AUTOMORPHIC FORMS ON THE EXPANDED SYMMETRIC DOMAIN OF TYPE IV

HIROKI AOKI

Abstract. we introduce a lifting from a given Jacobi form of index 1 to an automorphic form on an expanded domain of type IV, introduced by Saito [20, 22]. The method is a generalization of Gritsenko [10, 11] for symmetric domain of type IV. We construct a lifting function satisfying a certain translation formula on the expanded domain.

§0. Introduction

§1. Expanded symmetric domain of type IV §2. Expanded Jacobi forms §3. Lifting function of Jacobi forms §4. Arithmetic lifting on the expanded domain

0. Introduction

It is classically known as the Jacobi’s inversion problem to find a description of the coordination of a Riemann surface in terms of integrals on the Riemann surface. In case of an elliptic curve, this was solved by the use of elliptic modular functions. In general for higher genus case, this was solved by the use of theta functions (cf. Siegel [23]). One may consider similar problem for the integrals on higher dimensional varieties. In this case, the main problem is to construct automorphic forms on the domain of periods. The case for family of polarized K3 surfaces was studied by Pyatetski-Shapiro [18], where the domain of periods is the classical symmetric domain of type IV. Even in this case, we do not have a full description of the ring of automorphic forms.

Saito [19] introduced another theory of period integrals of primitive forms for any universal unfolding of a singularity, where the period domain is no more the classical domain. On the other hand, the inversion problem for this case gets a precise structure, because of some particular generator system, called the flat generator system on the theory of primitive forms. Except for a few cases, called simple singularities and simply elliptic singularities (cf. Satake [24]), there is almost no answer known on the flat inversion problem.

The next nontrivial case is the 14 exceptional unimodular singularities, that is introduced by Arnold [2]. The unfolding theory of these singularities was studied by several authors (cf. Brieskorn [5], Dolgacev-Nikulin [6], Looijenga [13, 14] and Pinkham [17]), where there is 1-codimensional subspace of the deformation space which is identified with the deformation space of certain polarized K3 surfaces. Hence the period domain should be a 1-dimensional expansion of the symmetric domain of type IV. Looijenga [13] suggested the expansion in connection with the triangle singularity. In fact, both domains studied by Saito and Looijenga coincide for these cases. Hence it is a natural problem to find flat generator system on the ring of automorphic forms on the expanded domain, where the weights of the generators are prescribed. The present article is to contribute for the construction of nontrivial automorphic forms on the expanded domain.

1991 Mathematics Subject Classification. 11F55, 14J17.

The construction of nontrivial automorphic forms on the classical domain of type IV was studied by several authors (cf. Oda [15], Zagier [26], Sugano [25], Borcherds [4] and Gritsenko [10, 11]). Following the idea of Gritsenko [11], in present article, we construct automorphic forms on the expanded domain by lifting a Jacobi form of index 1.

In section 1, we define the expanded domain and the group action on it, according to Saito [19, 20].

In section 2, we define and survey a Jacobi form on the expanded domain.

In section 3, we construct a function f_{m} : ℂ × ℍ → ℂ (m ∈ ℕ), which has the translation formula

f_{m}(t, τ) = exp (−π\sqrt{−1}\frac{mct}{cτ + d}) f_{m} (\frac{t}{(cτ + d)^{2}} , \frac{aτ + b}{cτ + d})

for any (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ). By multiplying f_{m}, a Jacobi form of index m on the classical

domain is lifted to a Jacobi form of index m on the expanded domain.

In section 4, we construct a formal power series with an automorphic property, from a Jacobi form of index 1 on the expanded domain. In case of a lifted Jacobi form, the formal power series gives a holomorphic automorphic form on the expanded domain.

1. Expanded symmetric domain of type IV

Let L be an even integral lattice of rank l + 4, with an inner product defined by a quadratic form S of signature (2, l + 2), where l ∈ ℕ = {1, 2, 3, … }, and V := L⊗_{ℤ}ℝ be a base vector space of L. We suppose the case L = L_{1} ⊕ II_{1,1} and L_{1} = L_{0} ⊕ II_{1,1}, where II_{1,1} is a hyperbolic plane. We fix one coordinate, such that

S = (\begin{matrix}& & 1 \\ & S_{1} \\ 1 \end{matrix}) , S_{1} = (\begin{matrix}& & 1 \\ & −S_{0} \\ 1 \end{matrix}) ,

where S_{0} is an even integral positive definite matrix. We denote A[B] := ^{t}BAB and A{B} := tB̄AB for matrixes A, B. Let X^{+} be one of the connected component of X, where X is a topological space. If X is a topological group, we choose X^{+} with the identity element.

The classical symmetric domain of type IV is defined by

D_{V} := {V′ ⊂ V | dim_{ℝ}V′ = l + 2, V′ < 0},

where V′ < 0 means S|_{V′} < 0. We define a left action of g ∈ G := O^{+}(S, ℝ) on D_{V} by V′ ↦ gV′. We fix one orthogonal decomposition of V = V_{+} ⊕ V_{−} such that V_{+} > 0 and V_{−} < 0, and let K := SO(V_{+}) × SO(V_{−}) be subgroup of G. Then the group G acts on D_{V} transitively and the group K is the stabilizer of V_{−}. Hence D_{V} is isomorphic to G/K as a real manifold, and D_{V} is a homogeneous domain.

We define a complex domain

H := {z ∈ L⊗_{ℤ}ℂ | S{z} > 0, S[z] = 0},

and let P_{ℂ}H be the ℂ-projective space of H. We remark that H and P_{ℂ}H have two connected components. Then the domain D_{V} is isomorphic to (P_{ℂ}H)^{+} as a real manifold. The induced action of G on (P_{ℂ}H)^{+} is z ↦ gz (g ∈ G, z ∈ (P_{ℂ}H)^{+}). This action preserves the complex structure of (P_{ℂ}H)^{+}. We define a complex domain

H_{S} := {w = ^{t} (w_{1}, w_{2}, …, w_{l+2}) ∈ L_{1}⊗_{ℤ}ℂ | S_{1}[Im(w)] > 0}

which has two connected components. The domain H_{S}^{+} is isomorphic to (P_{ℂ}H)^{+} as a complex manifold. For simplicity, we choose the connected component of H_{S}^{+} by Im(w_{1}) > 0. The induced action of g = (g_{i,j})_{0≤i,j≤l+3} ∈ G on H_{S}^{+}, induced by the action on (P_{ℂ}H)^{+} is

w ↦ g(w) = (\frac{−\frac{1}{2}g_{i,0}S_{1}[w] + ∑_{j=1}^{l+2} g_{i,j}w_{j} + g_{i,l+3}}{j(g, w)} )_{1≤i≤l+2}

where the denominator

j(g, w) := −\frac{1}{2}g_{l+3,0}S_{1}[w] + ∑_{j=1}^{l+2} g_{l+3,j}w_{j} + g_{l+3,l+3}

becomes an automorphic factor. This action preserves the complex structure of H_{S}^{+}. For a holomorphic function F on H_{S}^{+}, k ∈ ℤ, and g ∈ G, we define F |_{k}g by (F |_{k}g)(w) := j(g, w)^{−k}F (g(w)). This gives a right action of G on the set of all holomorphic functions on H_{S}^{+}. Let Γ be a discrete subgroup of G. The action of Γ on H_{S}^{+} is properly discontinuous. A holomorphic function F on H_{S}^{+} is called an automorphic form of weight k on H_{S}^{+} with respect to the group Γ if F satisfies the equation F |_{k}Γ = F. The set of all automorphic forms of weight k on H_{S}^{+} with respect to Γ is a ℂ-vector space and written as 𝕄_{k}(H_{S}^{+}, Γ).

The expanded domain introduced by Saito [20] is defined by

B_{V} := {φ ∈ Hom_{ℝ}(V, ℂ) | Ker(φ) < 0},

which G acts by φ ↦ φ ∘ g^{−1}. However, this action is not transitive on B_{V}^{+}. To make the action transitive, we consider another action. We define a left action of GL_{2}^{+}(ℝ) on ℂ by \binom{Re(Az)}{Im(Az)} := (_{c}^{a} _{d}^{b})\binom{Re(z)}{Im(z)}, where A = (_{c}^{a} _{d}^{b}) ∈ GL_{2}^{+}(ℝ). Hence we can consider the left action of G × GL_{2}^{+}(ℝ) on B_{V}^{+} by φ ↦ A ∘ φ ∘ g^{−1}. This action is transitive. Now, we fix one ℝ-isomorphism ι : V_{+} → ℂ and define ι̃ ∈ B_{V}^{+} by ι̃|_{V_{+}} = ι and ι̃|_{V_{−}} = 0. Let ΔSO(V_{+}) := {(g, g̃) | g ∈ SO(V_{+}), g̃ = ι ∘ g ∘ ι^{−1}}, ΔSO(V_{−}) := {(g, E_{2}) | g ∈ SO(V_{−})}, and K̃ := ΔSO(V_{+}) × ΔSO(V_{−}) be subgroups of G × GL_{2}^{+}(ℝ). We can easily show that the group K̃ is the stabilizer of ι̃ ∈ B_{V}^{+}. Hence B_{V}^{+} is isomorphic to (G × GL_{2}^{+}(ℝ))/K̃ as a real manifold, and B_{V}^{+} is the homogeneous domain.

The domain, considered by Looijenga [13], is

B := {z ∈ L⊗_{ℤ}ℂ | S{z} − |S[z]| > 0},

and let P_{ℂ}B be the ℂ-projective space of B. We remark that B and P_{ℂ}B have two connected components. Saito [20] showed that B_{V} is isomorphic to B as a complex manifold. The induced action of G×GL_{2}^{+}(ℝ) on B^{+} is given by z ↦ gz and \binom{Re(z)}{Im(z)} ↦ (_{c}^{a} _{d}^{b})\binom{Re(z)}{Im(z)}. Of course, this action is transitive, but it does not preserve the complex structure of B^{+}. The subgroup whose action preserves the complex structure is G × ℂ^{×}, where we recognize ℂ^{×} ↪ GL_{2}^{+}(ℝ) by a + b\sqrt{−1} ↦ (_{−b}^{a} _{a}^{b}). By the restriction to this subgroup, Saito [20] induced the action of G on (P_{ℂ}B)^{+}. The induced action is [z] ↦ [gz], which preserves the complex structure of (P_{ℂ}B)^{+}. This action is the expansion of the action of G on (P_{ℂ}H)^{+}, hence not transitive. But we remark that the value inv(z) := \frac{|S[z]|}{S\{z\}} (0 ≤ inv(z) < 1) is invariant with respect to the action of G.

Now, we realize this expanded domain (P_{ℂ}B)^{+} as a subset of complex Euclidean space. Let

B_{S} := \{Z = (\begin{matrix}t \\ w \end{matrix}) ∈ ℂ ⊕ (L_{1}⊗_{ℤ}ℂ) | S_{1}[Im(w)] > \frac{|t| − Re(t)}{2}\},

which has two connected components. The correspondence

B_{S} ∋ (\begin{matrix}t \\ w \end{matrix}) ↔ [(\frac{t−S_{1}[w]}{2} ; w ; 1)] ∈ P_{ℂ}B

gives the isomorphism B_{S} ≅ P_{ℂ}B as a complex manifolds For simplicity, we choose the connected component of B_{S}^{+} by Im(w_{1}) > 0, where w = ^{t}(w_{1}, …, w_{l+2}). We remark

inv(Z) = \frac{|t|}{Re(t) + 2S_{1}[Im(w)]}. (1)

By an easy calculation, we can write the induced action of G on B_{S}^{+} as follows,

Z ↦ g(Z) = ( \frac{t}{J(g,Z)^{2}} ; ( \frac{\frac{1}{2} g_{i,0}(t−S_{1}[w])+∑_{j=1}^{l+2} g_{i,j}w_{j}+g_{i,l+3}}{J(g,Z)} )1≤i≤l+2) ,

where the denominator

J(g, Z) := \frac{1}{2}g_{l+3,0}(t − S_{1}[w]) + ∑_{j=1}^{l+2} g_{l+3,j}w_{j} + g_{l+3,l+3}

becomes an automorphic factor. For a holomorphic function F on B_{S}^{+}, k ∈ ℤ, and g ∈ G, we define F |_{k}g by (F |_{k}g)(Z) := J(g, Z)^{−k}F (g(Z)). This gives a right action of G on the set of all holomorphic functions on B_{S}^{+}. The action of Γ on B_{S}^{+} is properly discontinuous. A holomorphic function F on B_{S}^{+} is called an automorphic form of weight k on B_{S}^{+} with respect to the group Γ if F satisfies F |_{k}Γ = F. The set of all automorphic forms of weight k on B_{S}^{+} with respect to Γ is a ℂ-vector space and written as 𝕄_{k}(B_{S}^{+}, Γ). We remark t ∈ 𝕄_{−2}(B_{S}^{+}, Γ) for any Γ. The following diagram means the relation of the spaces in this article.

classical domain  expanded domain
D_{V} ⋯ B_{V}^{+}
∥  ∥
(P_{ℂ}H)^{+} ⋯ B^{+}
∥  ↓ P_{ℂ}
H_{S}^{+}  ⟵^{t=0}  B_{S}^{+}

Remark The classical symmetric domain with respect to the quadratic form S̃ := S ⊕ (−2)

HS̃ := \{(\begin{matrix}w \\ t \end{matrix}) | S_{1}[Im(w)] − 2(Im(t))^{2} > 0\}

is the double covering space of the expanded symmetric domain B_{S}^{+} by

π : H+ ; S̃ ∋ (\begin{matrix}w \\ t \end{matrix}) ↦ (\begin{matrix}2t^{2} \\ w \end{matrix}) ∈ B_{S}^{+}.

Let σ : \binom{w}{t} ↦ \binom{w}{−t} be an involutive isomorphism of H+ ; S̃. Then we have the induced isomorphism

ι : H+ ; S̃/⟨σ⟩ ≃ B_{S}^{+}.

We define a G-action on H+ ; S̃ by e : G = O^{+}(S, ℝ) ↪ O^{+}(S̃, ℝ). It follows that π and ι are equivariant with the group action of G. Hence for any finite index subgroup Γ̃ ⊂ GL(l+5, ℤ)∩ O^{+}(S̃, ℝ), we have an injective map

ι̃: 𝕄_{k}(H+ ; S̃, Γ̃)^{σ} ↪ 𝕄_{k}(B_{S}^{+}, Γ),

where Γ := e^{−1} (Γ̃ ∩ e (O^{+}(S, ℝ))). The image of ι̃ is the finite dimensional ℂ-vector space. But indeed, 𝕄_{k}(B_{S}^{+}, Γ) is not finite dimensional. In this article, we construct automorphic forms on B_{S}^{+}, not via H+ ; S̃. The relation between automorphic forms in this article and the image of ι̃ has not been clear.

2. Expanded Jacobi forms

From now on, we fix the discrete subgroup Γ := G ∩ O(S, ℤ). For simplicity, we write w = ^{t}(ω,^{t}ξ, τ), where ω = w_{1}, ξ = ^{t}(w_{2}, …, w_{l+1}) and τ = w_{l+2}. To investigate automorphic forms and Jacobi forms, we define the following matrixes as elements of G,

[A] := (\begin{matrix}A^{∗} \\ & E_{l} \\ & & A \end{matrix})

( A = (\begin{matrix}a & b \\ c & d \end{matrix}) , A^{∗} := \frac{1}{ad − bc} (\begin{matrix}a & −b \\ −c & d \end{matrix}) ) ,

[U] := (\begin{matrix}E_{2} \\ & U \\ & & E_{2} \end{matrix})

( U ∈ SO(S_{0}, ℝ) ) ,

[x, y] := (\begin{matrix}1 & 0 & ^{t}yS_{0} & 0 & \frac{1}{2}S_{0}[y] \\ 0 & 1 & ^{t}xS_{0} & \frac{1}{2}S_{0}[x] & ^{t}xS_{0}y \\ 0 & 0 & E_{l} & x & y \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

( x, y ∈ ℝ^{l} ) ,

[r] := (\begin{matrix}1 & 0 & 0 & −r & 0 \\ 0 & 1 & 0 & 0 & r \\ 0 & 0 & E_{l} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) .

( r ∈ ℝ ) ,

Using these symbols, we can write the parabolic groups and a Jacobi group as follows,

P_{ℝ}^{+} = ⟨[GL_{2}^{+}(ℝ)], [SO(S_{0}, ℝ)], [ℝ^{l}, ℝ^{l}], [ℝ]⟩,
P_{ℝ}^{J} = ⟨[SL_{2}(ℝ)], [SO(S_{0}, ℝ)], [ℝ^{l}, ℝ^{l}], [ℝ]⟩,
P_{ℤ}^{+} := P_{ℝ}^{+} ∩ Γ = ⟨[SL_{2}(ℤ)], [SO(S_{0}, ℤ)], [ℤ^{l}, ℤ^{l}], [ℤ]⟩.

For any k, m ∈ ℤ, we can define an action of P_{ℝ}^{J} on {φ : ℂ × ℂ^{l} × ℍ → ℂ} as follows,

(φ|_{k,m}[A])(t, ξ, τ) = (cτ + d)^{−k}e (\frac{−mc(t + S_{0}[ξ])}{2(cτ + d)}) φ (\frac{t}{(cτ + d)^{2}} , \frac{ξ}{cτ + d}, \frac{aτ + b}{cτ + d}) ,
(φ|_{k,m}[U])(t, ξ, τ) = φ(t, Uξ, τ),
(φ|_{k,m}[x, y])(t, ξ, τ) = e (m(^{t}xS_{0}ξ + \frac{1}{2}S_{0}[x]τ +^{t}xS_{0}y)) φ(t, ξ + xτ + y, τ),
(φ|_{k,m}[r])(t, ξ, τ) = e(mr)φ(t, ξ, τ),

where e(z) := exp(2π\sqrt{−1}z).

A holomorphic function φ is called an expanded Jacobi form of weight k and index m with respect to the group P_{ℤ}^{+} if φ satisfies the equation φ|_{k,m}P_{ℤ}^{+} = φ and if φ has the Fourier expansion

φ(t, ξ, τ) = ∑_{{n∈ℕ∪\{0\}, u∈L′_{0} \atop 2nm≥S_{0}[u]}} A_{(n,u)}(t)e(−^{t}uS_{0}ξ + nτ),

where L′_{0} is the dual lattice with respect to L_{0}. The set of all expanded Jacobi forms of weight k and index m with respect to the group P_{ℤ}^{+} is a ℂ-vector space and written as 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}). Of course, restricting this definition within t = 0, we have the definition of 𝕁_{k,m}(H_{S}^{+}, P_{ℤ}^{+}).

Lemma 2.1. Let

φ(t, ξ, τ) = ∑_{{n∈ℕ∪\{0\}, u∈L′_{0} \atop 2nm≥S_{0}[u]}} A_{(n,u)}(t)e(−^{t}uS_{0}ξ + nτ) ∈ 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}).

The function A_{(n,u)} only depends on the values u mod mL_{0} and 2nm − S_{0}[u].

Proof. It is obvious from the action of [x, 0], where x ∈ ℤ^{l}.

The first important matter is to show that any Fourier coefficient of an automorphic form on B_{S}^{+} with respect to ω is a Jacobi form.

Lemma 2.2. Let p ∈ ℕ and F be a holomorphic function on B_{S}^{+}. If F is [(pℤ)^{l}, (pℤ)^{l}]- invariant, we have a Fourier expansion

F (Z) = ∑_{m=0}^{∞} φ_{m}(t, ξ, τ)e(mω)

and φ_{0}(t, ξ, τ) does not depend on ξ.

Proof. This proof is similar to the one of the book of Eichler-Zagier [8, Theorem 1.2].

Corollary 2.3. If m < 0, then 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}) = 0.

Proof. It is obvious from Lemma 2.2.

Proposition 2.4. If F (Z) ∈ 𝕄_{k}(B_{S}^{+}, Γ), we have the Fourier expansion

F (Z) = ∑_{m=0}^{∞} φ_{m}(t, ξ, τ)e(mω)

= ∑_{λ=^{t}(n,^{t}u,m)∈Λ} ψ_{λ}(t)e(^{t}λS_{1}w),

where Λ := {λ = ^{t}(n,^{t}u, m) ∈ L′_{1} | n ≥ 0, m ≥ 0, S_{1}[λ] ≥ 0}. Hence φ_{m} ∈ 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}).

Proof. Let F (Z) = ∑_{λ∈L′_{1}} ψ_{λ}(t)e(^{t}λS_{1}w). By Lemma 2.2 and (ω ↔ τ)-invariance of F, it is enough to show that ψ_{λ}(t) = 0 if S_{1}[λ] < 0 and n, m > 0. We assume that there exists a triple λ′ = (α, β, γ) such that ψ_{λ′}(t) ≠ 0, α, γ > 0 and S_{1}[λ′] < 0. To show the contradiction, we put

P := (\begin{matrix}1 & 0 & 0 \\ −v & E_{l} & 0 \\ \frac{1}{2}S_{0}[v] & ^{t}vS_{0} & 1 \end{matrix}) and P̃ := (\begin{matrix}1 & 0 & 0 \\ 0 & P & 0 \\ 0 & 0 & 1 \end{matrix}) ∈ G,

where v := −\frac{β}{α}. The function

F̃(Z) := F |_{k}P̃^{−1}(Z) = F (\begin{matrix}t \\ P^{−1}w \end{matrix})

is P̃ΓP̃^{−1}-invariant. By a direct calculation of P̃^{−1}[x, y]P̃, we have [(α^{2}ℤ)^{l}, (α^{2}ℤ)^{l}] ⊂ P̃ΓP̃^{−1}. Hence F̃ is [(α^{2}ℤ)^{l}, (α^{2}ℤ)^{l}]-invariant. By a similar calculation, we know that F̃ has some integer period for any component of w. Therefore we have the Fourier expansion

F̃ (\begin{matrix}t \\ w \end{matrix}) = ∑_{λ∈L′_{1}} ψ_{λ}(t)e (^{t}λS_{1}P^{−1}w)

= ∑_{λ∈L′_{1}} ψ_{λ}(t)e (^{t}(P λ)S_{1}w)

= ∑_{λ∈P L′_{1}} ψ_{P^{−1}λ}(t)e (λS_{1}w) .

Therefore, by the assumption, ψ_{λ′}(t), the Fourier coefficient of the term e(P λ′S_{1}w) of F̃, is not zero. However, F̃ is {[x, y] | x, y ∈ (α^{2}ℤ)^{l}}-invariant. Hence from the Lemma 2.2 the Fourier coefficient of the term e(P λ′S_{1}w) is zero. This completes the proof.

Fact 2.5. Let φ ∈ 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}) and λ, μ ∈ ℚ^{l}. Then there exists some group Γ′, which is a finite index subgroup of SL_{2}(ℤ), such that

ψ(τ) := e (\frac{m}{2}S_{0}[λ]τ ) φ(0, λτ + μ, τ)

is a modular form of weight k with respect to Γ′. This Γ′ depends only on λ and μ. Hence have the following results.

(1) If m = 0 and k < 0, then φ|_{t=0} = 0 for any φ ∈ 𝕁_{k,0}(B_{S}^{+}, P_{ℤ}^{+}).

(2) If m = 0 and k = 0, then φ|_{t=0} is a constant function for any φ ∈ 𝕁_{0,0}(B_{S}^{+}, P_{ℤ}^{+}).

(3) If m > 0 and k ≤ 0, then φ|_{t=0} is a constant function for any φ ∈ 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}).

It is easily induced by Eichler-Zagier [8, Theorem 1.3].

Corollary 2.6. We have the following results.

(1) If k < 0 and F ∈ 𝕄_{k}(B_{S}^{+}, Γ), then F |_{t=0} = 0. Hence t^{−1}F ∈ 𝕄_{k+2}(B_{S}^{+}, Γ).

(2) If k = 0 and F ∈ 𝕄_{0}(B_{S}^{+}, Γ), then F |_{t=0} is a constant function. Hence there exists some c ∈ ℂ such that t^{−1}(F − c) ∈ 𝕄_{2}(B_{S}^{+}, Γ).

(3) If k < 0, then 𝕄_{k}(H_{S}^{+}, Γ) = 0. If k = 0, then 𝕄_{0}(H_{S}^{+}, Γ) = ℂ.

Proof. It is obvious from previous propositions.

Proposition 2.7. If l is even and k is odd, then 𝕄_{k}(B_{S}^{+}, Γ) = {0}, and 𝕄_{k}(H_{S}^{+}, Γ) = {0}.

Proof. If l is even, then −E_{l+4} ∈ Γ. Hence F = (−1)^{k}F for any F ∈ 𝕄_{k}(B_{S}^{+}, Γ) and F ∈ 𝕄_{k}(H_{S}^{+}, Γ).

Let

A(H_{S}^{+}, Γ) := ⨁_{k=−∞}^{∞} 𝕄_{k}(H_{S}^{+}, Γ) = ⨁_{k=0}^{∞} 𝕄_{k}(H_{S}^{+}, Γ)

and

A(B_{S}^{+}, Γ) := ⨁_{k=−∞}^{∞} 𝕄_{k}(B_{S}^{+}, Γ)

be graded rings.

Theorem 2.8. Automorphic forms of positive weight and t generate A(B_{S}^{+}, Γ) over ℂ. If l is even, automorphic forms of positive even weight and t generate A(B_{S}^{+}, Γ) over ℂ.

Proof. It is obvious from Corollary 2.6 and Proposition 2.7.

Theorem 2.9. There exists a subring R of A(H_{S}^{+}, Γ), such that

A(B_{S}^{+}, Γ)/tA(B_{S}^{+}, Γ) ≅ R.

Proof. It is obvious from the homomorphism p : A(B_{S}^{+}, Γ) ∋ F ↦ F |_{t=0} ∈ A(H_{S}^{+}, Γ).

3. Lifting function of Jacobi forms

In this section, we construct a function f_{m}, which gives the map

𝕁_{k,m}(H_{S}^{+}, P_{ℤ}^{+}) ⟶^{×f_{m}} 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}).

The following fact is called Cauchy’s estimate. Fact 3.1. Let f(z) be a holomorphic function on {|z − a| ≤ r} and |f(z)| ≤ M. Then we have

|f^{(n)}(a)| ≤ Mn!r^{−n}.

The proof is, for example, in the book of Ahlfors [1, p.122].

For any m ∈ ℤ, let

f_{m,j}(τ) := \{ \begin{matrix}\frac{2(mπ\sqrt{−1})^{j}}{j!(j−1)!} (\frac{d^{j}}{dτ^{j}} log η(τ)) & (j ∈ ℕ) \\ 1 & (j = 0) \end{matrix}

and

f_{m}(t, τ) := ∑_{j=0}^{∞} f_{m,j}(τ)t^{j}, (2)

where η(τ) is the Dedekind’s eta function

η(τ) := e (\frac{τ}{24}) ∏_{n=1}^{∞} (1 − e(nτ)) (τ ∈ ℍ).

We remark

f_{m,j+1}(τ) = \frac{mπ\sqrt{−1}}{j(j + 1)}f′_{m,j}(τ) (3)

for any j ∈ ℕ. Proposition 3.2. The function f_{m} is holomorphic on ℂ × ℍ. Proof. Let a ∈ ℍ, 0 < r < \frac{Im(a)}{2}, and M := max_{|τ−a|≤2r} |log η(τ)|. By Fact 3.1, we have

|\frac{d^{j}}{dτ^{j}} log η(τ)| ≤ Mj!r^{−j} ( |τ − a| ≤ r ).

Hence the series (2) converges absolutely and locally uniformly on ℂ × ℍ, by Hadamard’s formula for the radius of convergence. Proposition 3.3. We have the Fourier expansion

f_{m}(t, τ) = 1 − \frac{mπ^{2}}{6} t + ∑_{n=1}^{∞} (∑_{j=1}^{∞} \frac{(−2)^{j+1}m^{j}π^{2j}n^{j−1}σ_{1}(n)}{j!(j − 1)!}t^{j}) e(nτ), (4)

where σ_{k}(n) = ∑_{a|n} a^{k}.

Proof. It is obvious from the following q-expansion

\frac{d}{dτ} log η(τ) = \frac{π\sqrt{−1}}{12} − 2π\sqrt{−1} ∑_{n=1}^{∞} σ_{1}(n)q^{n}, (5)

where q := e(τ).

Lemma 3.4. We have the translation formula

f_{m,j}(τ) = ∑_{n=0}^{j} \frac{(−mcπ\sqrt{−1})^{j−n}}{(j − n)!(cτ + d)^{j+n}} f_{m,n} (\frac{aτ + b}{cτ + d}) (6)

for any (_{c}^{a} _{d}^{b}) ∈ SL_{2}(ℤ).

Proof. The case j = 0, the equation (6) is obvious. The case j = 1, we have the equation (6) from the equation (η′/η)(τ) = (cτ + d)^{−2}(η′/η)(\frac{aτ+b}{cτ+d}) − \frac{c}{2(cτ+d)}. The case j ≥ 2, we use the induction on j. We have the equation (6) by the assumption of the induction and the equation (3).

Theorem 3.5. Let φ ∈ 𝕁_{k,m}(H_{S}^{+}, P_{ℤ}^{+}) be a Jacobi form on H_{S}^{+} and φ̃(t, ξ, τ) := φ(ξ, τ)f_{m}(t, τ) be a holomorphic function on B_{S}^{+}. Then φ̃ ∈ 𝕁_{k,m}(B_{S}^{+}, P_{ℤ}^{+}) and φ̃|_{t=0} = φ.

Proof. By Proposition 3.2 and Proposition 3.3, it is enough to show the translation formula

f_{m}(t, τ) = e (\frac{−mct}{2(cτ + d)}) f_{m} (\frac{t}{(cτ + d)^{2}} , \frac{aτ + b}{cτ + d})

for any (_{c}^{a} _{d}^{b}) ∈ SL_{2}(ℤ). By a direct calculation, we have this translation formula from Lemma 3.4.

4. Arithmetic lifting on the expanded domain

In this section, we construct an automorphic form from an expanded Jacobi form of index 1. About the case of the classical domain of type IV, see Gritsenko [11].

Let P_{ℚ}^{+} := P_{ℝ}^{+} ∩ GL_{l+4}(ℚ). We define an action of the Hecke ring H (P_{ℤ}^{+}, P_{ℚ}^{+}) on the set of all P_{ℤ}^{+}-invariant functions on B_{S}^{+} by

F |_{k}P_{ℤ}^{+}gP_{ℤ}^{+} := ∑_{i} (F |_{k}g_{i})

( P_{ℤ}^{+}gP_{ℤ}^{+} = ∑_{i} P_{ℤ}^{+}g_{i} ∈ H (P_{ℤ}^{+}, P_{ℚ}^{+}) ) ,

and we define an embedding of H (SL_{2}(ℤ), M_{2}^{+}(ℤ)) into H (P_{ℤ}^{+}, P_{ℚ}^{+}) by

p : SL_{2}(ℤ) (\begin{matrix}α & 0 \\ 0 & β \end{matrix}) SL_{2}(ℤ) ↦ P_{ℤ}^{+} [(\begin{matrix}β^{−1} & 0 \\ 0 & α^{−1} \end{matrix})] P_{ℤ}^{+},

where M_{2}^{+}(ℤ) := {A ∈ M_{2}(ℤ) | det A > 0}. Let

T_{−}(m) := p ( ∑_{{α,β∈ℕ \atop αβ=m, α|β}} SL_{2}(ℤ) (\begin{matrix}α & 0 \\ 0 & β \end{matrix}) SL_{2}(ℤ))

for any m ∈ ℕ. Easily we have

T_{−}(m) = ∑_{{a,d∈ℕ \atop ad=m}} ∑_{b=0}^{d−1} P_{ℤ}^{+} [(\begin{matrix}d^{−1} & m^{−1}b \\ 0 & a^{−1} \end{matrix})] . (7)

(See Gritsenko [10].)

Proposition 4.1. If φ̃ ∈ 𝕁_{k,m̃}(B_{S}^{+}, P_{ℤ}^{+}), then φ̃|_{k}T_{−}(m) ∈ 𝕁_{k,m̃m}(B_{S}^{+}, P_{ℤ}^{+}), where

(φ̃|_{k}T_{−}(m)) (t, ξ, τ) := {(φ̃(t, ξ, τ)e(m̃ω)) |_{k}T_{−}(m)} e(−m̃mω).

Proof. Because T_{−}(m) sends the term e(m̃ω) to e(m̃mω), φ̃|_{k}T_{−}(m) satisfies the translation formulas of 𝕁_{k,m̃m}(B_{S}^{+}, P_{ℤ}^{+}). Let

φ̃(t, ξ, τ) = ∑_{{n∈ℕ∪\{0\}, u∈L′_{0} \atop 2nm̃≥S_{0}[u]}} A_{(n,u)}(t)e(−^{t}uS_{0}ξ + nτ).

Then we have the Fourier expansion

φ̃|_{k}T_{−}(m) = ∑_{{n∈ℕ∪\{0\}, u∈L′_{0} \atop 2nm̃m≥S_{0}[u]}} ∑_{{a∈ℕ, a|(m,n) \atop a^{−1}u∈L′_{0}}} a^{k−1}mA_{( \frac{mn}{a^{2}} , \frac{u}{a})}(a^{2}t)e(−^{t}uS_{0}ξ + nτ) (8)

by a direct calculation.

Now, let

φ̃(t, ξ, τ) = ∑_{{n∈ℕ∪\{0\}, u∈L′_{0} \atop 2n≥S_{0}[u]}} A_{(n,u)}(t)e(−^{t}uS_{0}ξ + nτ) ∈ 𝕁_{k,1}(B_{S}^{+}, P_{ℤ}^{+}),

and A_{(0,0)}(t) = ∑_{n=0}^{∞} b_{n}t^{n}. We consider the formal power series

F (Z) := (Ã_{(0,0)}(t) + ∑_{n=1}^{∞} ∑_{{a∈ℕ \atop a|n}} a^{k−1}A_{(0,0)}(a^{2}t)e(nτ)) + ∑_{m=1}^{∞} \frac{1}{m} (φ̃|_{k}T_{−}(m)) (t, ξ, τ)e(mω),

where

Ã_{(0,0)}(t) := ∑_{n=0}^{∞} b_{n} \frac{(k + 2n − 1)!ζ(k + 2n)}{(2π\sqrt{−1})^{k+2n}}t^{n}.

Let

E \binom{}{Ã_{(0,0)}} (t, τ) := \frac{1}{2} ∑_{(_{c}^{a} _{d}^{b})∈\binom{1 ℤ}{0 1}\SL_{2}(ℤ)}(cτ + d)^{−k}Ã_{(0,0)} (\frac{t}{(cτ + d)^{2}} ) .

If the series Ã_{(0,0)}(t) is a holomorphic function on some domain D and if k ≥ 4, E \binom{}{Ã_{(0,0)}} (t, τ) is a holomorphic function on D × ℍ.

Lemma 4.2. If k is odd, we have A_{(0,0)}(t) = 0, hence the first part of F (Z) is zero. If k is

even and k ≥ 4, Ã_{(0,0)}(t) + ∑_{n=1}^{∞} ∑_{{a∈ℕ \atop a|n}} a^{k−1}A_{(0,0)}(a^{2}t)e(nτ) = E \binom{}{Ã_{(0,0)}} (t, τ). Hence F (Z) is

P_{ℤ}^{+}-invariant, formally.

Proof. From the action of [−E_{2}], we have φ̃(t, ξ, τ) = (−1)^{k}φ̃(t, −ξ, τ). Hence if k is odd, we have φ̃(t, 0, τ) = 0 and A_{(0,0)}(t) = 0. If k is even,

E \binom{}{Ã_{(0,0)}} (t, τ) = Ã_{(0,0)}(t) + ∑_{c=1}^{∞} ∑_{{d∈ℤ \atop (d,c)=1}}(cτ + d)^{−k}Ã_{(0,0)} (\frac{t}{(cτ + d)^{2}} ) .

AUTOMORPHIC FORMS ON THE EXPANDED SYMMETRIC DOMAIN OF TYPE IV 11

We calculate the Fourier coefficient of it. We have

∫_{0}^{1} ∑_{c=1}^{∞} ∑_{{d∈ℤ \atop (d,c)=1}}(cτ + d)^{−k}Ã_{(0,0)} (\frac{t}{(cτ + d)^{2}} ) e(−mτ)dRe(τ)

= ∑_{c=1}^{∞} \frac{1}{c}\{ ∑_{{dmodc \atop (d,c)=1}} e (\frac{md}{c})\}\{∫_{−∞}^{∞} τ^{−k}Ã_{(0,0)} (\frac{t}{τ^{2}} ) e (−\frac{mτ}{c} ) dRe(τ)\}
= ∑_{c=1}^{∞} \frac{1}{c}\{ ∑_{{dmodc \atop (d,c)=1}} e (\frac{md}{c})\}\{∑_{n=0}^{∞} b_{n} \frac{(k + 2n − 1)!ζ(k + 2n)}{(2π\sqrt{−1})^{k+2n}}t^{n}\{∫_{−∞}^{∞} τ^{−(k+2n)}e (−\frac{mτ}{c} ) dRe(τ)\}\}
= ∑_{n=0}^{∞} b_{n}t^{n}ζ(k + 2n)\{∑_{c=1}^{∞} \frac{m^{k+2n−1}}{c^{k+2n}} ∑_{{dmodc \atop (d,c)=1}} e (\frac{md}{c})\}
= ∑_{n=0}^{∞} b_{n}t^{n}σ_{k+2n−1}(m)
= ∑_{{a∈ℤ \atop a|m}} a^{k−1}Ã_{(0,0)}(a^{2}t).

This completes the proof. Lemma 4.3. The formal power series F (Z) is invariant with respect to exchange variables ω and τ. Proof. By the equation (8), we have

F (Z) = Ã_{(0,0)}(t) + ∑_{{n,m∈ℕ∪\{0\} \atop (n,m)≠(0,0)}} ∑_{{u∈L′_{0} \atop 2nm≥S_{0}[u]}} ∑_{{a∈ℕ, a|(m,n) \atop a^{−1}u∈L′_{0}}} a^{k−1}A_{( \frac{mn}{a^{2}} , \frac{u}{a})}(a^{2}t)e(−^{t}uS_{0}ξ + nτ + mω). (9)

Let

F̃_{ℤ}^{+} := \{(\begin{matrix}1 & 0 & 0 \\ 0 & T & 0 \\ 0 & 0 & 1 \end{matrix}) ∈ Γ | (T − E_{l+2})L′_{1} ⊂ L_{1}\} .

Lemma 4.4. Formally, F (Z) is F̃_{ℤ}^{+}-invariant. Proof. By Lemma 2.1, the equation (9) is F̃_{ℤ}^{+}-invariant. Fact 4.5. The group, generated by the group P_{ℤ}^{+}, F̃_{ℤ}^{+} and the exchange variables ω and τ, includes the group

Γ̃ := {g ∈ Γ | (g − E_{l+4})L′ ⊂ L}.

It is easily induced by Sugano [25, Proposition 6.3]. Remark If the lattice L is unimodular i.e. L′ = L, then Γ̃ = Γ. Proposition 4.6. Formally, F (Z) is Γ̃-invariant. Proof. It is obvious from previous lemmas.

From now on, we suppose that φ̃ can be written as φf_{1} for some φ ∈ 𝕁_{k,1}(H_{S}^{+}, P_{ℤ}^{+}). From Proposition 3.3, we have b_{n} = 0 for any n ≥ 2. Hence we have

Ã_{(0,0)}(t) := \frac{(k − 1)!ζ(k)}{(2π\sqrt{−1})^{k}}b_{0} + \frac{(k + 1)!ζ(k + 2)}{(2π\sqrt{−1})^{k+2}}b_{1}t

and E \binom{}{Ã_{(0,0)}} (t, τ) is holomorphic on ℂ × ℍ.

Lemma 4.7. If k ≥ 4 or A_{(0,0)}(t) = 0, the series F (Z) is a holomorphic function on the domain

B̃_{S}^{+} := \{Z = (\begin{matrix}t \\ w \end{matrix}) ∈ B_{S}^{+} | S_{1}[Im(w)] > |t|\}.

Proof. It is enough to show that

∑_{m=1}^{∞} \frac{1}{m} (φ̃|_{k}T_{−}(m)) (t, ξ, τ)e(mω)

= ∑_{m=1}^{∞} \{\frac{1}{m} ∑_{{a,d∈ℕ \atop ad=m}} ∑_{b=0}^{d−1} a^{k}φ̃ (a^{2}t, aξ, \frac{aτ + b}{d})\} e(mω)

is holomorphic on B̃_{S}^{+}. To use Hadamard’s formula for the radius of convergence with respect to e(ω), we estimate

|φ̃ (a^{2}t, aξ, \frac{aτ + b}{d})| = |f_{1} (a^{2}t, \frac{aτ + b}{d})| |φ (aξ, \frac{aτ + b}{d})| .

First, we estimate |φ (aξ, \frac{aτ+b}{d})|. Let

C := sup \{|φ(ξ, τ)| | Im(ξ) > 0 , Im(τ) ≥ \frac{\sqrt{3}}{2}\} ,

where Im(ξ) > 0 means that any component of Im(ξ) is positive. If Im (\frac{aτ+b}{d}) ≥ \frac{\sqrt{3}}{2}, we can

choose x ∈ ℤ^{l} such that Im (aξ + x\frac{aτ+b}{d}) > 0. Then we have

|φ (aξ, \frac{aτ + b}{d})|

= |e (^{t}xS_{0}aξ + \frac{1}{2}S_{0}[x]\frac{aτ + b}{d})| |φ (aξ + x\frac{aτ + b}{d}, \frac{aτ + b}{d})|
= exp \{−π (\frac{aIm(τ)}{d}S_{0} [x + \frac{dIm(ξ)}{Im(τ)}] − \frac{mS_{0}[Im(ξ)]}{Im(τ)})\} |φ (aξ + x\frac{aτ + b}{d}, \frac{aτ + b}{d})|
≤C exp (\frac{πm}{Im(τ)}S_{0}[Im(ξ)]) .

If Im (\frac{aτ+b}{d}) < \frac{\sqrt{3}}{2}, we can choose \binom{α β}{γ δ} ∈ SL_{2}(ℤ) and p ∈ ℤ such that Im (\frac{α \frac{aτ+b}{d} +β}{γ \frac{aτ+b}{d} +δ} ) ≥ \frac{\sqrt{3}}{2}

and Im (\frac{aξ+p}{γ \frac{aτ+b}{d} +δ} ) > 0. We remark that Im (\frac{aτ+b}{d}) < \frac{\sqrt{3}}{2} induces γ ≠ 0. Then we have

|φ (aξ, \frac{aτ + b}{d})| = |γ \frac{aτ + b}{d} + δ|^{−k} |e (\frac{−γS_{0}[aξ + p]}{2(γ \frac{aτ+b}{d} + δ)})| |φ (\frac{aξ + p}{γ \frac{aτ+b}{d} + δ} , \frac{α\frac{aτ+b}{d} + β}{γ \frac{aτ+b}{d} + δ} )| .

Because

|γ \frac{aτ + b}{d} + δ| ≥ Im (\frac{aτ + b}{d}) ≥ \frac{Im(τ)}{m}

AUTOMORPHIC FORMS ON THE EXPANDED SYMMETRIC DOMAIN OF TYPE IV 13

and

|e (\frac{−γS_{0}[aξ + p]}{2(γ \frac{aτ+b}{d} + δ)})| = exp \{πmIm (\frac{S_{0}[ξ + \frac{p}{a}]}{τ + \frac{b}{a} + \frac{dδ}{aγ}} )\} ≤ exp (πm\frac{S_{0}[Im(ξ)]}{Im(τ)}) ,

we have

|φ (aξ, \frac{aτ + b}{d})| ≤ C(\frac{m}{Im(τ)})^{k} exp (\frac{πm}{Im(τ)}S_{0}[Im(ξ)]) .

Second, we estimate |f_{1} (a^{2}t, \frac{aτ+b}{d})|. Let

M(T) := max \{|\frac{η′(z)}{η(z)}| | Im(z) ≥ T \} .

If Im(τ) > T > 0, we have

|\frac{d^{j}}{dτ^{j}} log η(τ)| ≤ M(T)(j − 1)!(Im(τ) − T)^{−j}.

Hence we have

|f_{1}(t, τ)| ≤ 1 + ∑_{j=1}^{∞} \frac{2M(T)}{j!}(\frac{π|t|}{Im(τ) − T} )^{j} = 1 + 2M(T) (e^{\frac{π|t|}{Im(τ)−T}} − 1) .

By the equation (5), we have

M(T) ≤ \frac{π}{12} + 2π ∑_{n=1}^{∞} \frac{n(n + 1)}{2}(e^{−2πT} )^{n}

< 2π ∑_{n=0}^{∞} \frac{(n + 1)(n + 2)}{2}(e^{−2πT} )^{n}

= \frac{2π}{(1 − e^{−2πT})^{3}} .

Let T = \frac{aϵ}{d}Im(τ) for 0 < ϵ < 1. Because T ≥ \frac{ϵ}{m}Im(τ), we have M(T) ≤ M (\frac{ϵ}{m}Im(τ)). Hence we have

|f_{1} (a^{2}t, \frac{aτ + b}{d})| ≤ 1 + 2M (\frac{ϵ}{m}Im(τ)) (e^{\frac{mπ|t|}{(1−ϵ)Im(τ)}} − 1)

< 1 + \frac{4π}{(1 − e^{− \frac{2πϵIm(τ)}{m}})^{3}} (e^{\frac{mπ|t|}{(1−ϵ)Im(τ)}} − 1)

and

|\frac{1}{m} (φ̃|_{k}T_{−}(m)) (t, ξ, τ)| < Cm^{k+2}\{max (1, \frac{m}{Im(τ)})\}^{k} exp (\frac{πm}{Im(τ)}S_{0}[Im(ξ)])

× \{\frac{4π}{(1 − e^{− \frac{2πϵIm(τ)}{m}})^{3}} (e^{\frac{mπ|t|}{(1−ϵ)Im(τ)}} − 1)\} (10)

for any 0 < ϵ < 1. Therefore we have

lim sup_{m→∞} |\frac{1}{m} (φ̃|_{k}T_{−}(m)) (t, ξ, τ)|^{\frac{1}{m}} ≤ exp \{\frac{πS_{0}[Im(ξ)]}{Im(τ)} + \frac{π|t|}{(1 − ϵ)Im(τ)}\}

for any 0 < ϵ < 1, that is,

lim sup_{m→∞} |\frac{1}{m} (φ̃|_{k}T_{−}(m)) (t, ξ, τ)|^{\frac{1}{m}} ≤ exp \{\frac{π(S_{0}[Im(ξ)] + |t|)}{Im(τ)}\} .

Because the right hand of (10) is monotone decreasing with respect to Im(τ) and monotone increasing with respect to S_{0}[Im(ξ)] and |t|, the series F (Z) converges absolutely and locally uniformly on B̃_{S}^{+}.

Corollary 4.8. If k ≥ 4 or A_{(0,0)}(t) = 0, F (Z) is a holomorphic Γ̃-invariant function on B̃_{S}^{+}.

Proof. It is obvious from Proposition 4.6 and Lemma 4.7.

Lemma 4.9. The domain B̃_{S}^{+} contains the fundamental domain of B_{S}^{+} with respect to the group P_{ℤ}^{+}. Hence we have an open covering

B_{S}^{+} = ⋃_{γ∈Γ} γB̃_{S}^{+}.

Proof. By the equation (1), the condition of B̃_{S}^{+} is

(\frac{1}{inv(Z)} − 2) |t| > Re(t) ⇔ \frac{1}{inv(Z)} − 2 > cos(arg t).

Let A = (_{c}^{a} _{d}^{b}) ∈ SL_{2}(ℤ), then [A] maps t to \frac{t}{(cτ+d)^{2}} . Let Z ∈ B_{S}^{+}. Because Im(τ) ≠ 0, we can choose A such that

\frac{1}{inv(Z)} − 2 > cos (arg \frac{t}{(cτ + d)^{2}} )

Hence [A]Z ∈ B̃_{S}^{+}. This completes the proof.

By Corollary 4.8 and Lemma 4.9, we can define F (Z) := J(g, Z̃)^{k}F (Z̃) for any Z ∈ B_{S}^{+}, where Z = gZ̃, g ∈ Γ̃ and Z̃ ∈ B̃_{S}^{+}. This definition does not depend on the choice of g and Z̃, because F (Z) is Γ̃-invariant on B̃_{S}^{+}.

Now, we have our main theorem.

Theorem 4.10. If k ≥ 4 or if A_{(0,0)}(t) = 0, F (Z) is a holomorphic Γ̃-invariant function on B_{S}^{+}. Moreover, if the lattice L is unimodular, then F (Z) ∈ 𝕄_{k}(B_{S}^{+}, Γ).

Proof. It is obvious from the definition of F (Z) on B_{S}^{+}.

Remark Now we have the following commutative diagram.

𝕁_{k,1}(H_{S}^{+}, P_{ℤ}^{+}) ∋ φ ⟶

Gritsenko [10, 11]

∗ ∈ 𝕄_{k}(H_{S}^{+}, Γ̃)
×f_{1}↓  ^{t=0}↑
𝕁_{k,1}(B_{S}^{+}, P_{ℤ}^{+}) ∋ φ̃ = φ × f_{1} ⟶_{Theorem 4.10} F (Z) ∈ 𝕄_{k}(B_{S}^{+}, Γ̃)

Acknowledgment I thank Professor Kyoji Saito and Doctor Yoshiki Hayashi for useful advice and discussions. I thank Professor Valeri A. Gritsenko for his instructive lectures at RIMS in the fall of 1996. I thank Professor Takashi Sugano, Professor Masaru Ueda and Professor Atsushi Murase for useful advice at the fifth summer school of number theory in

References

1. Ahlfors, L. V., Complex Analysis, Third Edition, McGraw-Hill, 1966, 1979.

2. Arnold, V. I., Classification of Unimodular Critical Points of Functions, Functional Anal. Appl., 7 (1973), 230–231.

3. Baily, W. L., Introductory Lectures on Automorphic Forms, Princeton-Univ. Press, Iwanami-Syoten, Tokyo, 1973.

4. Borcherds, R. E., Automorphic Forms on O_{s+2,2}(ℝ) and Infinite Products, Inv. Math., 120 (1995), 161–213.

5. Brieskorn, E., The Unfolding of Exceptional Singularities, Nova Acta Leopoldina, NF 52 Nr. 240 (1981), 65–93.

6. Dolgacev, I. V., Nikulin V. V., The Exceptional Singularities of V. I. Arnol’d, and K3 Surfaces (Russian), Seventh All-Union Topology Conf. (Minsk, 1977), Abstracts of Lectures and Communications, Inst. Mat. Akad. Nauk BSSR, Minsk, 1977.

7. Eguchi, T., Topological Field Theory to Tokuiten Riron (Japanese), Seminar note, 1993.

8. Eichler, M., Zagier D., The Theory of Jacobi Forms, Prog. in Math. 55, 1985.

9. Freitag, E., Siegelsche Modulfunktionen (German), GMW254, Springer Verlag, Berlin, 1983.

10. Gritsenko, V. A., Fourier-Jacobi Functions of n Variables, Zap. Nauk. Sem. LOMI, 168 (1988), 32-45; English transl. in J. Soviet Math., 53 (1991), 243–252.

11. , Modular Forms and Moduli Spaces of Abelian and K3 Surfaces, Algebra i Analyz, 6:6 (1994), 65–102; English transl. in St. Petersburg Math., 6:6 (1995), 1179–1208.

12. Kawai, T., n = 2 Heterotic String Threshold Correction, K3 Surface and Generalized Kac-Moody Superalgebra, hep-th 9512046, 1995.

13. Looijenga, E., The Smoothing Components of a Triangle Singularity. I, Proc. of Sympos. in Pure Math., 40-2 (1983), 173–183.

14. , The Smoothing Components of a Triangle Singularity. II, Math. Ann., 269 (1984), 357–387.

15. Oda, Takayuki, On Modular Forms Associated with Infinite Quadratic Forms of Signature (2, n+2), Math. Ann., 231 (1977), 97–144.

16. Oda, Tadao, K.Saito’s Period Mapping for Holomorphic Functions with Isolated Critical Points, Adv. Stud. in Pure Math., 10 (1987) (Alg. Geom. Sendai 1985), 591–648.

17. Pinkham, H. C., Singularités exceptionelles, la dualité étrange d’Arnold et les surfaces K3 (French), C. R. Acad. Sc. Paris, 284 (1977), 615–618.

18. Pyatetski-Shapiro, I. I., Automorphic Functions and the Geometry of Classical Domains, Gordon and Breach, New York, 1969.

19. Saito, K., Period Mapping Associated to a Primitive Form, Publ. RIMS, Kyoto Univ., 19 (1983), 1231– 1264.

20. , Futei Nijikeishiki I ni Taiousuru Syukisyazo B_{v} (Japanese), Seminar memo, 1991.

21. , Around the Theory of the Generalized Weight System: Relations with Singularity Theory, the Generalized Weyl Group and Its Invariant Theory, Etc. Amer. Math. Soc. Transl., 183-2 (1998), 101–143.

22. , Period Domain for Primitive Forms of Witt Index ≤ 2, In preparation.

23. Siegl, C. L., Topics in complex function theory, I, II, III, Wiley-Interscience, 1969, 1971, 1973.

24. Satake, I., Flat structure and the prepotential for the elliptic root system of type D_{4}^{(1,1)}, Topological Field Theory, Primitive Form and Related Topics, Birkhauser, 1998.

25. Sugano, T., Jacobi Forms and the Theta Lifting, Comment. Math. Univ. St. Paul., 44-1 (1995), 1–58.

26. Zagier, D., Sur la conjecture de Saito-Kurokawa (d’apres H. Maass) Seminaire Delange-Pisot-Poitou 1979-1980, Progr. Math., 5 (1980), Birkhäuser, 371–394.

Graduate School of Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

E-mail address: hiroki@kurims.kyoto-u.ac.jp