MIN HO LEE
Received 4 November 2005; Accepted 26 March 2006
We study mixed Jacobi-like forms of several variables associated to equivariant maps of the Poincar´e upper half-plane in connection with usual Jacobi-like forms, Hilbert mod-ular forms, and mixed automorphic forms. We also construct a lifting of a mixed auto-morphic form to such a mixed Jacobi-like form.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Jacobi-like forms of one variable are formal power series with holomorphic coefficients satisfying a certain transformation formula with respect to the action of a discrete sub-groupΓof SL(2,R), and they are related to modular forms forΓ, which of course play a major role in number theory. Indeed, by using this transformation formula, it can be shown that that there is a one-to-one correspondence between Jacobi-like forms whose coefficients are holomorphic functions on the Poincar´e upper half-plane and certain se-quences of modular forms of various weights (cf. [1,12]). More precisely, each coefficient of such a Jacobi-like form can be expressed in terms of derivatives of a finite number of modular forms in the corresponding sequence. Jacobi-like forms are also closely linked to pseudodifferential operators, which are formal Laurent series for the formal inverse ∂−1of the differentiation operator∂with respect to the given variable (see, e.g., [1]). In addition to their natural connections with number theory and pseudodifferential oper-ators, Jacobi-like forms have also been found to be related to conformal field theory in mathematical physics in recent years (see [2,10]).
The generalization of Jacobi-like forms to the case of several variables was studied in [8] in connection with Hilbert modular forms, which are essentially modular forms of several variables. As it is expected, Jacobi-like forms of several variables correspond to sequences of Hilbert modular forms. Another type of generalization can be provided by considering mixed Jacobi-like forms of one variable for a discrete subgroupΓ⊂SL(2,R), which are associated to a holomorphic map of the Poincar´e upper half-plane that is equi-variant with respect to a homomorphism ofΓinto SL(2,R) (cf. [7,9]). Mixed Jacobi-like
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 31542, Pages1–14
forms are related to mixed automorphic forms, and examples of mixed automorphic forms include holomorphic forms of the highest degree on the fiber product of elliptic surfaces (see [6]).
In this paper, we study mixed Jacobi-like forms of several variables associated to equi-variant maps of the Poincar´e upper half-plane in connection with usual Jacobi-like forms, Hilbert modular forms, and mixed automorphic forms. We also construct a lifting of a mixed automorphic form to such a mixed Jacobi-like form.
2. Jacobi-like forms
In this section, we review Jacobi-like forms of several variables and describe some of their properties. We also describe Hilbert modular forms, which are closely linked to such Jacobi-like forms.
Throughout this paper, we fix a positive integern. Let (z1,...,zn) be the standard co-ordinate system forCn, and denote the associated partial differentiation operators by
∂1= ∂
∂z1,...,∂n= ∂
∂zn. (2.1)
We will often use the multi-index notation. Thus, given α=(α1,...,αn)∈Zn andu= (u1,...,un)∈Cn, we have
∂α=∂α1
1 ...∂αnn, uα=uα11...uαnn, (2.2) and forβ=(β1,...,βn)∈Zn, we writeα≤βifαi≤βifor eachi=1,...,n. Furthermore, we also writec=(c,...,c)∈Znifc∈Z, and denote byZ
+the set of nonnegative integers. Givenα∈Znandβ∈Zn
+, we writeβ!=β1!...βn! and
α β
=
α1 β1
···
αn βn
, (2.3)
where for 1≤i≤n, we haveαi
0
=1 and
αi βi
=αi(αi−1)···(αi−βi+ 1)
βi! (2.4)
forβi>0.
LetᏴ⊂Cbe the Poincar´e upper half-plane. Then the usual action of SL(2,R) onᏴ by linear fractional transformations induces an action of SL(2,R)non the productᏴnof ncopies ofᏴ. Thus, ifγ∈SL(2,R)nandz=(z1,...,zn)∈Ᏼnwith
γ=γ1,...,γn
, γi=
ai bi ci di
∈SL(2,R) (1≤i≤n), (2.5)
then we have
γz=γ1z1,...,γnzn
=a1z1+b1 c1z1+d1,...,
anzn+bn cnzn+dn ∈Ᏼ
For suchγandz, we set
J(γ,z)=jγ1,z1
,...,jγn,zn
∈Cn, jγ i,zi
=cizi+di (2.7)
for 1≤i≤n. We denote byJ(γ,z) the diagonal matrix with diagonal entries j(γi,zi) with 1≤i≤n, that is,
J(γ,z)=diagjγ1,z1
,...,jγn,zn. (2.8)
Then the map (γ,z)→J(γ,z) satisfies the cocycle condition
J(γγ,z)=J(γ,γ z)J(γ ,z) (2.9)
for allγ,γ∈SL(2,R)nandz∈Ᏼn. Given an elementη=η1,...,η
n)∈Znand a map f :Ᏼn→C, we set
f|ηγ
(z)=J(γ,z)−ηf(γz) (2.10)
for allz∈Ᏼnandγ∈SL(2,R)n. LetΓbe a discrete subgroup of SL(2,R)n.
Definition 2.1. Givenη=(η1,...,ηn)∈Zn+, a Hilbert modular form of weightηforΓis a holomorphic function f :Ᏼn→Csuch that
f|ηγ= f (2.11)
for allγ∈Γ, where f|ηγis as in (2.10). Denote byᏹη(Γ) the space of all Hilbert modular forms of weightηforΓ.
Remark 2.2. The usual definition of Hilbert modular forms also includes the regularity condition at the cusps, which is satisfied automatically forn >1 according to Koecher’s principle (cf. [3,4]).
We denote byRthe ring of holomorphic functionsf(z1,...,zn) onᏴnand byR[[X]]= R[[X1,...,Xn]] the set of all formal power series inX1,...,Xnwith coefficients inR. Thus, using the multi-index notation, an element ofR[[X]] can be written in the form
Φ(z,X)= α≥0
fα(z)Xα (2.12)
withz=(z1,...,zn)∈ᏴnandXα=X1α1...Xnαnforα=(α1,...,αn)∈Zn+.
LetC×=C− {0}be the set of nonzero complex numbers. Givenλ=(λ1,...,λn)∈ (C×)n, we denote byλ=diag(λ
1,...,λn) the associatedn×ndiagonal matrix, and set
C×X=Xλ|λ∈C×n=
λ1X1,...,λnXn
|λ1,...,λn∈C×
, (2.13)
whereX=(X1,...,Xn) is regarded as a row vector. Using (2.9), we see that SL(2,R)nacts onᏴn×C×Xby
for allz∈Ᏼn,λ∈(C×)n, andγ∈SL(2,R)n, whereJ(γ,z) is as in (2.8) so that
XJ(γ,z) −2λ=jγ1,z1
−2
λ1X1,...,j
γn,zn
−2 λnXn
. (2.15)
We now set
Kξ,ηγ,z,Xλ=J(γ,z)ξexp n
i=1
ciηijγi,zi−1λiXi
(2.16)
forz∈Ᏼn,γas in (2.5), andλ∈(C×)n. Then it can be shown that
Kξ,η
γγ, (z,Xλ)=Kξ,η
γ,γ·(z,Xλ)Kξ,η
γ, (z,Xλ) (2.17)
for allγ,γ∈SL(2,R)n, whereγ·(z,Xλ) is as in ( 2.14).
Definition 2.3. Givenξ,η∈Zn, a Jacobi-like form forΓofnvariables of weightξ, and index ηis an element,
Φ(z,X)=Φz,X1,...,Xn
(2.18)
ofR[[X]] satisfying
Φγz,XJ(γ,z) −2=Kξ,ηγ, (z,X)Φ(z,X) (2.19)
for allγ∈Γandz∈Ᏼn. Denote by
ξ,η(Γ) the space of all Jacobi-like forms ofnvariables forΓof weightξand indexη.
Remark 2.4. Jacobi-like forms of several variables inξ,η(Γ) withξ=0 andη=1 were considered in [8], while Jacobi-like forms of one variable with index 0 were studied in [12].
Proposition 2.5. Givenε∈Zn
+, consider a formal power series
Φ(z,X)= α≥ε
φα(z)Xα∈R[X]. (2.20)
Then the following conditions are equivalent.
(i) The power seriesΦ(z,X) is a Jacobi-like form belonging toξ,η(Γ). (ii) The coefficient functionsφα:Ᏼ→Csatisfy
φα|2α+ξγ
(z)=α−ε δ=0
1 δ!
cδηδ
J(γ,z)δφα−δ(z) (2.21)
for allz∈Ᏼnandα≥ε, whereγ∈Γis as in (2.5) withc=(c
1,...,cn). (iii) There exist modular formsfν∈ᏹ2ν+ξ(Γ) forν≥εsuch that
φα(z)= α−ε
β=0
ηβ
β!(2α+ξ−β−ε)!∂βfα−β(z) (2.22)
Proof. The proposition can be proved by slightly modifying the proofs of [8, Lemma 4.2
and Theorem 4.4].
IfΦ(z,X)=α≥εφα(z)Xα∈ξ,η(Γ), then (2.21) implies that
φε|2ε+ξγ=φε (2.23)
for allγ∈Γ; hence the initial coefficientφε(z) of the formal power series Φ(z,X) is a Hilbert modular form of weight 2ε+ξforΓ. We set
ξ,η(Γ)ε=Xεξ,η(Γ), (2.24)
which is a subspace ofξ,η(Γ) consisting of the elements of the formα≥εφα(z)Xα. Then we see that there is a linear map
F:ξ,η(Γ)ε−→ᏹ2ε+ξ(Γ) (2.25)
sending an element ofξ,η(Γ)εto its coefficient ofXε.
3. Mixed Jacobi-like forms
In this section, we discuss Jacobi-like forms of several variables associated to holomorphic maps of the Poincar´e upper half-planeᏴthat are equivariant with respect to a discrete subgroup of SL(2,R). Such Jacobi-like forms are related to mixed automorphic forms.
LetΓbe a discrete subgroup of SL(2,R), and for eachk∈ {1,...,n}, let ωk:Ᏼ→Ᏼ andχk:Γ→SL(2,R) be a holomorphic map and a group homomorphism, respectively, satisfying
ωk(γζ)=χk(γ)ωk(ζ) (3.1)
for allζ∈Ᏼandγ∈Γ. By setting
ω=ω1,...,ωn, χ=χ1,...,χn, (3.2)
we obtain a holomorphic mapω:Ᏼ→Cnand a homomorphismχ:Γ→SL(2,R)n. Given η=(η1,...,ηn)∈Zn, we define the mapJω,χ: SL(2,R)×Ᏼ→Cnby
Jω,χ(γ,ζ)=jχ1(γ),ω1(ζ),...,jχn(γ),ωn(ζ) (3.3)
for allγ∈SL(2,R) andζ∈Ᏼ, wherej: SL(2,R)×Ᏼ→Cis as in (2.7).
Definition 3.1. Givenξ=(ξ1,...,ξn)∈Zn, a mixed automorphic form of typeξ associated toΓ,ω, andχis a holomorphic map f :Ᏼ→Csatisfying
f(γζ)=Jω,χ(γ,ζ)ξf(ζ)=jχ1(γ),ω1(ζ)ξ1···jχn(γ),ωn(ζ)ξnf(ζ) (3.4)
Definition 3.2. LetᏲbe the set of holomorphic functions onᏴ, and letᏲ[[X]] be the space of formal power series inX=(X1,...,Xn). Givenξ=(ξ1,...,ξn),η=(η1,...,ηn)∈
Zn, a formal power series F(ζ,X)∈Ᏺ[[X]] is a mixed Jacobi-like form of weightξ and indexηassociated toΓ,ω, andχif it satisfies
Fγζ,XJω,χ(γ,ζ)−2
=Jω,χ(γ,ζ)ξexp
n
k=1
cχ,kηkXk jχk(γ),ωk(ζ)
F(ζ,X) (3.5)
for allζ∈Ᏼandγ∈Γ, whereJω,χ(γ,ζ) denotes the diagonal matrix
diagjχ1(γ),ω1(ζ)
,...,jχn(γ),ωn(ζ) (3.6)
andcχ,kis the (2, 1)-entry of the matrixχk(γ)∈SL(2,R). Denote byξ,η(Γ,ω,χ) the space of mixed Jacobi-like forms of weightξand indexηassociated toΓ,ω, andχ.
Givenμ∈Znand a functionh:Ᏼ→C, set
h|ωμ,χγ(ζ)=h(γζ)Jω,χ(γ,ζ)−μ (3.7)
for allζ∈Ᏼandγ∈Γ.
Lemma 3.3. A formal power seriesF(ζ,X)=α≥εfα(ζ)Xα∈Ᏺ[[X]] withε∈Zn+is an element ofξ,η(Γ,ω,χ) if and only if
fα|ω2α,χ+ξγ
(ζ)=α
−ε
δ=0 1 δ!
cδ χηδ
Jω,χ(γ,ζ)δ fα−δ(ζ) (3.8)
for allγ∈Γwithcχ=(cχ,1,...,cχ,n),ζ∈Ᏼn, andα≥ε, wherecχ,j denotes the (2, 1)-entry of the matrix χj(γ)∈SL(2,R) for 1≤j≤n. In particular, the initial coefficient fε(ζ) of F(ζ,X) is an element ofᏹ2ε+ξ(Γ,ω,χ) ifF(ζ,X)∈ξ,η(Γ,ω,χ).
Proof. Givenγ∈Γas described by (3.4) and (3.5), the formal power series F(ζ,X)=
α≥εfα(ζ)Xαis an element ofξ,η(Γ,ω,χ) if and only if
α≥ε
fα(γζ)Jω,χ(γ,ζ)−2α−ξXα= n
i=1
∞
μi=0
1 μi!
cμi
χ,iημ
i
i Xμ
i
i jχi(γ),ωi(z)μi
·
ν≥ε fν(ζ)Xν
=
μ≥0
ν≥ε 1 μ!
cμημ
Jω,χ(γ,ζ)μfν(ζ)X μ+ν
(3.9)
for allζ∈Ᏼ. Thus by comparing the coefficients ofXα, we obtain
fα(γζ)Jω,χ(γ,ζ)−2α−ξ= α−ε
δ=0 1 δ!
cδηδ
Jω,χ(γ,ζ)δ fα−δ(ζ), (3.10)
For eachε∈Znwithε≥0, we set
ξ,η(Γ,ω,χ)ε=Xεξ,η(Γ,ω,χ). (3.11)
Then byLemma 3.3, we see that there is a linear map
Ᏺω,χ:ξ,η(Γ,ω,χ)ε−→ᏹ2ε+ξ(Γ,ω,χ) (3.12)
sending an elementα≥εfα(ζ)Xαofξ,η(Γ,ω,χ) to its initial coefficient fε(ζ). IfRis the set of holomorphic functions onᏴnas inSection 2, we define the maps
Δω:R−→Ᏺ, Δω X:R
[X]−→Ᏺ[X] (3.13)
associated to the mapω:Ᏼ→Ᏼnas in (3.2) by
Δωh(ζ)=hω(ζ), Δω XF
(ζ,X)=Fω(ζ),X (3.14)
for allζ∈Ᏼ,h∈R, andF∈R[[X]]. Given a discrete subgroupΓof SL(2,R), letΓχbe a discrete subgroup of SL(2,R)nsuch that
χ(Γ)=χ1(Γ)× ··· ×χn(Γ)⊂Γχ, (3.15)
whereχ=(χ1,...,χn) is as in (3.2). Theorem 3.4. (i) IfΔω:R→ᏲandΔω
X:R[[X]]→Ᏺ[[X]] are as in (3.14), then Δωᏹ
ξΓχ
⊂ᏹξ(Γ,ω,χ), ΔωX
ξ,ηΓχ
ε
⊂ξ,η(Γ,ω,χ)ε (3.16)
for allξ,η∈Zn.
(ii) IfᏲandᏲω,χare the linear maps in (2.25) and (3.12), respectively, then the diagram
ξ,ηΓχ
ε
Ᏺ
Δω X
ᏹ2ε+ξΓχ
Δω
ξ,η(Γ,ω,χ)ε
Ᏺω,χ
ᏹ2ε+ξ(Γ,ω,χ)
(3.17)
is commutative.
Proof. If f :Ᏼn→Cis an element ofᏹ
ξ(Γχ), then by (3.14) we have
Δωf(γζ)=fω(γζ)=fχ
1(γ)ω1(ζ),...,χn(γ)ωn(ζ)
for allζ∈Ᏼandγ∈Γ; henceΔωf is an element ofᏹ
ξ(Γχ,ω,χ). On the other hand, ifF is an element ofξ,η(Γχ) by (3.5) and (3.14), we see that
Δω
X(F)
γζ,XJω,χ(γ,ζ)−2=Fχ1(γ)ω1(ζ),...,χn(γ)ωn(ζ),XJω,χ(γ,ζ)−2
=Jχ(γ),ω(ζ)ξexp
n
k=1
cχ,kηkXk jχk(γ),ωk(ζ)
Fω(ζ),X
(3.19)
for allζ∈Ᏼandγ∈Γ. ThusΔω
XFis an element ofξ,η(Γ,ω,χ), andΔωXF∈ξ,η(Γ,ω,χ)ε ifF∈ξ,η(Γχ)ε, which proves (i). In order to verify (ii), consider an elementΦ(ζ,X)=
α≥εφα(ζ)Xα∈ξ,η(Γχ)ε. Then we have
Δω◦Ᏺ(Φ)(ζ)=Δωφε(ζ)=φεω1(ζ),...,ωn(ζ) (3.20)
forζ∈Ᏼ. On the other hand, we have
Δω XΦ
(ζ,X)=Φω(ζ),X=Φω1(ζ),...,ωn(ζ),X
=
α≥ε
φαω1(ζ),...,ωn(ζ)
Xα. (3.21)
Thus we see that
Ᏺω,χ◦ΔωX
(Φ)(ζ)=φε
ω1(ζ),...,ωn(ζ)
=Δω◦Ᏺ(Φ)(ζ), (3.22)
which implies (ii); hence the proof of the theorem is complete.
4. Examples
In this section, we discuss two examples related to mixed Jacobi-like forms. The first one involves a fiber bundle over a Riemann surface whose generic fiber is the product of elliptic curves, and the second one is linked to solutions of linear ordinary differential equations.
Example 4.1. LetEbe an elliptic surface (cf. [5]). ThusEis a compact surface overCthat is the total space of an elliptic fibrationπ:E→X over a Riemann surfaceX. LetE0 be the union of the regular fibers ofπ, and letΓ⊂PSL(2,R) be the fundamental group of X0=π(E0). Then the universal covering space ofX0may be identified with the Poincar´e upper half-planeᏴ, and we haveX0=Γ\Ᏼ, whereΓis regarded as a subgroup of SL(2,R) and the quotient is taken with respect to the action given by linear fractional transfor-mations. Givenz∈Ᏼ0, letΦbe a holomorphic 1-form onEz=π−1(z), and choose an ordered basis{α1(z),α2(z)}forH1(Ez,Z) which depends on the parameterzin a contin-uous manner. If we set
ω1(z)=
α1(z)Φ, ω2(z)=
thenω1/ω2is a many-valued function fromX0toᏴwhich can be lifted to a single-valued functionω:Ᏼ→Ᏼon the universal coverᏴofX0. Then it can be shown that there is a group homomorphismχ:Γ→SL(2,R), called the monodromy representation for the elliptic surfaceE, such that
ω(γz)=χ(γ)ω(z) (4.2)
for allγ∈Γandz∈Ᏼ. Thus the mapsχandωform an equivariant pair.
Let (χj,ωj) be an equivariant pair associated to an elliptic surfaceEof the type de-scribed above for eachj∈ {1,...,p}, and set
χ=1,χ1,...,χp, ω=1,ω1,...,ωp. (4.3)
Then, given a positive integerpand an element m=(m1,...,mp)∈Zqwithm1,...,mp> 0, the semidirect productΓχ(Z2)|m|pwith|m| =m1+···+mpassociated toχacts on
Ᏼ×C|m|pby
γ,1,...,p
·z,ζ1,...,ζp
=γz,ζ1,...,ζp
(4.4)
for allγ∈Γandz∈Ᏼ, where
j=μ1,j,ν1,j,...,μmj,j,νmj,j
∈Z2mj
,
ζj=
ζ1,j,...,ζmj,j
,ζj=ζ1,j,...,ζm j,j
∈Cmj (4.5)
for 1≤j≤pwith
ζr,j=
ζr,j+ωj(z)μr,j+νr,j cχjωj(z) +dχj
(4.6)
for eachr∈ {1,...,mj}if
χj(γ)=
aχj bχj
cχj dχj
∈SL(2,R). (4.7)
We denote byE|0m|pthe associated quotient space, that is,
E|m|0 p=Γ×
Z2|m|p\Ᏼ×C|m|p. (4.8)
Givenε∈Zp+1, we setξ=(2,m
1,...,mp)−2ε, and letF(z,X)∈ξ,η(Γ,ω,χ)ε. Then by Lemma 3.3, we see thatᏲω,χ(F(z,X)) is an element ofᏹ(2,m1,...,mp)(Γ,ω,χ), and it can be
shown that the associated holomorphic form
ωF(z)=Ᏺω,χ
F(z,X)dz∧dζ1∧ ··· ∧dζp (4.9)
on Ᏼ×C|m|p with z=(z,ζ
1,...,ζp)∈Ᏼ×C|m|p is invariant under the action ofΓ× (Z2)p. HenceωF(z) can be regarded as a holomorphic (|m|p+ 1)-form onE|m|p
therefore we obtain a canonical map
ξ,η(Γ,ω,χ)ε−→Ωp+1
E|0m|p
(4.10)
fromξ,η(Γ,ω,χ)εto the spaceΩp+1
E|m|0 p
of holomorphic (|m|p+ 1)-forms onE0|m|p. Example 4.2. LetΓbe a Fuchsian group of the first kind, and letK(X) be the function field of the smooth complex algebraic curveX=Γ\Ᏼ∪ {cusps}. Consider a second-order linear differential equation
d2 dx2+P(x)
d
dx+Q(x) f=0 (4.11)
forx∈XandP(x), Q(x) ∈K(X) with regular singular points, whose singular points are contained inΓ\{cusps} ⊂X. Let
Λ f =
d2 dz2+P(z)
d
dz+Q(z) f =0, (4.12)
forz∈Ᏼ, be the differential equation obtained by pulling back (4.11) via the natural projectionᏴ→Γ\Ᏼ⊂X. Letσ1andσ2be linearly independent solutions of (4.12), and letSm(Λ) be the linear ordinary differential operator of orderm+ 1 such that them+ 1 functions
σ1m,σ1m−1σ2,...,σ1σ2m−1,σ2m (4.13)
are linearly independent solutions of the corresponding linear homogeneous equation Sm(Λ)f =0. Letχ:Γ→SL(2,R) be the monodromy representation ofΓfor the second-order equationΛ f =0. Then the period mapω:Ᏼ→Ᏼdefined byω(z)=σ1(z)/σ2(z) for allz∈Ᏼis equivariant with respect toχ. Letψ:Ᏼ→Cbe a function correspond-ing to an element ofK(X) satisfying the parabolic residue condition in the sense of [11, Definition 3.20], and let fψbe a solution of the nonhomogeneous equationSm(Λ)f =ψ. Then the function
dm+1 dω(z)m+1
fψ(z)
σ2(z)m (4.14)
is a mixed automorphic form of type (0,m+ 2) associated toΓ,ω, andχ(cf. [11, page 32]).
Given a positive integerpand m=(m1,...,mp)∈Zpwithm1,...,mp>0, we consider a system of ordinary differential equations
SmjΛjfjz
j
=ψjzj, 1≤j≤p, (4.15)
of the type described above and for each j∈ {1,...,p}, choose a solution fψj
set
χ=χ1,...,χp
, ω=ω1,...,ωp
, Γ=Γ1∩ ··· ∩Γp. (4.16)
Then we see that the function f:Ᏼ→Cdefined by
f(z)=f1(z)···fp(z) (4.17)
for allz∈Ᏼis a mixed automorphic form belonging toᏹm(Γ,ω, χ).
5. Liftings of mixed automorphic forms
Letω=(ω1,...,ωn) andχ=(χ1,...,χn) be as inSection 3. Thusωi:Ᏼ→Ᏼis a holo-morphic map equivariant with respect to the homomorphismχi:Γ→SL(2,R) for each i∈ {1,...,n}, whereΓis a discrete subgroup of SL(2,R). In this section, we construct liftings of mixed automorphic forms associated toΓ,ω, andχof certain types to mixed Jacobi-like forms associated toΓ,ω, andχ.
We first consider discrete subgroupsΓ1,...,Γnof SL(2,R) satisfying
χi(Γ)⊂Γi (5.1)
for alli∈ {1,...,n}. Givenξ=(ξ1,...,ξn)∈Znandμ=(μ1,...,μn)∈Zn+, letM2μi+ξi(Γi)
denote the space of automorphic forms of one variable forΓiof weight 2μi+ξi. IfΔωi is the map in (3.14) associated toωi:Ᏼ→Ᏼin the case ofn=1, then we see that
ΔωiM
2μi+ξi(Γi)
=h◦ωi|h∈M2μi+ξi
Γi (5.2)
for 1≤i≤n. We denote the tensor product of these spaces by
ᏹ0
2μ+ξ(Γ,ω,χ)= n
i=1 ΔωiM
2μi+ξi
Γi, (5.3)
and consider an element of the form
h= p
k=1 Ck
n
i=1
hi,k◦ωi
∈ᏹ0
2μ+ξ(Γ,ω,χ) (5.4)
withCk∈Candhi,k∈M2μi+ξi(Γi) for 1≤i≤nand 1≤k≤p. Then we have
h(γz)= p
k=1 Ck
n
i=1
hi,k
ωi(γz)= p
k=1 Ck
n
i=1
hi,k
χi(γ)ωi(z)
= p k=1 Ck n
i=1
jχi(γ),ωi(z) 2μi+ξi
hi,k
χi(γ)ωi(z)
=
n
i=1
jχi(γ),ωi(z) 2μi+ξi
h(z)
for allz∈Ᏼandγ∈Γ; hencehis a mixed automorphic form belonging toᏹ2μ+ξ(Γ,ω,χ). Thus we see thatᏹ0
2μ+ξ(Γ,ω,χ) is a subspace ofᏹ2μ+ξ(Γ,ω,χ).
We now discuss a lifting of an element ofᏹ02μ+ξ(Γ,ω,χ) to a Jacobi-like form belonging toξ,η(Γ,ω,χ)ε withε=(ε1,...,εn)∈Zn+ andη=(η1,...,ηn)∈Zn. Given i∈ {1,...,n} andk∈ {1,...,p}, assuming thatμ≥ε, we set
hi,k,=
η−μi
i h (−μi)
i,k
−μi!+ξi+μi−εi! (5.6)
for≥μiand
hωk,α(z)=h1,k,α1
ω1(z),...,hn,k,αn
ωn(z) (5.7)
for all z∈Ᏼ and α=(α1,...,αn)≥μ. We define the formal power series Φh(z,X)∈
R[[X]] associated tohby
Φh(z,X)=
p
k=1
Ck
α≥μ hω
k,α(z) 1
Xα, (5.8)
where 1=(1,..., 1)∈Znso that hω
k,α(z) 1
=h1,k,α1
ω1(z)
···hn,k,αn
ωn(z)
(5.9)
forα=(α1,...,αn).
Theorem 5.1. The maph→Φhdetermines a lifting of an element ofᏹ02μ+ξ(Γ,ω,χ) to a Jacobi-like form belonging toξ,η(Γ,ω,χ)μ⊂ξ,η(Γ,ω,χ)εsuch that
Ᏺω,χΦh= h
(2μ+ξ−ε)! (5.10)
for allh∈ᏹ0
2μ+ξ(Γ,ω,χ), whereᏲω,χis the map sendingΦh(z,X) to the coefficient ofXμas in (3.12).
Proof. For 1≤i≤n, applyingProposition 2.5to the case ofn=1, we see that the formal power series
Φiz,Xi= ≥εi
φ(z)Xi (5.11)
inXiis a Jacobi-like form of one variable belonging toξi,ηi(Γi)εi if and only if there is a
sequence of modular forms{fr}r≥0with fr∈M2r+ξi(Γi) satisfying
φ=
−εi
j=0
ηijf(−j)j
for all≥εi. We now consider an elementh∈ᏹ0
ε(Γ,ω,χ) given by (5.4). Giveniandk, let{fr}r≥0be the sequence of functions onᏴdefined by
fr=
⎧ ⎨ ⎩
hi,k ifr=μi,
0 otherwise. (5.13)
Then clearly fr∈M2r+ξi(Γi) for eachr≥εi. IfΦi,k(z,Xi)=
≥εiφi,k,(z)X
i is the corre-sponding Jacobi-like form belonging toξi,ηi(Γi), then by (5.12) the coefficient function
φi,k,coincides withhi,k,in (5.6). Thus for eachk, we see that the product
Φω
k(z,X)=Φ1,k
ω1(z),X1
···Φn,k
ωn(z),Xn
=
α1≥μ1 ···
αn≥μn
h1,k,α1
ω1(z)
···hn,k,αn
ωn(z)
Xα1
1 ···Xnαn
=
α≥μ hω
k,α(z) 1
Xα
(5.14)
is a Jacobi-like form belonging toξ,η(Γ,ω,χ)μ⊂ξ,η(Γ,ω,χ)ε. From this and the fact that the formal power series in (5.8) can be written in the form
Φh(z,X)=
p
k=1 CkΦω
k(z,X), (5.15)
we see thatΦh(z,X) is a Jacobi-like form belonging toξ,η(Γ,ω,χ)μ. On the other hand, from (5.6) we have
hi,k,μi=
hi,k
2μi+ξi−εi
! (5.16)
for 1≤i≤nand 1≤k≤p, which implies that
Ᏺω,χΦωk(z,X)
= h1,kω1(z)
···hn,kωn(z)
2μ1+ξ1−ε1
!···2μn+ξn−εn!
=h1,k
ω1(z)
···hn,k
ωn(z)
(2μ+ξ−ε)! .
(5.17)
Combining this with (5.15), we obtain
Ᏺω,χ
Φh(z,X)=
p
k=1 Ck
h
1,k
ω1(z)
···hn,k
ωn(z)
(2μ+ξ−ε)! =
h(z)
(2μ+ξ−ε)!, (5.18)
where we identified the tensor product with the usual product inC; hence the proof of
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Min Ho Lee: Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, USA
