# Pullback of arithmetic theta series and its modularity for unitary Shimura
curves

Qiao He  Department of Mathematics, Columbia University, MC 4406, 2990
Broadway New York, NY 10027, USA<EMAIL_ADDRESS>,  Yousheng Shi  School
of Mathematical Sciences, Zhejiang University, 866 Yuhangtang Rd, Hangzhou,
Zhejiang Province 310058, China<EMAIL_ADDRESS>and  Tonghai Yang
Department of Mathematics, University of Wisconsin Madison, Van Vleck Hall,
Madison, WI 53706, USA<EMAIL_ADDRESS>
###### Abstract.

This paper is a complement of the modularity result of [BHK+20] for the
special case $U(1,1)$ not considered there. The main idea to embed a $U(1,1)$
Shimura curve to many $U(n-1,1)$ Shimura varieties for big $n$, and prove a
precise pullback formula of the generating series of arithmetic divisors.
Afterwards, we use the modularity result of [BHK+20] together with existence
of non-vanishing of classical theta series at any given point in the upper
half plane to prove the modulartiy result on $U(1,1)$ Shimura curves.

###### 2020 Mathematics Subject Classification:

11G18, 11F46, 14G40, 14G35

QH was partially supported by Columbia’s Faculty Research Allocation Program,
YS is partially supported by Zhejiang University’s start-up fund, and TY was
partially supported by UW-Madison Kellet Mid-career award

###### Contents

  1. 1 Introduction
  2. 2 Unitary Shimura varieties and special divisors
  3. 3 Comparison of two line bundles
  4. 4 Deformation of special divisors
  5. 5 Pullback of algebraic divisors
  6. 6 Pullback of arithmetic divisors
  7. 7 Modularity on unitary Shimura curves

##  1\. Introduction

Modularity is a beautiful way to organize a sequence of objects (say numbers)
with a lot of symmetry. A classical example is the theta function

(1.1) $\theta_{m}(\tau)=\sum_{n=0}^{\infty}r_{m}(n)q^{n},\quad q=e^{2\pi
i\tau},$

where $r_{m}(n)$ is the number of ways to express $n$ as sum of $m$ integers.
The modularity of $\theta_{m}$ implies in particular that if we know a few
$r_{m}(n)$ for “small” $n$, then we know all $r_{m}(n)$. Modularity has played
an important role in number theory and other fields, see for example [HZ76],
[KM90], [Kud97b], [Zha21], [BWR15], and [BHK+20].

In [BHK+20], the authors proved the modularity of a generating series of
special divisors on a compactified integral model of a Shimura variety
associated to a unitary group (Krämer model for a unimodular lattice) of
signature $(n-1,1)$ for $n>2$, and left the special case $n=2$ unresolved for
technical reasons. This paper finishes the job for $n=2$ by the so-called
embedding trick. We now describe the main results in a little more detail. For
undefined terms, see Section 2 for definition.

Let $\boldsymbol{k}=\mathbb{Q}(\sqrt{-D})$ be an imaginary quadratic field
with ring of integers $\mathcal{O}_{\boldsymbol{k}}$ and odd discriminant
$-D$. Let $\mathfrak{a}_{0}$ and $\mathfrak{a}$ be two unimodular Hermitian
$\mathcal{O}_{\boldsymbol{k}}$-lattices of signature $(1,0)$ and $(n-1,1)$
respectively. Let $\mathcal{S}^{*}$ be the associated integral model of the
Shimura variety with toroidal compactification (see Section 2). For each
$m>0$, Kudla and Rapoport ([KR14]) constructed divisors $\mathcal{Z}(m)$ in
the open Shimura variety $\mathcal{S}\subset\mathcal{S}^{*}$ over
$\mathcal{O}_{\boldsymbol{k}}$. There are two Green functions for
$Z(m)=\mathcal{Z}(m)(\mathbb{C})$, $\mathrm{Gr}_{B}(m)$ constructed by J.
Bruinier in his thesis ([Bru02], [BF04]), and $\mathrm{Gr}_{K}(m,v)$
constructed by S. Kudla with an extra parameter $v>0$ ([Kud97a]). By studying
their behavior at the boundary, one obtains two arithmetic divisors ([BHY15],
[How15], [ES18], see Sections 5 and 6)

$\widehat{\mathcal{Z}}_{B}^{\mathrm{tot}}(m)=(\mathcal{Z}_{B}^{\mathrm{tot}}(m),\mathrm{Gr}_{B}(m)),\hbox{
and
}\widehat{\mathcal{Z}}_{K}^{\mathrm{tot}}(m,v)=(\mathcal{Z}_{K}^{\mathrm{tot}}(m,v),\mathrm{Gr}_{K}(m,v))\in\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*}),$

with $\mathcal{Z}_{i}^{\mathrm{tot}}(m)=\mathcal{Z}^{*}(m)+\hbox{ boundary
components}$ for $i=B,K$. Here $\mathcal{Z}^{*}(m)$ is the Zariski closure of
$\mathcal{Z}(m)$ in $\mathcal{S}^{*}$, and
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$ is the
arithmetic Chow group with $\mathbb{C}$-coefficients and log-log singularities
near the boundary (see [BGKK07] and [How15]). Actually, $\mathrm{Gr}_{K}(m,v)$
is defined for all integers $m$ and induces an arithmetic divisor (for all
integers $m$)

$\widehat{\mathcal{Z}}_{K}^{\mathrm{tot}}(m,v)=(\mathcal{Z}_{K}^{\mathrm{tot}}(m),\mathrm{Gr}_{K}(m,v))\in\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*}).$

One also defines arithmetic divisors
$\widehat{\mathcal{Z}}_{B}^{\mathrm{tot}}(0)$ and
$\widehat{\mathcal{Z}}^{\mathrm{tot}}_{K}(0,v)$ in
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$. Now we have
two arithmetic theta functions with values in
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$.

(1.2) $\displaystyle\widehat{\Theta}_{B}(\tau)$ $\displaystyle=\sum_{m\geq
0}\widehat{\mathcal{Z}}_{B}^{\mathrm{tot}}(m)q^{m}\in\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})[[q]],$
(1.3) $\displaystyle\widehat{\Theta}_{K}(\tau)$
$\displaystyle=\sum_{m\in\mathbb{Z}}\widehat{\mathcal{Z}}_{K}^{\mathrm{tot}}(m,v)q^{m}\in\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})[[q,q^{-1}]].$

Ehlen and Sankran proved in [ES18, Theorem 1.4] that
$\widehat{\Theta}_{B}(\tau)-\widehat{\Theta}_{K}(\tau)$ is a modular form for
$\Gamma_{0}(D)$ of weight $n$, character $\chi_{-D}^{n}$ (where
$\chi_{-D}(\cdot)$ is the Kronecker symbol $\left(\frac{-D}{\cdot}\right)$)
and with values in
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$ (see Section 7
for detail and some subtlety). In [BHK+20], the authors proved the modularity
of $\widehat{\Theta}_{B}(\tau)$ for $n\geq 3$, and thus the modularity of
$\widehat{\Theta}_{K}(\tau)$ for $n\geq 3$. In this paper, we prove the
modularity of $\widehat{\Theta}_{K}(\tau)$ for $n=2$, which in turn implies
the modularity of $\widehat{\Theta}_{B}(\tau)$. We record it as

######  Theorem 1.1.

When $n=2$, $\widehat{\Theta}_{B}(\tau)$ and $\widehat{\Theta}_{K}(\tau)$ are
modular forms for $\Gamma_{0}(D)$ of weight $2$, trivial character, and with
values in $\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$.

We remark that the modularity of $\widehat{\Theta}_{B}$ could be used to prove
a complement of the original Gross-Zagier formula (the case when every $p|N$
is ramified in $\boldsymbol{k}$, in comparison to that every $p|N$ is split in
$\boldsymbol{k}$). Since
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$ is infinite
dimensional, we refer to [ES18, Definition 4.11] for the precise meaning of
the modularity. To prove the modularity of $\widehat{\Theta}_{K}(\tau)$ for
$n=2$, we use the following embedding trick (see, for example, [Li23],
[HMP20]). Let $\Lambda$ be a positive definite unimodular
$\mathcal{O}_{\boldsymbol{k}}$-lattice of rank $m$. Replacing $\mathfrak{a}$
by $\mathfrak{a}\obot\Lambda$, we obtain a Shimura variety
$\mathcal{S}^{\diamond,*}$ together with a canonical embedding:

(1.4)
$\varphi_{\Lambda}:\mathcal{S}^{*}\rightarrow\mathcal{S}^{\diamond,*},(A_{0},A,...)\mapsto(A_{0},A\times(A_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}\Lambda),...).$

It induces a homomorphism
$\varphi_{\Lambda}^{*}:\widehat{\operatorname{CH}}^{1}(\mathcal{S}^{\diamond,*})\rightarrow\widehat{\operatorname{CH}}^{1}(\mathcal{S}^{*})$.
The following is the main technical theorem of this paper, which should be of
independent interest.

######  Theorem 1.2.

Let notations be as above. Then

$\varphi_{\Lambda}^{*}(\widehat{\Theta}_{K}^{\diamond}(\tau))=\theta_{\Lambda}(\tau)\widehat{\Theta}_{K}(\tau).$

Here $\widehat{\Theta}_{K}^{\diamond}(\tau)$ is the arithmetic theta function
(with Kudla Green functions) associated to $\mathcal{S}^{\diamond,*}$, and

(1.5) $\theta_{\Lambda}(\tau)=\sum_{x\in\Lambda}q^{(x,x)}=\sum_{n\geq
0}r_{\Lambda}(n)q^{n}$

is a classical holomorphic modular form for $\Gamma_{0}(D)$ of weight $m$ and
character $\chi_{-D}^{m}$ .

By an analogue of a result of Yingkun Li [Li23, Lemma 3.2], for every $\tau$,
there is some $\Lambda$ with $\theta_{\Lambda}(\tau)\neq 0$. So
$\widehat{\Theta}_{K}(\tau)$ is everywhere defined and the modularity of
$\widehat{\Theta}_{K}(\tau)$ follows from the modularity result of [BHK+20].
Now Theorem 1.1 for $\widehat{\Theta}_{B}(\tau)$ follows from the modularity
for $\widehat{\Theta}_{K}(\tau)$ and the main result of [ES18]. Note that the
analogue of Theorem 1.2 does not hold for $\widehat{\Theta}_{B}$.

The main work to prove Theorem 1.2 is to understand the pullback of special
divisors on $\mathcal{S}^{\diamond,*}$ that intersect with $\mathcal{S}^{*}$
improperly and various subtleties about the boundary components. On the
generic fiber, this pullback is controlled by the line bundle of modular form
of weight $1$ (descended from the tautological line bundle), which can be
regarded as an analogue of the adjunction formula. The line bundle of modular
form of weight $1$ has two well-defined integral models: $\omega$ considered
in [BHK+20] and $\Omega$ considered in [How19]. Although $\omega$ is used in
the definition of $\widehat{\mathcal{Z}}_{K}^{\mathrm{tot}}(0,v)$ and
$\widehat{\mathcal{Z}}_{B}^{\mathrm{tot}}(0)$ in [BHK+20], it turns out that
$\Omega$ is the one that controls the pullback of special divisors that
intersect with $\mathcal{S}^{*}$ improperly. So it is important to figure out
their precise relation. We show in Section 3 (Theorem 3.2):

(1.6) $\Omega=\omega\otimes(\mathcal{O}_{\mathrm{Exc}})^{-1}.$

We believe that this relation is of independent interest. It is discovered by
[How19] that the line bundle $\Omega$ controls the deformation theory of
special divisors in the open Shimura variety $\mathcal{S}$. Another main
technical point of this paper is to extend this observation to the toroidal
compactification $\mathcal{S}^{*}$, see Theorem 4.4.

This paper is organized as follows. In Section 2, we review basics on integral
model, toroidal compactification and special divisors which are needed in this
paper. In Section 3, we prove the precise relation (1.6) between the two well-
defined integral models of the line bundle of modular forms of weight $1$
(Theorem 3.2). In Section 4, we extend the controlling deformation property of
$\Omega$ of Ben Howard to boundaries (Theorem 4.4). Section 5 is dedicated to
proving Theorem 5.5: Theorem 1.2 without Green functions. In Section 6, we
deal with pullbacks of Green functions and metrics on line bundles, and finish
the proof of Theorem 1.2. In the first part of Section 7, we use Li’s
embedding trick ([Li23]) to prove the modularity of
$\widehat{\Theta}_{K}(\tau)$. In the second part of Section 7, we describe two
slightly different Green functions for special divisors, one defined by
Bruinier in [BHK+20] and the other one defined by Ehlen and Sankaran in
[ES18]. We show that $\widehat{\Theta}_{B}(\tau)$ and
$\widehat{\Theta}_{ES}(\tau)$ are essentially the same (7.21), which will be
used in the modularity of difference
$\widehat{\Theta}_{K}(\tau)-\widehat{\Theta}_{B}(\tau)$. We remark that
Bruinier’s Green functions are a little easier to define than those of Ehlen
and Sankaran.

Acknowledgements: We thank Jan Bruinier, Ben Howard, Yingkun Li, Keerthi
Madapusi Pera, and Sid Sankaran their helpful discussions during the
preparation of this paper.

###  1.1. Notations

Let $\boldsymbol{k}=\mathbb{Q}(\sqrt{-D})$ be an imaginary quadratic field
with ring of integers $\mathcal{O}_{\boldsymbol{k}}$ and odd fundamental
discriminant $-D$. Let $\chi_{-D}(\cdot)$ be the Kronecker symbol
$\left(\frac{-D}{\cdot}\right)$. Fix a $\pi\in\mathcal{O}_{\boldsymbol{k}}$
such that $\mathcal{O}_{\boldsymbol{k}}=\mathbb{Z}+\pi\mathbb{Z}$. For any
$\mathcal{O}_{\boldsymbol{k}}$-scheme $S$, define

(1.7) $\displaystyle\epsilon_{S}=$ $\displaystyle\pi\otimes
1-1\otimes\varphi(\bar{\pi})\in\mathcal{O}_{\boldsymbol{k}}\otimes_{\mathbb{Z}}\mathcal{O}_{S},$
(1.8) $\displaystyle\bar{\epsilon}_{S}=$ $\displaystyle\bar{\pi}\otimes
1-1\otimes\varphi(\bar{\pi})\in\mathcal{O}_{\boldsymbol{k}}\otimes_{\mathbb{Z}}\mathcal{O}_{S},$

where $\varphi:\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{S}$ is the
structure map. We denote the Galois conjugate of $\varphi$ by $\bar{\varphi}$.
The ideal sheaf generated by these sections are independent of the choice of
$\pi$. The $\mathcal{O}_{\boldsymbol{k}}$-scheme $S$ will be usually clear
from the context and we often abbreviate $\epsilon_{S}$ and
$\bar{\epsilon}_{S}$ to $\epsilon$ and $\bar{\epsilon}$ respectively.

##  2\. Unitary Shimura varieties and special divisors

###  2.1. Unitary Shimura varieties

In this section, we review the theory of unitary Shimura variety following
[BHK+20]. Let $W_{0}$ and $W$ be $\boldsymbol{k}$-vector spaces endowed with
hermitian forms $H_{0}$ and $H$ of signatures $(1,0)$ and $(n-1,1)$,
respectively. We always assume that $n\geqslant 2$. Abbreviate

$W(\mathbb{R})=W\otimes_{\mathbb{Q}}\mathbb{R},\quad
W(\mathbb{C})=W\otimes_{\mathbb{Q}}\mathbb{C},\quad
W\left(\mathbb{A}_{f}\right)=W\otimes_{\mathbb{Q}}\mathbb{A}_{f},$

and similarly for $W_{0}$. In particular, $W_{0}(\mathbb{R})$ and
$W(\mathbb{R})$ are hermitian spaces over
$\mathbb{C}=\boldsymbol{k}\otimes_{\mathbb{Q}}\mathbb{R}$.

We assume the existence of $\mathcal{O}_{\boldsymbol{k}}$-lattices
$\mathfrak{a}_{0}\subset W_{0}$ and $\mathfrak{a}\subset W$, self-dual with
respect to the hermitian forms $H_{0}$ and $H$. As
$\delta=\sqrt{-D}\in\boldsymbol{k}$ generates the different of
$\boldsymbol{k}/\mathbb{Q}$, this is equivalent to self-duality with respect
to the symplectic forms

(2.1)
$\quad\psi_{0}\left(w,w^{\prime}\right)=\operatorname{Tr}_{\boldsymbol{k}/\mathbb{Q}}H_{0}\left(\delta^{-1}w,w^{\prime}\right),\quad\psi\left(w,w^{\prime}\right)=\operatorname{Tr}_{\boldsymbol{k}/\mathbb{Q}}H\left(\delta^{-1}w,w^{\prime}\right).$

Let $G\subset\mathrm{GU}\left(W_{0}\right)\times\mathrm{GU}(W)$ be the
subgroup of pairs for which the similitude factors are equal. We denote by
$\nu:G\rightarrow\mathbb{G}_{m}$ the common similitude character, and note
that $\nu(G(\mathbb{R}))\subset\mathbb{R}^{>0}$. Let
$\mathcal{D}\left(W_{0}\right)=\left\\{y_{0}\right\\}$ be a one-point set, and

(2.2) $\mathcal{D}(W)=\\{\text{negative definite }\mathbb{C}\text{-lines
}y\subset W(\mathbb{R})\\},$

so that $G(\mathbb{R})$ acts on the connected hermitian domain

$\mathcal{D}=\mathcal{D}\left(W_{0}\right)\times\mathcal{D}(W).$

The symplectic forms (2.1) determine a $\boldsymbol{k}$-conjugate-linear
isomorphism

(2.3)
$\operatorname{Hom}_{k}\left(W_{0},W\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Hom}_{k}\left(W,W_{0}\right),\
x\mapsto x^{\vee},$

characterized by
$\psi\left(xw_{0},w\right)=\psi_{0}\left(w_{0},x^{\vee}w\right)$. The
$\boldsymbol{k}$-vector space

$V=\operatorname{Hom}_{\boldsymbol{k}}\left(W_{0},W\right)$

carries a hermitian form of signature $(n-1,1)$ defined by

(2.4) $(x_{1},x_{2})=x_{2}^{\vee}\circ
x_{1}\in\operatorname{End}_{\boldsymbol{k}}\left(W_{0}\right)\cong\boldsymbol{k}.$

Let

(2.5)
$L\coloneqq\operatorname{Hom}_{\mathcal{O}_{k}}(\mathfrak{a}_{0},\mathfrak{a}),$

which is a unimodular hermitian $\mathcal{O}_{\boldsymbol{k}}$-lattice of
signature $(n-1,1)$. The group $G$ acts on $V$ in a natural way, defining an
exact sequence

(2.6)
$1\rightarrow\mathrm{Res}_{\boldsymbol{k}/\mathbb{Q}}\mathbb{G}_{m}\rightarrow
G\rightarrow\mathrm{U}(V)\rightarrow 1.$

The lattices $\mathfrak{a}_{0}$ and $\mathfrak{a}$ determine a compact open
subgroup

(2.7) $\displaystyle K=\left\\{g\in
G\left(\mathbb{A}_{f}\right):g\widehat{\mathfrak{a}}_{0}=\widehat{\mathfrak{a}}_{0}\text{
and }g\widehat{\mathfrak{a}}=\widehat{\mathfrak{a}}\right\\}\subset
G\left(\mathbb{A}_{f}\right),$

and the orbifold quotient

$\operatorname{Sh}(G,\mathcal{D})(\mathbb{C})=G(\mathbb{Q})\backslash\mathcal{D}\times
G\left(\mathbb{A}_{f}\right)/K$

is the set of complex points of a smooth $\boldsymbol{k}$-stack of dimension
$n-1$, denoted $\operatorname{Sh}(G,\mathcal{D})$.

###  2.2. Integral model

We recall the integral model defined by [BHK+20] originated from the work of
Krämer [Krä03]. For $(r,s)=(n-1,1)$ or $(n,0)$, let

$\mathcal{M}_{(r,s)}\rightarrow\operatorname{Spec}(\mathcal{O}_{\boldsymbol{k}})$

be the stack so that for an $\mathcal{O}_{\boldsymbol{k}}$-scheme $S$,
$\mathcal{M}_{(r,s)}(S)$ is the groupoid of quadruples
$\left(A,\iota,\lambda,\mathcal{F}_{A}\right)$ where

  1. (1)

$A\rightarrow S$ is an abelian scheme of relative dimension $n$,

  2. (2)

$\iota:\mathcal{O}_{\boldsymbol{k}}\rightarrow\operatorname{End}(A)$ is an
action of $\mathcal{O}_{\boldsymbol{k}}$,

  3. (3)

$\lambda:A\rightarrow A^{\vee}$ is a principal polarization whose induced
Rosati involution $\dagger$ on $\operatorname{End}^{0}(A)$ satisfies
$\iota(\alpha)^{\dagger}=\iota(\bar{\alpha})$ for all
$\alpha\in\mathcal{O}_{\boldsymbol{k}}$,

  4. (4)

$\mathcal{F}_{A}\subset\operatorname{Lie}(A)$ is an
$\mathcal{O}_{\boldsymbol{k}}$-stable $\mathcal{O}_{S}$-module local direct
summand of rank $r$ satisfying Krämer’s condition:
$\mathcal{O}_{\boldsymbol{k}}$ acts on $\mathcal{F}_{A}$ via the structure map
$\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{S}$, and acts on quotient
$\operatorname{Lie}(A)/\mathcal{F}_{A}$ via the complex conjugate of the
structure map.

In particular, when $(r,s)=(n,0)$, the $\mathcal{F}_{A}$ in the condition
$(4)$ above is simply $\operatorname{Lie}(A)$. In this case,
$\mathcal{M}(n,0)\rightarrow\mathrm{Spec}\,\mathcal{O}_{\boldsymbol{k}}$ is
proper and smooth of relative dimension $0$ by [How15, Proposition 2.1.2]. For
$\mathcal{M}_{(n-1,1)}$, we have the following theorem.

######  Theorem 2.1.

[BHK+20, Theorem 2.3.3] The $\mathcal{O}_{\boldsymbol{k}}$-stack
$\mathcal{M}_{(n-1,1)}$ is regular and flat with reduced fibers.

Finally, for the genus class $[[L]]$ of $L$, we define

$\mathcal{S}\subset\mathcal{M}_{(1,0)}\times\mathcal{M}_{(n-1,1)}$

to be the open and closed substack such that $\mathcal{S}(S)$ is the groupoid
of tuples

$\left(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda\right)\in\mathcal{M}_{(1,0)}(S)\times\mathcal{M}_{(n-1,1)}(S)$

such that at every geometric point $s\rightarrow S$, there exists an
isomorphism of hermitian $\mathcal{O}_{\boldsymbol{k},\ell}$-modules

(2.8)
$\displaystyle\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}\left(T_{\ell}A_{0,s},T_{\ell}A_{s}\right)\cong\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}\left(\mathfrak{a}_{0},\mathfrak{a}\right)\otimes\mathbb{Z}_{\ell}=L_{\ell},$

for every $\ell\neq\mathrm{char}(k(s))$. Here the hermitian form on the right
hand side is defined as in (2.4). For the hermitian form on the left hand
side, we define it similarly by replacing the symplectic forms (2.1) on
$W_{0}$ and $W$ with the Weil pairings on the Tate modules $T_{\ell}A_{0,s}$
and $T_{\ell}A_{s}$ ($A_{s}$ being the pullback of the universal object
$\mathbf{A}$ over $\mathcal{S}$ to $s$) induced by their polarizations. The
generic fiber
$\mathcal{S}\times_{\mathrm{Spec}\,\mathcal{O}_{\boldsymbol{k}}}\mathrm{Spec}\,\boldsymbol{k}$
is the Shimura variety $\operatorname{Sh}(G,\mathcal{D})$.

Now assume that $p\in\mathbb{Z}$ is a prime ramified in $\boldsymbol{k}$ (i.e.
dividing $D$). Following [How19, Appendix A] we define the exceptional divisor
$\mathrm{Exc}_{p}$ to be the locus of
$\mathcal{S}\times_{\mathrm{Spec}\,\mathcal{O}_{\boldsymbol{k},p}}\mathrm{Spec}\,\mathbb{F}_{p}$
where a geometric point $s\in\mathcal{S}(\bar{\mathbb{F}}_{p})$ is in
$\mathrm{Exc}_{p}$ if the action
$\mathcal{O}_{\boldsymbol{k}}\rightarrow\operatorname{Lie}A_{s}$ factor
through the reduction homomorphism
$\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathbb{F}_{p}$. Then
$\mathrm{Exc}_{p}$ can be given the structure of a reduced substack of
$\mathcal{S}$ and is in fact a Cartier divisor consisting of disjoint unions
of $\mathbb{P}^{n-1}$ over $\mathbb{F}_{p}$ ([BHK+20, Theorem 2.3.4], [How19,
Proposition A.2]). Finally, we define

$\mathrm{Exc}=\bigsqcup_{p|D}\mathrm{Exc}_{p}.$

###  2.3. Special divisors

For a connected $\mathcal{O}_{k}$-scheme $S$ and

$\left(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda\right)\in\mathcal{S}(S),$

we can define a positive definite hermitian form on
$\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}\left(A_{0},A\right)$ by

(2.9) $\left\langle
x_{1},x_{2}\right\rangle=\iota_{0}^{-1}(\lambda_{0}^{-1}\circ
x_{2}^{\vee}\circ\lambda\circ x_{1})\in\mathcal{O}_{\boldsymbol{k}}$

where $x_{2}^{\vee}:A^{\vee}\rightarrow A_{0}^{\vee}$ is the dual homomorphism
of $x_{2}$. By [KR14, Lemma 2.7], the form $\left\langle,\right\rangle$ is
positive-definite.

######  Definition 2.2.

([KR14, Definition 2.8]) For any $m\in\mathbb{Z}_{>0}$, define
$\mathcal{Z}(m)$ to be the moduli stack assigning to a connected
$\mathcal{O}_{\boldsymbol{k}}$-scheme $S$ the groupoid of tuples
$\left(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,x\right)$, where

  1. (1)

$\left(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda\right)\in\mathcal{S}(S)$,

  2. (2)

$x\in\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}\left(A_{0},A\right)$
satisfies $\langle x,x\rangle=m$.

According to the discussion in [BHK+20, §2.5], we may regard $\mathcal{Z}(m)$
as a Cartier divisor on $\mathcal{S}$.

###  2.4. Toroidal compactification

In this subsection, we briefly review the theory of toroidal compatification
of $\mathcal{S}$ following [BHK+20, §3].

A proper cusp label is an isomorphism class of pairs
$\Phi=(\mathfrak{n},L_{\Phi})$ in which $\mathfrak{n}$ is a projective
$\mathcal{O}_{\boldsymbol{k}}$-module of rank one, and $L_{\Phi}$ is a
unimodular Hermitian lattice of signature $(n-2,0)$. Consider a pair
$\mathfrak{m}\subset L$ where $L$ is a unimodular hermitian lattice of
signature $(n-1,1)$ and $\mathfrak{m}$ is an isotropic direct summand of rank
one. A normal decomposition of $\mathfrak{m}\subset L$ is an
$\mathcal{O}_{\boldsymbol{k}}$-module direct sum decomposition

(2.10) $\displaystyle L=(\mathfrak{m}\oplus\mathfrak{n})\obot L_{\Phi}$

where $L_{\Phi}=(\mathfrak{m}\oplus\mathfrak{n})^{\perp}$ and $\mathfrak{n}$
is an isotropic direct summand of rank one and can be identified as
$\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{m},\mathcal{O}_{\boldsymbol{k}})$.
The hermitian form on $L$ makes $L_{\Phi}\cong$
$\mathfrak{m}^{\perp}/\mathfrak{m}$ into a unimodular hermitian lattice of
signature $(n-2,0)$.

######  Lemma 2.3.

[How15, Proposition 2.6.3] Every pair $\mathfrak{m}\subset L$ as above admits
a normal decomposition. The rule

$\mathfrak{m}\subset
L\mapsto\left(L/\mathfrak{m}^{\perp},\mathfrak{m}^{\perp}/\mathfrak{m}\right)\cong(\mathfrak{n},L_{\Phi})$

establishes a bijection between the isomorphism classes of pairs
$\mathfrak{m}\subset L$ as above, and the set of cusp labels.

Now let $L$ be the unimodular lattice defined in 2.5.

######  Definition 2.4.

We define $\mathrm{Cusp}(L)$ to be the set of proper cusp labels
$\Phi=(\mathfrak{n},L_{\Phi})$ such that there exists a $\mathfrak{m}$ with
$L=\mathfrak{m}\oplus\mathfrak{n}\obot L_{\Phi}$. Moreover, we denote
$\mathrm{Cusp}([[L]])=\coprod_{L^{\prime}\in[[L]]}\mathrm{Cusp}(L^{\prime})$.

Because of Lemma 2.3, we also denote a proper cusp label as
$\Phi=(\mathfrak{m},L)$. The definition of proper cusp labels in [BHK+20] is
different with Definition 2.4. However, according to [BHK+20, Lemma 3.1.4] and
Lemma 2.3, there is a natural bijection between the set of equivalence classes
of proper cusp labels defined in [BHK+20] and the one defined in Definition
2.4.

We describe the boundary more explicitly following [BHK+20, §3.3]. For
$\left(A_{0},B,\ldots\right)\in\mathcal{M}_{(1,0)}(S)\times_{\mathcal{O}_{\boldsymbol{k}}}\mathcal{M}_{(n-2,0)}(S)$,
the étale sheaf
$\underline{\operatorname{Hom}}_{\mathcal{O}_{\boldsymbol{k}}}\left(B,A_{0}\right)$
is locally constant by [BHY15, Theorem 5.1]. For a fixed cusp label
$\Phi=(\mathfrak{n},L_{\Phi})$, let
$\Lambda_{\Phi}=\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(L_{\Phi},\mathcal{O}_{\boldsymbol{k}})$
be the hermitian dual of $L_{\Phi}$. Let $\mathcal{A}_{\Phi}$ be the moduli
space of triples $\left(A_{0},B,\varrho\right)$ over
$\mathcal{O}_{\boldsymbol{k}}$-schemes $S$, where

$\left(A_{0},B,\ldots\right)\in\mathcal{M}_{(1,0)}(S)\times_{\mathcal{O}_{\boldsymbol{k}}}\mathcal{M}_{(n-2,0)}(S),$

and

$\varrho:\underline{\Lambda_{\Phi}}\cong\underline{\operatorname{Hom}}_{\mathcal{O}_{\boldsymbol{k}}}\left(B,A_{0}\right)$

is an isomorphism of étale sheaves of hermitian
$\mathcal{O}_{\boldsymbol{k}}$-modules. Then
$\mathcal{A}_{\Phi}\rightarrow\mathrm{Spec}\,\mathcal{O}_{\boldsymbol{k}}$ is
smooth of relative dimension $0$. Now we define $\mathcal{B}_{\Phi}$ to be the
moduli space of quadruples $\left(A_{0},B,\varrho,c\right)$ over the
$\mathcal{O}_{\boldsymbol{k}}$-schemes $S$, where
$\left(A_{0},B,\varrho\right)\in\mathcal{A}_{\Phi}(S)$ and
$c:\mathfrak{n}\rightarrow B$ is an $\mathcal{O}_{\boldsymbol{k}}$-linear
homomorphism of group schemes over $S$. In other words,

(2.11)
$\displaystyle\mathcal{B}_{\Phi}=\underline{\operatorname{Hom}}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{n},B)$

where $\left(A_{0},B,\varrho\right)$ is the universal object over
$\mathcal{A}_{\Phi}$. More explicitly, according to [BHK+20, Proposition
3.4.4], we have

(2.12) $\mathcal{B}_{\Phi}\cong
A_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}L_{\Phi},$

where $A_{0}$ is the universal object of $\mathcal{M}_{(1,0)}$ and $\otimes$
is Serre’s tensor construction. The forgetful morphism
$\mathcal{B}_{\Phi}\rightarrow\mathcal{A}_{\Phi}$ is smooth of relative
dimension $n-2$.

Suppose we fix $\mu,\nu\in\mathfrak{n}$. For any scheme $U$ and any morphism
$U\rightarrow\mathcal{B}_{\Phi}$, there is a corresponding quadruple
$(A_{0},B,\varrho,c)$ over $U$. Evaluating the morphism of $U$-group schemes
$c:\mathfrak{n}\rightarrow B$ at $\mu$ and $\nu$ determines $U$-points
$c(\mu),c(\nu)\in B(U)$, and hence determines a morphism of $U$-schemes

$U\xrightarrow[]{c(\mu)\times c(\nu)}B\times B\cong B\times B^{\vee}.$

Denote by $\mathcal{L}(\mu,\nu)_{U}$ the pullback of the Poincaré line bundle
under this morphism. As $U$ varies, these line bundles are obtained as the
pullback of a single line bundle $\mathcal{L}(\mu,\nu)$ over
$\mathcal{B}_{\Phi}$.

It follows from standard bilinear properties of Poincaré bundles and
$\mathcal{O}_{\boldsymbol{k}}$-linearality of the polarization of $B$ that
$\mathcal{L}(\mu,\nu)$, up to canonical isomorphism, only depends on the image
of $\mu,\nu$ in

$\mathrm{Sym}_{\Phi}=\mathrm{Sym}^{2}_{\mathbb{Z}}(\mathfrak{n})/\langle(x\mu)\otimes\nu-\mu\otimes(\bar{x}\nu):x\in\mathcal{O}_{\boldsymbol{k}},\mu,\nu\in\mathfrak{n}\rangle.$

Thus for each $\chi\in\mathrm{Sym}_{\Phi}$, we may associate a line bundle
$\mathcal{L}(\chi)$ on $\mathcal{B}_{\Phi}$, such that there are canonical
isomorphisms

$\mathcal{L}(\chi)\otimes\mathcal{L}(\chi^{\prime})\cong\mathcal{L}(\chi+\chi^{\prime}).$

Our assumption that $D$ is odd implies that $\mathrm{Sym}_{\Phi}$ is a free
$\mathbb{Z}$-module of rank $1$. Moreover there is a positive cone in
$\mathrm{Sym}_{\Phi}\otimes_{\mathbb{Z}}\mathbb{R}$ uniquely determined by the
condition $\mu\otimes\mu\geq 0$ for all $\mu\in\mathfrak{n}$. Thus all the
line bundles $\mathcal{L}(\chi)$ are powers of the distinguished line bundle

$\mathcal{L}_{\Phi}=\mathcal{L}(\chi_{0}),$

determined by the unique positive generator $\chi_{0}\in\mathrm{Sym}_{\Phi}$.

Now define $\mathcal{B}_{\Phi}$-stacks

$\mathcal{C}_{\Phi}=\underline{\mathrm{Iso}}(\mathcal{L}_{\Phi},\mathcal{O}_{\mathcal{B}_{\Phi}}),\
\mathcal{C}_{\Phi}^{*}=\underline{\mathrm{Hom}}(\mathcal{L}_{\Phi},\mathcal{O}_{\mathcal{B}_{\Phi}}).$

In other words, $\mathcal{C}_{\Phi}^{*}$ is the total space of the line bundle
$\mathcal{L}_{\Phi}^{-1}$, and $\mathcal{C}_{\Phi}$ is the complement of the
zero section $\mathcal{B}_{\Phi}\hookrightarrow\mathcal{C}_{\Phi}^{*}$.

Finally define $\Delta_{\Phi}$ to be the finite group

$\Delta_{\Phi}=\mathrm{U}\left(\Lambda_{\Phi}\right)\times\mathrm{GL}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{n}).$

The group $\Delta_{\Phi}$ acts on $\mathcal{B}_{\Phi}$ by (see [BHK+20, Remark
3.3.2])

$(u,t)\cdot(A_{0},B,\varrho,c)=(A_{0},B,\varrho\circ u^{-1},c\circ t^{-1}),\
(u,t)\in\mathrm{U}\left(\Lambda_{\Phi}\right)\times\mathrm{GL}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{n}).$

The line bundle $\mathcal{L}_{\Phi}$ is invariant under $\Delta_{\Phi}$, hence
the action lifts to both $\mathcal{C}_{\Phi}$ and $\mathcal{C}_{\Phi}^{*}$.

The following theorem cites results that we need from [BHK+20, Theorem 3.7.1]
which ultimately depends on results from [FC13] and [Lan13].

######  Theorem 2.5.

There is a canonical toroidal compactification
$\mathcal{S}\hookrightarrow\mathcal{S}^{*}$ such that $\mathcal{S}^{*}$ is
flat over $\mathcal{O}_{\boldsymbol{k}}$ of relative dimension $n-1$. It
admits a stratification

$\mathcal{S}^{*}=\mathcal{S}\bigsqcup_{\Phi\in\mathrm{Cusp}([[L]])}\mathcal{S}^{*}(\Phi)$

as a disjoint union of locally closed substacks.

  1. (1)

The $\mathcal{O}_{\boldsymbol{k}}$-stack $\mathcal{S}^{*}$ is regular.

  2. (2)

The boundary divisor

$\partial\mathcal{S}^{*}=\bigsqcup_{\Phi\in\mathrm{Cusp}([[L]])}\mathcal{S}^{*}(\Phi)$

is a smooth divisor, flat over $\mathcal{O}_{\boldsymbol{k}}$.

  3. (3)

For each $\Phi\in\mathrm{Cusp}([[L]])$ the stratum $\mathcal{S}^{*}(\Phi)$ is
closed. All components of $\mathcal{S}^{*}(\Phi)_{/\mathbb{C}}$ are defined
over the Hilbert class field $\boldsymbol{k}^{\text{Hilb }}$, and they are
permuted simply transitively by
$\operatorname{Gal}\left(\boldsymbol{k}^{\mathrm{Hilb}}/\boldsymbol{k}\right)$.
Moreover there is a canonical identification of
$\mathcal{O}_{\boldsymbol{k}}$-stacks

${\Delta_{\Phi}\backslash\mathcal{B}_{\Phi}}$${\mathcal{S}^{*}(\Phi)}$${\Delta_{\Phi}\backslash\mathcal{C}_{\Phi}^{*}}$${\mathcal{S}^{*}}$$\scriptstyle{\pi_{\Phi}}$

such that $\pi_{\Phi}$ is an isomorphism, and the two stacks in the bottom row
become isomorphic after completion along their common closed substack in the
top row. In other words, we have

$\Delta_{\Phi}\backslash(\mathcal{C}_{\Phi}^{*})_{\mathcal{B}_{\Phi}}^{\wedge}\cong(\mathcal{S}^{*})^{\wedge}_{\mathcal{S}^{*}(\Phi)}.$

  4. (4)

The Zariski closure of the Cartier divisor $\mathcal{Z}(m)$ in
$\mathcal{S}^{*}$ is a Cartier divisor $\mathcal{Z}^{*}(m)$.

  5. (5)

The boundary divisor $\partial\mathcal{S}^{*}$ does not intersect with the
exceptional divisor $\mathrm{Exc}$.

We needs to extend the universal object over $\mathcal{S}$ to
$\mathcal{S}^{*}$. The following is due to [How15, Theorem 2.5.2].

######  Proposition 2.6.

The universal abelian scheme $\mathbf{A}$ over $\mathcal{S}$ extends to a
semi-abelian group scheme $\mathbf{G}$ over $\mathcal{S}^{*}$ with
$\mathcal{O}_{\boldsymbol{k}}$-action such that
$\mathbf{G}|_{\mathcal{S}}=\mathbf{A}$. At a geometric point
$z=\mathrm{Spec}\,\mathbb{F}\in\mathcal{S}(\Phi)$ where
$\Phi=(\mathfrak{m},L)\in\mathrm{Cusp}(L)$ , the semi-abelian group scheme
$\mathbf{G}_{z}$ is an extension

(2.13)
$0\rightarrow\mathfrak{m}\otimes_{\mathbb{Z}}\mathbb{G}_{m}\rightarrow\mathbf{G}_{z}\rightarrow
B\rightarrow 0,$

where $B$ is an abelian variety which sits in a triple
$(B,\iota_{B},\lambda_{B})\in\mathcal{M}_{(n-2,0)}(\mathbb{F})$. The flag of
bundles $\mathcal{F}_{\mathbf{A}}\subset\operatorname{Lie}\mathbf{A}$ over
$\mathcal{S}$ has a canonical extension to
$\mathcal{F}_{\mathbf{G}}\subset\operatorname{Lie}\mathbf{G}$ over
$\mathcal{S}^{*}$ which satisfies Krämer’s condition. On the complement of
$\mathrm{Exc}$, we have

(2.14)
$\mathcal{F}_{\mathbf{G}}=\mathrm{ker}(\bar{\epsilon}:\operatorname{Lie}{\mathbf{G}}\rightarrow\operatorname{Lie}{\mathbf{G}}).$

##  3\. Comparison of two line bundles

The goal of this section is to compare two line bundles $\boldsymbol{\omega}$
and $\Omega$ defined respectively in [BHY15] (or [BHK+20]) and [How19], and
reinterpret the constant term of the generating series of special divisors
(see (3.11) below) in [BHK+20].

We define the line bundle of modular form $\boldsymbol{\omega}$ on
$\mathcal{S}$ following [BHK+20, §2.4]. Let
$\left(\mathbf{A}_{0},\mathbf{A}\right)$ be the pair of universal abelian
schemes over $\mathcal{S}$, let
$\mathcal{F}_{\mathbf{A}}\subset\operatorname{Lie}(\mathbf{A})$ be the
universal subsheaf of Krämer’s moduli problem. Recall that for any abelian
scheme $A\rightarrow S$, we have the following exact sequence of locally free
$\mathcal{O}_{\mathcal{S}}$ sheaves

(3.1) $0\rightarrow\mathrm{Fil}(A)\rightarrow
H_{1}^{\mathrm{dR}}(A)\rightarrow\operatorname{Lie}A\rightarrow 0,$

where $H_{1}^{\mathrm{dR}}(A)$ can be defined as the Lie algebra of the
universal vector extension of $A$, and $\operatorname{Fil}(A)$ is canonically
isomorphic to the $\mathcal{O}_{S}$-dual of $\operatorname{Lie}A^{\vee}$. The
principal polarization of $\mathbf{A}$ induces a non-degenerate alternating
form $\langle,\rangle$ on $H_{1}^{\mathrm{dR}}(\mathbf{A})$, satisfying

$\langle\iota(a)x,y\rangle=\langle x,\iota(\bar{a})y\rangle,\forall
a\in\mathcal{O}_{\boldsymbol{k}},x,y\in H_{1}^{\mathrm{dR}}(\mathbf{A}).$

Moreover $\mathrm{Fil}(\mathbf{A})$ is totally isotropic with respect to
$\langle,\rangle$, hence $\langle,\rangle$ induces a perfect pairing

$\operatorname{Lie}\mathbf{A}\times\mathrm{Fil}({\mathbf{A}})\rightarrow\mathcal{O}_{S}.$

Let

$\mathcal{F}_{\mathbf{A}}^{\perp}\subset\mathrm{Fil}(\mathbf{A})$

be the orthogonal to $\mathcal{F}_{\mathbf{A}}$ under the pairing
$\langle,\rangle$. It is a rank one $\mathcal{O}_{\mathcal{S}}$-module local
direct summand on which $\mathcal{O}_{\boldsymbol{k}}$ acts through the
structure morphism
$\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{\mathcal{S}}$. Define the
line bundle $\boldsymbol{\omega}$ on $\mathcal{S}$ by

$\boldsymbol{\omega}\coloneqq\operatorname{Hom}\left(\operatorname{Lie}\left(\mathbf{A}_{0}\right),\mathcal{F}_{\mathbf{A}}^{\perp}\right).$

Equivalently, we have

(3.2)
$\boldsymbol{\omega}^{-1}=\operatorname{Hom}(\mathrm{Fil}(\mathbf{A}_{0}),\operatorname{Lie}(\mathbf{A})/\mathcal{F}_{\mathbf{A}}).$

We introduce another line bundle $\Omega$ which controls the deformation
theory of special divisors following [How19]. The following is essentially
[How19, Proposition 3.3]. Although [How19, Proposition 3.3] is about
$p$-divisible groups, the same proof carries over to abelian varieties.

######  Proposition 3.1.

There are inclusions of $\mathcal{O}_{\mathcal{S}}$-module local direct
summands $F_{\mathbf{A}}^{\perp}\subset\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A})\subset H_{1}^{\mathrm{dR}}(\mathbf{A})$. The
morphism $\epsilon:H_{1}^{\mathrm{dR}}(\mathbf{A})\rightarrow\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A})$ (see (1.7)) descends to a surjection

$\operatorname{Lie}\mathbf{A}\stackrel{{\scriptstyle\epsilon}}{{\rightarrow}}\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A})/\mathcal{F}_{\mathbf{A}}^{\perp}$

whose kernel $L_{\mathbf{A}}\subset\operatorname{Lie}\mathbf{A}$ is an
$\mathcal{O}_{\mathcal{S}}$-module local direct summand of rank one. It is
stable under $\mathcal{O}_{\boldsymbol{k}}$, which acts on
$\operatorname{Lie}\mathbf{A}/L_{\mathbf{A}}$ and $L_{\mathbf{A}}$ via
$\varphi$ and $\bar{\varphi}$ respectively, where
$\varphi:\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{\mathcal{S}}$ is
the structure map $\bar{\varphi}$ is its Galois conjugate.

We define the line bundle $\Omega$ on $\mathcal{S}$ by

(3.3)
$\Omega^{-1}\coloneqq\operatorname{Hom}\left(\mathrm{Fil}(\mathbf{A}_{0}),L_{\mathbf{A}}\right).$

The goal of this section is to prove the following comparison theorem.

######  Theorem 3.2.

We have the following isomorphism of line bundles on $\mathcal{S}^{*}$.

(3.4) $\Omega=\boldsymbol{\omega}\otimes\mathcal{O}(\mathrm{Exc})^{-1},$

where $\mathcal{O}(\mathrm{Exc})$ is the line bundle over $\mathcal{S}$
associated to the effective Cartier divisor $\mathrm{Exc}$ defined in Section
2.2.

###### Proof.

The theorem follows directly from Corollary 3.8 and equation (3.7) below. ∎

Assume that $p$ is an odd prime of $\mathbb{Q}$ ramified in $\boldsymbol{k}$.
Following [Krä03, Theorem 4.5, Step 2], we define a substack $\mathcal{Z}_{p}$
of the special fiber $\mathcal{S}_{p}$ of $\mathcal{S}$ over $p$ as follows.
Let $(\mathbf{A}_{0,p},\mathbf{A}_{p})$ be the universal abelian schemes over
$\mathcal{S}_{p}$. Define on $\epsilon H_{1}^{\mathrm{dR}}(\mathbf{A}_{p})$
(on $\mathcal{S}_{p}$ we have $\epsilon=\pi\otimes 1$) an
$\mathcal{O}_{\mathcal{S}_{p}}$-bilinear form $\\{,\\}$ by

(3.5) $\\{\epsilon x,\epsilon y\\}=\langle\epsilon x,y\rangle.$

As $\epsilon H_{1}^{\mathrm{dR}}(\mathbf{A}_{p})$ is isotropic with respect to
$\langle,\rangle$, the form $\\{,\\}$ is well-defined. Moreover $\\{,\\}$ is
symmetric:

$\\{\epsilon x,\epsilon y\\}=\langle(\pi\otimes 1)x,y\rangle=-\langle
y,(\pi\otimes 1)x\rangle=\langle(\pi\otimes 1)y,x\rangle=\\{\epsilon
y,\epsilon x\\}.$

######  Definition 3.3.

Let $\mathcal{Z}_{p}$ be the substack of $\mathcal{S}_{p}$ such that for any
$\mathcal{O}_{\boldsymbol{k}}/(p)$-scheme $S$, $\mathcal{Z}_{p}(S)$ is the
groupoid of isomorphism classes
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F})\in\mathcal{S}_{p}(S)$
such that $\mathcal{F}_{A}^{\bot}$ is isotropic with respect to $\\{,\\}$.

######  Lemma 3.4.

$\mathcal{Z}_{p}$ is a regular scheme and a Cartier divisor of
$\mathcal{S}_{p}$. Moreover we have the following equation of Cartier
divisors.

(3.6) $\mathcal{S}_{p}=\mathcal{Z}_{p}+\mathrm{Exc}_{p}.$

###### Proof.

First we show that

$\mathcal{S}_{p}^{\mathrm{red}}=\mathcal{Z}_{p}^{\mathrm{red}}\cup\mathrm{Exc}_{p}.$

Let $z$ be a geometric point in $\mathcal{S}_{p}\setminus\mathcal{Z}_{p}$. By
the same argument of [Krä03, Theorem 4.5, Step 3], we know that both
$\mathrm{Fil}(\mathbf{A}_{z})$ and $\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A}_{z})$ are equal to

$\mathcal{F}_{\mathbf{A}_{z}}^{\bot}\oplus(\epsilon^{-1}\mathcal{F}_{\mathbf{A}_{z}}^{\bot})^{\bot}.$

Here $(\epsilon^{-1}\mathcal{F}_{\mathbf{A}_{z}}^{\bot})^{\bot}$ is the
perpendicular complement of $\epsilon^{-1}\mathcal{F}_{\mathbf{A}_{z}}^{\bot}$
in $H_{1}^{dR}(\mathbf{A}_{z})$ with respect to $\langle,\rangle$. This
implies that $\epsilon=\pi\otimes 1$ acts trivially on
$\operatorname{Lie}\mathbf{A}_{z}$, so $z$ lies in $\mathrm{Exc}_{p}$ by
definition.

By the same calculation as in [Krä03, Theorem 4.5, Step 4], we can conclude
the fact that $\mathcal{Z}_{p}$ is a regular Cartier divisor and (3.6). ∎

######  Proposition 3.5.

The natural map
$L_{\mathbf{A}}\rightarrow\operatorname{Lie}\mathbf{A}/\mathcal{F}_{\mathbf{A}}$
defines a section
$s\in\operatorname{Hom}(\Omega^{-1},\boldsymbol{\omega}^{-1})$. We have the
following equation on the zero locus of $s$.

$(s)=\sum_{p\mid D}\mathcal{Z}_{p},$

where the summation is over all finite primes ramified in $\boldsymbol{k}$.

###### Proof.

Step 1. Let $z$ be a geometric point of $(s)$ with residue field $\kappa$. If
the characteristic of $\kappa$ does not divide the discriminant $D$ of
$\boldsymbol{k}$, then the structural morphism
$\varphi:\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{\mathcal{S}}$ and
its conjugate $\bar{\varphi}$ are not the same. Since
$\mathcal{O}_{\boldsymbol{k}}$ acts on $\mathcal{F}_{\mathbf{A}_{z}}$ via
$\varphi$, and on $L_{\mathbf{A}_{z}}$ via $\bar{\varphi}$,
$L_{\mathbf{A}_{z}}\not\subset\mathcal{F}_{\mathbf{A}_{z}}$, which means the
map $s_{z}$ has to be nonzero, a contradiction. Hence $z$ is in
$\mathcal{S}_{p}$ for some ramified $p$.

Step 2. Assume $z$ is a geometric point of $\mathcal{S}_{p}$ for a ramified
$p$. As $\mathcal{F}_{\mathbf{A}_{z}}^{\bot}\subset\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A}_{z})$, we can assume
$\mathcal{F}_{\mathbf{A}_{z}}^{\bot}$ is spanned by $\epsilon x$ for some
$x\in H_{1}^{\mathrm{dR}}(\mathbf{A}_{z})$. Notice that $\epsilon^{2}=0$ on
$\mathcal{S}_{p}$, hence

$\epsilon^{-1}\mathcal{F}_{\mathbf{A}_{z}}^{\bot}=\mathrm{Span}\\{x\\}\oplus\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A}_{z}).$

The line $L_{\mathbf{A}_{z}}$ by definition is
$\epsilon^{-1}\mathcal{F}_{\mathbf{A}_{z}}^{\bot}/\mathrm{Fil}(\mathbf{A}_{z})\subset\operatorname{Lie}\mathbf{A}_{z}$.
Hence

$L_{\mathbf{A}_{z}}\subset\mathcal{F}_{\mathbf{A}_{z}}\Leftrightarrow\langle
x,\mathcal{F}_{\mathbf{A}_{z}}^{\bot}\rangle=0\text{ and }\langle\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A}_{z}),\mathcal{F}_{\mathbf{A}_{z}}^{\bot}\rangle=0.$

The last condition is automatic as
$\mathcal{F}_{\mathbf{A}_{z}}^{\bot}\subset\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A}_{z})$ by Proposition 3.1 and $\epsilon
H_{1}^{\mathrm{dR}}(\mathbf{A}_{z})$ is isotropic. Hence

$L_{\mathbf{A}_{z}}\subset\mathcal{F}_{\mathbf{A}_{z}}\Leftrightarrow\langle
x,\mathcal{F}_{\mathbf{A}_{z}}^{\bot}\rangle=0\Leftrightarrow\langle
x,\epsilon x\rangle=0.$

By the definition of $\\{,\\}$, the last condition is true if and only if
$\\{x,x\\}=0$, i.e., $\mathcal{F}_{\mathbf{A}_{z}}^{\bot}$ is isotropic with
respect to $\\{,\\}$. Hence

$z\in(s)\cap\mathcal{S}_{p}\Leftrightarrow z\in\mathcal{Z}_{p}.$

In fact in the above argument we can replace $z$ by any Artinian scheme over
$\mathcal{S}_{p}$ and conclude by Nakayama’s Lemma that

$(s)|_{\mathcal{S}_{p}}=\mathcal{Z}_{p}.$

Step 3: We have shown that $(s)$ is supported on the primes ramified in
$\boldsymbol{k}$ and its special fiber over such a prime $p$ is
$\mathcal{Z}_{p}$. Since $\mathcal{Z}_{p}$ is regular, by [RTZ13, Lemma 10.3],
in order to prove the proposition it remains to show that $(s)$ has no
$\mathcal{O}_{\boldsymbol{k},p}/(\pi^{2})$-point for any ramified prime $p$.
Assume $\tilde{z}$ is such a point. Then on one hand $\pi$ acts on
$\mathcal{F}_{\mathbf{A}_{\tilde{z}}}$ by $\varphi(\pi)$. On the other hand
$\pi$ acts on
$L_{\mathbf{A}_{\tilde{z}}}\subset\mathcal{F}_{\mathbf{A}_{\tilde{z}}}$ by
$-\varphi(\pi)$. Since $p\neq 2$, we know that $\pi\neq-\pi$ in
$\mathcal{O}_{\boldsymbol{k}}/(\pi^{2})$. This is a contradiction. This
finishes the proof of the proposition. ∎

###  3.1. Extensions of the line bundles to the boundary

Let $\mathcal{Z}_{p}^{*}$ be the Zariski closure of $\mathcal{Z}_{p}$ in
$\mathcal{S}_{p}^{*}$, then
$\mathcal{Z}_{p}^{*}=\mathcal{S}_{p}^{*}-\hbox{Exc}_{p}$ is a Cartier divisor
of $\mathcal{S}_{p}^{*}$. Since $\mathcal{S}_{p}^{*}$ is the principal divisor
defined by $p$, we have the following equation in
$\mathrm{CH}^{1}(\mathcal{S}^{*})$.

(3.7) $\mathcal{Z}_{p}^{*}+\mathrm{Exc}_{p}=0.$

By [BHK+20, Theorem 3.7.1(6)], the line bundle $\boldsymbol{\omega}$ admits a
canonical extension to the compactification $\mathcal{S}^{*}$ which is still
denoted by $\boldsymbol{\omega}$ such that

(3.8)
$\boldsymbol{\omega}^{-1}|_{z}=\operatorname{Hom}(\mathrm{Fil}(\mathbf{A}_{0}),\operatorname{Lie}(\mathbf{G}_{z})/\mathcal{F}_{\mathbf{G}_{z}}).$

Here we recall from Proposition 2.6 that $\mathbf{G}$ is the extension of the
universal abelian scheme $\mathbf{A}$ over $\mathcal{S}$ to $\mathcal{S}^{*}$.
We have a similar result for $\Omega$.

######  Lemma 3.6.

We have the equality of vector bundles over
$\mathcal{S}\setminus\mathrm{Exc}$.

$L_{\mathbf{A}}=\bar{\epsilon}\operatorname{Lie}{\mathbf{A}}.$

###### Proof.

By the fact that $\epsilon\bar{\epsilon}=0$, we clearly have

(3.9) $\bar{\epsilon}\operatorname{Lie}{\mathbf{A}}\subset L_{\mathbf{A}}.$

Now let $z$ be any geometric close point of
$\mathcal{S}\setminus\mathrm{Exc}$. Since $L_{\mathbf{A}}$ has rank $1$, if
the inclusion $\bar{\epsilon}\operatorname{Lie}{\mathbf{A}_{z}}\subset
L_{\mathbf{A}_{z}}$ is strict, we must have
$\bar{\epsilon}\operatorname{Lie}{\mathbf{A}_{z}}=0$. In other words
$\bar{\epsilon}\operatorname{Lie}\mathbf{A}_{z}=\\{0\\}$, which forces $z$ to
be a point in $\mathrm{Exc}$ by the definition of $\mathrm{Exc}$. This is a
contradiction. So we know that the lemma is true on the level of geometric
close points. Since
$\bar{\epsilon}:\operatorname{Lie}\mathbf{A}\rightarrow\operatorname{Lie}\mathbf{A}$
is a section of the coherent sheaf
$\operatorname{End}(\operatorname{Lie}\mathbf{A})$, the lemma is true by
Nakayama’s lemma. ∎

######  Corollary 3.7.

The line bundle $L_{\mathbf{A}}$ can be extended to a line bundle over
$\mathcal{S}^{*}$ denoted by $L_{\mathbf{G}}$ such that for a geometric point
$z=\mathrm{Spec}\,\mathbb{F}\in\partial\mathcal{S}^{*}$ on the boundary,

$L_{\mathbf{G}_{z}}=\bar{\epsilon}\operatorname{Lie}\mathbf{G}_{z}\subset\operatorname{Lie}\mathbf{G}_{z}.$

###### Proof.

We first claim that $\bar{\epsilon}\operatorname{Lie}{\mathbf{G}}$ is a line
bundle over $\mathcal{S}^{*}\setminus\mathrm{Exc}$. By Nakayama’s lemma, it
suffices to check all geometric close points. For points in
$\mathcal{S}\setminus\mathrm{Exc}$, the claim follows from Lemma 3.6. For a
geometric point $z$ on $\mathcal{B}_{\Phi}$, we have the exact sequence of Lie
algebras

$0\rightarrow\mathfrak{m}\otimes_{\mathbb{Z}}\operatorname{Lie}\mathbb{G}_{m}\rightarrow\operatorname{Lie}\mathbf{G}_{z}\rightarrow\operatorname{Lie}B\rightarrow
0,$

from the exact sequence (2.13). Now since
$(B,\iota_{B},\lambda_{B})\in\mathcal{M}_{(n-2,0)}(\mathbb{F})$, we have
$\bar{\epsilon}\operatorname{Lie}B=0$. So
$\bar{\epsilon}\operatorname{Lie}\mathbf{G}_{z}=\bar{\epsilon}\operatorname{Lie}(\mathfrak{m}\otimes_{\mathbb{Z}}\mathbb{G}_{m})$
is of dimension $1$.

Now by Lemma 3.6 and the (5) of Theorem 2.5, over
$\mathcal{S}^{*}\setminus\mathrm{Exc}$, we can define $L_{\mathbf{G}}$ to be
$\bar{\epsilon}\operatorname{Lie}\mathbf{G}$. The corollary follows. ∎

As a consequence, $\Omega$ can be canonically extended to a line bundle over
$\mathcal{S}^{*}$ which is defined by

(3.10)
$\Omega^{-1}\coloneqq\operatorname{Hom}\left(\mathrm{Fil}(\mathbf{A}_{0}),L_{\mathbf{G}}\right).$

The natural map
$L_{\mathbf{A}}\rightarrow\operatorname{Lie}\mathbf{A}/\mathcal{F}_{\mathbf{A}}$
also has a canonical extension to
$L_{\mathbf{G}}\rightarrow\operatorname{Lie}\mathbf{G}/\mathcal{F}_{\mathbf{G}}$
which defines a section in
$\operatorname{Hom}(\Omega^{-1},\boldsymbol{\omega}^{-1})$ still denoted by
$s$.

######  Corollary 3.8.

We have the following equation on the zero locus of $s$ on $\mathcal{S}^{*}$.

$(s)=\sum_{p}\mathcal{Z}^{*}_{p},$

where the summation is over all finite primes ramified in $\boldsymbol{k}$.

###### Proof.

The argument in Step 1 of the proof of Proposition 3.5 extends to the
boundary. Hence none of the boundary component belongs to $(s)$. Since both
sides of the equation are Cartier divisors, the corollary follows from
Proposition 3.5. ∎

###  3.2. Constant term of the generating series

For $m\leq 0$, using the correspondence between line bundles and Cartier
divisors, we define the the class
$\mathcal{Z}^{*}(m)\in\mathrm{Pic}(\mathcal{S}^{*})\cong\mathrm{CH}^{1}(\mathcal{S}^{*})$
by

(3.11) $\mathcal{Z}^{*}(m)\coloneqq\begin{cases}\Omega^{-1}&\text{if }m=0,\\\
\mathcal{O}_{\mathcal{S}}&\text{if }m<0.\end{cases}$

By Theorem 3.2, the definition of $\mathcal{Z}^{*}(0)$ (the constant term in
the generating series of special divisors) agrees with that of [BHK+20,
Equation (1.1.4)]. We will see in the next section that as $\Omega$ controls
the deformation of special divisors, this definition of the constant term is
also compatible with pullbacks, see §5.

##  4\. Deformation of special divisors

It is discovered by [How19] that the line bundle $\Omega$ controls the
deformation theory of special divisors on the open Shimura variety
$\mathcal{S}$. The main goal of this section is to extend this observation to
$\mathcal{S}^{*}$. First we review some deformation theory that we will need
following [How19, §4]. Throughout this section we will repeatedly encounter
the following setting. Let $S$ be a stack and $Z$ be another stack with a
finite unramified morphism $Z\rightarrow S$. We assume there is an étale
morphism of stacks $U\rightarrow S$ such that $Z|_{U}\rightarrow S|_{U}$ is a
closed immersion of schemes on every connected component of its domain. Let
$\mathcal{Z}$ be such a connected component and $I_{\mathcal{Z}}$ be its ideal
sheaf in $S|_{U}$. Then $I_{\mathcal{Z}}^{2}$ defines a larger closed formal
subscheme

$\displaystyle\mathcal{Z}\hookrightarrow\tilde{\mathcal{Z}}\hookrightarrow
S|_{U}.$

The image of $I_{\mathcal{Z}}$ in the $\mathcal{O}_{\tilde{\mathcal{Z}}}$ can
be equipped with the trivial P.D. structure. The stack $\tilde{\mathcal{Z}}$
is called the first order thickening of $\mathcal{Z}$ in $S|_{U}$.

Now we specialize to the case $S=\mathcal{S}$ and $Z=\mathcal{Z}(m)$. By
[KR14, Proposition 2.9], for any geometric point $z\in\mathcal{S}$, there is
an étale neighborhood $U$ of $z$ in $\mathcal{S}$ such that
$\mathcal{Z}(m)|_{U}\rightarrow\mathcal{S}|_{U}$ is a closed immersion of
schemes on every connected component of its domain. Let $\mathcal{Z}$ be such
a connected component as in the previous paragraph. The universal morphism
$x:\mathbf{A}_{0}|_{\mathcal{Z}}\rightarrow\mathbf{A}|_{\mathcal{Z}}$ induces
a morphism of vector bundles over $\mathcal{Z}$.

$H_{1}^{\mathrm{dR}}(\mathbf{A}_{0}|_{\mathcal{Z}})\xrightarrow{x}H_{1}^{\mathrm{dR}}(\mathbf{A}|_{\mathcal{Z}}),$

which maps $\mathrm{Fil}(\mathbf{A}_{0}|_{\mathcal{Z}})$ to
$\mathrm{Fil}(\mathbf{A}|_{\mathcal{Z}})$. By Grothendieck-Messing theory
([Mes72]), this morphism admits a canonical extension to a morphism between
vector bundles over $\tilde{\mathcal{Z}}$,

$H_{1}^{\mathrm{dR}}(\mathbf{A}_{0}|_{\tilde{\mathcal{Z}}})\xrightarrow{\tilde{x}}H_{1}^{\mathrm{dR}}(\mathbf{A}|_{\tilde{\mathcal{Z}}}),$

which determines a morphism (still denoted by $\tilde{x}$)

(4.1)
$\mathrm{Fil}(\mathbf{A}_{0}|_{\tilde{\mathcal{Z}}})\xrightarrow{\tilde{x}}\operatorname{Lie}(\mathbf{A}|_{\tilde{\mathcal{Z}}}).$

The following is an analogue of [How19, Proposition 4.1].

######  Proposition 4.1.

The morphism (4.1) takes values in the rank one local direct summand

$L_{\mathbf{A}}|_{\tilde{\mathcal{Z}}}\subset\operatorname{Lie}\mathbf{A}|_{\tilde{\mathcal{Z}}},$

and so can be viewed as a morphism of line bundles

(4.2)
$\operatorname{Fil}\left(\mathbf{A}_{0}\right)|_{\tilde{\mathcal{Z}}}\stackrel{{\scriptstyle\tilde{x}}}{{\rightarrow}}L_{\mathbf{A}}|_{\tilde{\mathcal{Z}}}.$

The Kudla-Rapoport divisor $\mathcal{Z}$ is the largest closed formal
subscheme of $\tilde{\mathcal{Z}}$ over which (4.2) is trivial.

###### Proof.

The proof is identical to that of [How19, Proposition 4.1] if we replace
$p$-divisible groups by abelian schemes. ∎

We will extend Proposition 4.1 to the toroidal compactification
$\mathcal{S}^{*}$, see Theorem 4.4.

###  4.1. Special divisors on boundary charts

We would like to study special divisors on the boundary chart
$\mathcal{C}_{\Phi}^{*}$. As in §2.4, assume $\Phi=(\mathfrak{n},L_{\Phi})$
and $\mathfrak{m}$ be the hermitian dual of $\mathfrak{n}$. First let us
construct a semi-abelian scheme $G^{\sharp}$ over $\mathcal{B}_{\Phi}$ and a
$1$-motive $M$ over $\mathcal{C}_{\Phi}$ as follows.

By the definition of $\mathcal{B}_{\Phi}$, we get a tautological homomorphism
of fppf sheaves
$c^{\vee}\in\underline{\operatorname{Hom}}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{n},B)$
over $\mathcal{B}_{\Phi}$. Since

$\underline{\operatorname{Hom}}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{n},B)\cong\underline{\mathrm{Ext}}^{1}_{\mathcal{O}_{\boldsymbol{k}}}(B^{\vee},\mathfrak{n}^{\vee}\otimes_{\mathbb{Z}}\mathbb{G}_{m})$

(see for example [Lan13, Proposition 3.1.5.1]), $c^{\vee}$ determines a semi-
abelian scheme $(G^{\sharp})^{\vee}$ over $\mathcal{B}_{\Phi}$, such that
there is an exact sequence of fppf sheaves of
$\mathcal{O}_{\boldsymbol{k}}$-modules

(4.3)
$0\rightarrow\mathfrak{m}\otimes_{\mathbb{Z}}\mathbb{G}_{m}\rightarrow(G^{\sharp})^{\vee}\xrightarrow{\pi^{\prime}}B^{\vee}\rightarrow
0.$

Similarly define $c=\psi\circ c^{\vee}$ where $\psi:B\rightarrow B^{\vee}$ is
the principal polarization of $B$, then $c$ determines a semi-abelian scheme
$G^{\sharp}$ over $\mathcal{B}_{\Phi}$, such that there is an exact sequence
of fppf sheaves of $\mathcal{O}_{\boldsymbol{k}}$-modules

(4.4) $0\rightarrow\mathfrak{m}\otimes_{\mathbb{Z}}\mathbb{G}_{m}\rightarrow
G^{\sharp}\xrightarrow{\pi}B\rightarrow 0.$

We remark here that $G^{\sharp}$ agree with the pullback of
$\mathbf{G}|_{\mathcal{S}^{*}(\Phi)}$ under the morphism $\pi_{\Phi}$ in
Theorem 2.5 (3). By [Lan13, Lemma 3.4.2], the condition $c=\psi\circ c^{\vee}$
guarantees that there is an $\mathcal{O}_{\boldsymbol{k}}$-linear isomorphism
$\lambda^{\sharp}:G^{\sharp}\rightarrow(G^{\sharp})^{\vee}$. Since
$\mathcal{C}_{\Phi}$ and $\mathcal{C}_{\Phi}^{*}$ are
$\mathcal{B}_{\Phi}$-stacks, we can pullback $(G^{\sharp})^{\vee}$ and
$G^{\sharp}$ to semi-abelian schemes on $\mathcal{C}_{\Phi}$ and
$\mathcal{C}_{\Phi}^{*}$ (denoted by the same notations).

The definition of $\mathcal{C}_{\Phi}$ and $\mathcal{C}_{\Phi}^{*}$ gives us
in addition a positive, symmetric, and $\mathcal{O}_{\boldsymbol{k}}$-linear
isomorphism

$\tau:1_{(\underline{\mathfrak{n}}\times\underline{\mathfrak{n}})|_{\mathcal{C}_{\Phi}}}\rightarrow(c^{\vee}\times
c)^{*}(\mathcal{P}^{-1})|_{(\underline{\mathfrak{n}}\times\underline{\mathfrak{n}})|_{\mathcal{C}_{\Phi}}}$

of $\mathbb{G}_{m}$-biextensions of
$(\underline{\mathfrak{n}}\times\underline{\mathfrak{n}})|_{\mathcal{C}_{\Phi}}$.
Here $\mathcal{P}$ is the Poincare line bundle on $B\times B^{\vee}$. We refer
to [How15, §2.3] and [Lan13, §4.2.2] for the precise description of $\tau$. By
[Lan13, §4.2], this gives us $1$-motives
$M=[\underline{\mathfrak{n}}\xrightarrow{u}G^{\sharp}]$, and
$M^{\vee}=[\underline{\mathfrak{n}}\xrightarrow{v}(G^{\sharp})^{\vee}]$ over
$\mathcal{C}_{\Phi}$, where $u,v$ are morphisms of fppf sheaves of
$\mathcal{O}_{\boldsymbol{k}}$-modules such that

$\pi\circ u=c^{\vee},\ \pi^{\prime}\circ v=c.$

By the proof of [BHK+20, Proposition 3.3.3], the morphism $M\rightarrow
M^{\vee}$ induced by the identity map of $\mathfrak{n}$ and $\lambda^{\sharp}$
is a principal polarization of $M$ in the sense of [Del74, §10.2.11], which is
compatible with the given polarization $\psi:B\rightarrow B^{\vee}$ and with
the isomorphism $\mathfrak{m}\cong\mathfrak{n}^{\vee}$.

Suppose we have a morphism of group schemes $x:\mathbf{A}_{0}\rightarrow
G^{\sharp}$ over $\mathcal{B}_{\Phi}$. Its dual morphism
$x^{\vee}:(G^{\sharp})^{\vee}\rightarrow\mathbf{A}_{0}^{\vee}$ is defined as
follows. By the definition of the dual $1$-motive $M^{\vee}$, we have

$(G^{\sharp})^{\vee}=\underline{\mathrm{Ext}}^{1}([\underline{\mathfrak{n}}\xrightarrow{c^{\vee}}B],\mathbb{G}_{m}).$

For any $\mathcal{B}_{\Phi}$-scheme $S$, an $S$-point $\xi$ of
$(G^{\sharp})^{\vee}$ is a commutative diagram of group schemes over $S$

${0}$${\mathfrak{n}}$${\mathfrak{n}}$${0}$${0}$${\mathbb{G}_{m}}$${\tilde{B}}$${B}$${0}$$\scriptstyle{c^{\vee}}$

where the rows are exact. Then $x^{\vee}(\xi)\in
A_{0}^{\vee}(S)=\underline{\mathrm{Ext}}^{1}(A_{0},\mathbb{G}_{m})(S)$ is the
extension $\tilde{A_{0}}\rightarrow A_{0}$ which pulls back the extension
$\tilde{B}\rightarrow B$ along $\pi\circ x$. Namely we have the Cartesion
diagram

${\tilde{A}_{0}}$${A_{0}}$${\tilde{B}}$${B.}$$\scriptstyle{\pi\circ x}$

If we view $B^{\vee}$ as $\underline{\mathrm{Ext}}^{1}(B,\mathbb{G}_{m})$, we
immediately see that $x^{\vee}=(\pi\circ x)^{\vee}\circ\pi^{\prime}$ where
$\pi$ and $\pi^{\prime}$ are defined in (4.4) and (4.3). Now we are ready to
define special divisors on $\mathcal{B}_{\Phi}$, $\mathcal{C}_{\Phi}$ and
$\mathcal{C}_{\Phi}^{*}$. The following definition is due to [BHK+20, Remark
3.6.3].

######  Definition 4.2.

For $m\in\mathbb{Z}_{>0}$, define $\mathcal{Z}_{\mathcal{B}_{\Phi}}(m)$ to be
the subscheme of $\mathcal{B}_{\Phi}$ such that given a morphism of
$\mathcal{O}_{\boldsymbol{k}}$-scheme $S\rightarrow\mathcal{C}_{\Phi}$,

$\mathcal{Z}_{\mathcal{B}_{\Phi}}(m)(S)=\\{x\in\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(\mathbf{A}_{0,S}\
,G^{\sharp})\mid\langle x,x\rangle=m\\}.$

Here $\langle,\rangle$ is defined as in (2.9):

(4.5) $\left\langle
x_{1},x_{2}\right\rangle=\iota_{0}^{-1}(\lambda_{0}^{-1}\circ
x_{2}^{\vee}\circ\lambda^{\sharp}\circ x_{1})\in\mathcal{O}_{\boldsymbol{k}}$

We also define

(4.6)
$\mathcal{Z}_{\Phi}(m)=\mathcal{Z}_{\mathcal{B}_{\Phi}}(m)\times_{\mathcal{B}_{\Phi}}\mathcal{C}_{\Phi},\
\mathcal{Z}_{\Phi}^{*}(m)=\mathcal{Z}_{\mathcal{B}_{\Phi}}(m)\times_{\mathcal{B}_{\Phi}}\mathcal{C}_{\Phi}^{*}.$

By [BHK+20, Proposition 3.6.2], $\mathcal{Z}_{\mathcal{B}_{\Phi}}(m)$ is an
effective Cartier divisor of $\mathcal{B}_{\Phi}$ that is flat over
$\mathcal{A}_{\Phi}$ and $\mathcal{O}_{\boldsymbol{k}}$. As
$\mathcal{C}_{\Phi}$ and $\mathcal{C}_{\Phi}^{*}$ are smooth over
$\mathcal{B}_{\Phi}$, $\mathcal{Z}_{\Phi}(m)$ and $\mathcal{Z}_{\Phi}^{*}(m)$
are smooth over $\mathcal{Z}_{\mathcal{B}_{\Phi}}(m)$, and are effective
Cartier divisors of $\mathcal{C}_{\Phi}$ and $\mathcal{C}_{\Phi}^{*}$
respectively, flat over $\mathcal{A}_{\Phi}$ and
$\mathcal{O}_{\boldsymbol{k}}$.

###  4.2. Deformation of special divisors on boundary charts

Now consider $G^{\sharp}$ as the $1$-motive $[0\rightarrow G^{\sharp}]$ over
$\mathcal{B}_{\Phi}$. Let $H_{1}^{\mathrm{dR}}(G^{\sharp})$ be the De Rham
realization of $G^{\sharp}$ (see [Del74, §10.1.7]) defined by the Lie algebra
of its universal vector extension, which is a rank $2n-2$ vector bundle over
$\mathcal{B}_{\Phi}$, locally free of rank $n-1$ over
$\mathcal{O}_{\boldsymbol{k}}\otimes_{\mathbb{Z}}\mathcal{O}_{\mathcal{B}_{\Phi}}$.
Then $\mathrm{Fil}(G^{\sharp})=F^{0}H_{1}^{\mathrm{dR}}(G^{\sharp})$ is a
locally free direct summand of $H_{1}^{\mathrm{dR}}(G^{\sharp})$ of
$\mathcal{O}_{\mathcal{B}_{\Phi}}$-rank $n-2$. We have the following exact
sequence of Lia algebras

(4.7)
$0\rightarrow\mathfrak{m}\otimes_{\mathbb{Z}}\mathcal{O}_{\mathcal{B}_{\Phi}}\rightarrow\operatorname{Lie}G^{\sharp}\rightarrow\operatorname{Lie}B\rightarrow
0.$

Define

(4.8) $L_{G^{\sharp}}=\bar{\epsilon}\operatorname{Lie}G^{\sharp}.$

Then by the exact sequence (4.7) and an argument similar to that of [How15,
Lemma 2.3.6], we can easily see that $L_{G^{\sharp}}$ is an
$\mathcal{O}_{\mathcal{B}_{\Phi}}$-module local direct summand of rank one. It
is stable under $\mathcal{O}_{\boldsymbol{k}}$, which acts on
$\operatorname{Lie}G^{\sharp}/L_{G^{\sharp}}$ and $L_{G^{\sharp}}$ via
$\varphi,\bar{\varphi}$ :
$\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{\mathcal{B}_{\Phi}}$,
respectively. Pulling back to $\mathcal{C}_{\Phi}^{*}$, we get vector bundles
$H_{1}^{\mathrm{dR}}(G^{\sharp})$, $\mathrm{Fil}(G^{\sharp})$,
$\operatorname{Lie}G^{\sharp}$, and $L_{G^{\sharp}}$ over
$\mathcal{C}_{\Phi}^{*}$ denoted by the same notation.

Similarly let $H_{1}^{\mathrm{dR}}(M)$ be the De Rham realization of the
$1$-motive $M=[\underline{\mathfrak{n}}\xrightarrow{u}G^{\sharp}]$ defined in
§4.1. It is a rank $2n$ vector bundle over $\mathcal{C}_{\Phi}$, locally free
of rank $n$ over
$\mathcal{O}_{\boldsymbol{k}}\otimes_{\mathbb{Z}}\mathcal{O}_{\mathcal{C}_{\Phi}}$.
Then $\mathrm{Fil}(M)=F^{0}H_{1}^{\mathrm{dR}}(M)$ is a locally free direct
summand of $H_{1}^{\mathrm{dR}}(M)$ of $\mathcal{O}_{\mathcal{C}_{\Phi}}$-rank
$n$. We have a commutative diagram (see [BVS01, §1.4])

(4.9)
${0}$${\mathrm{Fil}(G^{\sharp})}$${H_{1}^{\mathrm{dR}}(G^{\sharp})}$${\operatorname{Lie}G^{\sharp}}$${0}$${0}$${\mathrm{Fil}(M)}$${H_{1}^{\mathrm{dR}}(M)}$${\operatorname{Lie}M}$${0}$$\scriptstyle{\cong}$

with exact rows and $H_{1}^{\mathrm{dR}}(M)$ is the pushout of
$H_{1}^{\mathrm{dR}}(G^{\sharp})$ along
$\mathrm{Fil}(G^{\sharp})\rightarrow\mathrm{Fil}(M)$. Define

(4.10) $L_{M}=\bar{\epsilon}\operatorname{Lie}M.$

Again we can easily see that $L_{M}$ is an
$\mathcal{O}_{\mathcal{C}_{\Phi}}$-module local direct summand of rank one. It
is stable under $\mathcal{O}_{\boldsymbol{k}}$, which acts on
$\operatorname{Lie}M/L_{M}$ and $L_{M}$ via $\varphi,\bar{\varphi}$ :
$\mathcal{O}_{\boldsymbol{k}}\rightarrow\mathcal{O}_{\mathcal{C}_{\Phi}}$,
respectively. Using the canonical isomorphism
$\operatorname{Lie}G^{\sharp}\cong\operatorname{Lie}M$, we can identify
$L_{G^{\sharp}}$ with $L_{M}$ over $\mathcal{C}_{\Phi}$.

We now revisit the setting at the beginning of this section with
$S=\mathcal{C}_{\Phi}^{*}$ and $Z=\mathcal{Z}_{\Phi}^{*}(m)$. For any
geometric point $z\in\mathcal{C}_{\Phi}$, by results in [BHK+20, §3.6], there
is an étale neighborhood $U$ of $z$ in $\mathcal{C}_{\Phi}^{*}$ such that
$\mathcal{Z}_{\Phi}^{*}(m)|_{U}\rightarrow\mathcal{C}_{\Phi}^{*}|_{U}$ is a
closed immersion of schemes on every connected component of its domain. Let
$\mathcal{Z}$ be such a connected component. The universal morphism
$x:\mathbf{A}_{0}|_{\mathcal{Z}}\rightarrow G^{\sharp}|_{\mathcal{Z}}$ induces
a morphism of vector bundles over $\mathcal{Z}$.

$H_{1}^{\mathrm{dR}}(\mathbf{A}_{0}|_{\mathcal{Z}})\xrightarrow{x}H_{1}^{\mathrm{dR}}(G^{\sharp}|_{\mathcal{Z}}),$

which maps $\mathrm{Fil}(\mathbf{A}_{0}|_{\mathcal{Z}})$ to
$\mathrm{Fil}(G^{\sharp}|_{\mathcal{Z}})$. By Grothendieck-Messing theory for
$1$-motives, this morphism admits a canonical extension to a morphism between
vector bundles over $\tilde{\mathcal{Z}}$,

$H_{1}^{\mathrm{dR}}(\mathbf{A}_{0}|_{\tilde{\mathcal{Z}}})\xrightarrow{\tilde{x}}H_{1}^{\mathrm{dR}}(G^{\sharp}|_{\tilde{\mathcal{Z}}}),$

which determines a homorphism

(4.11)
$\mathrm{Fil}(\mathbf{A}_{0}|_{\tilde{\mathcal{Z}}})\xrightarrow{\tilde{x}}\operatorname{Lie}(G^{\sharp}|_{\tilde{\mathcal{Z}}}).$

The following is an analogue of Proposition 4.1.

######  Proposition 4.3.

The homorphism (4.11) takes values in the rank one local direct summand

$L_{G^{\sharp}}|_{\tilde{\mathcal{Z}}}\subset\operatorname{Lie}G^{\sharp}|_{\tilde{\mathcal{Z}}},$

and so can be viewed as a morphism of line bundles

(4.12)
$\operatorname{Fil}\left(\mathbf{A}_{0}\right)|_{\tilde{\mathcal{Z}}}\stackrel{{\scriptstyle\tilde{x}}}{{\rightarrow}}L_{G^{\sharp}}|_{\tilde{\mathcal{Z}}}.$

The divisor $\mathcal{Z}$ is the largest closed formal subscheme of
$\tilde{\mathcal{Z}}$ over which (4.12) is trivial.

###### Proof.

By the signature condition of $\mathbf{A}_{0}$, we know that
$\bar{\epsilon}H_{1}^{\mathrm{dR}}(\mathbf{A}_{0})\subset\mathrm{Fil}(\mathbf{A}_{0})$.
Since both of them are locally free of rank $1$, we know they are equal. Hence
(4.11) takes values in
$\bar{\epsilon}\operatorname{Lie}G^{\sharp}|_{\tilde{\mathcal{Z}}}$ which is
exactly $L_{G^{\sharp}}|_{\tilde{\mathcal{Z}}}$ by definition. This proves the
first claim of the proposition. The second claim follows from Grothendieck-
Messing theory for $1$-motives, see for example [MP11, Lemma 1.3.4.1]. ∎

###  4.3. Deformation of $\mathcal{Z}^{*}(m)$ in $\mathcal{S}^{*}$

We now revisit the setting at the beginning of this section. Recall that for
any geometric point $z\in\mathcal{S}$, we choose an étale neighborhood $U$ of
$z$ in $\mathcal{S}$ such that
$\mathcal{Z}(m)|_{U}\rightarrow\mathcal{S}|_{U}$ is a closed immersion of
schemes on every connected component of its domain. Let $\mathcal{Z}$ be such
a connected component. Let $U^{*}$ be the Zariski closure of $U$ in
$\mathcal{S}^{*}$, and $\mathcal{Z}^{*}$ be the Zariski closure of
$\mathcal{Z}$ in $\mathcal{S}^{*}$. Recall that $\tilde{\mathcal{Z}}$ is the
first order thickening of $\mathcal{Z}$ in $\mathcal{S}|_{U}$ as defined at
the beginning of the section. Analogously let $\tilde{\mathcal{Z}}^{*}$ be the
first order thickening of $\mathcal{Z}^{*}$ in $\mathcal{S}^{*}|_{U^{*}}$. The
following is the main result of this section.

######  Theorem 4.4.

The morphism (4.2) can be extended to a morphism between line bundles over
$\mathcal{S}^{*}$:

(4.13)
$\operatorname{Fil}\left(\mathbf{A}_{0}\right)|_{\tilde{\mathcal{Z}}^{*}}\stackrel{{\scriptstyle\tilde{x}}}{{\rightarrow}}L_{\mathbf{G}}|_{\tilde{\mathcal{Z}}^{*}}.$

The Kudla-Rapoport divisor $\mathcal{Z}^{*}$ is the largest closed formal
subscheme of $\tilde{\mathcal{Z}}^{*}$ over which (4.13) is trivial.

###### Proof.

Without loss of generality, by shrinking $U$ if necessary we can assume that
$U^{*}$ intersects with exactly one boundary component $\mathcal{B}_{\Phi}$
nontrivially. As before assume assume $\Phi=(\mathfrak{n},L_{\Phi})$ and
$\mathfrak{m}$ be the hermitian dual of $\mathfrak{n}$. Recall from Theorem
2.5 that we have a morphism of stacks

$\pi_{\Phi}:(\mathcal{C}_{\Phi}^{*})^{\wedge}_{\mathcal{B}_{\Phi}}\rightarrow(\mathcal{S}^{*})^{\wedge}_{\mathcal{S}^{*}(\Phi)},$

which is a $\Delta_{\Phi}$-torsor and maps $\mathcal{B}_{\Phi}$ to
$\mathcal{S}^{*}(\Phi)$. By Proposition 4.1, it suffices to assume that
$U^{*}$ is an étale neighborhood of
$(\mathcal{S}^{*})^{\wedge}_{\mathcal{S}^{*}(\Phi)}$ and
$U=U^{*}\setminus\mathcal{S}^{*}(\Phi)|_{U^{*}}$. Let
$\mathcal{C}_{\Phi}^{\wedge}=(\mathcal{C}_{\Phi}^{*})^{\wedge}_{\mathcal{B}_{\Phi}}\setminus\mathcal{B}_{\Phi}$.

As explained in §4.1, we can associate a semi-abelian scheme $G^{\sharp}$ with
$\mathcal{O}_{\boldsymbol{k}}$-action over
$(\mathcal{C}_{\Phi}^{*})^{\wedge}_{\mathcal{B}_{\Phi}}$, and a principally
polarized $1$-motive $M$ with $\mathcal{O}_{\boldsymbol{k}}$-action over
$\mathcal{C}_{\Phi}^{\wedge}$. The relation between $G^{\sharp}$, $M$ and the
pullback of $\mathbf{G}$ over $\mathcal{S}^{*}$ under $\pi_{\Phi}$ is as
follows. Recall that we have an exact sequence of fppf sheaves of
$\mathcal{O}_{\boldsymbol{k}}$-modules over
$(\mathcal{C}_{\Phi}^{*})^{\wedge}_{\mathcal{B}_{\Phi}}$

$0\rightarrow\mathfrak{m}\otimes_{\mathbb{Z}}\mathbb{G}_{m}\rightarrow
G^{\sharp}\rightarrow B\rightarrow 0,$

together with a period map
$\underline{\mathfrak{n}}\xrightarrow[]{u}G^{\sharp}|_{\mathcal{C}_{\Phi}^{\wedge}}$
which determines the $1$-motive $M$. Then over $\mathcal{C}_{\Phi}^{\wedge}$
we get a principally polarized abelian scheme with
$\mathcal{O}_{\boldsymbol{k}}$-action
$(A,\ldots)\in\mathcal{M}_{(n-1,1)}^{\mathrm{naive}}(\mathcal{C}_{\Phi}^{\wedge})$
which is the quotient $\underline{\mathfrak{n}}\backslash
G^{\sharp}|_{\mathcal{C}_{\Phi}^{\wedge}}$111See [FC13, Proposition 4.8] for
the precise meaning of the quotient here.. On the boundary
$\mathcal{B}_{\Phi}$, $\mathbf{G}|_{\mathcal{B}_{\Phi}}$ is by definition
$G^{\sharp}|_{\mathcal{B}_{\Phi}}$. Moreover by [How15, Lemma 2.3.6], we get a
submodule $\mathcal{F}_{A}\subset\operatorname{Lie}A$ such that
$(A_{0},\ldots,A,\ldots,\mathcal{F}_{A})\in\mathcal{S}(\mathcal{C}_{\Phi}^{\wedge})$.

By construction the polarizations of $A$ and $M$ are compatible, hence
$\mathcal{Z}(m)$ pulls back to $\mathcal{Z}_{\Phi}(m)$ under the morphism
$\pi_{\Phi}$, see [BHK+20, Theorem 3.7.1(8)]. Taking Zariski closure, we see
that $\mathcal{Z}^{*}(m)$ pulls back to $\mathcal{Z}_{\Phi}^{*}(m)$. Suppose
that $\mathcal{Z}^{*}$ pulls back under $\pi_{\Phi}$ to $\mathcal{Y}^{*}$ on
$(\mathcal{C}_{\Phi}^{*})^{\wedge}_{\mathcal{B}_{\Phi}}|_{U^{*}}$. Denote by
$\mathcal{Y}$ the restriction of $\mathcal{Y}^{*}$ to
$\mathcal{C}_{\Phi}^{\wedge}|_{U}$ and let their first order thickening in
$(\mathcal{C}_{\Phi}^{*})^{\wedge}_{\mathcal{B}_{\Phi}}|_{U^{*}}$ be
$\tilde{\mathcal{Y}}$ and $\tilde{\mathcal{Y}^{*}}$ respectively. By Lemma 4.5
below, we have a canonical isomorphism preserving weight filtration

$H_{1}^{\mathrm{dR}}(M)\cong H_{1}^{\mathrm{dR}}(A).$

Hence we can identify $\operatorname{Lie}A$ with $\operatorname{Lie}M$,
$L_{A}$ with $L_{M}$ which is further identified with $L_{G^{\sharp}}$. Under
these identifications, the obstruction morphism in (4.2)

$\operatorname{Fil}\left(\mathbf{A}_{0}\right)|_{\tilde{\mathcal{Z}}}\stackrel{{\scriptstyle\tilde{x}}}{{\rightarrow}}L_{\mathbf{A}}|_{\tilde{\mathcal{Z}}}$

pull backs under $\pi_{\Phi}$ to a morphism

$\operatorname{Fil}\left(\mathbf{A}_{0}\right)|_{\tilde{\mathcal{Y}}}\stackrel{{\scriptstyle\tilde{y}}}{{\rightarrow}}L_{M}|_{\tilde{\mathcal{Y}}}\cong
L_{G^{\sharp}}|_{\tilde{\mathcal{Y}}}.$

By Proposition 4.3, this obstruction homomorphism extends to
$\tilde{\mathcal{Y}}^{*}$, and $\mathcal{Y}^{*}$ is the vanishing locus of
$\tilde{y}$ in $\tilde{\mathcal{Y}}^{*}$. The extended homomorphism is
obviously equi-variant under the action of $\Delta_{\Phi}$, so descends to a
homomorphism

$\operatorname{Fil}\left(\mathbf{A}_{0}\right)|_{\tilde{\mathcal{Z}}^{*}}\stackrel{{\scriptstyle\tilde{x}}}{{\rightarrow}}L_{\mathbf{G}}|_{\tilde{\mathcal{Z}}^{*}}.$

And $\mathcal{Z}^{*}$ is the vanishing locus of $\tilde{x}$ in
$\tilde{\mathcal{Z}}^{*}$. This finishes the proof of the proposition. ∎

######  Lemma 4.5.

Let the notation be as above. Then we have a canonical identification of de
Rham realizations

$H_{1}^{\mathrm{dR}}(M)\cong H_{1}^{\mathrm{dR}}(A)$

preserving weight filtrations.

###### Proof.

Let $E(A)$ (resp. $E(G^{\sharp})$) be the universal vector extension of $A$
(resp. $G^{\sharp}$). So we have an exact sequence of fppf-sheaves of
$\mathcal{O}_{\boldsymbol{k}}$-modules

$0\rightarrow\mathrm{Fil}(A)=\underline{\mathrm{Ext}}^{1}(A,\mathbb{G}_{a})^{\vee}\rightarrow
E(A)\rightarrow A\rightarrow 0.$

Consider the fiber product $E:=E(A)\times_{A}G^{\sharp}$ where we use the
natural quotient map $G^{\sharp}\rightarrow A$. Then $E$ is an extension of
$G^{\sharp}$ by the vector group
$\underline{\mathrm{Ext}}^{1}(A,\mathbb{G}_{a})^{\vee}$. Moreover the natural
morphism $E\rightarrow E(A)$ is étale (as $G^{\sharp}\rightarrow A$ is) so
induces an isomorphism on Lie algebras

$\operatorname{Lie}E\cong\operatorname{Lie}E(A)=H_{1}^{\mathrm{dR}}(A).$

By the universal property of $E(G^{\sharp})$, we have a commutative diagram of
fppf-sheaves of $\mathcal{O}_{\boldsymbol{k}}$-modules

${0}$${\mathrm{Fil}(G^{\sharp})}$${E(G^{\sharp})}$${G^{\sharp}}$${0}$${0}$${\mathrm{Fil}(A)}$${E}$${G^{\sharp}}$${0.}$$\scriptstyle{id}$

Taking Lie algebra, we have a commutative diagram of locally free sheaves

(4.14)
${0}$${\mathrm{Fil}(G^{\sharp})}$${H_{1}^{\mathrm{dR}}(G^{\sharp})}$${\operatorname{Lie}G^{\sharp}}$${0}$${0}$${\mathrm{Fil}(A)}$${H_{1}^{\mathrm{dR}}(A)}$${\operatorname{Lie}G^{\sharp}\cong\operatorname{Lie}A}$${0.}$$\scriptstyle{id}$

We would like to identify (4.14) with (4.9). In fact by the main result of
[Ber05], we know that

$\mathrm{Fil}(A)\cong(\operatorname{Lie}A^{\vee})^{\vee}\cong(\operatorname{Lie}(G^{\sharp})^{\vee})^{\vee}=(\operatorname{Lie}M^{\vee})^{\vee}\cong\mathrm{Fil}(M).$

Moreover the maps $\mathrm{Fil}(G^{\sharp})\rightarrow\mathrm{Fil}(A)$ and
$\mathrm{Fil}(G^{\sharp})\rightarrow\mathrm{Fil}(M)$ can be identified under
this isomorphism. Hence $H_{1}^{\mathrm{dR}}(A)$ and $H_{1}^{\mathrm{dR}}(M)$
have to agree, both are pushout of $H_{1}^{\mathrm{dR}}(G^{\sharp})$ along
$\mathrm{Fil}(G^{\sharp})\rightarrow\mathrm{Fil}(M)$. ∎

##  5\. Pullback of algebraic divisors

In this section we fix the hermitian lattice $L$ of signature $(n-1,1)$ as in
(2.5) and a unimodular hermitian $\mathcal{O}_{\boldsymbol{k}}$-lattice
$\Lambda$ with signature $(m,0)$. Then the map

(5.1) $[[L]]\rightarrow[[L^{\diamond}]],\quad M\mapsto
M^{\diamond}=M\obot\Lambda$

is a well-defined bijection between the genus of $L$ and the genus of
$L^{\diamond}$. Indeed, the above map is injective by Witt cancellation law
(we assume $D$ is odd so $2$ is not ramified in $\boldsymbol{k}$). Using
strong approximation for $\mathrm{SU}(V)$, we know that there are exactly
$2^{1-o(D)}h$ isometry classes in $[[L]]$ and $[[L^{\diamond}]]$. Here $o(D)$
is the number of prime divisors of $D$ and $h$ is the class number of
$\boldsymbol{k}$. So the above map is also a surjection.

We denote by $\mathcal{S}^{\diamond}$ the integral model of the Shimura
variety determined by $L^{\diamond}$ and by $\mathcal{S}^{\diamond,*}$ its
toroidal compactification as defined in Section 2.4. We also use
$\mathcal{Z}^{\diamond,*}(m)$ (resp. $\mathcal{Z}^{\diamond}(m)$) to denote
special divisors on $\mathcal{S}^{\diamond,*}$ (resp.
$\mathcal{S}^{\diamond}$).

Let $S$ be an $\mathcal{O}_{\boldsymbol{k}}$-scheme. Consider
$x_{0}=(A_{0},\iota_{0},\lambda_{0})\in\mathcal{M}_{(1,0)}(S)$. Then Serre’s
tensor construction carried out in [AK18, Theorem A] induces a morphism

(5.2)
$T_{\Lambda}:\mathcal{M}_{(1,0)}\rightarrow\mathcal{M}_{(m,0)},\quad(A_{0},\iota_{0},\lambda_{0})\mapsto(A_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}\Lambda,\iota_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}\Lambda,\lambda_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}\Lambda).$

This induces a morphism

(5.3)
$\varphi_{\Lambda}:\mathcal{S}\rightarrow\mathcal{S}^{\diamond},\quad(x_{0},y)\mapsto(x_{0},y\times
T_{\Lambda}(x_{0})).$

Since $\mathcal{S}^{\diamond,*}$ is proper over
$\mathrm{Spec}\,\mathcal{O}_{\boldsymbol{k}}$ ([How15, Theorem 2.5.2]) and
$\mathcal{S}^{*}$ is the Zariski closure of $\mathcal{S}$, the morphism
$\varphi_{\Lambda}$ extends canonically to a morphism
$\mathcal{S}^{*}\rightarrow\mathcal{S}^{\diamond,*}$, denoted by the same
notation.

######  Lemma 5.1.

The morphism $\varphi_{\Lambda}$ constructed above is finite.

###### Proof.

It suffices to show that $\varphi_{\Lambda}$ is proper and quasi-finite. By
the Néron property of abelian schemes we know $\varphi_{\Lambda}$ is proper.
The quasi-finiteness of $\varphi_{\Lambda}$ comes from the finiteness of
$\mathcal{M}_{(1,0)}$ over $\mathcal{O}_{\boldsymbol{k}}$. ∎

Given a proper cusp label $\Phi=(\mathfrak{m},L)\in\mathrm{Cusp}(L)$ and a
hermitian $\mathcal{O}_{\boldsymbol{k}}$-lattice $\Lambda$ of signature
$(m,0)$, we may produce a new cusp label

$\Phi^{\diamond}\coloneqq(\mathfrak{m},L\obot\Lambda)\in\mathrm{Cusp}(L\obot\Lambda).$

Then the image of the boundary divisor $\mathcal{S}^{*}(\Phi)$ under
$\varphi_{\Lambda}$ lies in
$\mathcal{S}^{\diamond,*}(\Phi^{\diamond})\subset\mathcal{S}^{\diamond,*}$.
The morphism near the boundary can be described using $\mathcal{C}_{\Phi}^{*}$
as follows. Let $L_{\Phi}=\mathfrak{m}^{\bot}/\mathfrak{m}$ as in (2.10). By
(2.12), we have

$\mathcal{B}_{\Phi}\cong\mathbf{A}_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}L_{\Phi},\
\mathcal{B}_{\Phi^{\diamond}}\cong\mathbf{A}_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}L_{\Phi}^{\diamond},$

where $\mathbf{A}_{0}$ is the universal elliptic curve over
$\mathcal{M}_{(1,0)}$ and $\otimes$ means Serre tensor. We then have the
obvious closed immersion

$\mathcal{B}_{\Phi}\rightarrow\mathcal{B}_{\Phi^{\diamond}},\quad\sum z\otimes
s\mapsto\sum z\otimes(s,0),$

where $z\in\mathcal{B}_{\Phi}(S)$ for any $\mathcal{B}_{\Phi}$-scheme $S$.
Recall from §2.4 that $\mathcal{C}_{\Phi^{\diamond}}^{*}$ (resp.
$\mathcal{C}_{\Phi}^{*}$) is the total space of the line bundle
$\mathcal{L}_{\Phi^{\diamond}}^{-1}$ (resp. $\mathcal{L}_{\Phi}^{-1}$). By
their definitions, we see immediately that

$\mathcal{L}_{\Phi^{\diamond}}|_{\mathcal{B}_{\Phi}}=\mathcal{L}_{\Phi}.$

Hence we get a natural closed immersion
$\mathcal{C}_{\Phi}^{*}\rightarrow\mathcal{C}_{\Phi^{\diamond}}^{*}$. The
closed immersions $\mathcal{B}_{\Phi}\rightarrow\mathcal{C}_{\Phi}^{*}$ and
$\mathcal{B}_{\Phi^{\diamond}}\rightarrow\mathcal{C}_{\Phi^{\diamond}}^{*}$
defined by zero sections make the following diagram Cartesian.

(5.4)
${\mathcal{B}_{\Phi}}$${\mathcal{B}_{\Phi^{\diamond}}}$${\mathcal{C}_{\Phi}^{*}}$${\mathcal{C}_{\Phi^{\diamond}}^{*}}$

To summarize we have the following proposition.

######  Proposition 5.2.

Let $\varphi_{\Lambda}$ be the finite morphism
$\mathcal{S}^{*}\rightarrow\mathcal{S}^{\diamond,*}$ defined above. It has the
following properties.

  1. (1)

We have the equality of line bundles over $\mathcal{S}^{*}$

(5.5) $\varphi_{\Lambda}^{*}(\Omega^{\diamond})\cong\Omega.$

  2. (2)

For a cusp label $\Phi^{\prime}$ of $\mathcal{S}^{\diamond}$, we have the
following equation of Cartier divisors.

$\displaystyle\varphi_{\Lambda}^{*}(\mathcal{S}^{\diamond,*}(\Phi^{\prime}))=\begin{cases}\mathcal{S}^{*}(\Phi)&\text{
if $\Phi^{\prime}=\Phi^{\diamond}$ for some cusp label $\Phi$ of
$\mathcal{S}^{*}$},\\\ 0&\text{ otherwise}.\end{cases}$

###### Proof.

Equation (5.5) is obvious from the definition of $\Omega$. Recall from §2.4
that $\mathcal{S}^{*}(\Phi)=\Delta_{\Phi}\backslash\mathcal{B}_{\Phi}$, and
the completion of $\mathcal{S}^{*}$ along $\mathcal{S}^{*}(\Phi)$ is
isomorphic to the completion of
$\Delta_{\Phi}\backslash\mathcal{C}_{\Phi}^{*}$ along
$\Delta_{\Phi}\backslash\mathcal{B}_{\Phi}$. Similar statements hold for
$\mathcal{S}^{\diamond,*}$ and $\mathcal{S}^{\diamond,*}(\Phi^{\diamond})$.
From (5.4) we get a Cartesian diagram of quotient stacks

(5.6)
${\Delta_{\Phi}\backslash\mathcal{B}_{\Phi}}$${\Delta_{\Phi^{\diamond}}\backslash\mathcal{B}_{\Phi^{\diamond}}}$${\Delta_{\Phi}\backslash\mathcal{C}_{\Phi}^{*}}$${\Delta_{\Phi^{\diamond}}\backslash\mathcal{C}_{\Phi^{\diamond}}^{*}.}$

By unraveling the construction of Mumford explained in [Lan13, §4.5] or [FC13,
Chapter 3], the above commutative diagram gives rise to the Cartesian diagram

(5.7)
${\mathcal{S}^{*}(\Phi)}$${\mathcal{S}^{\diamond,*}(\Phi^{\diamond})}$${\mathcal{S}^{*}}$${\mathcal{S}^{\diamond,*}.}$$\scriptstyle{\varphi_{\Lambda}}$

The proposition follows. ∎

### 5.1.

In the following discussion we use the correspondence between line bundles and
Cartier divisors. For a Cartier divisor $D$ on $\mathcal{S}^{\diamond,*}$, we
use $D|_{\mathcal{S}^{*}}$ to denote $\varphi_{\Lambda}^{*}(D)$, the pullback
of the corresponding line bundle via $\varphi_{\Lambda}$.

######  Proposition 5.3.

Assume $m\in\mathbb{Z}$. Then

(5.8)
$\mathcal{Z}^{\diamond,*}(m)|_{\mathcal{S}}\cong\displaystyle\bigotimes_{\begin{subarray}{c}m_{1}+m_{2}=m\\\
m_{1}\neq 0\end{subarray}}\mathcal{Z}^{*}(m_{1})^{\otimes
r_{\Lambda}(m_{2})}\bigotimes(\Omega^{-1})^{\otimes r_{\Lambda}(m)},$

where $r_{\Lambda}(m)$ is the cardinality of

$\displaystyle
R_{\Lambda}(m)\coloneqq\\{\lambda\in\Lambda\mid(\lambda,\lambda)=m\\}.$

###### Proof.

The proof is similar to that of [HMP20, Proposition 6.6.3]. If $m<0$, then
both sides of (5.8) are trivial line bundles. If $m=0$, then (5.8) is simply
(5.5) by §3.2. For any scheme $S\rightarrow\mathcal{S}$, let $\mathbf{A}_{S}$
be the pullback of the universal abelian scheme over $\mathcal{S}$, and
$\mathbf{A}_{S}^{\diamond}$ be the pullback of the universal abelian scheme
over $\mathcal{S}^{\diamond}$ via the morphism
$S\rightarrow\mathcal{S}\xrightarrow[]{\varphi_{\Lambda}}\mathcal{S}^{\diamond}$.
Define the following $\mathcal{O}_{\boldsymbol{k}}$-modules of special
homomorphisms

$V(\mathbf{A}_{S})=\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(\mathbf{A}_{0,S},\mathbf{A}_{S}),\quad
V(\mathbf{A}_{S}^{\diamond})=\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(\mathbf{A}_{0,S},\mathbf{A}_{S}^{\diamond}).$

By the definition of $\varphi_{\Lambda}$, we know
$\mathbf{A}_{S}^{\diamond}=\mathbf{A}_{S}^{\diamond}\oplus(\mathbf{A}_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}\Lambda)_{S}$.
By [AK18, Proposition 2], we know that
$\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(\mathbf{A}_{0},\mathbf{A}_{0}\otimes_{\mathcal{O}_{\boldsymbol{k}}}\Lambda)\cong\Lambda$.
Hence we have

(5.9) $V(\mathbf{A}_{S}^{\diamond})\cong V(\mathbf{A}_{S})\oplus\Lambda.$

Equation (5.9) induces the following isomorphism of $\mathcal{S}$-stacks.

$\mathcal{Z}^{\diamond}(m)|_{\mathcal{S}}\cong\bigsqcup_{\begin{subarray}{c}m_{1}+m_{2}=m\\\
m_{1}\neq 0\\\ \lambda\in
R_{\Lambda}(m_{2})\end{subarray}}\mathcal{Z}(m_{1})\sqcup\bigsqcup_{\lambda\in
R_{\Lambda}(m)}\mathcal{S}.$

By taking Zariski closure we have the the following isomorphism of
$\mathcal{S}^{*}$-stacks.

(5.10)
$\mathcal{Z}^{\diamond,*}(m)|_{\mathcal{S}^{*}}\cong\bigsqcup_{\begin{subarray}{c}m_{1}+m_{2}=m\\\
m_{1}\neq 0\\\ \lambda\in
R_{\Lambda}(m_{2})\end{subarray}}\mathcal{Z}^{*}(m_{1})\sqcup\bigsqcup_{\lambda\in
R_{\Lambda}(m)}\mathcal{S}^{*}.$

There is a corresponding canonical decomposition of
$\mathcal{S}^{\diamond,*}$-stack

(5.11)
$\mathcal{Z}^{\diamond,*}(m)=\mathcal{Z}^{\diamond}_{\mathrm{prop}}\sqcup\mathcal{Z}^{\diamond}_{\mathrm{im}},$

such that each connected component of $\mathcal{Z}^{\diamond}_{\mathrm{prop}}$
intersects the image of $\varphi_{\Lambda}$ properly and contributes to the
first term on the right hand side of (5.10), while each connected component of
$\mathcal{Z}^{\diamond}_{\mathrm{im}}$ contains (intersects “improperly” with)
a component of the image of $\varphi_{\Lambda}$ and contributes to the second
term on the right hand side of (5.10). Notice that the first term on the right
hand side of (5.10) corresponds to exactly the first term of on the right hand
side of (5.8). To prove the proposition, it suffices to prove the following
lemma. ∎

######  Lemma 5.4.

Let $\mathcal{Z}^{\diamond}$ be a connected component of
$\mathcal{Z}^{\diamond}_{\mathrm{im}}$ and $U^{\diamond}$ a connected étale
neighborhood of $\mathcal{S}^{\diamond,*}$ such that
$\mathcal{Z}^{\diamond}|_{U^{\diamond}}\rightarrow U^{\diamond}$ is a closed
immersion. Let $U$ be a connected étale neighborhood of $\mathcal{S}^{*}$
whose image under $\varphi_{\Lambda}$ is contained in
$\mathcal{Z}^{\diamond}|_{U^{\diamond}}$. Then we have the following equality
of line bundles.

$\mathcal{O}(\mathcal{Z}^{\diamond})|_{U}\cong\Omega^{-1}|_{U}.$

###### Proof.

The proof is identical to that of [BHY15, Theorem 7.10] or [HMP20, Lemma
6.6.4], given Theorem 4.4. We leave it to the reader. ∎

For $v>0$ and $m\geq 0$, define

(5.12) $\mathcal{B}(m,v)\coloneqq\frac{1}{4\pi
v}\sum_{\Phi\in\mathrm{Cusp}([[L]])}\\#\left\\{x\in L_{\Phi}:\langle
x,x\rangle=m\right\\}\cdot\mathcal{S}^{*}(\Phi)\in\mathrm{CH}^{1}_{\mathbb{C}}(\mathcal{S}^{*}),$

where $L_{\Phi}$ is the positive definite lattice in (2.10). Then we define
the total special divisor to be the following element in
$\mathrm{CH}^{1}_{\mathbb{C}}(\mathcal{S}^{*})$.

(5.13)
$\mathcal{Z}^{\mathrm{tot}}(m,v)\coloneqq\begin{cases}\mathcal{Z}^{*}(m)+\mathcal{B}(m,v)&\text{
if }m>0\\\ \Omega^{-1}+\mathcal{B}(0,v)&\text{ if }m=0\\\ 0&\text{ if
}m<0.\end{cases}$

######  Theorem 5.5.

Assume $m\in\mathbb{Z}$ and $v>0$. Then we have the following identity in
$\mathrm{CH}^{1}(\mathcal{S}^{*})$.

(5.14)
$\displaystyle\varphi_{\Lambda}^{*}(\mathcal{Z}^{\mathrm{tot},\diamond}(m,v))=\sum_{m_{1}+m_{2}=m}r_{\Lambda}(m_{2})\cdot\mathcal{Z}^{\mathrm{tot}}(m_{1},v).$

###### Proof.

By Proposition 5.2, we have

$\displaystyle\varphi_{\Lambda}^{*}(\mathcal{B}^{\diamond}(m,v))$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi
v}\sum_{\Phi^{\diamond}\in\mathrm{Cusp}([[L^{\diamond}]])}\\#\left\\{x\in
L_{\Phi^{\diamond}}:\langle
x,x\rangle=m\right\\}\cdot\varphi_{\Lambda}^{*}(\mathcal{S}^{\diamond,*}(\Phi^{\diamond}))$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi
v}\sum_{\Phi\in\mathrm{Cusp}([[L]])}\\#\left\\{x\in
L_{\Phi}\obot\Lambda:\langle x,x\rangle=m\right\\}\cdot\mathcal{S}^{*}(\Phi)$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi
v}\sum_{\Phi\in\mathrm{Cusp}([[L]])}\sum_{m_{1}+m_{2}=m}\\#\left\\{x\in\Lambda:\langle
x,x\rangle=m_{2}\right\\}\cdot\\#\left\\{x\in L_{\Phi}:\langle
x,x\rangle=m_{1}\right\\}\cdot\mathcal{S}^{*}(\Phi)$ $\displaystyle=$
$\displaystyle\sum_{m_{1}+m_{2}=m}\\#\left\\{x\in\Lambda:\langle
x,x\rangle=m_{2}\right\\}\cdot\frac{1}{4\pi
v}\sum_{\Phi\in\mathrm{Cusp}([[L]])}\\#\left\\{x\in L_{\Phi}:\langle
x,x\rangle=m_{1}\right\\}\cdot\mathcal{S}^{*}(\Phi)$ $\displaystyle=$
$\displaystyle\sum_{m_{1}+m_{2}=m}r_{\Lambda}(m_{2})\cdot\mathcal{B}(m_{1},v).$

Combining this with Proposition 5.3, the theorem is proved. ∎

##  6\. Pullback of arithmetic divisors

###  6.1. Arithmetic Chow groups

In this subsection, we review the theory of arithmetic Chow groups formed from
cycles with Green currents with certain log-log singularities along a fixed
normal crossing divisor developed by [BGKK07]. We follow [How15, §3.1]
closely.

######  Definition 6.1.

Let $M^{*}$ be a compact complex manifold of dimension $n-1,\partial
M^{*}\subset$ $M^{*}$ be a smooth codimension one submanifold,
$M=M^{*}\backslash\partial M^{*}$, and $z_{0}\in\partial M^{*}$. Suppose that
on some open neighborhood $V\subset M^{*}$ of $z_{0}$, there are coordinates
$q,u_{1},\ldots,u_{n-2}$ such that $\partial M^{*}$ is the vanishing locus of
$q$. After possibily shrinking $V$, we may always assume that
$\log\left|q^{-1}\right|>1$ on $V$. Then we call the open set $V$ and its
coordinates _adapted_ to $\partial M^{*}$.

######  Definition 6.2.

Suppose $f$ is a $C^{\infty}$ function on an open subset $U\subset M$. We say
that $f$ has $\log$-log growth along $\partial M^{*}$ if around any point of
$\partial M^{*}$ there is an open neighborhood $V\subset M^{*}$ and
coordinates $q,u_{1},\ldots,u_{n-2}$ adapted to $\partial M^{*}$ such that

(6.1) $\displaystyle f=O(\log\log\left|q^{-1}\right|)$

on $U\cap V$. A smooth differential form $\omega$ on $U$ has log-log growth
along $\partial M^{*}$ if around any point of $\partial M^{*}$ there is an
open neighborhood $V\subset M^{*}$ and coordinates $q,u_{1},\ldots,u_{n-2}$
adapted to $\partial M^{*}$ such that $\left.\omega\right|_{U\cap V}$ lies in
the subring (of the ring of all smooth forms on $U\cap V$ ) generated by

$\frac{dq}{q\log|q|},\frac{d\bar{q}}{\bar{q}\log|q|},du_{1},\ldots,du_{n-2},d\bar{u}_{1},\ldots,d\bar{u}_{n-2},$

and functions satisfying (6.1).

The work of [BGKK07] about arithmetic Chow group with log-log singularity
along a normal crossing divisor is for a flat, regular and proper
$\mathcal{O}_{\boldsymbol{k}}$-scheme of finite type. Since $\mathcal{S}^{*}$
is in fact a Deligne-Mumford stack, we need to adapt this theory to Deligne-
Mumford stack accordingly following [How15].

In particular, we extend the notion of log-log growth to the orbifold fibers
of $\mathcal{S}^{*}$ in the following way. We can write
$\mathcal{S}^{*}(\mathbb{C})$ as the quotient of a complex manifold $M^{*}$ by
the action of a finite group $H$. As a result, we can regard

$\partial\mathcal{S}^{*}(\mathbb{C})\rightarrow\mathcal{S}^{*}(\mathbb{C})\leftarrow\mathcal{S}(\mathbb{C})$

as the quotients of $H$-invariant morphisms of complex manifolds

$\partial M^{*}\rightarrow M^{*}\leftarrow M$

with $H$-actions. Then we say a smooth form on $\mathcal{S}^{*}(\mathbb{C})$
has log-log growth along the boundary $\partial\mathcal{S}^{*}(\mathbb{C})$ if
the corresponding $H$-invariant form on $M$ obtained via pullback has
$\log$-log growth along $\partial M^{*}$.

Let $\mathcal{Z}=\sum m_{i}\mathcal{Z}_{i}$ be a finite $\mathbb{R}$-linear
combination of pairwise distinct irreducible closed substacks of codimension
one where $m_{i}\in\mathbb{R}$, which is a divisor on $\mathcal{S}^{*}$ with
real coefficients.

######  Definition 6.3.

A Green function for $\mathcal{Z}$ consists of a smooth real-valued function
$\operatorname{Gr}(\mathcal{Z},\cdot)$ on
$\mathcal{S}(\mathbb{C})\backslash\mathcal{Z}(\mathbb{C})$ satisfying the
following properties:

  1. (1)

For every point of $\mathcal{S}^{*}(\mathbb{C})$, there is an open
neighborhood $V$ and local equations $\psi_{i}(z)=0$ for the divisors
$\mathcal{Z}_{i}(\mathbb{C})$ such that the function

$\mathcal{E}(z)=\operatorname{Gr}(\mathcal{Z},z)+\sum_{i}m_{i}\log||\psi_{i}(z)||^{2}$

on $V\cap(\mathcal{S}(\mathbb{C})\backslash\mathcal{Z}(\mathbb{C}))$ extends
smoothly to $V\cap\mathcal{S}(\mathbb{C})$.

  2. (2)

The forms $\mathcal{E},\partial\mathcal{E},\bar{\partial}\mathcal{E}$, and
$\partial\bar{\partial}\mathcal{E}$ on $V\cap\mathcal{S}(\mathbb{C})$ have
log-log growth along $\partial\mathcal{S}^{*}(\mathbb{C})$.

######  Definition 6.4.

We define an arithmetic divisor on $\mathcal{S}^{*}$ to be a pair
$(\mathcal{Z},\operatorname{Gr}(\mathcal{Z},\cdot))$, where $\mathcal{Z}$ is a
divisor on $\mathcal{S}^{*}$ with complex coefficients, and
$\operatorname{Gr}(\mathcal{Z},\cdot)$ is a Green function for $\mathcal{Z}$.
We define a principal arithmetic divisor to be

$\widehat{\operatorname{div}}(f)=\left(\operatorname{div}(f),-\log||f||^{2}\right)$

for some rational function $f$ on $\mathcal{S}^{*}$.

Given the definition of arithmetic divisor and principal arithmetic divisor,
we can define the first arithmetic Chow group as follows.

######  Definition 6.5.

We define the first arithmetic Chow group
$\widehat{\mathrm{CH}}_{\mathbb{C}}^{1}\left(\mathcal{S}^{*}\right)$ to be the
quotient

$\displaystyle\mathrm{Span}_{\mathbb{C}}\\{\text{arithmetic
divisors}\\}/\mathrm{Span}_{\mathbb{C}}\\{\text{principal arithmetic
divisors}\\}.$

Accordingly, we define $\widehat{\mathrm{Pic}}_{\mathbb{C}}(\mathcal{S}^{*})$
to be the isomorphism classes of metrized line bundles
$\widehat{\mathcal{L}}=(\mathcal{L},||\cdot||)$ such that there exists a
section $s$ of $\mathcal{L}$ with $-\log||s||^{2}=g$ on
$\mathcal{S}(\mathbb{C})$. Here $g$ is a Green function (in the sense of
Definition 6.3) for the divisor corresponding to $\mathcal{L}$. Then we have a
natural isomorphism

(6.2)
$i:\widehat{\mathrm{Pic}}_{\mathbb{C}}(\mathcal{S}^{*})\xrightarrow{\sim}\widehat{\mathrm{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$

such that under $i$, an element
$\widehat{\mathcal{L}}=(\mathcal{L},||\cdot||)$ goes to the class of
$(\hbox{div}(s),-\log||s||^{2})$ for some non-zero meromorphic section $s$ of
$\mathcal{L}$. Conversely, the preimage of the class of $(\mathcal{Z},g)$
under $i$ is represented by

$(\mathcal{O}(\mathcal{Z}),||\cdot||),$

where $-\log||{\bf 1}||^{2}=g$ with ${\bf 1}$ the canonical section of
$\mathcal{O}(\mathcal{Z})$ and $g$ a Green function for $\mathcal{Z}$. In
particular, we can define the pullback of arithmetic divisors to be the
pullback of the corresponding metrized line bundles.

###  6.2. Arithmetic divisors

The complex fiber of $\mathcal{S}$ is of the form

(6.3)
$\mathcal{S}(\mathbb{C})=\bigsqcup_{\mathrm{Cl}(\boldsymbol{k})}\bigsqcup_{L_{j}\in[[L]]}\Gamma_{j}\backslash\mathcal{D},$

where $\Gamma_{j}$ is the automorphism group of $L_{j}$. Each connected
component $\Gamma_{j}\backslash\mathcal{D}$ is a complex orbifold. For
$z\in\mathcal{D}$ and $x\in V_{\mathbb{R}}$, define

(6.4) $R(x,z)=-(\mathrm{pr}_{z}(x),\mathrm{pr}_{z}(x))\geq 0.$

Then the majorant hermitian form

$(x,x)_{z}:=(x,x)+2R(x,z)=(\mathrm{pr}_{z^{\perp}}(x),\mathrm{pr}_{z^{\perp}}(x))-(\mathrm{pr}_{z}(x),\mathrm{pr}_{z}(x))$

is positive definite. Define

(6.5) $\mathcal{D}(x)=\\{z\in\mathcal{D}\mid(z,x)=0\\}=\\{z\in\mathcal{D}\mid
R(x,z)=0\\}.$

Then $\mathcal{D}(x)$ is nonempty if and only if $(x,x)>0$. Also define

$\Gamma_{j,x}=\\{\gamma\in\Gamma_{j}\mid\gamma x=x\\},$

and $Z(x)_{j}$ to be the image of the natural complex immersion
$\Gamma_{j,x}\backslash\mathcal{D}(x)\rightarrow\Gamma_{j}\backslash\mathcal{D}$.

Let $Z(m)=\mathcal{Z}(m)(\mathbb{C})$. Then we know that ([KR14, §3]) for
$m\in\mathbb{Z}_{>0}$, we have

$Z(m)=\bigsqcup_{\mathrm{Cl}(\boldsymbol{k})}\bigsqcup_{L_{j}\in[[L]]}\bigsqcup_{\begin{subarray}{c}x\in
L_{j}\\\ (x,x)=m\end{subarray}}Z(x)_{j}.$

Define

$\beta_{1}(r)=\int_{1}^{\infty}e^{-rt}\frac{dt}{t}.$

Then there is a power series expansion

(6.6) $\beta_{1}(r)+\log
r=-\gamma-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\frac{r^{k}}{k!},$

where $\gamma$ is the Euler’s constant. In particular, $\beta_{1}(r)$ has log
singularity at $r=0$. For $x\in L_{j}$ and $v>0$, we define the Kudla’s Green
function on $\Gamma_{j}\backslash D$

(6.7) $\mathrm{Gr}(x,v)_{j}(z)=\beta_{1}(4\pi v\cdot R(x,z)).$

Then $\mathrm{Gr}(x,v)_{j}$ is a Green function for $Z(x)_{j}$ (c.f. [How12,
§3]). For $v>0$ and $m\in\mathbb{Z}$, define Kudla’s Green function
$\mathrm{Gr}(m,v)(z)$ such that on the component
$\Gamma_{j}\backslash\mathcal{D}$ it is of the form

(6.8)
$\mathrm{Gr}(m,v)(z)|_{\Gamma_{j}\backslash\mathcal{D}}=\sum_{\begin{subarray}{c}x\in
L_{j}\setminus\\{0\\}\\\ (x,x)=m\end{subarray}}\mathrm{Gr}(x,v)_{j}(z).$

Then $\mathrm{Gr}(m,v)$ is a Green function for $Z(m)$. Note that $Z(m)=0$ and
$\mathrm{Gr}(m,v)$ is smooth on $\mathcal{D}$ for $m\leq 0$. The following
theorem describes the behavior of $\mathrm{Gr}(m,v)$ near the boundary.

######  Theorem 6.6.

[How15, Theorem 3.7.4] Let $m$ be a non-zero integer, and $p$ be a complex
point of $\mathcal{S}^{*}(\Phi)$ for some cusp label $\Phi$. There exists an
open neighborhood $U\subset\mathcal{\mathcal{S}}^{*}(\mathbb{C})$ of $p$ such
that the smooth function

$\mathcal{E}(z)=\operatorname{Gr}(m,v)(z)+\log\left|\psi_{m}(z)\right|^{2}+\frac{\\#\left\\{x\in
L_{\Phi}:\langle x,x\rangle=m\right\\}}{4\pi v}\log|q(z)|^{2}$

on $U\setminus\mathcal{S}^{*}(\Phi)(\mathbb{C})$ is bounded, and the
differential forms $\partial\mathcal{E},\bar{\partial}\mathcal{E}$, and
$\partial\bar{\partial}\mathcal{E}$ have log-log growth along
$\mathcal{S}^{*}(\Phi)(\mathbb{C})$. Here $\psi_{m}(z)=0$ is a local equation
for $\mathcal{Z}^{*}(m)(\mathbb{C})$, and $q(z)=0$ is a local equation for the
boundary component $\mathcal{S}^{*}(\Phi)(\mathbb{C})$. In particular,
$\operatorname{Gr}(m,v)$ is a Green function of
$\mathcal{Z}^{\mathrm{tot}}(m,v)$.

By Theorem 6.6, we have a well-defined arithmetic Chow divisor for integers
$m\neq 0$

(6.9)
$\widehat{\mathcal{Z}}_{K}^{\mathrm{tot}}(m,v)\coloneqq(\mathcal{Z}^{\mathrm{tot}}(m,v),\mathrm{Gr}(m,v))\in\widehat{\operatorname{CH}}^{1}(\mathcal{S}^{*}).$

In the rest of this section, we often drop the subscript $K$ for simplicity.

Now we deal with the case $m=0$. Recall that on $\mathcal{S}(\mathbb{C})$ the
two line bundles $\Omega$ and $\omega$ agree, and can be identified the line
bundle (still denoted by $\Omega$) of modular forms on
$\mathcal{S}(\mathbb{C})$ by [BHY15, Proposition 6.2]. Moreover, this line
bundle is descended from the tautological line bundle on $\mathcal{D}$, again
denoted by $\Omega$. Let $s$ be a section of $\Omega$ and for
$z\in\mathcal{D}$, we let $s_{z}\in z\subset V_{\mathbb{R}}$ denote the value
of $s$ at $z$. Then we can define a metric on $\Omega$ by setting

(6.10) $\|s_{z}\|^{2}=-4\pi ve^{\gamma}(s_{z},s_{z}).$

Here $\gamma$ is the Euler constant and $v>0$ is an extra parameter
(eventually imaginary part of modular variable $\tau\in\mathbb{H}$). This
gives an element
$\widehat{\Omega}^{-1}\in\widehat{\hbox{Pic}}_{\mathbb{C}}(\mathcal{S})$. By
[BHY15, Proposition 6.3], this element extends naturally to an element, still
denoted by $\widehat{\Omega}^{-1}$ in
$\widehat{\mathrm{Pic}}_{\mathbb{C}}(\mathcal{S}^{*})\cong\widehat{\mathrm{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$.

Following [ES18, Page 34], We define
$\widehat{\mathcal{Z}}^{\mathrm{tot}}(0,v)$ to be

(6.11)
$\widehat{\mathcal{Z}}^{\mathrm{tot}}(0,v)=\widehat{\Omega}^{-1}+(\mathcal{B}(0,v),\mathrm{Gr}(0,v))\in\widehat{\mathrm{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*}),$

where $\mathcal{B}(0,v)$ is defined in (5.12). Although the metrized line
bundle $\widehat{\Omega}^{-1}$ depends on the number $v>0$, we omit it from
the notation for convenience.

###  6.3. Pullback formulas for arithmetic divisors and Proof of Theorem 1.2

As in §5, we use the superscript $\diamond$ to denote objects on
$\mathcal{S}^{\diamond}$ or $\mathcal{S}^{\diamond,*}$. For each
$L_{j}\in[[L]]$, define $L_{j}^{\diamond}=L_{j}\obot\Lambda$. The aim of this
subsection is to prove Theorem 1.2, which is equivalent to the following
theorem.

######  Theorem 6.7.

Assume $m\in\mathbb{Z}$ and $v>0$. Then we have the following equation in
$\widehat{\mathrm{CH}}^{1}_{\mathbb{C}}(\mathcal{S}^{*})$.

(6.12)
$\displaystyle\varphi_{\Lambda}^{*}(\widehat{\mathcal{Z}}^{\diamond,\mathrm{tot}}(m,v))=\sum_{m_{1}+m_{2}=m}r_{\Lambda}(m_{2})\cdot\widehat{\mathcal{Z}}^{\mathrm{tot}}(m_{1},v).$

###### Proof.

By Theorem 5.5, we have

$\varphi_{\Lambda}^{*}(\mathcal{Z}^{\mathrm{tot},\diamond}(m,v))=\sum_{m_{1}+m_{2}=m}r_{\Lambda}(m_{2})\cdot\mathcal{Z}^{\mathrm{tot}}(m_{1},v).$

It remains to check the Green functions of the divisors from the two sides
agree. Since both Green functions are analytic, it suffices to check this on
the open Shimura variety $\mathcal{S}(\mathbb{C})$. It is easy to see by
simply unfolding (the Green functions) that it suffices to prove the following
local identity for $z\in\mathcal{D}$

(6.13) $\mathrm{Gr}^{\diamond}(x,v)(z)=\mathrm{Gr}(x_{1},v)(z)$

for any $x=x_{1}+x_{2}$ with $x_{1}\in V_{\mathbb{R}}$ and
$x_{2}\in\Lambda_{\mathbb{R}}$, except that some modification is needed for
$x_{1}=0$. When $x_{1}\neq 0$, (6.13) is obvious as $R(x,z)=R(x_{1},z)$ for
$z\in\mathcal{D}$. This finishes the case with $m\neq 0$ and $x_{1}\neq 0$.
The case of $m=0$ follows basically from definition.

So we are left with the case $m>0$ and $x\in\Lambda_{\mathbb{R}}$ with
$(x,x)=m$. Viewing $\mathcal{D}^{\diamond}(x)$ as a line bundle over
$\mathcal{D}^{\diamond}$ with the metric given by
$\mathrm{Gr}^{\diamond}(x,v)$, we have a line bundle
$\mathcal{R}=\mathcal{D}^{\diamond}(x)\otimes\Omega$ with product metric. By
Lemma 5.4, $\mathcal{R}|_{\mathcal{D}}$ is a trivial line bundle. We need to
verify that the product metric on $\mathcal{R}$ is constant $1$ when
restricting to $\mathcal{D}$.

For $0\neq x\in\Lambda_{\mathbb{R}}$, we have $(x,x)>0$ and we can define a
section $s_{x}$ of $\Omega^{-1}$ via

(6.14) $s_{x}(a)=(x,a),\quad\text{where $a$ is a section of $\Omega|_{U}$}.$

Over a point $z\in\mathcal{D}^{\diamond}$, we have

$s_{x,z}(a)=(x,a)=(\mathrm{pr}_{z}(x),a),\quad\text{ for }a\in z.$

According to (6.10), we have

(6.15) $-\log\|s_{x,z}\|^{2}=-\gamma-\log(4\pi vR(x,z)).$

On the other hand, the canonical section ${\bf 1}$ of the line bundle
$\mathcal{D}^{\diamond}(x)$ has self inner product

$-\log\|{\bf 1}_{z}\|^{2}=\mathrm{Gr}^{\diamond}(x,v)=\beta_{1}(4\pi
vR(x,z))=-\gamma-\log(4\pi vR(x,z))+f_{x}(z)$

with a smooth function

$f_{x}(z)=-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\frac{(4\pi v\cdot
R(x,z))^{k}}{k!}.$

So the section $r={\bf 1}\otimes s_{x}^{-1}$ on $\mathcal{R}$ has the
following property

$-\log\|r_{z}\|^{2}=\mathrm{Gr}^{\diamond}(x,z)+\log\|s_{x,z}\|^{2}=f_{x}(z),$

which vanishes when $z\in\mathcal{D}\subset\mathcal{D}^{\diamond}(x)$. That is
$\|r_{z}\|=1$ for $z\in\mathcal{D}$ as expected. This proves (6.12) for Green
function part on $\mathcal{S}$. This proves the theorem. ∎

##  7\. Modularity on unitary Shimura curves

###  7.1. Eisenstein Series

In this subsection, we temporarily let $L$ be a unimodular positive definite
Hermitian lattice over $\mathcal{O}_{\boldsymbol{k}}$ of rank $\ell$. Let
$\psi=\prod_{p}\psi_{p}$ be the canonical additive character of
$\mathbb{Q}\backslash\mathbb{A}$ with $\psi_{\infty}(x)=e(x)=e^{2\pi ix}$.
Associated to it is a Weil representation $\omega=\omega_{L,\psi}$ of
$\hbox{SL}_{2}(\mathbb{A})\subset U(1,1)(\mathbb{A})$ on $S(V_{\mathbb{A}})$
with $V=L_{\mathbb{Q}}=L\otimes_{\mathbb{Z}}\mathbb{Q}$, and a
$\operatorname{SL}_{2}(\mathbb{A})$-equivariant “Rallis” map ($s_{0}=\ell-1$)

$\lambda:\,S(V_{\mathbb{A}})\rightarrow
I(s_{0},\chi),\quad\lambda(\phi)(g)=\omega(g)\phi(0).$

Here $\chi=\chi_{-D}^{\ell}$. Let
$\phi_{L}=\mathrm{Char}(\widehat{L})\otimes\phi_{\infty}\in S(V_{\mathbb{A}})$
with $\phi_{\infty}(x)=e^{-\pi(x,x)}$, and let
$\Phi=\Phi_{L}=\prod_{p\leq\infty}\Phi_{p}$ be the standard section in
$I(s,\chi)$ such that $\Phi_{L}(g,s_{0})=\lambda(\phi_{L})(g)$. Then it is
known that $\Phi_{\infty}=\gamma(V_{\infty})\Phi_{\infty}^{\ell}$ is the
standard section of weight $\ell$ (up to a scalar multiplication by a local
Weil index $\gamma(V_{\infty})$):

$\Phi_{\infty}^{\ell}(n(b)m(a)k_{\theta},s)=\chi_{\infty}(a)|a|^{s+1}e^{i\ell\theta},$

where

$n(b)=\left(\begin{smallmatrix}1&b\\\ 0&1\end{smallmatrix}\right),\quad
m(a)=\left(\begin{smallmatrix}a&0\\\
0&a^{-1}\end{smallmatrix}\right),\quad\hbox{ and }\quad
k_{\theta}=\left(\begin{smallmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{smallmatrix}\right).$

Let

(7.1) $E_{L}(\tau,s)=v^{-\ell/2}\sum_{\gamma\in
B\backslash\operatorname{SL}_{2}(\mathbb{Q})}\Phi_{L}(\gamma
g_{\tau},s),\quad\tau=u+iv\in\mathbb{H}$

be the associated Eisenstein series of weight $k$ (level $D$ and character
$\chi$), which depends only on the genus $[[L]]$ of $L$. Here

$B=NM=\\{n(b)m(a):\,a\in\mathbb{Q}^{\times},b\in\mathbb{Q}\\}.$

Let $\Gamma_{\infty}=B\cap\operatorname{SL}_{2}(\mathbb{Z})$, then
$B\backslash\operatorname{SL}_{2}(\mathbb{Q})=\Gamma_{\infty}\backslash\operatorname{SL}_{2}(\mathbb{Z})$.

For $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Q})$, let
$\gamma_{p}$ be its image in $\operatorname{SL}_{2}(\mathbb{Q}_{p})$. We first
record the following standard facts as lemmas and a theorem (see for example
[Yan05], [KRY99], [KY10]). They are needed in the next few subsections.

######  Lemma 7.1.

For $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{R})$, we have

$\Phi_{\infty}^{\ell}(\gamma
g_{\tau},s)=v^{\frac{s+1}{2}}j(\gamma,\tau)^{-\ell}|j(\gamma,\tau)|^{\ell-1-s}.$

Here $j(\gamma,\tau)=c\tau+d$. In particular,

$\Phi_{\infty}^{\ell}(\gamma
g_{\tau},s_{0})=v^{\frac{\ell}{2}}j(\gamma,\tau)^{-\ell}.$

######  Lemma 7.2.

Let $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{SL}_{2}(\mathbb{Z}_{p})$.

  1. (1)

When $p\nmid D$, $\Phi_{p}(\gamma,s_{0})=1$ and $\Phi_{p}$ is right
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$-invariant.

  2. (2)

When $p\mid D$ is odd, we have

$\Phi_{p}(\gamma,s_{0})=\begin{cases}\chi_{p}(a)&\hbox{if }p|c,\\\
\gamma(V_{p})\chi_{p}(c)p^{-\ell/2}&\hbox{if }p\nmid c.\end{cases}$

Moreover, $\Phi_{p}(g\gamma)=\Phi_{p}(g)\chi_{p}(a)$ for $p|c$, and
$\gamma(V_{p})$ is the local Weil index (a root of unity).

Notice that similar results hold when $s_{0}$ is replaced by general $s$
although we do not need it here. We have now the following theorem by the
above lemmas.

######  Theorem 7.3.

Let the notation be as above. Then for $\ell\geq 4$ and $s_{0}=\ell-1$,

$\displaystyle E_{L}(\tau,s_{0})$
$\displaystyle=\sum_{\gamma\in\Gamma_{\infty}\backslash\operatorname{SL}_{2}(\mathbb{Z})}a(\gamma,\ell)j(\gamma,\tau)^{-\ell}$
$\displaystyle=\sum_{\delta\in\Gamma_{\infty}\backslash\operatorname{SL}_{2}(\mathbb{Z})/\Gamma_{1}(D)}a(\delta,\ell)\sum_{\gamma\in\Gamma_{\infty}\backslash\delta\Gamma_{1}(N)}j(\gamma,\tau)^{-\ell}.$

Here $a(\gamma,\ell)=\prod_{p|D}a_{p}(\gamma,\ell)$ is right
$\Gamma_{1}(D)$-invariant with

$a_{p}(\gamma,\ell)=\begin{cases}\gamma(V_{p})\chi_{p}(a)&\hbox{if }p|c,\\\
\gamma(V_{p})\chi_{p}(c)p^{-\ell/2}&\hbox{if }p\nmid c,\end{cases}$

if $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$.

###  7.2. Modularity of arithmetic theta functions with Kudla Green functions

In this subsection, we keep the notation of previous sections but assume
$n=2$. In particular,
$L=\operatorname{Hom}_{\mathcal{O}_{\boldsymbol{k}}}(\mathfrak{a}_{0},\mathfrak{a})$
is a self-dual hermitian lattice of signature $(1,1)$. Let

(7.2)
$\widehat{\Theta}_{K}(\tau)=\sum_{m\in\mathbb{Z}}\widehat{\mathcal{Z}}^{\mathrm{tot}}(m,v)q^{m},\quad
v=\operatorname{Im}(\tau)$

be the arithmetic theta function with Kudla Green functions. The purpose of
this subsection is to prove the following theorem, which is part of Theorem
1.1.

######  Theorem 7.4.

Let the notation be as above. Then $\widehat{\Theta}_{K}(\tau)$ is a (non-
holomorphic) modular form of $\Gamma_{0}(D)$ with values in
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$.

###### Proof.

Let $L_{0}$ be a positive definite self-dual lattice of rank $1$ and
$V_{0}=L_{0}\otimes\mathbb{Q}$. Applying Theorem 1.2 to each lattice $\Lambda$
in the genus of $L_{0}^{\ell}$ and Siegel-Weil Formula for $L_{0}^{\ell}$, we
have

(7.3)
$\sum_{\Lambda\in[[L_{0}^{\ell}]]}\frac{1}{|\mathrm{Aut}(\Lambda)|}\varphi_{\Lambda}^{*}(\widehat{\Theta}_{K}^{\diamond}(\tau))=\sum_{\Lambda\in[[L_{0}^{\ell}]]}\frac{1}{|\mathrm{Aut}(\Lambda)|}\theta_{\Lambda}(\tau)\cdot\widehat{\Theta}_{K}(\tau)=E_{L_{0}^{\ell}}(\tau,s_{\ell})\widehat{\Theta}_{K}(\tau).$

Here $s_{\ell}=\ell-1$. By [BHK+20, Theorem B] and [ES18, Theorem 1.4], the
left hand side of (7.3) is a (non-)holomorphic modular form of $\Gamma_{0}(D)$
of weight $2+\ell$ character $\chi_{-D}^{\ell+2}$, we see that
$\widehat{\Theta}_{K}(\tau)$ transforms like a modular form of weight $2$ for
$\Gamma_{0}(D)$. We still need to prove that $\widehat{\Theta}_{K}(\tau)$ is
well-defined everywhere. By Theorem 7.3, we see that

$E_{L_{0}^{\ell}}(\tau,s_{\ell})=\sum_{\gamma\in\Gamma_{\infty}\backslash\mathrm{SL}_{2}(\mathbb{Z})}(\prod_{p|D}b_{p}(\gamma,\ell)\cdot
j(\gamma,\ell)^{-1})^{\ell},$

where

$b_{p}(\gamma,\ell)=\begin{cases}\gamma(V_{0})\chi_{p}(a)&\hbox{if }p|c,\\\
\gamma(V_{0})\chi_{p}(c)p^{-1/2}&\hbox{if }p\nmid c,\end{cases}$

if $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$. For every
$\tau_{0}\in\mathbb{H}$, [Fre90, Lemma 5.6, Chapter I] implies that there is
some $\ell>0$ such that $E_{L_{0}^{\ell}}(\tau_{0},s_{\ell})\neq 0$. So
$\widehat{\Theta}_{K}(\tau)$ is well-defined at $\tau_{0}$ and thus well-
defined everywhere. This proves that $\widehat{\Theta}_{K}(\tau)$ is modular.
∎

Now, the modularity for $\widehat{\Theta}_{B}(\tau)$ should follow directly
from Theorem 7.4 and [ES18, Theorem 1.4]. However, there is a little subtlety
involved: the Green function constructed by Ehlen and Sankaran is a little
different from that of Bruinier when $n=2$ as in our case. The next three
subsections explains the subtlety and proves Theorem 1.1.

###  7.3. A little preparation

To define the Ehlen and Sankaran’s Green functions, we need a little
preparation. In this subsection, let $L$ be an even dimensional integral non-
degenerate $\mathbb{Z}$-lattice of signature $(2,2)$ (in our case, view our
unimodular $\mathcal{O}_{\boldsymbol{k}}$-lattice $L$ of signature $(1,1)$ as
a quadratic lattice with quadratic form $q(x)=(x,x)$). Let $L^{\prime}$ be the
dual of $L$ with respect to the quadratic form, and
$S_{L}=\mathbb{C}[L^{\prime}/L]$ with Weil representation $\rho_{L}$. Let
$S_{L}^{\vee}$ be its dual with dual Weil representation $\rho_{L}^{\vee}$.
Let $\phi_{\mu}$ ($\mu\in L^{\prime}/L$) be the standard basis of $S_{L}$ and
$\phi_{\mu}^{\vee}$ be the dual basis of $S_{L}^{\vee}$.

######  Definition 7.5.

([BF04]) For an integer $k\in\mathbb{Z}$, Let $H_{k}(\rho_{L})$ be the space
of twice continuously differentiable functions $F:\mathbb{H}\rightarrow S_{L}$
such that

  1. (1)

$F(\gamma)(\tau)=(c\tau+d)^{k}\rho(\gamma)F(\tau)$ for
$\gamma\in\Gamma=\hbox{SL}_{2}(\mathbb{Z})$.

  2. (2)

(at most exponential growth) there is a constant $C>0$ such that
$F(u+iv)=O(e^{Cv})$ as $v\rightarrow\infty$.

  3. (3)

($k$-harmonic) We have $\Delta_{k}(F)=0$, where

$\Delta_{k}=-v^{2}(\frac{\partial^{2}}{\partial
u^{2}}+\frac{\partial^{2}}{\partial v^{2}})+ikv(\frac{\partial}{\partial
u}+i\frac{\partial}{\partial v})$

is the hyperbolic Laplace operator in weight $k$.

  4. (4)

$\xi_{k}(F)\in S_{2-k}(\rho_{L}^{\vee})$ is a cuspidal modular form values in
$S_{L}^{\vee}$. Here the $\xi$-operator is given by

$\xi_{k}(F)=2iv^{k}\overline{\frac{\partial
F}{\partial\bar{\tau}}}=v^{k-2}\overline{\mathbf{L}(F)},$

where $\mathbf{L}=-2iv^{2}\frac{\partial}{\partial\bar{\tau}}$ is the Maass
lowering operator.

An element in $H_{k}(\rho_{L})$ is called a harmonic Maass modular form (of
weight $k$ with values in $S_{L}$). It is called a weakly holomomorphic
modular form if $\xi_{k}(F)=0$. We denote the space of weakly holomomorphic
forms by $M_{k}^{!}(\rho_{L})$.

According to Bruinier and Funke ([BF04]). there is an exact sequence

(7.4) $0\rightarrow M_{k}^{!}(\rho_{L})\rightarrow H_{k}(\rho_{L})\rightarrow
S_{2-k}(\rho_{L}^{\vee})\rightarrow 0.$

Every $F\in H_{k}(\rho_{L})$ is smooth and admits a decomposition

(7.5) $F(\tau)=F^{+}(\tau)+F^{-}(\tau)$

into its holomorphic and non-holomorphic parts. Here its holomorphic part

(7.6) $F^{+}(\tau)=\sum_{m>-\infty}c_{F}^{+}(m)q^{m},\quad c_{F}^{+}(m)\in
S_{L}$

has only finitely many negative part, and its non-holomorphic part
$F^{-}(\tau)$ is of exponential decay as $v=\operatorname{Im}(\tau)$ goes to
infinity.

######  Lemma 7.6.

Let $F\in H_{0}(\rho_{L})$ with $c_{F}^{+}(m)=0$ for all $m<0$. Then $F\in
M_{0}(\rho_{L})=S_{L}^{\hbox{SL}_{2}(\mathbb{Z})}$ is a holomorphic modular
form of weight $0$, i.e., an element in $S_{L}$ fixed by
$\hbox{SL}_{2}(\mathbb{Z})$ via $\rho_{L}$.

###### Proof.

Since $c_{F}^{+}(m)=0$ for $m<0$, $F^{+}(\tau)$ and thus $F(\tau)$ is bounded
at the cusp $\infty$. So $F(\tau)$ is a bounded harmonic ($S_{L}$-valued)
function on the compact modular curve
$\hbox{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}^{*}=\mathbb{P}^{1}(\mathbb{C})$
and is thus a constant. ∎

The following is basically a special case of [ES18, Lemma 2.4].

######  Lemma 7.7.

For $m\in\mathbb{Q}$ and $\mu\in L^{\prime}/L$ with $Q(\mu)\equiv
m\mod\mathbb{Z}$, there is a unique $F_{m,\mu}\in H_{0}(\rho_{L})$ such that

  1. (1)

The holomorphic part of $F_{m,\mu}$ has the form

$F_{m,\mu}^{+}(\tau)=q^{-m}\tilde{\phi}_{\mu}+\sum_{n\geq
0}c_{m,\mu}^{+}(n)q^{n},\quad c_{m,\mu}^{+}(n)\in S_{L},$

where $\tilde{\phi}_{\mu}=\frac{1}{2}(\phi_{\mu}+\phi_{-\mu})$. We denote
$c_{m}^{+}(0)=c_{m,0}^{+}(0,0)$ for the $\phi_{0}$-component of
$c_{m,0}^{+}(0)$ for later use.

  2. (2)

(7.7) $\sum_{\nu\in
L^{\prime}/L}a_{f}(0,\nu)c_{m,\mu}^{+}(0,\nu)=a_{f}(m,\mu)$

for every $f\in M_{0}(\rho_{L}^{\vee})$. Here $a_{f}(n,\nu)$ is the
$(n,\nu)$-th coefficient of $f$, and $c_{m,\mu}^{+}(n,\nu)$ is the
$(n,\nu)$-th coefficient of $F_{m,\mu}^{+}$.

###### Proof.

It follows from [ES18, Lemma 2.4] together with the following explicit
construction of the splitting map $\eta$ in [ES18, (2.8)]. First notice that
$M_{0}(\rho_{L})=S_{L}^{\operatorname{SL}_{2}(\mathbb{Z})}$ is the maximal
subspace of $S_{L}$ on which $\operatorname{SL}_{2}(\mathbb{Z})$ acts
trivially. Secondly, we have a natural embedding

$M_{0}(\rho_{L}^{\vee})\rightarrow\hbox{Sing}_{2}(\rho_{L}^{\vee}),\quad
f=\sum c_{f}(m)q^{m}\mapsto P(f)=\sum_{m\leq 0}c_{f}(m)q^{m},$

where

$\hbox{Sing}_{2}(\rho_{L}^{\vee})=\\{P=\sum_{m\leq
0}a_{P}(m)q^{m}:\,a_{P}(m)\in S_{L}^{\vee}\\}.$

Now the natural non-degenerate bilinear form

$S_{L}\times S_{L}^{\vee}\rightarrow\mathbb{C},\quad\langle\sum
a(\mu)\phi_{\mu},\sum b(\mu)\phi_{\mu}^{\vee}\rangle=\sum a(\mu)b(\mu)$

gives rise to the splitting map

$\eta:M_{0}(\rho_{L})^{\vee}\cong
M_{0}(\rho_{L}^{\vee})\rightarrow\hbox{Sing}_{2}(\rho_{L}^{\vee}).$

It is clear from the definition that [ES18, Lemma 2.4 Condition(ii)] is the
same as (7.7). ∎

######  Corollary 7.8.

Let the notation be as in Lemma 7.7. Then $F_{m,\mu}=0$ for $m<0$, and
$F_{m,\mu}\in M_{0}(\rho_{L})$.

Following Ehlen and Sankaran [ES18], we can view $F_{m}$ as a harmonic modular
form of weight $0$ valued in $S_{L}\otimes S_{L}^{\vee}$ via

$F_{m}=\sum_{\mu\in L^{\prime}/L}F_{m,\mu}\phi_{\mu}^{\vee}.$

Let $V=L_{\mathbb{Q}}=L\otimes_{\mathbb{Z}}\mathbb{Q}$, and let
$\mathcal{D}^{o}$ be the negative $2$-planes in $V_{\mathbb{R}}$. For
$z\in\mathcal{D}^{o}$, decompose

$V_{\mathbb{R}}=z\oplus z^{\perp},\quad x=x_{z}+x_{z^{\perp}}.$

For $h\in\mathrm{SO}(V)(\mathbb{A}_{f})$ and $z\in\mathcal{D}^{o}$, one has
the Siegel-theta function

(7.8) $\theta(\tau,z,h;\phi)=v\sum_{x\in
V}\phi(h^{-1}x)\phi_{\infty}(\sqrt{v},x,z)$

which is a modular form of $\tau$ of weight $0$, and
$\mathrm{SO}(V)(\mathbb{Q})$-invariant. Here

$\phi_{\infty}(x,z)=e^{-\pi(x,x)_{z}}\in S(V_{\mathbb{R}}).$

Denote

(7.9) $\theta_{L}(\tau,z,h)=\sum_{\mu\in
L^{\prime}/L}\theta(\tau,z,h;\phi_{\mu})\phi_{\mu}^{\vee}.$

which is a (non-holomorphic) modular form of
$\operatorname{SL}_{2}(\mathbb{Z})$ of weight $0$ valued in $S_{L}^{\vee}$.
The Bruinier Green functions (the Ehlen and Sankaran version) are defined as

(7.10) $\displaystyle G_{ES}(m,\mu)$ $\displaystyle=\int^{reg}\langle
F_{m,\mu},\theta_{L}\rangle d\mu(\tau)$
$\displaystyle=\operatorname{CT}_{s=0}\lim_{T\rightarrow\infty}\int_{\mathcal{F}_{T}}\langle
F_{m,\mu},\theta_{L}\rangle v^{-s}d\mu(\tau).$

Here $\langle F_{m,\mu},\theta_{L}\rangle$ comes from the natural paring
between $S_{L}$ and $S_{L}^{\vee}$,

$\mathcal{F}_{T}=\\{\tau=u+iv\in\mathbb{H}:\,-\frac{1}{2}\leq
u\leq\frac{1}{2},0<v<T,|\tau|\geq 1\\}$

is the truncated fundamental domain for $\operatorname{SL}_{2}(\mathbb{Z})$,
and finally $\operatorname{CT}_{s=1}(f(s))$ is the constant term of the
Laurent series of $f(s)$ at $s=0$.

###  7.4. Arithmetic theta function with Bruinier Green functions I—Ehlen and
Sankaran’s normalization

Now we come back to the notation and restriction of Section 7.2. The harmonic
functions $F_{m}$ in Section 7.3 are now associated to the lattice $L$ with
quadratic form $Q(x)=(x,x)$.

For a cusp $\Phi$ associated to the genus of $L$ and $m\geq 0$, let

(7.11) $\eta_{\Phi}(m)=-2[\sigma_{1}(m)+c_{m}^{+}(0)].$

Here $c_{m}^{+}(0)$ is defined in Lemma 7.7 and $\sigma_{1}(0)=-\frac{1}{24}$.

Let

(7.12)
$\mathcal{Z}_{ES}^{\mathrm{tot}}(m)=\mathcal{Z}^{*}(m)+\sum_{\Phi\in\mathrm{Cusps}([[L]])}\eta_{\Phi}(m)[\mathcal{S}^{*}(\Phi)]\in{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*}),$

for $m\in\mathbb{Z}_{>0}$, and

(7.13) $\mathcal{Z}_{ES}^{\mathrm{tot}}(0)=\widehat{\Omega}^{-1}+(0,-\log
D)+\sum_{\Phi\in\mathrm{Cusps}([[L]])}\eta_{\Phi}(0)[\mathcal{S}^{*}(\Phi)]\in{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*}).$

Define

(7.14)
$\widehat{\mathbb{Z}}_{ES}^{\mathrm{tot}}(m)=(\mathcal{Z}_{ES}^{\mathrm{tot}}(m),G_{ES}(m))\in\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*}).$

Here the Ehlen-Sankaran Green function $G_{ES}(m)=G_{ES}(m,0)$ is the Green
function (7.10) restricting to $\mathcal{D}\times U(V)(\mathbb{A}_{f})$, under
the natural embedding $U(V)\subset\mathrm{SO}(V)$.

Finally define the arithmetic theta function with Ehlen-Sankran Green
functions

(7.15) $\widehat{\Theta}_{ES}(\tau)=\sum_{m\in D^{-1}\mathbb{Z}_{\geq
0}}\widehat{\mathcal{Z}}_{ES}^{\mathrm{tot}}(m)q^{m}.$

######  Theorem 7.9.

Let the notations be as above. Then $\widehat{\Theta}_{ES}(\tau)$ is a
(holomorphic) modular form of $\Gamma_{0}(D)$ of weight $2$ (trivial
character) with values in
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$.

###### Proof.

By [ES18, Theorem 1.4] (more precisely its $\phi_{0}$-component),
$\widehat{\Theta}_{K}(\tau)-\widehat{\Theta}_{ES}(\tau)$ is modular. Now
Theorem 7.4 implies that $\widehat{\Theta}_{ES}$ is modular. ∎

Since the Eisenstein series
$E_{2}(\tau)=\sum_{m=0}\sigma_{1}(m)q^{m}+\frac{1}{8\pi v}$ is a modular form
of $\operatorname{SL}_{2}(\mathbb{Z})$ of weight $2$, we have the following
corollary.

######  Corollary 7.10.

Let the notations be as above. Then

$\sum_{m\geq 0}c_{m}^{+}(0)q^{m}$

is a modular form for $\Gamma_{0}(D)$ of weight $2$.

###### Proof.

Looking at the generic fiber of $\widehat{\Theta}_{ES}$, we see that

$\Theta_{ES}(\tau)=\left(E_{2}(\tau)+\sum_{m\in\mathbb{Z}_{\geq
0}}c_{m}^{+}(0)q^{m}\right)\left(\sum_{\Phi\in\mathrm{Cusps}([[L]])}[B_{\Phi}]\right)+\Theta^{\mathrm{geo}}(\tau)$

is a modular form with values in ${\operatorname{CH}}_{\mathbb{C}}^{1}(S^{*})$
where $S^{*}$ is the generic fiber of $\mathcal{S}^{*}$. Here

$\Theta^{\mathrm{geo}}(\tau)=-\Omega-\frac{1}{8\pi
v}\sum_{\Phi\in\mathrm{Cusps}([[L]])}[B_{\Phi}]+\sum_{m\in\mathbb{Z}_{>0}}Z^{*}(m)q^{m}.$

Embedding $U(V)$ to $\hbox{SO}(V)$ and applying [EGT23, Theorem 1.2], we see
that $\Theta^{\mathrm{geo}}(\tau)$ is a modular form for $\Gamma_{0}(D)$ of
weight $2$ (trivial character) with values in
$\operatorname{CH}_{\mathbb{C}}^{1}(S^{*})$. Now the corollary is clear. ∎

###  7.5. Arithmetic theta function with Bruinier Green functions II-
Bruinier’s normalization

For each $m\geq 0$, there is a unique harmonic Maass (scalar valued) modular
form $f_{m}$ for $\Gamma_{0}(D)$ of weight $0$ such that

  1. (1)

(7.16) $f_{m}^{+}(\tau)=q^{-m}+\sum_{n\geq 0}c_{m}(n)q^{n},$

  2. (2)

the constant term of $f_{m}^{+}$ is zero,

  3. (3)

At another cusp $P=\gamma(\infty)\neq\infty$ for $\Gamma_{0}(D)$,

$f_{m}^{+}(\gamma\tau)=\sum_{n\geq 0}c_{m,\gamma}(n)q^{n}.$

For such an $f_{m}$, let

$\tilde{f}_{m}=\sum_{\gamma\in\Gamma_{0}(D)\backslash\operatorname{SL}_{2}(\mathbb{Z})}f(\gamma\tau)\rho_{L}(\gamma^{-1})\phi_{0}$

be the associated harmonic Maass modular form for
$\operatorname{SL}_{2}(\mathbb{Z})$ of weight $0$ and representation
$\rho_{L}$. By [BHK+20, Proposition 6.1.2], we also have

$\tilde{f}_{m}^{+}=q^{-m}\phi_{0}+\sum_{n\geq
0}\tilde{c}_{m}(n)q^{n},\quad\tilde{c}_{m}(n)\in S_{L}.$

Let

(7.17) $G_{B}(m)=\int^{reg}\tilde{f}_{m}\theta_{L}d\mu(\tau)$

be the regularized theta lifting of $\tilde{f}_{m}$ as in (7.10). Then similar
to [BHK+20, Section 7.3], we have an arithmetic divisor

(7.18)
$\widehat{\mathcal{Z}}_{B}^{\mathrm{tot}}(m)=(\mathcal{Z}_{B}^{\mathrm{tot}}(m),G_{B}(m))\in\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$

with

(7.19)
$\mathcal{Z}_{B}^{\mathrm{tot}}(m)=\mathcal{Z}^{*}(m)-2\sigma_{1}(m)\sum_{\Phi\in\hbox{Cusps}[[L]]}[\mathcal{S}^{*}(\Phi)]+\begin{cases}0&\mathrm{if}m>0,\\\
\widehat{\Omega}^{-1}+(0,-\log D)&\mathrm{if}m=0.\end{cases}$

We again form the arithmetic theta function

(7.20) $\widehat{\Theta}_{B}(\tau)=\sum_{m\geq
0}\widehat{\mathcal{Z}}_{B}^{\mathrm{tot}}(m)q^{m}.$

######  Theorem 7.11.

The arithmetic theta function $\widehat{\Theta}_{B}(\tau)$ is a modular form
for $\Gamma_{0}(D)$ of weight $2$ with values in
$\widehat{\operatorname{CH}}_{\mathbb{C}}^{1}(\mathcal{S}^{*})$.

###### Proof.

The issue will be stated at the end. The idea is to prove

(7.21) $\widehat{\Theta}_{B}(\tau)-\widehat{\Theta}_{ES}(\tau)=\sum_{m\geq
0}\left(c_{m}^{+}(0)\sum_{\Phi}\mathcal{S}^{*}(\Phi),-\log\|\Psi_{m}\|^{2}\right)q^{m}$

is modular. Here $\Psi_{m}$ is the Borcherds lifting of the vector valued
weakly holomorphic modular form $\tilde{f}_{m}-F_{m,0}$ and has weight $0$.
Over $\mathbb{C}$, we know
$\hbox{Div}(\Psi_{m})=c_{m}^{+}(0)\sum_{\Phi}\mathcal{S}^{*}(\Phi)$. By
Corollary 7.10, we see that $\sum c_{m}^{+}(0)q^{m}$ is a modular form of
weight $2$ for $\Gamma_{0}(D)$. Now the same proof as that in [ES18, Theorem
4.13] gives the modularity of (7.21).

∎

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