MIXED JACOBI-LIKE FORMS OF SEVERAL VARIABLES

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 subgroup Γ of SL(2,R), and they are related to modular forms for Γ, 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 Poincaré upper half-plane and certain sequences 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 ∂−1 of the differentiation operator ∂ with respect to the given variable (see, e.g., [1]). In addition to their natural connections with number theory and pseudodifferential operators, 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 subgroup Γ⊂ SL(2,R), which are associated to a holomorphic map of the Poincaré upper half-plane that is equivariant with respect to a homomorphism of Γ into SL(2,R) (cf. [7, 9]). Mixed Jacobi-like


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 subgroup Γ of SL(2, R), and they are related to modular forms for Γ, 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 Poincaré upper half-plane and certain sequences 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 ∂ −1 of the differentiation operator ∂ with respect to the given variable (see, e.g., [1]).In addition to their natural connections with number theory and pseudodifferential operators, 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 subgroup Γ ⊂ SL(2,R), which are associated to a holomorphic map of the Poincaré upper half-plane that is equivariant with respect to a homomorphism of Γ into SL(2,R) (cf.[7,9]).Mixed Jacobi-like 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 equivariant maps of the Poincaré 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.

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 integer n.Let (z 1 ,...,z n ) be the standard coordinate system for C n , and denote the associated partial differentiation operators by We will often use the multi-index notation.Thus, given α and for β = (β 1 ,...,β n ) ∈ Z n , we write α ≤ β if α i ≤ β i for each i = 1,...,n.Furthermore, we also write c = (c,...,c) ∈ Z n if c ∈ Z, and denote by Z + the set of nonnegative integers.Given α ∈ Z n and β ∈ Z n + , we write β! = β 1 !...β n ! and where for 1 ≤ i ≤ n, we have αi 0 = 1 and Let Ᏼ ⊂ C be the Poincaré upper half-plane.Then the usual action of SL(2, R) on Ᏼ by linear fractional transformations induces an action of SL(2,R) n on the product then we have Min Ho Lee 3 For such γ and z, we set for 1 ≤ i ≤ n.We denote by J(γ,z) the diagonal matrix with diagonal entries j(γ i ,z i ) with Then the map (γ,z) → J(γ,z) satisfies the cocycle condition for all γ,γ ∈ SL(2,R) n and z ∈ Ᏼ n .Given an element η = η 1 ,...,η n ) ∈ Z n and a map f : Ᏼ n → C, we set for all γ ∈ Γ, where f | η γ is as in (2.10).Denote by ᏹ η (Γ) the space of all Hilbert modular forms of weight η for Γ.
Remark 2.2.The usual definition of Hilbert modular forms also includes the regularity condition at the cusps, which is satisfied automatically for n > 1 according to Koecher's principle (cf.[3,4]).
We denote by R the ring of holomorphic functions f (z 1 ,...,z n ) on Ᏼ n and by R[[X]] = R[[X 1 ,...,X n ]] the set of all formal power series in X 1 ,...,X n with coefficients in R. Thus, using the multi-index notation, an element of R[[X]] can be written in the form be the set of nonzero complex numbers.Given λ = (λ 1 ,...,λ n ) ∈ (C × ) n , we denote by λ = diag(λ 1 ,...,λ n ) the associated n × n diagonal matrix, and set where X = (X 1 ,...,X n ) is regarded as a row vector.Using (2.9), we see that SL(2,R) (2.15) We now set for z ∈ Ᏼ n , γ as in (2.5), and λ ∈ (C × ) n .Then it can be shown that Definition 2.3.Given ξ,η ∈ Z n , a Jacobi-like form for Γ of n variables of weight ξ, and index η is an element, for all γ ∈ Γ and z ∈ Ᏼ n .Denote by ξ,η (Γ) the space of all Jacobi-like forms of n variables for Γ of weight ξ and index η.
Proposition 2.5.Given ε ∈ Z n + , consider a formal power series Then the following conditions are equivalent.
Min Ho Lee 5 Proof.The proposition can be proved by slightly modifying the proofs of [8, Lemma 4.2 and Theorem 4 for all γ ∈ Γ; hence the initial coefficient φ ε (z) of the formal power series Φ(z,X) is a Hilbert modular form of weight 2ε + ξ for Γ.We set which is a subspace of ξ,η (Γ) consisting of the elements of the form α≥ε φ α (z)X α .Then we see that there is a linear map sending an element of ξ,η (Γ) ε to its coefficient of X ε .

Mixed Jacobi-like forms
In this section, we discuss Jacobi-like forms of several variables associated to holomorphic maps of the Poincaré upper half-plane Ᏼ that are equivariant with respect to a discrete subgroup of SL(2, R).Such Jacobi-like forms are related to mixed automorphic forms.Let Γ be a discrete subgroup of SL(2,R), and for each k ∈ {1, ...,n}, let ω k : Ᏼ → Ᏼ and χ k : Γ → SL(2,R) be a holomorphic map and a group homomorphism, respectively, satisfying for all ζ ∈ Ᏼ and γ ∈ Γ.By setting we obtain a holomorphic map ω : Ᏼ → C n and a homomorphism χ : for all ζ ∈ Ᏼ and γ ∈ Γ. Denote by ᏹ ξ (Γ,ω,χ) the space of mixed automorphic forms of type ξ associated to Γ, ω, and χ.Definition 3.2.Let Ᏺ be the set of holomorphic functions on Ᏼ, and let Ᏺ[[X]] be the space of formal power series in ] is a mixed Jacobi-like form of weight ξ and index η associated to Γ, ω, and χ if it satisfies for all ζ ∈ Ᏼ and γ ∈ Γ, where J ω,χ (γ,ζ) denotes the diagonal matrix and c χ,k is the (2,1)-entry of the matrix χ k (γ) ∈ SL(2,R).Denote by ξ,η (Γ,ω,χ) the space of mixed Jacobi-like forms of weight ξ and index η associated to Γ, ω, and χ.Given μ ∈ Z n and a function h : for all ζ ∈ Ᏼ and γ ∈ Γ.
Min Ho Lee 7 For each ε ∈ Z n with ε ≥ 0, we set Then by Lemma 3.3, we see that there is a linear map If R is the set of holomorphic functions on Ᏼ n as in Section 2, we define the maps associated to the map ω : Ᏼ → Ᏼ n as in (3.2) by where χ = (χ 1 ,...,χ n ) is as in (3.2).
(ii) If Ᏺ and Ᏺ ω,χ are the linear maps in (2.25) and (3.12), respectively, then the diagram ), then by (3.14) we have for all ζ ∈ Ᏼ and γ ∈ Γ; hence Δ ω f is an element of ᏹ ξ ( Γ χ ,ω,χ).On the other hand, if F is an element of ξ,η ( Γ χ ) by (3.5) and (3.14), we see that for ζ ∈ Ᏼ.On the other hand, we have Thus we see that which implies (ii); hence the proof of the theorem is complete.

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.Let E be an elliptic surface (cf.[5]).Thus E is a compact surface over C that is the total space of an elliptic fibration π : E → X over a Riemann surface X.Let E 0 be the union of the regular fibers of π, and let Γ ⊂ PSL(2,R) be the fundamental group of X 0 = π(E 0 ).Then the universal covering space of X 0 may be identified with the Poincaré upper half-plane Ᏼ, and we have X 0 = Γ\Ᏼ, where Γ is regarded as a subgroup of SL(2,R) and the quotient is taken with respect to the action given by linear fractional transformations.Given z ∈ Ᏼ 0 , let Φ be a holomorphic 1-form on E z = π −1 (z), and choose an ordered basis {α 1 (z),α 2 (z)} for H 1 (E z ,Z) which depends on the parameter z in a continuous manner.If we set then ω 1 /ω 2 is a many-valued function from X 0 to Ᏼ which can be lifted to a single-valued function ω : Ᏼ → Ᏼ on the universal cover Ᏼ of X 0 .Then it can be shown that there is a group homomorphism χ : Γ → SL(2,R), called the monodromy representation for the elliptic surface E, such that for all γ ∈ Γ and z ∈ Ᏼ.Thus the maps χ and ω form an equivariant pair.Let (χ j ,ω j ) be an equivariant pair associated to an elliptic surface E of the type described above for each j ∈ {1, ..., p}, and set Then, given a positive integer p and an element m = (m 1 ,...,m p ) ∈ Z q with m 1 ,...,m p > 0, the semidirect product Γ χ (Z for all γ ∈ Γ and z ∈ Ᏼ, where for 1 ≤ j ≤ p with ζ r, j = ζ r, j + ω j (z)μ r, j + ν r, j c χj ω j (z) + d χj (4.6) for each r ∈ {1, ...,m j } if We denote by E |m|p 0 the associated quotient space, that is, Given ε ∈ Z p+1 , we set ξ = (2,m 1 ,...,m p ) − 2ε, and let F(z,X) ∈ ξ,η (Γ,ω,χ) ε .Then by Lemma 3.3, we see that Ᏺ ω,χ (F(z,X)) is an element of ᏹ (2,m1,...,mp) (Γ,ω,χ), and it can be shown that the associated holomorphic form Example 4.2.Let Γ be a Fuchsian group of the first kind, and let K(X) be the function field of the smooth complex algebraic curve X = Γ\Ᏼ ∪ {cusps}.Consider a second-order linear differential equation for x ∈ X and P(x), Q(x) ∈ K(X) with regular singular points, whose singular points are contained in Γ\{cusps} ⊂ X.Let for z ∈ Ᏼ, be the differential equation obtained by pulling back (4.11) via the natural projection Ᏼ → Γ\Ᏼ ⊂ X.Let σ 1 and σ 2 be linearly independent solutions of (4.12), and let S m (Λ) be the linear ordinary differential operator of order m + 1 such that the m + 1 functions are linearly independent solutions of the corresponding linear homogeneous equation S m (Λ) f = 0. Let χ : Γ → SL(2,R) be the monodromy representation of Γ for the secondorder equation Λ f = 0. Then the period map ω : Ᏼ → Ᏼ defined by ω(z) = σ 1 (z)/σ 2 (z) for all z ∈ Ᏼ is equivariant with respect to χ.Let ψ : Ᏼ → C be a function corresponding to an element of K(X) satisfying the parabolic residue condition in the sense of [11,Definition 3.20], and let f ψ be a solution of the nonhomogeneous equation S m (Λ) f = ψ.
Then the function is a mixed automorphic form of type (0,m + 2) associated to Γ, ω, and χ (cf.[11, page 32]).Given a positive integer p and m = (m 1 ,...,m p ) ∈ Z p with m 1 ,...,m p > 0, we consider a system of ordinary differential equations S mj Λ j f j z j = ψ j z j , 1≤ j ≤ p, (4.15) of the type described above and for each j ∈ {1, ..., p}, choose a solution f ψj j (z j ) for the jth equation.For 1 ≤ j ≤ p, let χ j : Γ j → SL(2,R) and ω j : Ᏼ → Ᏼ be the monodromy representation and the period map, respectively, associated to the operator S mj (Λ j ), and set Then we see that the function f : Ᏼ → C defined by for all z ∈ Ᏼ is a mixed automorphic form belonging to ᏹ m (Γ, ω, χ).
Theorem 5.1.The map h → Φ h determines a lifting of an element of ᏹ 0 2μ+ξ (Γ,ω,χ) to a Jacobi-like form belonging to ξ,η (Γ,ω,χ) μ ⊂ ξ,η (Γ,ω,χ) ε such that for all h ∈ ᏹ 0 2μ+ξ (Γ,ω,χ), where Ᏺ ω,χ is the map sending Φ h (z,X) to the coefficient of X μ as in (3.12).Proof.For 1 ≤ i ≤ n, applying Proposition 2.5 to the case of n = 1, we see that the formal power series Φ i z,X i = ≥εi φ (z)X i (5.11) in X i is a Jacobi-like form of one variable belonging to ξi,ηi (Γ i ) εi if and only if there is a sequence of modular forms { f r } r≥0 with f r ∈ M 2r+ξi (Γ i ) satisfying Min Ho Lee 13 for all ≥ ε i .We now consider an element h ∈ ᏹ 0 ε (Γ,ω,χ) given by (5.4).Given i and k, let { f r } r≥0 be the sequence of functions on Ᏼ defined by otherwise. (5.13) Then clearly f r ∈ M 2r+ξi (Γ i ) for each r ≥ ε i .If Φ i,k (z,X i ) = ≥εi φ i,k, (z)X i is the corresponding Jacobi-like form belonging to ξi,ηi (Γ i ), then by (5.12) the coefficient function φ i,k, coincides with h i,k, in (5.6).Thus for each k, we see that the product is a Jacobi-like form belonging to ξ,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 C k Φ ω k (z,X), (5.15) we see that Φ h (z,X) is a Jacobi-like form belonging to ξ,η (Γ,ω,χ) μ .On the other hand, from (5.6) we have h i,k,μi = h i,k 2μ i + ξ i − ε i !(5.16) for 1 ≤ i ≤ n and 1 ≤ k ≤ p, which implies that (5.17) Combining this with (5.15), we obtain where we identified the tensor product with the usual product in C; hence the proof of the theorem is complete.