid:169)··,,··(cid:170)(cid:170)obneloMw(,j,bku1+tkl2e)(tΓu2)s×fiMrstk3s(tΓa2t)eatnhde aadcvoannsttaagnetocf∈th(cid:67)is∗.effWoret.wWillegciavne
replace the modular form F that was defined via a RC-operator by any other modular form of
weight (j,k +k ). So for instance, we can take F=E and f =ϕ and if we then apply (cid:169)·,·(cid:170),
1 2 6 3 4
weget
(cid:169)E ,ϕ (cid:170)∈M (Γ ).
6 4 (6,11) 2
Wewillnowgiveanexplicitformulafor(cid:169)·,·(cid:170).
Definition2.4. Supposethat F∈M(j,k)(Γ2)andϕ∈M(cid:96)(Γ2). DefinethedeterminantW(r)by
(cid:175) (cid:175)
(cid:175) r11 r12 r22 (cid:175)
(cid:175) (cid:175)
W(r):=(cid:175)(cid:175) y2 −xy x2 (cid:175)(cid:175):Hj→Hj⊗(cid:67)R1.
(cid:175) ∂2 ∂2 ∂2 (cid:175)
(cid:175) ∂x2 ∂x∂y ∂y2 (cid:175)
6
Wenowdefinethebrackets(cid:169)·,·(cid:170)asfollows.
(cid:169)F,ϕ(cid:170):= 1 (cid:179)(k+j/2−1)W(cid:161)dϕ(cid:162)F−(cid:96)ϕW(cid:161) d (cid:162)F(cid:180).
(j−1)2πi dτ dτ
Byourconsiderationsabove,wethenhavethefollowingresult.
Proposition 2.5. Take F andϕasinDefinition2.4. Theform(cid:169)F,ϕ(cid:170)isthenaSiegelmodular
formofweight(j,k+(cid:96)+1).
Inordertobeabletoworkwiththeforms(cid:169)F,ϕ(cid:170),weneedtoknowhowtocomputeFouriercoef-
ficients. Wecaneasilyfindaformulasimilarto(6). Writeϕ=(cid:80)n(cid:186)0a(n)qn andF=(cid:80)n(cid:186)0b(n)qn
andlet(cid:169)F,ϕ(cid:170)=(cid:80)n(cid:186)0c(n)qn,thenweget
(cid:88)
(j−1)c(n)= (k+j/2−1)a(n )W(n )b(n )−(cid:96)a(n )W(n )b(n ). (7)
1 1 2 1 2 2
(nn1,n+2n)∈(=Sn+2)2
1 2
Remark. A modular form that is defined using the brackets (cid:169)·,·(cid:170) will always be a cusp form.
Supposethatweuse F∈M(j,k)(Γ2)andϕ∈M(cid:96)(Γ2)inordertoconstruct(cid:169)F,ϕ(cid:170)∈M(j,k+(cid:96)+1)(Γ2),
then either k+(cid:96)+1 is odd and hence (cid:169)F,ϕ(cid:170) must be a cusp form, or one of the integers (cid:96) or k
isodd,implyingthat F orϕisacuspform.
We can also see directly from Formula (7) that the brackets map to S(Γ ) by considering the
2
Fouriercoefficientsatsingularindicesn. Supposethatn=n +n wheren,n ,n ∈S+ andnis
1 2 1 2 2
singular,thenwecanfinda u∈SL(2,(cid:90))suchthat unu(cid:48)=(ν,0,0), un u(cid:48)=(ν ,0,0)and un u(cid:48)=
1 1 2
(ν ,0,0). Hence, byFormula(1)wecanassumewithoutlossofgeneralitythat n isoftheform
2
n=(ν,0,0). The Fourier coefficients b(n) of F for n=(ν,0,0) always have the form α(ν)·xj for
someconstantα(ν). Therefore,wegetfor s=1,2that
(cid:175) (cid:175)
(cid:175) −xy x2 (cid:175)
W(n )b(n )=ν (cid:175) (cid:175)α(ν )xj=0.
s 2 s(cid:175) ∂2 ∂2 (cid:175) 2
(cid:175) ∂x∂y ∂y2 (cid:175)
Thisshowsthat c(n)inFormula(7)vanishesforsingular n.
2.3 Generators for M1
(6,∗)
As promised, we will now give explicit generators for M1 . We first use the brackets (cid:169)·,·(cid:170) to
(6,∗)
defineformsofweight(6,11)and(6,13):
F :=(cid:169)E ,ϕ (cid:170)/1152, F :=(cid:169)Θ ,ϕ (cid:170)/4
11 6 4 13 8 4
andfortheremainderofthegenerators,weusetheconstructionwithM . Choosethefollowing
k
ellipticRC-polynomials p andtypes k :
i i
p polynomial type k
i i
p (r ,r ) = (5r2−14r r +7r2)/160 (5,4,5)
15 1 2 1 1 2 2
p (r ,r ) = (4r2−8r r +3r2)/192 (5,6,5)
17 1 2 1 1 2 2
p (r ,r ) = (22r2−24r r +5r2)/1920 (4,10,4)
19 1 2 1 1 2 2
p (r ,r ) = (22r2−24r r +5r2)/2880 (4,10,6)
21 1 2 1 1 2 2
p (r ,r ) = (13r2−14r r +3r2)/16 (5,12,5)
23 1 2 1 1 2 2
7
k λ(2)on N (Γ ) λ(2)on S (Γ )
(6,k) 2 (6,k) 2
6 −24·(1+24) —
8 — 0
10 216·(1+28) 1680
11 — −11616
12 −528·(1+210) X2−22368X+57231360
13 — −24000
15 — X2+68256X+593510400
17 — X3+363264X2+136028160X−4603543289856000
19 — X4+1202400X3−1311202861056X2
−179858880190218240X−1566691549034368204800
Table1: EigenvaluesoftheHeckeoperatorT(2)onM (Γ )forsomevaluesofk. Ifapolynomialin X
(6,k) 2
is given, the eigenvalues λ(2) are the roots of this polynomial. The space N (Γ ) is the orthogonal
(6,k) 2
complementofS (Γ )withrespecttothePeterssonproduct.
(6,k) 2
Wethenwrite P =M p for i=15,17,...,23anddefinemodularforms F asfollows.
i ki i i
F modularform weight
i
F = D[P ](χ ,ϕ ,χ ) (6,15)
15 15 5 4 5
F = D[P ](χ ,ϕ ,χ ) (6,17)
17 17 5 6 5
F = D[P ](ϕ ,χ ,ϕ ) (6,19)
19 19 4 10 4
F = D[P ](ϕ ,χ ,ϕ ) (6,21)
21 21 4 10 6
F = D[P ](χ ,χ ,χ ) (6,23)
23 23 5 12 5
The form χ denotes the square root of χ in the ring of holomorphic functions on H (see
5 10 2
e.g.[8]). Ourmainresultcannowbeformulatedasfollows:
MainTheorem. The M0-module M1 isfreeandcanbewrittenasthefollowingdirectsum:
∗ (6,∗)
M1 =(cid:77)F ·M0,
(6,∗) i ∗
i∈I
where F aredefinedaboveandthedirectsumistakenovertheset I={11,13,15,17,19,21,23}.
i
2.4 Eigenvalues of the Hecke operators
Since we now know all modular forms of weight (6,k) with k∈(cid:90), we can calculate the eigen-
values of the Hecke operators T(p) (cf. [2, 1]). We did this for p=2,3 and some k (Table 1
and2).
At our request, G. van der Geer computed some of these eigenvalues using a completely
independent method that is based on counting points on hyperelliptic curves over finite
fields [9, 6, 7]. The values listed here agree with these. Also note that the characteristic
polynomialsof T(2)and T(3)on S (Γ )with k=12and15havethefollowingdiscriminant:
(6,k) 2
k ∆(det(T(2)−X)) ∆(det(T(3)−X))
12 2103272601 2143672132601
15 2103229·83·103 2123829·53283·103
8
k λ(3)on S (Γ )
(6,k) 2
8 −27000
10 −6120
11 −106488
12 X2+335664X−14832719455680
13 −8505000
15 X2+228022128X+8319716602228800
17 X3+1086146712X2−341960280255362880X−188775313801934579676864000
Table2: EigenvaluesoftheHeckeoperatorT(3)onM (Γ )forsomevaluesofk.
(6,k) 2
This shows for p=2,3 that the eigenvalues of T(p) on S (Γ ) and S (Γ ) are elements
(cid:112) (cid:112) (6,12) 2 (6,15) 2
ofthesamequadraticnumberfield(cid:81)( 601)and(cid:81)( 29·83·103)respectively. Wealsoverified
thatthecharacteristicpolynomialsofT(2)andT(3)onS (Γ )definethesamenumberfield.
(6,17) 2
3 Proof of the main theorem
We will only give a sketch of the full proof since it involves many elementary computations.
Let U denote the Hodge bundle corresponding to the factor of automorphy j. The forms F
i
definedabovearesectionsofSym6(U)⊗L⊗ki,whereLdenotesthelinebundledet(U). Inorder
to show that the forms F are independent over M0, we have to show that χ :=F ∧F ∧
i ∗ 140 11 13
···∧F ∈L⊗140 isnon-vanishing. Theformχ isanelementof M (Γ )andwecancompute
23 140 140 2
theFouriercoefficientsofthisform. Wewillthereforeprovethemaintheorembycomputinga
non-zeroFouriercoefficientofχ . Writeχ =(cid:80) c(n)qn and F =(cid:80) a (n)qn. Thefollowing
140 140 n i n i
formulaholdsfortheFouriercoefficients c(n):
(cid:88)
c(n)= det(a (n ),a (n ),...,a (n )). (8)
11 1 13 2 23 7
(cid:80)ini=n
As mentioned above, this can be hard to compute if σ(n) is large. In order to find a non-zero
Fourier coefficient c(n) of χ , we need at least σ(n)≥14 since the forms F are cusp forms.
140 i
Indeed,undertheSiegeloperatorΦ,theformsF maptoellipticmodularformsofoddweight.
i
Theseformsallvanish.
Using algorithms provided by Resnikoff and Saldaña [16], we calculated Fourier coefficients
of the classical Siegel modular forms ϕ ,ϕ ,χ and χ . The Fourier coefficients of χ can be
4 6 10 12 5
computedusingχ2=χ .1 WethenwereabletocomputeFouriercoefficientsoftheforms F .
5 10 i
In order to find Fourier coefficients of F and F , we first had to compute Fourier coeffi-
11 13
cients of E and Θ . The modular form ϕ E is an element of M (Γ ) and this space has
6 8 4 6 (6,10) 2
dimension 2. Using a RC-polynomial of weight (6,0) and the modular forms ϕ and ϕ , we
4 6
can find a modular forms F in the space M (Γ ) (cf. Ibukiyama [13]) and this form is not
10 (6,10) 2
an eigenform. Hence, we can find Fourier coefficients of F and T(2)F . For some α,β∈(cid:81),
10 10
we must have αF +βT(2)F =ϕ E . The form E is an eigenform and by a theorem due to
10 10 4 6 6
Arakawa [2] we know the corresponding eigenvalue of T(2) (cf. Table 1). This gives a relation
1WhilecomputingtheFouriercoefficientsofχ5 weencounteredanerrorinTableIVof[16]. Thecoefficientsat
calculationclasses(3,3,3)and(2,6,0)shouldhaveoppositesign.
9
i 11 13 15 17 19 21 23
n (1,1,0) (1,1,1) (2,1,0) (2,1,0) (2,1,1) (2,1,1) (2,2,1)
a (n) 0 0 0 0 1 −5 3
i
−20 −2 312 0 14 −10 −37
0 −5 0 0 36 −6 −50
0 0 180 −300 24 −24 0
0 5 0 0 0 −30 50
20 2 −102 354 0 −12 37
0 0 0 0 0 0 −3
Table 3: A few Fourier coefficients of the forms F . The Fourier coefficients are written as column
i
vectors. The determinant of the above matrix occurs in the sum (8) for c(12,8,4). This determinant
equals214355311.
for the Fourier coefficients of E and using this relation, we were able to find α and β. This
6
alsoallowedustocomputetheFouriercoefficientsof E .
6
AsimilarmethodcanbeusedtocomputeFouriercoefficientsofΘ . AgainusingaRC-operator
8
ofweight(6,2)andthemodularformsϕ andϕ ,wecanfindamodularformF inS (Γ )
4 6 12 (6,12) 2
(cf. Ibukiyama [13]). This form is again not an eigenform. The form ϕ Θ ∈S (Γ ) must
4 8 (6,12) 2
be a linear combination αF +βT(2)F and since Ibukiyama has computed a few Fourier
12 12
coefficientsofΘ (cf. [10]),wewereabletodetermineαandβ.
8
Wethenusedformulas(6)and(7)tocomputeFouriercoefficientsoftheformsF . Afewexam-
i
plesaregiveninTable3.
We wrote a script to compute the Fourier coefficient of χ at n =(12,8,4) and found that
140
c(12,8,4)=−2183752(cid:54)=0. We checked our computations by also computing the Fourier coeffi-
cientat(12,8,−4)andfoundthat c(12,8,−4)=c(12,8,4)whichisinlinewithequation(1). This
provesourmainresult.
References
[1] A. N. Andrianov. Euler products corresponding to Siegel modular forms of genus 2. Rus-
sianMath.Surveys,29(3):45–116,1974.
[2] T. Arakawa. Vector valued Siegel’s modular forms of degree two and the associated An-
drianovL-functions. Manuscriptamath.,44:155–185,1983.
[3] Y.ChoieandW.Eholzer. Rankin-CohenoperatorsforJacobiandSiegelforms. Journalof
NumberTheory,68:160–177,1998.
[4] H. Cohen. Sums involving the values at negative integers of L-functions of quadratic
characters. MathematischeAnnalen,217:271–285,1975.
[5] W.EholzerandT.Ibukiyama.Rankin-CohentypedifferentialoperatorsforSiegelmodular
forms. InternationalJournalofMathematics,9(4):442–463,1998.
10