2.1 Tube domain model

Throughout this section, we fix a rank $1$ primitive isotropic sublattice I of L. The zero-dimensional cusp corresponding to I is the point ${{\mathbb P}}I_{{\mathbb {C}}}$ of Q. We abbreviate $I^{\perp }=I^{\perp }\cap L$ and write

$$ \begin{align*} L(I) = (I^{\perp} / I) \otimes I. \end{align*} $$

Twisting by I, not choosing its generator, will be rather essential. The quadratic form on $I^{\perp }/I$ and an isomorphism $I\simeq {\mathbb {Z}}$ define a hyperbolic quadratic form on $L(I)$ . This is independent of the choice of $I\simeq {\mathbb {Z}}$ . We denote by ${\mathcal {C}_{I}}$ the positive cone in $L(I)_{{\mathbb {R}}}$ , namely a chosen connected component of $\{ w \in L(I)_{{\mathbb {R}}} \: | \: (w, w)>0 \}$ .

We write ${\mathcal {D}(I)}=Q - Q\cap {{\mathbb P}}I_{{\mathbb {C}}}^{\perp }$ . Then ${\mathcal {D}}$ is contained in ${\mathcal {D}(I)}$ . Indeed, if $[\omega ]\in {\mathcal {D}}\cap {{\mathbb P}}I_{{\mathbb {C}}}^{\perp }$ , the positive-definite plane $\langle \mathrm {Re}(\omega ), \mathrm {Im}(\omega ) \rangle $ would be contained in $I_{{\mathbb {R}}}^{\perp }/I_{{\mathbb {R}}}$ , which contradicts the hyperbolicity of $I_{{\mathbb {R}}}^{\perp }/I_{{\mathbb {R}}}$ . The linear projection ${{\mathbb P}}L_{{\mathbb {C}}}\dashrightarrow {{\mathbb P}}(L/I)_{{\mathbb {C}}}$ from the point ${{\mathbb P}}I_{{\mathbb {C}}}\in Q$ defines an isomorphism

$$ \begin{align*} {\mathcal{D}(I)} \: \stackrel{\simeq}{\to} \: {{\mathbb P}}(L/I)_{{\mathbb{C}}} - {{\mathbb P}}(I^{\perp}/I)_{{\mathbb{C}}}. \end{align*} $$

If we choose a rank $1$ sublattice $I'$ of L with $(I, I')\not \equiv 0$ , this defines the base point ${{\mathbb P}}(\langle I, I' \rangle _{{\mathbb {C}}}/I_{{\mathbb {C}}})$ of the affine space ${{\mathbb P}}(L/I)_{{\mathbb {C}}} - {{\mathbb P}}(I^{\perp }/I)_{{\mathbb {C}}}$ , and hence an isomorphism

$$ \begin{align*} {{\mathbb P}}(L/I)_{{\mathbb{C}}} - {{\mathbb P}}(I^{\perp}/I)_{{\mathbb{C}}} \simeq (I^{\perp}/I)_{{\mathbb{C}}}\otimes (I')_{{\mathbb{C}}}^{\vee} \simeq L(I)_{{\mathbb{C}}}. \end{align*} $$

The image of ${\mathcal {D}}\subset {\mathcal {D}(I)}$ by this series of isomorphisms is the tube domain in $L(I)_{{\mathbb {C}}}$ defined by

$$ \begin{align*} \mathcal{D}_{I} = \{ \: Z\in L(I)_{{\mathbb{C}}} \: | \: \mathrm{Im}(Z)\in {\mathcal{C}_{I}} \: \}. \end{align*} $$

In this way, we obtain the tube domain realization

(2.1) $$ \begin{align} {\mathcal{D}} \stackrel{\simeq}{\to} \mathcal{D}_{I} \subset L(I)_{{\mathbb{C}}} \end{align} $$

depending on the choice of $I'$ .

If we change $I'$ , the base point is changed, and the tube domain realization (2.1) is shifted by the corresponding translation of $L(I)_{{\mathbb {C}}}$ . For a given $I'$ , we can always find a (unique) isotropic line $\ne I_{{\mathbb {Q}}}$ from the hyperbolic plane $\langle I, I'\rangle _{{\mathbb {Q}}}$ . (Explicitly, if we take vectors $l\in I_{{\mathbb {Q}}}, l'\in I^{\prime }_{{\mathbb {Q}}}$ with $(l, l')=1$ , the vector $l'-2^{-1}(l', l')l$ generates this isotropic line.) This means that we can replace the given $I'$ to be isotropic without changing the base point. When $I'$ is isotropic, the inverse of (2.1) is given by

(2.2) $$ \begin{align} \mathcal{D}_{I} \to {\mathcal{D}}, \qquad z\otimes l\mapsto {\mathbb{C}}(l'+z-2^{-1}(z, z)l), \end{align} $$

where $z\in (I^{\perp }/I)_{{\mathbb {C}}}$ , $l\in I$ , and $l'\in I^{\prime }_{{\mathbb {Q}}}$ with $(l, l')=1$ , and we identify $(I^{\perp }/I)_{{\mathbb {C}}}\simeq \langle I, I' \rangle _{{\mathbb {C}}}^{\perp }\subset L_{{\mathbb {C}}}$ in the right side.

2.2 Stabilizer over ${\mathbb {Q}}$

Let ${\Gamma (I)_{\mathbb {Q}}}$ be the stabilizer of I in $\mathrm {O}^{+}(L_{{\mathbb {Q}}})$ . Note that we are not considering the stabilizer of $I_{{\mathbb {Q}}}$ , but of I. This is not restrictive when restricting to subgroups of ${\mathrm {O}^{+}(L)}$ . We put

$$ \begin{align*} {U(I)_{\mathbb{Q}}} = {\operatorname{Ker}}({\Gamma(I)_{\mathbb{Q}}} \to \mathrm{ O}(I_{{\mathbb{Q}}}^{\perp}/I_{{\mathbb{Q}}})\times \mathrm{GL}(I)). \end{align*} $$

This is the unipotent radical of ${\Gamma (I)_{\mathbb {Q}}}$ and can be explicitly described as follows. For a vector $m\otimes l$ of $L(I)_{{\mathbb {Q}}}$ , the Eichler transvection $E_{m\otimes l}\in {\Gamma (I)_{\mathbb {Q}}}$ is defined by (cf. [Reference Gritsenko, Hulek and Sankaran9, Reference Scattone17])

$$ \begin{align*} E_{m\otimes l}(v) = v - (\tilde{m}, v)l + (l, v)\tilde{m} - \frac{1}{2}(m, m)(l, v)l, \qquad v\in L_{{\mathbb{Q}}}, \end{align*} $$

where $\tilde {m}\in I^{\perp }_{{\mathbb {Q}}}=I^{\perp }_{{\mathbb {Q}}}\cap L_{{\mathbb {Q}}}$ is an arbitrary lift of $m\in (I^{\perp }/I)_{{\mathbb {Q}}}$ . This does not depend on the choice of $\tilde {m}$ . In particular, $E_{m\otimes l}(v) = v - (m, v)l$ when $v\in I^{\perp }_{{\mathbb {Q}}}$ . We have $E_{w}\circ E_{w'}=E_{w+w'}$ for $w, w'\in L(I)_{{\mathbb {Q}}}$ . Then we have the canonical isomorphism

$$ \begin{align*} L(I)_{{\mathbb{Q}}} \to {U(I)_{\mathbb{Q}}}, \qquad m\otimes l \mapsto E_{m\otimes l}. \end{align*} $$

We identify ${U(I)_{\mathbb {Q}}}$ with $L(I)_{{\mathbb {Q}}}$ in this way. We also identify $\textrm {O}(I_{{\mathbb {Q}}}^{\perp }/I_{{\mathbb {Q}}})\times \mathrm {GL}(I)$ with $\textrm {O}(L(I)_{{\mathbb {Q}}})\times \mathrm {GL}(I)$ by the canonical twisted isomorphism

$$ \begin{align*} \textrm{O}(I_{{\mathbb{Q}}}^{\perp}/I_{{\mathbb{Q}}})\times \mathrm{GL}(I) \to \textrm{O}(L(I)_{{\mathbb{Q}}})\times \mathrm{GL}(I), \qquad (\gamma_{1}, \gamma_{2})\mapsto (\gamma_{1}\otimes \gamma_{2}, \gamma_{2}). \end{align*} $$

We thus have the canonical exact sequence

(2.3) $$ \begin{align} 0 \to L(I)_{{\mathbb{Q}}}\to {\Gamma(I)_{\mathbb{Q}}} \stackrel{\pi}{\to} \mathrm{ O}^{+}(L(I)_{{\mathbb{Q}}})\times \mathrm{GL}(I) \to 1. \end{align} $$

If we choose a lift $(I^{\perp }/I)_{{\mathbb {Q}}}\hookrightarrow I^{\perp }_{{\mathbb {Q}}}$ of $(I^{\perp }/I)_{{\mathbb {Q}}}$ , or equivalently, a rank $1$ sublattice $I'$ of L with $(I, I')\not \equiv 0$ , the exact sequence (2.3) splits:

(2.4) $$ \begin{align} {\Gamma(I)_{\mathbb{Q}}} \simeq (\mathrm{O}^{+}(L(I)_{{\mathbb{Q}}})\times \mathrm{GL}(I))\ltimes L(I)_{{\mathbb{Q}}}. \end{align} $$

Here the lifted group $\mathrm {O}^{+}(L(I)_{{\mathbb {Q}}})$ acts on the lifted component $(I^{\perp }/I)_{{\mathbb {Q}}}\subset L_{{\mathbb {Q}}}$ through the natural isomorphism $\mathrm {O}(L(I)_{{\mathbb {Q}}})\simeq \mathrm { O}((I^{\perp }/I)_{{\mathbb {Q}}})$ , and $\mathrm {GL}(I)$ corresponds to $\{ \pm \mathrm {id}_{L} \}$ . Since $\gamma \circ E_{w} \circ \gamma ^{-1} =E_{\gamma w}$ for $\gamma \in {\Gamma (I)_{\mathbb {Q}}}$ , the adjoint action of ${\Gamma (I)_{\mathbb {Q}}}$ on ${U(I)_{\mathbb {Q}}}$ coincides with the natural action of ${\Gamma (I)_{\mathbb {Q}}}$ on $L(I)_{{\mathbb {Q}}}$ . Therefore, in the induced action of ${\mathrm {O}^{+}}(L(I)_{{\mathbb {Q}}})\times \mathrm {GL}(I)$ on ${U(I)_{\mathbb {Q}}}$ , $\mathrm {GL}(I)=\{ \pm 1 \}$ acts trivially, and ${\mathrm {O}^{+}}(L(I)_{{\mathbb {Q}}})$ acts by its natural action on $L(I)_{{\mathbb {Q}}}$ .

We take the tube domain realization ${\mathcal {D}}\to {\mathcal {D}}_{I}$ associated with (the same) $I'$ . Then the action of ${\Gamma (I)_{\mathbb {Q}}}$ on ${\mathcal {D}}$ is translated to the action of the right side of (2.4) on ${\mathcal {D}}_{I}$ . This is described as follows.

2.3 Stabilizer over ${\mathbb {Z}}$

Now let ${\Gamma }$ be a subgroup of ${\mathrm {O}^{+}(L)}$ of finite index. We write

$$ \begin{align*} {\Gamma(I)_{\mathbb{Z}}} = {\Gamma(I)_{\mathbb{Q}}}\cap {\Gamma}, \quad {U(I)_{\mathbb{Z}}} = {U(I)_{\mathbb{Q}}}\cap {\Gamma}, \quad {\overline{\Gamma(I)}_{\mathbb{Z}}} = {\Gamma(I)_{\mathbb{Z}}}/{U(I)_{\mathbb{Z}}}. \end{align*} $$

Then ${U(I)_{\mathbb {Z}}}$ is a lattice on ${U(I)_{\mathbb {Q}}}$ . By definition, we have the exact sequence

(2.5) $$ \begin{align} 0 \to {U(I)_{\mathbb{Z}}} \to {\Gamma(I)_{\mathbb{Z}}} \to {\overline{\Gamma(I)}_{\mathbb{Z}}} \to 1. \end{align} $$

Although (2.3) splits, this does not mean that (2.5) splits. We write ${U(I)_{\mathbb {Q}/\mathbb {Z}}}={U(I)_{\mathbb {Q}}}/{U(I)_{\mathbb {Z}}}$ . This is the group of torsion points of the algebraic torus $T(I)=U(I)_{{\mathbb {C}}}/{U(I)_{\mathbb {Z}}}$ . We also put

$$ \begin{align*} {\overline{\Gamma(I)}_{\mathbb{Q}}} = \pi^{-1}(\mathrm{O}^{+}({U(I)_{\mathbb{Z}}})\times \mathrm{GL}(I)) / {U(I)_{\mathbb{Z}}}, \end{align*} $$

which makes sense because ${U(I)_{\mathbb {Z}}}$ is normal in $\pi ^{-1}(\mathrm {O}^{+}({U(I)_{\mathbb {Z}}})\times \mathrm {GL}(I))$ by definition. This group has the canonical exact sequence

(2.6) $$ \begin{align} 0 \to {U(I)_{\mathbb{Q}/\mathbb{Z}}} \to {\overline{\Gamma(I)}_{\mathbb{Q}}} \to \mathrm{ O}^{+}({U(I)_{\mathbb{Z}}})\times \mathrm{GL}(I) \to 1. \end{align} $$

Then ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ is a subgroup of ${\overline {\Gamma (I)}_{\mathbb {Q}}}$ naturally. We have

(2.7) $$ \begin{align} {\overline{\Gamma(I)}_{\mathbb{Z}}}\cap {U(I)_{\mathbb{Q}/\mathbb{Z}}} = \{ 0 \} \end{align} $$

by the definition ${U(I)_{\mathbb {Z}}}={\Gamma (I)_{\mathbb {Z}}}\cap {U(I)_{\mathbb {Q}}}$ of ${U(I)_{\mathbb {Z}}}$ .

We choose a rank $1$ sublattice $I'\subset L$ with $(I, I')\not \equiv 0$ and accordingly take a tube domain realization of ${\mathcal {D}}$ and a splitting of ${\Gamma (I)_{\mathbb {Q}}}$ . Dividing by ${U(I)_{\mathbb {Z}}}$ and writing ${\mathcal {X}(I)}={\mathcal {D}}/{U(I)_{\mathbb {Z}}}$ , we obtain isomorphisms

$$ \begin{align*} {\mathcal{X}(I)} \simeq \mathcal{D}_{I}/{U(I)_{\mathbb{Z}}} \quad \subset \quad {\mathcal{D}(I)}/{U(I)_{\mathbb{Z}}} \simeq T(I), \end{align*} $$

(2.8) $$ \begin{align} {\overline{\Gamma(I)}_{\mathbb{Q}}} \: \simeq \: (\mathrm{O}^{+}({U(I)_{\mathbb{Z}}})\times \mathrm{ GL}(I))\ltimes {U(I)_{\mathbb{Q}/\mathbb{Z}}}, \end{align} $$

both depending on the choice of $I'$ . By Lemma 2.1, the natural action of ${\overline {\Gamma (I)}_{\mathbb {Q}}}$ on ${\mathcal {X}(I)}$ is translated to the standard action of $(\mathrm {O}^{+}({U(I)_{\mathbb {Z}}})\times \mathrm {GL}(I))\ltimes {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ on $T(I)$ . Here $\mathrm {O}^{+}({U(I)_{\mathbb {Z}}})$ acts by torus automorphisms fixing the identity, $\mathrm {GL}(I)$ acts trivially, and ${U(I)_{\mathbb {Q}/\mathbb {Z}}}$ acts by translation.

By (2.7), the projection ${\overline {\Gamma (I)}_{\mathbb {Z}}}\to \mathrm {O}^{+}({U(I)_{\mathbb {Z}}})\times \mathrm {GL}(I)$ is injective. But this does not mean that ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ as a subgroup of ${\overline {\Gamma (I)}_{\mathbb {Q}}}$ is contained in the lifted subgroup $\mathrm {O}^{+}({U(I)_{\mathbb {Z}}})\times \mathrm {GL}(I)$ in (2.8). Thus, the action of ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ on ${\mathcal {X}(I)}$ may have translation component.

2.4 Partial toroidal compactification

We recall partial toroidal compactification of ${\mathcal {X}(I)}={\mathcal {D}}/{U(I)_{\mathbb {Z}}}$ following [Reference Ash, Mumford, Rapoport and Tai1]. We put a ${\mathbb {Q}}$ -structure on $U(I)_{{\mathbb {R}}}$ by ${U(I)_{\mathbb {Q}}}\simeq L(I)_{{\mathbb {Q}}}$ . We write ${\mathcal {C}_{I}^{+}}={\mathcal {C}_{I}}\cup \bigcup _{w}{\mathbb {R}}_{\geq 0}w$ , where w ranges over all isotropic vectors of $L(I)_{{\mathbb {Q}}}$ in the closure of ${\mathcal {C}_{I}}$ . A rational polyhedral cone decomposition (fan) $\Sigma =(\sigma _{\alpha })_{\alpha }$ in $U(I)_{{\mathbb {R}}}$ is called ${\Gamma (I)_{\mathbb {Z}}}$ -admissible [Reference Ash, Mumford, Rapoport and Tai1] if the support of $\Sigma $ is ${\mathcal {C}_{I}^{+}}$ , $\Sigma $ is preserved under the adjoint (= natural) action of ${\Gamma (I)_{\mathbb {Z}}}$ on $U(I)_{{\mathbb {R}}}=L(I)_{{\mathbb {R}}}$ , and there are only finitely many cones up to the action of ${\Gamma (I)_{\mathbb {Z}}}$ . Isotropic rays in $\Sigma $ correspond to rational isotropic lines in $L(I)_{{\mathbb {Q}}}$ (hence independent of $\Sigma $ ), which in turn correspond to rank $2$ primitive isotropic sublattices J of L containing I.

The fan $\Sigma $ defines a torus embedding $T(I)\hookrightarrow T(I)^{\Sigma }$ of the torus $T(I)=U(I)_{{\mathbb {C}}}/{U(I)_{\mathbb {Z}}}$ . Each ray $\sigma $ of $\Sigma $ defines a sub-torus embedding $T(I)\hookrightarrow T(I)^{\sigma }\subset T(I)^{\Sigma }$ , isomorphic to $({\mathbb {C}}^{\times })^{b} \hookrightarrow {\mathbb {C}}\times ({\mathbb {C}}^{\times })^{b-1}$ , whose unique boundary divisor is the quotient torus defined by the quotient lattice ${U(I)_{\mathbb {Z}}}/({\mathbb {R}}\sigma \cap {U(I)_{\mathbb {Z}}})$ . The character group of this boundary torus is $\sigma ^{\perp }\cap U(I)_{{\mathbb {Z}}}^{\vee }$ . Here we regard $U(I)_{{\mathbb {Z}}}^{\vee }$ as a lattice on ${U(I)_{\mathbb {Q}}}$ by the quadratic form on ${U(I)_{\mathbb {Q}}}=L(I)_{{\mathbb {Q}}}$ , which gives the pairing between ${U(I)_{\mathbb {Z}}}$ and $U(I)_{{\mathbb {Z}}}^{\vee }$ .

We take a tube domain realization of ${\mathcal {D}}$ by choosing $I'\subset L$ with $(I, I')\not \equiv 0$ . Then let ${\mathcal {X}(I)^{\Sigma }}$ be the interior of the closure of ${\mathcal {X}(I)}\simeq \mathcal {D}_{I}/{U(I)_{\mathbb {Z}}}$ in $T(I)^{\Sigma }$ . This embedding ${\mathcal {X}(I)}\hookrightarrow {\mathcal {X}(I)^{\Sigma }}$ is the partial toroidal compactification over I defined by the fan $\Sigma $ . It is ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ -equivariant, and does not depend on the choice of $I'$ . We can think of ${\mathcal {X}(I)^{\Sigma }}$ as giving a local chart for the boundary points of a full toroidal compactification lying over the I-cusp (see Section 7), like ${\mathcal {D}}$ gives a local chart for the interior points in ${\Gamma }\backslash {\mathcal {D}}$ .

3.1 Irregularity

We give several equivalent definitions of irregularity of a zero-dimensional cusp in the following form.

Proposition 3.1 The following conditions are equivalent.

1. (1) ${U(I)_{\mathbb {Z}}}\ne {U(I)_{\mathbb {Z}}^{\prime }}$ where ${U(I)_{\mathbb {Z}}^{\prime }}={U(I)_{\mathbb {Q}}}\cap {\langle \Gamma , -\mathrm {id} \rangle }$ .

2. (2) $-{\mathrm {id}}\not \in {\Gamma }$ and $-E_{w}\in {\Gamma (I)_{\mathbb {Z}}}$ for some $w\in L(I)_{{\mathbb {Q}}}$ .

3. (3) $-{\mathrm {id}}\not \in {\Gamma }$ and ${\overline {\Gamma (I)}_{\mathbb {Z}}}\to {\mathrm { O}^{+}}({U(I)_{\mathbb {Z}}})$ is not injective.

4. (4) ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ contains an element which acts by a nonzero translation on ${{\mathcal {X}(I)}={\mathcal {D}}/{U(I)_{\mathbb {Z}}}}$ .

When these hold, we have ${U(I)_{\mathbb {Z}}^{\prime }}/{U(I)_{\mathbb {Z}}} = \langle E_{w} \rangle \simeq {\mathbb {Z}}/2$ and

$$ \begin{align*} {\operatorname{Ker}}({\overline{\Gamma(I)}_{\mathbb{Z}}}\to {\mathrm{O}^{+}}({U(I)_{\mathbb{Z}}})) = \langle -E_{w} \rangle \simeq {\mathbb{Z}}/2, \end{align*} $$

and the translation in $(4)$ is given by $[w]\in {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ and is unique.

Proof $(1) \Rightarrow (2)$ : Since ${\Gamma }\ne {\langle \Gamma , -\mathrm {id} \rangle }$ , we have $-{\mathrm {id}}\not \in {\Gamma }$ . Let $E_{w}\in {U(I)_{\mathbb {Z}}^{\prime }}$ , but $E_{w}\not \in {U(I)_{\mathbb {Z}}}$ . Since ${\langle \Gamma , -\mathrm {id} \rangle }={\Gamma }\sqcup -{\Gamma }$ , we have $E_{w}\in -{\Gamma }$ , and so $-E_{w}\in {\Gamma }$ . Note that ${U(I)_{\mathbb {Z}}^{\prime }}/{U(I)_{\mathbb {Z}}}\simeq {\langle \Gamma , -\mathrm {id} \rangle }/{\Gamma }$ is of order $2$ , and so ${U(I)_{\mathbb {Z}}^{\prime }}/{U(I)_{\mathbb {Z}}} = \langle E_{w} \rangle $ .

$(2) \Rightarrow (1)$ : If $-E_{w}\in {\Gamma (I)_{\mathbb {Z}}}$ and $-{\mathrm {id}}\not \in {\Gamma }$ , then $E_{w}\not \in {U(I)_{\mathbb {Z}}}$ , but $E_{w}\in {U(I)_{\mathbb {Z}}^{\prime }}$ .

$(2) \Rightarrow (3)$ : Since $-E_{w}$ acts on $L(I)_{{\mathbb {Q}}}={U(I)_{\mathbb {Q}}}$ trivially, its image in ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ is contained in the kernel of ${\overline {\Gamma (I)}_{\mathbb {Z}}}\to {\mathrm {O}^{+}}({U(I)_{\mathbb {Z}}})$ .

$(3) \Rightarrow (2)$ : Recall from (2.6) that the kernel of ${\overline {\Gamma (I)}_{\mathbb {Q}}}\to {\mathrm {O}^{+}}({U(I)_{\mathbb {Z}}})$ is

$$ \begin{align*} {\mathrm{GL}}(I)\times {U(I)_{\mathbb{Q}/\mathbb{Z}}} = {U(I)_{\mathbb{Q}/\mathbb{Z}}} \sqcup (-{\mathrm{ id}})\cdot {U(I)_{\mathbb{Q}/\mathbb{Z}}}. \end{align*} $$

Since ${\overline {\Gamma (I)}_{\mathbb {Z}}}\cap {U(I)_{\mathbb {Q}/\mathbb {Z}}} = \{ 0 \}$ by (2.7), a nontrivial element of the kernel of ${\overline {\Gamma (I)}_{\mathbb {Z}}}\to {\mathrm { O}^{+}}({U(I)_{\mathbb {Z}}})$ must be contained in $(-{\mathrm {id}})\cdot {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ , hence, is the image of $-E_{w}$ for some $w\in L(I)_{{\mathbb {Q}}}$ . This also shows that the kernel is ${\mathbb {Z}}/2$ generated by $-E_{w}$ .

$(2) \Rightarrow (4)$ : The element $-E_{w}$ of ${\overline {\Gamma (I)}_{\mathbb {Z}}}$ acts on ${\mathcal {X}(I)}$ by the translation by $[w] \in {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ . Since $-{\mathrm {id}}\not \in {\Gamma }$ , we have $E_{w}\not \in {U(I)_{\mathbb {Z}}}$ . This means that $[w]\ne 0 \in {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ .

$(4) \Rightarrow (2), (3)$ : We choose a splitting of ${\overline {\Gamma (I)}_{\mathbb {Q}}}$ as in (2.8) and express an element of ${\overline {\Gamma (I)}_{\mathbb {Z}}}\subset {\overline {\Gamma (I)}_{\mathbb {Q}}}$ as $\gamma =(\gamma _{1}, \gamma _{2}, [w])$ accordingly, where $\gamma _{1}\in {\mathrm {O}^{+}}({U(I)_{\mathbb {Z}}})$ , $\gamma _{2}\in \mathrm {GL}(I)$ and $[w]\in {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ . If $\gamma $ acts on ${\mathcal {X}(I)}$ by a nonzero translation, we must have $\gamma _{1}={\mathrm {id}}_{L(I)}$ and the translation is given by $[w]\in {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ . Therefore, $\gamma $ is contained in the kernel of the projection to ${\mathrm {O}^{+}}({U(I)_{\mathbb {Z}}})$ . Since ${\overline {\Gamma (I)}_{\mathbb {Z}}}\cap {U(I)_{\mathbb {Q}/\mathbb {Z}}} = \{ 0 \}$ by (2.7) and $[w] \ne 0$ , we have $\gamma _{2}=-{\mathrm {id}}_{I}$ . Thus, $\gamma =-E_{w}$ . Finally, we have $-{\mathrm {id}}\not \in {\Gamma }$ , for otherwise $E_{w}=-\gamma $ would be contained in ${U(I)_{\mathbb {Z}}}$ and then $[w]=0\in {U(I)_{\mathbb {Q}/\mathbb {Z}}}$ .

The condition (2) is useful for explicit calculation (Section 4). We give some immediate consequences.

The lattice ${U(I)_{\mathbb {Z}}^{\prime }}={U(I)_{\mathbb {Q}}}\cap {\langle \Gamma , -\mathrm {id} \rangle }$ is the projection image of

$$ \begin{align*} U(I)_{{\mathbb{Z}}}^{\star} = (\{ \pm {\mathrm{id}} \} \cdot {U(I)_{\mathbb{Q}}}) \cap {\Gamma} = {\operatorname{Ker}}({\Gamma(I)_{\mathbb{Z}}}\to {\mathrm{O}^{+}}({U(I)_{\mathbb{Z}}})) \end{align*} $$

in ${U(I)_{\mathbb {Q}}}$ . Thus, ${U(I)_{\mathbb {Z}}^{\prime }}$ is the lattice of translation in the ${\Gamma (I)_{\mathbb {Z}}}$ -action on the tube domain model. We have

(3.1) $$ \begin{align} U(I)_{{\mathbb{Z}}}^{\star}/{U(I)_{\mathbb{Z}}} = \begin{cases} \langle -{\mathrm{id}} \rangle \simeq {\mathbb{Z}}/2, & -{\mathrm{id}}\in {\Gamma}, \\ \{ 1 \}, & -{\mathrm{id}}\not\in {\Gamma}, I \; \textrm{regular}, \\ \langle -E_{w} \rangle \simeq {\mathbb{Z}}/2, & I \; \textrm{irregular}. \end{cases} \end{align} $$

This gives yet another characterization of irregularity: $-{\mathrm {id}}\not \in {\Gamma }$ and ${U(I)_{\mathbb {Z}}}\ne U(I)_{{\mathbb {Z}}}^{\star }$ .

As we will explain in Section 8.1, ${U(I)_{\mathbb {Z}}}$ is the lattice for Fourier expansion of ${\Gamma }$ -modular forms around I. Thus, irregular zero-dimensional cusps are those cusps whose lattice of translation is larger than the lattice for Fourier expansion.

Remark 3.6 In the case $b=1$ , we have an accidental isomorphism $\mathrm {SO}^{+}(2, 1)\simeq \mathrm {PSL}(2, {\mathbb {R}})$ which induces an isomorphism between the type IV domain here and the upper half-plane. However, $\mathrm {O}^{+}(2, 1)=\mathrm {SO}^{+}(2, 1) \times \{ \pm \mathrm {id} \}$ and $\mathrm {SL}(2, {\mathbb {R}})$ are different double covers of $\mathrm {SO}^{+}(2, 1)\simeq \mathrm {PSL}(2, {\mathbb {R}})$ . Therefore, although we have the perfect analogy

$$ \begin{align*} {U(I)_{\mathbb{Z}}} \leftrightarrow \left\{ \begin{pmatrix} 1 & \ast \\ 0 & 1 \end{pmatrix} \in {\Gamma} \right\}, \end{align*} $$

$$ \begin{align*} {U(I)_{\mathbb{Z}}^{\prime}} \leftrightarrow \left\{ \begin{pmatrix} 1 & \ast \\ 0 & 1 \end{pmatrix} \in \langle {\Gamma}, -1 \rangle \right\}, \end{align*} $$

subgroups of $\mathrm {O}^{+}(2, 1)$ which have irregular cusps never correspond to subgroups of $\mathrm {SL}(2, {\mathbb {R}})$ which have irregular cusps (in the classical sense [Reference Diamond and Shurman3]): they live in different covers of $\mathrm {SO}^{+}(2, 1)\simeq \mathrm {PSL}(2, {\mathbb {R}})$ . Subgroups of $\mathrm {SO}^{+}(2, 1)\simeq \mathrm {PSL}(2, {\mathbb {R}})$ have no irregular cusp anyway.

3.2 Irregular boundary divisors

Let $\Sigma =(\sigma _{\alpha })$ be a ${\Gamma (I)_{\mathbb {Z}}}$ -admissible fan in $U(I)_{{\mathbb {R}}}$ , and let ${\mathcal {X}(I)}\hookrightarrow {\mathcal {X}(I)^{\Sigma }}$ be the partial compactification defined in Section 2.4. For a ray $\sigma $ in $\Sigma $ , we denote by $D(\sigma )\subset {\mathcal {X}(I)^{\Sigma }}$ the corresponding boundary divisor. When I is irregular, these boundary divisors are divided into two types as follows.

4.1 Stable orthogonal groups

Let L be an even lattice of signature $(2, b)$ . Let ${\widetilde {\mathrm {O}}^{+}(L)}<{\mathrm {O}^{+}(L)}$ be the kernel of the reduction map ${\mathrm { O}^{+}(L)}\to \mathrm {O}(A_{L})$ , called the stable orthogonal group or the discriminant kernel. The following was asserted in [Reference Ma13, p. 901]. We supplement the proof for the sake of completeness.

We obtain a first example of regular cusps.

For ${\Gamma }={\widetilde {\mathrm {O}}^{+}(L)}$ , we have the following constraints for existence of irregular cusp.

Proof Let $I={\mathbb {Z}}l$ and assume that $-E_{m\otimes l}\in {\widetilde {\mathrm {O}}^{+}(L)}$ for a vector m of $I^{\perp }_{{\mathbb {Q}}}/I_{{\mathbb {Q}}}$ . Then, for any $v\in I^{\perp }\cap L^{\vee }$ , the vector $-E_{m\otimes l}(v)=-v+(m, v)l$ must be contained in $v+L$ . This implies that

(4.1) $$ \begin{align} 2v \in (m, v)l + L. \end{align} $$

If we substitute $v=l/\mathrm {div}(I)$ , we find that $(2/\mathrm {div}(I))l \in L$ , and so $\mathrm {div}(I)=1$ or $2$ . The case $\mathrm {div}(I)=1$ is excluded by Lemma 4.2. Thus, $\mathrm {div}(I)=2$ .

If $[v]\in (I^{\perp }/I)^{\vee }$ denotes the image of $v\in I^{\perp }\cap L^{\vee }$ , then (4.1) means that $2[v]\in I^{\perp }/I$ . This shows that $A_{I^{\perp }/I}$ is 2-elementary. Finally, (4.1) implies that $(m, v)\in {\mathbb {Z}}$ for every $v\in I^{\perp }\cap L$ , and so $m\in (I^{\perp }/I)^{\vee }$ .

This determines the structure of $A_{L}$ when ${\widetilde {\mathrm {O}}^{+}(L)}$ has an irregular zero-dimensional cusp.

4.2 O’Grady 10

In this subsection, we let L be an even lattice of the form $L= M \oplus \langle -2d \rangle $ with M of signature $(2, b-1)$ . We consider the group

$$ \begin{align*} {\Gamma} = \{ \: \gamma \in {\mathrm{O}^{+}(L)} \: | \: \gamma|_{A_{M}}=\pm{\mathrm{id}}, \: \gamma|_{A_{\langle -2d \rangle}}={\mathrm{id}} \; \}. \end{align*} $$

Then ${\Gamma }$ contains ${\widetilde {\mathrm {O}}^{+}(L)}$ with index $\leq 2$ , with ${\Gamma }={\widetilde {\mathrm {O}}^{+}(L)}$ if and only if $A_{M}$ is 2-elementary. We have $-{\mathrm {id}}\in {\Gamma }$ if and only if $d=1$ . When $M=2U\oplus 2E_{8} \oplus A_{2}$ , ${\Gamma }$ is the modular group for the moduli space of polarized O’Grady $10$ manifolds with polarization of split type and degree $2d$ [Reference Gritsenko, Hulek and Sankaran6].

Proof Assume that $I={\mathbb {Z}}l$ is an irregular cusp for ${\Gamma }$ and $-E_{m\otimes l}\in {\Gamma }$ for $m\in I^{\perp }_{{\mathbb {Q}}}/I_{{\mathbb {Q}}}$ . The case $d=1$ is excluded by $-{\mathrm {id}}\not \in {\Gamma }$ . We shall show that $d\mid 4$ . Since $E_{2m\otimes l}=(-E_{m\otimes l})\circ (-E_{m\otimes l}) \in {\widetilde {\mathrm {O}}^{+}(L)}$ , we see that $2m\otimes l \in L(I)$ by Lemma 4.1. Hence, we can take a lift $\tilde {m}$ of m from $I^{\perp }\cap \frac {1}{2}L$ . Let v be a generator of $\langle -2d \rangle ^{\vee }$ . The vector

$$ \begin{align*} -E_{m\otimes l}(v) = -v + (\tilde{m}, v)l - (l, v)\tilde{m} + \frac{1}{2}(m, m)(l, v)l \end{align*} $$

must be contained in $v+L$ , and hence

(4.2) $$ \begin{align} 2v \in (\tilde{m}, v)l - (l, v)\tilde{m} + \frac{1}{2}(m, m)(l, v)l + L. \end{align} $$

Since $(\tilde {m}, v)\in \frac {1}{2}{\mathbb {Z}}$ , $(l, v)\in {\mathbb {Z}}$ and $(m, m)\in \frac {1}{2}{\mathbb {Z}}$ , we find that $2v\in \frac {1}{4}L$ . Hence, $2d\mid 8$ .

The case $d=2$ does not occur when $|A_{M}|$ is square-free, because then $A_{L}$ is anisotropic and hence $\mathrm {div}(I)=1$ . In Proposition 4.14, we show that ${\Gamma }$ indeed has an irregular cusp when $d=4$ and M contains U.

4.3 Generalized Kummer

In this subsection, we let $L=M\oplus \langle -2d \rangle $ be as in Section 4.2 and consider the group

$$ \begin{align*} {\Gamma} = \{ \: \gamma \in {\mathrm{O}^{+}(L)} \: | \: \gamma|_{A_{M}}=\det (\gamma) {\mathrm{id}}, \: \gamma|_{A_{\langle -2d \rangle}}={\mathrm{id}} \; \}. \end{align*} $$

This is an index $\leq 2$ subgroup of the group considered in Section 4.2. When $M=2U \oplus \langle -2t \rangle $ with $t\geq 3$ , ${\Gamma }$ is the modular group for the moduli space of polarized deformation generalized Kummer varieties of $A^{[t]}$ -type with polarization of split type and degree $2d$ [Reference Dawes2].

This shows that when $M=2U \oplus \langle -2t \rangle $ , ${\Gamma }$ has no irregular cusp when $(t, d)\ne (4, 1), (2, 2), (1, 4)$ . In Section 4.5, we show that ${\Gamma }$ indeed has an irregular cusp in these exceptional cases.

4.4 Special cubic fourfolds

In this subsection, we let L be an even lattice of the form $L=M\oplus K$ with $|A_{K}|>1$ odd. (K may be either negative-definite or hyperbolic or of signature $(2, \ast )$ .) We consider the group

$$ \begin{align*} {\Gamma} = \{ \: \gamma \in {\mathrm{O}^{+}(L)} \: | \: \gamma|_{A_{M}}= \pm{\mathrm{id}}, \: \gamma|_{A_{K}}={\mathrm{id}} \; \}. \end{align*} $$

When $M=\langle 2n \rangle \oplus U \oplus 2E_{8}$ and $K=A_{2}$ , ${\Gamma }$ is the modular group for the moduli space of special cubic fourfolds of discriminant $6n$ [Reference Tanimoto and Várilly-Alvarado18].

4.5 Examples of irregular cusps

In this subsection, we present two series of examples of irregular cusps, infinitely many in every dimension. We will denote by $e, f$ the standard basis of U.

As the first series of examples, we consider even lattices of the form $L=U\oplus \langle -8 \rangle \oplus M$ with M hyperbolic. We define the group ${\Gamma }$ by

$$ \begin{align*} {\Gamma} = \{ \: \gamma \in {\mathrm{O}^{+}(L)} \: | \: \gamma|_{A_{M}}=\pm{\mathrm{id}}, \: \gamma|_{A_{\langle -8 \rangle}}={\mathrm{id}} \; \}. \end{align*} $$

This is the group considered in Section 4.2 with $d=4$ . The group ${\Gamma }$ contains ${\widetilde {\mathrm {O}}^{+}(L)}$ with index $\leq 2$ , and we have ${\Gamma }={\widetilde {\mathrm {O}}^{+}(L)}$ if and only if $A_{M}$ is 2-elementary.

Proof First, note that $-{\mathrm {id}}\not \in {\Gamma }$ by the condition $\gamma |_{A_{\langle -8 \rangle }}={\mathrm { id}}$ . Let v be a generator of $\langle -8 \rangle $ . We take the vectors

$$ \begin{align*} l = 2e + 2f + v, \quad m = e/2 - f/2, \end{align*} $$

and show that $-E_{m\otimes l}\in {\Gamma }$ . This amounts to checking the following:

$$ \begin{align*} E_{m\otimes l}(L) \subset L, \quad E_{m\otimes l}|_{A_{M}} = \pm{\mathrm{id}}, \quad E_{m\otimes l}(v/8) \in -v/8+L. \end{align*} $$

Since $M\perp \langle l, m \rangle $ , $E_{m\otimes l}$ acts trivially on M. By direct calculation, we see that

$$ \begin{align*} E_{m\otimes l}(e)=4e+f+v, \quad E_{m\otimes l}(f)=e, \quad E_{m\otimes l}(v)=-v-8e. \end{align*} $$

This proves our assertion.

As the second series of examples, we consider even lattices of the form $L=U\oplus \langle -4 \rangle ^{\oplus 2} \oplus M$ with M hyperbolic, and the group ${\Gamma }$ defined by

$$ \begin{align*} {\Gamma} = \{ \: \gamma \in {\mathrm{O}^{+}(L)} \: | \: \gamma|_{A_{M}}=\pm{\mathrm{id}}, \: \gamma|_{A_{\langle -4 \rangle^{\oplus 2}}}={\mathrm{id}} \; \}. \end{align*} $$

The group ${\Gamma }$ contains ${\widetilde {\mathrm {O}}^{+}(L)}$ with index $\leq 2$ , and ${\Gamma }={\widetilde {\mathrm {O}}^{+}(L)}$ if and only if $A_{M}$ is 2-elementary.

Proof This is similar to the first example. We let $v_{1}, v_{2}$ be the standard basis of $\langle -4 \rangle ^{\oplus 2}$ and show that $-E_{m\otimes l}\in {\Gamma }$ for the vectors

$$ \begin{align*} l = 2e + 2f + v_{1} + v_{2}, \quad m = e + v_{1}/2. \end{align*} $$

The detail is left to the reader.

5.1 Siegel domain model

We consider the two-step linear projection

$$ \begin{align*} {{\mathbb P}}L_{{\mathbb{C}}} \dashrightarrow {{\mathbb P}}(L/J)_{{\mathbb{C}}} \dashrightarrow {{\mathbb P}}(L/J^{\perp})_{{\mathbb{C}}} \end{align*} $$

and restrict it to ${\mathcal {D}}\subset Q \subset {{\mathbb P}}L_{{\mathbb {C}}}$ . The center of the first projection ${{\mathbb P}}L_{{\mathbb {C}}} \dashrightarrow {{\mathbb P}}(L/J)_{{\mathbb {C}}}$ is the line ${{\mathbb P}}J_{{\mathbb {C}}}$ , and its fibers are planes containing ${{\mathbb P}}J_{{\mathbb {C}}}$ (minus ${{\mathbb P}}J_{{\mathbb {C}}}$ ). Since Q contains ${{\mathbb P}}J_{{\mathbb {C}}}$ , a plane containing ${{\mathbb P}}J_{{\mathbb {C}}}$ either

• • intersects with Q at two distinct lines (one is ${{\mathbb P}}J_{{\mathbb {C}}}$ ), or

• • intersects with Q at ${{\mathbb P}}J_{{\mathbb {C}}}$ with multiplicity $2$ , or

• • is contained in Q.

The first case occurs exactly when the plane is not contained in ${{\mathbb P}}J^{\perp }_{{\mathbb {C}}}$ . In that case, we can write the plane as ${{\mathbb P}}\langle J, v \rangle _{{\mathbb {C}}}$ with $(v, v)=0$ and $(v, J)\not \equiv 0$ . Then we have

$$ \begin{align*} {{\mathbb P}}\langle J, v \rangle_{{\mathbb{C}}} \cap Q = {{\mathbb P}}J_{{\mathbb{C}}} \cup {{\mathbb P}}\langle l, v \rangle_{{\mathbb{C}}}\,, \end{align*} $$

where ${\mathbb {C}}l=v^{\perp }\cap J_{{\mathbb {C}}}$ . This shows that the restriction of the first projection ${{\mathbb P}}L_{{\mathbb {C}}} \dashrightarrow {{\mathbb P}}(L/J)_{{\mathbb {C}}}$ to $Q-Q\cap {{\mathbb P}}J_{{\mathbb {C}}}^{\perp }$

$$ \begin{align*} \pi_{1} : Q - Q\cap {{\mathbb P}}J^{\perp}_{{\mathbb{C}}} \to {{\mathbb P}}(L/J)_{{\mathbb{C}}} - {{\mathbb P}}L(J)_{{\mathbb{C}}} \end{align*} $$

is an affine line bundle, with the fiber over the point ${{\mathbb P}}(\langle J, v \rangle _{{\mathbb {C}}}/J_{{\mathbb {C}}})$ being the affine line ${{\mathbb P}}\langle l, v \rangle _{{\mathbb {C}}} - [l]$ . The inequality $(\omega , \bar {\omega })>0$ defines a (shifted) upper half-plane in this affine line.

We identify $(L/J^{\perp })_{{\mathbb {C}}}=J^{\vee }_{{\mathbb {C}}}$ by the pairing. The second projection

$$ \begin{align*} \pi_{2} : {{\mathbb P}}(L/J)_{{\mathbb{C}}} - {{\mathbb P}}L(J)_{{\mathbb{C}}} \to {{\mathbb P}}(L/J^{\perp})_{{\mathbb{C}}} = {{\mathbb P}}J^{\vee}_{{\mathbb{C}}} \end{align*} $$

is an affine space bundle. It is (non-canonically) isomorphic to the vector bundle $L(J)_{{\mathbb {C}}} \otimes \mathcal {O}_{{{\mathbb P}}J^{\vee }_{{\mathbb {C}}}}(1)$ where, by abuse of notation, $\mathcal {O}_{{{\mathbb P}}J^{\vee }_{{\mathbb {C}}}}(1)$ stands for the line bundle corresponding to this sheaf (the dual of the tautological line bundle). To be more specific, if we choose a lift $L(J)_{{\mathbb {C}}}\hookrightarrow J_{{\mathbb {C}}}^{\perp }$ of $L(J)_{{\mathbb {C}}}$ , this determines a splitting $(L/J)_{{\mathbb {C}}}\simeq L(J)_{{\mathbb {C}}}\oplus (L/J^{\perp })_{{\mathbb {C}}}$ where $(L/J^{\perp })_{{\mathbb {C}}}$ is mapped to $L(J)_{{\mathbb {C}}}^{\perp }/J_{{\mathbb {C}}}$ . This splitting defines an isomorphism

$$ \begin{align*} L(J)_{{\mathbb{C}}} \otimes \mathcal{O}_{{{\mathbb P}}J^{\vee}_{{\mathbb{C}}}}(1) \: \simeq \: {{\mathbb P}}(L/J)_{{\mathbb{C}}} - {{\mathbb P}}L(J)_{{\mathbb{C}}} \end{align*} $$

over ${{\mathbb P}}J_{{\mathbb {C}}}^{\vee }={{\mathbb P}}(L/J^{\perp })_{{\mathbb {C}}}$ . At the fiber over each point $[v]$ of ${{\mathbb P}}(L/J^{\perp })_{{\mathbb {C}}}$ , this isomorphism is written as

(5.1) $$ \begin{align} {\operatorname{Hom}}({\mathbb{C}}v, L(J)_{{\mathbb{C}}}) \to {{\mathbb P}}({\mathbb{C}}v \oplus L(J)_{{\mathbb{C}}}) - {{\mathbb P}}L(J)_{{\mathbb{C}}}, \end{align} $$

where to a linear map ${\mathbb {C}}v\to L(J)_{{\mathbb {C}}}$ we associate its graph.

The orientation of J determines a connected component ${\mathbb {H}_{J}}$ of ${{\mathbb P}}J^{\vee }_{{\mathbb {C}}} - {{\mathbb P}}J^{\vee }_{{\mathbb {R}}}$ . We write ${\mathcal {V}_{J}} = \pi _{2}^{-1}({\mathbb {H}_{J}})$ and ${\mathcal {D}}(J) = \pi _{1}^{-1}({\mathcal {V}_{J}})$ . By definition, ${\mathcal {D}}(J)$ consists of points ${\mathbb {C}}\omega \in Q$ such that the map $(\cdot , \omega )\colon J_{{\mathbb {R}}}\to {\mathbb {C}}$ is an orientation-preserving ${\mathbb {R}}$ -isomorphism. We thus have the enlarged two-step fibration

$$ \begin{align*} {\mathcal{D}} \subset {\mathcal{D}}(J) \stackrel{\pi_{1}}{\to} {\mathcal{V}_{J}} \stackrel{\pi_{2}}{\to} {\mathbb{H}_{J}}. \end{align*} $$

This is the Siegel domain realization of ${\mathcal {D}}$ with respect to J. Here ${\mathcal {D}}(J) \to {\mathcal {V}_{J}}$ is an affine line bundle, inside which ${\mathcal {D}}\to {\mathcal {V}_{J}}$ is a fibration of upper half-planes. Over ${\mathbb {H}_{J}}\subset {{\mathbb P}}J_{{\mathbb {C}}}^{\vee }$ , we have the Hodge line bundle in $\underline {J_{{\mathbb {C}}}}=J_{{\mathbb {C}}} \otimes \mathcal {O}_{{\mathbb {H}_{J}}}$ defined by

$$ \begin{align*} F := \mathcal{O}_{{\mathbb{H}_{J}}}(-1)^{\perp} \subset \underline{J_{{\mathbb{C}}}}, \end{align*} $$

where we view $\mathcal {O}_{{\mathbb {H}_{J}}}(-1)$ as a sub-line bundle of $J_{{\mathbb {C}}}^{\vee } \otimes \mathcal {O}_{{\mathbb {H}_{J}}}$ naturally. Then $\mathcal {O}_{{\mathbb {H}_{J}}}(1)$ is naturally isomorphic to $\underline {J_{{\mathbb {C}}}}/F$ . To summarize, we have an isomorphism

(5.2) $$ \begin{align} {\mathcal{V}_{J}} \simeq L(J)_{{\mathbb{C}}}\otimes \mathcal{O}_{{\mathbb{H}_{J}}}(1) \simeq L(J)_{{\mathbb{C}}} \otimes (\underline{J_{{\mathbb{C}}}}/F). \end{align} $$

The relation with the tube domain model is as follows. We choose a rank $1$ primitive sublattice I of J. This corresponds to a zero-dimensional cusp in the closure of the one-dimensional cusp for J. The filtration $I\subset J \subset J^{\perp } \subset L$ determines the projections ${{\mathbb P}}(L/I)_{{\mathbb {C}}}\dashrightarrow {{\mathbb P}}(L/J)_{{\mathbb {C}}}\dashrightarrow {{\mathbb P}}(L/J^{\perp })_{{\mathbb {C}}}$ . Then the composition of this with the tube domain realization ${\mathcal {D}}\subset {\mathcal {D}(I)} \hookrightarrow {{\mathbb P}}(L/I)_{{\mathbb {C}}}$ is the Siegel domain realization above.

5.2 Stabilizer over ${\mathbb {Q}}$

Let ${\Gamma (J)_{\mathbb {Q}}}$ be the subgroup of the stabilizer of $J_{{\mathbb {Q}}}$ in $\mathrm {O}^{+}(L_{{\mathbb {Q}}})$ that acts on $J_{{\mathbb {Q}}}$ with determinant $1$ . The determinant $1$ condition is not restrictive when restricting to subgroups of ${\mathrm {O}^{+}(L)}$ . We write

$$ \begin{align*} {W(J)_{\mathbb{Q}}} = {\operatorname{Ker}}({\Gamma(J)_{\mathbb{Q}}} \to \mathrm{O}(L(J)_{{\mathbb{Q}}}) \times \mathrm{SL}(J_{{\mathbb{Q}}})), \end{align*} $$

$$ \begin{align*} {U(J)_{\mathbb{Q}}} = {\operatorname{Ker}}({\Gamma(J)_{\mathbb{Q}}} \to \mathrm{GL}(J_{{\mathbb{Q}}}^{\perp})), \end{align*} $$

$$ \begin{align*} {V(J)_{\mathbb{Q}}} = {W(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Q}}}. \end{align*} $$

By definition, we have the canonical exact sequences

(5.3) $$ \begin{align} 1 \to {W(J)_{\mathbb{Q}}} \to {\Gamma(J)_{\mathbb{Q}}} \to \mathrm{O}(L(J)_{{\mathbb{Q}}}) \times \mathrm{ SL}(J_{{\mathbb{Q}}}) \to 1, \end{align} $$

(5.4) $$ \begin{align} 0 \to {U(J)_{\mathbb{Q}}} \to {W(J)_{\mathbb{Q}}} \to {V(J)_{\mathbb{Q}}} \to 0. \end{align} $$

The group ${W(J)_{\mathbb {Q}}}$ is the unipotent radical of ${\Gamma (J)_{\mathbb {Q}}}$ . If we choose a splitting $L_{{\mathbb {Q}}}\simeq 2U_{{\mathbb {Q}}}\oplus L(J)_{{\mathbb {Q}}}$ for $J_{{\mathbb {Q}}}$ , the first exact sequence (5.3) splits (non-canonically):

(5.5) $$ \begin{align} {\Gamma(J)_{\mathbb{Q}}} \: \simeq \: (\mathrm{O}(L(J)_{{\mathbb{Q}}}) \times \mathrm{SL}(J_{{\mathbb{Q}}})) \ltimes {W(J)_{\mathbb{Q}}}. \end{align} $$

Here $\mathrm {SL}(J_{{\mathbb {Q}}})$ acts on the component $2U_{{\mathbb {Q}}}\simeq J_{{\mathbb {Q}}}\oplus J_{{\mathbb {Q}}}^{\vee }$ , and $\mathrm {O}(L(J)_{{\mathbb {Q}}})$ acts on the component $L(J)_{{\mathbb {Q}}}$ .

On the other hand, the second exact sequence (5.4) never splits. Indeed, ${W(J)_{\mathbb {Q}}}$ is a Heisenberg group as follows. We have a canonical $\wedge ^{2}J$ -valued symplectic form on $L(J)\otimes J$ as the tensor product of the quadratic form on $L(J)$ and the canonical symplectic form $J\times J \to \wedge ^{2}J$ on J. This gives a Heisenberg group structure on $\wedge ^{2}J_{{\mathbb {Q}}}\times (L(J)_{{\mathbb {Q}}}\otimes J_{{\mathbb {Q}}})$ . Explicitly, we take a bijection $L(J)_{{\mathbb {Q}}}\otimes J_{{\mathbb {Q}}} \simeq L(J)_{{\mathbb {Q}}} \times L(J)_{{\mathbb {Q}}}$ by choosing a positive basis of J, and define a product on $\wedge ^{2}J_{{\mathbb {Q}}}\times L(J)_{{\mathbb {Q}}} \times L(J)_{{\mathbb {Q}}}$ by

$$ \begin{align*} (\alpha, v_{1}, v_{2})\cdot (\beta, w_{1}, w_{2}) = (\alpha+\beta+(v_{2}, w_{1}), v_{1}+w_{1}, v_{2}+w_{2}). \end{align*} $$

The center is $\wedge ^{2}J_{{\mathbb {Q}}}\times \{ 0 \} \times \{ 0 \}$ .

Lemma 5.1 ${W(J)_{\mathbb {Q}}}$ is isomorphic to the Heisenberg group for $L(J)_{{\mathbb {Q}}}\otimes J_{{\mathbb {Q}}}$ with center ${U(J)_{\mathbb {Q}}}$ , and we have the canonical isomorphisms

$$ \begin{align*} \wedge^{2}J_{{\mathbb{Q}}} \to {U(J)_{\mathbb{Q}}}, \quad l\wedge l' \mapsto E_{l\otimes l'}, \end{align*} $$

$$ \begin{align*} L(J)_{{\mathbb{Q}}}\otimes J_{{\mathbb{Q}}} \to {V(J)_{\mathbb{Q}}}, \quad m\otimes l \mapsto E_{\tilde{m}\otimes l} \; \mathrm{mod} \; {U(J)_{\mathbb{Q}}}. \end{align*} $$

Proof This should be well-known (see, e.g., [Reference Looijenga12]), but we provide a proof in the present context for the convenience of the readers. We choose a rank $1$ primitive sublattice I of J and put $\bar {J}=(J/I)\otimes I \subset L(I)$ . Note that $\bar {J}\simeq \wedge ^{2}J$ naturally. We restrict the sequence (2.3) for ${\Gamma (I)_{\mathbb {Q}}}$ to ${W(J)_{\mathbb {Q}}}\subset {\Gamma (I)_{\mathbb {Q}}}$ . It is clear that ${W(J)_{\mathbb {Q}}} \cap {U(I)_{\mathbb {Q}}} = \bar {J}_{{\mathbb {Q}}}^{\perp }\cap L(I)_{{\mathbb {Q}}}$ , which contains ${U(J)_{\mathbb {Q}}}$ with

(5.6) $$ \begin{align} {U(J)_{\mathbb{Q}}} = (\bar{J}_{{\mathbb{Q}}}^{\perp})^{\perp} = \bar{J}_{{\mathbb{Q}}} \simeq \wedge^{2}J_{{\mathbb{Q}}} \quad \subset {U(I)_{\mathbb{Q}}}. \end{align} $$

The image of ${W(J)_{\mathbb {Q}}}\to {\mathrm {O}^{+}}(L(I)_{{\mathbb {Q}}})$ is the subgroup of the stabilizer of $\bar {J}_{{\mathbb {Q}}}$ that acts trivially on $\bar {J}_{{\mathbb {Q}}}$ and $\bar {J}_{{\mathbb {Q}}}^{\perp }/\bar {J}_{{\mathbb {Q}}}$ . This consists of Eichler transvections of $L(I)_{{\mathbb {Q}}}$ with respect to $\bar {J}_{{\mathbb {Q}}}$ , hence isomorphic to $(\bar {J}_{{\mathbb {Q}}}^{\perp }/\bar {J}_{{\mathbb {Q}}})\otimes \bar {J}_{{\mathbb {Q}}} \simeq L(J)_{{\mathbb {Q}}}\otimes (J/I)_{{\mathbb {Q}}}$ . In this way, we obtain the exact sequence

(5.7) $$ \begin{align} 0 \to \bar{J}_{{\mathbb{Q}}}^{\perp}\cap L(I)_{{\mathbb{Q}}} \to {W(J)_{\mathbb{Q}}} \to L(J)_{{\mathbb{Q}}}\otimes (J/I)_{{\mathbb{Q}}} \to 0. \end{align} $$

We choose lifts $L(J)_{{\mathbb {Q}}}\hookrightarrow J_{{\mathbb {Q}}}^{\perp }$ and $(J/I)_{{\mathbb {Q}}}\hookrightarrow J_{{\mathbb {Q}}}$ . This induces a section of (5.7) which consists of the Eichler transvections $E_{w}$ of $L_{{\mathbb {Q}}}$ with $w\in L(J)_{{\mathbb {Q}}}\otimes (J/I)_{{\mathbb {Q}}}$ . Together with the splitting $\bar {J}_{{\mathbb {Q}}}^{\perp }\cap L(I)_{{\mathbb {Q}}} \simeq \bar {J}_{{\mathbb {Q}}} \oplus (L(J)_{{\mathbb {Q}}}\otimes I_{{\mathbb {Q}}})$ , we obtain a bijection

$$ \begin{align*} {W(J)_{\mathbb{Q}}} \simeq \bar{J}_{{\mathbb{Q}}}\times (L(J)_{{\mathbb{Q}}}\otimes I_{{\mathbb{Q}}}) \times (L(J)_{{\mathbb{Q}}}\otimes (J/I)_{{\mathbb{Q}}}). \end{align*} $$

This gives an isomorphism with the Heisenberg group.

Note that ${U(J)_{\mathbb {Q}}}$ is not just the center of ${W(J)_{\mathbb {Q}}}$ , but also the center of ${\Gamma (J)_{\mathbb {Q}}}$ . This is the reason we put the determinant $1$ condition in the definition of ${\Gamma (J)_{\mathbb {Q}}}$ .

The action of ${\Gamma (J)_{\mathbb {Q}}}$ on the Siegel domain model can be described through the filtration ${U(J)_{\mathbb {Q}}}\subset {W(J)_{\mathbb {Q}}}\subset {\Gamma (J)_{\mathbb {Q}}}$ . By definition ${U(J)_{\mathbb {Q}}}$ acts on ${\mathcal {V}_{J}}\subset {{\mathbb P}}(J_{{\mathbb {C}}}^{\perp })^{\vee }$ trivially and ${W(J)_{\mathbb {Q}}}$ acts on ${\mathbb {H}_{J}}\subset {{\mathbb P}}J_{{\mathbb {C}}}^{\vee }$ trivially. We let $U(J)_{{\mathbb {C}}}\subset \mathrm {O}(L_{{\mathbb {C}}})$ be the group of Eichler transvections $E_{l\otimes l'}$ with $l, l'\in J_{{\mathbb {C}}}$ . Then $U(J)_{{\mathbb {C}}}\simeq \wedge ^{2}J_{{\mathbb {C}}}$ preserves ${\mathcal {D}}(J)$ and acts on ${\mathcal {V}_{J}}$ trivially. The following descriptions should be well-known, but we provide a proof in the present setting for the convenience of the readers.

Proof (1) Recall from Section 5.1 that a fiber of ${\mathcal {D}}(J)\to {\mathcal {V}_{J}}$ is an affine line ${{\mathbb P}}\langle l, v \rangle _{{\mathbb {C}}}-[l]$ where $v\in L_{{\mathbb {C}}}$ is an isotropic vector with $(v, J)\not \equiv 0$ and ${\mathbb {C}}l=v^{\perp }\cap J_{{\mathbb {C}}}$ . We take $l'\in J_{{\mathbb {C}}}$ with $(l', v)=1$ . Then $E_{\alpha l\wedge l'}\in U(J)_{{\mathbb {C}}}$ , $\alpha \in {\mathbb {C}}$ , sends a point ${\mathbb {C}}(v+\beta l)$ of ${{\mathbb P}}\langle l, v \rangle _{{\mathbb {C}}}-[l]$ to

$$ \begin{align*} {\mathbb{C}}(v+\beta l) \mapsto {\mathbb{C}}(v+(\alpha l', v)l + \beta l) = {\mathbb{C}}(v+(\alpha+\beta)l). \end{align*} $$

This shows that $U(J)_{{\mathbb {C}}}$ acts on each fiber of ${\mathcal {D}}(J)\to {\mathcal {V}_{J}}$ freely and transitively.

(2) We choose a splitting $L_{{\mathbb {Q}}}\simeq J_{{\mathbb {Q}}}\oplus L(J)_{{\mathbb {Q}}} \oplus (L/J^{\perp })_{{\mathbb {Q}}}$ for $J_{{\mathbb {Q}}}$ where the lift of $(L/J^{\perp })_{{\mathbb {Q}}}$ is perpendicular to the lift of $L(J)_{{\mathbb {Q}}}$ . Let $[v]$ be a point of ${\mathbb {H}_{J}}\subset {{\mathbb P}}(L/J^{\perp })_{{\mathbb {C}}}$ . By (5.1), the fiber of ${\mathcal {V}_{J}}\to {\mathbb {H}_{J}}$ over $[v]$ is the affine line

$$ \begin{align*} {{\mathbb P}}({\mathbb{C}}v\oplus L(J)_{{\mathbb{C}}}) - {{\mathbb P}}L(J)_{{\mathbb{C}}} \simeq {\operatorname{Hom}}({\mathbb{C}}v, L(J)_{{\mathbb{C}}}) \end{align*} $$

in ${{\mathbb P}}(L/J)_{{\mathbb {C}}}$ . Here the point corresponding to $f\in {\operatorname {Hom}}({\mathbb {C}}v, L(J)_{{\mathbb {C}}})$ is its graph ${\mathbb {C}}(v+f(v))$ . We take $E_{m\otimes l}\in {W(J)_{\mathbb {Q}}}/{U(J)_{\mathbb {Q}}}$ where $m\in L(J)_{{\mathbb {Q}}}$ and $l\in J_{{\mathbb {Q}}}$ . Then $E_{m\otimes l}$ sends ${\mathbb {C}}(v+f(v))$ to

$$ \begin{align*} {\mathbb{C}}(v+f(v)) & \mapsto {\mathbb{C}}(v + (l, v)m - 2^{-1}(m, m)(l, v)l + f(v) - (m, f(v))l) \\ & = {\mathbb{C}}(v + f(v) + (l, v)m) \: \: \in {{\mathbb P}}(L/J)_{{\mathbb{C}}}. \end{align*} $$

This means that $E_{m\otimes l}$ acts on ${\operatorname {Hom}}({\mathbb {C}}v, L(J)_{{\mathbb {C}}})$ by the translation $f\mapsto f + (l, \cdot )m$ . Finally, we notice that the ${\mathbb {R}}$ -isomorphism

$$ \begin{align*} L(J)_{{\mathbb{R}}}\otimes_{{\mathbb{R}}}J_{{\mathbb{R}}} \to L(J)_{{\mathbb{C}}}\otimes_{{\mathbb{C}}}(J_{{\mathbb{C}}}/F_{[v]}) = L(J)_{{\mathbb{C}}}\otimes_{{\mathbb{C}}}(J_{{\mathbb{C}}}/v^{\perp}) \simeq L(J)_{{\mathbb{C}}}\otimes_{{\mathbb{C}}}({\mathbb{C}}v)^{\vee} \end{align*} $$

sends $m\otimes l$ to $m\otimes (l, \cdot )$ . This proves the assertion (2).

The proof of (3) is straightforward and is left to the readers.

5.3 Stabilizer over ${\mathbb {Z}}$

Now let ${\Gamma }$ be a finite-index subgroup of ${\mathrm {O}^{+}(L)}$ . We write

$$ \begin{align*} {\Gamma(J)_{\mathbb{Z}}} = {\Gamma(J)_{\mathbb{Q}}}\cap {\Gamma}, \quad {W(J)_{\mathbb{Z}}} = {W(J)_{\mathbb{Q}}} \cap {\Gamma}, \quad {U(J)_{\mathbb{Z}}} = {U(J)_{\mathbb{Q}}} \cap {\Gamma}, \end{align*} $$

and consider the quotients

$$ \begin{align*} {\overline{\Gamma(J)}_{\mathbb{Z}}} = {\Gamma(J)_{\mathbb{Z}}}/{U(J)_{\mathbb{Z}}}, \quad {V(J)_{\mathbb{Z}}} = {W(J)_{\mathbb{Z}}}/{U(J)_{\mathbb{Z}}}, \quad \Gamma_{J} = {\Gamma(J)_{\mathbb{Z}}}/{W(J)_{\mathbb{Z}}}, \end{align*} $$

$$ \begin{align*} {\overline{\Gamma(J)}_{\mathbb{Q}}} = {\Gamma(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Z}}}, \; \; {W(J)_{\mathbb{Q}/\mathbb{Z}}} = {W(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Z}}}, \; \; {U(J)_{\mathbb{Q}/\mathbb{Z}}} = {U(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Z}}}. \end{align*} $$

By definition, we have the canonical exact sequences

(5.8) $$ \begin{align} 0 \to {V(J)_{\mathbb{Z}}} \to {\overline{\Gamma(J)}_{\mathbb{Z}}} \to \Gamma_{J} \to 1, \end{align} $$

(5.9) $$ \begin{align} 0 \to {W(J)_{\mathbb{Q}/\mathbb{Z}}} \to {\overline{\Gamma(J)}_{\mathbb{Q}}} \to \mathrm{ O}(L(J)_{{\mathbb{Q}}})\times \mathrm{SL}(J_{{\mathbb{Q}}}) \to 1, \end{align} $$

$$ \begin{align*} 0 \to {U(J)_{\mathbb{Q}/\mathbb{Z}}} \to {W(J)_{\mathbb{Q}/\mathbb{Z}}} \to {V(J)_{\mathbb{Q}}} \to 0. \end{align*} $$

Then (5.8) is canonically embedded in (5.9). We have

(5.10) $$ \begin{align} {V(J)_{\mathbb{Z}}}\cap {U(J)_{\mathbb{Q}/\mathbb{Z}}} = \{ 0 \} \end{align} $$

as subgroups of ${W(J)_{\mathbb {Q}/\mathbb {Z}}}$ because ${W(J)_{\mathbb {Z}}} \cap {U(J)_{\mathbb {Q}}}={U(J)_{\mathbb {Z}}}$ by definition. Note that ${U(J)_{\mathbb {Q}/\mathbb {Z}}}$ is the group of torsion points of the one-dimensional torus $T(J)=U(J)_{{\mathbb {C}}}/{U(J)_{\mathbb {Z}}}$ . If we choose a splitting for $J_{{\mathbb {Q}}}$ , the induced splitting (5.5) of ${\Gamma (J)_{\mathbb {Q}}}$ defines a splitting of (5.9):

(5.11) $$ \begin{align} {\overline{\Gamma(J)}_{\mathbb{Q}}} \: \simeq \: (\mathrm{O}(L(J)_{{\mathbb{Q}}}) \times \mathrm{ SL}(J_{{\mathbb{Q}}})) \ltimes {W(J)_{\mathbb{Q}/\mathbb{Z}}}. \end{align} $$

But this does not mean that (5.8) splits.

5.4 Partial toroidal compactification

We denote ${\mathcal {T}(J)}={\mathcal {D}}(J)/{U(J)_{\mathbb {Z}}}$ and ${\mathcal {X}(J)}={\mathcal {D}}/{U(J)_{\mathbb {Z}}}$ . By Lemma 5.2, ${\mathcal {T}(J)}\to {\mathcal {V}_{J}}$ is a principal $T(J)$ -bundle acted on equivariantly by ${\overline {\Gamma (J)}_{\mathbb {Q}}}$ . The projection ${\mathcal {X}(J)\to {\mathcal {V}_{J}}}$ is a punctured disk bundle in ${\mathcal {T}(J)}\to {\mathcal {V}_{J}}$ . By Lemma 5.2, the action of ${\overline {\Gamma (J)}_{\mathbb {Q}}}$ on ${\mathcal {V}_{J}}\to {\mathbb {H}_{J}}$ is described as follows.

Thus, ${\mathcal {V}_{J}}/{V(J)_{\mathbb {Z}}}$ is a fibration of abelian varieties over ${\mathbb {H}_{J}}$ isogenous to the self-fiber product of the universal elliptic curve. The group $\Gamma _{J}$ acts on ${\mathcal {V}_{J}}/{V(J)_{\mathbb {Z}}}$ by the equivariant action plus some possible translation.

Now let $\overline {T(J)}\simeq {\mathbb {C}}$ be the canonical partial compactification of the torus $T(J)$ . We take the relative torus embedding

$$ \begin{align*} {\overline{\mathcal{T}(J)}}= ({\mathcal{T}(J)} \times \overline{T(J)})/T(J). \end{align*} $$

This is the line bundle associated with the principal $T(J)$ -bundle ${\mathcal {T}(J)}\to {\mathcal {V}_{J}}$ and the standard character of $T(J)$ . Let ${\overline {\mathcal {X}(J)}}$ be the interior of the closure of ${\mathcal {X}(J)}$ in ${\overline {\mathcal {T}(J)}}$ . This is the partial toroidal compactification of ${\mathcal {X}(J)}$ over the one-dimensional cusp J. Note that no choice of fan is required: this is canonical. The boundary divisor of ${\overline {\mathcal {X}(J)}}$ is canonically isomorphic to ${\mathcal {V}_{J}}$ .

The relation with a partial toroidal compactification over an adjacent zero-dimensional cusp $I \subset J$ is as follows. Recall that $\bar {J}=(J/I)\otimes I \simeq \wedge ^{2}J$ is an isotropic sublattice of $L(I)$ , oriented by the orientation of J. The ray $\sigma _{J}=(\bar {J}_{{\mathbb {R}}})_{\geq 0}$ is in the closure of the positive cone, and it is contained in any ${\Gamma (I)_{\mathbb {Z}}}$ -admissible fan $\Sigma $ . The torus embedding $T(I)\hookrightarrow T(I)^{\sigma _{J}}$ defined by $\sigma _{J}$ is a Zariski open set of $T(I)^{\Sigma }$ . By (5.6), we have $U(J)_{{\mathbb {R}}}=\bar {J}_{{\mathbb {R}}}\subset U(I)_{{\mathbb {R}}}$ and

(5.12) $$ \begin{align} {U(J)_{\mathbb{Z}}} = \bar{J}_{{\mathbb{R}}} \cap {U(I)_{\mathbb{Z}}} = {\mathbb{R}}\sigma_{J} \cap {U(I)_{\mathbb{Z}}}. \end{align} $$

Therefore, the inclusion ${\mathcal {D}}(J)\subset {\mathcal {D}(I)}$ induces the etale map

(5.13) $$ \begin{align} {\overline{\mathcal{T}(J)}}\to T(I)^{\sigma_{J}}\subset T(I)^{\Sigma}, \end{align} $$

which maps the boundary divisor of ${\overline {\mathcal {T}(J)}}$ to the unique boundary divisor of $T(I)^{\sigma _{J}}$ . We note that ${U(I)_{\mathbb {Z}}}\subset {\Gamma (J)_{\mathbb {Z}}}$ .

8.1 Modular forms

Let $\mathcal {L}=\mathcal {O}_{{{\mathbb P}}L_{{\mathbb {C}}}}(-1)|_{{\mathcal {D}}}$ be the restriction of the tautological line bundle to ${\mathcal {D}}\subset {{\mathbb P}}L_{{\mathbb {C}}}$ . Let $\chi $ be a character of ${\Gamma }$ . By our assumption $b\geq 3$ , $\chi ({\Gamma })\subset {\mathbb {C}}^{\times }$ is finite [Reference Margulis14]. We assume that $\chi |_{{U(I)_{\mathbb {Z}}}}\equiv 1$ for every zero-dimensional cusp I. This holds, e.g., for $\chi =1, \det $ . A  ${\Gamma }$ -invariant section of the ${\Gamma }$ -linearized line bundle $\mathcal {L}^{\otimes k}\otimes \chi $ over ${\mathcal {D}}$ is called a modular form of weight k and character $\chi $ with respect to ${\Gamma }$ .

Let I be a rank $1$ primitive isotropic sublattice of L. We choose a generator $l_{I}$ of I. This defines a frame $s_{I}$ of $\mathcal {L}$ determined by the condition $(s_{I}([\omega ]), l_{I})=1$ , where we view $s_{I}([\omega ])\in \mathcal {L}_{[\omega ]}={\mathbb {C}}\omega \subset L_{{\mathbb {C}}}$ . The factor of automorphy with respect to $s_{I}$ is given by

(8.1) $$ \begin{align} j(\gamma, [\omega]) = \frac{(\gamma\omega, l_{I})}{(\omega, l_{I})} = \frac{(\omega, \gamma^{-1}l_{I})}{(\omega, l_{I})} , \qquad \gamma\in {\Gamma}, \; [\omega]\in {\mathcal{D}}. \end{align} $$

Let $1_{\chi }$ be a nonzero vector in the representation line of $\chi $ . Then $s_{I}^{\otimes k}\otimes 1_{\chi }$ is a frame of the line bundle $\mathcal {L}^{\otimes k}\otimes \chi $ , via which modular forms $F=f s_{I}^{\otimes k}\otimes 1_{\chi }$ of weight k and character $\chi $ are identified with holomorphic functions f on ${\mathcal {D}}$ satisfying