# Irregular cusps of orthogonal modular varieties

Shouhei Ma  Department of Mathematics, Tokyo Institute of Technology, Tokyo
152-8551, Japan<EMAIL_ADDRESS>
###### Abstract.

Irregular cusp of an orthogonal modular variety is a cusp where the lattice
for Fourier expansion is strictly smaller than the lattice of translation.
Presence of such a cusp affects the study of pluricanonical forms on the
modular variety using modular forms. We study toroidal compactification over
an irregular cusp, and clarify there the cusp form criterion for the
calculation of Kodaira dimension. At the same time, we show that irregular
cusps do not arise frequently: besides the cases when the group is neat or
contains $-1$, we prove that the stable orthogonal groups of most (but not
all) even lattices have no irregular cusp.

###### Key words and phrases:

Supported by JSPS KAKENHI 17K14158 and 20H00112.

##  1\. Introduction

Irregular cusp of a modular curve is a cusp where the width of translation is
strictly smaller than the width for Fourier expansion. It does not arise
frequently, but does exist. At such a cusp, vanishing order of cusp forms has
to be considered carefully, especially when compared with that of
pluricanonical forms (cf. [3] §3.2 – §3.3). In this article we study and
classify irregular cusps for orthogonal groups of signature $(2,b)$, and
clarify the effect of such cusps on the study of Kodaira dimension of
orthogonal modular varieties.

Let $L$ be a lattice of signature $(2,b)$. Let
${\mathcal{D}}={\mathcal{D}_{L}}$ be the Hermitian symmetric domain attached
to $L$, which is defined as either of the two connected components of the
space

$\\{{\mathbb{C}}\omega\in{{\mathbb{P}}}L_{{\mathbb{C}}}\>|\>(\omega,\omega)=0,\>(\omega,\bar{\omega})>0\\}.$

We write ${{\rm O}^{+}(L)}$ for the subgroup of the orthogonal group ${\rm
O}(L)$ that preserves the component ${\mathcal{D}}$.

The domain ${\mathcal{D}}$ has $0$-dimensional and $1$-dimensional cusps. For
simplicity of exposition, we speak only of $0$-dimensional cusps for the
moment: in fact, the case of $1$-dimensional cusps can be reduced to that of
adjacent $0$-dimensional cusps. A $0$-dimensional cusp of ${\mathcal{D}}$
corresponds to a rank $1$ primitive isotropic sublattice $I$ of $L$. Let
${U(I)_{\mathbb{Q}}}$ be the unipotent part of the stabilizer of $I$ in ${\rm
O}^{+}(L_{{\mathbb{Q}}})$. Then ${U(I)_{\mathbb{Q}}}$ is already abelian: it
is a ${\mathbb{Q}}$-vector space of dimension $b$ (with a hyperbolic quadratic
form). Let ${\Gamma}$ be a finite-index subgroup of ${{\rm O}^{+}(L)}$. The
cusp $I$ is called an irregular cusp for ${\Gamma}$ if
${U(I)_{\mathbb{Q}}}\cap{\Gamma}\neq{U(I)_{\mathbb{Q}}}\cap{\langle\Gamma,-{\rm
id}\rangle}$. As we will explain,
${U(I)_{\mathbb{Z}}}={U(I)_{\mathbb{Q}}}\cap{\Gamma}$ is the lattice for
Fourier expansion of ${\Gamma}$-modular forms around $I$, while
${U(I)_{\mathbb{Z}}^{\prime}}={U(I)_{\mathbb{Q}}}\cap{\langle\Gamma,-{\rm
id}\rangle}$ is the lattice of translation around $I$ in the
${\Gamma}$-action. We give several characterizations of irregularity
(Proposition 3.1), including one suitable for explicit calculation.

Irregular cusps are rather rare: they do not exist when $-{\rm id}\in{\Gamma}$
or when ${\Gamma}$ is neat or when ${\Gamma}\subset{\rm SO}^{+}(L)$ with $b$
odd. But they do exist, in infinitely many examples in every dimension (§4.5).
Our particular interest is in the so-called stable orthogonal groups
${\widetilde{{\rm O}}^{+}(L)}$ of even lattices $L$, defined as the kernel of
the reduction map ${{\rm O}^{+}(L)}\to{\rm O}(L^{\vee}/L)$. This is the group
that most frequently appear in the moduli problem related to orthogonal
modular varieties. Our calculation concerning ${\widetilde{{\rm O}}^{+}(L)}$
can be summarized as follows.

######  Proposition 1.1 (§4.1, §4.5).

The stable orthogonal group ${\widetilde{{\rm O}}^{+}(L)}$ of an even lattice
$L$ has no irregular cusp unless when
$L^{\vee}/L\simeq{\mathbb{Z}}/8\oplus({\mathbb{Z}}/2)^{\oplus a}$ or
$L^{\vee}/L\simeq({\mathbb{Z}}/4)^{\oplus 2}\oplus({\mathbb{Z}}/2)^{\oplus a}$
as abelian groups. Conversely, if $L=U\oplus\langle-8\rangle\oplus M$ or
$L=U\oplus\langle-4\rangle^{\oplus 2}\oplus M$ with $M^{\vee}/M$ 2-elementary,
then ${\widetilde{{\rm O}}^{+}(L)}$ has an irregular $0$-dimensional cusp.

Consequently, we obtain classification for the following examples from moduli
spaces (§4):

  * •

The modular group for $K3$ surfaces of degree $2d$ has an irregular cusp
exactly when $d=4$.

  * •

The modular group for irreducible symplectic manifolds of $K3^{[t+1]}$-type
with polarization of split type and degree $2d$ ([8]) has an irregular cusp
exactly when $(t,d)=(1,4),(2,2),(4,1)$.

  * •

The modular group for O’Grady10 manifolds with polarization of split type and
degree $2d$ ([9]), which is larger than ${\widetilde{{\rm O}}^{+}(L)}$, has an
irregular cusp exactly when $d=4$.

  * •

Similarly, the modular group for deformation generalized Kummer varieties with
polarization of split type and degree $2d$ ([2]) has an irregular cusp exactly
when $(t,d)=(4,1)$.

  * •

We will also cover the groups considered in [17], [15], [4].

A subtle issue concerning irregular cusps, which is the main object of this
article, is comparison of vanishing order between cusp forms and
pluricanonical forms. We take a toroidal compactification
${\mathcal{F}(\Gamma)^{\Sigma}}$ of the modular variety
${\mathcal{F}(\Gamma)}={\Gamma}\backslash{\mathcal{D}}$, which is defined by
choosing a finite collection $\Sigma=(\Sigma_{I})$ of suitable fans, one for
each ${\Gamma}$-equivalence class of rank $1$ primitive isotropic sublattices
$I$ of $L$. A ray $\sigma$ in $\Sigma_{I}$ corresponds to a boundary divisor
$D(\sigma)$ of the torus embedding
$\overline{{\mathcal{D}}/{U(I)_{\mathbb{Z}}}}$, and thus determines a boundary
divisor $\Delta(\sigma)$ of ${\mathcal{F}(\Gamma)^{\Sigma}}$ as the image of
$D(\sigma)$. Then the projection
$\overline{{\mathcal{D}}/{U(I)_{\mathbb{Z}}}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$
is ramified along $D(\sigma)$, with index $2$, exactly when $I$ is irregular
and the ray $\sigma$ is irregular in the sense of Definition 3.8.

Vanishing order $\nu_{\sigma}(F)$ of a ${\Gamma}$-modular form $F$ at
$D(\sigma)\subset\overline{{\mathcal{D}}/{U(I)_{\mathbb{Z}}}}$ can be measured
by Fourier expansion (§8.2): this is done with ${U(I)_{\mathbb{Z}}}$. On the
other hand, vanishing order of a pluricanonical form $\omega$ on
${\mathcal{F}(\Gamma)}$ should be measured at the level of
$\Delta(\sigma)\subset{\mathcal{F}(\Gamma)^{\Sigma}}$: this is essentially
done with ${U(I)_{\mathbb{Z}}^{\prime}}$. When $\omega$ is $m$-canonical and
corresponds to $F$ (of weight $k=mb$ and character $\chi=\det^{m}$), we have
the relation (Proposition 8.7)

$\nu_{\Delta(\sigma)}(\omega)=a_{\sigma}\cdot\nu_{\sigma}(F)-m,$

where $a_{\sigma}=1$ if $\sigma$ is regular but $a_{\sigma}=1/2$ if $\sigma$
is irregular due to the boundary ramification. If we are involved only with
modular forms of specific parity of weight $k$, namely $k$ even for $\chi=1$
or $k\equiv b$ mod $2$ for $\chi=\det$, we do not need to worry about
irregular cusps because we can enlarge ${\Gamma}$ to ${\langle\Gamma,-{\rm
id}\rangle}$ without any loss. However, when we use product with a cusp form
of weight in the remaining parity in the construction of a modular form
corresponding to a pluricanonical form, we cannot add $-{{\rm id}}$ to
${\Gamma}$, and have to be careful about the coefficient $a_{\sigma}=1/2$ at
irregular rays $\sigma$.

Gritsenko-Hulek-Sankaran [6] gave a criterion, called the low weight cusp form
trick, for ${\mathcal{F}(\Gamma)}$ to be of general type in terms of existence
of a certain cusp form. It appears that irregular cusps are not covered in
[6], essentially by assuming $-{\rm id}\in{\Gamma}$, explicitly for
$1$-dimensional cusps ([6] p.539) and implicitly for $0$-dimensional cusps
(see the remark below). In view of the coefficient $a_{\sigma}=1/2$ at
irregular $\sigma$, it seems that the criterion needs to be modified at such
boundary divisors. The result is summarized as follows (compare with [6]
Theorem 1.1).

######  Theorem 1.2 (Theorem 8.9).

Let $L$ be a lattice of signature $(2,b)$ with $b\geq 9$ and ${\Gamma}$ be a
subgroup of ${{\rm O}^{+}(L)}$ of finite index. We take a
${\Gamma}$-admissible collection $\Sigma=(\Sigma_{I})$ of fans so that
$\Sigma_{I}$ is basic with respect to
${U(I)_{\mathbb{Q}}}\cap{\langle\Gamma,-{\rm id}\rangle}$ at every
$0$-dimensional cusp $I$. Assume that there exists a ${\Gamma}$-cusp form $F$
of weight $k<b$ and some character satisfying the following:

  1. (1)

$F$ vanishes at the ramification divisor of
${\mathcal{D}}\to{\mathcal{F}(\Gamma)}$.

  2. (2)

$\nu_{\sigma}(F)\geq 2$ at every irregular ray $\sigma$ at every irregular
$I$.

Then ${\mathcal{F}(\Gamma)}$ is of general type.

The condition on $\Sigma$ is put to ensure that
${\mathcal{F}(\Gamma)^{\Sigma}}$ has canonical singularities ([6], [12]), and
this can always be satisfied. When ${\Gamma}$ has no irregular cusp, the
condition (2) is vacuous, and this is the criterion in [6]; the choice of
$\Sigma$ does not matter with $F$ and can be dropped (or hidden) from the
criterion. Even when ${\Gamma}$ has an irregular cusp, if the weight $k$ is
even for $\chi=1$ or $k\equiv b$ mod $2$ for $\chi=\det$, the condition (2) is
still automatically satisfied by the cuspidality of $F$ (Proposition 8.3).
However, when ${\Gamma}$ has an irregular cusp and $k$ belongs to the
remaining parity, the condition (2) arises, and the choice of $\Sigma_{I}$ is
then involved with $F$. Practically it would not be very easy to check (or
achieve) $\nu_{\sigma}(F)\geq 2$ for specific $F$ and $\Sigma_{I}$. Probably
the most plausible scenario would be to expect and check that the group
${\Gamma}$ in question has no irregular cusp. We could say that this is a
small cost for using cusp forms of arbitrary weight.

By the examples discussed after Proposition 1.1, the general-type results in
[6], [8], [9], [2], [17], [15], [4] are not affected. This is our essential
purpose.

As a final remark concerning irregular cusps, it should be remembered that, in
[1], subgroups of ${\rm O}^{+}(L_{{\mathbb{R}}})/\pm{\rm id}$ are considered,
rather than of ${\rm O}^{+}(L_{{\mathbb{R}}})$. This means that the given
group ${\Gamma}<{{\rm O}^{+}(L)}$ is replaced by ${\langle\Gamma,-{\rm
id}\rangle}/\pm{{\rm id}}$. In this situation, it is not
${U(I)_{\mathbb{Z}}}={U(I)_{\mathbb{Q}}}\cap{\Gamma}$ but rather
${U(I)_{\mathbb{Z}}^{\prime}}={U(I)_{\mathbb{Q}}}\cap{\langle\Gamma,-{\rm
id}\rangle}$ that is written as $U(F)_{{\mathbb{Z}}}$ in the notation of [1].
This is a subtle difference that may arise when working with [1], and that is
related with irregularity.

To conclude, irregular cusps are cusps where the lattice for Fourier expansion
is smaller than the lattice of translation. It is the central element $-{\rm
id}$ in the Lie group ${\rm O}^{+}(L_{{\mathbb{R}}})$ that is eventually
responsible for the presence of such cusps. We need to be careful about such
cusps when using a cusp form of odd weight with $\chi=1$ or weight
$k\not\equiv b$ mod $2$ with $\chi=\det$ to construct a pluricanonical form on
${\mathcal{F}(\Gamma)^{\Sigma}}$, as a benefit of not dividing ${\rm
O}^{+}(L_{{\mathbb{R}}})$ by $-{{\rm id}}$.

I would like to thank Valery Gritsenko, Klaus Hulek, Shigeyuki Kondo and
Gregory Sankaran for their valuable comments.

###### Contents

  1. 1 Introduction
  2. 2 $0$-dimensional cusps
  3. 3 Irregular $0$-dimensional cusps
  4. 4 Examples
  5. 5 $1$-dimensional cusps
  6. 6 Irregular $1$-dimensional cusps
  7. 7 Toroidal compactification
  8. 8 Modular forms and pluricanonical forms

This article is organized as follows. In §2 we recall the structure of the
stabilizer of a $0$-dimensional cusp. In §3 we define and study irregular
$0$-dimensional cusps. In §4 we give examples of groups ${\Gamma}$
with/without irregular cusp. In §5 we recall the structure of the stabilizer
of a $1$-dimensional cusp. In §6 we study irregular $1$-dimensional cusps. In
§7 we study some basic properties of a toroidal compactification of
${\mathcal{F}(\Gamma)}$. In §8 we prove Theorem 1.2. The main contents of this
article are contained in §3, §4, §6 and §8. §2 and §5 are expository, but we
tried to be rather self-contained because of the subtle nature of irregular
cusps and for calculation of explicit examples in §4. The logical dependence
of these sections is as follows.

$\textstyle{\S\ref{sec: 0dim
cusp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\S\ref{sec:
irr 0dim
cusp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\S\ref{sec:
example}}$$\textstyle{\S\ref{sec: 1dim
cusp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\S\ref{sec:
irr 1dim
cusp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\S\ref{sec:
toroidal
cpt}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\S\ref{sec:
cusp form criterion}}$

Throughout the article, a lattice usually means a free ${\mathbb{Z}}$-module
of finite rank endowed with a nondegenerate integral symmetric bilinear form
$(\cdot,\cdot):L\times L\to{\mathbb{Z}}$. In a few occasions, we use the word
”lattice” just for a free ${\mathbb{Z}}$-module of finite rank, but no
confusion will likely to occur. The dual lattice
${\operatorname{Hom}}(L,{\mathbb{Z}})$ of $L$ will be denoted by $L^{\vee}$. A
sublattice $I\subset L$ is called primitive when $L/I$ is free, and isotropic
when $(I,I)\equiv 0$. A lattice $L$ is called even if $(l,l)\in 2{\mathbb{Z}}$
for every $l\in L$, but this is not assumed except in §4. We write $U$ for the
even unimodular lattice of signature $(1,1)$ given by the Gram matrix
$\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$.

##  2\. $0$-dimensional cusps

Let $L$ be a lattice of signature $(2,b)$. We write $Q=Q_{L}$ for the
isotropic quadric in ${{\mathbb{P}}}L_{{\mathbb{C}}}$ defined by
$(\omega,\omega)=0$. The Hermitian symmetric domain attached to $L$ is the
open set of $Q$

${\mathcal{D}}={\mathcal{D}_{L}}=\\{{\mathbb{C}}\omega\in
Q|(\omega,\bar{\omega})>0\\}^{+},$

where $+$ means the choice of a connected component. The domain
${\mathcal{D}}$ has two types of rational boundary components (cusps):
$0$-dimensional and $1$-dimensional cusps. They correspond to primitive
isotropic sublattices of $L$ of rank $1$ and $2$ respectively. In this section
we recall the structure of the stabilizer of a $0$-dimensional cusp and
partial toroidal compactification over it. Although the contents of this
section are quite standard (cf. [16], [10], [6], [11]), we tried to be rather
self-contained and explicit for two reasons: because of the subtle nature of
irregular cusps (§3), and for the sake of calculation of explicit examples
(§4).

###  2.1. Tube domain model

Throughout this section we fix a rank $1$ primitive isotropic sublattice $I$
of $L$. The $0$-dimensional cusp corresponding to $I$ is the point
${{\mathbb{P}}}I_{{\mathbb{C}}}$ of $Q$. We write

$L(I)=(I^{\perp}\cap L/I)\otimes I.$

Twist 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 quadratic form on $L(I)$, independent of the choice of
$I\simeq{\mathbb{Z}}$, which has signature $(1,b-1)$. We denote by
${\mathcal{C}_{I}}$ the positive cone in $L(I)_{{\mathbb{R}}}$, namely a
chosen 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 ${\mathcal{D}(I)}$. The 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

${\mathcal{D}(I)}\>\stackrel{{\scriptstyle\simeq}}{{\to}}\>{{\mathbb{P}}}(L/I)_{{\mathbb{C}}}-{{\mathbb{P}}}(I^{\perp}/I)_{{\mathbb{C}}}.$

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

${{\mathbb{P}}}(L/I)_{{\mathbb{C}}}-{{\mathbb{P}}}(I^{\perp}/I)_{{\mathbb{C}}}\simeq(I^{\perp}/I)_{{\mathbb{C}}}\otimes(I^{\prime})_{{\mathbb{C}}}^{\vee}\simeq
L(I)_{{\mathbb{C}}}.$

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

$\mathcal{D}_{I}=\\{\>Z\in L(I)_{{\mathbb{C}}}\>|\>{\rm
Im}(Z)\in{\mathcal{C}_{I}}\>\\}.$

This is the tube domain realization of ${\mathcal{D}}$ with respect to $I$.
Note that this is not canonical, depending on the choice of $I^{\prime}$ (=
choice of base point). Change of $I^{\prime}$ acts by translation on
$L(I)_{{\mathbb{C}}}$.

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

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

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

This is the unipotent part of ${\Gamma(I)_{\mathbb{Q}}}$ and can be explicitly
described as follows. For $m\otimes l\in L(I)_{{\mathbb{Q}}}$ the Eichler
transvection $E_{m\otimes l}\in{\Gamma(I)_{\mathbb{Q}}}$ is defined by (cf.
[16], [7])

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

where $\tilde{m}\in I^{\perp}_{{\mathbb{Q}}}=I^{\perp}_{{\mathbb{Q}}}\cap
L_{{\mathbb{Q}}}$ is an arbitrary lift of $m\in(I^{\perp}/I)_{{\mathbb{Q}}}$.
In particular, $E_{m\otimes l}(v)=v-(m,v)l$ when $v\in
I^{\perp}_{{\mathbb{Q}}}$. We have $E_{w}\circ
E_{w^{\prime}}=E_{w+w^{\prime}}$ for $w,w^{\prime}\in L(I)_{{\mathbb{Q}}}$.
Then we have the canonical isomorphism

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

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

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

We thus have the canonical exact sequence

(2.1) $0\to
L(I)_{{\mathbb{Q}}}\to{\Gamma(I)_{\mathbb{Q}}}\stackrel{{\scriptstyle\pi}}{{\to}}{\rm
O}^{+}(L(I)_{{\mathbb{Q}}})\times{\rm GL}(I)\to 1.$

This sequence non-canonically splits if we choose a lift
$I^{\perp}_{{\mathbb{Q}}}/I_{{\mathbb{Q}}}\hookrightarrow
I^{\perp}_{{\mathbb{Q}}}$ of $I^{\perp}_{{\mathbb{Q}}}/I_{{\mathbb{Q}}}$, or
equivalently, a rank $1$ sublattice $I^{\prime}$ of $L$ with
$(I,I^{\prime})\not\equiv 0$. In any such a splitting of
${\Gamma(I)_{\mathbb{Q}}}$, the element $-{\rm
id}_{L}\in{\Gamma(I)_{\mathbb{Q}}}$ is expressed as $-{\rm id}_{L}=({\rm
id}_{L(I)},-{\rm id}_{I},0)$ where $0\in L(I)_{{\mathbb{Q}}}$. 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 $L(I)_{{\mathbb{Q}}}$. In the induced
adjoint action of ${{\rm O}^{+}}(L(I)_{{\mathbb{Q}}})\times{\rm GL}(I)$ on
${U(I)_{\mathbb{Q}}}$, ${\rm GL}(I)=\\{\pm 1\\}$ acts trivially, and ${{\rm
O}^{+}}(L(I)_{{\mathbb{Q}}})$ acts by its natural action on
$L(I)_{{\mathbb{Q}}}$.

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

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

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

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

(2.2)
$0\to{U(I)_{\mathbb{Z}}}\to{\Gamma(I)_{\mathbb{Z}}}\to{\overline{\Gamma(I)}_{\mathbb{Z}}}\to
1.$

Although (2.1) splits, this does not necessarily mean that (2.2) 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

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

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

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

Then ${\overline{\Gamma(I)}_{\mathbb{Z}}}$ is canonically a subgroup of
${\overline{\Gamma(I)}_{\mathbb{Q}}}$ with
${\overline{\Gamma(I)}_{\mathbb{Z}}}\cap{U(I)_{\mathbb{Q}/\mathbb{Z}}}=\\{0\\}$.
Although the projection ${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{\rm
O}^{+}({U(I)_{\mathbb{Z}}})\times{\rm GL}(I)$ is injective, we should not view
${\overline{\Gamma(I)}_{\mathbb{Z}}}$ as a subgroup of ${\rm
O}^{+}({U(I)_{\mathbb{Z}}})\times{\rm GL}(I)$ when considering its action on
${\mathcal{D}}/{U(I)_{\mathbb{Z}}}$ (cf. [12] Appendix).

We choose a rank $1$ sublattice $I^{\prime}\subset L$ with
$(I,I^{\prime})\not\equiv 0$. As explained, this induces 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

${\mathcal{X}(I)}\simeq\mathcal{D}_{I}/{U(I)_{\mathbb{Z}}}\quad\subset\quad{\mathcal{D}(I)}/{U(I)_{\mathbb{Z}}}\simeq
T(I),$ (2.3) ${\overline{\Gamma(I)}_{\mathbb{Q}}}\>\simeq\>({\rm
O}^{+}({U(I)_{\mathbb{Z}}})\times{\rm
GL}(I))\ltimes{U(I)_{\mathbb{Q}/\mathbb{Z}}},$

both depending on the choice of $I^{\prime}$. Then the canonical action of
${\overline{\Gamma(I)}_{\mathbb{Z}}}$ on ${\mathcal{X}(I)}$ is induced from
the natural action of $({\rm O}^{+}({U(I)_{\mathbb{Z}}})\times{\rm
GL}(I))\ltimes{U(I)_{\mathbb{Q}/\mathbb{Z}}}$ on $T(I)$ via these (non-
canonical but compatible) isomorphisms. Here ${\rm
O}^{+}({U(I)_{\mathbb{Z}}})$ acts by torus automorphism fixing the identity,
${\rm GL}(I)$ acts trivially, and ${U(I)_{\mathbb{Q}/\mathbb{Z}}}$ acts by
translation. The action of ${\overline{\Gamma(I)}_{\mathbb{Z}}}$ on
${\mathcal{X}(I)}$ may have translation component, unless
${\overline{\Gamma(I)}_{\mathbb{Z}}}$ is contained in the section of ${\rm
O}^{+}({U(I)_{\mathbb{Z}}})\times{\rm GL}(I)$.

######  Remark 2.1.

Let $I={\mathbb{Z}}l$ and $\Gamma(l)_{{\mathbb{Z}}}<{\Gamma(I)_{\mathbb{Z}}}$
be the kernel of ${\Gamma(I)_{\mathbb{Z}}}\to{\rm GL}(I)$. When $-{{\rm
id}}\in{\Gamma}$, we have
${\Gamma(I)_{\mathbb{Z}}}=\Gamma(l)_{{\mathbb{Z}}}\times\\{\pm{{\rm id}}\\}$,
so we may replace ${\Gamma(I)_{\mathbb{Z}}}$ by $\Gamma(l)_{{\mathbb{Z}}}$
when considering action on ${\mathcal{D}}$, as was done in [12] Appendix. (The
last sentence of [12] Remark A.8 for ${\Gamma}={\widetilde{{\rm O}}^{+}(L)}$
is valid under the condition
$\Gamma(l)_{{\mathbb{Z}}}={\Gamma(I)_{\mathbb{Z}}}$ (e.g. ${\rm div}(I)>2$) or
$A_{L}$ 2-elementary, or ${\rm div}(I)=1$. It is also valid when ${\rm
div}(I)=2$ if $[l/2]\in A_{L}$ is indivisible.)

###  2.4. Partial toroidal compactification

We recall partial toroidal compactification of
${\mathcal{X}(I)}={\mathcal{D}}/{U(I)_{\mathbb{Z}}}$ following [1]. 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 ([1]) 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
$I^{\prime}\subset L$ with $(I,I^{\prime})\not\equiv 0$. 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 $I^{\prime}$. 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 §7), like ${\mathcal{D}}$ gives a local chart for
the interior points in ${\Gamma}\backslash{\mathcal{D}}$.

##  3\. Irregular $0$-dimensional cusps

We now study irregular $0$-dimensional cusps. Let ${\Gamma}$ be a finite-index
subgroup of ${{\rm O}^{+}(L)}$ and $I$ be a rank $1$ primitive isotropic
sublattice of $L$. We keep the notation from §2. We will define irregularity
in two stages: irregularity of a cusp (§3.1), and irregularity of a toroidal
boundary divisor over (or adjacent to) an irregular cusp (§3.2). The first
stage is concerned only with ${\Gamma}$, but the second stage is also involved
with an ${\Gamma(I)_{\mathbb{Z}}}$-admissible fan.

###  3.1. Irregularity

We give several equivalent definitions of irregularity of a $0$-dimensional
cusp in the following form.

######  Proposition 3.1.

The following conditions are equivalent.

  1. (1)

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

  2. (2)

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

  3. (3)

$-{{\rm id}}\not\in{\Gamma}$ and ${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{{\rm
O}^{+}}({U(I)_{\mathbb{Z}}})$ 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

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

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

######  Definition 3.2.

We say that the $0$-dimensional cusp $I$ is irregular for ${\Gamma}$ when
these properties hold, and regular otherwise.

###### Proof.

$(1)\Rightarrow(2)$: We see that $-{{\rm id}}\not\in{\Gamma}$ by
${\Gamma}\neq{\langle\Gamma,-{\rm id}\rangle}$. Let
$E_{w}\in{U(I)_{\mathbb{Z}}^{\prime}}$ but $E_{w}\not\in{U(I)_{\mathbb{Z}}}$.
Since ${\langle\Gamma,-{\rm id}\rangle}={\Gamma}\sqcup-{\Gamma}$, we have
$E_{w}\in-{\Gamma}$, and so $-E_{w}\in{\Gamma}$. Note that
${U(I)_{\mathbb{Z}}^{\prime}}/{U(I)_{\mathbb{Z}}}\simeq{\langle\Gamma,-{\rm
id}\rangle}/{\Gamma}$ 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 $-{{\rm
id}}\not\in{\Gamma}$, 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
${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{{\rm O}^{+}}({U(I)_{\mathbb{Z}}})$.

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

${{\rm
GL}}(I)\times{U(I)_{\mathbb{Q}/\mathbb{Z}}}={U(I)_{\mathbb{Q}/\mathbb{Z}}}\sqcup(-{{\rm
id}})\cdot{U(I)_{\mathbb{Q}/\mathbb{Z}}}.$

Since
${\overline{\Gamma(I)}_{\mathbb{Z}}}\cap{U(I)_{\mathbb{Q}/\mathbb{Z}}}=\\{0\\}$,
a nontrivial element of the kernel of
${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{{\rm O}^{+}}({U(I)_{\mathbb{Z}}})$
must be contained in $(-{{\rm id}})\cdot{U(I)_{\mathbb{Q}/\mathbb{Z}}}$, 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 $-{{\rm
id}}\not\in{\Gamma}$, we have $E_{w}\not\in{U(I)_{\mathbb{Z}}}$. This means
that $[w]\neq 0\in{U(I)_{\mathbb{Q}/\mathbb{Z}}}$.

$(4)\Rightarrow(2),(3)$: We choose a splitting of
${\overline{\Gamma(I)}_{\mathbb{Q}}}$ as in (2.3) 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 $\gamma_{1}\in{{\rm
O}^{+}}({U(I)_{\mathbb{Z}}})$, $\gamma_{2}\in{\rm GL}(I)$ and
$[w]\in{U(I)_{\mathbb{Q}/\mathbb{Z}}}$. If $\gamma$ acts on ${\mathcal{X}(I)}$
by a nonzero translation, we must have $\gamma_{1}={{\rm id}}_{L(I)}$ 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 ${{\rm
O}^{+}}({U(I)_{\mathbb{Z}}})$. Since
${\overline{\Gamma(I)}_{\mathbb{Z}}}\cap{U(I)_{\mathbb{Q}/\mathbb{Z}}}=\\{0\\}$
and $[w]\neq 0$, we have $\gamma_{2}=-{{\rm id}}_{I}$. Thus $\gamma=-E_{w}$.
Finally, we have $-{{\rm id}}\not\in{\Gamma}$, for otherwise $E_{w}=-\gamma$
would be contained in ${U(I)_{\mathbb{Z}}}$ and then
$[w]=0\in{U(I)_{\mathbb{Q}/\mathbb{Z}}}$. ∎

######  Remark 3.3.

Let $\Gamma(l)_{{\mathbb{Z}}}<{\Gamma(I)_{\mathbb{Z}}}$ be as in Remark 2.1.
By the condition (3), $I$ is irregular if and only if $-{{\rm
id}}\not\in{\Gamma}$, $\Gamma(l)_{{\mathbb{Z}}}\neq{\Gamma(I)_{\mathbb{Z}}}$,
and $\Gamma(l)_{{\mathbb{Z}}}$ and ${\Gamma(I)_{\mathbb{Z}}}$ have the same
image in ${{\rm O}^{+}}({U(I)_{\mathbb{Z}}})$. We do not use this
characterization.

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

######  Corollary 3.4.

The group ${\Gamma}$ has no irregular cusp when $-{{\rm id}}\in{\Gamma}$ or
when ${\Gamma}$ is neat or when ${\Gamma}<{\rm SO}^{+}(L)$ with $b$ odd.

###### Proof.

The case $-{{\rm id}}\in{\Gamma}$ is obvious. When ${\Gamma}$ is neat, the
subquotient ${\overline{\Gamma(I)}_{\mathbb{Z}}}$ is torsion-free, so it does
not contain an element of finite order like $-E_{w}$. When $b$ is odd,
$-E_{w}$ has determinant $(-1)^{b+2}=-1$, so a subgroup of ${\rm SO}^{+}(L)$
never contains such an element. ∎

######  Corollary 3.5.

When $b$ is even, ${\Gamma}$ has an irregular cusp if and only if
${\Gamma}\cap{\rm SO}^{+}(L)$ has an irregular cusp.

###### Proof.

When $b$ is even, both $-{{\rm id}}$ and $-E_{w}$ are contained in ${\rm
SO}^{+}(L)$. ∎

######  Corollary 3.6.

If ${\Gamma}$ has an irregular cusp, any $\Gamma^{\prime}<{{\rm O}^{+}(L)}$
with $\Gamma^{\prime}\supset{\Gamma}$ and $-{{\rm id}}\not\in\Gamma^{\prime}$
has an irregular cusp. Equivalently, if $-{{\rm id}}\not\in{\Gamma}$ and
${\Gamma}$ has no irregular cusp, any subgroup of ${\Gamma}$ of finite index
has no irregular cusp.

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

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

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

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

As we will explain in §8.1, ${U(I)_{\mathbb{Z}}}$ is the lattice for Fourier
expansion of ${\Gamma}$-modular forms around $I$. Thus irregular
$0$-dimensional cusps are those cusps whose lattice of translation is larger
than the lattice for Fourier expansion.

###  3.2. Irregular boundary divisors

Let $\Sigma=(\sigma_{\alpha})$ be a ${\Gamma(I)_{\mathbb{Z}}}$-admissible fan
in $U(I)_{{\mathbb{R}}}$, and
${\mathcal{X}(I)}\hookrightarrow{\mathcal{X}(I)^{\Sigma}}$ be the partial
compactification defined in §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.

######  Proposition 3.7.

Let $I$ be an irregular $0$-dimensional cusp for ${\Gamma}$. Let
$-E_{w}\in{\Gamma(I)_{\mathbb{Z}}}$. The following conditions for a ray
$\sigma$ in $\Sigma$ are equivalent:

  1. (1)

$\sigma\cap{U(I)_{\mathbb{Z}}}\neq\sigma\cap{U(I)_{\mathbb{Z}}^{\prime}}$.

  2. (2)

$-E_{w}$ acts trivially on the boundary divisor $D(\sigma)$.

  3. (3)

$D(\sigma)$ is fixed by some nontrivial element of
${\overline{\Gamma(I)}_{\mathbb{Z}}}$.

When these hold, the element in $(3)$ is given by $-E_{w}$. In particular, it
is unique, independent of $\sigma$, and of order $2$.

######  Definition 3.8.

When these properties hold, we call $\sigma$ an irregular ray and $D(\sigma)$
an irregular boundary divisor. Otherwise we call $\sigma$ regular. For the
sake of completeness, we call any ray $\sigma$ regular when $I$ is regular.

###### Proof.

$(1)\Leftrightarrow(2)$: Recall that $-E_{w}$ acts on ${\mathcal{X}(I)}\subset
T(I)$ as the translation by $[w]\in{U(I)_{\mathbb{Q}/\mathbb{Z}}}$. A Zariski
open set of $D(\sigma)$ is the quotient torus (or its analytic open set)
defined by the quotient lattice ${U(I)_{\mathbb{Z}}}/\Lambda_{\sigma}$ where
$\Lambda_{\sigma}={\mathbb{R}}\sigma\cap{U(I)_{\mathbb{Z}}}$. Hence $-E_{w}$
acts on $D(\sigma)$ as the translation by the image of $[w]$ in
${U(I)_{\mathbb{Q}}}/({U(I)_{\mathbb{Z}}}+(\Lambda_{\sigma})_{{\mathbb{Q}}})$.
This is trivial if and only if
$w\in{U(I)_{\mathbb{Z}}}+(\Lambda_{\sigma})_{{\mathbb{Q}}}$, which in turn is
equivalent to
$\Lambda_{\sigma}\neq{\mathbb{R}}\sigma\cap{U(I)_{\mathbb{Z}}^{\prime}}$. In
this case $-E_{w}$ acts by $-1$ on the normal torus
$(\Lambda_{\sigma})_{{\mathbb{C}}}/\Lambda_{\sigma}\simeq{\mathbb{C}}^{\times}$.

$(2)\Rightarrow(3)$ is obvious.

$(3)\Rightarrow(2)$: Suppose that
$\gamma\in{\overline{\Gamma(I)}_{\mathbb{Z}}}$ acts trivially on $D(\sigma)$.
Let $\gamma_{1}$ be the image of $\gamma$ in ${{\rm
O}^{+}}({U(I)_{\mathbb{Z}}})$. Then $\gamma_{1}$ must preserve
$\sigma\cap{U(I)_{\mathbb{Z}}}$ and act trivially on
${U(I)_{\mathbb{Z}}}/\Lambda_{\sigma}$. Hence $\gamma_{1}$ acts trivially on
$\Lambda_{\sigma}$ and $\Lambda_{\sigma}^{\perp}$, and so $\gamma_{1}={{\rm
id}}$. This implies that $\gamma$ is contained in the kernel of
${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{{\rm O}^{+}}({U(I)_{\mathbb{Z}}})$,
whence $\gamma=-E_{w}$ by Proposition 3.1. Therefore $-E_{w}$ acts trivially
on $D(\sigma)$. ∎

######  Corollary 3.9.

When $I$ is irregular, the quotient map
${\mathcal{X}(I)^{\Sigma}}\to{\mathcal{X}(I)^{\Sigma}}/{\overline{\Gamma(I)}_{\mathbb{Z}}}$
is ramified along the irregular boundary divisors with ramification index $2$,
caused by the common subgroup $\langle-E_{w}\rangle\simeq{\mathbb{Z}}/2$ of
${\overline{\Gamma(I)}_{\mathbb{Z}}}$, and not ramified along other boundary
divisors. When $I$ is regular,
${\mathcal{X}(I)^{\Sigma}}\to{\mathcal{X}(I)^{\Sigma}}/{\overline{\Gamma(I)}_{\mathbb{Z}}}$
is not ramified along any boundary divisor.

###### Proof.

It remains to supplement the argument in the case $I$ is regular. If
$\gamma\in{\overline{\Gamma(I)}_{\mathbb{Z}}}$ fixes a boundary divisor, we
see that $\gamma$ acts trivially on ${U(I)_{\mathbb{Z}}}$ by the same argument
as $(3)\Rightarrow(2)$ above. When $-{{\rm id}}\not\in{\Gamma}$,
${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{{\rm O}^{+}}({U(I)_{\mathbb{Z}}})$ is
injective by the condition (3) of Proposition 3.1, so we find that
$\gamma={{\rm id}}$. When $-{{\rm id}}\in{\Gamma}$, the kernel of
${\overline{\Gamma(I)}_{\mathbb{Z}}}\to{{\rm O}^{+}}({U(I)_{\mathbb{Z}}})$ is
$\\{\pm{{\rm id}}\\}$, so $\gamma=\pm{{\rm id}}$, which acts trivially on
${\mathcal{X}(I)}$. ∎

##  4\. Examples

In this section we study some examples of groups with/without irregular cusp.
Logically this section should be read after §6 where we complete the
discussion of irregular $1$-dimensional cusps. But we encourage the reader to
read this section just after §3 for the following two reasons. First, most
part of §5 and §6 is designed for §7 and 8, while the only result from §5 and
§6 we need in this section is Corollary 6.5, which just says that ${\Gamma}$
has no irregular $1$-dimensional cusp if it has no irregular $0$-dimensional
cusp. Second, it is Proposition 3.1 (2) that is frequently used in this
section, so we do not want to put this section too far from it.

We assume that the lattice $L$ is even in this section. The quotient
$A_{L}=L^{\vee}/L$ is called the discriminant group of $L$, equipped with a
canonical quadratic form $A_{L}\to{\mathbb{Q}}/2{\mathbb{Z}}$ called the
discriminant form. If $I$ is a rank $1$ primitive isotropic sublattice of $L$,
we write ${\rm div}(I)$ for the positive generator of the ideal
$(I,L)\subset{\mathbb{Z}}$. Then $I^{\ast}={\rm div}(I)^{-1}I$ is primitive in
$L^{\vee}$, and we have a canonical isometry $I^{\perp}\cap
L^{\vee}/I^{\ast}\simeq(I^{\perp}/I)^{\vee}$.

###  4.1. Stable orthogonal groups

Let $L$ be an even lattice of signature $(2,b)$. Let ${\widetilde{{\rm
O}}^{+}(L)}<{{\rm O}^{+}(L)}$ be the kernel of the reduction map ${{\rm
O}^{+}(L)}\to{\rm O}(A_{L})$, called the stable orthogonal group or the
discriminant kernel. The following was asserted in [12] p.901. We supplement
the proof for the sake of completeness.

######  Lemma 4.1.

Let $I$ be a rank $1$ primitive isotropic sublattice of $L$. For
${\Gamma}={\widetilde{{\rm O}}^{+}(L)}$ we have ${U(I)_{\mathbb{Z}}}=L(I)$.

###### Proof.

We take a generator $l$ of $I$. The inclusion $L(I)\subset{U(I)_{\mathbb{Z}}}$
can be checked by testing the definition of $E_{m\otimes l}(v)$ for $v\in
L^{\vee}$ and $m\in I^{\perp}/I$, taking a lift of $m$ from $I^{\perp}\cap L$.
Conversely, if $E_{m\otimes l}\in{\widetilde{{\rm O}}^{+}(L)}$ for a vector
$m\in I^{\perp}_{{\mathbb{Q}}}/I_{{\mathbb{Q}}}$, then $E_{m\otimes
l}(v)=v-(m,v)l$ must be contained in $v+L$ for $v\in I^{\perp}\cap L^{\vee}$.
This implies that $(m,v)\in{\mathbb{Z}}$ for every $v\in I^{\perp}\cap
L^{\vee}$, and so $m\in(I^{\perp}/I)^{\vee\vee}=I^{\perp}/I$. ∎

We obtain a first example of regular cusps.

######  Lemma 4.2.

If ${\Gamma}\supset{\widetilde{{\rm O}}^{+}(L)}$ and ${\rm div}(I)=1$, then
$I$ is a regular cusp for ${\Gamma}$.

###### Proof.

We take a generator $l$ of $I$. Since ${\rm div}(I)=1$, we can take an
isotropic vector $l^{\prime}\in L$ with $(l,l^{\prime})=1$. We can and do
identify $I^{\perp}/I$ with $\langle l,l^{\prime}\rangle^{\perp}\cap L$. We
have the splitting $L=\langle l,l^{\prime}\rangle\oplus(\langle
l,l^{\prime}\rangle^{\perp}\cap L)$. Suppose $-E_{m\otimes l}\in{\Gamma}$ for
a vector $m\in\langle l,l^{\prime}\rangle^{\perp}\cap L_{{\mathbb{Q}}}$. Then
$E_{m\otimes l}$ preserves $L$, so we find that the vector $E_{m\otimes
l}(l^{\prime})=l^{\prime}+m-\frac{1}{2}(m,m)l$ is contained in $L$. This
implies that $m\in\langle l,l^{\prime}\rangle^{\perp}\cap L=I^{\perp}/I$.
Since ${\Gamma}\supset{\widetilde{{\rm O}}^{+}(L)}$, we have
${U(I)_{\mathbb{Z}}}\supset L(I)$ by Lemma 4.1, and so $m\otimes
l\in{U(I)_{\mathbb{Z}}}$. This means that $E_{m\otimes l}\in{\Gamma}$, and
then $-{{\rm id}}\in{\Gamma}$. ∎

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

######  Lemma 4.3.

If $I$ is an irregular $0$-dimensional cusp for ${\Gamma}={\widetilde{{\rm
O}}^{+}(L)}$, then ${\rm div}(I)=2$, $A_{L(I)}$ is 2-elementary, and
${U(I)_{\mathbb{Z}}^{\prime}}/{U(I)_{\mathbb{Z}}}\simeq{\mathbb{Z}}/2$ is a
subgroup of $A_{L(I)}$.

###### Proof.

Let $I={\mathbb{Z}}l$ and assume that $-E_{m\otimes l}\in{\widetilde{{\rm
O}}^{+}(L)}$ 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) $2v\in(m,v)l+L.$

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

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 ${\widetilde{{\rm O}}^{+}(L)}$
has an irregular $0$-dimensional cusp.

######  Proposition 4.4.

If ${\widetilde{{\rm O}}^{+}(L)}$ has an irregular cusp, then
$A_{L}\simeq{\mathbb{Z}}/8\oplus({\mathbb{Z}}/2)^{\oplus a}$ or
$A_{L}\simeq({\mathbb{Z}}/4)^{\oplus 2}\oplus({\mathbb{Z}}/2)^{\oplus a}$ as
abelian groups.

###### Proof.

Let $I={\mathbb{Z}}l$ be as in Lemma 4.3. Let $x=[l/2]\in A_{L}$. Since
$A_{L(I)}\simeq x^{\perp}/x$ is 2-elementary, $A_{L}$ must be isomorphic to
either $({\mathbb{Z}}/2)^{\oplus a}$ or
${\mathbb{Z}}/4\oplus({\mathbb{Z}}/2)^{\oplus a}$ or
${\mathbb{Z}}/8\oplus({\mathbb{Z}}/2)^{\oplus a}$ or $({\mathbb{Z}}/4)^{\oplus
2}\oplus({\mathbb{Z}}/2)^{\oplus a}$. The first case
$A_{L}\simeq({\mathbb{Z}}/2)^{\oplus a}$ cannot occur because then $-{{\rm
id}}\in{\widetilde{{\rm O}}^{+}(L)}$. The second case
$A_{L}\simeq{\mathbb{Z}}/4\oplus({\mathbb{Z}}/2)^{\oplus a}$ does not occur
too, because then $x^{\perp}/x$ would contain an element of order $4$. ∎

######  Remark 4.5.

Further calculation shows that $x=[l/2]\in A_{L}$ is divisible by $4$ (and
hence unique) in the case
$A_{L}\simeq{\mathbb{Z}}/8\oplus({\mathbb{Z}}/2)^{\oplus a}$, and divisible by
$2$ in the case $A_{L}\simeq({\mathbb{Z}}/4)^{\oplus
2}\oplus({\mathbb{Z}}/2)^{\oplus a}$.

######  Example 4.6.

Let $L=2U\oplus mE_{8}\oplus\langle-2d\rangle$. Then ${\widetilde{{\rm
O}}^{+}(L)}$ has no irregular cusp when $d\neq 4$. We show in Proposition 4.14
that ${\widetilde{{\rm O}}^{+}(L)}$ indeed has an irregular cusp when $d=4$.
When $m=2$, ${\widetilde{{\rm O}}^{+}(L)}$ is the modular group for the moduli
space of polarized $K3$ surfaces of degree $2d$.

######  Example 4.7.

Let $L=2U\oplus mE_{8}\oplus\langle-2t\rangle\oplus\langle-2d\rangle$. Then
${\widetilde{{\rm O}}^{+}(L)}$ has no irregular cusp when
$(t,d)\neq(4,1),(2,2),(1,4)$. We show in §4.5 that ${\widetilde{{\rm
O}}^{+}(L)}$ indeed has an irregular cusp in these exceptional cases. When
$m=2$, ${\widetilde{{\rm O}}^{+}(L)}$ is the modular group for the moduli
space of polarized irreducible symplectic manifolds of $K3^{[t-1]}$-type with
polarization of split type and degree $2d$ ([8]).

######  Example 4.8.

When $L=U\oplus 2E_{8}\oplus M$, where $M$ is a certain lattice of signature
$(1,2)$ and discriminant $d\equiv 2$ mod $6$, ${\widetilde{{\rm O}}^{+}(L)}$
is the modular group for the moduli space of special cubic fourfolds of
discriminant $d$ ([17]). Since $A_{L}$ has length $\leq 3$ and order $d$, we
find that ${\widetilde{{\rm O}}^{+}(L)}$ has no irregular cusp when $d\neq
8,32$.

######  Example 4.9.

Similarly, when $L=U\oplus 2E_{8}\oplus M$, where $M$ is a certain lattice of
signature $(1,2)$ and discriminant $d\equiv 0,2,4$ mod $8$, ${\widetilde{{\rm
O}}^{+}(L)}$ is the modular group for the moduli space of special
$K3^{[2]}$-fourfolds of degree $2$ and discriminant $d$ ([15]). This group has
no irregular cusp when $d\neq 32$.

######  Example 4.10.

When $L=U\oplus 2E_{8}\oplus\langle 2d\rangle$, ${\widetilde{{\rm O}}^{+}(L)}$
is the modular group for the moduli space of
$U\oplus\langle-2d\rangle$-polarized $K3$ surfaces studied in [4]. This group
has no irregular cusp when $d\neq 4$.

###  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

${\Gamma}=\\{\>\gamma\in{{\rm O}^{+}(L)}\>|\>\gamma|_{A_{M}}=\pm{{\rm
id}},\>\gamma|_{A_{\langle-2d\rangle}}={{\rm id}}\;\\}.$

Then ${\Gamma}$ contains ${\widetilde{{\rm O}}^{+}(L)}$ with index $\leq 2$,
with ${\Gamma}={\widetilde{{\rm O}}^{+}(L)}$ if and only if $A_{M}$ is
2-elementary. We have $-{{\rm id}}\in{\Gamma}$ 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 $2d$ ([9]).

######  Proposition 4.11.

The group ${\Gamma}$ has no irregular cusp when $d\neq 2,4$.

###### 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
$-{{\rm id}}\not\in{\Gamma}$. We shall show that $d\mid 4$. Since
$E_{2m\otimes l}=(-E_{m\otimes l})\circ(-E_{m\otimes l})\in{\widetilde{{\rm
O}}^{+}(L)}$, 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

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

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

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

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 ${\rm div}(I)=1$. 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 §4.2 and
consider the group

${\Gamma}=\\{\>\gamma\in{{\rm O}^{+}(L)}\>|\>\gamma|_{A_{M}}=\det(\gamma){{\rm
id}},\>\gamma|_{A_{\langle-2d\rangle}}={{\rm id}}\;\\}.$

This is an index $\leq 2$ subgroup of the group considered in §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 $2d$ ([2]).

######  Proposition 4.12.

The group ${\Gamma}$ has no irregular cusp when $d\nmid 4$. Moreover, when $b$
is even, ${\Gamma}$ has no irregular cusp unless when $A_{L}$ is isomorphic to
${\mathbb{Z}}/8\oplus({\mathbb{Z}}/2)^{\oplus a}$ or $({\mathbb{Z}}/4)^{\oplus
2}\oplus({\mathbb{Z}}/2)^{\oplus a}$ as abelian groups.

###### Proof.

The assertion $d\nmid 4$ follows from Corollary 3.6 and Proposition 4.11.
Since ${\Gamma}\cap{\rm SO}^{+}(L)={\widetilde{{\rm O}}^{+}(L)}\cap{\rm
SO}^{+}(L)$, Corollary 3.5 shows that when $b$ is even, ${\Gamma}$ has an
irregular cusp if and only if ${\widetilde{{\rm O}}^{+}(L)}$ has an irregular
cusp. Then our assertion follows from Proposition 4.4. ∎

This shows that when $M=2U\oplus\langle-2t\rangle$, ${\Gamma}$ has no
irregular cusp when $(t,d)\neq(4,1),(2,2),(1,4)$. In §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

${\Gamma}=\\{\>\gamma\in{{\rm O}^{+}(L)}\>|\>\gamma|_{A_{M}}=\pm{{\rm
id}},\>\gamma|_{A_{K}}={{\rm id}}\;\\}.$

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 $6n$ ([17]).

######  Proposition 4.13.

The group ${\Gamma}$ has no irregular cusp.

###### Proof.

Suppose to the contrary that $I={\mathbb{Z}}l$ is an irregular cusp and
$-E_{m\otimes l}\in{\Gamma}$. As in the proof of Proposition 4.11, we can take
a lift of $m$ from $I^{\perp}\cap\frac{1}{2}L$. We take a vector $v\in
K^{\vee}-K$. Then $-E_{m\otimes l}(v)$ must be contained in $v+L$. The same
calculation as (4.2) tells us that $8v\in L$. Therefore $[v]\in A_{K}\subset
A_{L}$ satisfies $8[v]=0$, but this contradicts the assumption that $|A_{K}|$
is odd. ∎

###  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

${\Gamma}=\\{\>\gamma\in{{\rm O}^{+}(L)}\>|\>\gamma|_{A_{M}}=\pm{{\rm
id}},\>\gamma|_{A_{\langle-8\rangle}}={{\rm id}}\;\\}.$

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

######  Proposition 4.14.

The group ${\Gamma}$ has an irregular cusp.

###### Proof.

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

$l=2e+2f+v,\quad m=e/2-f/2,$

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

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

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

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

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

${\Gamma}=\\{\>\gamma\in{{\rm O}^{+}(L)}\>|\>\gamma|_{A_{M}}=\pm{{\rm
id}},\>\gamma|_{A_{\langle-4\rangle^{\oplus 2}}}={{\rm id}}\;\\}.$

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

######  Proposition 4.15.

The group ${\Gamma}$ has an irregular cusp.

###### 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

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

The detail is left to the reader. ∎

##  5\. $1$-dimensional cusps

In this section we recall, following [16], [10], [6], [11], the structure of
the stabilizer of a $1$-dimensional cusp of ${\mathcal{D}}={\mathcal{D}_{L}}$
with its action on the Siegel domain model, and the canonical partial toroidal
compactification over the cusp. This is a long preliminary for the next §6.
Although this section is no more than expository, we need to keep the rather
self-contained style of §2, for the same reasons as in §2 and for consistency.

Throughout this section we fix a rank $2$ primitive isotropic sublattice $J$
of $L$. The choice of the component ${\mathcal{D}}$ determines a connected
component of ${{\mathbb{P}}}J_{{\mathbb{C}}}-{{\mathbb{P}}}J_{{\mathbb{R}}}$,
which is the cusp corresponding to $J$. This in turn determines an orientation
of $J$. We write

$L(J)=J^{\perp}\cap L/J,$

which is a negative-definite lattice of rank $b-2$. We will call an embedding
$2U_{{\mathbb{Q}}}\hookrightarrow L_{{\mathbb{Q}}}$ a splitting for
$J_{{\mathbb{Q}}}$ if it sends the standard $2$-dimensional isotropic subspace
of $2U_{{\mathbb{Q}}}$ to $J_{{\mathbb{Q}}}$. This defines a lift
$L(J)_{{\mathbb{Q}}}\hookrightarrow J^{\perp}_{{\mathbb{Q}}}$ of
$L(J)_{{\mathbb{Q}}}$ as $2U_{{\mathbb{Q}}}^{\perp}$.

###  5.1. Siegel domain model

We consider restriction of the two-step projection
${{\mathbb{P}}}L_{{\mathbb{C}}}\dashrightarrow{{\mathbb{P}}}(L/J)_{{\mathbb{C}}}\dashrightarrow{{\mathbb{P}}}(L/J^{\perp})_{{\mathbb{C}}}$
to ${\mathcal{D}}\subset Q\subset{{\mathbb{P}}}L_{{\mathbb{C}}}$. Since $Q$
contains the line ${{\mathbb{P}}}J_{{\mathbb{C}}}$, the center of the first
projection, a plane containing ${{\mathbb{P}}}J_{{\mathbb{C}}}$ either
intersects with $Q$ at two lines (one ${{\mathbb{P}}}J_{{\mathbb{C}}}$) or is
contained in $Q$. The latter occurs when the plane is contained in
${{\mathbb{P}}}J^{\perp}_{{\mathbb{C}}}$. This shows that the first projection

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

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

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

is an affine space bundle isomorphic to
$L(J)_{{\mathbb{C}}}\otimes\mathcal{O}_{{{\mathbb{P}}}J^{\vee}_{{\mathbb{C}}}}(1)$.
Here, 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).
A choice of a splitting for $J_{{\mathbb{C}}}$ determines a section of
$\pi_{2}$.

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 ${\mathbb{C}}\omega\in Q$ 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 for ${\mathcal{D}}$:

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

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 an upper half space bundle. Over
${\mathbb{H}_{J}}\subset{{\mathbb{P}}}J_{{\mathbb{C}}}^{\vee}$ we have the
Hodge bundles

$(F^{\vee})^{1,0}=\mathcal{O}_{{\mathbb{H}_{J}}}(-1)\subset\underline{J_{{\mathbb{C}}}^{\vee}},\quad
F^{1,0}=((F^{\vee})^{1,0})^{\perp}\subset\underline{J_{{\mathbb{C}}}},$

where we write
$\underline{J_{{\mathbb{C}}}^{\vee}}=J_{{\mathbb{C}}}^{\vee}\otimes\mathcal{O}_{{\mathbb{H}_{J}}}$
and
$\underline{J_{{\mathbb{C}}}}=J_{{\mathbb{C}}}\otimes\mathcal{O}_{{\mathbb{H}_{J}}}$.
Then
$\mathcal{O}_{{\mathbb{H}_{J}}}(1)\simeq\underline{J_{{\mathbb{C}}}}/F^{1,0}$
canonically. This shows that ${\mathcal{V}(J)}\to{\mathbb{H}_{J}}$ is an
affine space bundle isomorphic to
$L(J)_{{\mathbb{C}}}\otimes(\underline{J_{{\mathbb{C}}}}/F^{1,0})$.

The relationship with the tube domain model is as follows. We choose a rank
$1$ primitive sublattice $I$ of $J$. This corresponds to a $0$-dimensional
cusp in the closure of the $1$-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 ${\rm O}^{+}(L_{{\mathbb{Q}}})$ that acts on
$J_{{\mathbb{Q}}}$ with determinant $1$. The determinant $1$ condition is not
restrictive when restricting to subgroups of ${{\rm O}^{+}(L)}$. We write

${W(J)_{\mathbb{Q}}}={\operatorname{Ker}}({\Gamma(J)_{\mathbb{Q}}}\to{\rm
O}(L(J)_{{\mathbb{Q}}})\times{\rm SL}(J_{{\mathbb{Q}}})),$
${U(J)_{\mathbb{Q}}}={\operatorname{Ker}}({\Gamma(J)_{\mathbb{Q}}}\to{\rm
GL}(J_{{\mathbb{Q}}}^{\perp})),$
${V(J)_{\mathbb{Q}}}={W(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Q}}}.$

By definition we have the canonical exact sequences

(5.1) $1\to{W(J)_{\mathbb{Q}}}\to{\Gamma(J)_{\mathbb{Q}}}\to{\rm
O}(L(J)_{{\mathbb{Q}}})\times{\rm SL}(J_{{\mathbb{Q}}})\to 1,$
$0\to{U(J)_{\mathbb{Q}}}\to{W(J)_{\mathbb{Q}}}\to{V(J)_{\mathbb{Q}}}\to 0.$

The group ${W(J)_{\mathbb{Q}}}$ is the unipotent part of
${\Gamma(J)_{\mathbb{Q}}}$. The first sequence splits non-canonically if we
choose a splitting for $J_{{\mathbb{Q}}}$, while the second never splits,
being a Heisenberg group.

We have a $\wedge^{2}J$-valued symplectic form on $L(J)\otimes J$ induced from
the quadratic form on $L(J)$ and the canonical symplectic form $J\times
J\to\wedge^{2}J$. This gives a Heisenberg group structure on
$\wedge^{2}J_{{\mathbb{Q}}}\times(L(J)_{{\mathbb{Q}}}\otimes
J_{{\mathbb{Q}}})$. Explicitly, we identify
$\wedge^{2}J_{{\mathbb{Q}}}\simeq{\mathbb{Q}}$ and 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 product
on ${\mathbb{Q}}\times L(J)_{{\mathbb{Q}}}\times L(J)_{{\mathbb{Q}}}$ by

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

The center is ${\mathbb{Q}}\times\\{0\\}\times\\{0\\}$.

######  Lemma 5.1 (cf. [11]).

${W(J)_{\mathbb{Q}}}$ 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

$\wedge^{2}J_{{\mathbb{Q}}}\to{U(J)_{\mathbb{Q}}},\quad l\wedge
l^{\prime}\mapsto E_{l\otimes l^{\prime}},$ $L(J)_{{\mathbb{Q}}}\otimes
J_{{\mathbb{Q}}}\to{V(J)_{\mathbb{Q}}},\quad m\otimes l\mapsto
E_{\tilde{m}\otimes l}\;{\rm mod}\;{U(J)_{\mathbb{Q}}}.$

###### Proof.

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 (2.1) 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

(5.2)
${U(J)_{\mathbb{Q}}}=(\bar{J}_{{\mathbb{Q}}}^{\perp})^{\perp}=\bar{J}_{{\mathbb{Q}}}\simeq\wedge^{2}J_{{\mathbb{Q}}}\quad\subset{U(I)_{\mathbb{Q}}}.$

The image of ${W(J)_{\mathbb{Q}}}\to{{\rm O}^{+}}(L(I)_{{\mathbb{Q}}})$ 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

(5.3) $0\to\bar{J}_{{\mathbb{Q}}}^{\perp}\cap
L(I)_{{\mathbb{Q}}}\to{W(J)_{\mathbb{Q}}}\to
L(J)_{{\mathbb{Q}}}\otimes(J/I)_{{\mathbb{Q}}}\to 0.$

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 (5.3) 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

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

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 $U(J)_{{\mathbb{C}}}\subset{\rm O}(L_{{\mathbb{C}}})$ be the group of
Eichler transvections $E_{l\otimes l^{\prime}}$ with $l,l^{\prime}\in
J_{{\mathbb{C}}}$. Then $U(J)_{{\mathbb{C}}}\simeq\wedge^{2}J_{{\mathbb{C}}}$
acts on ${\mathcal{D}}(J)$ leaving ${\mathcal{V}(J)}$ fixed. One observes that

  * •

${\mathcal{D}}(J)\to{\mathcal{V}(J)}$ is a principal
$U(J)_{{\mathbb{C}}}$-bundle.

  * •

${V(J)_{\mathbb{Q}}}\simeq L(J)_{{\mathbb{Q}}}\otimes J_{{\mathbb{Q}}}$ acts
on ${\mathcal{V}(J)}\to{\mathbb{H}_{J}}$ by relative translation through
${\mathcal{V}(J)}\simeq
L(J)_{{\mathbb{C}}}\otimes(\underline{J_{{\mathbb{C}}}}/F^{1,0})$.

  * •

If we choose a splitting for $J_{{\mathbb{Q}}}$, which induces a splitting of
(5.1) and the above isomorphism for ${\mathcal{V}(J)}$, then the section of
${\rm O}(L(J)_{{\mathbb{Q}}})\times{\rm SL}(J_{{\mathbb{Q}}})$ acts on
${\mathcal{V}(J)}\to{\mathbb{H}_{J}}$ by the equivariant action of ${\rm
SL}(J_{{\mathbb{Q}}})$ on $\underline{J_{{\mathbb{C}}}}/F^{1,0}$ and the
linear action of ${\rm O}(L(J)_{{\mathbb{Q}}})$ on $L(J)_{{\mathbb{C}}}$.

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

Let ${\Gamma}$ be a finite-index subgroup of ${{\rm O}^{+}(L)}$. We set

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

and consider the quotients

${\overline{\Gamma(J)}_{\mathbb{Z}}}={\Gamma(J)_{\mathbb{Z}}}/{U(J)_{\mathbb{Z}}},\quad{V(J)_{\mathbb{Z}}}={W(J)_{\mathbb{Z}}}/{U(J)_{\mathbb{Z}}},\quad\Gamma_{J}={\Gamma(J)_{\mathbb{Z}}}/{W(J)_{\mathbb{Z}}},$
${\overline{\Gamma(J)}_{\mathbb{Q}}}={\Gamma(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Z}}},\;\;{W(J)_{\mathbb{Q}/\mathbb{Z}}}={W(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Z}}},\;\;{U(J)_{\mathbb{Q}/\mathbb{Z}}}={U(J)_{\mathbb{Q}}}/{U(J)_{\mathbb{Z}}}.$

By definition we have the canonical exact sequences

(5.4)
$0\to{V(J)_{\mathbb{Z}}}\to{\overline{\Gamma(J)}_{\mathbb{Z}}}\to\Gamma_{J}\to
1,$ (5.5)
$0\to{W(J)_{\mathbb{Q}/\mathbb{Z}}}\to{\overline{\Gamma(J)}_{\mathbb{Q}}}\to{\rm
O}(L(J)_{{\mathbb{Q}}})\times{\rm SL}(J_{{\mathbb{Q}}})\to 1,$
$0\to{U(J)_{\mathbb{Q}/\mathbb{Z}}}\to{W(J)_{\mathbb{Q}/\mathbb{Z}}}\to{V(J)_{\mathbb{Q}}}\to
0.$

Then (5.4) is canonically embedded in (5.5). We have
${V(J)_{\mathbb{Z}}}\cap{U(J)_{\mathbb{Q}/\mathbb{Z}}}=\\{0\\}$ as subgroups
of ${W(J)_{\mathbb{Q}/\mathbb{Z}}}$ because the projection
${V(J)_{\mathbb{Z}}}\to{V(J)_{\mathbb{Q}}}$ is injective. Note that
${U(J)_{\mathbb{Q}/\mathbb{Z}}}$ is the group of torsion points of the
$1$-dimensional torus $T(J)=U(J)_{{\mathbb{C}}}/{U(J)_{\mathbb{Z}}}$. The
sequence (5.5) splits (non-canonically) if we choose a splitting for
$J_{{\mathbb{Q}}}$, though this does not mean that (5.4) 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 the description so
far, ${\mathcal{T}(J)}\to{\mathcal{V}(J)}$ is a principal $T(J)$-bundle acted
on equivariantly by
${\overline{\Gamma(I)}_{\mathbb{Z}}}<{\overline{\Gamma(I)}_{\mathbb{Q}}}$, in
which ${\mathcal{X}(J)}$ is a punctured disc bundle. The action of
${\overline{\Gamma(I)}_{\mathbb{Z}}}$ on ${\mathcal{V}(J)}\to{\mathbb{H}_{J}}$
is described as follows. We choose a splitting for $J_{{\mathbb{Q}}}$ to give
a zero section of ${\mathcal{V}(J)}$ and a splitting of
${\overline{\Gamma(J)}_{\mathbb{Q}}}$ in (5.5). We express an element $\gamma$
of
${\overline{\Gamma(J)}_{\mathbb{Z}}}\subset{\overline{\Gamma(J)}_{\mathbb{Q}}}$
as $\gamma=(\gamma_{1},\gamma_{2},\alpha)$ accordingly, where
$\gamma_{1}\in{\rm O}(L(J))$, $\gamma_{2}\in{\rm SL}(J)$ and
$\alpha\in{W(J)_{\mathbb{Q}/\mathbb{Z}}}$. Then $\gamma$ acts on
${\mathcal{V}(J)}\simeq
L(J)_{{\mathbb{C}}}\otimes(\underline{J_{{\mathbb{C}}}}/F^{1,0})$ by the
equivariant action by $(\gamma_{1},\gamma_{2})$ and the translation by
$[\alpha]\in{V(J)_{\mathbb{Q}}}\simeq L(J)_{{\mathbb{Q}}}\otimes
J_{{\mathbb{Q}}}$. 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 compactification
of the torus $T(J)$. We take the relative torus embedding
${\overline{\mathcal{T}(J)}}={\mathcal{T}(J)}\times_{T(J)}\overline{T(J)}$,
and let ${\overline{\mathcal{X}(J)}}$ be the interior of the closure of
${\mathcal{X}(J)}$ in ${\overline{\mathcal{T}(J)}}$. This is the partial
compactification of ${\mathcal{X}(J)}$ over the $1$-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 relationship with a partial compactification over an adjacent
$0$-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 (5.2) we have
$U(J)_{{\mathbb{R}}}=\bar{J}_{{\mathbb{R}}}\subset U(I)_{{\mathbb{R}}}$ and

(5.6)
${U(J)_{\mathbb{Z}}}=\bar{J}_{{\mathbb{R}}}\cap{U(I)_{\mathbb{Z}}}={\mathbb{R}}\sigma_{J}\cap{U(I)_{\mathbb{Z}}}.$

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

(5.7) ${\overline{\mathcal{T}(J)}}\to T(I)^{\sigma_{J}}\subset T(I)^{\Sigma},$

which maps the boundary divisor of ${\overline{\mathcal{T}(J)}}$ to the unique
boundary divisor of $T(I)^{\sigma_{J}}$. It may be worth noting that
${U(I)_{\mathbb{Z}}}\subset{\Gamma(J)_{\mathbb{Z}}}$.

##  6\. Irregular $1$-dimensional cusps

In this section we define and study irregular $1$-dimensional cusps. For
simplicity we assume $b\geq 3$ so that $L(J)\neq\\{0\\}$. Let ${\Gamma}$ be a
finite-index subgroup of ${{\rm O}^{+}(L)}$ and $J$ be a rank $2$ primitive
isotropic sublattice of $L$. We keep the notation from §5. Irregularity of the
$1$-dimensional cusp $J$ can be characterized as follows.

######  Proposition 6.1.

The following conditions are equivalent.

  1. (1)

${U(J)_{\mathbb{Z}}}\neq{U(J)_{\mathbb{Z}}^{\prime}}$ where
${U(J)_{\mathbb{Z}}^{\prime}}={U(J)_{\mathbb{Q}}}\cap{\langle\Gamma,-{\rm
id}\rangle}$.

  2. (2)

$-{{\rm id}}\not\in{\Gamma}$ and $-E_{w}\in{\Gamma(J)_{\mathbb{Z}}}$ for some
$w\in\wedge^{2}J_{{\mathbb{Q}}}$.

  3. (3)

$-{{\rm id}}\not\in{\Gamma}$ and ${\overline{\Gamma(J)}_{\mathbb{Z}}}$
contains an element $\gamma$ of finite order whose image in ${\rm
O}(L(J))\times{\rm SL}(J)$ is $(-{{\rm id}}_{L(J)},-{{\rm id}}_{J})$.

  4. (4)

${\overline{\Gamma(J)}_{\mathbb{Z}}}$ contains an element $\gamma$ which acts
trivially on ${\mathcal{V}(J)}$ but nontrivially on ${\mathcal{X}(J)}$.

When these hold, the element $\gamma$ of ${\overline{\Gamma(J)}_{\mathbb{Z}}}$
in (3), (4) is given by $-E_{w}$ in (2), has order $2$, and is unique.

######  Definition 6.2.

We say that the $1$-dimensional cusp $J$ is irregular when these properties
hold, and regular otherwise.

###### Proof.

The equivalence $(1)\Leftrightarrow(2)$ is similar to $(1)\Leftrightarrow(2)$
in Proposition 3.1. The quotient
${U(J)_{\mathbb{Z}}^{\prime}}/{U(J)_{\mathbb{Z}}}\simeq{\mathbb{Z}}/2$ is
generated by $E_{w}$ in (2).

$(2)\Rightarrow(4)$: Since $E_{w}$ for $w\in\wedge^{2}J_{{\mathbb{Q}}}$ acts
trivially on ${\mathcal{V}(J)}$, so does $-E_{w}$.

$(2)\Rightarrow(3)$: The element $\gamma=[-E_{w}]$ of
${\overline{\Gamma(J)}_{\mathbb{Z}}}$ is of order $2$ and acts on $J$, $L(J)$
by $-1$.

$(3)\Rightarrow(4)$: By the description of the
${\overline{\Gamma(J)}_{\mathbb{Z}}}$-action on ${\mathcal{V}(J)}$ in §5.4, we
find that the element $\gamma$ of (3) acts on ${\mathcal{V}(J)}$ by some
translation. Since $\gamma$ is of finite order by assumption, this translation
must be trivial.

$(4)\Rightarrow(2),(3)$: Suppose that
$\gamma\in{\overline{\Gamma(J)}_{\mathbb{Z}}}$ acts trivially on
${\mathcal{V}(J)}$ but nontrivially on ${\mathcal{X}(J)}$. We take a splitting
of ${\overline{\Gamma(J)}_{\mathbb{Q}}}$ and express $\gamma$ as
$(\gamma_{1},\gamma_{2},\alpha)$ as in §5.4. Since $\gamma$ acts on
$\mathbb{H}_{J}$ trivially, we must have $\gamma_{2}={{\rm id}}_{J}$ or
$-{{\rm id}}_{J}$. Then, since $\gamma$ acts on ${\mathcal{V}(J)}\simeq
L(J)_{{\mathbb{C}}}\otimes(\underline{J_{{\mathbb{C}}}}/F^{1,0})$ trivially,
we see that $(\gamma_{1},\gamma_{2})=({{\rm id}}_{L(J)},{{\rm id}}_{J})$ or
$(-{{\rm id}}_{L(J)},-{{\rm id}}_{J})$, and the image of
$\alpha\in{W(J)_{\mathbb{Q}/\mathbb{Z}}}$ in ${V(J)_{\mathbb{Q}}}$ must be
$0$, namely $\alpha\in{U(J)_{\mathbb{Q}/\mathbb{Z}}}$. The case
$(\gamma_{1},\gamma_{2})=({{\rm id}}_{L(J)},{{\rm id}}_{J})$ cannot occur
because then
$\gamma\in{U(J)_{\mathbb{Q}/\mathbb{Z}}}\cap{V(J)_{\mathbb{Z}}}=\\{0\\}$.
Therefore $\gamma=(-{{\rm id}}_{L(J)},-{{\rm id}}_{J},E_{w})$ for some
$w\in\wedge^{2}J_{{\mathbb{Q}}}$. Since $-{{\rm id}}_{L}=(-{{\rm
id}}_{L(J)},-{{\rm id}}_{J},0)$ with respect to this (and any) splitting, we
find that $\gamma=-E_{w}$. Thus $-E_{w}\in{\Gamma}$. Finally, we have $-{{\rm
id}}\not\in{\Gamma}$, for otherwise $E_{w}=-\gamma$ would be contained in
${U(J)_{\mathbb{Z}}}$, which in turn implies that $\gamma$ acts trivially on
${\mathcal{X}(J)}$. ∎

As in the case of $0$-dimensional cusps, $U(J)_{{\mathbb{Z}}}^{\prime}$ is the
projection image of $U(J)_{{\mathbb{Z}}}^{\star}=(\\{\pm{{\rm
id}}\\}\cdot{U(J)_{\mathbb{Q}}})\cap{\Gamma}$ in ${U(J)_{\mathbb{Q}}}$, and we
have $U(J)_{{\mathbb{Z}}}^{\star}/{U(J)_{\mathbb{Z}}}=\langle-E_{w}\rangle$
when $J$ is irregular.

Since the boundary divisor of ${\overline{\mathcal{X}(J)}}$ is naturally
isomorphic to ${\mathcal{V}(J)}$, the condition $(4)$ can be restated as
follows.

######  Corollary 6.3.

A $1$-dimensional cusp $J$ is irregular if and only if
${\overline{\mathcal{X}(J)}}\to{\overline{\mathcal{X}(J)}}/{\overline{\Gamma(J)}_{\mathbb{Z}}}$
is ramified along the boundary divisor of ${\overline{\mathcal{X}(J)}}$. In
that case, the ramification index is $2$, and the unique nontrivial element of
${\overline{\Gamma(J)}_{\mathbb{Z}}}$ fixing the boundary divisor is given by
$-E_{w}$.

By the condition (1), irregularity of a $1$-dimensional cusp reduces to that
of an adjacent $0$-dimensional cusp as follows.

######  Proposition 6.4.

Let $I\subset J$ be a rank $1$ primitive sublattice and $\sigma_{J}\subset
U(I)_{{\mathbb{R}}}$ be the isotropic ray corresponding to $J$. Then $J$ is
irregular if and only if $I$ is irregular and $\sigma_{J}$ is an irregular
ray.

###### Proof.

Recall that $\sigma_{J}$ is called irregular when
${\mathbb{R}}\sigma_{J}\cap{U(I)_{\mathbb{Z}}}\neq{\mathbb{R}}\sigma_{J}\cap{U(I)_{\mathbb{Z}}^{\prime}}$.
By (5.6) we have
${\mathbb{R}}\sigma_{J}\cap{U(I)_{\mathbb{Z}}}={U(J)_{\mathbb{Z}}}$, and
similarly
${\mathbb{R}}\sigma_{J}\cap{U(I)_{\mathbb{Z}}^{\prime}}={U(J)_{\mathbb{Z}}^{\prime}}$.
This proves our assertion. ∎

######  Corollary 6.5.

If ${\Gamma}$ has no irregular $0$-dimensional cusp, it has no irregular
$1$-dimensional cusp.

##  7\. Toroidal compactification

In this section we study singularities and ramification divisors in the
boundary of a toroidal compactification of the modular variety. These are
studied in [6], [12] under the condition $-{{\rm id}}\in{\Gamma}$, and we
explain what modification is necessary in the general case, especially at the
irregular cusps.

Let $L$ be a lattice of signature $(2,b)$ and ${\Gamma}$ be a subgroup of
${{\rm O}^{+}(L)}$ of finite index. The input data for constructing a toroidal
compactification of ${\mathcal{F}(\Gamma)}={\Gamma}\backslash{\mathcal{D}}$ is
a collection $\Sigma=(\Sigma_{I})_{I}$ of
${\Gamma(I)_{\mathbb{Z}}}$-admissible fans (§2.4), one for each
${\Gamma}$-equivalence class of rank $1$ primitive isotropic sublattices $I$
of $L$. No choice is required for $1$-dimensional cusps. Thus $\Sigma$ is a
finite collection of independent fans.

The toroidal compactification associated to $\Sigma$ is defined as ([1] p.163)

${\mathcal{F}(\Gamma)^{\Sigma}}=\left({\mathcal{D}}\sqcup\bigsqcup_{I}\mathcal{X}(I)^{\Sigma_{I}}\sqcup\bigsqcup_{J}{\overline{\mathcal{X}(J)}}\right)/\sim,$

where $I$ (resp. $J$) ranges over all primitive isotropic sublattices of $L$
of rank $1$ (resp. $2$), and $\sim$ is the equivalence relation generated by
the following:

  * •

Action of $\gamma\in{\Gamma}$ giving ${\mathcal{D}}\to{\mathcal{D}}$,
$\mathcal{X}(I)^{\Sigma_{I}}\to\mathcal{X}(\gamma I)^{\Sigma_{\gamma I}}$ and
${\overline{\mathcal{X}(J)}}\to\overline{\mathcal{X}(\gamma J)}$.

  * •

The natural maps ${\mathcal{D}}\to\mathcal{X}(I)^{\Sigma_{I}}$ and
${\mathcal{D}}\to{\overline{\mathcal{X}(J)}}$.

  * •

The etale gluing maps
${\overline{\mathcal{X}(J)}}\to\mathcal{X}(I)^{\Sigma_{I}}$ for $I\subset J$
given by (5.7).

######  Theorem 7.1 ([1]).

The space ${\mathcal{F}(\Gamma)^{\Sigma}}$ is a compact Moishezon space
containing ${\mathcal{F}(\Gamma)}$ as a Zariski open set, and we have a
morphism from ${\mathcal{F}(\Gamma)^{\Sigma}}$ to the Baily-Borel
compactification of ${\mathcal{F}(\Gamma)}$. For each cusp $I$, $J$, the
natural map

$\mathcal{X}(I)^{\Sigma_{I}}/{\overline{\Gamma(I)}_{\mathbb{Z}}}\to{\mathcal{F}(\Gamma)^{\Sigma}},\qquad{\overline{\mathcal{X}(J)}}/{\overline{\Gamma(J)}_{\mathbb{Z}}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$

is locally isomorphic in an open neighborhood of boundary points lying over
that cusp.

Perhaps a word might be in order because, strictly speaking, the theory of [1]
is applied to the image of ${\Gamma}$ in ${\rm
O}^{+}(L_{{\mathbb{R}}})/\pm{{\rm id}}$, which is ${\langle\Gamma,-{\rm
id}\rangle}/\pm{{\rm id}}$, rather than ${\Gamma}$ itself. Then
${U(I)_{\mathbb{Z}}}$ should be replaced by ${U(I)_{\mathbb{Z}}^{\prime}}$,
${\mathcal{X}(I)}={\mathcal{D}}/{U(I)_{\mathbb{Z}}}$ by
${\mathcal{X}(I)^{\prime}}={\mathcal{D}}/{U(I)_{\mathbb{Z}}^{\prime}}$,
${\Gamma(I)_{\mathbb{Z}}}$ by
$\Gamma^{\prime}(I)_{{\mathbb{Z}}}=\langle{\Gamma(I)_{\mathbb{Z}}},-{{\rm
id}}\rangle/\pm{{\rm id}}$, and similarly for $1$-dimensional cusps $J$. But
since ${\mathcal{X}(I)^{\prime}}={\mathcal{X}(I)}$ or
${\mathcal{X}(I)^{\prime}}={\mathcal{X}(I)}/\langle-E_{w}\rangle$ with
$-E_{w}\in{\Gamma(I)_{\mathbb{Z}}}$ (and similarly for $J$), we have naturally

$\left({\mathcal{D}}\sqcup\bigsqcup_{I}\mathcal{X}(I)^{\Sigma_{I}}\sqcup\bigsqcup_{J}{\overline{\mathcal{X}(J)}}\right)/\sim\;\;=\;\;\left({\mathcal{D}}\sqcup\bigsqcup_{I}({\mathcal{X}(I)^{\prime}})^{\Sigma_{I}}\sqcup\bigsqcup_{J}\overline{\mathcal{X}(J)^{\prime}}\right)/\sim^{\prime},$

where $\sim^{\prime}$ is the equivalence relation similar to $\sim$. The last
statement of Theorem 7.1 ([1] p.175) is justified because we have

$\mathcal{X}(I)^{\Sigma_{I}}/{\overline{\Gamma(I)}_{\mathbb{Z}}}=({\mathcal{X}(I)^{\prime}})^{\Sigma_{I}}/(\Gamma^{\prime}(I)_{{\mathbb{Z}}}/{U(I)_{\mathbb{Z}}^{\prime}})$

(see also (7)), and similarly for $J$.

The reason we prefer to work with ${U(I)_{\mathbb{Z}}}$ rather than
${U(I)_{\mathbb{Z}}^{\prime}}$ is that Fourier expansion of ${\Gamma}$-modular
forms of arbitrary weight can be done with ${U(I)_{\mathbb{Z}}}$ (see §8).

If $D(\sigma)\subset\mathcal{X}(I)^{\Sigma_{I}}$ is the boundary divisor
corresponding to a ray $\sigma\in\Sigma_{I}$, general points of $D(\sigma)$
lie over the $I$-cusp if and only if $\sigma$ is positive-definite. When
$\sigma=\sigma_{J}$ is isotropic corresponding to a $1$-dimensional cusp
$J\supset I$, $D(\sigma_{J})$ is glued with the boundary divisor of
${\overline{\mathcal{X}(J)}}$, and its general points lie over the $J$-cusp.
By combining the last statement of Theorem 7.1 with Corollaries 3.9 and 6.3,
we obtain the following.

######  Proposition 7.2.

(1) The projection
$\mathcal{X}(I)^{\Sigma_{I}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$ is ramified
along irregular boundary divisors of $\mathcal{X}(I)^{\Sigma_{I}}$ with
ramification index $2$, and not ramified along other boundary divisors. If we
take quotient by $U(I)_{{\mathbb{Z}}}^{\star}/{U(I)_{\mathbb{Z}}}$, then
$({\mathcal{D}}/U(I)_{{\mathbb{Z}}}^{\star})^{\Sigma_{I}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$
is not ramified along the boundary divisors.

(2) The projection
${\overline{\mathcal{X}(J)}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$ is ramified
along the unique boundary divisor (with index $2$) if and only if $J$ is
irregular. If we take quotient by
$U(J)_{{\mathbb{Z}}}^{\star}/{U(J)_{\mathbb{Z}}}$, then
$\overline{{\mathcal{D}}/U(J)_{{\mathbb{Z}}}^{\star}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$
is not ramified along the boundary divisor.

###### Proof.

What remains is to show that (1) is still true even when a ray
$\sigma=\sigma_{J}$ is isotropic. Since the map
${\overline{\mathcal{X}(J)}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$ in (2)
factorizes as
${\overline{\mathcal{X}(J)}}\to\mathcal{X}(I)^{\Sigma_{I}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$
and the gluing map ${\overline{\mathcal{X}(J)}}\to\mathcal{X}(I)^{\Sigma_{I}}$
is etale, our assertion for
$\mathcal{X}(I)^{\Sigma_{I}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$ follows from
(2) and Proposition 6.4. ∎

When ${\Gamma}$ contains $-{{\rm id}}$, Proposition 7.2 is proved in [6],
[12]. In that case, we have no irregular cusp, so no ramification divisor in
the boundary.

######  Remark 7.3.

It appears that in some literatures, the “no ramification boundary divisor”
property is used to claim that
$\mathcal{F}(\Gamma^{\prime})^{\Sigma}\to{\mathcal{F}(\Gamma)^{\Sigma}}$ is
not ramified along the boundary divisors for neat subgroups
$\Gamma^{\prime}<{\Gamma}$. This seems not true already in the case of modular
curves: for example, $\Gamma(N)<{\rm SL}_{2}({\mathbb{Z}})$. The point is that
$U(I)_{{\mathbb{Z}},{\Gamma}}={U(I)_{\mathbb{Q}}}\cap{\Gamma}$ depends on
${\Gamma}$, so
$U(I)_{{\mathbb{Z}},{\Gamma}^{\prime}}={U(I)_{\mathbb{Q}}}\cap{\Gamma}^{\prime}$
is in general smaller than $U(I)_{{\mathbb{Z}},{\Gamma}}$. If $\sigma$ is a
ray in $\Sigma_{I}$, assumed regular for simplicity, we have ramification
index

$[{\mathbb{R}}\sigma\cap U(I)_{{\mathbb{Z}},{\Gamma}}:{\mathbb{R}}\sigma\cap
U(I)_{{\mathbb{Z}},{\Gamma}^{\prime}}]$

at the corresponding boundary divisor. It seems that so far, all argument
using the above claim can be avoided: see the proof of Theorem 8.9.

Next we study singularities. A fan $\Sigma_{I}=(\sigma_{\alpha})$ is called
basic with respect to a lattice $\Lambda\subset{U(I)_{\mathbb{Q}}}$ if each
cone $\sigma_{\alpha}$ is generated by a part of a basis of $\Lambda$. The
singularity theorem ([6], [12]) is still true, if we require the fan
$\Sigma_{I}$ to be basic with respect to ${U(I)_{\mathbb{Z}}^{\prime}}$,
rather than ${U(I)_{\mathbb{Z}}}$.

######  Proposition 7.4 (cf. [6], [12]).

(1) We choose the fans $\Sigma=(\Sigma_{I})$ so that each $\Sigma_{I}$ is
basic with respect to ${U(I)_{\mathbb{Z}}^{\prime}}$. Then
${\mathcal{F}(\Gamma)^{\Sigma}}$ has canonical singularities at the boundary
points lying over the $0$-dimensional cusps.

(2) When $b\geq 9$, ${\mathcal{F}(\Gamma)^{\Sigma}}$ has canonical
singularities at the boundary points lying over the $1$-dimensional cusps.

###### Proof.

When ${\Gamma}$ contains $-{{\rm id}}$, this is proved in [6], [12] for
$0$-dimensional cusps, and in [6] for $1$-dimensional cusps. We show that the
general case is reduced to this case. We consider $0$-dimensional cusps. The
case of $1$-dimensional cusps is similar. It suffices to show that
${\mathcal{X}(I)}^{\Sigma_{I}}/{\overline{\Gamma(I)}_{\mathbb{Z}}}$ has
canonical singularities.

Let $\Gamma^{\prime}={\langle\Gamma,-{\rm id}\rangle}$ and
$\Gamma^{\prime}(I)_{{\mathbb{Z}}}=\Gamma^{\prime}\cap{\Gamma(I)_{\mathbb{Q}}}$.
Then ${U(I)_{\mathbb{Z}}^{\prime}}={U(I)_{\mathbb{Q}}}\cap\Gamma^{\prime}$ and
$\Gamma^{\prime}(I)_{{\mathbb{Z}}}=\langle{\Gamma(I)_{\mathbb{Z}}},-{{\rm
id}}\rangle$. Since the fan $\Sigma_{I}$ is also rational with respect to
${U(I)_{\mathbb{Z}}^{\prime}}$, it defines a toroidal embedding
$({\mathcal{D}}/{U(I)_{\mathbb{Z}}^{\prime}})^{\Sigma_{I}}$ of
${\mathcal{D}}/{U(I)_{\mathbb{Z}}^{\prime}}$. This is the quotient of
$({\mathcal{D}}/{U(I)_{\mathbb{Z}}})^{\Sigma_{I}}$ by the translation by
${U(I)_{\mathbb{Z}}^{\prime}}/{U(I)_{\mathbb{Z}}}$ (which is nontrivial
exactly when $I$ is irregular). Since
${U(I)_{\mathbb{Z}}^{\prime}}/{U(I)_{\mathbb{Z}}}\subset\Gamma^{\prime}(I)_{{\mathbb{Z}}}/{U(I)_{\mathbb{Z}}}$,
we have

$\displaystyle({\mathcal{D}}/{U(I)_{\mathbb{Z}}})^{\Sigma_{I}}/{\overline{\Gamma(I)}_{\mathbb{Z}}}$
$\displaystyle=$
$\displaystyle({\mathcal{D}}/{U(I)_{\mathbb{Z}}})^{\Sigma_{I}}/(\Gamma^{\prime}(I)_{{\mathbb{Z}}}/{U(I)_{\mathbb{Z}}})$
$\displaystyle\simeq$
$\displaystyle({\mathcal{D}}/{U(I)_{\mathbb{Z}}^{\prime}})^{\Sigma_{I}}/(\Gamma^{\prime}(I)_{{\mathbb{Z}}}/{U(I)_{\mathbb{Z}}^{\prime}}).$

Since $\Sigma_{I}$ is basic with respect to ${U(I)_{\mathbb{Z}}^{\prime}}$ and
$-{{\rm id}}\in\Gamma^{\prime}$, we can apply the result of [12] to the last
quotient to see that this has canonical singularities. ∎

##  8\. Modular forms and pluricanonical forms

Let $L$ be a lattice of signature $(2,b)$ and ${\Gamma}$ be a subgroup of
${{\rm O}^{+}(L)}$ of finite index. For simplicity we assume $b\geq 3$. In
this section we compare vanishing order of cusp forms and pluricanonical
forms, and explain how the low weight cusp form trick of Gritsenko-Hulek-
Sankaran [6] is modified at irregular boundary divisors. We take this occasion
to generalize “low weight” to “low slope”, for possible future use.

###  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 ([13]). We assume that
$\chi|_{{U(I)_{\mathbb{Z}}}}\equiv 1$ for every $0$-dimensional cusp $I$. This
holds, e.g., for $\chi=1$ and $\chi=\det$. 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

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

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
$F=fs_{I}^{\otimes k}\otimes 1_{\chi}$ of weight $k$ and character $\chi$ are
identified with holomorphic functions $f$ on ${\mathcal{D}}$ satisfying

$f(\gamma[\omega])=\chi(\gamma)j(\gamma,[\omega])^{k}f([\omega]),\quad\gamma\in{\Gamma},\;[\omega]\in{\mathcal{D}}.$

Since $s_{I}^{\otimes k}\otimes 1_{\chi}$ is invariant under
${U(I)_{\mathbb{Z}}}$ by our assumption, $f$ is
${U(I)_{\mathbb{Z}}}$-invariant, hence descends to a function on
${\mathcal{D}}/{U(I)_{\mathbb{Z}}}$. By the tube domain realization
${\mathcal{D}}\to{\mathcal{D}}_{I}\subset U(I)_{{\mathbb{C}}}$ (after a choice
of $I^{\prime}\subset L$ with $(I,I^{\prime})\not\equiv 0$), $f$ is identified
with a function on ${\mathcal{D}}_{I}$ invariant under translation by the
lattice ${U(I)_{\mathbb{Z}}}$. Then it admits a Fourier expansion

(8.1) $f(Z)=\sum_{l\in U(I)_{{\mathbb{Z}}}^{\vee}}a(l)q^{l},\quad q^{l}={\rm
exp}(2\pi i(l,Z)),\;Z\in\mathcal{D}_{I}.$

By the Koecher principle, we have $a(l)\neq 0$ only when
$l\in\overline{{\mathcal{C}_{I}}}$. The modular form $F$ is called a cusp form
if $a(l)=0$ for every $l\in U(I)_{{\mathbb{Z}}}^{\vee}$ with $(l,l)=0$ at
every rank $1$ primitive isotropic sublattice $I$ of $L$. ($a(0)$ is the value
of $f$ at the $0$-dimensional cusp for $I$, and $\sum_{\sigma\cap
U(I)_{{\mathbb{Z}}}^{\vee}}a(l)q^{l}$ for an isotropic ray $\sigma=\sigma_{J}$
gives the restriction of $f$ to the $1$-dimensional cusp for $J\supset I$.)

Fourier expansion at an irregular cusp satisfies the following.

######  Lemma 8.1.

Suppose that $I$ is irregular and $-E_{w}\in{\Gamma(I)_{\mathbb{Z}}}$. When
the weight $k$ satisfies $\chi(-E_{w})=(-1)^{k+1}$, e.g., $k$ odd for $\chi=1$
or $k\not\equiv b$ mod $2$ for $\chi=\det$, then we have $a(l)=0$ for
$l\in(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$. In particular, $a(0)=0$ in this
case. When $\chi(-E_{w})=(-1)^{k}$, we have $a(l)=0$ for
$l\not\in(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$.

###### Proof.

Since the factor of automorphy of $-E_{w}$ is $-1$, we find that
$f(Z+w)=\chi(-E_{w})(-1)^{k}f(Z)$. On the other hand, we have
$(l,w)\in{\mathbb{Z}}$ if $l\in(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$ and
$(l,w)\in 1/2+{\mathbb{Z}}$ if $l\in
U(I)_{{\mathbb{Z}}}^{\vee}-(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$. Therefore,
if we substitute $Z\to Z+w$ into $q^{l}={\rm exp}(2\pi i(l,Z))$, then
$q^{l}\to q^{l}$ if $l\in(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$ and
$q^{l}\to-q^{l}$ if $l\in
U(I)_{{\mathbb{Z}}}^{\vee}-(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$. This
implies our assertion. ∎

###  8.2. Vanishing order

In this subsection we study vanishing order of modular forms along boundary
divisors. We will define two types of vanishing order: $\nu_{\sigma}(F)$ and
$\nu_{\sigma,geom}(F)$. $\nu_{\sigma}(F)$ is defined by Fourier expansion and
is always an integer. On the other hand, $\nu_{\sigma,geom}(F)$ can be
strictly half-integral, and measures the vanishing order at the level of
${\mathcal{F}(\Gamma)^{\Sigma}}$.

Let $I$ be a rank $1$ primitive isotropic sublattice of $L$. Let
$\Sigma=\Sigma_{I}=(\sigma_{\alpha})$ be a
${\Gamma(I)_{\mathbb{Z}}}$-admissible fan in $U(I)_{{\mathbb{R}}}$ and
$\sigma$ be a ray in $\Sigma$. Let $w_{\sigma}$ be the generator of
$\sigma\cap{U(I)_{\mathbb{Z}}}$. Let $f(Z)=\sum_{l\in
U(I)_{{\mathbb{Z}}}^{\vee}}a(l)q^{l}$ be the Fourier expansion of a
${\Gamma}$-modular form $F=fs_{I}^{\otimes k}\otimes 1_{\chi}$ around $I$. We
define the vanishing order of $F$ along $\sigma$ as

$\nu_{\sigma}(F)=\min\\{\>(l,w_{\sigma})\>|\>l\in
U(I)_{{\mathbb{Z}}}^{\vee},\>a(l)\neq 0\>\\}.$

This is a nonnegative integer. Clearly $\nu_{\sigma}(F)$ depends on
${U(I)_{\mathbb{Z}}}$ and hence on ${\Gamma}$. If we shrink ${\Gamma}$ without
changing $F$ and $\sigma$, then $\nu_{\sigma}(F)$ will be multiplied in
general.

When $\sigma$ is positive-definite, we have $\nu_{\sigma}(F)>0$ if and only if
$a(0)=0$, because $\sigma^{\perp}\cap\overline{{\mathcal{C}_{I}}}=\\{0\\}$.
When $\sigma$ is isotropic, we have $\nu_{\sigma}(F)>0$ if and only if
$a(l)=0$ for all $l\in\sigma\cap U(I)_{{\mathbb{Z}}}^{\vee}$, because
$\sigma^{\perp}\cap\overline{{\mathcal{C}_{I}}}=\sigma$. Thus, $F$ is a cusp
form if and only if $\nu_{\sigma}(F)>0$ at every ray $\sigma$ at every
$0$-dimensional cusp $I$.

The following criterion is trivial but perhaps might be sometimes useful in
view of Theorem 8.9. Compare with [1], [5] in related cases.

######  Corollary 8.2.

Assume the following holds: if $a(l)\neq 0$, then $(l,w)\geq r$ for every
$w\in{U(I)_{\mathbb{Z}}}\cap\overline{{\mathcal{C}_{I}}}$. Then we have
$\nu_{\sigma}(F)\geq r$ for every ray $\sigma\in\Sigma$.

###### Proof.

Take $w$ to be the generator of $\sigma\cap{U(I)_{\mathbb{Z}}}$. ∎

When $\sigma$ is irregular, $\nu_{\sigma}(F)$ belongs to the following parity.

######  Proposition 8.3.

Suppose that the ray $\sigma$ is irregular and
$-E_{w}\in{\Gamma(I)_{\mathbb{Z}}}$. Then $\nu_{\sigma}(F)$ is odd when
$\chi(-E_{w})=(-1)^{k+1}$, and even when $\chi(-E_{w})=(-1)^{k}$.

###### Proof.

Let $w_{\sigma}$ be the generator of $\sigma\cap{U(I)_{\mathbb{Z}}}$. Since
${U(I)_{\mathbb{Z}}^{\prime}}=\langle{U(I)_{\mathbb{Z}}},w_{\sigma}/2\rangle$,
a vector $l$ of $U(I)_{{\mathbb{Z}}}^{\vee}$ belongs to
$(U(I)_{{\mathbb{Z}}}^{\prime})^{\vee}$ if and only if $(l,w_{\sigma})$ is
even. Then our assertion follows from Lemma 8.1. ∎

We also define the geometric vanishing order of $F$ along $\sigma$ as

$\nu_{\sigma,geom}(F)=\begin{cases}\nu_{\sigma}(F)&\sigma:\textrm{regular}\\\
\frac{1}{2}\nu_{\sigma}(F)&\sigma:\textrm{irregular}\end{cases}$

If $w_{\sigma}^{\prime}$ is the generator of
$\sigma\cap{U(I)_{\mathbb{Z}}^{\prime}}$, we can write uniformly as

(8.2) $\nu_{\sigma,geom}(F)=\min\\{\>(l,w_{\sigma}^{\prime})\>|\>l\in
U(I)_{{\mathbb{Z}}}^{\vee},\>a(l)\neq 0\>\\}.$

Note that $\nu_{\sigma,geom}(F)$ is in $1/2+{\mathbb{Z}}$ when $\sigma$ is
irregular and the weight $k$ satisfies $\chi(-E_{w})=(-1)^{k+1}$ so that
$\nu_{\sigma}(F)$ is odd.

Geometric interpretation of $\nu_{\sigma}(F)$ is as follows. Recall that the
ray $\sigma$ corresponds to a boundary divisor $D(\sigma)$ of the partial
compactification ${\mathcal{X}(I)^{\Sigma}}$ of
${\mathcal{X}(I)}={\mathcal{D}}/{U(I)_{\mathbb{Z}}}$. The line bundle
$\mathcal{L}^{\otimes k}\otimes\chi$ descends to a line bundle over
${\mathcal{X}(I)}$, again denoted by $\mathcal{L}^{\otimes k}\otimes\chi$. The
point is that, since $s_{I}^{\otimes k}\otimes 1_{\chi}$ is
${U(I)_{\mathbb{Z}}}$-invariant, it descends to a frame of
$\mathcal{L}^{\otimes k}\otimes\chi$ over ${\mathcal{X}(I)}$, and we use this
frame to extend $\mathcal{L}^{\otimes k}\otimes\chi$ to a line bundle over
${\mathcal{X}(I)^{\Sigma}}$, still denoted by the same notation. Namely,
$s_{I}^{\otimes k}\otimes 1_{\chi}$ extends to a frame of the extended line
bundle by definition. The property
$l\in\overline{{\mathcal{C}_{I}}}=\overline{{\mathcal{C}_{I}}}^{\vee}$ in the
Fourier expansion implies that a modular form $F$ extends holomorphically over
${\mathcal{X}(I)^{\Sigma}}$ as a section of $\mathcal{L}^{\otimes
k}\otimes\chi$.

######  Proposition 8.4.

$\nu_{\sigma}(F)$ is equal to the vanishing order of $F$ as a section of
$\mathcal{L}^{\otimes k}\otimes\chi$ over ${\mathcal{X}(I)^{\Sigma}}$ along
the boundary divisor $D(\sigma)$.

###### Proof.

Recall that $\sigma$ defines a sub toroidal embedding
$\mathcal{X}(I)^{\sigma}\subset{\mathcal{X}(I)^{\Sigma}}$, the unique boundary
divisor of which is a Zariski open set of $D(\sigma)$ and is the quotient
torus (or its analytic open set) defined by the quotient lattice
${U(I)_{\mathbb{Z}}}/{\mathbb{Z}}w_{\sigma}$. The character group of this
boundary torus is $\sigma^{\perp}\cap U(I)_{{\mathbb{Z}}}^{\vee}$. We choose a
vector $l_{\sigma}\in U(I)_{{\mathbb{Z}}}^{\vee}$ such that
$(l_{\sigma},w_{\sigma})=1$ and put $q=q^{l_{\sigma}}$, which is a character
of $T(I)$. Then $q$ extends holomorphically over $\mathcal{X}(I)^{\sigma}$
with $D(\sigma)=(q=0)$. The Fourier expansion (8.1) can be arranged as
$f=\sum_{m\geq 0}\varphi_{m}q^{m}$ where

$\varphi_{m}=\sum_{l\in\sigma^{\perp}\cap
U(I)_{{\mathbb{Z}}}^{\vee}}a(l+ml_{\sigma})q^{l}.$

This is a Taylor expansion of $f$ along the divisor $D(\sigma)$. Since
$(l+ml_{\sigma},w_{\sigma})=m$ for $l\in\sigma^{\perp}\cap
U(I)_{{\mathbb{Z}}}^{\vee}$, we find that

$\nu_{\sigma}(F)=\min\\{\>m\>|\>\varphi_{m}\not\equiv 0\>\\}.$

This proves our assertion. ∎

We can also give a geometric interpretation of $\nu_{\sigma,geom}(F)$ when

(8.3) $s_{I}^{\otimes k}\otimes 1_{\chi}\;\textrm{is invariant
under}\;U(I)_{{\mathbb{Z}}}^{\star}=(\\{\pm{{\rm
id}}\\}\cdot{U(I)_{\mathbb{Q}}})\cap{\Gamma}.$

This holds, e.g., when $k$ is even with $\chi=1$ and when $k\equiv b$ mod $2$
with $\chi=\det$. Recall that ${U(I)_{\mathbb{Z}}^{\prime}}$ is the image of
$U(I)_{{\mathbb{Z}}}^{\star}$ in ${U(I)_{\mathbb{Q}}}$. Under the condition
(8.3), the function $f(Z)$ on the tube domain $\mathcal{D}_{I}$ is invariant
under translation by ${U(I)_{\mathbb{Z}}^{\prime}}$, so the index lattice in
the Fourier expansion reduces to $({U(I)_{\mathbb{Z}}^{\prime}})^{\vee}\subset
U(I)_{{\mathbb{Z}}}^{\vee}$. In other words, $a(l)=0$ if
$l\not\in({U(I)_{\mathbb{Z}}^{\prime}})^{\vee}$, so $\nu_{\sigma,geom}(F)$ is
an integer. The frame $s_{I}^{\otimes k}\otimes 1_{\chi}$ descends to a frame
of $\mathcal{L}^{\otimes k}\otimes\chi$ over

${\mathcal{X}(I)^{\prime}}={\mathcal{D}}/U(I)_{{\mathbb{Z}}}^{\star}={\mathcal{D}}/{U(I)_{\mathbb{Z}}^{\prime}},$

using which we can extend $\mathcal{L}^{\otimes k}\otimes\chi$ to a line
bundle over ${(\mathcal{X}(I)^{\prime})^{\Sigma}}$. The ray $\sigma$
corresponds to a boundary divisor $D(\sigma)^{\prime}$ of
${(\mathcal{X}(I)^{\prime})^{\Sigma}}$. We have

  * •

$D(\sigma)^{\prime}=D(\sigma)$ in
${\mathcal{X}(I)^{\Sigma}}={(\mathcal{X}(I)^{\prime})^{\Sigma}}$ when $I$ is
regular.

  * •

$D(\sigma)^{\prime}\simeq D(\sigma)$ with
${\mathcal{X}(I)^{\Sigma}}\to{(\mathcal{X}(I)^{\prime})^{\Sigma}}$ doubly
ramified along $D(\sigma)^{\prime}$ when $\sigma$ is irregular.

  * •

$D(\sigma)^{\prime}$ is the quotient of $D(\sigma)$ by
${U(I)_{\mathbb{Z}}^{\prime}}/{U(I)_{\mathbb{Z}}}\simeq{\mathbb{Z}}/2$ with
${\mathcal{X}(I)^{\Sigma}}\to{(\mathcal{X}(I)^{\prime})^{\Sigma}}$ unramified
along $D(\sigma)^{\prime}$ when $I$ is irregular but $\sigma$ is regular.

Then we see, either from Proposition 8.4 or by a similar argument, the
following.

######  Proposition 8.5.

When (8.3) holds, $\nu_{\sigma,geom}(F)$ is equal to the vanishing order of
$F$ as a section of $\mathcal{L}^{\otimes k}\otimes\chi$ over
${(\mathcal{X}(I)^{\prime})^{\Sigma}}$ along the boundary divisor
$D(\sigma)^{\prime}$.

Vanishing order at a $1$-dimensional cusp $J$ is reduced to the case
considered above. We choose a rank $1$ primitive sublattice $I\subset J$ and
let $\sigma_{J}$ be the isotropic ray in $U(I)_{{\mathbb{R}}}$ corresponding
to $J$. Then we define

$\nu_{J}(F)=\nu_{\sigma_{J}}(F),\qquad\nu_{J,geom}(F)=\nu_{\sigma_{J},geom}(F).$

The Taylor expansion $f=\sum_{m}\varphi_{m}q^{m}$ in this case is nothing but
the Fourier-Jacobi expansion, and $\varphi_{m}$ is essentially the $m$-th
Fourier-Jacobi coefficient. Thus $\nu_{J}(F)$ is the minimal degree of nonzero
Fourier-Jacobi coefficients.

We also have the following geometric interpretation of $\nu_{J}(F)$. We use
the ${U(J)_{\mathbb{Z}}}$-invariant frame $s_{I}^{\otimes k}\otimes 1_{\chi}$
to extend $\mathcal{L}^{\otimes k}\otimes\chi$ to a line bundle over
${\overline{\mathcal{X}(J)}}$. This is the pullback of the extended line
bundle $\mathcal{L}^{\otimes k}\otimes\chi$ over ${\mathcal{X}(I)^{\Sigma}}$
by the etale gluing map
${\overline{\mathcal{X}(J)}}\to{\mathcal{X}(I)^{\Sigma}}$. This extension does
not depend on the choice of $I$ up to isomorphism. Then $\nu_{J}(F)$ is the
vanishing order of $F$ as a section of the extended line bundle
$\mathcal{L}^{\otimes k}\otimes\chi$ over ${\overline{\mathcal{X}(J)}}$ along
the boundary divisor. Similarly, when $s_{I}^{\otimes k}\otimes 1_{\chi}$ is
invariant under $U(J)_{{\mathbb{Z}}}^{\star}$, $\nu_{J,geom}(F)$ equals to the
vanishing order of $F$ along the boundary divisor of
$\overline{{\mathcal{D}}/U(J)_{{\mathbb{Z}}}^{\star}}=\overline{{\mathcal{D}}/{U(J)_{\mathbb{Z}}^{\prime}}}$.

###  8.3. Pluricanonical forms

In this subsection we compare vanishing order of modular forms and
pluricanonical forms along the boundary divisors. Recall that we have a
canonical isomorphism

$\mathcal{L}^{\otimes b}\otimes\det\simeq K_{\mathcal{D}}$

over ${\mathcal{D}}$, as a consequence of the isomorphism
$K_{{{\mathbb{P}}}L_{{\mathbb{C}}}}\simeq\mathcal{O}_{{{\mathbb{P}}}L_{{\mathbb{C}}}}(-b-2)\otimes\det$
and the adjunction formula. Let $I$ be a rank $1$ primitive isotropic
sublattice of $L$. The above isomorphism descends to $\mathcal{L}^{\otimes
b}\otimes\det\simeq K_{{\mathcal{X}(I)^{\prime}}}$ over
${\mathcal{X}(I)^{\prime}}={\mathcal{D}}/U(I)_{{\mathbb{Z}}}^{\star}$. Both
line bundles are extended over the partial compactification
${(\mathcal{X}(I)^{\prime})^{\Sigma}}$ in the respective manner:
$\mathcal{L}^{\otimes b}\otimes\det$ is extended by the frame $s_{I}^{\otimes
b}\otimes 1_{\det}$, while $K_{{\mathcal{X}(I)^{\prime}}}$ is extended to
$K_{{(\mathcal{X}(I)^{\prime})^{\Sigma}}}$.

######  Proposition 8.6 (cf. [14]).

Over ${(\mathcal{X}(I)^{\prime})^{\Sigma}}$ the above isomorphism extends to

$\mathcal{L}^{\otimes
b}\otimes\det\>\simeq\>K_{{(\mathcal{X}(I)^{\prime})^{\Sigma}}}(\sum_{\sigma}D(\sigma)^{\prime}),$

where $\sigma$ ranges over all rays in $\Sigma$ and $D(\sigma)^{\prime}$ is
the boundary divisor of ${(\mathcal{X}(I)^{\prime})^{\Sigma}}$ corresponding
to $\sigma$.

###### Proof.

By the isomorphism $\mathcal{L}^{\otimes b}\otimes\det\simeq
K_{{\mathcal{D}}}$, the frame $s_{I}^{\otimes b}\otimes 1_{\det}$ of
$\mathcal{L}^{\otimes b}\otimes\det$ corresponds to a flat canonical form
$\omega_{I}$ on the tube domain ${\mathcal{D}}_{I}\subset
U(I)_{{\mathbb{C}}}$, because both extend over ${\mathcal{D}}(I)\simeq
U(I)_{{\mathbb{C}}}$ and are $U(I)_{{\mathbb{C}}}$-invariant. Let $\sigma$ be
a ray in $\Sigma$ and $w_{\sigma}^{\prime}$ be the generator of
$\sigma\cap{U(I)_{\mathbb{Z}}^{\prime}}$. We take a vector
$l_{\sigma}\in({U(I)_{\mathbb{Z}}^{\prime}})^{\vee}$ with
$(l_{\sigma},w_{\sigma}^{\prime})=1$ and extend it to a basis of
$({U(I)_{\mathbb{Z}}^{\prime}})^{\vee}$. This defines a coordinate
$Z_{1}=(l_{\sigma},\cdot),Z_{2},\cdots,Z_{b}$ on $U(I)_{{\mathbb{C}}}$. We
have $\omega_{I}=dZ_{1}\wedge\cdots\wedge dZ_{b}$ up to constant. Then
$q=q^{l_{\sigma}},Z_{2},\cdots,Z_{b}$ define a local coordinate around a point
of $D(\sigma)^{\prime}\subset{(\mathcal{X}(I)^{\prime})^{\Sigma}}$ with
$D(\sigma)^{\prime}=(q=0)$. Since we have

$s_{I}^{\otimes b}\otimes 1_{\det}=dZ_{1}\wedge\cdots\wedge
dZ_{b}=\frac{dq}{q}\wedge dZ_{2}\wedge\cdots\wedge dZ_{b}$

around a point of $D(\sigma)^{\prime}$, this proves our assertion. ∎

This is the situation at a local chart for the boundary. We pass to the global
situation.

######  Proposition 8.7.

Let $F$ be a modular form of weight $mb$ and character $\det^{m}$ with respect
to ${\Gamma}$ and $\omega_{F}$ be the corresponding rational $m$-canonical
form on ${\mathcal{F}(\Gamma)^{\Sigma}}$. Let $I$ be a $0$-dimensional cusp,
$\sigma$ be a ray in $\Sigma_{I}$, and $\Delta(\sigma)$ be the corresponding
boundary divisor of ${\mathcal{F}(\Gamma)^{\Sigma}}$. Then the vanishing order
$\nu_{\Delta(\sigma)}(\omega_{F})$ of $\omega_{F}$ along $\Delta(\sigma)$ is
given by

$\nu_{\Delta(\sigma)}(\omega_{F})=\nu_{\sigma,geom}(F)-m=\begin{cases}\nu_{\sigma}(F)-m&\sigma:\textrm{regular}\\\
\frac{1}{2}\nu_{\sigma}(F)-m&\sigma:\textrm{irregular}\end{cases}$

###### Proof.

Let
$\pi\colon(\mathcal{X}(I)^{\prime})^{\Sigma_{I}}\to{\mathcal{F}(\Gamma)^{\Sigma}}$
be the projection. By Propositions 8.5 and 8.6, we have

$\nu_{D(\sigma)^{\prime}}(\pi^{\ast}\omega_{F})=\nu_{\sigma,geom}(F)-m.$

By Proposition 7.2 (1), $\pi$ is not ramified along $D(\sigma)^{\prime}$,
regardless of whether $\sigma$ is positive-definite or isotropic. This implies
that
$\nu_{D(\sigma)^{\prime}}(\pi^{\ast}\omega_{F})=\nu_{\Delta(\sigma)}(\omega_{F})$.
∎

When $\sigma=\sigma_{J}$ is isotropic, the above equality can be written as

$\nu_{\Delta(\sigma_{J})}(\omega_{F})=\nu_{J,geom}(F)-m,$

where $\Delta(\sigma_{J})$ is the boundary divisor of
${\mathcal{F}(\Gamma)^{\Sigma}}$ over $J$.

By Gritsenko-Hulek-Sankaran [6], every irreducible component of the
ramification divisor of ${\mathcal{D}}\to{\mathcal{F}(\Gamma)}$ has
ramification index $2$ (and is defined by a reflection). Since every boundary
divisor of ${\mathcal{F}(\Gamma)^{\Sigma}}$ is of the form $\Delta(\sigma)$
for some ray $\sigma$ at some $0$-dimensional cusp $I$, Proposition 8.7
implies the following.

######  Corollary 8.8.

The $m$-canonical form $\omega_{F}$ extends holomorphically over the regular
locus of ${\mathcal{F}(\Gamma)^{\Sigma}}$ if and only if the following hold:

  1. (1)

$\nu_{R}(F)\geq m$ at every irreducible component $R$ of the ramification
divisor of ${\mathcal{D}}\to{\mathcal{F}(\Gamma)}$.

  2. (2)

$\nu_{\sigma}(F)\geq m$ at every regular ray $\sigma$ for every
$0$-dimensional cusp.

  3. (3)

$\nu_{\sigma}(F)\geq 2m$ at every irregular ray $\sigma$ for every irregular
$0$-dimensional cusp.

Note that extendability at the boundary divisors over the $1$-dimensional
cusps is encoded in the conditions (2), (3) at isotropic rays $\sigma$ for
adjacent $0$-dimensional cusps.

###  8.4. Low slope cusp form criterion

We now arrive at our principal purpose. Theorem 1.2 follows from the case
$k<b$ in the following.

######  Theorem 8.9.

Let $L$ be a lattice of signature $(2,b)$ with $b\geq 9$. Let ${\Gamma}$ be a
subgroup of ${{\rm O}^{+}(L)}$ of finite index. We take a
${\Gamma}$-admissible collection of fans $\Sigma=(\Sigma_{I})$ such that
$\Sigma_{I}$ is basic with respect to
${U(I)_{\mathbb{Z}}^{\prime}}={U(I)_{\mathbb{Q}}}\cap{\langle\Gamma,-{\rm
id}\rangle}$ at each $0$-dimensional cusp $I$. Assume that we have a cusp form
$F$ of some weight $k$ and character with respect to ${\Gamma}$ satisfying the
following:

  1. (1)

At every irreducible component $R$ of the ramification divisor of
${\mathcal{D}}\to{\mathcal{F}(\Gamma)}$, we have $\nu_{R}(F)/k>1/b$.

  2. (2)

At every regular ray $\sigma$ of $\Sigma_{I}$ at every $0$-dimensional cusp
$I$, we have $\nu_{\sigma}(F)/k>1/b$.

  3. (3)

At every irregular ray $\sigma$ of $\Sigma_{I}$ at every irregular
$0$-dimensional cusp $I$, we have $\nu_{\sigma}(F)/k>2/b$.

Then ${\mathcal{F}(\Gamma)}$ is of general type.

###### Proof.

The following argument is a slight modification of the proof of [6] Theorem
1.1, avoiding the use of a neat cover.

Replacing $F$ with its power, which does not change the slopes
$\nu_{\ast}(F)/k$, we may assume that the character $\chi$ is trivial. We
first consider the case $b\nmid k$. By further replacing $F$ with its power
$F^{2^{N}}$, where $N$ is determined by $[k/b]+2^{-N-1}\leq k/b<[k/b]+2^{-N}$,
we may assume that $k/b\geq[k/b]+1/2$ so that $[2k/b]=2[k/b]+1$. We write
$N_{0}=[k/b]+1$. Then $F$ has vanishing order $\geq N_{0}$ at the ramification
divisors of ${\mathcal{D}}\to{\mathcal{F}(\Gamma)}$ and at the regular
boundary divisors, and vanishing order $\geq 2N_{0}$ at the irregular boundary
divisors. We denote by $M_{l}({\Gamma})$ the space of ${\Gamma}$-modular forms
of weight $l$ with trivial character. For an even number $m$ we consider the
subspace $V_{m}=F^{m}\cdot M_{(bN_{0}-k)m}({\Gamma})$ of
$M_{bN_{0}m}({\Gamma})$. Modular forms in $V_{m}$ have vanishing order $\geq
mN_{0}$ at the interior ramification divisors and at the regular boundary
divisors, and vanishing order $\geq 2mN_{0}$ at the irregular boundary
divisors. Thus the corresponding $mN_{0}$-canonical forms extend
holomorphically over the regular locus of ${\mathcal{F}(\Gamma)^{\Sigma}}$ by
Corollary 8.8. By our choice of $\Sigma$, ${\mathcal{F}(\Gamma)^{\Sigma}}$ has
canonical singularities at the boundary points by Proposition 7.4, and the
interior ${\mathcal{F}(\Gamma)}$ has canonical singularities by Gritsenko-
Hulek-Sankaran [6]. Therefore these $mN_{0}$-canonical forms extend
holomorphically over a desingularization $X$ of
${\mathcal{F}(\Gamma)^{\Sigma}}$. Since $bN_{0}>k$, we have

$\dim V_{m}=\dim M_{(bN_{0}-k)m}({\Gamma})\sim c\cdot m^{b}\qquad(m\to\infty)$

for some $c>0$, so we find that $K_{X}$ is big.

When $b\mid k$, we replace $F$ with the product of a sufficiently large power
of $F$ and a modular form of weight indivisible by $b$. This perturbs the
slopes $\nu_{\ast}(F)/k$ only by $\varepsilon$, so the inequalities in (1) –
(3) still hold. Then the same argument works. ∎

######  Remark 8.10.

If we replace ”$>$” in the conditions (1) – (3) by ”$\geq$”, then the
conclusion will be weakened to ”${\mathcal{F}(\Gamma)}$ has nonnegative
Kodaira dimension”. A power of $F$ gives a nonzero pluricanonical form.

Geometric explanation of Theorem 8.9 is as follows. We have the
${\mathbb{Q}}$-linear equivalence

$K_{{\mathcal{F}(\Gamma)^{\Sigma}}}\sim_{{\mathbb{Q}}}b\mathcal{L}-B/2-\Delta_{reg}-\Delta_{irr}$

over ${\mathcal{F}(\Gamma)^{\Sigma}}$, where $B$ is the interior branch
divisor and $\Delta_{reg},\Delta_{irr}$ are the regular and irregular boundary
divisors respectively. The coefficients of $B$ and $\Delta_{irr}$ will be
multiplied by $2$ when pulled back to local charts. The existence of the cusp
form $F$ means that $b^{\prime}\mathcal{L}-B/2-\Delta_{reg}-\Delta_{irr}$ is
${\mathbb{Q}}$-effective for some $b^{\prime}<b$, $b^{\prime}\in{\mathbb{Q}}$.
(To be explicit, $b^{\prime}=k/N_{0}$ in the case $b\nmid k$ in the proof.)
Thus we have

$K_{{\mathcal{F}(\Gamma)^{\Sigma}}}\sim({\mathbb{Q}}\textrm{-effective})+(b-b^{\prime})\mathcal{L}=({\mathbb{Q}}\textrm{-effective})+(\textrm{big})=(\textrm{big}),$

and the singularities do not impose obstruction.

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