Orthogonal Eisenstein series at harmonic points and modular forms of singular weight

Abstract

We investigate the behaviour of orthogonal non-holomorphic Eisenstein series at their harmonic points by using theta lifts. In the case of singular weight, we show that the orthogonal non-holomorphic Eisenstein series that can be written as a theta lift have a simple pole at $s = 1$ whose residues yield holomorphic orthogonal modular forms that are Eisenstein series on the boundary. Moreover, we will investigate the image of this construction and give sufficient conditions for the surjectivity.

1 Introduction

Let L be an even lattice in a rational quadratic space V of signature (2, l). There is an index 2 subgroup of the corresponding orthogonal group $O^+(2, l) \subseteq O(2, l)$ acting on the orthogonal upper half-plane ${\mathbb {H}}_l$. Similar to the case of elliptic modular forms, it is possible to define orthogonal modular forms, which can be seen to be global sections of a hermitian line bundle. For $l = 1$ we obtain the classical case of elliptic modular forms and for $l = 2$ we obtain Hilbert modular forms for real quadratic number fields.

An important problem is the construction of such modular forms, in particular for low weight. It turns out that there is a minimal weight for non-zero holomorphic modular forms, which is given by $\kappa = \frac{l}{2} - 1$ for $l > 2$, see [7]. The weight $\kappa = \frac{l}{2} - 1$ is called the singular weight, and modular forms of singular weight have many vanishing Fourier coefficients. In particular, there are no cusp forms of singular weight (for an analogous theory in the Siegel case see [13], where it is shown that every holomorphic modular form of singular weight is a linear combination of theta functions). In contrast to the symplectic case, there are no holomorphic theta series for the orthogonal group (except for some low dimensional examples where we have an exceptional isomorphism from the orthogonal group to the symplectic group). Moreover, the usual constructions of holomorphic modular forms do not work in low weight. For example Eisenstein series of low weight do not converge. On the other hand, using the celebrated multiplicative Borcherds lift of [5, Theorem 13.3], examples of holomorphic modular forms of singular weight can be constructed. For results in this direction see [9, 17, 19]. Another method is to use the additive Borcherds lift of [5, Theorem 14.3] using holomorphic modular forms of weight 0, i.e. invariant vectors, as input functions.

Our aim is to investigate Eisenstein series ${\mathcal {E}}_{\kappa , \lambda }(Z)$ of low weight, in particular of singular weight $\kappa = \frac{l}{2} - 1$ by considering the non-holomorphic Eisenstein series ${\mathcal {E}}_{\kappa , \lambda }(Z, s)$, get a meromorphic continuation to all $s \in {\mathbb {C}}$ and hope that they yield holomorphic modular forms for special values of s. This will be done using the results of [14].

We will give more details now. As above, let L be an even lattice of signature (2, l) and $L'$ the corresponding dual lattice. Throughout we will assume that L has Witt rank 2 and that $l > 2$. For a primitive isotropic vector $z \in L$ and $z' \in L'$ with $(z, z') = 1$ consider $K = L \cap z^\perp \cap z'^\perp$. Let ${\mathbb {H}}_l = K \otimes {\mathbb {R}}+ i C$, where C is a fixed connected component of

$$\begin{aligned} \{Y \in K \otimes {\mathbb {R}}\mid q(Y) > 0 \}. \end{aligned}$$

Then ${\mathbb {H}}_l$ is a hermitian symmetric domain and there is an index 2 subgroup $O^+(L \otimes {\mathbb {R}}) \subseteq O(L \otimes {\mathbb {R}})$ acting on ${\mathbb {H}}_l$. Let $\Gamma (L)$ be the kernel of the natural map $O^+(L) \rightarrow O(L' / L)$. By the theory of Baily–Borel (see [1, 2]) we can add 0-dimensional boundary components (points) and 1-dimensional boundary components (isomorphic to ${\mathbb {H}}$) to obtain ${\mathbb {H}}_l^*$, such that the quotient $\Gamma (L) \backslash {\mathbb {H}}_l^*$ is a compact complex analytic space, which can be shown to be projective and hence, by Chow’s theorem, algebraic. For an holomorphic orthogonal modular form $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ of weight $\kappa \in {\mathbb {Z}}$ with respect to $\Gamma (L)$ one can now define its restriction $F \vert _I : {\mathbb {H}}\rightarrow {\mathbb {C}}$ to a 1-dimensional boundary component I. One easily sees that $F \vert _I$ is a holomorphic modular form of weight $\kappa$ for an appropriate arithmetic subgroup of ${\text {SL}}_2({\mathbb {Q}})$. In particular, it makes sense to talk about the space ${\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$ of holomorphic orthogonal modular forms that are linear combinations of Eisenstein series on the boundary. If $F \vert _I$ vanishes for every boundary component, we say that F is a cusp form and we write ${\text {S}}_\kappa (\Gamma (L))$ for the space of cusp forms.

For an easier exposition, we assume throughout the introduction that L splits two hyperbolic planes over ${\mathbb {Z}}$. Then the 0-dimensional cusps of $\Gamma (L) \backslash {\mathbb {H}}_l$ are in bijective correspondence to the isotropic elements in $L' / L$ (which we denote by ${\text {Iso}}(L' / L)$) up to $\pm 1$. For an appropriate automorphic form $f : {\mathbb {H}}\rightarrow {\mathbb {C}}[L' / L]$ of weight k consider the additive Borcherds lift defined by [5]

$$\begin{aligned} \Phi (Z, f) := \int _{{\text {SL}}_2({\mathbb {Z}}) \backslash {\mathbb {H}}}^{\text {reg}} \langle f(\tau ), \Theta _L(\tau , Z) \rangle v^k \frac{{\mathrm {d}}u {\mathrm {d}}v}{v^2}, \end{aligned}$$

(1)

where $\langle \cdot , \cdot \rangle$ denotes the inner product ${\mathbb {C}}[L' / L]$ which is anti-linear in the second component and $\Theta _L$ is a certain Siegel theta function of weight k in $\tau$ (see Sect. 6). In [14] we have seen that the additive Borcherds lift $\Phi _{k, \beta }(Z, s)$ of a vector-valued non-holomorphic Eisenstein series $E_{k, \beta }(\tau , s), \beta \in {\text {Iso}}(L' / L)$ of weight k for the Weil representation $\rho _L$ defined by [4] (in [4] the dual Weil representation $\rho _L^*$ is considered) is a linear combination of orthogonal non-holomorphic Eisenstein series ${\mathcal {G}}_{\kappa , \delta }(Z, s)$ of weight $\kappa = \frac{l}{2} - 1 + k$ with respect to $\Gamma (L)$ corresponding to a 0-dimensional cusp $\delta \in {\text {Iso}}(L' / L)$. To obtain holomorphic Eisenstein series one would try to evaluate this at the harmonic point $s = 0$. By calculating the Fourier expansion, one can split up $\Phi _{k,\beta }(Z) := \Phi _{k, \beta }(Z, 0)$ into a holomorphic part $\Phi _{k,\beta }^+(Z)$ and a non-holomorphic part $\Phi _{k,\beta }^-(Z)$. We will show

Proposition 1.1

(See Proposition 8.6) The holomorphic part $\Phi _{k,\beta }^+(Z)$ is given by

$$\begin{aligned} \Phi _{k,\beta }^{+}(Z)&= \frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \delta _{\beta , \frac{bz}{N_z}} \zeta ^b(\kappa ) \\&\quad + \sum _{\begin{array}{c} \lambda \in K \\ q(\lambda ) = 0 \\ (\lambda , Y)> 0 \\ \lambda \text { primitive} \end{array}} \sum _{\begin{array}{c} b\in {\mathbb {Z}}/ N_z {\mathbb {Z}}\\ c \in {\mathbb {Z}}/ N_\lambda {\mathbb {Z}} \end{array}} \delta _{\beta , \frac{c \lambda }{N_\lambda } - \frac{c (\lambda , \zeta )}{N_\lambda N_z} z + \frac{bz}{N}} \sum _{m = 1}^\infty {\tilde{\sigma }}_{\kappa - 1}^{c, b}(m) e\left( \frac{m (\lambda , Z - \frac{\zeta _K}{N_z})}{N_\lambda }\right) \\&\quad + \sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda )> 0 \\ (\lambda , Y) > 0 \end{array}} b(\lambda ) e\left( \frac{(\lambda , Z - \frac{\zeta _K}{N_z})}{N_\lambda }\right) \end{aligned}$$

where $e(x) = e^{2 \pi i x},$ $N_z$ and $N_\lambda$ are the levels of z and $\lambda ,$

$$\begin{aligned} {\tilde{\sigma }}_{\kappa - 1}^{c, b}(m) = \sum _{\begin{array}{c} n \mid m \\ \frac{m}{n} \equiv c \bmod {N_\lambda } \end{array}} {\text {sgn}}(n) n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \end{aligned}$$

is a divisor sum over positive and negative divisors,  $\zeta ^b(\kappa )$ is a modified Riemann zeta function (see Sect. 7) and $b(\lambda )$ are certain Fourier coefficients.

Since only the terms with $q(\lambda ) = 0$ in the Fourier expansion contribute to the restrictions to boundary components, one easily sees that the holomorphic part restricted to a boundary component is given by an Eisenstein series, see [10, Theorem 4.2.3] for their Fourier expansions. By inspecting the non-holomorphic part and using that vector-valued non-holomorphic Eisenstein series converge at $s = 0$ for $k > 2$ we obtain

Theorem 1.2

(See Theorem 8.9) The non-holomorphic part $\Phi _{k,\beta }^-(Z)$ vanishes for $k > 2$ and hence $\Phi _{k,\beta }(Z) = \Phi _{k,\beta }^+(Z)$ is a holomorphic orthogonal modular form that is an Eisenstein series on the boundary. Moreover,  every holomorphic orthogonal modular form that is a linear combination of Eisenstein series on the boundary is obtained as a theta lift (up to cusp forms).

For $k = 0$ and hence singular weight $\kappa = \frac{l}{2} - 1$ this does not work immediately, since the non-holomorphic part does not vanish in general. Using the functional equation relating the values at s and $1 - s$ we will instead inspect the behaviour at $s = 1$. We obtain

Theorem 1.3

(See Theorem 8.10) The additive Borcherds lift $\Phi _{0,\beta }(Z, s)$ has a simple pole at $s = 1$ with residue given by

$$\begin{aligned}&\frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa }\sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\text {res}}_{s = 1} c_{0,\beta }\left( \frac{bz}{N_z}, 0, s\right) \zeta ^b(\kappa ) \\&\quad + \sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda ) = 0 \\ (\lambda , Y) > 0 \end{array}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\quad \times {\text {res}}_{s = 1} c_{0,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, 0, s\right) e(\lambda , Z). \end{aligned}$$

where $c_{0,\beta }(0, 0, s)$ is a Fourier coefficients of the vector-valued non-holomorphic Eisenstein series $E_{0, \beta }(\tau , s)$. In particular,  it is a holomorphic orthogonal modular form of singular weight that is an Eisenstein series on the boundary.

In fact this is just the Borcherds lift of the invariant vector ${\text {res}}_{s = 1} E_{0, \beta }(\tau , s)$ so that the theorem reads ${\text {res}}_{s = 1} \Phi _{0, \beta }(Z, s) = \Phi (Z, {\text {res}}_{s = 1} E_{0, \beta }(\cdot , s))$.

As in the higher weight case, it is natural to ask whether all holomorphic orthogonal modular forms of singular weight that are linear combinations of Eisenstein series on the boundary can be obtained in this way. Therefore we consider the adjoint theta lift $\Phi ^*$ mapping holomorphic orthogonal modular forms of singular weight to modular forms of weight 0. For an orthogonal modular form $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ it is given by

$$\begin{aligned} \Phi ^*(\tau , F) := \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l}. \end{aligned}$$

(2)

We will show that if $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ is holomorphic of singular weight, then $\Phi ^*(\tau , F)$ is a harmonic function of weight 0 (see Lemma 9.2). In Sect. 9 we prove the following main theorem.

Theorem 1.4

(See Theorem 9.3) For an orthogonal modular form of singular weight $\kappa = \frac{l}{2} - 1 > 0$ the theta lift $\Phi ^*(\tau , F)$ is an invariant vector for the Weil representation given by

$$\begin{aligned} \frac{\Gamma (l/2)}{2(2 \pi )^{l / 2}} \sum _{\gamma \in {\text {Iso}}(L' / L)} \sum _{\begin{array}{c} \delta \in {\text {Iso}}(L' / L) \\ \gamma = k_\delta \delta \end{array}} \zeta _+^{k_\delta }(l - \kappa ) a_{F, \delta }(0) C(\delta ) {\mathfrak {e}}_\gamma , \end{aligned}$$

where $a_{F, \delta }(0)$ is the value of F in the 0-dimensional cusp corresponding to $\delta$ and $C(\delta )$ is a non-zero constant. Again,  $\zeta _+^{k_\delta }$ is a modified Riemann zeta function (see Sect. 7).

As a corollary one obtains

Corollary 1.5

(See Corollary 9.6) Assume that L splits two hyperbolic planes. The theta lift $\Phi ^*(\tau , F)$ vanishes if and only if F vanishes in every 0-dimensional cusp.

Since the theta lifts (1) and (2) are adjoint to each other we obtain

Corollary 1.6

(See Corollary 9.8) Assume that L splits two hyperbolic planes. Then every holomorphic orthogonal modular form of singular weight for $\Gamma (L)$ that is an Eisenstein series on the boundary is the residue at $s = 1$ of some non-holomorphic Eisenstein series and the additive Borcherds lift is an isomorphism

$$\begin{aligned} {\text {Inv}}({\mathbb {C}}[L' / L]) \rightarrow {\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L)). \end{aligned}$$

If L is a maximal lattice, then ${\text {Inv}}({\mathbb {C}}[L' / L])$ is either 1-dimensional (if L is unimodular) or 0-dimensional (if L is not unimodular). Since maximal lattices of Witt rank 2 always split two hyperbolic planes over ${\mathbb {Z}}$, we obtain for singular weight $\kappa = \frac{l}{2} - 1$.

Corollary 1.7

(See Corollary 9.9) If L is a maximal lattice of Witt rank 2,  then the space ${\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$ is either 1-dimensional (if L is unimodular) or 0-dimensional (if L is not unimodular). Moreover,  if $\kappa = 2, 4, 6, 8, 10, 14,$ i.e. $l = 6, 10, 14, 18, 22, 30,$ then we have ${\text {M}}_\kappa (\Gamma (L)) = {\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$ and the space of holomorphic modular forms of singular weight is either 0 or 1-dimensional depending on L being unimodular or not.

If L does not split two hyperbolic planes over ${\mathbb {Z}}$, then we can still fully determine the image of the theta lift (see Corollary 9.7).

2 Vector-valued non-holomorphic Eisenstein series

We will now introduce the Weil representation and vector-valued modular forms. Therefore, let ${\mathbb {H}}:= \{\tau = u + iv \in {\mathbb {C}}\mid v > 0 \}$ be the usual upper half-plane. For $z \in {\mathbb {C}}$ we write $e(z) := e^{2 \pi i z}$ and we denote by $\sqrt{z} = z^{\frac{1}{2}}$ the principal branch of the square-root, i.e. $\arg (\sqrt{z}) \in (- \frac{\pi }{2}, \frac{\pi }{2}]$.

Definition 2.1

We denote by ${\text {Mp}}_2({\mathbb {R}})$ the metaplectic cover of ${\text {SL}}_2({\mathbb {R}})$. It is realized as pairs $(M, \phi )$, where $M \in {\text {SL}}_2({\mathbb {R}})$ and $\phi : {\mathbb {H}}\rightarrow {\mathbb {C}}$ is a holomorphic square root of $\tau \mapsto c \tau + d$. The product for $(M_1, \phi _1), (M_2, \phi _2) \in {\text {Mp}}_2({\mathbb {R}})$ is given by

$$\begin{aligned} (M_1, \phi _1(\tau ))(M_2, \phi _2(\tau )) := (M_1 M_2, \phi _1(M_2 \tau ) \phi _2(\tau )), \end{aligned}$$

where $\left( {\begin{matrix} a &{} b \\ c &{} d\end{matrix}}\right) \tau = \frac{a \tau + d}{c \tau + d}$ is the usual action of ${\text {SL}}_2({\mathbb {R}})$.

By ${\text {Mp}}_2({\mathbb {Z}})$ we denote the inverse image of ${\text {SL}}_2({\mathbb {Z}})$ under the covering map. It is generated by

$$\begin{aligned} T = \left( \begin{pmatrix}1 &{} 1 \\ 0 &{} 1\end{pmatrix}, 1\right) \quad \text {and} \quad S = \left( \begin{pmatrix}0 &{} -1 \\ 1 &{} 0\end{pmatrix}, \sqrt{\tau }\right) . \end{aligned}$$

We have the relation $S^2 = (ST)^3 = Z$, where $Z = \left( \left( {\begin{matrix}-1 &{} 0 \\ 0 &{} -1\end{matrix}}\right) , i\right)$ is the standard generator of the center of ${\text {Mp}}_2({\mathbb {Z}})$. Furthermore we will write $\Gamma _\infty := \{\left( {\begin{matrix}1 &{} n \\ 0 &{} 1\end{matrix}}\right) \mid n \in {\mathbb {Z}}\}$ and ${\tilde{\Gamma }}_\infty := \{\left( \left( {\begin{matrix}1 &{} n \\ 0 &{} 1\end{matrix}}\right) , 1\right) \mid n \in {\mathbb {Z}}\}$. For an even non-degenerate lattice L of signature $(b^+, b^-)$ consider the group ring ${\mathbb {C}}[L' / L]$ with standard basis $({\mathfrak {e}}_\gamma )_{\gamma \in L' / L}$. For ${\mathfrak {v}}= \sum _{\gamma \in L' / L} {\mathfrak {v}}_\gamma {\mathfrak {e}}_\gamma \in {\mathbb {C}}[L' / L]$ we write ${\mathfrak {v}}^* := \sum _{\gamma \in L' / L} {\mathfrak {v}}_\gamma {\mathfrak {e}}_{-\gamma }$. Moreover we write $\langle \cdot , \cdot \rangle$ for the standard inner product on ${\mathbb {C}}[L' / L]$ which is anti-linear in the second variable. Write ${\text {Iso}}(L' / L) \subseteq L' / L$ for the set of isotropic elements and denote the subspace of vectors that are supported on isotropic elements by ${\text {Iso}}({\mathbb {C}}[L' / L])$. Moreover we introduce the notation ${\mathfrak {e}}_\gamma (\tau ) := e(\tau ) {\mathfrak {e}}_\gamma = e^{2 \pi i \tau } {\mathfrak {e}}_\gamma$.

Definition 2.2

The Weil representation is the unitary representation $\rho _L$ of ${\text {Mp}}_2({\mathbb {Z}})$ on ${\mathbb {C}}[L' / L]$ defined by

$$\begin{aligned} \rho _L(T) {\mathfrak {e}}_\gamma := {\mathfrak {e}}_\gamma (q(\gamma )) \quad \text {and} \quad \rho _L(S) {\mathfrak {e}}_\gamma := \frac{\sqrt{i}^{b^- - b^+}}{\sqrt{L' / L}} \sum _{\delta \in L' / L} {\mathfrak {e}}_\delta (-(\gamma , \delta )). \end{aligned}$$

The Weil representation factors through a finite quotient of ${\text {Mp}}_2({\mathbb {Z}})$. The space of invariant vectors under the Weil representation is denoted by ${\text {Inv}}({\mathbb {C}}[L' / L])$.

A short calculation using orthogonality of characters shows for example $\rho _L(Z) {\mathfrak {e}}_\gamma = i^{b^- - b^+} {\mathfrak {e}}_{-\gamma }$. For $\beta , \gamma \in L' / L$ we define the coefficients

$$\begin{aligned} \rho _{\beta , \gamma }(M, \phi ) := \langle \rho _L(M, \phi ) {\mathfrak {e}}_\gamma , {\mathfrak {e}}_\beta \rangle \quad \text {and} \quad \rho _{\beta , \gamma }^{-1}(M, \phi ) := \langle \rho _L^{-1}(M, \phi ) {\mathfrak {e}}_\gamma , {\mathfrak {e}}_\beta \rangle . \end{aligned}$$

Theorem 2.3

[21, Proposition 1.6] For $M \in {\text {SL}}_2({\mathbb {Z}})$ the coefficient $\rho _{\beta , \gamma }({\tilde{M}})$ is given by

$$\begin{aligned} \sqrt{i}^{(b^- - b^+)(1 - {\text {sgn}}(d))} \delta _{\beta , a\gamma } e(ab q(\beta )) \end{aligned}$$

if $c = 0$ and by

$$\begin{aligned} \frac{\sqrt{i}^{(b^- - b^+) {\text {sgn}}(c)}}{|c |^{(b^- + b^+)/2}\sqrt{|L' / L |}} \sum _{r \in L / cL} e\left( \frac{a(\beta + r, \beta + r) - 2 (\gamma , \beta + r) + d (\gamma , \gamma )}{2c}\right) \end{aligned}$$

if $c \ne 0$.

For a vector-valued function $f : {\mathbb {H}}\rightarrow {\mathbb {C}}[L' / L]$ we write $f_\gamma : {\mathbb {H}}\rightarrow {\mathbb {C}}$ for its components, i.e. $f = \sum _{\gamma \in L' / L} f_\gamma {\mathfrak {e}}_\gamma$. For $k \in \frac{1}{2} {\mathbb {Z}}$ we define the Petersson slash operator $f \mapsto f\vert _{k, L}(M, \phi )$ by

$$\begin{aligned} (f \vert _{k, L}(M, \phi ))(\tau ) = \phi (\tau )^{-2 k} \rho _L^{-1}(M, \phi ) f(M \tau ). \end{aligned}$$

If $f : {\mathbb {H}}\rightarrow {\mathbb {C}}[L' / L]$ is smooth and invariant under the action of T, i.e. $f \vert _{k, L} T = f$, then we have a Fourier expansion

$$\begin{aligned} f(\tau ) = \sum _{\gamma \in L' / L} \sum _{n \in {\mathbb {Z}}+ q(\gamma )} c(\gamma , n, v) {\mathfrak {e}}_\gamma (nu). \end{aligned}$$

Definition 2.4

A function $f : {\mathbb {H}}\rightarrow {\mathbb {C}}[L' / L]$ is said to be a modular form of weight k with respect to the Weil representation if $f \vert _{k, L}(M, \phi ) = f$ for all $(M, \phi ) \in {\text {Mp}}_2({\mathbb {Z}})$. We call f a holomorphic modular form if f is holomorphic with Fourier expansion

$$\begin{aligned} f(\tau ) = \sum _{\gamma \in L' / L} \sum _{\begin{array}{c} n \in {\mathbb {Z}}+ q(\gamma ) \\ n \ge 0 \end{array}} c(\gamma , n) {\mathfrak {e}}_\gamma (n\tau ). \end{aligned}$$

Obviously, non-trivial modular forms only exist for weights with $2k + b^- - b^+ = 0 \bmod 2$.

Assume now that $b^+ - b^-$ is even and let $k \in {\mathbb {Z}}$. Moreover set $\kappa = \frac{b^- - b^+}{2} + k$. Let $\beta \in {\text {Iso}}(L' / L)$ and similar to [4] define the vector-valued non-holomorphic Eisenstein series of weight k by

$$\begin{aligned} E_{k, \beta }(\tau , s) = \frac{1}{2} \sum _{M \in {\tilde{\Gamma }}_\infty \backslash {\text {Mp}}_2({\mathbb {Z}})} v^s {\mathfrak {e}}_\beta \vert _{k, L} M. \end{aligned}$$

Observe that [4] consider the dual Weil representation $\rho _L$. For $\beta \in {\text {Iso}}(L' / L)$ of order $N_\beta$ and a character $\chi : ({\mathbb {Z}}/ N_\beta {\mathbb {Z}})^\times$ we define

$$\begin{aligned} E_{k, \beta , \chi }(\tau , s) := \sum _{n \in ({\mathbb {Z}}/ N_\beta {\mathbb {Z}})^\times } \chi (n) E_{k, n \beta }(\tau , s). \end{aligned}$$

More generally, for ${\mathfrak {v}}\in {\text {Iso}}({\mathbb {C}}[L' / L])$ we define

$$\begin{aligned} E_{k, {\mathfrak {v}}}(\tau , s)&= \frac{1}{2} \sum _{M \in {\tilde{\Gamma }}_\infty \backslash {\text {Mp}}_2({\mathbb {Z}})} v^s {\mathfrak {v}}\vert _{k, L} M \\&= \sum _{\beta \in {\text {Iso}}(L' / L)} {\mathfrak {v}}_\beta E_{k, \beta }(\tau , s). \end{aligned}$$

We have $E_{k, {\mathfrak {v}}^*} = (-1)^\kappa E_{k, {\mathfrak {v}}}$ and a Fourier expansion of the form

$$\begin{aligned} E_{k, {\mathfrak {v}}} =&({\mathfrak {v}}+ (-1)^\kappa {\mathfrak {v}}^*) v^s + \sum _{\gamma \in {\text {Iso}}(L' / L)} c_{k, {\mathfrak {v}}}(\gamma , 0, s) v^{1 - s - k} \\&+ \sum _{\gamma \in L' / L} \sum _{\begin{array}{c} n \in {\mathbb {Z}}+ q(\gamma ) \\ n \ne 0 \end{array}} c_{k, {\mathfrak {v}}}(\gamma , n, s) {\mathcal {W}}_s(4 \pi n y) {\mathfrak {e}}_\gamma (n u), \end{aligned}$$

where ${\mathcal {W}}_s$ is a special Whittaker function. For the precise coefficients see [4, 22] (or [3, 18, 20] for the holomorphic case), we will not need them here. It can be easily seen that the vector-valued Eisenstein series are ${\text {Mp}}_2({\mathbb {Z}})$-translates of usual scalar-valued Eisenstein series. They are normalized such that their meromorphic continuation to ${\mathbb {C}}$ are holomorphic in $s = 0$ and have a simple pole for $k = 0$ at $s = 1$ whose residue is an invariant vector. For an invariant vector ${\mathfrak {v}}\in {\text {Inv}}({\mathbb {C}}[L' / L])$ we have

$$\begin{aligned} E_k(\tau , s) {\mathfrak {v}}= E_{k, {\mathfrak {v}}}(\tau , s), \end{aligned}$$

where $E_k(\tau , s)$ is the suitably normalized Eisenstein series for ${\text {SL}}_2({\mathbb {Z}})$. For $k > 2$ the series converges at $s = 0$ and defines a holomorphic Eisenstein series $E_{k, {\mathfrak {v}}}(\tau ) = E_{k, {\mathfrak {v}}}(\tau , 0)$ with Fourier expansion

$$\begin{aligned} E_{k, {\mathfrak {v}}}(\tau ) = {\mathfrak {v}}+ (-1)^\kappa {\mathfrak {v}}^* + \sum _{\gamma \in L' / L} \sum _{\begin{array}{c} n \in {\mathbb {Z}}+ q(\gamma ) \\ n > 0 \end{array}} c_{k, {\mathfrak {v}}}(\gamma , n) {\mathfrak {e}}_\gamma (n \tau ). \end{aligned}$$

For $k = 2$ they have a Fourier expansion of the form [10, 15]

$$\begin{aligned} E_{2, {\mathfrak {v}}}(\tau ) = {\mathfrak {w}}v^{-1} + {\mathfrak {v}}+ (-1)^\kappa {\mathfrak {v}}^* + \sum _{\gamma \in L' / L} \sum _{\begin{array}{c} n \in {\mathbb {Z}}+ q(\gamma ) \\ n > 0 \end{array}} c_{2, {\mathfrak {v}}}(\gamma , n) {\mathfrak {e}}_\gamma (n \tau ). \end{aligned}$$

Applying the lowering operator shows that ${\mathfrak {w}}\in {\text {Inv}}({\mathbb {C}}[L' / L])$ is an invariant vector. For $k = 0$ their residue at $s = 1$ yields an invariant vector and if ${\mathfrak {v}}\in {\text {Inv}}({\mathbb {C}}[L' / L])$ we obtain

$$\begin{aligned} {\text {res}}_{s = 1} E_{0, {\mathfrak {v}}}(\tau , s) = {\text {res}}_{s = 1} E_0(\tau , s) {\mathfrak {v}}, \end{aligned}$$

in particular these residues span the space of invariants.

3 Orthogonal modular forms

From now on let L be an even lattice of signature (2, l) and let $V = L \otimes {\mathbb {Q}}, V({\mathbb {R}}) = V \otimes {\mathbb {R}}$. Write $P(V({\mathbb {C}}))$ for the corresponding projective space. For elements $Z_L = X_L + i Y_L \in V({\mathbb {C}}) \setminus \{0\}$ we write $[Z_L]$ for the canonical projection onto $P(V({\mathbb {C}}))$. The subset

$$\begin{aligned} {\mathcal {K}}= \{[Z_L] \in P(V({\mathbb {C}})) \mid (Z_L, Z_L) = 0, (Z_L, \overline{Z_L}) > 0 \} \end{aligned}$$

is the hermitian symmetric domain associated to O(V). It has two connected components which are interchanged by $[Z_L] \mapsto [\overline{Z_L}]$. We choose one of them and call it ${\mathcal {K}}^+$. The action of $O(V({\mathbb {R}}))$ on $V({\mathbb {R}})$ induces an action on ${\mathcal {K}}$. Let $O^+(V({\mathbb {R}}))$ be the subgroup which preserves the connected components of ${\mathcal {K}}$ and let

$$\begin{aligned} {\tilde{{\mathcal {K}}}}^+ = \{Z_L \in V({\mathbb {C}}) \setminus \{0\} \mid [Z_L] \in {\mathcal {K}}^+ \} \end{aligned}$$

be the preimage of ${\mathcal {K}}^+$ under the projection. Write ${\text {Iso}}_0(L)$ for the set of primitive isotropic elements of L. For $z \in {\text {Iso}}_0(L)$ and $z' \in L'$ with $(z, z') = 1$ write $K = L \cap z^\perp \cap z'^\perp$. Let $d \in {\text {Iso}}_0(K)$, $d' \in K'$ with $(d, d') = 1$ and $D = K \cap d^\perp \cap d'^\perp$. Moreover let ${\tilde{z}} = z' - q(z') z$ and ${\tilde{d}} = d' - q(d') d$. Let $d_3, \ldots , d_l$ be a basis of D. Then ${\tilde{d}}, d, d_3, \ldots , d_l$ is a basis of $K \otimes {\mathbb {R}}$. We define the orthogonal upper half-plane as

$$\begin{aligned} {\mathbb {H}}_l := \{Z = X + iY \in W({\mathbb {C}}) = K \otimes {\mathbb {C}}\mid q(Y)> 0, (Y, d) > 0\}. \end{aligned}$$

We will write $Z = z_1 {\tilde{d}} + z_2 d + Z_D$ with $Z_D \in D \otimes {\mathbb {C}}$ and analogously for $X, Y \in W({\mathbb {R}})$. For $Z \in {\mathbb {H}}_l$ we define

$$\begin{aligned} Z_L := Z - q(Z) z + {\tilde{z}}. \end{aligned}$$

If the component ${\mathcal {K}}^+$ is chosen properly, this yields a biholomorphic map $Z \mapsto [Z_L]$. By setting $i C = {\mathbb {H}}_l \cap i (K \otimes {\mathbb {R}})$ we see that ${\mathbb {H}}_l = K \otimes {\mathbb {R}}+ iC$ is a tube domain. The action of $O^+(V({\mathbb {R}}))$ on ${\mathcal {K}}^+$ induces an action of $O^+(V({\mathbb {R}}))$ on ${\mathbb {H}}_l$. Let

$$\begin{aligned} j : O^+(V) \times {\mathbb {H}}_l \rightarrow {\mathbb {C}}^\times , \quad j(\sigma , Z) := (\sigma (Z_L), z) \end{aligned}$$

be the factor of automorphy, so that we have

$$\begin{aligned} j(\sigma , Z)(\sigma Z)_L = \sigma (Z_L) \end{aligned}$$

and the cocycle relation

$$\begin{aligned} j(\sigma _1 \sigma _2, Z) = j(\sigma _1, \sigma _2 Z) j(\sigma _2, Z). \end{aligned}$$

According to [6, Lemma 3.20] we have

$$\begin{aligned} q({\text {Im}}(\sigma Z)) = \frac{q({\text {Im}}(Z))}{|j(\sigma , Z) |^2}. \end{aligned}$$

Definition 3.1

A function $F : {\tilde{{\mathcal {K}}}}^+ \rightarrow {\mathbb {C}}$ is called modular form of weight $\kappa \in {\mathbb {Z}}$ with respect to $\Gamma \subseteq \Gamma (L)$ if it satisfies

• (i) $F(t Z_L) = t^{-\kappa } F(Z_L)$ for all $t \in {\mathbb {C}}^\times$.

• (ii) $F(\sigma Z_L) = F(Z_L)$ for all $\sigma \in \Gamma$.

For a modular form $F : {\tilde{{\mathcal {K}}}}^+ \rightarrow {\mathbb {C}}$ of weight $\kappa \in {\mathbb {Z}}$ with respect to $\Gamma$ define

$$\begin{aligned} F_z : {\mathbb {H}}_l \rightarrow {\mathbb {C}}, \quad F_z(Z) := F(Z_L) = F(Z - q(Z) z + {\tilde{z}}). \end{aligned}$$

Then $F_z$ satisfies

$$\begin{aligned} F_z(\sigma Z) = j(\sigma , Z)^\kappa F_z(Z) \end{aligned}$$

for all $\sigma \in \Gamma$ and we have a bijective correspondence between modular forms and functions with this transformation property. If we define the weight $\kappa$ slash operator $F_z \vert _\kappa \sigma$ for $\sigma \in O^+(V)$ by

$$\begin{aligned} (F_z \vert _\kappa \sigma )(Z) := j(\sigma , Z)^{-\kappa } F_z(\sigma Z), \end{aligned}$$

then the modular forms on ${\mathbb {H}}_l$ are exactly the functions that are invariant under the slash operator for all $\sigma \in \Gamma$. If $F_z$ is holomorphic on ${\mathbb {H}}_l$ it has a Fourier expansion of the form

$$\begin{aligned} F_z(Z) = \sum _{\lambda \in K'} a_z(\lambda ) e(\lambda , Z), \end{aligned}$$

where $e(\lambda , Z) := e((\lambda , Z)) = e^{2 \pi i (\lambda , Z)}$.

Definition 3.2

We say that a modular form f is a holomorphic modular form if f is holomorphic and for all 0 dimensional cusps $z \in {\text {Iso}}_0(L)$ we have $a_z(\lambda ) = 0$ for $\lambda \notin {\overline{C}}$. We write ${\text {M}}_\kappa (\Gamma )$ for the space of modular forms.

Remark 3.3

For $l \ge 4$ the weight $\frac{l}{2} - 1$ is called singular weight. By the theory of singular weights, there are no holomorphic modular forms for $\Gamma (L)$ for weight $\frac{l}{2} - 1> \kappa > 0$. Moreover, if

$$\begin{aligned} F_z(Z) = \sum _{\lambda \in K'} a_z(\lambda ) e(\lambda , Z) \end{aligned}$$

is a holomorphic modular form of singular weight, then $a_z(\lambda ) \ne 0$ implies $\lambda \in C$.

Definition 3.4

For $t > 0$ we define the Siegel domain ${\mathcal {S}}_t$ as the set of $Z = X + iY \in {\mathbb {H}}_l$ satisfying

$$\begin{aligned} x_1^2 + x_2^2 + |q(X_D) |&< t^2, \\ 1 / t&< y_1, \\ y_1^2&< t^2 q(Y), \\ |q(Y_D) |&< t^2 y_1^2. \end{aligned}$$

The set of $Y \in C$ satisfying the last three inequalities is denoted by ${\mathcal {R}}_t$.

We have the following

Proposition 3.5

[6, Proposition 4.10] Let $\Gamma \subseteq \Gamma (L)$ be a subgroup of finite index.

• (i) For any $t > 0$ and any $g \in O^+(V)$ the set

  $$\begin{aligned} \{\sigma \in \Gamma \mid \sigma g {\mathcal {S}}_t \cap {\mathcal {S}}_t \ne \emptyset \} \end{aligned}$$

  is finite.

• (ii) There exists a $t > 0$ and finitely many $g_1, \ldots , g_n \in O^+(V)$ such that for

  $$\begin{aligned} {\mathcal {S}}= g_1 {\mathcal {S}}_t \cup \cdots \cup g_n {\mathcal {S}}_t \end{aligned}$$

  we have $\Gamma {\mathcal {S}}= {\mathbb {H}}_l$.

The invariant volume element on ${\mathbb {H}}_l$ is given by

$$\begin{aligned} \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l}. \end{aligned}$$

The previous proposition means in particular, that for a $\Gamma$-invariant measurable function $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ we have $F \in L^p({\mathbb {H}}_l / \Gamma )$ if and only if $\int _{\mathcal {S}}|F(Z) |^p \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} < \infty$. This again is the case if for every choice of $z, z', d, d', t > 0$ the integral $\int _{{\mathcal {S}}_t} |F(Z) |^p \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l}$ is finite. We will need the following estimates.

Lemma 3.6

[6, Lemma 4.13] Let $t > 0$. Then there exists $\varepsilon > 0$ such that for all $\lambda = (\lambda _1, \lambda _2, \lambda _D) \in K'$ and $Y \in {\mathcal {R}}_t$ we have

$$\begin{aligned} \frac{(\lambda , Y)^2}{Y^2} - q(\lambda ) \ge \varepsilon (y_2^2 \lambda _1^2 / 2 + y_1^2 \lambda _2^2 / 2 + Y^2 \lambda _D^2) / Y^2. \end{aligned}$$

Using the inequality

$$\begin{aligned} \frac{y_1 y_2}{1 + t^4}< q(Y) < y_1 y_2 \end{aligned}$$

this yields

$$\begin{aligned} \frac{(\lambda , Y)^2}{Y^2} - q(\lambda ) > \varepsilon (y_2 / y_1 \lambda _1^2 / 2 + y_1 / y_2 \lambda _2^2 / 2 + \lambda _D^2). \end{aligned}$$

The following lemma is a slight generalisation of [6, Lemma 4.11, 4.12]

Lemma 3.7

Suppose $p_2 + p < l - 1$ and $p_1 + p_2 + 2p < l$. Then

$$\begin{aligned} \int _{{\mathcal {S}}_t} y_1^{p_1} y_2^{p_2} q(Y)^p \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} < 0. \end{aligned}$$

4 Siegel operator

For an isotropic line $I \subseteq V({\mathbb {R}})$ generated by some isotropic vector $\lambda \in I$ the corresponding point $[\lambda ]$ is in the closure of ${\mathcal {K}}^+$ in ${\mathcal {N}}$. To see this, take two sequences $x_n, y_n \in V({\mathbb {R}})$ with positive norm which are orthogonal and converge to $\lambda$. Then $[x_n + i y_n] \in {\mathcal {K}}^+$ with limit $[(1 + i)\lambda ] = [\lambda ] \in {\mathcal {N}}$ for $n \rightarrow \infty$.

Definition 4.1

A boundary point of ${\mathcal {K}}^+$ of the form $[\lambda ] \in {\mathcal {N}}$ which is represented by a real isotropic line is called special boundary point. A set consisting of one special boundary points is called 0-dimensional boundary component. A non-special boundary point is called generic boundary point.

Let now $I \subseteq V({\mathbb {R}})$ be an isotropic plane and consider the set of all boundary points which can be represented by elements of $I \otimes {\mathbb {C}}$.

Definition 4.2

For a totally isotropic plane $I \subseteq V({\mathbb {R}})$ the set of all generic boundary points which can be represented by an element of $I \otimes {\mathbb {C}}$ is called 1-dimensional boundary component attached to I. By a boundary component we mean a 0-dimensional or 1-dimensional boundary component.

Lemma 4.3

The 1-dimensional boundary components are isomorphic to usual upper half-planes. Moreover,  there is a bijective correspondence between boundary components and non-zero isotropic subspaces of $V({\mathbb {R}})$.

Proof

Let $I \subseteq V({\mathbb {R}})$ be an isotropic plane. Take a basis z, d of I and consider ${\tilde{z}}, {\tilde{d}}$ isotropic such that $(z, {\tilde{z}}) = (d, {\tilde{d}}) = 1$ and all other products vanish. We will use the shorthand notation $(z_1, z_2, z_3, z_4)$ for $z_1 {\tilde{z}} + z_2 z + z_3 {\tilde{d}} + z_4 d$. Then the elements of I have the form $(0, z_2, 0, z_4)$. Assume that this is not a multiple of a real point, i.e. it is not a special boundary point. Then $z_2 \ne 0 \ne z_4$ and we can normalize it such that $z_4 = 1$, i.e. we have a point of the form $(0, \tau , 0, 1)$ for some $\tau \in {\mathbb {C}}\setminus {\mathbb {R}}$. Making suitable choices of the basis and ${\mathcal {K}}^+$ we can assume that $[1, 1, i, i] \in {\mathcal {K}}^+$. Then we have an embedding

$$\begin{aligned} {\mathbb {H}}\times {\mathbb {H}}\rightarrow {\mathcal {K}}^+, \quad (\tau _1, \tau _2) \mapsto [1, - \tau _1 \tau _2, \tau _1, \tau _2]. \end{aligned}$$

In the projective space we have

$$\begin{aligned} \lim _{t \rightarrow \infty } [1, - \tau it, \tau , it] = \lim _{t \rightarrow \infty } \left[ \frac{1}{it}, -\tau , \frac{\tau }{it}, 1\right] = [0, -\tau , 0, 1]. \end{aligned}$$

Thus the point $[0, -\tau , 0, 1]$ is in the boundary of ${\mathcal {K}}^+$ if and only if $v = {\text {Im}}(\tau ) > 0$. The set of all boundary points represented by elements in $I \otimes {\mathbb {C}}$ can be identified with ${\mathbb {H}}\cup {\mathbb {R}}\cup \infty$. In particular, if we let $\tau = iv$ with $v \rightarrow \infty$, we obtain the special boundary point [z]. $\square$

Definition 4.4

A boundary component is called rational boundary component if the corresponding isotropic subspace is defined over ${\mathbb {Q}}$. Write ${\mathbb {H}}_l^*$ for the union of ${\mathbb {H}}_l \simeq {\mathcal {K}}^+$ with all rational boundary components. Then the rational orthogonal group $O^+(V) := O^+(V({\mathbb {R}})) \cap O(V)$ acts on ${\mathbb {H}}_l^*$.

By the theory of Baily–Borel (see [1, 2]), there is a topology on ${\mathbb {H}}_l^*$ such that for congruence subgroups $\Gamma \subseteq O^+(V)$ the quotient $X_\Gamma = {\mathbb {H}}_l^* / \Gamma$ carries the structure of a projective variety, which contains ${\mathbb {H}}_l / \Gamma$ as a Zariski open subvariety.

Let now $I \subseteq L \otimes {\mathbb {Q}}$ be an isotropic plane. Let $z \in L \cap I$ be primitive and $z' \in L$ with $(z, z') = 1$. Let $K = L \cap z^\perp \cap z'^\perp$ and take $d \in K \cap I$ primitive, $d' \in K'$ with $(d, d') = 1$ and let $D = K \cap d^\perp \cap d'^\perp$. As before, write ${\tilde{z}} = z' - q(z') z, {\tilde{d}} = d' - q(d') d$ and let $d_3, \ldots , d_l$ be a basis of D. Then we obviously have $I = \langle z, d \rangle$. Recall the orthogonal upper half-plane corresponding to z

$$\begin{aligned} {\mathbb {H}}_l := \{Z = X + iY \in K \otimes {\mathbb {C}}\mid q(Y)> 0, (Y, d) > 0 \} \end{aligned}$$

and that we write $Z = z_1 {\tilde{d}} + z_2 d + Z_D$. Let $C \subseteq K \otimes {\mathbb {R}}$ be the positive cone such that

$$\begin{aligned} {\mathbb {H}}_l = K \otimes {\mathbb {R}}+ i C \end{aligned}$$

and write ${\overline{C}}$ for its closure. For $\tau \in {\mathbb {H}}$ and $t \in {\mathbb {R}}_{>0}$ we have $\tau {\tilde{d}} + it d \in {\mathbb {H}}_l$, which corresponds to $[{\tilde{z}} - \tau it z + \tau {\tilde{d}} + it d] \in {\mathcal {K}}^+$. For a function $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ we define the Siegel operator corresponding to the boundary component I as (if it exists)

$$\begin{aligned} F \vert _I : {\mathbb {H}}\rightarrow {\mathbb {C}}, \quad \tau \mapsto \lim _{t \rightarrow \infty } F(\tau {\tilde{d}} + it d). \end{aligned}$$

For $\lambda = (\lambda _1, \lambda _2, \lambda _D) \in K'$ we have

$$\begin{aligned} (\lambda , \tau {\tilde{d}} + itd) = it \lambda _1 + \tau \lambda _2. \end{aligned}$$

Assume that F has a Fourier expansion of the form

$$\begin{aligned} F(Z) = \sum _{\begin{array}{c} \lambda \in K' \\ \lambda \in {\overline{C}} \end{array}} a(\lambda ) e(\lambda , Z), \end{aligned}$$

then the Siegel operator exists and we have

$$\begin{aligned} F \vert _I(\tau )&= \lim _{t \rightarrow \infty } \sum _{\begin{array}{c} (\lambda _1, \lambda _2, \lambda _D) \in K' \\ (\lambda _1, \lambda _2, \lambda _D) \in {\overline{C}} \end{array}} a(\lambda _1, \lambda _2, \lambda _D) e(\lambda _1 {\tilde{d}} + \lambda _2 d + \lambda _D, \tau d' + itd) \\&= \lim _{t \rightarrow \infty } \sum _{\begin{array}{c} (\lambda _1, \lambda _2, \lambda _D) \in K' \\ (\lambda _1, \lambda _2, \lambda _D) \in {\overline{C}} \end{array}} a(\lambda _1, \lambda _2, \lambda _D) \exp (-2 \pi t \lambda _1) e(\lambda _2 \tau ) \\&= \sum _{\begin{array}{c} (0, \lambda _2, 0) \in K' \\ \lambda _2 \ge 0 \end{array}} a(0, \lambda _2, 0) e(\lambda _2 \tau ) \\&= \sum _{m = 0}^\infty a(0, m / N_d, 0) e(m \tau / N_d). \end{aligned}$$

The value in the 0-dimensional cusp z is given by the constant term of the Fourier expansion, i.e.

$$\begin{aligned} a(0) = \lim _{t \rightarrow \infty } F(t Z) \end{aligned}$$

for arbitrary $Z \in {\mathbb {H}}_l$.

Let now $F : {\tilde{{\mathcal {K}}}}^+ \rightarrow {\mathbb {C}}$ be a holomorphic modular form of weight $\kappa$ for some congruence subgroup $\Gamma$. Then its restriction to the boundary component $F_z \vert _I : {\mathbb {H}}\rightarrow {\mathbb {C}}$ is a holomorphic modular form of weight $\kappa$ for some subgroup of ${\text {SL}}_2({\mathbb {Q}})$. One easily sees that for the constant Fourier coefficient we have $a_{-z}(0) = (-1)^\kappa a_z(0)$.

Definition 4.5

We call a holomorphic modular form $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ of weight $\kappa$ for some congruence subgroup $\Gamma$ a cusp form if $F \vert _I$ vanishes identically for all boundary components I. We say that F is an Eisenstein series on the boundary if $F \vert _I$ is an Eisenstein series for all boundary components I. We write ${\text {S}}_\kappa (\Gamma )$ for the space of cusp forms and ${\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma )$ for the space of holomorphic modular forms that are Eisenstein series on the boundary. In particular we have ${\text {S}}_\kappa (\Gamma ) \subseteq {\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma )$.

Remark 4.6

By the theory of singular weights, there are no cusp forms of singular weight. In particular, a holomorphic modular form of singular weight is completely determined by its values at the boundary and a holomorphic modular form of singular weight that is an Eisenstein series on the boundary is completely determined by its values in the 0-dimensional cusps.

Remark 4.7

If a holomorphic modular form F is an Eisenstein series on the boundary, then its restrictions to the boundary are fully determined by the values in the 0-dimensional cusps, i.e. the constant Fourier coefficients.

Definition 4.8

For $\Gamma = \Gamma (L)$ recall the map

$$\begin{aligned} \pi _L : \{0\text {-dimensional cusps of } \Gamma (L) \backslash {\mathbb {H}}_l \} \simeq \Gamma (L) \backslash {\text {Iso}}_0(L') \rightarrow {\text {Iso}}(L' / L). \end{aligned}$$

We write ${\text {M}}_\kappa ^{\pi }(\Gamma (L))$ for the subspace of ${\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$ that consists of holomorphic modular forms whose values in the 0-dimensional cusps only depend on their image in $L' / L$. In particular, if $\delta = -\delta \in L' / L$, the value in the 0-dimensional cusps corresponding to $\delta$ vanish if $\kappa$ is odd. We have ${\text {S}}_\kappa (\Gamma (L)) \subseteq {\text {M}}_\kappa ^\pi (\Gamma (L))$ and if $\pi _L$ is injective we have ${\text {M}}_\kappa ^\pi (\Gamma (L)) = {\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$.

We want to mention the following result which is sometimes called Eichler criterion.

Lemma 4.9

[12, Lemma 4.4] If L splits two hyperbolic planes,  then $\pi _L$ is bijective.

5 Differential operators

Let M be a hermitian manifold and E a holomorphic vector bundle. We write ${\mathcal {E}}(E) = {\mathcal {E}}(M, E)$ for the space of (global) sections of E and, more generally, for $U \subseteq M$ open, we write ${\mathcal {E}}(U, E)$ for the space of sections over U. Assume now that E carries a hermitian metric. Then the hermitian metric on E induces a hermitian metric on E-valued differential forms ${\mathcal {A}}^*(M, E)$ locally given by

$$\begin{aligned} (\varphi \otimes \sigma , \varphi ' \otimes \sigma ') = (\varphi , \varphi ') (\sigma , \sigma ') \end{aligned}$$

on decomposable forms, where the first factor is the hermitian metric on differential forms coming from the hermitian metric on M and the second factor is the hermitian metric on the vector bundle E. Moreover, the hermitian metric on E together with the Hodge-$*$-operator $* : {\mathcal {A}}^*(M) \rightarrow {\mathcal {A}}^*(M)$ induces a Hodge-$*$-operator

$$\begin{aligned} \overline{*}_E : {\mathcal {A}}^*(M, E) \rightarrow {\mathcal {A}}^*(M, E^*). \end{aligned}$$

The evaluation map $E \otimes E^* \rightarrow {\mathbb {C}}$ induces a wedge product $\wedge$ which satisfies

$$\begin{aligned} \alpha \wedge \overline{*}_E \beta = (\alpha , \beta ) {\text {vol}}, \end{aligned}$$

where ${\text {vol}}$ is the volume form induced from the hermitian metric on M. Moreover we have a hermitian inner product on E-valued differential forms given by

$$\begin{aligned} (\alpha , \beta )_2 = \int _M \alpha \wedge \overline{*}_E \beta = \int _M (\alpha , \beta ) {\text {vol}}. \end{aligned}$$

Let ${\overline{\partial }}^*_E = \overline{*}_{E^*} {\overline{\partial }}_{E^*} \overline{*}_E$ and $\Delta _E = {\overline{\partial }}^*_E {\overline{\partial }}_E$ be the Laplace operator on sections of E. We will need the following

Theorem 5.1

[8, Section 3, Example (B)] Let M be a complete connected hermitian manifold and let E be a hermitian vector bundle. If u, v are smooth square integrable sections of E such that $\Delta _E u, \Delta _E v$ are also square integrable,  then

$$\begin{aligned} (\Delta _E u, v) = (u, \Delta _E v). \end{aligned}$$

Let now L be an even lattice of signature (2, l). For $z \in {\text {Iso}}_0(L)$ and $z' \in L'$ with $(z, z') = 1$ write $K = L \cap z^\perp \cap z'^\perp$. Let $b_1, \ldots , b_l$ be a basis of $K \otimes {\mathbb {R}}$ such that

$$\begin{aligned} q(y_1 b_1 + \cdots + y_l b_l) = y_1^2 - y_2^2 - \cdots - y_l^2. \end{aligned}$$

If $Z = z_1 b_1 + z_2 b_2 + z_3 b_3 + \cdots + z_l b_l \in K \otimes {\mathbb {C}}$, we write $Z = (z_1, \ldots , z_l)$ and similarly $X = (x_1, \ldots , x_l), Y = (y_1, \ldots , y_l)$ if $Z = X + i Y$ with $X, Y \in K \otimes {\mathbb {R}}$. Denote by ${\mathbb {H}}_l = K \otimes {\mathbb {R}}+ i C$ the corresponding tube domain model, where

$$\begin{aligned} C = \{Y = (y_1, \ldots , y_l) \in K_z \otimes {\mathbb {R}}\mid y_1> 0, q(Y) > 0\}. \end{aligned}$$

The Kähler form is given by

$$\begin{aligned} w = -\frac{i}{2} \partial {\overline{\partial }} \log (q(Y))) = \frac{i}{2} \sum _{i, j} h_{ij}(Z) {\mathrm {d}}z_i \wedge {\mathrm {d}}{\overline{z}}_j, \end{aligned}$$

where $h(Z) = h(Y) = (h_{ij})_{1 \le i, j \le l}$ is the associated hermitian form given by

$$\begin{aligned} \frac{1}{q(Y)^2} \begin{pmatrix} y_1^2 &{} -y_1 y_2 &{} -y_1 y_3 &{} \ldots &{} -y_1 y_l \\ -y_1 y_2 &{} y_2^2 &{} y_2 y_3 &{} \ldots &{} y_2 y_l \\ -y_1 y_3 &{} y_2 y_3 &{} y_3^2 &{} \ddots &{} \vdots \\ \vdots &{} \vdots &{} \ddots &{} \ddots &{} y_{l-1} y_l \\ -y_1 y_l &{} y_2 y_l &{} \ldots &{} y_{l-1} y_l &{} y_l^2 \end{pmatrix} + \frac{1}{2q(Y)} \begin{pmatrix} -1 &{} &{} &{} \\ &{} 1 &{} &{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} 1 \end{pmatrix} \end{aligned}$$

and its inverse is

$$\begin{aligned} h^{-1}(Y) = 4 \begin{pmatrix} y_1^2 &{} y_1 y_2 &{} y_1 y_3 &{} \ldots &{} y_1 y_l \\ y_1 y_2 &{} y_2^2 &{} y_2 y_3 &{} \ldots &{} y_2 y_l \\ y_1 y_3 &{} y_2 y_3 &{} y_3^2 &{} \ddots &{} \vdots \\ \vdots &{} \vdots &{} \ddots &{} \ddots &{} y_{l-1} y_l \\ y_1 y_l &{} y_2 y_l &{} \ldots &{} y_{l-1} y_l &{} y_l^2 \end{pmatrix} + 2 q(Y) \begin{pmatrix} -1 &{} &{} &{} \\ &{} 1 &{} &{} &{} \\ &{} &{} \ddots &{} &{} \\ &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} 1 \end{pmatrix}. \end{aligned}$$

Its determinant is $\det (h) = \frac{1}{2^{l} q(Y)^l}$ and hence the volume form is given by

$$\begin{aligned} \omega _g&= \frac{1}{(4i q(Y))^l} {\mathrm {d}}z_1 \wedge {\mathrm {d}}{\overline{z}}_1 \wedge \cdots \wedge {\mathrm {d}}z_l \wedge {\mathrm {d}}{\overline{z}}_l \\&= \frac{1}{2^l q(Y)^{l}} {\mathrm {d}}x_1 \wedge {\mathrm {d}}y_1 \wedge \cdots \wedge {\mathrm {d}}x_l \wedge {\mathrm {d}}y_l. \end{aligned}$$

We write

$$\begin{aligned} \widehat{{\mathrm {d}}z_i} = {\mathrm {d}}z_1 \wedge {\mathrm {d}}{\overline{z}}_1 \wedge \cdots \wedge {\mathrm {d}}\overline{z_{i-1}} \wedge {\mathrm {d}}{\overline{z}}_i \wedge \cdots \wedge {\mathrm {d}}z_l \wedge {\mathrm {d}}{\overline{z}}_l \end{aligned}$$

and similarly for $\widehat{{\mathrm {d}}{\overline{z}}_i}$. Then we have

$$\begin{aligned} {\mathrm {d}}z_i \wedge \widehat{{\mathrm {d}}z_i} = (4i q(Y))^l \omega _g, \quad {\mathrm {d}}{\overline{z}}_i \wedge \widehat{{\mathrm {d}}{\overline{z}}_i} = -(4i q(Y))^l \omega _g. \end{aligned}$$

Now the Hodge-$\overline{*}$-operator is defined by the equality

$$\begin{aligned} \alpha \wedge \overline{*} \beta = \langle \alpha , \beta \rangle \omega _g, \end{aligned}$$

and thus we have

$$\begin{aligned} \overline{*} {\mathrm {d}}{\overline{z}}_i = -\frac{1}{2} \sum _{j = 1}^l \frac{h^{ji}(Y)}{(4 i q(Y))^l} \widehat{{\mathrm {d}}{\overline{z}}_j}. \end{aligned}$$

Modular forms of weight $\kappa$ form a line bundle ${\mathcal {L}}_\kappa$ which carries a hermitian metric given by the Petersson metric, i.e. $F(Z) \overline{G(Z)} q(Y)^\kappa$ on the fiber. The dual bundle ${\mathcal {L}}_\kappa ^*$ can be identified using the hermitian metric with the line bundle ${\mathcal {L}}_{-\kappa }$ of modular forms of weight $-\kappa$ and the mapping $F(Z) \mapsto q(Y)^\kappa \overline{F(Z)}$ defines an anti-linear bundle isomorphism. This gives us the Hodge-$\overline{*}$-operator

$$\begin{aligned} \overline{*}_{\kappa } : {\mathcal {A}}^{p, q}(\Gamma \backslash {\mathbb {H}}_l, {\mathcal {L}}_\kappa ) \rightarrow {\mathcal {A}}^{n - p, n - q}(\Gamma \backslash {\mathbb {H}}_l, {\mathcal {L}}_{-\kappa }), \quad \overline{*}_{\kappa }(\phi \otimes F) := (\overline{*} \phi ) \otimes (q(Y)^\kappa {\overline{F}}). \end{aligned}$$

The weight $\kappa$ Laplace operator is then given by

$$\begin{aligned} \Omega _\kappa = \overline{*}_{-\kappa } {\overline{\partial }} \overline{*}_\kappa {\overline{\partial }}. \end{aligned}$$

Theorem 5.2

For a modular form F of weight $\kappa$ the weight $\kappa$ Laplace operator is given by

$$\begin{aligned} \Omega _\kappa F(Z)&= \frac{q(Y)^{l-\kappa }}{2} \sum _{j = 1}^l \sum _{i = 1}^l \frac{\partial }{\partial z_j}\left( h^{ji}(Y) q(Y)^{\kappa - l} \frac{\partial F(Z)}{\partial {\overline{z}}_i} \right) \\&= 2 \sum _{j = 1}^l \sum _{i = 1}^l y_i y_j \frac{\partial ^2 F(Z)}{\partial z_j \partial {\overline{z}}_i} - q(Y) \left( \frac{\partial ^2 F(Z)}{\partial z_1 \partial {\overline{z}}_1} - \sum _{i = 2}^l \frac{\partial ^2 F(Z)}{\partial z_i \partial {\overline{z}}_i} \right) - i \kappa \sum _{i = 1}^l y_i \frac{\partial F(Z)}{\partial {\overline{z}}_i}. \end{aligned}$$

Proof

We have

$$\begin{aligned} \overline{*}_\kappa {\overline{\partial }} F(Z)&= \sum _{i = 1}^l q(Y)^k \frac{\partial \overline{F(Z)}}{\partial z_i} \overline{*} {\mathrm {d}}{\overline{z}}_i \\&= -\frac{1}{2 (4i)^l} \sum _{j = 1}^l \sum _{i = 1}^l h^{ji}(Y) q(Y)^{\kappa - l} \frac{\partial \overline{F(Z)}}{\partial z_i} \widehat{{\mathrm {d}}{\overline{z}}_j}. \end{aligned}$$

Applying $\overline{*}_{-\kappa } {\overline{\partial }}$ yields

$$\begin{aligned} \Omega _\kappa F(Z)&= -\frac{1}{2 (4i)^l} \sum _{j = 1}^l \sum _{i = 1}^l \overline{*}_{-\kappa } {\overline{\partial }} \left( h^{ji}(Y) q(Y)^{\kappa - l} \frac{\partial \overline{F(Z)}}{\partial z_i} \widehat{{\mathrm {d}}{\overline{z}}_j}\right) \\&= -\frac{1}{2 (4i)^l} \sum _{j = 1}^l \sum _{i = 1}^l \overline{*}_{-\kappa } \frac{\partial }{\partial {\overline{z}}_j}\left( h^{ji}(Y) q(Y)^{\kappa - l} \frac{\partial \overline{F(Z)}}{\partial z_i} \right) {\mathrm {d}}{\overline{z}}_j \wedge \widehat{{\mathrm {d}}{\overline{z}}_j} \\&= \frac{q(Y)^l}{2} \sum _{j = 1}^l \sum _{i = 1}^l \overline{*}_{-\kappa } \frac{\partial }{\partial {\overline{z}}_j}\left( h^{ji}(Y) q(Y)^{\kappa - l} \frac{\partial \overline{F(Z)}}{\partial z_i} \right) \omega _g \\&= \frac{q(Y)^{l-\kappa }}{2} \sum _{j = 1}^l \sum _{i = 1}^l \frac{\partial }{\partial z_j}\left( h^{ji}(Y) q(Y)^{\kappa - l} \frac{\partial F(Z)}{\partial {\overline{z}}_i} \right) \omega _g \\&= \frac{1}{2} \sum _{j = 1}^l \sum _{i = 1}^l h^{ij}(Y) \frac{\partial ^2 F(Z)}{\partial z_j \partial {\overline{z}}_i} \\&\quad + \frac{q(Y)^{l - \kappa }}{2} \sum _{j = 1}^l \sum _{i = 1}^l \left( \frac{\partial }{\partial z_j} h^{ij}(Y) q(Y)^{\kappa - l}\right) \frac{\partial F(Z)}{\partial {\overline{z}}_i}. \end{aligned}$$

A short calculation yields

$$\begin{aligned} \frac{\partial q(Y)}{\partial y_1} = 2y_1, \quad \frac{\partial q(Y)}{\partial y_j} = -2y_j, j > 1, \quad \frac{\partial h^{ij}(Y)}{\partial y_1} = 4 y_i. \end{aligned}$$

Thus, for $i = 1$ we have

$$\begin{aligned}&\sum _{j = 1}^l \frac{\partial }{\partial z_j} h^{ij}(Y) q(Y)^{\kappa - l} \\&\quad = -\frac{i}{2} \sum _{j = 1}^l \left( q(Y)^{\kappa - l} \frac{\partial h^{ij}(Y)}{\partial y_j} + (\kappa - l) h^{ij}(Y) q(Y)^{\kappa - l - 1} \frac{\partial q(Y)}{\partial y_j} \right) \\&\quad = -\frac{i}{2} \left( 4q(Y)^{\kappa - l} l y_1 + 2(\kappa - l) q(Y)^{\kappa - l - 1} \left( (4 y_1^2 - 2 q(Y)) y_1 - 4 y_1 \sum _{j = 2}^l y_j^2\right) \right) \\&\quad = -\frac{i}{2} \left( 4q(Y)^{\kappa - l} l y_1 + 2 y_1 (\kappa - l) q(Y)^{\kappa - l - 1} \left( 4 y_1^2 - 2 q(Y) + 4 \sum _{j = 2}^l y_j^2\right) \right) \\&\quad = -2 i \kappa q(Y)^{\kappa - l} y_1 \end{aligned}$$

and similarly for $i > 1$

$$\begin{aligned}&\sum _{j = 1}^l \frac{\partial }{\partial z_j} h^{ij}(Y) q(Y)^{\kappa - l} \\&\quad = -\frac{i}{2} \left( 4q(Y)^{\kappa - l} l y_i + 2(\kappa - l) q(Y)^{\kappa - l - 1} \left( 4 y_i y_1^2 - 4 y_i \sum _{j = 2}^l y_j^2 - 2q(Y) y_i\right) \right) \\&\quad = -2 i \kappa q(Y)^{\kappa - l} y_i \end{aligned}$$

and hence

$$\begin{aligned} \Omega _\kappa F(Z)= & {} \frac{1}{2} \sum _{j = 1}^l \sum _{i = 1}^l h^{ij}(Y) \frac{\partial ^2 F(Z)}{\partial z_j \partial {\overline{z}}_i} - i \kappa \sum _{i = 1}^l y_i \frac{\partial F(Z)}{\partial {\overline{z}}_i} \\= & {} 2 \sum _{j = 1}^l \sum _{i = 1}^l y_i y_j \frac{\partial ^2 F(Z)}{\partial z_j \partial {\overline{z}}_i} \\&- q(Y) \left( \frac{\partial ^2 F(Z)}{\partial z_1 \partial {\overline{z}}_1} - \sum _{i = 2}^l \frac{\partial ^2 F(Z)}{\partial z_i \partial {\overline{z}}_i} \right) - i \kappa \sum _{i = 1}^l y_i \frac{\partial F(Z)}{\partial {\overline{z}}_i}. \square \end{aligned}$$

Remark 5.3

There is an ad hoc definition of the weight (m, n) Laplace operator given by [23, 24] which coincides with $4 \Omega _\kappa$ for $m = \kappa , n = 0$. In particular $\Omega _\kappa$ commutes with the weight $\kappa$ slash operator and satisfies

$$\begin{aligned} \Omega _\kappa q(Y)^s = s (s + \kappa - l / 2) q(Y)^s. \end{aligned}$$

6 Siegel theta function

Let p be a polynomial on ${\mathbb {R}}^{(b^+, b^-)}$ which is homogeneous of degree $\kappa$ in the positive definite variables and of degree 0 in the negative definite variables. For an isometry $\nu : L \otimes {\mathbb {R}}\rightarrow {\mathbb {R}}^{(b^+, b^-)}$ we write $\nu ^+$ and $\nu ^-$ for the inverse image of ${\mathbb {R}}^{(b^+, 0)}$ and ${\mathbb {R}}^{(0, b^-)}$. For an element $\lambda \in L \otimes {\mathbb {R}}$ we write $\lambda _{\nu ^\pm }$ for the projection of $\lambda$ onto $\nu ^\pm$. The positive definite majorant associated to $\nu$ is then given by $q_\nu (\lambda ) = q(\lambda _{\nu ^+}) - q(\lambda _{\nu ^-})$. For $\gamma \in L' / L, \tau \in {\mathbb {H}}$ and an isometry $\nu : L \otimes {\mathbb {R}}\rightarrow {\mathbb {R}}^{(b^+, b^-)}$ we define the Siegel theta function

$$\begin{aligned} \theta _\gamma (\tau , \alpha , \beta , \nu , p)&:= \sum _{\lambda \in \gamma + L} \exp \left( \frac{\Delta }{8 \pi v}\right) (p)(\nu (\lambda + \beta )) \\&\quad \times e(\tau q((\lambda + \beta )_{\nu ^+}) + {\overline{\tau }} q((\lambda + \beta )_{\nu ^-} - (\lambda + \beta / 2, \alpha ))), \end{aligned}$$

where $\Delta$ is the usual Laplace operator on ${\mathbb {R}}^{b^+ + b^-}$ and $\alpha , \beta \in L \otimes {\mathbb {R}}$. Moreover we define

$$\begin{aligned} \Theta _L(\tau , \alpha , \beta , \nu , p)&:= \sum _{\gamma \in L' / L} \theta _\gamma (\tau , \alpha , \beta , \nu , p) {\mathfrak {e}}_\gamma . \end{aligned}$$

For $\alpha = \beta = 0$ we write $\theta _\gamma (\tau , \nu , p) := \theta _\gamma (\tau , 0, 0, \nu , p)$ and $\Theta _L(\tau , \nu , p) := \Theta _L(\tau , 0, 0, \nu , p)$. Using Poisson summation one obtains

Theorem 6.1

[5, Theorem 4.1] For $(M, \phi ) \in {\text {Mp}}_2({\mathbb {Z}}), M = \left( {\begin{matrix}a &{} b \\ c &{} d\end{matrix}}\right)$ we have

$$\begin{aligned} \Theta _L(M\tau , a \alpha + b \beta , c \alpha + d \beta , \nu , p) = \phi (\tau )^{b^+ + 2 \kappa } \overline{\phi (\tau )}^{b^-} \rho _L(M, \phi ) \Theta _L(\tau , \alpha , \beta , \nu , p). \end{aligned}$$

In particular,  for $\alpha = \beta = 0,$ the theta function $\Theta _L(\tau , \nu , p)$ has weight $(\frac{b^+}{2} + \kappa , \frac{b^-}{2})$.

Borcherds shows in [5] that the theta function can be written as a Poincaré series. We will indicate the construction. Write ${\text {Iso}}_0(L)$ for the primitive isotropic elements of L and let $z \in {\text {Iso}}_0(L), z' \in L'$ with $(z, z') = 1$. Let $N_z$ be the level of z and define the lattice

$$\begin{aligned} K = L \cap z^\perp \cap z'^\perp . \end{aligned}$$

Then K has signature $(b^+ - 1, b^- - 1)$. For a vector $\lambda \in L \otimes {\mathbb {R}}$ we write $\lambda _K$ for its orthogonal projection to $K \otimes {\mathbb {R}}$, which is given by

$$\begin{aligned} \lambda _K = \lambda - (\lambda , z) z' + (\lambda , z)(z', z') z - (\lambda , z') z. \end{aligned}$$

Let $\zeta \in L$ such that $(z, \zeta ) = N_z$ and write

$$\begin{aligned} \zeta = \zeta _K + N_z z' + B z \end{aligned}$$

for some $B \in {\mathbb {Q}}$. Then we have

$$\begin{aligned} L = K \oplus {\mathbb {Z}}\zeta + {\mathbb {Z}}z. \end{aligned}$$

Consider the sublattice

$$\begin{aligned} L_0' = \{ \lambda \in L' \vert (\lambda , z) \equiv 0 \bmod {N_z} \} \subseteq L' \end{aligned}$$

and the projection

$$\begin{aligned} \pi : L_0' \rightarrow K', \lambda \mapsto \pi (\lambda ) = \lambda _K + \frac{(\lambda , z)}{N_z} \zeta _K. \end{aligned}$$

This projection induces a surjective map $L_0' / L \rightarrow K' / K$ which we also denote by $\pi$ and we have

$$\begin{aligned} L_0' / L = \{ \lambda \in L' / L \vert (\lambda , z) \equiv 0 \bmod {N_z} \}. \end{aligned}$$

For an isometry $\nu : L \otimes {\mathbb {R}}\rightarrow {\mathbb {R}}^{(b^+, b^-)}$ we write $\omega ^{\pm }$ for the orthogonal complement of $z_{\nu ^\pm }$ in $\nu ^\pm$. This yields a decomposition

$$\begin{aligned} L \otimes {\mathbb {R}}= \omega ^+ \oplus {\mathbb {R}}z_{\nu ^+} \oplus \omega ^- \oplus {\mathbb {R}}z_{\nu ^-} \end{aligned}$$

and for $\lambda \in L \otimes {\mathbb {R}}$ we write $\lambda _{\omega ^\pm }$ for the corresponding projections of $\lambda$ onto $\omega ^\pm$. Additionally the map

$$\begin{aligned} \omega : L \otimes {\mathbb {R}}\rightarrow {\mathbb {R}}^{(b^+, b^-)}, \lambda \mapsto \nu (\lambda _{\omega ^+} + \lambda _{\omega ^-}) \end{aligned}$$

can be seen to be an isometry $K \otimes {\mathbb {R}}\rightarrow {\mathbb {R}}^{(b^+ - 1, b^- - 1)}$ by restriction. For a polynomial p on ${\mathbb {R}}^{(b^+, b^-)}$ as above we now define the homogeneous polynomials $p_{\omega , h}$ of degree $\kappa - h$ in the positive definite variables by

$$\begin{aligned} p(\nu (\lambda )) = \sum _{h} (\lambda , z_{\nu ^+})^{h} p_{\omega , h}(\omega (\lambda )). \end{aligned}$$

We have the following

Theorem 6.2

[5, Theorem 5.2] Let $\mu = -z' + \frac{z_{\nu ^+}}{2 z_{\nu ^+}^2} + \frac{z_{\nu ^-}}{2 z_{\nu ^-}^2} \in L \otimes {\mathbb {R}}$. Then

$$\begin{aligned} \theta _{\gamma + L}(\tau , \nu , p)&= \frac{1}{\sqrt{2 v z_{\nu ^+}^2}} \sum _{\begin{array}{c} c, d \in {\mathbb {Z}}\\ c \equiv (\gamma , z) \bmod N_z \end{array}} \sum _{h} (-2iv)^{-h} \\&\quad \times (c {\overline{\tau }} + d)^{h} e\left( -\frac{|c \tau + d |^2}{4 i v z_{\nu ^+}^2} - (\gamma , z') d + q(z') cd \right) \\&\quad \times \theta _{K + \pi (\gamma - cz')}(\tau , d \mu _K, -c \mu _K, \omega , p_{\omega , h}). \end{aligned}$$

We will need the following

Lemma 6.3

Let $\gamma \in K' / K$. Then

$$\begin{aligned}&\rho _L(M) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\gamma + \frac{mz}{N_z}}\left( -\frac{mn}{N_z}\right) \\&\quad = (\rho _K(M) {\mathfrak {e}}_\gamma ) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z} - nc z'}\left( -\frac{amn}{N_z} + q(z') acn^2 \right) . \end{aligned}$$

Proof

Write $\lambda = \gamma + \frac{mz}{N_z}$ We first consider the case $M = \left( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}}\right)$ with $c = 0$. Then by Shintani’s formula 2.3

$$\begin{aligned} \rho _L(M) {\mathfrak {e}}_{\lambda }&= \sqrt{i}^{(b^- - b^+)(1 - {\text {sgn}}(d))} \sum _{\beta \in L' / L} \delta _{\beta , a \lambda } {\mathfrak {e}}_\beta (ab q(\beta )) \\&= \sqrt{i}^{(b^- - b^+)(1 - {\text {sgn}}(d))} {\mathfrak {e}}_{a \lambda }(ab q(a\lambda )) \\&= \sqrt{i}^{((b^- - 1) - (b^+ - 1))(1 - {\text {sgn}}(d))} {\mathfrak {e}}_{a \gamma }(ab q(a \gamma )) {\mathfrak {e}}_{\frac{amz}{N_z}} = (\rho _K(M) {\mathfrak {e}}_\gamma ) {\mathfrak {e}}_{\frac{am z}{N_z}}. \end{aligned}$$

Multiplying by $e\left( -\frac{mn}{N_z}\right)$ and summing over $m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}$ yields the result. Let now $c \ne 0$. Again, Shintani’s formula yields

$$\begin{aligned} \rho _L(M) {\mathfrak {e}}_\lambda = C_L(c) \sum _{\beta \in L' / L} \sum _{r \in L / cL} {\mathfrak {e}}_\beta \left( \frac{a(\beta + r, \beta + r) - 2(\lambda , \beta + r) + d(\lambda , \lambda )}{2c}\right) , \end{aligned}$$

where

$$\begin{aligned} C_L(c) = \frac{\sqrt{i}^{(b^- - b^+) {\text {sgn}}(c)}}{|c |^{\frac{b^- + b^+}{2}} \sqrt{|L' / L |}} = \frac{\sqrt{i}^{((b^- - 1) - (b^+ - 1) {\text {sgn}}(c)}}{|c |^{\frac{(b^- - 1) + (b^+ - 1)}{2}} \sqrt{|K' / K |}} \cdot \frac{1}{c N_z} = \frac{C_K(c)}{c N_z}. \end{aligned}$$

Since $L = K \oplus {\mathbb {Z}}\zeta \oplus {\mathbb {Z}}z$ we can instead sum over $r + k \zeta + k' z, r \in K / cK, k, k' \in {\mathbb {Z}}/ c {\mathbb {Z}}$ to obtain

$$\begin{aligned} \rho _L(M) {\mathfrak {e}}_\lambda&= C_L(c) \sum _{\beta \in L' / L} \sum _{\begin{array}{c} r \in K / cK \\ k \in {\mathbb {Z}}/ c {\mathbb {Z}} \end{array}} e\left( \frac{a(\beta + r, \beta + r) - 2(\gamma , k \zeta ) + d (\gamma , \gamma )}{2c}\right) \\&\quad \times e\left( \frac{2a (\beta + r, k\zeta ) + a(k \zeta , k' \zeta ) - 2(\gamma , k \zeta ) - 2(\frac{mz}{N_z}, \beta + k \zeta )}{2c}\right) \\&\quad \times \sum _{k' \in {\mathbb {Z}}/ c {\mathbb {Z}}} e\left( \frac{2a (\beta , k' z) + 2a (k \zeta , k' \zeta )}{2c}\right) {\mathfrak {e}}_\beta \\&= C_L(c) \sum _{\beta \in L' / L} \sum _{\begin{array}{c} r \in K / cK \\ k \in {\mathbb {Z}}/ c {\mathbb {Z}} \end{array}} e\left( \frac{a(\beta + r, \beta + r) - 2(\gamma , k \zeta ) + d (\gamma , \gamma )}{2c}\right) \\&\quad \times e\left( \frac{2a (\beta + r, k\zeta ) + a(k \zeta , k' \zeta ) - 2(\gamma , k \zeta ) - 2(\frac{mz}{N_z}, \beta + k \zeta )}{2c}\right) \\&\quad \times \sum _{k' \in {\mathbb {Z}}/ c {\mathbb {Z}}} e \left( k' \frac{a (\beta + k \zeta , k' z)}{2c}\right) {\mathfrak {e}}_\beta . \end{aligned}$$

Now the last sum vanishes by orthogonality of characters unless

$$\begin{aligned} (\beta + k \zeta , z) = (\beta , z) + kN_z \equiv 0 \bmod {c}, \end{aligned}$$

in which case it sums to c. This yields

$$\begin{aligned} \rho _L(M) {\mathfrak {e}}_\lambda&= C_L(c) c \sum _{\beta \in L' / L} \sum _{\begin{array}{c} r \in K / cK \\ k \in {\mathbb {Z}}/ c {\mathbb {Z}}\\ c \mid kN_z + (\beta , z) \end{array}} e\left( \frac{a(\beta + r, \beta + r) - 2(\gamma , k \zeta ) + d (\gamma , \gamma )}{2c}\right) \\&\quad \times e\left( \frac{2a (\beta + r, k\zeta ) + a(k \zeta , k' \zeta ) - 2(\gamma , k \zeta ) - 2(\frac{mz}{N_z}, \beta + k \zeta )}{2c}\right) {\mathfrak {e}}_\beta . \end{aligned}$$

We now multiply this by $e\left( -\frac{mn}{N_z}\right)$ and sum over $m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}$ to obtain

$$\begin{aligned}&\rho _L(M) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\gamma + \frac{mz}{N_z}}\left( -\frac{mn}{N_z}\right) \\&\quad = C_L(c) c \sum _{\beta \in L' / L} \sum _{\begin{array}{c} r \in K / cK \\ k \in {\mathbb {Z}}/ c {\mathbb {Z}}\\ c \mid kN_z + (\beta , z) \end{array}} e\left( \frac{a(\beta + r, \beta + r) - 2(\gamma , k \zeta ) + d (\gamma , \gamma )}{2c}\right) \\&\qquad \times e\left( \frac{2a (\beta + r, k\zeta ) + a(k \zeta , k' \zeta ) - 2(\gamma , k \zeta )}{2c}\right) \\&\qquad \times \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} e\left( -\frac{(\frac{mz}{N_z}, \beta + k \zeta + nc z')}{c}\right) {\mathfrak {e}}_\beta . \end{aligned}$$

The latter sum again vanishes unless

$$\begin{aligned} (z, \beta + k \zeta + ncz') = (z, \beta + ncz') + kN_z \equiv 0 \bmod {cN_z}, \end{aligned}$$

in which case it is equal to $N_z$. In particular we have $(z, \beta + ncz') \equiv 0 \bmod {N_z}$ and thus $\beta + ncz' \in L_0' / L$. But every element in $L_0' / L$ can be written as $\alpha + \frac{mz}{N_z}$ for some $\alpha \in K' / K$ and $m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}$, which shows that $(\beta + ncz', \frac{z}{N_z}) = 0$. But this implies $k = 0$. Hence we obtain

$$\begin{aligned}&\rho _L(M) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\gamma + \frac{mz}{N_z}}\left( -\frac{mn}{N_z}\right) \\&\quad = C_K(c) \sum _{\beta \in K' / K} \sum _{r \in K / cK} {\mathfrak {e}}_\beta \left( \frac{a(\beta + r, \beta + r) - 2(\gamma , \beta + r) + d(\gamma , \gamma )}{2c}\right) \\&\qquad \times \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z} - ncz'}\left( -\frac{amn}{N_z} + q(z') acn^2\right) \\&\quad = (\rho _K(M) {\mathfrak {e}}_\gamma ) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z} - ncz'}\left( -\frac{amn}{N_z} + q(z') acn^2\right) , \end{aligned}$$

which shows the claim. $\square$

Theorem 6.4

We have

$$\begin{aligned} \Theta _L(\tau , \nu , p)&= \frac{1}{\sqrt{2 v z_{\nu ^+}^2}} \Theta _{K}(\tau , \omega , p_{\omega , 0}) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z}}\\&\quad + \frac{1}{\sqrt{2 z_{\nu ^+}^2}} \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{h} (-2i)^{-h} \sum _{n = 1}^\infty n^{h} \\&\quad \times \frac{(c {\overline{\tau }} + d)^{-\frac{b^-}{2} - \kappa ^-} (c \tau + d)^{ -\frac{b^+}{2} - \kappa ^+}}{{\text {Im}}(M\tau )^{h + \frac{1}{2}}} \exp \left( -\frac{\pi n^2}{2{\text {Im}}(M \tau ) z_{\nu ^+}^2}\right) \\&\quad \times \rho _L(M)^{-1} \left( \Theta _K(\tau , n \mu _K, 0, \omega , p_{\omega , h}) \sum _{m \in {\mathbb {Z}}/ N {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N}}\left( -\frac{mn}{N}\right) \right) . \end{aligned}$$

Proof

We have

$$\begin{aligned} \Theta _L(\tau , \nu , p)&= \frac{1}{\sqrt{2 v z_{\nu ^+}^2}} \sum _{c, d \in {\mathbb {Z}}} \sum _{\begin{array}{c} \gamma \in L' / L \\ c \equiv (\gamma , z) \bmod {N_z} \end{array}} \sum _{h} (-2iv)^{-h} \\&\quad \times (c {\overline{\tau }} + d)^{h} e\left( -\frac{|c \tau + d |^2}{4 i v z_{\nu ^+}^2} - (\gamma , z') d + q(z') cd \right) \\&\quad \times \theta _{K + \pi (\gamma - cz')}(\tau , d \mu _K, -c \mu _K, \omega , p_{\omega , h}) {\mathfrak {e}}_\gamma . \end{aligned}$$

We make the change $\gamma \mapsto \gamma + cz'$ and sum over coprime c, d to obtain

$$\begin{aligned}&\frac{1}{\sqrt{2 v z_{\nu ^+}^2}} \sum _{\gamma \in L_0' / L} \theta _{K + \pi (\gamma )}(\tau , \omega , p_{\omega , 0}) {\mathfrak {e}}_\gamma \\&\quad + \frac{1}{\sqrt{2 v z_{\nu ^+}^2}} \sum _{(c, d) = 1} \sum _{h} (-2iv)^{-h} \sum _{n = 1}^\infty n^{h} (c {\overline{\tau }} + d)^{h} e\left( -\frac{n^2 |c \tau + d |^2}{4 i v z_{\nu ^+}^2}\right) \\&\quad \times \sum _{\gamma \in L_0' / L} \theta _{K + \pi (\gamma )}(\tau , nd \mu _K, -nc \mu _K, \omega , p_{\omega , h}) {\mathfrak {e}}_{\gamma + ncz'}\left( -(\gamma , z') nd - q(z') n^2cd\right) . \end{aligned}$$

The elements $\gamma \in L_0' / L$ are represented by $\gamma + \frac{mz}{N_z}$ for $\gamma \in K' / K, m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}$. Hence we obtain using the transformation formula for the theta function

$$\begin{aligned}&\sum _{\gamma \in L_0' / L} \theta _{K + \pi (\gamma )}(\tau , nd \mu _K, -nc \mu _K, \omega , p_{\omega , h}) {\mathfrak {e}}_{\gamma + ncz'}\left( -(\gamma , z') nd - q(z') n^2cd\right) \\&\quad = \sum _{\gamma \in K' / K} \theta _{K + \gamma }(\tau , nd \mu _K, -nc \mu _K, \omega , p_{\omega , h}) {\mathfrak {e}}_\gamma \\&\qquad \times \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z} + ncz'}\left( -\frac{mnd}{N_z} - q(z') n^2cd\right) \\&\quad = \Theta _K(\tau , nd \mu _K, -nc \mu _K, \omega , p_{\omega , h}) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z} + ncz'}\left( -\frac{mnd}{N_z} - q(z') n^2cd\right) \\&\quad = (c \tau + d)^{-\frac{b^+ - 1}{2} - \kappa + h} (c {\overline{\tau }} + d)^{-\frac{b^- - 1}{2}} \\&\qquad \times \left( \rho _K^{-1}(M) \Theta _K(\tau , n \mu _K, 0, \omega , p_{\omega , h})\right) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z} + ncz'}\left( -\frac{mnd}{N_z} - q(z') n^2cd\right) \\&\quad = (c \tau + d)^{-\frac{b^+ - 1}{2} - \kappa + h} (c {\overline{\tau }} + d)^{-\frac{b^- - 1}{2}} \\&\qquad \times \rho _L(M)^{-1} \left( \Theta _K(\tau , n \mu _K, 0, \omega , p_{\omega , h}) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z}}\left( -\frac{mn}{N_z}\right) \right) , \end{aligned}$$

where $M = \left( {\begin{matrix}* &{} * \\ c &{} d\end{matrix}}\right) \in {\text {SL}}_2({\mathbb {Z}})$. This yields

$$\begin{aligned} \Theta _L(\tau , \nu , p)&= \frac{1}{\sqrt{2 v z_{\nu ^+}^2}} \Theta _{K}(\tau , \omega , p_{\omega , 0}) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z}}\\&\quad + \frac{1}{\sqrt{2 z_{\nu ^+}^2}} \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{h} (-2i)^{-h} \sum _{n = 1}^\infty n^{h} \\&\quad \times \frac{(c {\overline{\tau }} + d)^{-\frac{b^-}{2}} (c \tau + d)^{ -\frac{b^+}{2} - \kappa }}{{\text {Im}}(M\tau )^{h + \frac{1}{2}}} e\left( -\frac{n^2}{4 i {\text {Im}}(M \tau ) z_{\nu ^+}^2}\right) \\&\quad \times \rho _L(M)^{-1} \left( \Theta _K(\tau , n \mu _K, 0, \omega , p_{\omega , h}) \sum _{m \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{mz}{N_z}}\left( -\frac{mn}{N_z}\right) \right) . \end{aligned}$$

$\square$

Let now L be an even lattice of signature (2, l) with $l \equiv 0 \bmod {2}$. Let $\kappa = \frac{l}{2} - 1 + k$ and $p(x_1, x_2) = (x_1 + ix_2)^\kappa$. Recall the identification $\nu : {\mathcal {K}}^+ \rightarrow {\text {Gr}}(L), [Z_L] = [X_L + iY_L] \mapsto \nu (Z_L) = {\mathbb {R}}X_L + {\mathbb {R}}Y_L$, where ${\text {Gr}}(L)$ is the Grassmannian of positive definite 2-dimensional subspaces of $L \otimes {\mathbb {R}}$. For $Z_L = X_L + iY_L \in {\tilde{{\mathcal {K}}}}^+$ the map

$$\begin{aligned} a X_L + b Y_L \mapsto |Y_L |\begin{pmatrix}a \\ b\end{pmatrix} \end{aligned}$$

defines an isometry $\nu _{Z_L} : \nu (Z_L) \rightarrow {\mathbb {R}}^{(2, 0)}$ and by abuse of notation we will write $\nu _{Z_L}$ for every isometry which equals $\nu _{Z_L}$ on $\nu (Z_L)$. For $\lambda \in V({\mathbb {R}})$ we write $\lambda _{Z_L}$ and $\lambda _{Z_L^\perp }$ for the projection of $\lambda$ to $\nu (Z_L)$ and $\nu (Z_L)^\perp$. Then $q(\lambda ) = q(\lambda _{Z_L}) + q(\lambda _{Z_L^\perp })$ and we denote by $q_{Z_L}(\lambda ) = q(\lambda _{Z_L}) - q(\lambda _{Z_L^\perp })$ the positive definite majorant. Now, the map

$$\begin{aligned} \lambda \mapsto p(\nu _{Z_L}(\lambda )) = \frac{(\lambda , Z_L)^\kappa }{|Y_L |^\kappa } \end{aligned}$$

is well-defined, since p only depends on the positive definite variables. Thus we define

$$\begin{aligned} \Theta _L(\tau , Z)&:= \frac{i^\kappa v^{\frac{l}{2}}}{2 |Y_L |^\kappa } \Theta _L(\tau , \nu _{Z_L}, p) \\&= \frac{v^{\frac{l}{2}}}{2 (-2i)^{\kappa }} \sum _{\lambda \in L'} \frac{(\lambda , Z_L)^\kappa }{q(Y_L)^\kappa } {\mathfrak {e}}_\lambda (\tau q(\lambda _{Z_L}) + {\overline{\tau }} q(\lambda _{Z_L^\perp })) \end{aligned}$$

which is modular of weight $k = 1 - \frac{l}{2} + \kappa$ in $\tau = u + i v \in {\mathbb {H}}$ and weight $\kappa$ in $Z = X + iY \in {\mathbb {H}}_l$. Let $z \in {\text {Iso}}_0(L)$ of level $N_z$ and $z' \in L'$ with $(z, z') = 1$ write $K = L \cap z^\perp \cap z'^\perp$. Further, let $d \in {\text {Iso}}_0(K)$ of level $N_d$ and $d' \in K'$ with $(d, d') = 1$ and $D = K \cap d^\perp \cap d'^\perp$. Moreover we let ${\tilde{z}} = z' - q(z') z, {\tilde{d}} = d' - q(d') d$. We expand $\Theta _L(\tau , Z)$ with respect to $z, z', K$ to obtain

$$\begin{aligned} \Theta _L(\tau , Z) =&\frac{v^{\frac{l - 1}{2}}}{2 \sqrt{2} i^\kappa |Y |^{\kappa - 1}} \Theta _K(\tau , Y / |Y |, p_{Y, 0}) \sum _{m_z \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {e}}_{\frac{m_zz}{N_z}} \\&+ \frac{i^\kappa }{2\sqrt{2} |Y |^{\kappa - 1}} \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{h = 0}^\kappa \sum _{n = 1}^\infty n^h \\&\times \frac{(c \tau + d)^{-k}}{(-2i)^h {\text {Im}}(M\tau )^{h + \frac{1}{2} - \frac{l}{2}}} \exp \left( -\frac{q(Y) n^2 \pi }{{\text {Im}}(M\tau )}\right) \\&\times \rho _L^{-1}(M) \left( \Theta _K(M \tau , nX, 0, Y / |Y |, p_{Y, h}) \sum _{m_z \in {\mathbb {Z}}/ N_z Z} {\mathfrak {e}}_{\frac{m_zz}{N_z}}\left( -\frac{m_zn}{N_z} \right) \right) . \end{aligned}$$

and we expand $\Theta _K(\tau , Y / |Y |, p_{Y, 0})$ in the first summand with respect to $d, d', D$ to obtain

$$\begin{aligned} \Theta _L(\tau , Z)&= \delta _{\kappa , 0} \frac{v^{\frac{l}{2} - 1} q(Y)}{2 y_1} \Theta _D(\tau ) \sum _{\begin{array}{c} m_d \in {\mathbb {Z}}/ N_d {\mathbb {Z}}\\ m_z \in {\mathbb {Z}}/ N_z {\mathbb {Z}} \end{array}} {\mathfrak {e}}_{\frac{m_d d}{N_d} + \frac{m_z z}{N_z}} \\&+ \frac{q(Y)}{|Y |^{\kappa } 2^{\kappa + 1} y_1} \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{n = 1}^\infty n^\kappa \frac{(c \tau + d)^{-k}}{{\text {Im}}(M\tau )^{k}} \exp \left( -\frac{q(Y) \pi n^2}{{\text {Im}}(M \tau ) y_1^2}\right) \\&\quad \times \rho _L(M)^{-1} \left( \Theta _D(M \tau , n Y_D / y_1, 0) \sum _{\begin{array}{c} m_d \in {\mathbb {Z}}/ N_d {\mathbb {Z}}\\ m_z \in {\mathbb {Z}}/ N_z {\mathbb {Z}} \end{array}} {\mathfrak {e}}_{\frac{m_d d}{N_d} + \frac{m_z z}{N_z}}\left( -\frac{m_d n}{N_d}\right) \right) \\&+ \frac{i^\kappa }{2\sqrt{2} |Y |^{\kappa - 1}} \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{h = 0}^\kappa \sum _{n = 1}^\infty n^h \\&\quad \times \frac{(c \tau + d)^{-k}}{(-2i)^h {\text {Im}}(M\tau )^{h + \frac{1}{2} - \frac{l}{2}}} \exp \left( -\frac{q(Y) n^2 \pi }{{\text {Im}}(M\tau )}\right) \\&\quad \times \rho _L^{-1}(M) \left( \Theta _K(M \tau , nX, 0, Y / |Y |, p_{Y, h}) \sum _{m_z \in {\mathbb {Z}}/ N_z Z} {\mathfrak {e}}_{\frac{m_zz}{N_z}}\left( -\frac{m_zn}{N_z} \right) \right) . \end{aligned}$$

Observe that the second and third summand can be rewritten using the weight k slash operator $\vert _{k, L}$. We want to get bounds for $\Theta _L(\tau , Z)$ and $\Omega _k \overline{\Theta _L(\tau , Z)}$. According to [23, Proposition 2.5] we have

$$\begin{aligned} \Omega _\kappa \overline{\Theta _L(\tau , Z)} = \overline{\Delta _k \Theta _L(\tau , Z)}, \end{aligned}$$

where $\Delta _k = -v^2 \left( \frac{\partial ^2}{\partial u^2} + \frac{\partial ^2}{\partial v^2}\right) + i k v \left( \frac{\partial }{\partial u} + i \frac{\partial }{\partial v}\right)$ is the weight k Laplace operator in $\tau$. Let

$$\begin{aligned} f(\tau , Z, n) := \frac{\sqrt{q(Y) \pi } n}{v^\kappa y_1} \exp \left( - \frac{q(Y) \pi n^2}{v y_1^2}\right) \Theta _D(\tau , n Y_D /(|Y |y_1), 0) \end{aligned}$$

and

$$\begin{aligned} g(\tau , Z, n) := |Y |^{1 - \kappa } v^{-h} \exp \left( - \frac{q(Y) n^2 \pi }{v}\right) \Theta _K(\tau , nX, 0, Y / |Y |, p_{Y, h}) \end{aligned}$$

so that $\Theta _L(\tau , Z)$ is given by

$$\begin{aligned}&\delta _{\kappa , 0} \frac{q(Y)}{2 y_1} \Theta _D(\tau ) \sum _{\begin{array}{c} m_d \in {\mathbb {Z}}/ N_d {\mathbb {Z}}\\ m_z \in {\mathbb {Z}}/ N_z {\mathbb {Z}} \end{array}} {\mathfrak {e}}_{\frac{m_d d}{N_d} + \frac{m_z z}{N_z}} + \frac{|Y |^{1 - \kappa }}{2^{\kappa + 1} \sqrt{2}} \\&\quad \times \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{n = 1}^\infty n^{\kappa - 1} \left( f(\tau , Z, n) \sum _{\begin{array}{c} m_d \in {\mathbb {Z}}/ N_d {\mathbb {Z}}\\ m_z \in {\mathbb {Z}}/ N_z {\mathbb {Z}} \end{array}} {\mathfrak {e}}_{\frac{m_d d}{N_d} + \frac{m_z z}{N_z}}\left( -\frac{m_d n}{N_d}\right) \right) \bigg \vert _{k, L} M \\&\quad + \frac{i^\kappa }{2 \sqrt{2}} \sum _{M \in \Gamma _\infty \backslash \Gamma } \sum _{h = 0}^\kappa \sum _{n = 1}^\infty \frac{n^h}{(-2i)^h} \left( g(\tau , Z, n) \sum _{m_z \in {\mathbb {Z}}/ N_z Z} {\mathfrak {e}}_{\frac{m_zz}{N_z}}\left( -\frac{m_zn}{N_z} \right) \right) \bigg \vert _{k, L} M. \end{aligned}$$

Since the Laplace operator commutes with the slash operator, we only have to find bounds for $f, \Delta _k f$ and $g, \Delta _k g$. We will need the following elementary lemma.

Lemma 6.5

Let $a > 0, b, c \ge 0, n \in {\mathbb {N}}$.

• 1. The function $x^n \exp (- ax^2)$ has a maximum given by $\left( \frac{n}{2 a e}\right) ^{\frac{n}{2}}$.

• 2. The function $x^n \exp (- ax)$ has a maximum on $x \ge 0$ given by $\left( \frac{n}{ae}\right) ^n$.

• 3. The function $bx + \frac{c}{x}$ has a minimum on $x > 0$ given by $2 \sqrt{bc}$.

We start with a bound for

$$\begin{aligned} f(\tau , Z, n) = \frac{\sqrt{q(Y)} n}{v^\kappa y_1} \exp \left( - \frac{q(Y) \pi n^2}{v y_1^2}\right) \Theta _D(\tau , n Y_D /(|Y |y_1), 0). \end{aligned}$$

Observe that the absolute value of f and $\Delta _k f$ can be bounded by finite sums of the form

$$\begin{aligned} v^{-i} (y n)^j \exp \left( - \frac{(y n)^2}{v}\right) \sum _{\lambda \in D'} q(\lambda )^l \exp (- 2 \pi v q(\lambda )). \end{aligned}$$

We have

Lemma 6.6

Let

$$\begin{aligned} f_{i, j, l}(v, y, n) := v^{-i} (y n)^j \exp \left( - \frac{3 (y n)^2}{v}\right) \sum _{\lambda \in D'} q(\lambda )^l \exp (- 2 \pi v q(\lambda )). \end{aligned}$$

Assume that $v \in {\mathbb {R}}_{>0}, y> C > 0$ and $n \ge 1$. Then we have the bound

$$\begin{aligned} f_{i, j, l}(v, y, n) \ll v^{-i+j} \exp \left( - \frac{C^2 n^2}{3v}\right) , \end{aligned}$$

where the implied constant is independent of v, y, n. In particular,  we obtain with $y^2 = \frac{q(Y) \pi }{y_1^2}, C^2 = \frac{\pi }{t^2}$

$$\begin{aligned} f(\tau , Z, n) \ll v^{s_1} \exp \left( - \frac{\pi n^2}{3v t^2}\right) \end{aligned}$$

and

$$\begin{aligned} \Delta _k f(\tau , Z, n) \ll v^{s_2} \exp \left( - \frac{\pi n^2}{3v t^2}\right) \end{aligned}$$

for some $s_1, s_2 \in {\mathbb {R}}$.

Proof

We split up the exponential term and use Lemma 6.5 to obtain

$$\begin{aligned} (yn)^j \exp \left( -\frac{(yn)^2}{v}\right)&= (yn)^j \exp \left( -\frac{(yn)^2}{3v}\right) \exp \left( -\frac{(yn)^2}{3v}\right) \exp \left( -\frac{(yn)^2}{3v}\right) \\&\quad \ll v^j \exp \left( -\frac{(yn)^2}{3v}\right) \exp \left( -\frac{C^2}{3v}\right) . \end{aligned}$$

This yields

$$\begin{aligned}&f_{i, j, l}(v, y, n) \\&\quad \ll v^{-i + j} \exp \left( - \frac{(y n)^2}{3v}\right) \sum _{\lambda \in D'} q(\lambda )^l \exp \left( - 2 \pi v q(\lambda ) - \frac{C^2}{3v}\right) \\&\quad \ll v^{-i+j} \exp \left( - \frac{(y n)^2}{3v}\right) \sum _{\lambda \in D'} q(\lambda )^{l} \exp \left( - 2 C \sqrt{2 \pi q(\lambda )}\right) \ll \exp \left( - \frac{(yn)^2}{3v}\right) \end{aligned}$$

where we have used Lemma 6.5 again. $\square$

Next consider

$$\begin{aligned} g_h(\tau , Z, n) = |Y |^{1 - \kappa } v^{-h} \exp \left( - \frac{q(Y) n^2 \pi }{v}\right) \Theta _K(\tau , nX, 0, Y / |Y |, p_{Y, h}). \end{aligned}$$

The theta function is given by $v^{\frac{l - 1}{2}}$ times

$$\begin{aligned}&\sum _{\lambda \in K'} \exp \left( -\frac{\Delta }{8 \pi v}\right) (p_{Y, h})(\omega _Y(\lambda )) {\mathfrak {e}}_\lambda \left( u q(\lambda ) - (\lambda , nX)\right) \\&\quad \times \exp \left( - 2 \pi v\left( \frac{(\lambda , Y)^2}{Y^2} - q(\lambda )\right) \right) . \end{aligned}$$

The terms

$$\begin{aligned} \exp \left( - \frac{\Delta }{8 \pi v}\right) (p_{Y, h})(\omega _Y(\lambda )) \end{aligned}$$

are given by a finite sum of constants times terms of the form

$$\begin{aligned} v^{-j} |Y |^h (\lambda , Y / |Y |)^{\kappa - h - 2j}. \end{aligned}$$

Again, we see that the absolute value of $g_h$ and $\Delta _k g_h$ can be bounded by a finite sum of terms of the form

$$\begin{aligned}&|Y |^{h} v^{- i} \exp \left( -\frac{q(Y) n^2 \pi }{v}\right) \\&\quad \times \sum _{\lambda \in K'} \left( \frac{(\lambda , Y)^2}{Y^2}\right) ^j q(\lambda )^l \exp \left( - 2 \pi v\left( \frac{(\lambda , Y)^2}{Y^2} - q(\lambda )\right) \right) . \end{aligned}$$

Lemma 6.7

Let $g_{h, i, j, l}(v, Y, n)$ denote the function

$$\begin{aligned} |Y |^h v^{- i} \exp \left( -\frac{q(Y) n^2 \pi }{v}\right) \sum _{\lambda \in K'} \left( \frac{(\lambda , Y)^2}{Y^2}\right) ^j q(\lambda )^l \exp \left( - 2 \pi v\left( \frac{(\lambda , Y)^2}{Y^2} - q(\lambda )\right) \right) \end{aligned}$$

for $v \in {\mathbb {R}}_{>0}, Y \in {\mathcal {R}}_t$ and $n \ge 1$. Then

$$\begin{aligned} g_{h, i, j, l}(v, Y, n) \ll v^{\frac{h}{2} - i - j} \exp \left( - \frac{q(Y) n^2 \pi }{4v}\right) , \end{aligned}$$

where the implied constant is independent of v, Y, n. In particular,  we have

$$\begin{aligned} g_h(\tau , Z, n) \ll v^{s_1}\exp \left( - \frac{q(Y) n^2 \pi }{4v}\right) \end{aligned}$$

and

$$\begin{aligned} \Delta _k g_h(\tau , Z, n) \ll v^{s_2} \exp \left( - \frac{q(Y) n^2 \pi }{4v}\right) \end{aligned}$$

for some $s_1, s_2 \in {\mathbb {R}}$ independent of v, Y, n, h.

Proof

We have

$$\begin{aligned} \left( \frac{(\lambda , Y)^2}{Y^2}\right) ^{j} \le (y_2 / y_1 \lambda _1^2 + y_1 / y_2 \lambda _2^2 + q(\lambda _D))^{j} \end{aligned}$$

and by Lemma 3.6 the exponential term can be bounded by

$$\begin{aligned} \exp \left( - 2 \pi \varepsilon v\left( y_2 / y_1 \lambda _1^2 + y_1 / y_2 \lambda _2^2 + q(\lambda _D) \right) \right) \end{aligned}$$

on ${\mathcal {R}}_t$, which yields the bound

$$\begin{aligned}&g_{h, i, j, l}(v, Y, n) \\&\quad \le |Y |^h v^{- i} \exp \left( -\frac{q(Y) n^2 \pi }{v}\right) \sum _{\lambda \in K'} q(\lambda )^l \exp \left( - \pi \varepsilon v\left( y_2 / y_1 \lambda _1^2 + y_1 / y_2 \lambda _2^2 + q(\lambda _D) \right) \right) \\&\qquad \times (y_2 / y_1 \lambda _1^2 + y_1 / y_2 \lambda _2^2 + q(\lambda _D))^{j} \exp \left( - \pi \varepsilon v\left( y_2 / y_1 \lambda _1^2 + y_1 / y_2 \lambda _2^2 + q(\lambda _D) \right) \right) . \end{aligned}$$

Again we split up the exponential term Lemma 6.5 to obtain

$$\begin{aligned} |Y |^h \exp \left( -\frac{q(Y) n^2 \pi }{v}\right)&= |Y |^h \exp \left( -\frac{q(Y) n^2 \pi }{4v}\right) \exp \left( -\frac{3q(Y) n^2 \pi }{4v}\right) \\&\ll v^{\frac{h}{2}} \exp \left( -\frac{3q(Y) n^2 \pi }{4v}\right) . \end{aligned}$$

We obtain

$$\begin{aligned}&g_{h, i, j, l}(v, Y, n) \\&\quad \ll v^{- i - j + \frac{h}{2}} \exp \left( -\frac{3q(Y) n^2 \pi }{4v}\right) \\&\qquad \times \sum _{\lambda \in K'} q(\lambda )^l \exp \left( - \pi \varepsilon v\left( y_2 / y_1 \lambda _1^2 + y_1 / y_2 \lambda _2^2 + q(\lambda _D) \right) \right) , \end{aligned}$$

where we have used Lemma 6.5 again. Now use $y_1> t^{-1}, y_2 / y_1> t^{-2}, q(Y) > \frac{y_1 y_2}{1 + t^4}$ and $q(Y) > t^{-4}$ on every Siegel domain ${\mathcal {S}}_t$ to obtain

$$\begin{aligned}&g_{h, i, j, l}(v, Y, n) \\&\quad \ll v^{- i - j + \frac{h}{2}} \exp \left( -\frac{q(Y) n^2 \pi }{4 v}\right) \\&\qquad \times \sum _{\lambda \in K'} q(\lambda )^l \exp \left( - \pi \varepsilon v\left( \frac{y_2 \lambda _1^2}{y_1} + q(\lambda _D) \right) - \frac{q(Y) n^2 \pi }{4 v} - 2 \pi \varepsilon v \frac{y_1 \lambda _2^2}{y_2} - \frac{q(Y) n^2 \pi }{4 v} \right) \\&\quad \le v^{- i - j + \frac{h}{2}} \exp \left( -\frac{q(Y) n^2 \pi }{4 v}\right) \\&\qquad \times \sum _{\lambda \in K'} q(\lambda )^l \exp \left( - \pi \varepsilon v\left( \frac{\lambda _1^2}{t^2} + q(\lambda _D) \right) - \frac{\pi }{4 t^4 v} - 2 \pi \varepsilon v \frac{y_1 \lambda _2^2}{y_2} - \frac{y_1 y_2 \pi }{4 v (1 + t^4)} \right) \\&\quad \ll v^{- i - j + \frac{h}{2}} \exp \left( -\frac{q(Y) n^2 \pi }{4 v}\right) \sum _{\lambda \in K'} q(\lambda )^l \exp \left( - \pi \sqrt{\varepsilon } \left( |\lambda _1 |+ t |\lambda _D |+ \frac{\sqrt{2} |\lambda _2 |}{t \sqrt{1 + t^4}}\right) \right) \\&\quad \ll v^{- i - j + \frac{h}{2}} \exp \left( -\frac{q(Y) n^2 \pi }{4 v}\right) . \end{aligned}$$

$\square$

Proposition 6.8

Let $C \in {\mathbb {R}}_{>0}$. The theta function $\Theta _L(\tau , Z)$ is bounded by a constant times

$$\begin{aligned} v^s |Y |^{1 - \kappa } \left( 1 + \delta _{\kappa , 0} \frac{|Y |}{y_1}\right) \end{aligned}$$

for all $Z \in {\mathcal {S}}_t$ and $\tau \in {\mathbb {H}}$ with ${\text {Im}}(\tau ) > C$ and some $s \in {\mathbb {R}}$. The constant only depends on the constant C. Similarly,  the function $\Omega _\kappa \overline{\Theta _L} = \overline{\Delta _k \Theta _L}$ is bounded by

$$\begin{aligned} q(Y)^{\frac{1 - \kappa }{2}}. \end{aligned}$$

Proof

This is now a direct consequence of the previous two Lemmas together with the fact that the Laplace operator $\Delta _k$ commutes with the slash operator $\vert _{k, L}$. $\square$

Corollary 6.9

Both,  the theta function $\Theta _L$ and $\Omega _\kappa \overline{\Theta _L}$ are square-integrable for $l \ge 3$ and $\kappa = \frac{l}{2} - 1 + k > 0$.

Proof

The square is bounded by

$$\begin{aligned} q(Y)^{1 - \kappa } \end{aligned}$$

and hence we have to show that

$$\begin{aligned} \int _{{\mathcal {S}}_t} q(Y) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} < \infty . \end{aligned}$$

By Lemma 3.7 this is the case if $l \ge 3$. $\square$

7 Orthogonal Eisenstein series

Let L be an even lattice of signature (2, l) and let $\kappa \in {\mathbb {Z}}$. Let $\lambda \in {\text {Iso}}_0(L')$ and recall the tube domain representation ${\mathbb {H}}_l$ corresponding to a fixed 0-dimensional cusp z. Denote by $\Gamma (L)_\lambda \subseteq \Gamma (L)$ the stabilizer of $\lambda$ in $\Gamma (L)$ and write $\sigma _{\lambda } \in O^+(V)$ for an element satisfying $\sigma _{\lambda } N_\lambda \lambda = z$, where $N_\lambda$ is the order of $\lambda$ in $L' / L$. Then $q(Y)^s \vert _\kappa \sigma _{\lambda }$ has weight $\kappa$ with respect to $\Gamma (L)_\lambda$. Hence we define the non-holomorphic Eisenstein series

$$\begin{aligned} {\mathcal {E}}_{\kappa , \lambda }(Z, s)&:= \sum _{\sigma \in \Gamma (L)_\lambda \backslash \Gamma (L)} q(Y)^s \vert _\kappa \sigma _{\lambda }\sigma \\&= \sum _{\sigma \in \Gamma (L)_\lambda \backslash \Gamma (L)} j(\sigma _{\lambda } \sigma , Z)^{-\kappa } \left( \frac{q(Y)}{|j(\sigma _{\lambda } \sigma , Z) |^2}\right) ^s \\&= \sum _{\sigma \in \Gamma (L)_\lambda \backslash \Gamma (L)} (N_\lambda \lambda , \sigma (Z_L))^{-\kappa } \left( \frac{q(Y)}{|(N_\lambda \lambda , \sigma (Z_L)) |^2}\right) ^s \end{aligned}$$

for $Z \in {\mathbb {H}}_l$ and ${\text {Re}}(s) \gg 0$. The Eisenstein series does not depend on the choice of $\sigma _\lambda$. We have $\Gamma (L)_{-\lambda } = \Gamma (L)_\lambda$ and ${\mathcal {E}}_{\kappa , -\lambda }(Z, s) = (-1)^\kappa {\mathcal {E}}_{\kappa , \lambda }$. We have

$$\begin{aligned} \Omega _\kappa {\mathcal {E}}_{\kappa , \lambda }(Z, s) = s\left( s + \kappa - \frac{l}{2}\right) {\mathcal {E}}_{\kappa , \lambda }(Z, s), \end{aligned}$$

i.e. the harmonic points of ${\mathcal {E}}_{\kappa , \lambda }$ are $s = 0$ and $s = \frac{l}{2} - \kappa$. Consider the projection map

$$\begin{aligned} \pi _L : \Gamma (L) \backslash {\text {Iso}}_0(L') \rightarrow {\text {Iso}}(L' / L). \end{aligned}$$

For $\delta \in {\text {Iso}}(L' / L)$ we let

$$\begin{aligned} {\mathcal {G}}_{\kappa , \delta } := \sum _{\lambda \in \pi _L^{-1}(\delta )} {\mathcal {E}}_{\kappa , \lambda }, \end{aligned}$$

in particular, ${\mathcal {G}}_{\kappa , \delta } = 0$ if the preimage is empty and if $\delta = -\delta$ for odd $\kappa$.

For a vector-valued modular form $f : {\mathbb {H}}\rightarrow {\mathbb {C}}[L ' / L]$ of weight k consider the additive Borcherds lift is defined as

$$\begin{aligned} \Phi (Z, f) := \int _{{\text {SL}}_2({\mathbb {Z}}) \backslash {\mathbb {H}}}^{\text {reg}} \langle f(\tau ), \Theta _L(\tau , Z) \rangle v^k \frac{{\mathrm {d}}u {\mathrm {d}}v}{v^2}, \end{aligned}$$

if it exists. Here, the regularization is defined by the constant term at $t = 0$ of the Laurent expansion of

$$\begin{aligned} \lim _{T \rightarrow \infty } \int _{{\mathcal {F}}_T} \langle f(\tau ), \Theta _L(\tau , Z) \rangle v^{k-t} \frac{{\mathrm {d}}u {\mathrm {d}}v}{v^2}, \end{aligned}$$

where ${\mathcal {F}}_T$ is a truncated fundamental domain. For the lift of the vector-valued non-holomorphic Eisenstein series $E_{k, \beta }(\tau , s)$ we write $\Phi _{k, \beta }(Z, s)$. For $N \in {\mathbb {Z}}$ and $b \in {\mathbb {Z}}/ N {\mathbb {Z}}$ define the modified Riemann zeta functions

$$\begin{aligned} \zeta _+^b(s) := \sum _{\begin{array}{c} n = 1 \\ n = b \mod N \end{array}}^{\infty } n^{-s} \quad \text {and} \quad \zeta ^b(s) := \sum _{\begin{array}{c} n \in {\mathbb {Z}}\\ n = b \mod N \end{array}} n^{-s}. \end{aligned}$$

See [10, Section 4.4] for their basic properties and the analytic continuation. We have the following

Theorem 7.1

[14, Theorem 8.1] The theta lift is equal to

$$\begin{aligned} \Phi _{k, \beta }(Z, s)&= \frac{\Gamma (s + \kappa )}{(-2\pi i)^{\kappa }\pi ^{s}} \sum _{\lambda \in \Gamma (L) \backslash {\text {Iso}}_0(L')} N_\lambda ^{2s + \kappa } \zeta _+^{k_{\lambda \beta }}(2s + \kappa ) {\mathcal {E}}_{\kappa , \lambda }(Z, s) \\&= \frac{\Gamma (s + \kappa )}{(-2\pi i)^{\kappa }\pi ^{s}} \sum _{\delta \in {\text {Iso}}(L' / L)} N_\delta ^{2s + \kappa } \zeta _+^{k_{\delta \beta }}(2s + \kappa ) {\mathcal {G}}_{\kappa , \delta }(Z, s), \end{aligned}$$

where $k_{\delta \beta } \in {\mathbb {Z}}/ N_\delta {\mathbb {Z}}$ with $\beta = k_{\delta \beta } \delta$ (and the corresponding summands vanish if such a $k_{\delta \beta }$ does not exist),  $N_\lambda$ is the order of $\lambda$ and $N_\delta$ is the order of $\delta$ in $L' / L$.

We also have the following

Theorem 7.2

[14, Theorem 8.2] The theta lifts $\Phi _{k,\beta }(Z, s)$ generate the space of Eisenstein series ${\mathcal {G}}_{\kappa , \delta }(Z, s)$ for $\Gamma (L)$. In particular,  if $\pi _L$ is injective (resp. surjective),  then the theta lift is surjective (resp. injective) onto (resp. on) non-holomorphic Eisenstein series.

Hence, instead of evaluating the Eisenstein series ${\mathcal {E}}_{\kappa , \lambda }$ at $s = 0$, we will consider the theta lifts $\Phi _{k, \beta }$. Their Fourier expansion is given by

Theorem 7.3

[14, Theorem 8.6] Recall that $c_{k, \beta }(\gamma , n, s)$ are the Fourier coefficients of the vector-valued Eisenstein series $_{k, \beta }(\tau , s)$ (see Sect. 2). Let $z \in {\text {Iso}}_0(L)$ of level $N_z$ and let $z' \in L'$ with $(z, z') = 1$. The theta lift $\Phi _{k,\beta }(Z, s)$ has the Fourier expansion in the 0-dimensional cusp z given by

$$\begin{aligned}&\frac{i^\kappa }{2 \sqrt{2}|Y |^{\kappa -1}} \Phi ^K_{k,\beta }\left( \frac{Y}{|Y |}, s\right) + \sum _{\lambda \in K'} b_{k,\beta }(\lambda , Y, s) e(\lambda , X), \end{aligned}$$

where $\Phi ^K_{k,\beta }\left( \frac{Y}{|Y |}, s\right)$ is a theta lift corresponding to the sublattice K. The Fourier coefficient $b_{k,\beta }(0, Y, s)$ is given by

$$\begin{aligned}&\sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \bigg (\frac{\Gamma (s + \kappa ) N_z^{2s + \kappa }}{(-2 \pi i)^\kappa \pi ^s} q(Y)^s (\delta _{\beta , \frac{bz}{N_z}} + (-1)^\kappa \delta _{- \beta , \frac{bz}{N_z}}) \zeta _+^b(2s + \kappa ) \\&\qquad + \frac{\Gamma (1 - s - k + \kappa ) N_z^{2 - 2s - 2k + \kappa }}{(-2 \pi i)^\kappa \pi ^{1 - s - k}} q(Y)^{1 - s - k} c_{k,\beta }\left( \frac{bz}{N_z}, 0, s\right) \zeta _+^b(2 - 2s - 2k + \kappa )\bigg ). \end{aligned}$$

For $q(\lambda ) = 0, \lambda \ne 0$ the coefficient $b_{k,\beta }(\lambda , Y, s)$ is given by

$$\begin{aligned}&\frac{2 |(\lambda , Y) |^{\frac{1}{2}}}{2^\kappa } \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \bigg (\frac{q(Y)^s}{|(\lambda , Y) |^{s}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \\&\quad \times \sum _{n \mid \lambda } n^{2s - 1 + \kappa } (\delta _{\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}}) e\left( \frac{nb}{N_z}\right) \\&\quad \times \sum _{h = 0}^\infty \sum _{j = 0}^\infty \frac{(-1)^{j}}{(4 \pi |(\lambda , Y)|)^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa - h)!}{(\kappa -h-2j)!} \\&\quad \times \left( \frac{(\lambda , Y)}{|(\lambda , Y) |}\right) ^{\kappa -h} K_{s - \frac{1}{2} + \kappa - h - j}(2 \pi |(\lambda , Y) |)\\&\quad + \frac{q(Y)^{1 - s - k}}{|(\lambda , Y) |^{1 - s - k}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \\&\quad \times \sum _{n \mid \lambda } n^{1 - 2s - 2k + \kappa } c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, 0, s\right) e\left( \frac{nb}{N_z}\right) \\&\quad \times \sum _{h = 0}^\infty \sum _{j = 0}^\infty \frac{(-1)^{j}}{(4 \pi |(\lambda , Y)|)^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa - h)!}{(\kappa -h-2j)!} \\&\quad \times \left( \frac{(\lambda , Y)}{|(\lambda , Y) |}\right) ^{\kappa -h} K_{\frac{1}{2} - s - k + \kappa - h - j}(2 \pi |(\lambda , Y ) |)\bigg ). \end{aligned}$$

For $q(\lambda ) \ne 0$ the coefficient $b_{k,\beta }(\lambda , Y, s)$ is given by

$$\begin{aligned}&\frac{1}{\sqrt{2}|Y |^{\kappa -1}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{h = 0}^\infty (2i)^{-h} \sum _{j = 0}^\infty \frac{(-1)^j i^{h}}{(8 \pi )^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa -h)!}{(\kappa -h-2j)!} |Y |^h \left( \frac{(\lambda , Y)}{|Y |}\right) ^{\kappa -h - 2j} \\&\quad \times e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{n \mid \lambda } n^{2j - \kappa + 2h} e\left( \frac{nb}{N_z}\right) c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, \frac{q(\lambda )}{n^2}, s\right) \\&\quad \times \int _0^\infty \exp \left( -\frac{\pi n^2}{2vz_Z^2} - \frac{2 \pi v q_\omega (\lambda )}{n^2}\right) {\mathcal {W}}_s\left( 4 \pi \frac{q(\lambda )}{n^2}v\right) v^{-\frac{3}{2} + \kappa - h - j} {\mathrm {d}}v. \end{aligned}$$

8 Theta lifts at harmonic points

We will now consider the theta lifts at their harmonic points $s = 0$ and for $k = 0$ at $s = 1$. We set $\Phi _{\mathfrak {v}}(Z) = \Phi _{\mathfrak {v}}(Z, 0)$ and $b_{k,\beta }(\lambda , Y) = b_{k,\beta }(\lambda , Y, 0)$. We need the following

Lemma 8.1

We have

$$\begin{aligned}&\sum _{h = 0}^\infty \sum _{j = 0}^\infty \frac{(-1)^{j}}{(4 \pi |y |)^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa -h)!}{(\kappa -h-2j)!} \left( \frac{y}{|y |}\right) ^{\kappa -h} K_{-\frac{1}{2} + \kappa - h - j}\left( 2 \pi |y |\right) \\&\quad = 2^{\kappa - 1} y^{-\frac{1}{2}} e^{-2 \pi y}. \end{aligned}$$

if $y > 0$ and the left hand side vanishes otherwise.

Proof

This is part of the proof of [5, Theorem 14.3]. $\square$

Now we can calculate the Fourier coefficient for $q(\lambda ) \ge 0$ at $s = 0$.

Lemma 8.2

For $q(\lambda ) > 0$ the coefficient $b_{k,\beta }(\lambda , Y)$ is equal to

$$\begin{aligned} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, \frac{q(\lambda )}{n^2}\right) e(\lambda , iY), \end{aligned}$$

if $(\lambda , Y) > 0$ and $b_{k,\beta }(\lambda , Y, 0) = 0$ if $(\lambda , Y) < 0$. For $q(\lambda ) = 0$ with $(\lambda , Y) > 0$ the Fourier coefficient $b_{k,\beta }(\lambda , Y)$ is given by

$$\begin{aligned}&\sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \bigg (e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\quad \times (\delta _{\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}}) e(\lambda , iY) \\&\quad + \frac{2 q(Y)^{1 - k}}{|(\lambda , Y) |^{\frac{1}{2} - k}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{n \mid \lambda } n^{1 - 2k + \kappa } c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, 0\right) e\left( \frac{nb}{N_z}\right) \\&\quad \times \sum _{h = 0}^\infty \sum _{j = 0}^\infty \frac{(-1)^{j}}{(4 \pi |(\lambda , Y)|)^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa - h)!}{(\kappa -h-2j)!} \\&\quad \times \left( \frac{(\lambda , Y)}{|(\lambda , Y) |}\right) ^{\kappa -h} K_{\frac{1}{2} - k + \kappa - h - j}(2 \pi (\lambda , Y ))\bigg ). \end{aligned}$$

For $q(\lambda ) = 0$ with $(\lambda , Y) < 0$, $b_{k,\beta }(\lambda , Y)$ is given by

$$\begin{aligned}&\frac{2 q(Y)^{1 - k}}{|(\lambda , Y) |^{\frac{1}{2} - k}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{n \mid \lambda } n^{1 - 2k + \kappa } c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, 0\right) e\left( \frac{nb}{N_z}\right) \\&\quad \times \sum _{h = 0}^\infty \sum _{j = 0}^\infty \frac{(-1)^{j}}{(4 \pi |(\lambda , Y)|)^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa - h)!}{(\kappa -h-2j)!} \\&\quad \times \left( \frac{(\lambda , Y)}{|(\lambda , Y) |}\right) ^{\kappa -h} K_{\frac{1}{2} - k + \kappa - h - j}(2 \pi |(\lambda , Y ) |). \end{aligned}$$

Proof

Recall that for $q(\lambda ) > 0$ the coefficient $b_{k,\beta }(\lambda , Y, s)$ is given by

$$\begin{aligned}&\sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \bigg (\frac{1}{\sqrt{2}|Y |^{\kappa -1}} \sum _{h = 0}^\infty (2i)^{-h} \sum _{j = 0}^\infty \frac{(-1)^j i^{h}}{(8 \pi )^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa -h)! |Y |^h}{(\kappa -h-2j)!} \left( \frac{(\lambda , Y)}{|Y |}\right) ^{\kappa -h - 2j} \\&\quad \times e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{n \mid \lambda } n^{2j - \kappa + 2h} e\left( \frac{nb}{N_z}\right) c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, \frac{q(\lambda )}{n^2}, s\right) \\&\quad \times \int _0^\infty \exp \left( -\frac{\pi n^2}{2vz_Z^2} - \frac{2 \pi v q_\omega (\lambda )}{n^2}\right) {\mathcal {W}}_s\left( 4 \pi \frac{q(\lambda )}{n^2}v\right) v^{-\frac{3}{2} + \kappa - h - j} {\mathrm {d}}v \bigg ). \end{aligned}$$

We now plug in $s = 0$. We have

$$\begin{aligned} {\mathcal {W}}_0(v) = e^{-\frac{v}{2}} \cdot {\left\{ \begin{array}{ll} 1 &{} \text{ if } v > 0, \\ \Gamma (1 - k, -v) &{} \text{ if } v < 0. \end{array}\right. } \end{aligned}$$

Now observe that

$$\begin{aligned} q_\omega (\lambda ) + q(\lambda ) = 2 q(\lambda _{\omega ^+}) = \frac{(\lambda , Y)^2}{2 q(Y)} \end{aligned}$$

and use the formula [11, p. 313, 6.3(17)]

$$\begin{aligned} \int _{v = 0}^\infty \exp \left( - \alpha v - \frac{\beta }{v}\right) v^{\gamma } {\mathrm {d}}v = 2 \left( \frac{\beta }{\alpha }\right) ^{\frac{1}{2} (\gamma + 1)} K_{\gamma + 1}(2\sqrt{\alpha \beta }) \end{aligned}$$

with $\alpha = \frac{\pi (\lambda , Y)^2}{n^2 q(Y)}, \beta = \pi n^2 q(Y), \gamma = -\frac{3}{2} + \kappa - h - j$ to obtain

$$\begin{aligned}&\int _0^\infty \exp \left( -\frac{\pi n^2 q(Y)}{v} - \frac{2 \pi v q_\omega (\lambda )}{n^2}\right) {\mathcal {W}}_0\left( 4 \pi \frac{q(\lambda )}{n^2}v\right) v^{-\frac{3}{2} + \kappa - h - j} {\mathrm {d}}v \\&\quad = 2 \left( \frac{n^2 q(Y)}{|(\lambda , Y) |}\right) ^{-\frac{1}{2} + \kappa - h - j} K_{-\frac{1}{2} + \kappa - h - j}\left( 2 \pi |(\lambda , Y) |\right) . \end{aligned}$$

This yields for the Fourier coefficient $b_{k,\beta }(\lambda , Y)$

$$\begin{aligned}&2^{1 - \kappa } |(\lambda , Y) |^{\frac{1}{2}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \\&\quad \times \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, \frac{q(\lambda )}{n^2}\right) \\&\quad \times \sum _{h = 0}^\infty \sum _{j = 0}^\infty \frac{(-1)^{j}}{(4 \pi |(\lambda , Y) |)^j j!} \left( {\begin{array}{c}\kappa \\ h\end{array}}\right) \frac{(\kappa -h)!}{(\kappa -h-2j)!} \\&\quad \times \left( \frac{(\lambda , Y)}{|(\lambda , Y) |}\right) ^{\kappa -h} K_{-\frac{1}{2} + \kappa - h - j}\left( 2 \pi |(\lambda , Y) |\right) . \end{aligned}$$

Now use Lemma 8.1. For $q(\lambda ) = 0$ plug in $s = 0$ and use Lemma 8.1. $\square$

Definition 8.3

Define the holomorphic part $\Phi _{k,\beta }^+(Z)$ of the theta lift by

$$\begin{aligned}&\frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa }\sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \delta _{\beta , \frac{bz}{N_z}}\zeta ^b(\kappa ) + \sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda ) = 0 \\ (\lambda , Y)> 0 \end{array}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\quad \times (\delta _{\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}}) e\left( \lambda , Z - \frac{\zeta }{N_z}\right) \\&\quad + \sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda )> 0 \\ (\lambda , Y) > 0 \end{array}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\quad \times c_{k,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, \frac{q(\lambda )}{n^2}\right) e\left( \lambda , Z - \frac{\zeta }{N_z}\right) \end{aligned}$$

and the non-holomorphic part $\Phi _{k,\beta }^-(Z) = \Phi _{k,\beta }(Z) - \Phi _{k,\beta }^+(Z)$.

Remark 8.4

Assume that $E_{k, {\mathfrak {v}}}(\tau ) := E_{k, {\mathfrak {v}}}(\tau , 0)$ is holomorphic. Then $\Phi _{\mathfrak {v}}^-(Z)$ vanishes identically and hence $\Phi _{\mathfrak {v}}(Z) = \Phi _{\mathfrak {v}}^+(Z)$ is a holomorphic modular form. See also [5, Theorem 14.3]. Write ${\text {M}}_\kappa ^\Phi (\Gamma (L))$ for the space of holomorphic modular forms of weight $\kappa$ that are given as a theta lift $\Phi _{\mathfrak {v}}(Z)$ for some ${\mathfrak {v}}\in {\text {Iso}}({\mathbb {C}}[L' / L])$.

Definition 8.5

Define the holomorphic boundary part $\Phi _{k,\beta }^{\partial +}(Z)$ of the theta lift to be

$$\begin{aligned}&\frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \delta _{\beta , \frac{bz}{N_z}} \zeta ^b(\kappa ) \\&\quad + \sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda ) = 0 \\ (\lambda , Y) > 0 \end{array}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\quad \times (\delta _{\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}}) e\left( \lambda , Z - \frac{\zeta }{N_z}\right) . \end{aligned}$$

Proposition 8.6

We have

$$\begin{aligned} \Phi _{k,\beta }^{\partial +}(Z)&= \frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \delta _{\beta , \frac{bz}{N_z}} \zeta ^b(\kappa ) \\&\quad + \sum _{\begin{array}{c} \lambda \in K \\ q(\lambda ) = 0 \\ (\lambda , Y) > 0 \\ \lambda \text { primitive} \end{array}} \sum _{\begin{array}{c} b\in {\mathbb {Z}}/ N_z {\mathbb {Z}}\\ c \in {\mathbb {Z}}/ N_\lambda {\mathbb {Z}} \end{array}} \delta _{\beta , \frac{c \lambda }{N_\lambda } - \frac{c (\lambda , \zeta )}{N_\lambda N_z} z + \frac{bz}{N}} \sum _{m = 1}^\infty {\tilde{\sigma }}_{\kappa - 1}^{c, b}(m) e\left( \frac{m (\lambda , Z - \frac{\zeta _K}{N_z})}{N_\lambda }\right) . \end{aligned}$$

In particular,  for an isotropic plane $I \subseteq L \otimes {\mathbb {Q}}$ with $I = \langle z, d \rangle ,$ we have,  writing $\beta = \frac{c_\beta d}{N_d} - \frac{c_\beta (d, \zeta )z}{N_d N_z} + \frac{b_\beta z}{N_z},$ (if such a decomposition exists it is unique and if it does not exist then $\Phi _{k,\beta }^{\partial +}\vert _I$ vanishes identically)

$$\begin{aligned} \Phi _{k,\beta }^{\partial +} \vert _I (\tau )&= \frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \delta _{c_\beta } \zeta ^{b_\beta }(\kappa ) + \sum _{m = 1}^\infty {\tilde{\sigma }}_{\kappa - 1}^{c_\beta , b_\beta }(m) e\left( \frac{m (\tau - (d, \frac{\zeta _K}{N_z}))}{N_d}\right) , \end{aligned}$$

which is the holomorphic part of an Eisenstein series on the boundary component I (see [10, 15]). Here for $c \in {\mathbb {Z}}/ N_\lambda {\mathbb {Z}}, b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}$ we have the divisor sum

$$\begin{aligned} {\tilde{\sigma }}_{\kappa - 1}^{c, b}(m) = \sum _{\begin{array}{c} n \mid m \\ \frac{m}{n} \equiv c \bmod {N_\lambda } \end{array}} {\text {sgn}}(n) n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) , \end{aligned}$$

where the sum is over positive and negative divisors. Observe that the constant term only depends on the image of $\frac{z}{N_z}$ in $L' / L$.

Proof

For $\lambda \in K$ primitive let $N_\lambda$ be its level. Then we can rewrite the second summand as

$$\begin{aligned}&\sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda ) = 0 \\ (\lambda , Y)> 0 \end{array}} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\qquad \times (\delta _{\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}}) e\left( \lambda , Z - \frac{\zeta }{N_z}\right) \\&\quad = \sum _{\begin{array}{c} \lambda \in K \\ q(\lambda ) = 0 \\ (\lambda , Y)> 0 \\ \lambda \text { primitive} \end{array}} \sum _{m = 1}^\infty \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid m} n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\qquad \times (\delta _{\beta , \frac{m \lambda }{n N_\lambda } - \frac{(m \lambda , \zeta )}{nN_\lambda N_z} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{m \lambda }{n N_\lambda } - \frac{(m \lambda , \zeta )}{nN_\lambda N_z} z + \frac{bz}{N_z}}) e\left( \frac{m(\lambda , Z - \frac{\zeta _K}{N_z})}{N_\lambda }\right) \\&\quad = \sum _{\begin{array}{c} \lambda \in K \\ q(\lambda ) = 0 \\ (\lambda , Y) > 0 \\ \lambda \text { primitive} \end{array}} \sum _{\begin{array}{c} b\in {\mathbb {Z}}/ N_z {\mathbb {Z}}\\ c \in {\mathbb {Z}}/ N_\lambda {\mathbb {Z}} \end{array}} (\delta _{\beta , \frac{c \lambda }{N_\lambda } - \frac{c (\lambda , \zeta )}{N_\lambda N} z + \frac{bz}{N_z}} + (-1)^\kappa \delta _{-\beta , \frac{c \lambda }{N_\lambda } - \frac{c (\lambda , \zeta )}{N_\lambda N_z} z + \frac{bz}{N_z}}) \\&\qquad \times \sum _{m = 1}^\infty \sum _{\begin{array}{c} n \mid m \\ \frac{m}{n} \equiv c \bmod {N_\lambda } \end{array}} n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) e\left( \frac{m(\lambda , Z - \frac{\zeta _K}{N_z})}{N_\lambda }\right) . \end{aligned}$$

Summing over positive and negative divisors in the divisor sum we can rewrite the holomorphic boundary part as

$$\begin{aligned} \Phi _{k,\beta }^{\partial +}(Z) =&\frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \delta _{\beta , \frac{bz}{N_z}} \zeta ^b(\kappa ) \\&+ \sum _{\begin{array}{c} \lambda \in K \\ q(\lambda ) = 0 \\ (\lambda , Y) > 0 \\ \lambda \text { primitive} \end{array}} \sum _{\begin{array}{c} b\in {\mathbb {Z}}/ N_z {\mathbb {Z}}\\ c \in {\mathbb {Z}}/ N_\lambda {\mathbb {Z}} \end{array}} \delta _{\beta , \frac{c \lambda }{N_\lambda } - \frac{c (\lambda , \zeta )}{N_\lambda N_z} z + \frac{bz}{N_z}} \sum _{m = 1}^\infty {\tilde{\sigma }}_{\kappa - 1}^{c, b}(m) e\left( \frac{m (\lambda , Z - \frac{\zeta _K}{N_z})}{N_\lambda }\right) . \end{aligned}$$

Let now $I \subseteq L \otimes {\mathbb {Q}}$ be an isotropic plane with $I = \langle z, d \rangle , d \in {\text {Iso}}_0(K)$ and consider the Siegel operator corresponding to this plane. Then

$$\begin{aligned} \Phi _{k,\beta }^{\partial +} \vert _I (\tau ) =&\frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \delta _{\beta , \frac{bz}{N_z}} \zeta ^b(\kappa ) \\&+ \sum _{\begin{array}{c} b\in {\mathbb {Z}}/ N_z {\mathbb {Z}}\\ c \in {\mathbb {Z}}/ N_d {\mathbb {Z}} \end{array}} \delta _{\beta , \frac{c d}{N_d} - \frac{c (d, \zeta )}{N_d N_z} z + \frac{bz}{N_z}} \sum _{m = 1}^\infty {\tilde{\sigma }}_{\kappa - 1}^{c, b}(m) e\left( \frac{m (\tau - (d, \frac{\zeta _K}{N_z}))}{N_d}\right) . \end{aligned}$$

Writing $\beta = \frac{c_\beta d}{N_d} - \frac{c_\beta (d, \zeta )z}{N_d N_z} + \frac{b_\beta z}{N_z}$ (if such a decomposition exists it is unique and if it does not exist then $\Phi _{k,\beta }^{\partial +}\vert _I$ vanishes identically), we obtain

$$\begin{aligned} \Phi _{k,\beta }^{\partial +} \vert _I (\tau )&= \frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa } \delta _{c_\beta } \zeta ^{b_\beta }(\kappa ) + \sum _{m = 1}^\infty {\tilde{\sigma }}_{\kappa - 1}^{c_\beta , b_\beta }(m) e\left( \frac{m (\tau - (d, \frac{\zeta _K}{N_z}))}{N_d}\right) , \end{aligned}$$

which shows the result. $\square$

Proposition 8.7

Assume that the projection map $\pi _L : \Gamma (L) \backslash {\text {Iso}}_0(L') \rightarrow {\text {Iso}}(L' / L)$ is surjective. Then the theta lift is injective.

Proof

Assume that there is some ${\mathfrak {v}}\in {\text {Iso}}({\mathbb {C}}[L' / L])$ with $\Phi _{\mathfrak {v}}(Z) = 0$. Then in particular, $\Phi _{\mathfrak {v}}^{\partial +}(Z) = 0$ for every 0-dimensional cusp z, i.e.

$$\begin{aligned} \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\mathfrak {v}}_{\frac{bz}{N_z}} \zeta ^b(\kappa ) = 0 \end{aligned}$$

for all 0-dimensional cusps z. We want to show that $E_{k, {\mathfrak {v}}}(\tau , 0) = 0$, which is equivalent to ${\mathfrak {v}}+ (-1)^\kappa {\mathfrak {v}}^* = 0$. Therefore, assume there is some $\delta \in {\text {Iso}}(L' / L)$ with ${\mathfrak {v}}_\delta \ne -(-1)^\kappa {\mathfrak {v}}_{-\delta }$. By surjectivity of $\pi$ there is a 0-dimensional cusp z corresponding to $\delta$. Choose such $\delta$ with minimal order. Then by assumption the value in the 0-dimensional cusp z is

$$\begin{aligned} \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } {\mathfrak {v}}_{b \delta } \zeta ^b(\kappa ) = 0. \end{aligned}$$

But of course this is also true for the 0-dimensional cusps corresponding to $c \delta$ for $c \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times$, i.e. we have

$$\begin{aligned} \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } {\mathfrak {v}}_{b c \delta } \zeta ^b(\kappa ) = \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } {\mathfrak {v}}_{b \delta } \zeta ^{bc^*}(\kappa ) = 0. \end{aligned}$$

Rewrite this using $\zeta ^{bc^*}(\kappa ) = \zeta _+^{bc^*}(\kappa ) + (-1)^\kappa \zeta _+^{-bc^*}(\kappa )$ to obtain

$$\begin{aligned} \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } ({\mathfrak {v}}_{b \delta } + (-1)^\kappa {\mathfrak {v}}_{-b \delta }) \zeta _+^{bc^*}(\kappa ) = 0. \end{aligned}$$

For a character $\chi : ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times \rightarrow {\mathbb {C}}^\times$ consider now

$$\begin{aligned} 0&= \sum _{c \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } \chi (c) \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } ({\mathfrak {v}}_{b \delta } + (-1)^\kappa {\mathfrak {v}}_{-b \delta }) \zeta _+^{bc^*}(\kappa ) \\&= \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } \chi (b) ({\mathfrak {v}}_{b \delta } + (-1)^\kappa {\mathfrak {v}}_{-b \delta }) \sum _{c \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } \chi (c) \zeta _+^{c^*}(\kappa ) \\&= L({\overline{\chi }}, \kappa ) \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } \chi (b) ({\mathfrak {v}}_{b \delta } + (-1)^\kappa {\mathfrak {v}}_{-b \delta }). \end{aligned}$$

Since $L({\overline{\chi }}, \kappa ) \ne 0$ we have

$$\begin{aligned} \sum _{b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times } \chi (b) ({\mathfrak {v}}_{b \delta } + (-1)^\kappa {\mathfrak {v}}_{-b \delta }) = 0 \end{aligned}$$

for all Dirichlet characters $\chi$. But that means ${\mathfrak {v}}_{b \delta } + (-1)^\kappa {\mathfrak {v}}_{-b\delta } = 0$ for all $b \in ({\mathbb {Z}}/ N_z {\mathbb {Z}})^\times$ contradicting the assumption ${\mathfrak {v}}_{\delta } \ne -(-1)^\kappa {\mathfrak {v}}_{-\delta }$. $\square$

Of course, if $\pi$ is not surjective, then the theta lift is not injective, i.e. the converse of the previous theorem is also true.

Proposition 8.8

For $\kappa$ even and every $\delta \in L' / L$ isotropic there is a theta lift $\Phi _{k, {\mathfrak {v}}}$ for some ${\mathfrak {v}}\in {\text {Iso}}({\mathbb {C}}[L' / L])$ such that the holomorphic part vanishes in all 0-dimensional cusps except for the 0-dimensional cusps corresponding to $\pm \delta$. For $\kappa$ odd this is true if $\delta \ne - \delta$. Moreover,  in this case we have $\Phi _{k, {\mathfrak {v}}} = {\mathcal {G}}_{\kappa , \delta }$.

Proof

If $\delta$ generates a maximal cyclic isotropic subgroup, then the holomorphic part of $\Phi _{c\delta }, c \in ({\mathbb {Z}}/ N_\delta {\mathbb {Z}})^\times$ vanishes in every 0-dimensional cusp except for the 0-dimensional cusps corresponding to the generators of $\langle \delta \rangle$. In the 0-dimensional cusps corresponding to $b \delta$ for $b \in ({\mathbb {Z}}/ N_\delta {\mathbb {Z}})^\times$ the value is given by

$$\begin{aligned} \frac{\Gamma (\kappa ) N_\delta ^\kappa }{(-2 \pi i)^\kappa } \zeta ^{c b^*}(\kappa ). \end{aligned}$$

Now consider the linear combination

$$\begin{aligned} \frac{(-2 \pi i)^\kappa }{2 \varphi (N_\beta ) \Gamma (\kappa ) N_\delta ^\kappa }\sum _{\chi } \frac{1}{L(\chi , \kappa )} \sum _{c \in ({\mathbb {Z}}/ N_\delta {\mathbb {Z}})^\times } \chi (c) \Phi _{c \delta }, \end{aligned}$$

whose value in the 0-dimensional cusp $b \delta$ for $b \in ({\mathbb {Z}}/ N_\delta {\mathbb {Z}})^\times$ is given by

$$\begin{aligned}&\frac{1}{2 \varphi (N_\beta )} \sum _{\chi } \frac{1}{L(\chi , \kappa )} \sum _{c \in ({\mathbb {Z}}/ N_\delta {\mathbb {Z}})^\times } \chi (c) \zeta ^{cb^*}(\kappa ) \\&\quad = \frac{1}{2 \varphi (N_\beta )} \sum _{\chi } \frac{\chi (b)}{L(\chi , \kappa )} \sum _{c \in ({\mathbb {Z}}/ N_\delta {\mathbb {Z}})^\times } \chi (c) \zeta ^{c}(\kappa ) = \frac{1}{2 \varphi (N_\beta )} \sum _{\chi } \left( \chi (b) + (-1)^\kappa \chi (-b) \right) , \end{aligned}$$

i.e. the value in the 0-dimensional cusps vanishes except for the 0-dimensional cusps corresponding to $\delta$, where the value is 1. Moreover, using Theorem 7.1 shows that this linear combination is in fact given by ${\mathcal {G}}_{\kappa , \delta }$. Now do induction over the maximal length of chains of cyclic isotropic subgroups containing $\delta$. $\square$

This means in particular that for $F \in {\text {M}}_\kappa ^\pi (\Gamma (L))$ there is a theta lift $\Phi _{\mathfrak {v}}$ for some ${\mathfrak {v}}\in {\text {Iso}}({\mathbb {C}}[L' / L])$ whose holomorphic boundary part is given by the boundary part of F (observe that for $\kappa$ odd the values in 0-dimensional cusps corresponding to $\delta \in {\text {Iso}}(L' / L)$ with $\delta = - \delta$ must be zero).

Theorem 8.9

Let $k > 2$ and thus $\kappa > \frac{l}{2} + 1$. Then $\Phi _{k,\beta }(Z) = \Phi _{k,\beta }^+(Z)$ is a holomorphic modular form of weight $\kappa$ which is an Eisenstein series on the boundary. In particular we have ${\text {M}}_\kappa ^\pi (\Gamma (L)) = {\text {S}}_\kappa (\Gamma (L)) + {\text {M}}_\kappa ^\Phi (\Gamma (L))$ in this case.

Proof

Using that the coefficients $c_{k,\beta }(\gamma , n, 0)$ vanish for $n \le 0$ one obtains the result using Theorem 7.3 and Lemma 8.2 together with [5, Theorem 10.3] for the term $\Phi _{k, \beta }^K(Y / |Y |)$. Of course, this reproduces the result of [5, Theorem 14.3]. $\square$

We obtain the Fourier expansion of the Eisenstein series ${\mathcal {E}}_{\kappa , {\mathfrak {v}}}(Z)$ and that the Eisenstein series ${\mathcal {E}}_{\kappa , {\mathfrak {v}}}(Z) = {\mathcal {E}}_{\kappa , {\mathfrak {v}}}(Z, 0)$ for ${\mathfrak {v}}\in {\text {Iso}}({\mathbb {C}}[L' / L])$ are holomorphic if $\kappa > \frac{l}{2} + 1$. Moreover, if

$$\begin{aligned} \pi _L : \Gamma (L) \backslash {\text {Iso}}_0(L') \rightarrow {\text {Iso}}(L' / L) \end{aligned}$$

is injective, then we obtain all holomorphic orthogonal Eisenstein series as a lift of vector-valued Eisenstein series $E_{k, {\mathfrak {v}}}(\tau )$.

If $k = 0$, the Eisenstein series $E_{k, {\mathfrak {v}}}(\tau ) = E_{k, {\mathfrak {v}}}(\tau , 0)$ are usually not holomorphic in $\tau$. Hence we can not expect that ${\mathcal {E}}_{\kappa , {\mathfrak {v}}}(Z) = {\mathcal {E}}_{\kappa ,{\mathfrak {v}}}(Z, 0)$ is holomorphic. Since ${\text {res}}_{s = 1} E_{k, {\mathfrak {v}}}(\tau , s)$ is always an invariant vector we have

Theorem 8.10

Let $k = 0, \kappa = \frac{l}{2} - 1 > 1,$ i.e. $l > 4$. In the 0-dimensional cusp z we have the expansion

$$\begin{aligned} {\text {res}}_{s = 1} \Phi _{0,\beta }(Z, s)&= \Phi (Z, {\text {res}}_{s = 1} E_{0, \beta }(\cdot , s)) \\&= \frac{\Gamma (\kappa ) N_z^\kappa }{(-2 \pi i)^\kappa }\sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} {\text {res}}_{s = 1} c_{0,\beta }\left( \frac{bz}{N_z}, 0, s\right) \zeta ^b(\kappa ) \\&\quad + \sum _{\begin{array}{c} \lambda \in K' \\ q(\lambda ) = 0 \\ (\lambda , Y) > 0 \end{array}} e\left( - \frac{(\lambda , \zeta )}{N_z}\right) \sum _{b \in {\mathbb {Z}}/ N_z {\mathbb {Z}}} \sum _{n \mid \lambda } n^{\kappa - 1} e\left( \frac{nb}{N_z}\right) \\&\quad \times {\text {res}}_{s = 1} c_{0,\beta }\left( \frac{\lambda }{n} - \frac{(\lambda , \zeta )}{nN_z} z + \frac{bz}{N_z}, 0, s\right) e(\lambda , Z). \end{aligned}$$

In particular,  every invariant vector yields a holomorphic modular form of singular weight which is an Eisenstein series on the boundary. If $\kappa = 1, l = 4$ we obtain the additional summand ${\text {res}}_{s = 1} \Phi _{k,\beta }^K(Y / |Y |, s),$ which can be shown to be a constant.

Proof

One observes that the terms with $q(\lambda ) \ne 0$ are holomorphic in $s = 1$ and hence their residue vanishes (this follows from the corresponding result for vector-valued Eisenstein series). The calculation for the other Fourier coefficients is analogous to the case for $s = 0$. For the term ${\text {res}}_{s = 1} \Phi _{k, \beta }^K(Y / |Y |, s)$ see [5, Theorem 10.3]. See also [5, Theorem 14.3]. $\square$

The question if this yields all holomorphic modular forms of singular weight which are Eisenstein series on the boundary will be answered in the next section. We want to mention that we can also construct these in a different way. For an invariant vector ${\mathfrak {v}}\in {\mathbb {C}}[L' / L]$ we have

$$\begin{aligned} E_k(\tau , s) {\mathfrak {v}}= \sum _{\begin{array}{c} \beta \in L' / L \\ q(\beta ) = 0 \end{array}} {\mathfrak {v}}_\beta E_{k, \beta }(\tau , s), \end{aligned}$$

where $E_k(\tau , s)$ is the usual suitably normalized scalar-valued Eisenstein series for ${\text {SL}}_2({\mathbb {Z}})$. Now the left-hand side is holomorphic in $s = 0$ and equal to a multiple of ${\mathfrak {v}}$ and the lift of $E_k(\tau , s) {\mathfrak {v}}$ is holomorphic in $s = 0$ and yields a holomorphic modular form for $s = 0$ as in the case $k > 2$.

9 Lifting holomorphic orthogonal modular forms

Let L be an even lattice of signature $(2, l), l \ge 3$. For $z \in {\text {Iso}}_0(L)$ and $z' \in L'$ with $(z, z') = 1$ write $K = K_z = L \cap z^\perp \cap z'^\perp$. Let $b_1, \ldots , b_l$ be a basis of $K \otimes {\mathbb {R}}$ with $b_1 \perp \langle b_2, \ldots , b_l \rangle$ and $q(b_1) > 0$. If $Z = z_1 b_1 + z_2 b_2 + z_3 b_3 + \cdots + z_l b_l \in K \otimes {\mathbb {C}}$, we write $Z = (z_1, \ldots , z_l)$ and similarly $X = (x_1, \ldots , x_l), Y = (y_1, \ldots , y_l)$ if $Z = X + i Y$ with $X, Y \in K \otimes {\mathbb {R}}$. Denote by ${\mathbb {H}}_l = K \otimes {\mathbb {R}}+ i C$ the corresponding tube domain model, where

$$\begin{aligned} C = \{Y = (y_1, \ldots , y_l) \in K \otimes {\mathbb {R}}\mid y_1> 0, q(Y) > 0\}. \end{aligned}$$

For $\lambda \in {\text {Iso}}_0(L')$ let $\sigma _\lambda \in O^+(V)$ with $\sigma _\lambda \lambda = z$ and set $\lambda ' = \sigma _\lambda ^{-1} z'$. Define

$$\begin{aligned} K_\lambda = \lambda ^\perp \cap \lambda '^\perp \cap L = L \cap \sigma _\lambda ^{-1}(K \otimes {\mathbb {R}}). \end{aligned}$$

Then

$$\begin{aligned} \sigma _\lambda \Gamma (L)_\lambda \sigma _\lambda ^{-1} = \Gamma (\sigma _\lambda L)_z \supseteq (\sigma _\lambda K_\lambda ) \rtimes \Gamma (\sigma _\lambda K_\lambda ), \end{aligned}$$

where $\sigma _\lambda K_\lambda$ acts via translation and $\Gamma (\sigma _\lambda K_\lambda )$ via multiplication on ${\mathbb {H}}_l = K_z \otimes {\mathbb {R}}+ iC$. A fundamental domain is given by ${\mathcal {F}}= {\mathcal {F}}_1 + i {\mathcal {F}}_2$, where ${\mathcal {F}}_1$ is a fundamental domain of the action $\sigma _\lambda K_\lambda$ on $K_z \otimes {\mathbb {R}}$ and ${\mathcal {F}}_2$ is a fundamental domain of the action $\Gamma (\sigma _\lambda K_\lambda )$ on C. By abuse of notation we will write $\sigma _\lambda K_\lambda \backslash K_z \otimes {\mathbb {R}}+ i \Gamma (\sigma _\lambda K_\lambda ) \backslash C$. Moreover, recall the map

$$\begin{aligned} \pi _L : \Gamma (L) \backslash {\text {Iso}}_0(L') \rightarrow {\text {Iso}}(L' / L). \end{aligned}$$

Let $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ be a modular form of weight $\kappa$. We define its theta lift to be (if it exists)

$$\begin{aligned} \Phi ^*(\tau , F) := \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l}. \end{aligned}$$

Proposition 6.8 and Lemma 3.7 show that the theta lift exists for holomorphic modular forms of singular weight (in fact the lift exists for weight $\kappa < l - 1$ and for arbitrary weight if F is a cusp form). A straight forward calculation yields

Lemma 9.1

Let $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ be a holomorphic modular form of singular weight $\kappa = \frac{l}{2} - 1$ and ${\mathfrak {v}}\in {\text {Inv}}({\mathbb {C}}[L' / L])$ an invariant vector. Then

$$\begin{aligned} \langle \Phi ^*(\tau , F), {\mathfrak {v}}\rangle = \langle F, \Phi (Z, {\mathfrak {v}}) \rangle , \end{aligned}$$

where the left hand side denotes the Petersson inner product on invariant vectors and the right hand side denotes the Petersson inner product on holomorphic modular forms of singular weight. In particular,  the theta lifts are adjoint to eachother.

Lemma 9.2

Let $l \ge 3, \kappa = \frac{l}{2} - 1$. If F and $\Omega _\kappa F$ are square-integrable,  then the theta lift exists and we have $\Phi ^*(\tau , \Omega _\kappa F) = \Delta _0 \Phi ^*(\tau , F)$. In particular,  if F is holomorphic,  then $\Phi ^*(\tau , F)$ exists and is harmonic.

Proof

Let $\Gamma \subseteq \Gamma (L)$ be a finite index subgroup which acts freely. Then $\Gamma \backslash {\mathbb {H}}_l$ is a complete connected hermitian manifold. By Corollary 6.9, $\Theta _L$ and $\Omega _\kappa \Theta _L$ are square-integrable. Using the assumption on F and $\Omega _\kappa F$, we can apply Theorem 5.1 to square-integrable sections of the hermitian line bundle of modular forms of weight $\kappa$. This yields, using $\overline{\Omega _\kappa \overline{\Theta _L}} = \Delta _0 \Theta _L$ and Theorem 5.1,

$$\begin{aligned} \Delta _0 \Phi ^*(\tau , F)&= \Delta _0 \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= [\Gamma (L) : \Gamma ] \Delta _0 \int _{\Gamma \backslash {\mathbb {H}}_l} F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= [\Gamma (L) : \Gamma ] \int _{\Gamma \backslash {\mathbb {H}}_l} F(Z) \Delta _0 \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= [\Gamma (L) : \Gamma ] \int _{\Gamma \backslash {\mathbb {H}}_l} F(Z) \overline{\Omega _\kappa \overline{\Theta _L(\tau , Z)}} q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= [\Gamma (L) : \Gamma ] \int _{\Gamma \backslash {\mathbb {H}}_l} \Omega _\kappa F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= \int _{\Gamma (L) \backslash {\mathbb {H}}_l} \Omega _\kappa F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} = \Phi ^*(\tau , \Omega _\kappa F). \end{aligned}$$

The other assertions follow immediately. $\square$

Theorem 9.3

Let F be a holomorphic modular form of singular weight $\kappa = \frac{l}{2} - 1 > 0$. Then its theta lift

$$\begin{aligned} \Phi ^*(\tau , F) = \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \end{aligned}$$

is an invariant vector given by

$$\begin{aligned} \frac{\Gamma (l/2)}{2(2 \pi )^{l / 2}} \sum _{\gamma \in {\text {Iso}}(L' / L)} \sum _{\begin{array}{c} \delta \in {\text {Iso}}(L' / L) \\ \gamma = k_\delta \delta \end{array}} \zeta _+^{k_\delta }(l - \kappa ) \sum _{\lambda \in \pi _L^{-1}(\delta )} a_{F, \lambda }(0) C(\lambda ) {\mathfrak {e}}_\gamma , \end{aligned}$$

where

$$\begin{aligned} C(\lambda )&= {\text {vol}}(\sigma _\lambda K_\lambda ) [\Gamma (\sigma _\lambda L)_z : \sigma _\lambda K_\lambda \rtimes \Gamma (\sigma _\lambda K_\lambda )] C(\Gamma (\sigma _\lambda K_\lambda )) \\&= {\text {vol}}(K_\lambda ) [\Gamma (L)_\lambda : K_\lambda \rtimes \Gamma (K_\lambda )] C(\Gamma (K_\lambda )) \end{aligned}$$

for $\sigma _\lambda \in O^+(V)$ with $\sigma _\lambda \lambda = z$ and $C(\Gamma (K_\lambda ))$ is a positive constant.

Proof

First observe that the integral converges since holomorphic modular forms of singular weight are square integrable. We have

$$\begin{aligned} \Phi ^*(\tau , F)&= \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) \Theta _L(\tau , Z) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= \frac{v^{\frac{l}{2}}}{2} \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) \sum _{\lambda \in L'} \frac{(\lambda , Z_L)^{\kappa }}{q(Y)^{\kappa }} \exp (-2 \pi v q_{Z_L}(\lambda )) {\mathfrak {e}}_\lambda (u q(\lambda )) q(Y)^\kappa \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&= \frac{v^{\frac{l}{2}}}{2} \sum _{\lambda \in L'} \int _{\Gamma (L) \backslash {\mathbb {H}}_l} F(Z) (\lambda , Z_L)^{\kappa } \exp (-2 \pi v q_{Z_L}(\lambda )) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} {\mathfrak {e}}_\lambda (u q(\lambda )) \end{aligned}$$

and remark that the integral is bounded for $v \rightarrow \infty$ and hence the lift grows polynomially. Moreover, by Lemma 9.2 it is harmonic of weight 0 and thus its growth comes from the constant Fourier coefficient. The constant Fourier coefficients are then given by (the $\lambda = 0$ term vanishes since $\kappa > 0$)

$$\begin{aligned}&\frac{v^{\frac{l}{2}}}{2} \sum _{\begin{array}{c} \lambda \in \Gamma (L) \backslash L' \\ q(\lambda ) = 0 \end{array}} \int _{\Gamma (L)_\lambda \backslash {\mathbb {H}}_l} F(Z) (\lambda , Z_L)^{\kappa } \exp (-2 \pi v q_{Z_L}(\lambda ))\frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} {\mathfrak {e}}_\lambda \\&\quad = \frac{v^{\frac{l}{2}}}{2} \sum _{\lambda \in \Gamma (L) \backslash {\text {Iso}}_0(L')} \sum _{m = 1}^\infty m^\kappa \\&\qquad \times \int _{\Gamma (L)_\lambda \backslash {\mathbb {H}}_l} F(Z) (\lambda , Z_L)^{\kappa } \exp (-2 \pi v m^2 q_{Z_L}(\lambda )) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} {\mathfrak {e}}_{m\lambda }. \end{aligned}$$

As above let $\sigma _{\lambda } \in O^+(V)$ such that $\sigma _{\lambda } \lambda = z$ and write $\lambda ' = \sigma _{\lambda }^{-1}(z')$. Then we can rewrite the integral to (observe that $(z, Z_L) = 1$ and $q_{Z_L}(z) = 1 / q(Y)$)

$$\begin{aligned} \int _{\Gamma (\sigma _\lambda L)_z \backslash {\mathbb {H}}_l} (F \mid _\kappa \sigma _\lambda )(Z) \exp (-2 \pi v m^2 / q(Y)) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l}. \end{aligned}$$

and hence, using the Fourier expansion of $F \mid _k \sigma _\lambda$

$$\begin{aligned}&\sum _{\delta \in \sigma _\lambda K_\lambda '} a_{F, \lambda }(\delta ) \int _{\Gamma (\sigma _\lambda L)_z \backslash {\mathbb {H}}_l} e(\delta , Z) \exp (-2 \pi v m^2 / q(Y)) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&\quad = [\Gamma (L)_\lambda : K_\lambda \rtimes \Gamma (K_\lambda )] \sum _{\delta \in \sigma _\lambda K_\lambda '} a_{F, \lambda }(\delta ) \\&\qquad \times \int _{\sigma _\lambda K_\lambda \rtimes \Gamma (\sigma _\lambda K_\lambda ) \backslash {\mathbb {H}}_l} e(\delta , Z) \exp (-2 \pi v m^2 / q(Y)) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l} \\&\quad = [\Gamma (L)_\lambda : K_\lambda \rtimes \Gamma (K_\lambda )] \sum _{\delta \in \sigma _\lambda K_\lambda '} a_{F, \lambda }(\delta ) \\&\qquad \times \int _{\sigma _\lambda K_\lambda \backslash K_z \otimes {\mathbb {R}}} e(\delta , X) {\mathrm {d}}X \int _{\Gamma (\sigma _\lambda K_\lambda ) \backslash C} e(\delta , iY)\exp (-2 \pi v m^2 / q(Y)) \frac{ {\mathrm {d}}Y}{q(Y)^l} \\&\quad = a_{F, \lambda }(0) {\text {vol}}(K_\lambda ) [\Gamma (L)_\lambda : K_\lambda \rtimes \Gamma (K_\lambda )] \int _{\Gamma (\sigma _\lambda K_\lambda ) \backslash C} \exp (-2 \pi v m^2 / q(Y)) \frac{{\mathrm {d}}X {\mathrm {d}}Y}{q(Y)^l}. \end{aligned}$$

Consider the diffeomorphism

$$\begin{aligned} \varphi : [0, \infty ) \times {\mathbb {R}}^{l - 1} \rightarrow C, \quad (r, y_2, \ldots , y_l) \mapsto \sqrt{r} (\sqrt{1 - q(0, y_2, \ldots , y_l)}, y_2, \ldots , y_l) \end{aligned}$$

with inverse

$$\begin{aligned} \varphi ^{-1} : C \rightarrow [0, \infty ) \times {\mathbb {R}}^{l - 1}, \quad Y \mapsto (q(Y), y_2 / \sqrt{q(Y)}, \ldots , y_l / \sqrt{q(Y)}). \end{aligned}$$

Then we have for integrable $f : C \rightarrow {\mathbb {C}}$

$$\begin{aligned} \int _{C} f(Y) {\mathrm {d}}Y&= \int _0^\infty \int _{{\mathbb {R}}^{l-1}} f(\varphi (r, y_2, \ldots , y_l)) |\det (\varphi '(r, y_1, \ldots , y_l)) |{\mathrm {d}}y_2 \ldots {\mathrm {d}}y_l {\mathrm {d}}r \\&= \int _{{\mathbb {R}}^{l-1}} \int _0^\infty r^{\frac{l}{2}} f(\varphi (r, y_2, \ldots , y_l)) \frac{{\mathrm {d}}r}{r} |\det (\varphi '(1, y_1, \ldots , y_l)) |{\mathrm {d}}y_2 \ldots {\mathrm {d}}y_l. \end{aligned}$$

Here $f(Y) = g(q(Y))$ for $g(r) = \exp (2 \pi v n^2 / r) r^{-l}$ and thus

$$\begin{aligned} f(\varphi (r, y_2, \ldots , y_l)) = g(r) = \exp (2 \pi v n^2 / r) r^{-l}. \end{aligned}$$

Hence we obtain

$$\begin{aligned}&a_{F, \lambda }(0) {\text {vol}}(K_\lambda ) [\Gamma (L)_\lambda : K_\lambda \rtimes \Gamma (K_\lambda )]\\&\qquad \times \int _0^\infty \exp (-2 \pi v m^2 / r) r^{-\frac{l}{2}} \frac{{\mathrm {d}}r}{r} \int _{\Gamma (\sigma _\lambda K_\lambda ) \backslash {\mathbb {R}}^{l-1}} |\det (\varphi '(1, y_2, \ldots , y_l)) |{\mathrm {d}}y_2, \ldots , y_l \\&\quad = a_{F, \lambda }(0) C(\lambda ) (2 \pi v)^{- \frac{l}{2}} m^{-l} \Gamma (l/2). \end{aligned}$$

Hence the constant Fourier coefficient is given by

$$\begin{aligned}&\frac{\Gamma (l/2)}{2(2 \pi )^{l / 2}} \sum _{\lambda \in \Gamma (L) \backslash {\text {Iso}}_0(L')} a_{F, \lambda }(0) C(\lambda ) \sum _{m = 1}^\infty m^{\kappa - l} {\mathfrak {e}}_{m \lambda } \\&\quad = \frac{\Gamma (l/2)}{2(2 \pi )^{l / 2}} \sum _{\gamma \in {\text {Iso}}(L' / L)} \sum _{\begin{array}{c} \delta \in {\text {Iso}}(L' / L) \\ \gamma = k_\delta \delta \end{array}} \zeta _+^{k_\delta }(l - \kappa ) \sum _{\lambda \in \pi _L^{-1}(\delta )} a_{F, \lambda }(0) C(\lambda ) {\mathfrak {e}}_\gamma . \end{aligned}$$

In particular, this is independent of v. Hence $\Phi ^*(\tau , F)$ is bounded and since it is harmonic of weight 0 it is an invariant vector. $\square$

Remark 9.4

Observe that $\Phi ^*(\tau , F)$ only depends on the values in the 0-dimensional cusps. In particular it vanishes on functions that are zero in every 0-dimensional cusp.

Remark 9.5

Let $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ be a cusp form of arbitrary weight. Then $\Phi ^*(\tau , F)$ exists and one can show as above that it is harmonic. Moreover, the same calculation as in the previous proof shows that the constant Fourier coefficient vanishes. Hence, $\Phi ^*(\tau , F)$ decays exponentially and hence is square-integrable and harmonic of weight k. Thus, $\Phi ^*(\tau , F)$ is a holomorphic cusp form. This reproduces the result of [16].

Corollary 9.6

For $\delta \in L' / L$ isotropic let $a_{F, \delta }(0) := \sum _{\lambda \in \pi _L^{-1}(\delta )} a_{F, \lambda }(0) C(\lambda )$. Then $\Phi ^*(\tau , F)$ vanishes if and only if $a_{F, \delta }(0)$ vanishes for all isotropic $\delta \in L' / L$. In particular,  if $\pi _L$ is injective,  then the theta lift $\Phi ^*(\tau , F)$ vanishes if and only if F vanishes in every 0-dimensional cusp.

Proof

If $\Phi ^*(\tau , F)$ vanishes, then the coefficient of ${\mathfrak {e}}_\gamma$, given by

$$\begin{aligned} \sum _{\begin{array}{c} \delta \in L' / L \\ q(\delta ) = 0 \\ \gamma = k_\delta \delta \end{array}} \zeta _+^{k_\delta }(l - \kappa ) a_{F, \delta }(0), \end{aligned}$$

vanishes for all isotropic $\gamma \in L' / L$. In particular, if $\gamma \in L' / L$ is isotropic with maximal order such that $a_{F, \gamma }(0) \ne 0$, then for $k' \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times$ the coefficient of ${\mathfrak {e}}_{k' \gamma }$

$$\begin{aligned} \sum _{k \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \zeta _+^{k^*}(l - \kappa ) a_{F, k k'\gamma }(0) = \sum _{k \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \zeta _+^{k'k^*}(l - \kappa ) a_{F, k \gamma }(0) \end{aligned}$$

vanishes. Thus, for all Dirichlet characters $\chi : ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times \rightarrow {\mathbb {C}}$, the sum

$$\begin{aligned}&\sum _{k' \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \chi (k') \sum _{k \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \zeta _+^{k'k^*}(l - \kappa ) a_{F, k \gamma }(0) \\&\quad = \sum _{k' \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \chi (k')\zeta _+^{k'}(l - \kappa ) \sum _{k \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \chi (k) a_{F, k \gamma }(0) \\&\quad = L(l - \kappa , \chi ) \sum _{k \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \chi (k) a_{F, k \gamma }(0) \end{aligned}$$

vanishes. Now $L(l - \kappa , \chi ) \ne 0$ and hence

$$\begin{aligned} \sum _{k \in ({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times } \chi (k) a_{F, k \gamma }(0) = 0 \end{aligned}$$

for all Dirichlet characters $\chi$. But then we must have $a_{F, k\gamma }(0) = 0$ for all k, since Dirichlet characters form an orthogonal basis of $({\mathbb {Z}}/ N_\gamma {\mathbb {Z}})^\times$. $\square$

Corollary 9.7

The theta lift $\Phi$ surjects onto the space of holomorphic modular forms of singular weight which are Eisenstein series on the boundary and whose value in a 0-dimensional cusp only depends on its image in $L' / L,$ i.e. we have ${\text {M}}_\kappa ^\Phi (\Gamma (L)) = {\text {M}}_\kappa ^\pi (\Gamma (L))$.

Proof

Let $F : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ be a holomorphic modular form which is an Eisenstein series on the boundary whose value in a 0-dimensional cusp only depends on its image in $L' / L$. Since $\Phi$ and $\Phi ^*$ are adjoint to each other by Lemma 9.1, we can write $F = \Phi _{0, {\mathfrak {v}}}(Z) + G$ for an invariant vector ${\mathfrak {v}}$ and a holomorphic modular form $G : {\mathbb {H}}_l \rightarrow {\mathbb {C}}$ of singular weight with $\Phi ^*(\tau , G) = 0$. By Corollary 9.6 we have $a_{G, \delta }(0) = 0$ for all isotropic $\delta \in L' / L$. Moreover, the value of $G = F - \Phi _{0, {\mathfrak {v}}}(Z)$ in a 0-dimensional cusp only depends on its image in $L' / L$. Hence we have

$$\begin{aligned} 0 = a_{G, \delta }(0) = \sum _{\lambda \in \pi _L^{-1}(\delta )} a_{G, \lambda }(0) C(\lambda ) = a_{G, {\tilde{\lambda }}}(0) \sum _{\lambda \in \pi _L^{-1}(\delta )} C(\lambda ) \end{aligned}$$

for some ${\tilde{\lambda }} \in \pi _L^{-1}(\delta )$. But then $a_{G, {\tilde{\lambda }}}(0) = 0$ and hence G vanishes in every 0-dimensional cusp and thus is a cusp form on the boundary. But since F and $\Phi _{0, {\mathfrak {v}}}(Z)$ are Eisenstein series on the boundary, $G = F - \Phi _{0, {\mathfrak {v}}}(Z)$ must vanish on the boundary and hence G is a cusp form. Since we are in singular weight, G must vanish and thus $F = \Phi _{0, {\mathfrak {v}}}(Z)$. $\square$

Corollary 9.8

If $\pi _L$ is injective,  then the theta lift $\Phi$ surjects onto the space of holomorphic modular forms of singular weight which are Eisenstein series on the boundary,  i.e. we have ${\text {M}}_\kappa ^\Phi (\Gamma (L)) = M_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$.

Proof

Since $\pi _L$ is injective we have $M_\kappa ^\pi (\Gamma (L)) = M_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$. $\square$

Corollary 9.9

If L is a maximal lattice of Witt rank 2,  then the space ${\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$ is either 1-dimensional (if L is unimodular) or 0-dimensional (if L is not unimodular). Moreover,  if $\kappa = 2, 4, 6, 8, 10, 14,$ i.e. $l = 6, 10, 14, 18, 22, 30,$ then we have ${\text {M}}_\kappa (\Gamma (L)) = {\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$. In the other cases there could be additional holomorphic modular forms that are cusp forms on the boundary.

Proof

Let L be maximal of signature $(2, l), l \ge 6$. Then L splits two hyperbolic planes over ${\mathbb {Z}}$ and hence the map $\pi _L$ is bijective. Thus the space of holomorphic modular forms of singular weight which are linear combinations of Eisenstein series on the boundary has the same dimension as the space of invariant vectors in ${\mathbb {C}}[L' / L]$. The latter is either 0-dimensional (if L is not unimodular) or 1-dimensional (if L is unimodular). Since we are in singular weight, there are no cusp forms. Since the 1-dimensional boundary components of $\Gamma (L) \backslash {\mathbb {H}}_l$ are given by ${\text {SL}}_2({\mathbb {Z}}) \backslash {\mathbb {H}}$ and since there are no cusp forms of weight 2, 4, 6, 8, 10, 14 for ${\text {SL}}_2({\mathbb {Z}})$, we obtain ${\text {M}}_\kappa (\Gamma (L)) = {\text {M}}_\kappa ^{\partial {\text {Eis}}}(\Gamma (L))$ if $l = 6, 10, 14, 18, 22, 30$. $\square$