MIXED JACOBI-LIKE FORMS OF SEVERAL VARIABLES MIN HO LEE Received 4 November 2005; Accepted 26 March 2006

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.

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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

DOI10.1155/IJMMS/2006/31542

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,...,z_n) 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,...,u_n)∈Cⁿ, we have

∂^α=∂^α1¹...∂^α_nⁿ, u^α=u^α1¹...u^α_nⁿ, (2.2) and forβ=(β1,...,β_n)∈Zⁿ, we writeα≤βifα_i≤β_ifor 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

···

α_n βn

, (2.3)

where for 1≤i≤n, we have^α₀ⁱ=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,...,γ_nz_n=

a1z1+b1

c1z1+d1,...,anzn+bn

c_nz_n+d_n ^∈Ᏼⁿ. (2.6)

For suchγandz, we set J(γ,z)=

jγ1,z1

,...,jγ_n,z_n∈Cⁿ, jγ_i,z_i=c_iz_i+d_i (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)X^α (2.12)

withz=(z1,...,z_n)∈ᏴⁿandX^α=X1^α1...Xn^αnforα=(α1,...,α_n)∈Zⁿ+.

LetC^×=C− {0}be the set of nonzero complex numbers. Givenλ=(λ1,...,λn)∈ (C^×)ⁿ, we denote byλ=diag(λ1,...,λ_n) the associatedn×ndiagonal matrix, and set

C^×X=

Xλ|λ∈ C^×_n

=

λ1X1,...,λ_nX_n|λ1,...,λ_n∈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

γ·(z,Xλ) =

γz,XJ(γ, z)⁻²λ (2.14)

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,...,jγ_n,z_n⁻²λ_nX_n. (2.15) We now set

Kξ,η

γ,z,Xλ=J(γ,z)^ξexp _n

i=1

ciηijγi,zi−1

λiXi

(2.16) forz∈Ᏼⁿ,γas in (2.5), andλ∈(C^×)ⁿ. Then it can be shown that

K_ξ,η

γγ, (z,Xλ)=K_ξ,η

γ,γ·(z,Xλ)K_ξ,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,η

γ, (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 in᏶ξ,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)X^α∈R[X]. (2.20)

Then the following conditions are equivalent.

(i) The power seriesΦ(z,X) is a Jacobi-like form belonging to᏶ξ,η(Γ).

(ii) The coefficient functionsφα:Ᏼ→Csatisfy φ_α|2α+ξγ(z)=^α−ε

δ=0

1 δ!

c^δη^δ

J(γ,z)^δφ_α−δ(z) (2.21) for allz∈Ᏼⁿandα≥ε, whereγ∈Γis as in (2.5) withc=(c1,...,cn).

(iii) There exist modular formsfν∈ᏹ2ν+ξ(Γ) forν≥εsuch that φ_α(z)=^α−ε

β=0

η^β

β!(2α+ξ−β−ε)!∂^βf_α−β(z) (2.22) for allα≥ε.

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)X^α∈᏶ξ,η(Γ), then (2.21) implies that

φ_ε|2ε+ξγ=φ_ε (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=X^ε᏶ξ,η(Γ), (2.24) which is a subspace of᏶ξ,η(Γ) consisting of the elements of the form_α≥εφα(z)X^α.

Then we see that there is a linear map

F:᏶ξ,η(Γ)_ε−→ᏹ2ε+ξ(Γ) (2.25) sending an element of᏶ξ,η(Γ)εto its coefficient ofX^ε.

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 ω_k:Ᏼ→Ᏼ andχk:Γ→SL(2,R) be a holomorphic map and a group homomorphism, respectively, satisfying

ω_k(γζ)=χ_k(γ)ω_k(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,...,η_n)∈Zⁿ, we define the mapJ_ω,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) for allζ∈Ᏼandγ∈Γ. Denote byᏹξ(Γ,ω,χ) the space of mixed automorphic forms of typeξassociated toΓ,ω, andχ.

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, whereJ_ω,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 by᏶ξ,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)J_ω,χ(γ,ζ)^−μ (3.7) for allζ∈Ᏼandγ∈Γ.

Lemma 3.3. A formal power seriesF(ζ,X)=

α≥εfα(ζ)X^α∈Ᏺ[[X]] withε∈Zⁿ+is an element of᏶ξ,η(Γ,ω,χ) if and only if

fα|^ω_2α^,χ_+ξγ(ζ)=^α

−ε

δ=0

1 δ!

c^δ_χη^δ

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 χ_j(γ)∈SL(2,R) for 1≤j≤n. In particular, the initial coefficient f_ε(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)X^αis an element of᏶ξ,η(Γ,ω,χ) if and only if

α≥ε

fα(γζ)Jω,χ(γ,ζ)^−2α−ξX^α= n i=1

_∞

μi=0

1 μi!

c^μ_χ,ⁱiη^μ_iⁱX_i^μi jχi(γ),ωi(z)^μi

·

ν≥ε

fν(ζ)X^ν

=

μ≥0

ν≥ε

1 μ!

c^μη^μ

Jω,χ(γ,ζ)^μfν(ζ)X^μ+ν

(3.9)

for allζ∈Ᏼ. Thus by comparing the coefficients ofX^α, we obtain fα(γζ)Jω,χ(γ,ζ)^−2α−ξ=^α

−ε

δ=0

1 δ!

c^δη^δ

Jω,χ(γ,ζ)^δ fα−δ(ζ), (3.10)

and therefore the lemma follows.

For eachε∈Zⁿwithε≥0, we set

᏶ξ,η(Γ,ω,χ)_ε=X^ε᏶ξ,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 element_α≥εfα(ζ)X^αof᏶ξ,η(Γ,ω,χ) to its initial coefficient fε(ζ).

IfRis the set of holomorphic functions onᏴⁿas inSection 2, we define the maps Δ^ω:R−→Ᏺ, Δ^ω_X:R[X]−→Ᏺ[X] (3.13) associated to the mapω:Ᏼ→Ᏼⁿas in (3.2) by

Δ^ωh(ζ)=hω(ζ), Δ^ω_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) IfΔ^ω:R→ᏲandΔ^ω_X:R[[X]]→Ᏺ[[X]] are as in (3.14), then Δ^ωᏹξΓχ

⊂ᏹξ(Γ,ω,χ), Δ^ω_X᏶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 Δ^ωX

ᏹ2ε+ξΓχ

Δ^ω

᏶ξ,η(Γ,ω,χ)ε Ᏺω,χ

ᏹ2ε+ξ(Γ,ω,χ)

(3.17)

is commutative.

Proof. If f :Ᏼⁿ→Cis an element ofᏹξ(Γχ), then by (3.14) we have Δ^ωf(γζ)=fω(γζ)=fχ1(γ)ω1(ζ),...,χ_n(γ)ω_n(ζ)

=Jχ(γ),ω(ζ)^ξfω(ζ)=Jχ(γ),ω(ζ)^ξΔ^ωf(ζ)

(3.18)

for allζ∈Ᏼandγ∈Γ; henceΔ^ωf is an element ofᏹξ(Γχ,ω,χ). On the other hand, ifF is an element of᏶ξ,η(Γχ) by (3.5) and (3.14), we see that

Δ^ω_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

c_χ,kη_kX_k jχ_k(γ),ω_k(ζ)

Fω(ζ),X (3.19) for allζ∈Ᏼandγ∈Γ. ThusΔ^ω_XFis an element of᏶ξ,η(Γ,ω,χ), andΔ^ω_XF∈᏶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)X^α∈᏶ξ,η(Γχ)_ε. Then we have Δ^ω◦Ᏺ(Φ)(ζ)=

Δ^ωφε

(ζ)=φε

ω1(ζ),...,ωn(ζ) (3.20) forζ∈Ᏼ. On the other hand, we have

Δ^ω_XΦ(ζ,X)=Φω(ζ),X=Φω1(ζ),...,ω_n(ζ),X

=

α≥ε

φα

ω1(ζ),...,ωn(ζ)X^α. (3.21)

Thus we see that

Ᏺω,χ◦Δ^ω_X(Φ)(ζ)=φ_εω1(ζ),...,ω_n(ζ)=

Δ^ω◦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(E_z,Z) which depends on the parameterzin a contin- uous manner. If we set

ω1(z)=

α1(z)Φ, ω2(z)=

α2(z)Φ, (4.1)

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)∈Z^qwithm1,...,mp>

0, the semidirect productΓχ(Z²)^|m|pwith|m| =m1+···+mpassociated toχacts on Ᏼ×C^|m|pby

γ,1,...,p

·

z,ζ1,...,ζ_p=

γz,ζ1,...,ζ_p (4.4) for allγ∈Γandz∈Ᏼ, where

j=

μ1,j,ν1,j

,...,μmj,j,νmj,j

∈ Z²_mj

, ζ_j=

ζ1,j,...,ζmj,j

,ζ_j=ζ1,j,...,ζmj,j

∈C^mj (4.5)

for 1≤j≤pwith

ζ_r,j=ζ_r,j+ω_j(z)μ_r,j+νr,j

c_χjω_j(z) +d_χj

(4.6) for eachr∈ {1,...,mj}if

χj(γ)=

aχj bχj

c_χj d_χj

∈SL(2,R). (4.7)

We denote byE^|0^m|pthe associated quotient space, that is, E^|0^m|p=Γ×

Z²_|m|p\Ᏼ×C^|m|p. (4.8) Givenε∈Z^p+1, we setξ=(2,m1,...,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

ω_F(z)=Ᏺω,χ

F(z,X)dz∧dζ1∧ ··· ∧dζ_p (4.9) on Ᏼ×C^|m|p with z=(z,ζ1,...,ζ_p)∈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^|0^m|p, and

therefore we obtain a canonical map

᏶ξ,η(Γ,ω,χ)_ε−→Ω^p+1E^|0^m|p

(4.10)

from᏶ξ,η(Γ,ω,χ)εto the spaceΩ^p+1E^|0^m|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 letS^m(Λ) be the linear ordinary differential operator of orderm+ 1 such that them+ 1 functions

σ1^m,σ1^m−1σ2,...,σ1σ2^m−1,σ2^m (4.13) are linearly independent solutions of the corresponding linear homogeneous equation S^m(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 f^ψbe a solution of the nonhomogeneous equationS^m(Λ)f =ψ.

Then the function

d^m+1 dω(z)^m+1

f^ψ(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)∈Z^pwithm1,...,mp>0, we consider a system of ordinary differential equations

S^mjΛj

fj

zj

=ψj

zj

, 1≤j≤p, (4.15)

of the type described above and for each j∈ {1,...,p}, choose a solution f_j^ψj(z_j) for the jth equation. For 1≤j≤p, letχj:Γj→SL(2,R) andωj:Ᏼ→Ᏼbe the monodromy representation and the period map, respectively, associated to the operatorS^mj(Λj), and

set

χ=

χ1,...,χ_p, ω=

ω1,...,ω_p, 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,...,ω_n) andχ=(χ1,...,χ_n) be as inSection 3. Thusω_i: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 2μ_i+ξ_i. IfΔ^ωi is the map in (3.14) associated toωi:Ᏼ→Ᏼin the case ofn=1, then we see that

Δ^ωiM2μi+ξi(Γi)=

h◦ω_i|h∈M2μi+ξi

Γi

(5.2)

for 1≤i≤n. We denote the tensor product of these spaces by ᏹ2⁰μ+ξ(Γ,ω,χ)=ⁿ

i=1

Δ^ωiM2μi+ξi

Γi

, (5.3)

and consider an element of the form h=

p k=1

Ck

n i=1

hi,k◦ωi

∈ᏹ⁰2μ+ξ(Γ,ω,χ) (5.4)

withC_k∈Candh_i,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

C_k n

i=1

jχ_i(γ),ω_i(z)^2μi+ξih_i,k

χ_i(γ)ω_i(z)

= _n

i=1

jχ_i(γ),ω_i(z)^2μi+ξi

h(z)

(5.5)