construct some interesting hermitian line bundles on it, and explain the relations between them.

\subsection{A special Shimura variety}
\label{ss:special shimura}

To attach a Shimura variety to our fixed  $V$,  choose  relevant (Definition \ref{def:relevant})  hermitian spaces  $(W_0,h_0)$ and $(W,h)$ of  signatures $(1,0)$ and $(n-1,1)$, respectively,  in such a way that
\begin{equation}\label{hermitian hom}
V \iso \Hom_\kk(W_0 , W)
\end{equation}
as $$-hermitian spaces, where the hermitian form  $⟨-  , - ⟩$ on the right  satisfies
\begin{equation}\label{basic hom hermitian}
\langle x,y\rangle  \cdot h_0(w_0,w'_0 ) = h( x(w_0) , y(w'_0) )
\end{equation}
for all $w_0 , w_0'∈W_0$ and $x,y∈V$.

\begin{remark}
Such $W_0$ and $W$ always exist:  take $W_0=\kk$ with its norm form, and $W=V$.
They are not uniquely determined by $V$, but their strict similarity classes are.
\end{remark}

As  in \S 2.2 of   [32], if  $S$ is a connected $_$-scheme  and
\[
(A_0,A) \in   \mathcal{M}_{W_0}(S)  \times \mathcal{M}_{W}  (S),
\]
then $__(A_0,A)$ carries a positive definite  hermitian form
\begin{equation}\label{KR hermitian}
\langle x,y\rangle =    \psi_0^{-1} \circ y^\vee \circ \psi\circ  x   \in \End_{\co_\kk}(A_0) \iso \co_\kk,
\end{equation}
where $ψ_0 :A_0 →A_0^∨$ and $ψ:A→A^∨$ are the principal polarizations.
%If  $S=\Spec(F)$ with $F$ an algebraically closed field and $\ell \neq\mathrm{char}(F)$ is prime, we can repeat this construction with $A_0$ and $A$ replaced by their $\ell$-adic Tate modules to endow  $\Hom_{\co_\kk} ( T_\ell(A_0)   , T_\ell(A)  )$ with an $\co_{\kk,\ell}$-valued hermitian form.
%If $S=\Spec(\C)$  may similarly endow $\Hom_{\co_\kk} ( H_1(A_0,\Z)   ,  H_1(A,\Z)  )$ with an $\co_\kk$-valued hermitian form, and then
%\Hom_{\co_\kk} ( T_\ell(A_0)   , T_\ell(A)  ) \iso \Hom_{\co_\kk} ( H_1(A_0,\Z)   ,  H_1(A,\Z)  ) \otimes_\Z \Z_\ell.

As in \S 2.3 of [13],  there is an open and closed substack
\begin{equation}\label{moduli inclusion}
\mathcal{S}_V \subset \mathcal{M}_{W_0} \times_{\co_\kk} \mathcal{M}_{W}
\end{equation}
characterized by  its  points valued in  algebraically closed fields,  which are those pairs
\[
(A_0,A)\in   \mathcal{M}_{W_0} (F) \times  \mathcal{M}_{W} (F)
\]
for which there exists an isometry
\[
V\otimes \Q_\ell \iso  \Hom_{\co_\kk} \big( T_\ell(A_0)   , T_\ell(A)  \big)  \otimes \Q_\ell
\]
of $_ℓ$-hermitian spaces for every $ℓ≠char(F) $.
Here the hermitian form on the right is defined as in \eqref{KR hermitian}.
When $F=$ this is equivalent to the existence of an isometry of $$-hermitian spaces
\[
V \iso \Hom_{\kk} \big( H_1(A_0 , \Q)  , H_1(A  , \Q) \big) .
\]

\begin{remark}
When $n$ is even the inclusion \eqref{moduli inclusion} is  an isomorphism.
\end{remark}

\begin{remark}\label{rem:projection fiber}
The projection
\mathcal{S}_V \to \mathcal{M}_W
is a finite \'etale surjection, and the fiber  over a geometric point $x \in  \mathcal{M}_W(\F)$ satisfies
\[
\sum_{  y \in \mathcal{S}_{V,x}(\F)   } \frac{1}{ |\Aut(y) | }  = \frac{ |\mathrm{CL}(\kk) |} {  | \co_\kk^\times|}
\cdot \begin{cases}
1 & \mbox{if $n$ is even} \\
2^{1-o(D)}  & \mbox{if $n$ is odd}.
\end{cases}
\]
\end{remark}

\begin{remark}\label{rem:honest shimura}
The   stack $\mathcal{S}_V$ is denoted $\mathcal{S}_\Kra$    in [13].
Later we will want to vary $V$, and so we have  included it in the notation to avoid confusion.
As explained in [loc.~cit.], the generic fiber of  $\mathcal{S}_V$ is a Shimura variety
for the subgroup $G \subset \mathrm{GU}(W_0) \times \mathrm{GU}(W)$ of pairs $(g_0,g)$ for which the similitude factors of the two components are equal.
\end{remark}

\begin{definition}\label{def:special exceptional}
If $n\ge 2$,  define the \emph{exceptional divisor}
\[
\mathrm{Exc}_V \subset \mathcal{S}_V
\]
as the pullback of the exceptional divisor \eqref{basic exceptional} via $ \mathcal{S}_V  \to \mathcal{M}_{(n-1,1)}$.  Equivalently, it is defined by the cartesian diagram
\begin{equation}\label{special exceptional}
\xymatrix{
{\mathrm{Exc}_V }  \ar[r] \ar[d]  &  { \mathcal{S}_V } \ar[d] \\
{\mathrm{Sing}_{(n-1,1)} } \ar[r]  & { \mathcal{M}^\Pap_{(n-1,1)} }.
\end{equation}
\end{definition}

\begin{definition}[{[32]}]
\label{def:KR}
For any positive $m\in \Z$, the \emph{Kudla-Rapoport divisor}   $\mathcal{Z}_V(m)$  is the $\co_\kk$-stack classifying triples $(A_0,A,x)$ consisting of a pair
\[
(A_0,A) \in \mathcal{S}_V(S)
\]
and an
x\in \Hom_{\co_\kk}(A_0,A)
satisfying $\langle x,x\rangle=m$.
\end{definition}

The natural forgetful morphism
\[
\mathcal{Z}_V(m) \to \mathcal{S}_V
\]
is finite and unramified, with image of codimension $1$. We denote again by $𝒵_V(m)$ the image of this morphism, viewed as a divisor on $𝒮_V$ in the usual way (that is to say, each irreducible component of  $𝒵_V(m)$ contributes an irreducible component of the image, counted with multiplicity equal to the length of the local ring of its generic point).
Denote by
\[
\bar{\mathcal{Z}}_V(m)\to \bar{\mathcal{S}}_V
\]
the normalization of $𝒵_V(m) →𝒮̅_V$, and  denote in the same way its image, viewed as a divisor on  $𝒮̅_V$. In other words, take the Zariski closure of  $𝒵_V(m)$.

Loosely speaking,  each Kudla-Rapoport divisor  is a union of  unitary Shimura varieties associated to  $$-hermitian spaces of signature $(n-2,1)$.  In  \S \ref{s:KR divisors} we will  make this more precise, at least  when $m$ is a prime split in $$.

\begin{definition}
Let $N$ be a positive integer, and let $S$ be an $\co_\kk$-scheme with $N\in \co_S^\times$.
A \emph{level $N$-structure} on a pair $(A_0,A) \in \mathcal{S}_V(S)$ consists of  level $N$-structures $(\eta_0,\xi_0)$ and $(\eta,\xi)$ on $A_0$ and $A$, in the sense of Definition \ref{def:kramer-pappas level},  such   that $ \xi_0 =   \xi$.
\end{definition}

Adding level structure to pairs $(A_0,A)$ defines  a finite \'etale cover
\[
\mathcal{S}_V(N) \to  \mathcal{S}_{ V /\co_\kk[1/N]  },
\]
and  \eqref{moduli inclusion} lifts to a canonical open and closed immersion
\[
\mathcal{S}_V (N) \subset \mathcal{M}_{W_0} (N) \times_{\co_\kk[1/N]} \mathcal{M}_W (N).
\]
Define a toroidal compactification
\begin{equation}\label{compact inclusion}
\bar{\mathcal{S}}_V (N) \subset \mathcal{M}_{W_0} (N) \times_{\co_\kk[1/N]}  \bar{\mathcal{M}}_W (N)
\end{equation}
as the Zariski closure of $𝒮_V(N)$, or, equivalently, as the normalization of
\[
\mathcal{S}_V (N) \to \bar{\mathcal{M}}_{W/ \co_\kk[1/N]}.
\]
When $N=1$ we abbreviate this to $𝒮̅_V$.

\begin{remark}\label{rem:special compact level}
As \eqref{compact inclusion} is an open and closed immersion, and $\mathcal{M}_{W_0} (N)$ is finite \'etale over $\co_\kk[1/N]$, the compactification $\bar{\mathcal{S}}_V (N)$ inherits all the nice properties of $\bar{\mathcal{M}}_W (N)$.  In particular, Proposition \ref{prop:full compactification} holds word-for-word with $\mathcal{M}_W$ replaced by $\mathcal{S}_V$, and the same is true of the entire discussion of arithmetic intersection theory in \S \ref{ss:chow} and \S \ref{ss:volumes}.
\end{remark}

\subsection{Construction of hermitian line bundles}

The construction \eqref{hodge metric} associates to the universal CM elliptic curve $A_0 →ℳ_W_0 = ℳ_(1,0)$ its metrized Hodge bundle
\begin{equation}\label{elliptic hodge}
\widehat{\omega}^\mathrm{Hdg}_{A_0 / \mathcal{M}_{W_0} } \in \widehat{\Pic}(\mathcal{M}_{W_0} ) .
\end{equation}
Similarly, the universal $A  →ℳ_W$  determines a metrized Hodge bundle
\begin{equation}\label{basic hodge}
\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W } \in \widehat{\Pic}(\mathcal{M}_W ) .
\end{equation}

Pulling back the universal objects via projection to the two factors in  \eqref{moduli inclusion} yields a universal pair $(A_0,A)$ of polarized abelian schemes (of  relative dimensions $1$ and $n$)  over  $𝒮_V$.
As in \S 2.4 of [13] there is a  \emph{metrized line bundle of modular forms}
\begin{equation}\label{metrized taut}
\widehat{\taut}_V \in \widehat{\Pic}(\mathcal{S}_V).
\end{equation}
The line bundle underlying \eqref{metrized taut} has  inverse
\begin{equation}\label{dual taut}
\taut_V^{-1} =   \Lie(A_0)  \otimes   \Lie(A) / \mathcal{F} ,
\end{equation}
where $ℱ ⊂(A)$ is the universal hyperplane satisfying Kr\"amer's signature condition (\S \ref{ss:basic moduli}).
The hermitian metric  is defined as in \S 7.2 of  [13]:
if we use  Proposition 2.4.2 of [13] to identify
\begin{equation}\label{taut realization}
\taut_{V,z}  \subset  \Hom_{\kk} ( H_1(A_{0,z} ,\Q) , H_1(A_z,\Q) )  \otimes_\Q\C \iso V\otimes_\Q\C
\end{equation}
at a complex point $z∈𝒮_V()$, the line  $_V,z$ is   isotropic  with respect to  the $$-bilinear extension of the $$-bilinear form  $ [ x,y] = Tr_/ ⟨x,y⟩$
on $V$, and
\begin{equation}\label{taut metric}
\| s \|^2 = - \frac{ [s,\overline{s}] }{ 4\pi e^\gamma }
\end{equation}
for any   $s∈_V,z$.
Note that our  $_V$ is denoted $ω$ in  [13, 14].

\begin{proposition}\label{prop:taut-hodge1}
The  hermitian line bundles   \eqref{basic hodge} and  \eqref{metrized taut}  lie in the subgroups
\begin{align*}
\widehat{\Pic}( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK )   \subset \widehat{\Pic}(\mathcal{M}_W  )
\quad \mbox{and} \quad
\widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK )   \subset \widehat{\Pic}(\mathcal{S}_V  ),
\end{align*}
respectively, of \eqref{pre-log injection}.
\end{proposition}

\begin{proof}
The extension  to $\bar{\mathcal{M}}_W$ of the line bundle
\[
\omega^\mathrm{Hdg}_{A / \mathcal{M}_W } \iso \det( \Lie(A) )^{-1}
\]
underlying \eqref{basic hodge}  is part of \eqref{hyperplane extension}.
The fact that the hermitian metric has a pre-log singularity along the boundary is a special case of
Theorem 6.16 of [12].
Note that this also uses   Proposition 3.2 of [16], which shows  that any rank one log-singular hermitian vector bundle in the sense of [12] is also a  pre-log-singular hermitian line bundle in the sense of Definition 1.20 of [6].
This proves the claim for  \eqref{basic hodge}, and the proof for \eqref{metrized taut} is the same.
\end{proof}

Pulling back the hermitian line bundles \eqref{elliptic hodge} and \eqref{basic hodge} via projection to the two  factors  in \eqref{compact inclusion}, we obtain three hermitian line bundles
\begin{equation}\label{three bundles}
\widehat{\omega}^\mathrm{Hdg}_{A_0 / \mathcal{S}_V },\,
\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{S}_V },\,
\widehat{\mathcal{L}}_V
\in  \widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK  ).
\end{equation}
In the remaining subsections we will make explicit the relations between them.
The reader may wish to skip directly to Theorem \ref{thm:taut-hodge compare} for the main results.

\subsection{An application of the Chowla-Selberg formula}

We will prove that, up to numerical equivalence (Definition \ref{def:numerical}), the first line bundle in \eqref{three bundles} is  just the trivial line bundle with a constant metric.
The constant defining the metric is an interesting quantity in its own right.
Recall  the Chowla-Selberg formula: the Faltings height,  normalized as in \S 5.2 of [14],
of any elliptic curve with CM by $_$ is
\begin{equation}\label{faltings}
h^\mathrm{Falt}_\kk   =
- \frac{1}{2} \frac{L'(0,\eps)}{L(0,\eps)}   - \frac{1}{4}   \log(4\pi^2 D )  .
\end{equation}

\begin{proposition}\label{prop:easy numerical}
The metrized line bundles
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0 / \mathcal{S}_V }  \in \widehat{\Pic}(\mathcal{S}_V )
\quad \mbox{and}\quad   ( 0 , C_1) \in  \widehat{\Pic}(\mathcal{S}_V )
\]
are  numerically equivalent, where $C_1 = \log(2\pi) + 2 h^\mathrm{Falt}_\kk$, and we are using the notation of  \eqref{constant metrics}.
\end{proposition}

\begin{proof}
Fix an $N\ge 3$.  As $\mathcal{M}_{W_0}(N)$ is finite \'etale over $\co_\kk[1/N]$, there is an isomorphism
\[
\mathcal{M}_{W_0}(N) \iso \bigsqcup_i  \mathcal{X}_i
\]
with each $\mathcal{X}_i \iso \Spec(\co_{\kk_i} [ 1/N])$ for a finite field extension $\kk_i/\kk$ unramified outside $N$.
For each $\mathcal{X}_i$, consider the metrized Hodge bundle
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{X}_i} \in \widehat{\Pic}(\mathcal{X}_i )
\]
of the universal $A_0 \to \mathcal{X}_i$.  The underlying line bundle can be identified with an element in the ideal class group of $\co_{\kk_i}[1/N]$, and hence some power of it admits a trivializing section
\[
s_i \in H^0\big( \mathcal{X}_i , ( \omega^\mathrm{Hdg}_{A_0/\mathcal{X}_i})^{\otimes d_i} \big)
\iso
H^0( A_0 , \Omega^{\otimes d_i} _{ A_0 / \mathcal{X}_i } ).
\]
Comparing \eqref{hodge metric} with the definition of the Faltings height, normalized as in \S 5.2 of [14], shows that
\begin{align*}
\frac{ -1}{  \# \mathcal{X}_i(\C) }  \sum_{ x\in \mathcal{X}_i(\C) } \log\| s_{i,x}\|^2
% & =  \frac{ -1}{  \# \mathcal{X}_i(\C) } \sum_{ x\in \mathcal{X}_i(\C) }  \log  \left|  \frac{1}{2\pi}   \int_{ A_{0,x} (\C)} s_{i,x} \wedge \overline{s}_{i,x}  \right| \\
& = d_i  \cdot C_1,
\end{align*}
up to a $\Q$-linear combination of $\{ \log(p) : p\mid N\}$.

Letting $d$ be the least common multiple of all $d_i$'s, we see that
\[
\big( \widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)} \big)^{\otimes d} \in \widehat{\Pic}( \bar{\mathcal{S}}_V(N)  )
\]
admits a trivializing section $s$  with  $-\log\| s\|^2$  constant on every connected component of $\bar{\mathcal{S}}_V(N)(\C)$, and such that  the average value of $-\log\| s\|^2$ over any $\Aut(\C/\kk)$-orbit of  components is $d\cdot C_1$,
up to a $\Q$-linear combination of $\{ \log(p) : p\mid N\}$.

If we represent
\[
d \cdot \widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)}  \in \widehat{\mathrm{CH}}^1(  \bar{\mathcal{S}}_V(N)  )
\]
by the arithmetic divisor
\widehat{\mathrm{div}}(s)  = ( 0 , - \log\| s\|^2),
then for any
\[
(\mathcal{Z}_N , g_N) \in \widehat{\mathrm{CH}}^{n-1}( \bar{\mathcal{S}}_V(N)  )
\]
the vanishing of the Chern form of $-\log\| s\|^2$ implies  the $*$-product formula
\[
[ - \log\| s\|^2] * g_N =      - \log\| s\|^2  \wedge \delta_{\mathcal{Z}_N},
\]
which implies the intersection formula
\[
d \cdot \widehat{\deg}_N\big(  (\mathcal{Z}_N , g_N) \cdot
\widehat{\omega}^\mathrm{Hdg} _{A_0/\mathcal{S}_V(N)} \big)
- \sum_{ x \in \mathcal{Z}_N(\C)  } \log\| s_x\|^2
\]
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$.

Let   $L\subset \C$ be a finite Galois extension of $\kk$ large enough that all complex points of $\mathcal{Z}_N$ are defined over $L$, and rewrite the equality above as
\[
d \cdot \widehat{\deg}_N \big(   (\mathcal{Z}_N , g_N) \cdot
\widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)}   \big)
\frac{-1}{ [ L:\kk] }  \sum_{  \substack{ x \in \mathcal{Z}_N(L)  \\  \sigma \in \Gal(L/\kk)  }  }   \log\| s_{x^\sigma} \|^2.
\]
The  right hand side is  $d C_1 \cdot \# \mathcal{Z}_N(\C)$, and hence
\[
\widehat{\deg}_N \big(  (\mathcal{Z}_N , g_N) \cdot \widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)}   \big)
= C_1 \cdot \# \mathcal{Z}_N(\C)
=   \widehat{\deg}_N  \big(   (\mathcal{Z}_N , g_N) \cdot (0,C_1)  \big)
\]
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$.
Varying  $N$ shows that
\[
\widehat{\deg}\big(  \widehat{\mathcal{Z}} \cdot \widehat{\omega}^\mathrm{Hdg} _{A_0/\mathcal{S}_V}   \big)
\widehat{\deg}\big(  \widehat{\mathcal{Z}} \cdot (0,C_1)  \big)
\]
for every  $\widehat{\mathcal{Z}}   \in \widehat{\mathrm{CH}}^{n-1}( \bar{\mathcal{S}}_V  )$,
and the claim follows.
\end{proof}

\subsection{Another hermitian line bundle}

In order to relate the  hermitian line bundles of \eqref{three bundles}, we recall from  \S 5.1 of  [14] a fourth hermitian line bundle.
Denote by
\[
H^1_\mathrm{dR}(A) = \mathbb{R}^1\pi_* \Omega^\bullet_{A/\mathcal{S}_V}
\]
the first relative algebraic deRham cohomology of $π: A→𝒮_V$,  a rank $2n$  vector bundle on $𝒮_V$.
The action of $_$ on $A$ induces an action on  $H_1^dR(A) = H^1_dR(A)^∨$, which is locally free of rank $n$ over $_⊗__S$.
\[
H_1^\mathrm{dR}(A)  \to    \mathcal{V}
\]
denotes the    largest quotient  on which the action of $_$ is through the structure morphism $_→_𝒮_V$, then $𝒱$ is a rank $n$ vector bundle on $𝒮_V$, equipped with
a    morphism
\[
\det(\mathcal{V})^{-1} \to \bigwedge\nolimits^n H^1_{\mathrm{dR}}(A)
\to H^n_{\mathrm{dR}}(A) .
\]
Given a complex point $z∈𝒮_V()$ and a vector
s_z ∈(𝒱_z)^-1,
we view $s_z$ as an element of $H^n(A_z,)$, and define
\[
\| s_z \|^2 = \left|      \int_{A_z (\C)} s_z \wedge \overline{s}_z  \right| .
\]
This defines a hermitian metric on $(𝒱)^-1$, and hence we obtain a hermitian line bundle
\begin{equation}\label{det bundle}
\det(\mathcal{V})    \in     \widehat{\Pic}(\mathcal{S}_V)  .
\end{equation}

\begin{proposition} \label{prop:full bundle compare}
Assume $n\ge 2$.
Recalling the exceptional divisor of Definition \ref{def:special exceptional} and the notation  \eqref{constant metrics},  we have the equality
\[
2  \widehat{\taut}_V
=  \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}
+  2  \widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V}
+  \det( \mathcal{V})    +   ( \mathrm{Exc}_V , C_2)
\]
in $\widehat{\Pic}(\mathcal{S}_V)$, where
\[
C_2 =  2  \log\left(   \frac{2 e^\gamma}{ D}  \right)   + (2-n)  \log(2\pi).
\]
In particular,  $\det(\mathcal{V}) \in \widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK )$ by  \eqref{three bundles}.
\end{proposition}

\begin{proof}
This is a restatement of Proposition 5.1.2 of [14], keeping in mind that the metric on
\[
\det(\Lie(A))^{-1} = \omega^\mathrm{Hdg}_{A / \mathcal{S}_V}
\]
used in [loc.~cit.] differs from \eqref{hodge metric} by a power of $2\pi$.
\end{proof}

It is an observation of Gross [18] that the hermitian line bundle $(𝒱)$
behaves in the generic fiber, for all arithmetic purposes, like the trivial bundle with a constant metric.
This observation was extended to integral models in
\S 5.3 of [14], whose results are the  basis of the following proposition.

\begin{proposition}\label{prop:gross numerical}
The Chern form of   $\det(\mathcal{V})$ is identically $0$.
If  $n>2$ then, up to numerical equivalence,
\[
\det(\mathcal{V})  =   ( 0 , C_3 ),
\]
\[
C_3 = (4-2n) h_\kk^\mathrm{Falt} + \log( 4\pi^2 D).
\]
\end{proposition}

\begin{proof}
Fix an integer  $N\ge 3$.
As in  Theorem 1 of  [18],  the $\C$-vector space $H^0 (  \mathcal{S}_V(N)_{/\C}  , \det(\mathcal{V}) )$ has dimension $1$,  and the norm of  any  global  section  is a locally constant function on     $\mathcal{S}_V(N)(\C)$. The vanishing of the Chern form follows.
Moreover, one can choose a nonzero   global section $t$  defined over a finite  Galois extension $\kk'/\kk$,
and then for every $\kk$-algebra embedding
$\sigma:\kk' \to \C$ the norm   $\| t^\sigma  \|$ must again be  locally constant.

If $n>2$ then\footnote{The assumption $n>2$ is mistakenly omitted in Theorem 5.3.1 of [14], whose proof requires that  $\mathcal{M}^\Pap_{(n-1,1)}$ has geometrically normal fibers.
The normality is a theorem of  Pappas  when $n>2$, but is false when $n=2$.}
Theorem 5.3.1 of [14] allows us to  choose $t$  so that it extends to
a  nowhere vanishing section
\[
t  \in H^0 \big(  \mathcal{S}_V(N)_{/\co_{\kk'}[1/N] }  , \det(\mathcal{V}) \big) .
\]
Setting $d=[\kk':\kk]$ and taking the  tensor product of all $\Gal(\kk'/\kk)$-conjugates of $t$, we obtain a section
\[
s \in H^0( \mathcal{S}_V(N) ,  \det(\mathcal{V})^{\otimes d})
\]
such that $\mathrm{div}(s) =0$, and such that  $-\log\| s\|$  is  locally constant.
\[
c_X =  -\log\| s\|^2
\]
denote its  value on the connected component  $X \subset \mathcal{S}_V(N)_{/\C}$.

Fix a finite Galois extension  $L/\kk$ contained $\C$ large enough that every component
$X \subset \mathcal{S}_V(N)_{/\C}$ is defined over $L$.
We may further enlarge $L$ to assume that each $X$ admits an $L$-point
\[
x \in X(L) \subset \mathcal{S}_V(N)(L)
\]
that extends to
\[
\underline{x}  : \Spec(  \co_L[1/N] )  \to  \mathcal{S}_V(N).
\]
For example, start by fixing a complex point  $x\in X(\C)$ corresponding to a pair $(A_0,A)$ for which $A$ has complex multiplication.  Then  enlarge $L$ so that both $A_0$ and $A$ (along with their level $N$-structures)  are defined over $L$ and have everywhere good reduction.

Applying Corollary 5.3.2 and Proposition 5.3.3 of [14]  to $\underline{x}$, we find
\[
\frac{-1}{ [ L:\kk] }  \sum_{ \sigma \in \Gal(L/\kk) }  \log\| s_{x^\sigma} \|^2
=  d  C_3
\]
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$, and hence
\[
\frac{1}{ [ L:\kk] }  \sum_{ \sigma \in \Gal(L/\kk) }  c_{X^\sigma} =  d  C_3
\]
up to the same ambiguity.

We have now shown that the average value of $-\log\|s\|^2$ over any $\Aut(\C/\kk)$ orbit of connected components  in $\mathcal{S}_V(N)(\C)$ is $d C_3$,
up to a $\Q$-linear combination of $\{ \log(p) : p \mid N\}$.  The  proposition follows from this, by the same argument used in the proof of Proposition \ref{prop:easy numerical}.
\end{proof}

\subsection{Comparison of hermitian line bundles}
%\label{ss:bundle comparisons}

We now come to the main results of \S \ref{s:special shimura bundles}.

The exceptional divisor of Definition \ref{def:special exceptional} is a recurring nuisance,
in part because it has nontrivial arithmetic intersection with  $_V$.
This will be explored more fully in \S \ref{ss:exceptional volume} below.
The following proposition allows us to avoid this nuisance by slightly modifiying $_V$, and also clarifies the relation between the second and third line bundles in \eqref{three bundles}.

\begin{theorem}\label{thm:taut-hodge compare}
Assume $n\ge 2$.
Recalling the notation \eqref{constant metrics},  the hermitian line bundle
\[
\widehat{\tautmod}_V \define 2 \widehat{\taut}_V - (\mathrm{Exc}_V,0)  \in \widehat{\Pic}( \bar{\mathcal{S}}_V, \mathscr{D}_\BKK )
\]
enjoys the following properties.
\begin{enumerate}
\item
Every irreducible component $E\subset\mathrm{Exc}_V$   satisfies
\[
(E,0) \cdot \widehat{\tautmod}_V =0
\]
in $ \widehat{\mathrm{CH}}^2( \bar{\mathcal{S}}_V , \mathscr{D}_\BKK )_\Q$, as well as the height relation
\[
\mathrm{ht}_{\widehat{\tautmod}_V  } (E)   =0 .
\]
The same equalities hold if we replace $\widehat{\tautmod}_V$ with $\widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}$ or  $\widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V}$.
\item
We have the equality of Chern forms
\[
\chern(   \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}   ) =2 \chern( \widehat{\taut}_V ) =
\chern( \widehat{\tautmod}_V ) .
\]
\item
If $n>2$ then, up to numerical equivalence,
\[
\widehat{\tautmod}_V
\widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}   +    ( 0  ,  C_0(n) )
\in \widehat{\mathrm{Pic}}(\mathcal{S}_V) ,
\]
where $C_0(n)$ is the constant of Theorem \ref{thm:intro main}.  In particular, up to numerical equivalence,
\[
2 \widehat{\taut}_V
\widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}   +    ( \mathrm{Exc}_V  ,  C_0(n) ) .
\]
\end{enumerate}
\end{theorem}

\begin{proof}
For (1),  they key observation is that  the abelian scheme $A \to \mathcal{S}_V$ is a pullback  via the vertical arrow on the right in \eqref{special exceptional}.
In particular,  $\omega^\mathrm{Hdg}_{A/\mathcal{S}_V}$ is isomorphic to the pullback of
\[
\omega^\mathrm{Hdg}_{A/\mathcal{M}^\Pap_{(n-1,1)}} \in \Pic ( \mathcal{M}^\Pap_{(n-1,1)}  ) .
\]
If  $\mathcal{M}^\Pap_{(n-1,1)}$ were a scheme we could  trivialize this latter line bundle  over a Zariski open neighborhood of  the ($0$-dimensional) singular locus.
This would pull-back to a trivialization of $\omega^\mathrm{Hdg}_{A/ \mathcal{S}_V}$  over  an open neighborhood of   $\mathrm{Exc}_V$, and in particular over an open neighborhood of
any irreducible component $E\subset \mathrm{Exc}_V$.

To account for the stackiness, simply fix an  integer $N\ge 3$
and apply the same reasoning with level $N$-structure  to see that
$\omega^\mathrm{Hdg}_{A/\mathcal{S}_V(N)}$ is trivial in some Zariski open neighborhood of
\[
E(N)  = E \times_{ \mathcal{S}_V} \mathcal{S}_V(N).
\]
This implies  the arithmetic intersection formula
( E(N) , 0 )  \cdot  \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V(N)} =0
and varying $N$ proves
\[
( E , 0 )  \cdot  \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V} =0 .
\]
The line bundle \eqref{det bundle}  is also a pullback via the vertical arrow on the right in \eqref{special exceptional},   hence the same argument  shows
\[
(E,0) \cdot \det(\mathcal{V}) =0.
\]

The proof of Proposition \ref{prop:easy numerical} shows that, for any $N\ge 3$,
there is a positive multiple of
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0/\mathcal{S}_V(N)}
\in \widehat{\mathrm{CH}}^1( \bar{ \mathcal{S}} _V (N) , \mathscr{D}_\BKK )
\]
that can be represented by a purely archimedean  arithmetic divisor $(0,g)$.
Any such arithmetic divisor satisfies
(E(N) ,0 ) \cdot (0,g) =0,
and  varying $N$ shows that
\[
(E ,0 ) \cdot \widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V} =0.
\]

Rewriting the  relation of Proposition \ref{prop:full bundle compare} as
\begin{equation}\label{Kalt}
\widehat{\tautmod}_V
=  \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}
+  2  \widehat{\omega}^\mathrm{Hdg}_{ A_0 / \mathcal{S}_V}
+  \det( \mathcal{V})    +   ( 0 , C_2)  ,
\end{equation}
we have shown that the right hand side  has trivial arithmetic intersection with $(E,0)$, and hence so does the left hand side.  This proves the first equality of (1).  The second  is a formal consequence of this and  \eqref{height degree}.

The first equality of (2) follows  from \eqref{Kalt},
as the Chern forms of the final three terms on the right vanish by    Lemma \ref{lem:numerical basics}, Proposition \ref{prop:easy numerical}, and  Proposition \ref{prop:gross numerical}.
The second equality of (2) is clear from the definitions.

Claim (3) also follows from \eqref{Kalt}, using  Proposition  \ref{prop:easy numerical},  Proposition \ref{prop:gross numerical},   and the equality
$ C_0 (n) = 2C_1 +  C_2+ C_3$.
\end{proof}

\begin{remark}\label{rem:taut-hodge volume}
If $n>2$ then part (3) of Theorem \ref{thm:taut-hodge compare} implies
\begin{align*}
2^n \cdot  \widehat{\mathrm{vol}} (  \widehat{\tautmod}_V   )
& = \widehat{\mathrm{vol}} \big(  \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}
+    ( 0 ,  C_0(n)  ) \big)  \\
& =
\widehat{\mathrm{vol}} \big(  \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}   \big) +  n C_0(n)    \int_{\mathcal{S}_V(\C) }  \chern( \widehat{\omega}^\mathrm{Hdg}_{ A / \mathcal{S}_V}    )^{n-1}
\end{align*}
where we have used  Lemma \ref{lem:numerical basics} for the first equality, and Lemma \ref{lem:trivial volume shift} for the second.
In Proposition \ref{prop:taut-hodge strong volume} we will show that this equality also holds when $n=2$.
\end{remark}

\subsection{Volume of the exceptional divisor}
\label{ss:exceptional volume}

In this subsection we assume $n ≥2$, and fix an irreducible ($=$ connected) component $E$ of the  exceptional divisor  of Definition \ref{def:special exceptional}.
Recalling the notation \eqref{constant metrics}, our goal is to compute  the arithmetic volume of
\[
(E,0) \in \widehat{\Pic}( \bar{\mathcal{S}}_V ).
\]

By definition of the exceptional divisor, there is a commutative diagram with cartesian squares
\begin{equation*}%\label{singular point}
\xymatrix{
{E} \ar[r] \ar[d] & { e } \ar[d] \\
{\mathrm{Exc}}_V  \ar[r] \ar[d] & {  \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathrm{Sing}_{(n-1,1)} }  \ar[d] \\
{\mathcal{S}_V}  \ar[r]  & {  \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{(n-1,1)}  }
\end{equation*}
in which $e$ is a connected component of the $0$-dimensional reduced and irreducible $_$-stack
$ℳ_(1,0) ×__ Sing_(n-1,1)$.  In particular, $e$ is supported in a single characteristic $p|D$,  and admits  a presentation as a stack quotient
\[
e \iso \Delta \backslash \Spec(\F_\mathfrak{p}'),
\]
in which $𝔭 ⊂_$ is the prime above $p$,   $_𝔭'$ is a finite extension of  $_𝔭=_/𝔭$, and $Δ$ is a finite group acting on  $_𝔭'$.
Define a rational number
\[
m_E \define \sum_{ z\in e(\F_\mathfrak{p}^\alg) } \frac{ 1 }{ |\Aut(z)|} = \frac{[  \F_\mathfrak{p}' : \F_\mathfrak{p}   ]  }{ |\Delta| }    .
\]

\begin{proposition}\label{prop:projective intersection}
The iterated intersection
\[
\widehat{\taut}_V^{-1} \cdots  \widehat{\taut}_V^{-1}  \cdot (E,0) \in \widehat{\mathrm{CH}}^n(\bar{\mathcal{S}}_V , \mathscr{D}_\BKK)
\]
has arithmetic degree $\widehat{m}_E = m_E \log(p)$
\end{proposition}

\begin{proof}
After Remark \ref{rem:singular description},  one might expect that $E$ is isomorphic to the projective space of hyperplanes in an $n$-dimensional vector space.
In particular, there should be a universal hyperplane $\mathcal{F} \subset \co_E^n$, with the property that the restriction of  $\taut_V^{-1}$ to $E$ is isomorphic to  $\co_E^n /\mathcal{F}$.

The obstruction to this being true is due  to stacky issues, which can be removed by fixing an integer $N\ge 3$ prime to $p$ and adding level $N$-structure to the universal pair $(A_0,A)$ over $e$.
Let $e(N) \to e$ be the scheme classifying such level structures, and consider the cartesian diagram
(this is the definition of the upper left corner)
\[
\xymatrix{
{ E(N) } \ar[r]\ar[d]  & {  e(N) }  \ar[d] \\
{ E } \ar[r] & { e .}
\]
The scheme $e(N)$, being a reduced scheme finite over $\F_\mathfrak{p}$,  is a disjoint union of finitely many spectra of finite extensions of $\F_\mathfrak{p}$.

Fix a connected component $E' \subset E(N)$, and let
\[
\Spec(\F_\mathfrak{p}') = e' \subset e(N)
\]
be the connected component  below it.
As in Remark \ref{rem:singular description}, $E'$ is precisely the projective space over  $e'$ classifying hyperplanes  $ \mathcal{F} \subset \Lie(A|_{e'})$.
Moreover,  after fixing a trivialization  $\Lie(A_0|_{e'}) \iso \F_\mathfrak{p}'$, the isomorphism  \eqref{dual taut} identifies
\[
\taut_V^{-1} |_{E'} =      \Lie(A|_{E'}) / \mathcal{F}  \in \Pic(E') \iso \mathrm{CH}^1(E')
\]
where $\mathcal{F} \subset \Lie(A|_{E'})$ is the universal hyperplane.
A routine exercise then shows that the iterated intersection
\[
( \taut_V^{-1} |_{E'}) \cdots (\taut_V^{-1} |_{E'}) \in \mathrm{CH}^{n-1}(E') \iso \Z
\]
is  represented by the cycle  class of  any $\F_\mathfrak{p}'$-valued point of $E'$.
In other words, the cycle class of any section to $E' \to e'$.

Now allow  the connected component $E' \subset E(N)$ to vary.  If we fix
any section to $E(N) \to e(N)$, and use it to view $e(N)$ as a $0$-cycle on $E(N)$, then
\[
e(N) = ( \taut_V^{-1} |_{E(N)}) \cdots (\taut_V^{-1} |_{E(N)}) \in \mathrm{CH}^{n-1}(E(N)) .
\]
This implies  the arithmetic intersection formula
\[
( e(N) , 0 ) = \widehat{\taut}_V^{-1} \cdots  \widehat{\taut}_V^{-1} \cdot ( E(N) ,  0 )
\in \widehat{\mathrm{CH}}^n( \bar{\mathcal{S}}_V(N) , \mathscr{D}_\BKK )_\Q.
\]

Finally, recalling \eqref{degree normalization}, we deduce
\[
\widehat{\deg}  \big(  \widehat{\taut}_V^{-1} \cdots  \widehat{\taut}_V^{-1} \cdot ( E ,  0 )  \big)
\frac{ \# e(N) (\F_\mathfrak{p}^\alg) }{d_N}  \log(p)
\sum_{ z\in e(\F_\mathfrak{p}^\alg) } \frac{\log(p)}{ |\Aut(z)|}
\]
up to a $\Q$-linear combination of $\{ \log(p) : p\mid N\}$.  Varying $N$ completes the proof.
\end{proof}

\begin{corollary}\label{cor:exceptional volume}
The hermitian line bundle $(E,0) \in \widehat{\Pic}(\bar{\mathcal{S}}_V )$ has arithmetic volume
\[
\widehat{\vol} ( E,0) = (-2)^{n-1} \cdot  \widehat{m}_E .
\]
\end{corollary}

\begin{proof}
Part (1) of Theorem \ref{thm:taut-hodge compare} implies the second equality in
\[
(E,0)\cdot (E,0)  = (\mathrm{Exc}_V,0) \cdot (E,0) = 2 \widehat{\taut}_V  \cdot (E,0)
\in \widehat{\mathrm{CH}}^2(\bar{\mathcal{S}}_V , \mathscr{D}_\BKK)_\Q.
\]
This implies the iterated  intersection formula
\[
(E,0)\cdots (E,0)
=  2^{n-1} \widehat{\taut}_V \cdots \widehat{\taut}_V   \cdot (E,0)
\in \widehat{\mathrm{CH}}^n(\bar{\mathcal{S}}_V , \mathscr{D}_\BKK)_\Q  ,
\]
and  claim follows using  Proposition \ref{prop:projective intersection}.
\end{proof}

\section{Kudla-Rapoport divisors at split primes}
\label{s:KR divisors}

Assume  $n > 2$, and let $𝒮_V$ be the Shimura variety \eqref{moduli inclusion} associated to a $$-hermitian space  $V$  of signature $(n-1,1)$ containing a self-dual lattice.
Fix a prime $p$ split in $$,  and factor
\[
p\co_\kk = \mathfrak{p} \overline{\mathfrak{p}}.
\]
We explain how the Kudla-Rapoport divisor  $𝒵_V(p) →𝒮_V$ of Definition \ref{def:KR}
is  related to  a Shimura variety $𝒮_V'$   with $V'$ of signature $(n-2,1)$.

\subsection{Statement of the results}
\label{ss:split divisors main}

Let $V'$  denote the $$-hermitian space of signature $(n-2,1)$ whose local invariants satisfy
\begin{equation}\label{V'}
\mathrm{inv}_\ell(V')  = ( p, -D)_\ell \cdot \mathrm{inv}_\ell(V)
\end{equation}
for all places $ℓ≤∞$.  Equivalently,  $V'$ is the orthogonal complement to the $$-span of any
$x∈V$ of hermitian norm $⟨x,x⟩=p$.

\begin{lemma}\label{lem:lower hermitian}
The hermitian space $V'$  admits a self-dual $\co_\kk$-lattice.
\end{lemma}

\begin{proof}
Suppose $\ell$ is a rational prime unramified in $\kk$.  If $\ell \neq p$ then $p\in \Z_\ell^\times$, and hence is a norm from the unramified extension $\kk_\ell$.  If $\ell =p$ then $p$ is  again a norm from $\kk_\ell \iso \Q_p \times \Q_p$.    In either case, $(p,-D)_\ell=1$, and so  the local invariants of $V$ and $V'$ agree at all unramified primes.

In general, a $\kk$-hermitian space  over $\kk$ admits a self-dual $\co_\kk$-lattice if and only  if it has local invariant $1$ at every prime  unramified  in $\kk$.
As $V$ satisfies this condition by hypothesis, so does $V'$.
\end{proof}

Lemma \ref{lem:lower hermitian} allows us to form the $_$-stacks
\[
\mathcal{S}_{V'} \subset  \bar{\mathcal{S}}_{V'} \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}_{(n-2,1)}
\]
analogous to
\[
\mathcal{S}_V \subset \bar{\mathcal{S}}_V\subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}_{(n-1,1)},
\]
but in one dimension lower.  The stack $𝒮_V'$ is  endowed with its own  hermitian line bundle $_V'$  as in \eqref{metrized taut},  its own exceptional divisor $Exc_V'$  as in Definition \ref{def:special exceptional}, and its own universal pair $(A_0',A')$ of polarized abelian schemes with $_$-actions.

The  following theorems, whose proofs will occupy the remainder of \S \ref{s:KR divisors}, lie at the core of the inductive arguments of \S \ref{s:volumes}.

\begin{theorem}\label{thm:height descent 1}
If we set
\[
\widehat{\tautmod}_V  = 2 \widehat{\taut}_V - ( \mathrm{Exc}_V , 0) \in \widehat{\Pic} ( \bar{\mathcal{S}}_V , \mathscr{D}_\BKK )
\]
as in Theorem \ref{thm:taut-hodge compare},  and similarly with $V$ replaced by $V'$, then
\[
\int_{ \mathcal{Z}_V(p) (\C)  }  \chern( \widehat{\tautmod}_V)^{n-2}
=  ( p^{n-1}+1)  \vol_\C( \widehat{\tautmod}_{V'} ).
\]
Moreover, there is a rational number $a(p)$  such that
\[
\frac{  \mathrm{ht}_{ \widehat{\tautmod}_V }  (\bar{\mathcal{Z}}_V(p))  } {    p^{n-1}+1    }
=    \widehat{\vol} (\widehat{\tautmod}_{V'} )  + a(p) \log(p) .
\]
\end{theorem}

\begin{theorem}\label{thm:height descent 2}
There is a  rational number $b(p)$ such that
\begin{align*}
\frac{ \mathrm{ht}_{\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}}  (\bar{\mathcal{Z}}_V(p))  }{  p^{n-1}+1   }
& =     \widehat{\vol} ( \widehat{\omega}^\mathrm{Hdg}_{A'/\mathcal{S}_{V'}} )   + b(p) \log(p)   \\
& \quad   +  (1-n)  \left(  \frac{L'(0,\eps)}{L(0,\eps)}   +\frac{\log(D)}{2}  \right)
\vol_\C(   \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}  )  .
\end{align*}
\end{theorem}

The proofs are rather long, so we summarize now the key steps.
The central idea  is to make precise the  impressionistic relation
\[
\mathcal{Z}_V(p)  ``="  (p^{n-1}+1)  \cdot \mathcal{S}_{V'}   ,
\]
by  decomposing
\begin{equation}\label{flavor decomp}
\mathcal{Z}_V(p)_{/\co_\kk[1/p]}  =
\mathcal{U}_0  \sqcup   \mathcal{U}_{\mathfrak{p}}  \sqcup   \mathcal{U}_{\overline{\mathfrak{p}}}
\end{equation}
as a disjoint union of open and closed substacks (Proposition \ref{prop:split divisor flavors} below).
The stacks on the right hand side are related to $𝒮_V'$ by  closed immersions
\[
i_\mathfrak{p} :  \mathcal{S}_{V' / \co_\kk[1/p]}   \to \mathcal{U}_{\mathfrak{p}}  ,\qquad
i_{\overline{\mathfrak{p}}} :  \mathcal{S}_{V' / \co_\kk[1/p]}   \to   \mathcal{U}_{\overline{\mathfrak{p}}},
\]
and  by a diagram
\[
\xymatrix{
{  \mathcal{S}_{V' / \co_\kk[1/p]}   }   & { \mathcal{T}_{V'} } \ar[l] \ar[r]^{i_0} &  { \mathcal{U}_0   }
\]
in which the leftward arrow is a finite \'etale surjection of degree $p^n-1-1$, and the rightward arrow is a closed immersion.
See  \eqref{tau cover} for the definition of the stack $ 𝒯_V' $.

Recalling the exceptional locus of Definition \ref{def:special exceptional},
denote by
\[
\mathcal{S}^{\nonexc}_V =  \mathcal{S}_V \smallsetminus \mathrm{Exc}_V
\]
the (open) nonexceptional locus of $𝒮_V$,  set
\[
\mathcal{Z}^{\nonexc}_{V}(p) =  \mathcal{Z}_{V}(p)  \times_{\mathcal{S}_{V}} \mathcal{S}^\nonexc_V ,
\]
and make the same definitions with $V$ replaced by $V'$.
\[
\mathcal{U}^\nonexc_\square =  \mathcal{U}_\square \cap \mathcal{Z}^\nonexc_V(p)
\]
for $□∈{ 0, 𝔭 ,𝔭 }$, and
\[
\mathcal{T}^\nonexc_{V'}  = \mathcal{T}_{V'} \times_{\mathcal{S}_{V'}} \mathcal{S}_{V'}^\nonexc .
\]

We will show that the closed immersions $i_0$, $i_𝔭$, and $i_𝔭$ are close to being isomorphisms.  More precisely,     $i_0$  restricts to an isomorphism
\[
i_0 : \mathcal{T}^\nonexc_{V'}    \iso \mathcal{U}^\nonexc_0,
\]
$i_𝔭$ and $i_𝔭$ restrict to isomorphisms
\[
i_\mathfrak{p} :  \mathcal{S}^\nonexc_{V' / \co_\kk[1/p]}   \iso \mathcal{U}^\nonexc_{\mathfrak{p}}
\quad \mbox{and}\quad
i_{\overline{\mathfrak{p}}} :  \mathcal{S}^\nonexc_{V' / \co_\kk[1/p]}   \iso   \mathcal{U}^\nonexc_{\overline{\mathfrak{p}}}.
\]
After taking  compactifications into account,  both theorems above will follow easily from these isomorphisms.
Note that it suffices to work only over the nonexceptional locus, as part (1) of Theorem \ref{thm:taut-hodge compare} guarantees that the hermitian line bundles in questions have trivial arithmetic intersection with all components of the exceptional divisor (which is why we work with $𝒦_V$ in Theorem \ref{thm:height descent 1} instead of $ℒ_V$).

%If $F$ is an algebraically closed field, it follows from the local model calculations of Pappas [45] that an $F$-point
% (A_0,A) \in \mathcal{S}_{V} (F)
%lies in  nonexceptional locus  if and only if the action of $\sqrt{-D} \in \End(A)$ on $\Lie(A)$ is nonzero.
%See \S 2.3, and especially Theorem 2.3.2, of [13] for a more thorough discussion.

\subsection{Decomposing the Kudla-Rapoport divisor}

The decomposition \eqref{flavor decomp} is a geometric reflection of a result of  linear algebra.

As in  \eqref{hermitian hom},  choose an isomorphism
\[
V \iso \Hom_\kk(W_0,W)
\]
in which $(W_0,h_0)$ and $(W,h)$ are  relevant hermitian spaces of signatures $(1,0)$ and $(n-1,1)$.
Fix self-dual $_$-lattices $𝔞_0 ⊂W_0$ and $𝔞 ⊂W$.
Any  vector
\[
x\in \Hom_{\co_\kk}(\mathfrak{a}_0, \mathfrak{a}) \subset V
\]
with $⟨x,x⟩=p$  determines an orthogonal decomposition
\[
W = \tilde{W}_0 \oplus W'
\]
with  $W̃_0 = x(W_0)$, and a corresponding decomposition
\[
V = \kk x \oplus V'
\]
with $V'=  _(W_0 ,W')$.

\begin{lemma}\label{lem:lattice trifurcation}
The $\co_\kk$-lattice
\tilde{\mathfrak{a}}_0=\mathfrak{a} \cap \tilde{W}_0
satisfies exactly one of
\[
\tilde{\mathfrak{a}}_0 = x(\mathfrak{a}_0) ,\qquad \tilde{\mathfrak{a}}_0 =\mathfrak{p}^{-1} x(\mathfrak{a}_0), \qquad
\tilde{\mathfrak{a}}_0=\overline{\mathfrak{p}}^{-1} x(\mathfrak{a}_0).
\]
\end{lemma}

\begin{proof}
Use $x$ to identify $W_0 = \tilde{W}_0$, and hence $\mathfrak{a}_0 \subset \tilde{\mathfrak{a}}_0 \subset \mathfrak{a}$.
Recalling the symplectic forms
\[
e_0: W_0 \times W_0 \to \Q ,\qquad e: W \times W \to \Q
\]
of   \eqref{symplectic}, the relation  \eqref{basic hom hermitian}  implies that  $e|_{W_0} = p \cdot e_0$.
The inclusion
\[
e_0 ( p \tilde{\mathfrak{a}}_0 , \mathfrak{a}_0)
= e ( \tilde{\mathfrak{a}}_0 , \mathfrak{a}_0) \subset e( \mathfrak{a} , \mathfrak{a}) =\Z,
\]
together with the self-duality of $\mathfrak{a}_0$ under $e_0$,
shows that $p \tilde{\mathfrak{a}}_0 \subset \mathfrak{a}_0$.   If equality held we would have
\[
e_0( \mathfrak{a}_0 , \mathfrak{a}_0 )  = p^{-1}  e(  \mathfrak{a}_0 , \mathfrak{a}_0)
\subset p e(  \tilde{\mathfrak{a}}_0,\tilde{\mathfrak{a}}_0)\subset p e(\mathfrak{a},\mathfrak{a}) \subset  p \Z,
\]
contradicting  $\mathfrak{a}_0$ being self-dual under $e_0$.

As $\tilde{\mathfrak{a}}_0$ is $\co_\kk$-stable with
$\mathfrak{a}_0 \subset \tilde{\mathfrak{a}}_0 \subsetneq p^{-1} \mathfrak{a}_0$, it is
$\mathfrak{a}_0$, $\mathfrak{p}^{-1} \mathfrak{a}_0$, or $\overline{\mathfrak{p}}^{-1} \mathfrak{a}_0$.
\end{proof}

\begin{proposition}\label{prop:split divisor flavors}
Let  $(A_0,A,x)$ be the universal triple over $\mathcal{Z}_V(p)_{/\co_\kk[1/p]}$, so that
x\in \Hom_{\co_\kk}(A_0,A)
satisfies $\langle x,x\rangle =p$.
There is a decomposition \eqref{flavor decomp} into open and closed substacks in which
\begin{itemize}
\item
$\mathcal{U}_0$ is the locus of points where $\ker(x)=0$,
\item
$\mathcal{U}_\mathfrak{p}$ is the locus of points where $\ker(x) = A_0[\mathfrak{p}]$,
\item
$ \mathcal{U}_{\overline{\mathfrak{p}}}$ is the locus of points where $\ker(x) = A_0[ \overline{\mathfrak{p}}]$.
\end{itemize}
\end{proposition}

\begin{proof}
Recalling \eqref{KR hermitian},  the relation $\langle x,x\rangle =p$ implies that  multiplication-by-$p$ factors as
\[
A_0 \map{x} A \iso A^\vee \map{x^\vee} A_0^\vee \iso A_0,
\]
and so  $\ker(x) \subset A_0[p]$.   Both of these group schemes are finite \'etale over $\mathcal{Z}_V(p)_{/\co_\kk[1/p]}$, which implies that each of $\mathcal{U}_0$, $\mathcal{U}_\mathfrak{p}$, and $\mathcal{U}_{\overline{\mathfrak{p}}}$ is open and closed.
It only remains to prove that every geometric point $s \to \mathcal{Z}_V(p)_{/\co_\kk[1/p]}$ is contained in one of them.

Abbreviate $T_0= T_p(A_{0 s})$ and $T=T_p(A_s)$.
Applying the snake lemma to the diagram
\[
\xymatrix{
0 \ar[r]  & {T_0 } \ar[r] \ar[d]_{x_s} & { T_0\otimes \Q_p } \ar[r] \ar[d]_{x_s} & { A_{0 s}[p^\infty]}  \ar[r] \ar[d]_{x_s} & 0  \\
0 \ar[r]  & {T  } \ar[r]  & { T \otimes \Q_p } \ar[r]   & { A_{s}[p^\infty]}  \ar[r] & 0  ,
\]
and using the vertical arrow on the left   to identify $T_0 \subset T$, we find that
\[
\ker(x_s) \iso \tilde{T}_0 / T_0,
\]
where $\tilde{T}_0 = T \cap  T_{0 \Q_p}$.

After fixing an isomorphism $\Z_p \iso \Z_p(1)$ of \'etale sheaves on $s$,  there are unique $\co_{\kk,p}$-valued hermitian forms $h_0$ and $h$ on $T_0$ and $T$, respectively, related to the  Weil pairings $e_0$ and $e$  by \eqref{symplectic}.  Thus we may apply  Lemma \ref{lem:lattice trifurcation} with $\mathfrak{a}_0$ and $\mathfrak{a}$ replaced by $T_0$ and $T$, to see that $\tilde{T}_0$ must be one of $T_0$, $\mathfrak{p}^{-1} T_0$, or $\overline{\mathfrak{p}}^{-1} T_0$.  These three cases correspond to $\ker(x_s)$ being  trivial,  $A_{0s}[\mathfrak{p}]$, or $A_{0s}[\overline{\mathfrak{p}}]$.
\end{proof}

\subsection{Analysis of $\mathcal{U}_\mathfrak{p}$}

In this subsection we study the structure of the substack $𝒰_𝔭$  of \eqref{flavor decomp},
and make explicit its relation to $𝒮_V'$.
The analogous analysis of $𝒰_𝔭$ is obtained by replacing $𝔭$ by $𝔭$ everywhere.

Return to the situation of Lemma \ref{lem:lattice trifurcation}, so that
$x∈__(𝔞_0,𝔞)$ with $⟨x,x⟩=p$ determines an orthogonal decomposition
\[
W = \tilde{W}_0 \oplus W' .
\]
Set $𝔞̃_0 = 𝔞∩W̃_0$.

\begin{lemma}\label{lem:pi linear algebra}
If  $\tilde{\mathfrak{a}}_0 =\mathfrak{p}^{-1} x(\mathfrak{a}_0)$,  there is an orthogonal  decomposition
\[
\mathfrak{a} = \tilde{\mathfrak{a}}_0 \oplus \mathfrak{a}' \subset W
\]
in which $\mathfrak{a}' = \mathfrak{a} \cap W'$.
\end{lemma}

\begin{proof}
If we use $x$ to identify $W_0 = \tilde{W}_0$,  the  assumption  $\tilde{\mathfrak{a}}_0=\mathfrak{p}^{-1}\mathfrak{a}_0$ implies that  $\tilde{\mathfrak{a}}_0$ is self-dual with respect to the hermitian form $h|_{ \tilde{W}_0 }= p h_0$.  The  desired decomposition  then  follows by elementary linear algebra.
\end{proof}

The lemma suggests that if $(A_0,A,x) ∈𝒰_𝔭(S)$ for  an $_[1/p]$-scheme $S$,  then $x:A_0 →A$ should determine an $_$-linear splitting
\begin{equation}\label{pi product}
A =  \tilde{A}_0 \times A'
\end{equation}
of principally polarized abelian schemes.   Indeed, this is the case.
If we set
\[
\tilde{A}_0 =  A_0 / A_0[\mathfrak{p}]
\]
and recall that $(x) = A_0[𝔭]$,  the morphism $x :A_0 →A$ factors  as
\[
A_0 \to \tilde{A}_0 \map{y} A
\]
for some $y∈__(Ã_0, A) $ satisfying   $⟨y,y⟩=1$.
In other words,  the composition
\[
\tilde{A}_0 \map{y} A \iso A^\vee \map{y^\vee} \tilde{A}_0^\vee \iso \tilde{A}_0
\]
is the identity.  This implies that the composition
\[
A \iso A^\vee \map{y^\vee} \tilde{A}_0^\vee \iso \tilde{A}_0 \map{y} A
\]
is a Rosati-fixed idempotent  in $__(A)$,  and $A$ admits  a unique  splitting \eqref{pi product}
of principally polarized abelian schemes over $S$ such that this idempotent is the projection to the first factor.

Apply the above construction to the universal triple $(A_0,A,x)$ over  $𝒰_𝔭$, and
recall that $A$ comes equipped with an $_$-stable  hyperplane  $ℱ ⊂(A)$ satisfying Kr\"amer's signature condition.
Using the  decomposition
\[
\Lie(A) =  \Lie(\tilde{A}_0) \oplus \Lie(A')
\]
of vector bundles on $S$, denote by
the largest  closed substack over which    $(Ã_0 ) ⊂ℱ$.

\begin{proposition}\label{prop:i_p construction}
There is a canonical isomorphism
\[
i_\mathfrak{p} : \mathcal{S}_{V'/\co_\kk[1/p]} \iso \mathcal{U}_\mathfrak{p}^\dagger.
\]
\end{proposition}

\begin{proof}
As above, let $S$ be an $\co_\kk[1/p]$-scheme.
Given a point $(A_0,A,x) \in \mathcal{U}_\mathfrak{p}^\dagger(S)$,  the splitting \eqref{pi product} determines
\[
A' \in \mathcal{M}_{( n-2,1)}(S),
\]
where we have endowed  $A'$ with the $\co_\kk$-stable hyperplane
\[
\mathcal{F}' = \mathcal{F} / \Lie(\tilde{A}_0 )  \subset \Lie(A)/  \Lie(\tilde{A}_0 ) \iso \Lie(A').
\]
satisfying Kr\"amer's condition.
The pair $(A_0,A')$ defines an $S$-point of
\[
\mathcal{S}_{V'} \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}_{( n-2,1)},
\]
and we have now constructed a morphism
\begin{equation}\label{i_p inverse}
\mathcal{U}^\dagger_\mathfrak{p} \map{  (A_0,A,x) \to (A_0,A')}  \mathcal{S}_{V'/\co_\kk[1/p]}.
\end{equation}

Conversely, start with an $S$-point
\[
(A'_0 , A') \in \mathcal{S}_{V'}(S) .
\]
First define  elliptic curves $A_0 = A_0'$ and  $\tilde{A}_0 = A_0 /A_0[\mathfrak{p}]$.
Then define an abelian scheme $A$ by \eqref{pi product}, and endow $A$ with  its product principal polarization and product $\co_\kk$-action.
Recalling that $A'$ comes equipped with a hyperplane $\mathcal{F}' \subset \Lie(A')$ satisfying Kr\"amer's  condition, we endow $A$ with the hyperplane
\[
\mathcal{F} = \Lie( \tilde{A}_0 ) \oplus \mathcal{F}' \subset \Lie(A).
\]
It is easy to check that $A$, with its extra data, defines an $S$-point of $\mathcal{M}_{(n-1,1)}$, and that
$(A_0,A)$ defines an $S$-point of the open and closed substack
\[
\mathcal{S}_V \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}_{(n-2,1)}.
\]
If we define  $x\in \Hom_{\co_\kk}(A_0,A)$  as the composition
\[
A_0 \to \tilde{A}_0  \hookrightarrow \tilde{A}_0 \times A' =A,
\]
where the first arrow is the quotient map,  then   $\langle x ,x \rangle =p$ and $\ker(x) =A_0[\mathfrak{p}]$.
The triple   $(A_0,A,x)$ defines an  $S$-point of $\mathcal{U}^\dagger_\mathfrak{p}$, and the morphism
\[
\mathcal{S}_{V'/\co_\kk[1/p]} \map{ (A'_0 , A')  \to  (A_0,A,x)  }    \mathcal{U}^\dagger_\mathfrak{p}
\]
is inverse to \eqref{i_p inverse}.
\end{proof}

Proposition \ref{prop:i_p construction} gives us a  commutative diagram
\begin{equation}\label{i_p pullback}
\xymatrix{
{ \mathcal{S}_{V'/\co_\kk[1/p] } }   \ar[r]\ar[d] & { \mathcal{S}_{V/\co_\kk[1/p] } }  \ar[d] \\
{ \big(  \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-2,1) }  \big)_{/\co_\kk[1/p]} } \ar[r] & { \big(    \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-1,1) }  \big)_{/\co_\kk[1/p]}    }
\end{equation}
in which the top horizontal arrow is the composition
\[
\mathcal{S}_{V'/\co_\kk[1/p]} \map{i_\mathfrak{p}} \mathcal{U}^\dagger_\mathfrak{p} \hookrightarrow   \mathcal{Z}_V(p)_{/\co_\kk[1/p]}  \to \mathcal{S}_{V/\co_\kk[1/p]} ,
\]
and the bottom horizontal arrow sends  $(A_0' , A') ↦(A_0,A)$, where
\begin{equation}\label{i_p pullback explicit}
A_0 = A_0' , \quad \tilde{A}_0 = A_0 / A_0[\mathfrak{p}] , \quad A = \tilde{A}_0 \times A'.
\end{equation}
Both horizontal arrows are finite and unramified.

\begin{proposition}\label{prop:i_p pullbacks}
The homomorphism
\[
\widehat{\Pic}  (    \mathcal{S}_{V/\co_\kk[1/p]}  )_\Q  \to  \widehat{\Pic}  (    \mathcal{S}_{V'/\co_\kk[1/p]}  )_\Q
\]
induced by  \eqref{i_p pullback} sends
\[
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{S}_V } \mapsto \widehat{\omega}^\mathrm{Hdg}_{A'_0/ \mathcal{S}_{V'} } +  \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'} } ,
\]
where  $(A_0' , A')$ is the universal pair over $\mathcal{S}_{V'}$.  The same map also sends
\[
\widehat{\taut}_V \mapsto \widehat{\taut}_{V'}
\quad \mbox{and}\quad
(\mathrm{Exc}_V,0) \mapsto (\mathrm{Exc}_{V'} ,0 ).
\]
\end{proposition}

\begin{proof}
It follows from \eqref{i_p pullback explicit} that the top horizontal arrow in \eqref{i_p pullback}  sends
\[
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{S}_V } \mapsto
\widehat{\omega}^\mathrm{Hdg}_{\tilde{A}_0/ \mathcal{S}_{V'} }
+  \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'} }.
\]
As we have inverted $p$ on the base, the degree $p$ quotient map $A_0' \to \tilde{A}_0$ induces an isomorphism on Lie algebras, and hence an isomorphism
\[
\omega^\mathrm{Hdg}_{\tilde{A}_0/ \mathcal{S}_{V'} }  \iso  \omega^\mathrm{Hdg}_{ A'_0/ \mathcal{S}_{V'} } .
\]
This isomorphism does not respect the metrics  \eqref{hodge metric}, but one can easily check that
\[
\widehat{\omega}^\mathrm{Hdg}_{\tilde{A}_0/ \mathcal{S}_{V'} }
\widehat{\omega}^\mathrm{Hdg}_{ A'_0/ \mathcal{S}_{V'} } + (0 , - \log(p) )
\]
as elements of $\widehat{\Pic}( \mathcal{S}_{V'/\co_\kk[1/p]})$.  The correction term $(0 , - \log(p) )$ is torsion in the arithmetic Picard group, as
\[
2 ( 0 , -\log(p) ) = (\mathrm{div}(p) , - \log(p^2) ) = 0,
\]
and the first claim follows.

For the second claim, let $S$ be an $\co_\kk[1/p]$-scheme, fix an $S$-point
$(A_0' , A') \in \mathcal{S}_{V'}(S)$, and let $(A_0,A) \in \mathcal{S}_V(S)$ be its image under the top horizontal arrow in \eqref{i_p pullback}.
Tracing through the construction of $i_\mathfrak{p}$ in Proposition \ref{prop:i_p construction}, we see that
\[
\Lie(A) / \mathcal{F} \iso  \Lie(A') / \mathcal{F}'.
\]
It follows immediately from this and \eqref{dual taut} that $\taut_V|_S \iso \taut_{V'}|_S$.
At a complex point $s \in S(\C)$ we have a decomposition
\[
H_1(A_s , \Q) = H_1( \tilde{A}_{0s} ,\Q) \oplus H_1( A_s' , \Q),
\]
which induces an isometric embedding
\[
\Hom_\kk( H_1(A_{0s} ,\Q )  , H_1( A_s' , \Q) ) \subset \Hom_\kk( H_1(A_{0s} ,\Q )  , H_1( A_s , \Q) )
\]
of $\kk$-hermitian spaces.  Recalling \eqref{taut realization}, this isometric embedding restricts to the isomorphism
$\taut_{V',s} \iso \taut_{V,s}$ just constructed, which therefore preserves the metrics defined by \eqref{taut metric}.

Finally, we show that under the top horizontal arrow in \eqref{i_p pullback}, the exceptional divisor $\mathrm{Exc}_V \subset \mathcal{S}_V$ pulls back to the exceptional divisor $\mathrm{Exc}_{V'} \subset \mathcal{S}_{V'}$.
These  divisors are, by Definition \ref{def:special exceptional}, the pullbacks of the singular loci
\begin{align}
\mathcal{M}_{(1,0)} \times_{\co_\kk} \mathrm{Sing}_{ (n-1,1) }
& \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-1,1) }  \label{first singular} \\
\mathcal{M}_{(1,0)} \times_{\co_\kk}  \mathrm{Sing}_{ (n-2,1) }
& \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}^\Pap_{ (n-2,1) } \label{second singular}
\end{align}
of \eqref{singular} under the vertical arrows in \eqref{i_p pullback}, and so it suffices to prove that the first singular locus  \eqref{first singular} pulls back to the second singular locus  \eqref{second singular} under the bottom horizontal arrow in \eqref{i_p pullback}.
Moreover,  as each of \eqref{first singular} and \eqref{second singular} is a reduced $\co_\kk$-stack of dimension $0$, and as   the bottom horizontal arrow in \eqref{i_p pullback} is finite and unramified, it suffices to check this on the level of geometric points.
This is an easy exercise in linear algebra, using the characterization of the singular locus found in
Remark \ref{rem:singular description}.
%So, let $F$ be an algebraically closed field $F$ of characteristic dividing $D$, fix an $F$-point
%(A_0',A') \in \mathcal{M}_{(1,0)}(F)  \times \mathcal{M}^\Pap_{ (n-2,1) }(F) ,
%and let
%(A_0,A) \in \mathcal{M}_{(1,0)}(F)  \times \mathcal{M}^\Pap_{ (n-1,1) }(F)
%be its image under the bottom horizontal arrow in \eqref{i_p pullback}.
%By  \eqref{i_p pullback explicit}, there in an $\co_\kk$-linear isomorphism of $F$ vector spaces
%\Lie(A) = \Lie( \tilde{A}_0 ) \oplus \Lie(A').
%Consider the endomorphism $\sqrt{-D} \in \End( \tilde{A}_0 )$.  As $F$ has characteristic dividing $D$,  the square of this endomorphism annihilates the $1$-dimensional  space $\Lie(\tilde{A}_0)$, which implies that $\sqrt{-D}$ also annihilates it.
%Now we use  Pappas's characterization of the singular locus [45] (see  also Theorem 2.3.2 of [13]):   the pair $(A_0',A')$ lies in \eqref{second singular} if and only if $\sqrt{-D} \in \End(A')$ annihilates the Lie algebra of $A'$.
%This hold if and only if    $\sqrt{-D} \in \End(A)$ annihilates the Lie algebra of $A$, which holds if and only if  $(A_0,A)$ lies in \eqref{first singular}.
% Thus \eqref{first singular} pulls back to \eqref{second singular} under \eqref{i_p pullback}, completing  the proof.
\end{proof}

\begin{proposition}\label{prop:i_p nonexceptional}
The nonexceptional locus $ \mathcal{U}_\mathfrak{p}^\nonexc \subset  \mathcal{U}_\mathfrak{p}$ satisfies
\begin{equation*}
%\label{dagger open}
\mathcal{U}_\mathfrak{p}^\nonexc \subset \mathcal{U}_\mathfrak{p}^\dagger,
\end{equation*}
and the  isomorphism   of Proposition \ref{prop:i_p construction}  restricts to an isomorphism
\[
\mathcal{S}^\nonexc_{V' / \co_\kk[1/p] }    \iso
\mathcal{U}^\nonexc_{\mathfrak{p}}  .
\]
\end{proposition}

\begin{proof}
Suppose $S$ is an $\co_\kk[1/p]$-scheme,
\[
(A_0,A,x) \in \mathcal{U}_\mathfrak{p}^\nonexc(S),
\]
and let $A'$ be the  abelian scheme  of \eqref{pi product}.
In particular, \[\Lie(A) = \Lie( \tilde{A}_0 ) \oplus \Lie(A').\]

The natural map
\[
\mathcal{U}_\mathfrak{p}^\nonexc \to  \mathcal{M}_{(n-1,1)} \smallsetminus \mathrm{Exc}_{(n-1,1)}
\]
endows $A$ with a hyperplane $\mathcal{F} \subset \Lie(A)$, and results of Kr\"amer, as summarized in Theorem 2.3.3 of [13], shows that this hyperplane is actually determined by the $\co_\kk$-action on $A$ using the following recipe:
If we fix  a $\pi \in \co_\kk$ such that $\co_\kk= \Z[\pi]$, and let
\[
\overline{\epsilon}_S = \overline{\pi} \otimes 1 -  1\otimes i_S(\overline{\pi}) \in \co_\kk \otimes_\Z \co_S,
\]
where $i_S : \co_\kk \to \co_S$ is the structure map,  then
\[
\mathcal{F} = \mathrm{ker}( \overline{\epsilon}_S : \Lie(A) \to \Lie(A) ).
\]

On the other hand,  the action of $\co_\kk$ on the Lie algebra of $A_0$ is through the structure morphism $i_S$, and so the same is true of  $\tilde{A}_0 = A_0 / A_0[\mathfrak{p}]$.
This implies that $\overline{\epsilon}_S$ annihilates $ \Lie( \tilde{A}_0 )$, and hence
\Lie( \tilde{A}_0 ) \subset \mathcal{F}.
Having shown
\[
(A_0,A,x) \in \mathcal{U}_\mathfrak{p}^\dagger(S),
\]
the first claim of the proposition is proved.
The second claim follows from the first and Proposition \ref{prop:i_p pullbacks}.
\end{proof}

\begin{proposition}\label{prop:i_p toroidal}
The closed immersion $i_\mathfrak{p} : \mathcal{S}_{V'/\co_\kk[1/p]} \to \mathcal{U}_\mathfrak{p}$ of Proposition \ref{prop:i_p construction}  extends uniquely to a proper morphism
\[
\bar{\mathcal{S}}_{V'/\co_\kk[1/p] } \to  \bar{\mathcal{U}}_\mathfrak{p},
\]
where the codomain is defined as the Zariski closure of
\[
\mathcal{U}_\mathfrak{p} \subset  \bar{\mathcal{Z}}_V(p)_{/\co_\kk[1/p]}.
\]
or, equivalently, as the normalization of
$ \mathcal{U}_\mathfrak{p} \to \bar{\mathcal{S}}_{V/\co_\kk[1/p]}.$
\end{proposition}

\begin{proof}
For ease of notation, we omit all subscripts $\co_\kk[1/p]$ in the proof.

As the exceptional locus of $\bar{\mathcal{S}}_{V'}$ does not meet the boundary, it suffices to construct the extension over its complement
\[
\bar{\mathcal{S}}_{V'}^\nonexc = \bar{\mathcal{S}}_{V'}  \smallsetminus \mathrm{Exc}_{V'} .
\]
Consider the universal pair $(A_0,A')$ over $\mathcal{S}_{V'}^\nonexc$,  and let
\[
x: A_0 \to A =  \tilde{A}_0 \times A'
\]
be as in the proof of Proposition \ref{prop:i_p construction}.   In other words,  the triple $(A_0,A,x)$ over $\mathcal{S}_{V'}^\nonexc$ determines the isomorphism
i_\mathfrak{p} :  \mathcal{S}_{V'}^\nonexc \iso  \mathcal{U}_\mathfrak{p}^\nonexc.
of Proposition \ref{prop:i_p nonexceptional}.

The discussion of \S \ref{ss:toroidal} shows that  $A'$ extends to a semi-abelian scheme over $\bar{\mathcal{S}}_{V'}^\nonexc$.
The elliptic curve  $A_0$  extends to an elliptic curve $\bar{\mathcal{S}}_{V'}^\nonexc$,  and so the same is true of its quotient $\tilde{A}_0$.
It follows  that $A$  extends to a semi-abelian scheme over $\bar{\mathcal{S}}_{V'}^\nonexc$,  and  the extension property used  in the proof of Lemma \ref{lem:normalized compactification} provides us with a morphism
\[
i_\mathfrak{p} :  \bar{\mathcal{S}}_{V'}^\nonexc \to  \mathcal{M}_{(1,0)} \times_{\co_\kk} \bar{\mathcal{M}}^\Pap_{(n-1,1)}
\]
taking values in the open subscheme
\[
\mathcal{M}_{(1,0)} \times_{\co_\kk} \big( \bar{\mathcal{M}}^\Pap_{(n-1,1)} \smallsetminus \mathrm{Sing}_{(n-1,1)} \big)
\iso
\mathcal{M}_{(1,0)} \times_{\co_\kk} \big( \bar{\mathcal{M}}_{(n-1,1)} \smallsetminus \mathrm{Exc}_{(n-1,1)} \big).
\]
This provides us with a morphism
\[
i_\mathfrak{p} :  \bar{\mathcal{S}}_{V'}^\nonexc \to  \mathcal{M}_{(1,0)} \times_{\co_\kk}  \bar{\mathcal{M}}_{(n-1,1)} ,
\]
taking values in the open and closed substack $\bar{\mathcal{S}}_V$.

We now have a commutative diagram of solid arrows
\[
\xymatrix{
{ \mathcal{S}_{V'}^\nonexc  } \ar[r]^{i_\mathfrak{p}}  \ar[d]  & { \bar{\mathcal{U}}_\mathfrak{p}   } \ar[d]  \\
{ \bar{\mathcal{S}}_{V'}^\nonexc  }  \ar[r]<EMAIL_ADDRESS> { \bar{\mathcal{S}}_V  }
\]
in which the vertical arrow on the right is integral, and hence affine, by its construction as a normalization; see Lemma 29.53.4 of [51].
To complete the proof of the proposition, it suffices to show that there is a unique dotted arrow making the diagram commute.
This immediately reduces to the corresponding claim for affine schemes, in which
we are given homomorphisms of rings
\[
\xymatrix{
{ R'  }   & { B } \ar[l]<EMAIL_ADDRESS>\\
{ \bar{R}' } \ar[u]  &   { A  } \ar[l] \ar[u]
\]
with $B$ integral over $A$,  and $\bar{R}' \subset R'$ is an inclusion of normal domains with the same field of fractions.
The image of any $b\in B$ under $B \to R'$ is integral over  $\bar{R}'$, hence contained in $\bar{R}'$.
Thus there is a unique dotted arrow making the diagram commute.
\end{proof}

\subsection{Two lemmas on abelian schemes}

For use in the next subsection, we prove  two lemmas on abelian schemes.
The first is a criterion for determining when a polarization descends to a quotient.
The second is a criterion for the existence of an extension as a semi-abelian scheme.

\begin{lemma}\label{lem:polarization descent}
Suppose  $f : X\to Y$ is an isogeny of abelian schemes over a Noetherian scheme $S$, and $\psi_X : X\to X^\vee$ is a polarization whose degree $d$ is invertible in $\co_S$.    The following are equivalent.
\begin{enumerate}
\item
The kernel of $f$ is contained in $\mathrm{ker}(\psi_X)$, and is totally isotropic with respect to the
Weil pairing
\[
e : \mathrm{ker}(\psi_X)  \times \mathrm{ker}(\psi_X) \to \mu_d.
\]
\item
There exists a (necessarily unique) polarization $\psi_Y : Y \to Y^\vee$ making the diagram
\[
\xymatrix{
{  X } \ar[r]^{\psi_X}  \ar[d]_f &  { X^\vee }  \\
{ Y } \ar[r]_{\psi_Y} & { Y^\vee } \ar[u]_{f^\vee}
\]
\end{enumerate}
\end{lemma}

\begin{proof}
This is routine.
See, for example, Proposition 10.4 of [46].
%We may assume that $S$ is connected, and fix a geometric point $s \to S$.  For each $p\mid d$ set
%V_p(X_s) = T_p(X_s) \otimes_{\Z_p} \Q_p.
%The polarization $\psi_X$ induces a $p$-adic Weil pairing
%e_p :  V_p(X_s) \times V_p(X_s) \to \Q_p(1),
%and identifies $T_p(X_s^\vee)$ with the dual lattice of $T_p(X_s)$ with respect to \eqref{p-weil}.
%Moreover, we may identify
%\mathrm{ker}(\psi_X)_s \iso T_p(X_s^\vee)/T_p(X_s)
%in such a way that the Weil pairing on the left hand side agrees with the pairing
% T_p(X_s^\vee)/T_p(X_s) \times T_p(X_s^\vee)/T_p(X_s) \to (\Q_p/\Z_p)(1).
%induced by \eqref{p-weil}.
%If we now use $f:X\to Y$  to identify $V_p(X_s) \iso V_p(Y_s)$, the claim is that   both properties of the proposition are  equivalent to
%\item [(3)]
%for every  prime $p\mid d$, the pairing \eqref{p-weil} is $\Z_p(1)$-valued on $T_p(Y_s)$.
\end{proof}

\begin{lemma}\label{lem:semiextension}
Suppose $S$ is a normal Noetherian  scheme, $U\subset S$ is a dense open subscheme, and
$f : X \to Y$ is an isogeny of abelian schemes over $U$ whose degree $d$ is invertible in $\co_S$.
If $X$ extends to a semi-abelian scheme  over $S$ then so does $Y$, and both extensions are unique.
\end{lemma}

\begin{proof}
The uniqueness claim follows from Proposition I.2.7 of [15], so we only need to prove the existence of a semi-abelian extension of $Y$.

Consider the morphisms
\mathrm{ker}(f) \to X[d] \to U.
The second arrow and the composition are both  \'etale, as $d\in \co_U^\times$, and so the first arrow is also \'etale by \cite[\href{https://stacks.math.columbia.edu/tag/02GW}{Tag 02GW}]{stacks-project}.
As the first arrow is  also a closed immersion, we deduce that  $\mathrm{ker}(f) \subset X[d]$ is a union of connected components.

If $X^* \to S$ denotes the semi-abelian extension of $X$, the multiplication map $[d] : X^* \to X^*$ is quasi-finite and flat. Indeed, by Lemma 37.16.4 of [51] flatness can be checked fiberwise on $S$, where it follows from  Proposition 3.11 of Expos\'e VI.B  of [50],  and the surjectivity of $[d]$ on any extension of an abelian variety by a torus.
The group scheme $X^*[d] \to S$ is therefore quasi-finite and flat.
Using our assumption that $d\in \co_S^\times$,  and  Lemma 29.36.8 of  [51], we deduce that  it is \'etale.

As we assume that $S$ is normal, so is  $X^*[d]$.   As the connected components of a normal scheme are the same as its irreducible components, it follows that the Zariski closure of $\mathrm{ker}(f)$ in $X^*[d]$  is  an open and closed subscheme $\mathrm{ker}(f)^* \subset X^*[d]$.  In particular  $\mathrm{ker}(f)^*\to S$ is quasi-finite \'etale.

By Lemma IV.7.1.2 of [44],  the fppf quotient  $Y^* = X^*/\mathrm{ker}(f)^*$  provides  a semi-abelian extension of $Y$ to $S$.
\end{proof}

\subsection{Analysis of $\mathcal{U}_0$}

In this subsection we study the substack $𝒰_0$ of \eqref{flavor decomp}, and make explicit its relation with $𝒮_V'$.

Return to the situation of Lemma \ref{lem:lattice trifurcation}, so that
$x∈__(𝔞_0,𝔞)$ with $⟨x,x⟩=p$ determines an orthogonal decomposition
\begin{equation}\label{W split}
W = \tilde{W}_0 \oplus W' .
\end{equation}
Set $𝔞̃_0 = 𝔞∩W̃_0$.

\begin{lemma}\label{lem:0 linear algebra}
Assume that  $\tilde{\mathfrak{a}}_0 = x(\mathfrak{a}_0)$.
If we set  $\mathfrak{b}  = \mathfrak{a} \cap W'$ and let $\mathfrak{b}^\vee \subset W' $ be its dual lattice with respect to $h|_{ W' }$,  the $\co_\kk$-lattice
\[
\mathfrak{a}'=  \mathfrak{p} \cdot  \mathfrak{b}^\vee + \mathfrak{b} ,
\]
is self-dual  with respect to $h|_{W'}$.  Moreover, there are inclusions of $\co_\kk$-lattices
\begin{equation}\label{lattice correspondence}
\tilde{\mathfrak{a}}_0 \oplus \mathfrak{a}'
\stackrel{p}{\supset}  \tilde{\mathfrak{a}}_0 \oplus \mathfrak{b}
\stackrel{p^2}{\subset}  \mathfrak{a}
\end{equation}
in $W$ of the indicated indices, and a  canonical  $\co_\kk$-linear injection
\begin{equation}\label{lattice t}
\mathfrak{p}^{-1} \tilde{\mathfrak{a}}_0/ \tilde{\mathfrak{a}}_0 \iso  \mathfrak{p}^{-1} \mathfrak{b} / \mathfrak{b}  \map{t} \mathfrak{p}^{-1} \mathfrak{a}'/ \mathfrak{a}' .
\end{equation}
\end{lemma}

\begin{proof}
Throughout the proof, we use  $x$ to identify $W_0 = \tilde{W}_0$, so that $h|_{W_0} = p h_0$.
The projections to the two factors in \eqref{W split} induce  isomorphisms
\[
\mathfrak{a} / ( \mathfrak{a}_0 \oplus \mathfrak{b} )  \to \mathfrak{a}_0^\vee / \mathfrak{a}_0,
\qquad
\mathfrak{a} /(  \mathfrak{a}_0 \oplus \mathfrak{b} )  \to \mathfrak{b}^\vee / \mathfrak{b},
\]
where  $\mathfrak{a}_0^\vee= p^{-1}  \mathfrak{a}_0$ is the dual lattice of $\mathfrak{a}_0$ with respect to $h|_{W_0}$.
Composing these isomorphisms yields an $\co_\kk$-linear isomorphism
\begin{equation}\label{p trivialization}
p^{-1} \mathfrak{a}_0 / \mathfrak{a}_0 \iso \mathfrak{b}^\vee / \mathfrak{b}
\end{equation}
respecting the $p^{-1}\co_\kk/\co_\kk$-valued hermitian forms induced  by $h|_{W_0}= ph_0$ and $h|_{W'}$.
It follows  that $\mathfrak{p} \cdot (  \mathfrak{b}^\vee / \mathfrak{b} )  \subset \mathfrak{b}^\vee / \mathfrak{b}$
is maximal isotropic with respect to $h|_{W'}$,  and so $\mathfrak{a}'$
is self-dual under $h|_{W'}$.

Consider the  inclusions  $\mathfrak{b} \subset  \mathfrak{a}' \subset \mathfrak{b}^\vee$ of $\co_\kk$-modules.
The first  is an isomorphism everywhere locally except at $\overline{\mathfrak{p}}$, while the second is an isomorphism    everywhere locally except at $\mathfrak{p}$.
In particular, the first  induces an isomorphism
\[
\mathfrak{p}^{-1} \mathfrak{b} / \mathfrak{b} \iso \mathfrak{p}^{-1} \mathfrak{a}'/\mathfrak{a}' .
\]
Restricting  \eqref{p trivialization} to an isomorphism of $\mathfrak{p}$-torsion submodules  defines  \eqref{lattice t}.
The inclusions of \eqref{lattice correspondence}, with the indicated indices,  are clear from what we have said.
\end{proof}

Just as Lemma \ref{lem:0 linear algebra} is  more complicated than Lemma \ref{lem:pi linear algebra}, the analysis of $𝒰_0$ is  more complicated than that of $𝒰_𝔭$.

Let $S$ be an $_[1/p]$-scheme.
Given  Lemma \ref{lem:0 linear algebra}, one expects  that   any  $S$-point $(A_0,A,x) ∈𝒰_0(S)$  should determine a diagram of  abelian schemes with $_$-actions
\begin{equation}\label{abelian correspondence}
\xymatrix{
{ \tilde{A}_0 \times A' } &  &{ \tilde{A}_0 \times B}  \ar[ll]_{\deg = p} \ar[rr]^{\deg = p^2} &  & { A}
\end{equation}
in which the arrows are $_$-linear isogenies of the indicated degrees.
There should also be a canonical $_$-linear closed immersion
\begin{equation}\label{t construct}
\tilde{A}_0[\mathfrak{p}] \iso B[\mathfrak{p}]  \map{t} A'[ \mathfrak{p} ],
\end{equation}
and the morphism $x:A_0 →A$ should factor as
\[
A_0 \iso \tilde{A}_0 \hookrightarrow \tilde{A}_0 \times B \to A.
\]

Moreover,  these abelian schemes should come with polarizations with the following properties.
The elliptic curve $Ã_0$ is equipped with its unique polarization of degree $p^2$,
$B$ is equipped with a polarization of degree $p^2$, and  $A'$ is equipped with a principal polarization.
The pullback  of the product polarization on $Ã_0 ×A'$ via the leftwards arrow in \eqref{abelian correspondence} is the product polarization on $Ã_0 ×B$, and similarly for the  the pullback of the principal  polarization on $A$  via the rightward arrow.

Here is how to construct the data above from $(A_0,A,x)$.
As in the proof of Proposition \ref{prop:split divisor flavors},  at any geometric point $s →S$ we  may apply Lemma   \ref{lem:0 linear algebra} with $𝔞_0$ and $𝔞$ replaced by the  $p$-adic Tate modules of $A_0s$ and $A_s$ to obtain $_$-lattices
\begin{equation}\label{tate correspondence}
T_p( \tilde{A}_{0s}) \oplus T_p( A_s')  \supset T_p( \tilde{A}_{0s}) \oplus T_p(B_s)  \subset T_p(A_s).
\end{equation}
analogous to \eqref{lattice correspondence}, along with an $_$-linear injection
\begin{equation}\label{tate-level t}
\mathfrak{p}^{-1} T_p(\tilde{A}_{0s})/T_p(\tilde{A}_{0s})  \iso \mathfrak{p}^{-1} T_p(B_s)/T_p(B_s)
\map{t}  \mathfrak{p}^{-1} T_p(A_s')/T_p(A_s').
\end{equation}
To be clear, we have not yet constructed  the abelian schemes $Ã_0$, $A'$, and $B$, only  lattices in $T_p(A_s)[1/p]$ that we choose to call  $T_p(Ã_s)$,  $T_p(A_s')$,  and $T_p(B_s)$.
However, as $p∈_S^×$ and both inclusions in \eqref{tate correspondence} have finite index, there are abelian schemes $C$ and $D$ over $S$ with $_$-actions and $_$-linear  isogenies
\[
\xymatrix{
{  D }  &  {  C    }  \ar[r]^{p^2} \ar[l]_{p} &  { A   }
\]
of the indicated degrees that identify
\begin{equation}\label{tate-level decomp}
T_p(C_s) = T_p(\tilde{A}_{0s}) \oplus T_p(B_s) ,\qquad T_p(D_s) = T_p(\tilde{A}_{0s}) \oplus T_p(A_s').
\end{equation}

The condition that $⟨x,x⟩=p$ implies that
\[
A_0 \map{x} A \iso A^\vee \map{x^\vee} A_0^\vee \iso A_0
\]
is multiplication by $p$, which implies that the composition
\[
A \iso A^\vee \map{x^\vee} A_0^\vee\iso A_0 \map{x} A
\]
is a Rosati-fixed element $ α∈__(A)[1/p]$ such that $α∘α= pα$.
In particular, we obtain an idempotent
\[
p^{-1}\alpha \in  \End_{\co_\kk}(A)[1/p]\iso  \End_{\co_\kk}(C)[1/p]\iso  \End_{\co_\kk}(D)[1/p],
\]
whose induced actions on $T_p(C_s)[1/p]$ and $T_p(D_s)[1/p]$ are just the projections to the first summands in \eqref{tate-level decomp}.  It follows that we in fact have
\[
p^{-1}\alpha \in      \End_{\co_\kk}(C) \iso  \End_{\co_\kk}(D).
\]
These idempotents are the projections to the first factors for unique decompositions
\[
C= \tilde{A}_0 \times B ,\qquad  D=\tilde{A}_0 \times A'
\]
of abelian schemes with $_$-actions, and the image of $x$ under
\[
\Hom_{\co_\kk}(A_0 , A) [1/p] \iso \Hom_{\co_\kk}(A_0 , C)[1/p] \to \Hom_{\co_\kk}(A_0 , \tilde{A}_0)[1/p]
\]
is an isomorphism  $A_0 Ã_0$.
In particular, we now have $_$-linear isogenies \eqref{abelian correspondence}
such that taking $p$-adic Tate modules  recovers \eqref{tate correspondence}, and \eqref{tate-level t} induces
the closed immersion \eqref{t construct}.

If we pullback the principal polarization via $Ã_0 ×B →A$, we obtain a  polarization on $Ã_0 ×B$ of degree $p^4$.
As the idempotent $p^-1α$ constructed above is Rosati-fixed,  this polarization of $Ã_0 ×B$ splits as the product of  polarizations $Ã_0 →Ã_0^∨$  and   $B→B^∨$.
By examining the induced Weil pairing on $p$-adic Tate modules, one can show that  each polarization has  kernel  isomorphic,  \'etale locally on $S$, to $_/(p)$, and that
\[
\ker(B\to A')  = \ker( B\to B^\vee )[\overline{\mathfrak{p}}]
\]
is totally isotropic under the Weil pairing on $(B→B^∨)$.
Hence, by Lemma \ref{lem:polarization descent},  $B→B^∨$  descends   to a principal polarization on $A'$.
This completes the construction of the abelian schemes \eqref{abelian correspondence} with all of their expected extra structure.

Now apply this construction to the universal triple $(A_0,A,x)$ over $𝒰_0$.
As $p∈_𝒰_0^×$, the $p$-power isogenies \eqref{abelian correspondence} induce an isomorphism
\begin{equation}\label{i_0 Lie}
\Lie(A) \iso \Lie( \tilde{A}_0) \times \Lie(A')
\end{equation}
of rank $n$ vector bundles on $𝒰_0$.
Recalling that $A$ comes equipped with an $_$-stable  hyperplane $ℱ ⊂(A)$, denote by
\[
\mathcal{U}_0^\dagger \subset  \mathcal{U}_0
\]
the largest closed substack over which  $(Ã_0) ⊂ℱ$

Define a finite \'etale cover
\begin{equation}\label{tau cover}
\mathcal{T}_{V'}   \to \mathcal{S}_{V'/\co_\kk[1/p]}
\end{equation}
of degree $p^n-1-1$ as follows:
for any $_[1/p]$-scheme $S$, let  $𝒯_V'(S)$ be the groupoid of triples $(A'_0,A',t)$ in which
\[
(A'_0,A') \in \mathcal{S}_{V'}(S) \quad
\mbox{and} \quad
t : A'_0[ \mathfrak{p} ] \to A'[\mathfrak{p}]
\]
is an $_$-linear closed immersion of  group schemes over $S$.

\begin{proposition}\label{prop:i_0 construction}
There is a canonical isomorphism $i_0 :  \mathcal{T}_{V'}  \iso \mathcal{U}_0^\dagger$.
\end{proposition}

\begin{proof}
The morphism $\mathcal{U}_0^\dagger \to  \mathcal{T}_{V'}$ is essentially described above.
Suppose $S$ is an $\co_\kk[1/p]$-scheme and
\[
(A_0,A,x) \in \mathcal{U}_0^\dagger(S).
\]
We can use the isomorphism \eqref{i_0 Lie} to endow the abelian scheme
$A'$ of \eqref{abelian correspondence} with the  rank one local direct summand
\[
\mathcal{F} ' = \mathcal{F} / \Lie( \tilde{A}_0) \subset \Lie(A) /\Lie( \tilde{A}_0) =  \Lie(A')
\]
to obtain a point $A' \in \mathcal{M}_{(n-2,1)}(S)$.  Setting $A_0'=\tilde{A}_0$,
this defines a morphism
\[
\mathcal{U}_0^\dagger \map{  (A_0,A,x) \mapsto (A'_0,A')  }  \mathcal{S}_{V' / \co_\kk[1/p] },
\]
and  the  closed immersion  \eqref{t construct} defines the desired lift to
$\mathcal{U}_0^\dagger \to  \mathcal{T}_{V'}$.

Now we construct the inverse.
Start with  a point
(A'_0, A' , t) \in \mathcal{T}_{V'}(S).
Denote by $B$ the abelian scheme dual to
\[
B^\vee = A' / \mathrm{Im}(t).
\]
Using the principal polarization to identify $A'$ with its dual, the quotient map $A' \to B^\vee$ dualizes to a morphism $B \to A'$, and  the composition
\[
B\to A' \to B^\vee
\]
is a degree $p^2$ polarization  (it is the pullback via $B \to A'$ of the principal polarization on $A'$).
It has  the property that each factor on the right hand side of
\[
\ker( B \to B^\vee ) = \ker(B \to B^\vee ) [\mathfrak{p}]  \times \ker(B \to B^\vee ) [ \overline{\mathfrak{p}} ]
\]
has order $p$, and  the second factor is the kernel of $B\to A'$.  In particular, the induced map
$B[\mathfrak{p}] \to A'[\mathfrak{p}]$ is an isomorphism, and the closed immersion
\[
A'_0[\mathfrak{p}] \map{t} A'[\mathfrak{p}] \iso  B[\mathfrak{p}]
\]
has image $\ker(B \to B^\vee) [\mathfrak{p}] $.
Setting  $\tilde{A}_0 = A'_0$, the resulting   isomorphism
\[
\tilde{A}_0[\mathfrak{p}] \iso \ker(B \to B^\vee) [\mathfrak{p}]
\]
admits a  unique $\co_\kk$-linear extension to an isomorphism
\[
t : \tilde{A}_0[ p ] \iso \ker(B \to B^\vee)
\]
identifying the Weil pairings on source and target.

The antidiagonal subgroup
\[
\Delta \define \tilde{A}_0[p]\map{  a_0 \mapsto ( - a_0 , t (a_0) ) }  \tilde{A}_0 \times B.
\]
is totally isotropic under the Weil pairing induced by the product polarization (where $\tilde{A}_0$ is given the polarization of degree $p^2$), which therefore descends, by Lemma \ref{lem:polarization descent},  to a principal polarization on the quotient
\[
A = (\tilde{A}_0\times B)/\Delta.
\]
As  $p \in \co_S^\times$,  there is an  induced  isomorphism of Lie algebras \eqref{i_0 Lie},
which allows us to define a corank one local direct summand $\mathcal{F} \subset \Lie(A)$ as the product
\[
\mathcal{F}  = \Lie(\tilde{A}_0) \times \mathcal{F}'.
\]
This defines  a point $A \in \mathcal{M}_{(n-1,1)}(S)$.  Setting $A_0 = \tilde{A}_0$,  the composition
\[
A_0=\tilde{A}_0 \hookrightarrow  \tilde{A}_0 \times B \to A
\]
defines  $x \in \Hom_{\co_\kk}(A_0,A)$ with $\langle x,x\rangle=p$

The above construction  determines  a morphism
\[
\mathcal{T}_{V'} \map{ (A'_0,A',t) \to (A_0,A,x)    }  \mathcal{Z}_V(p)
\]
taking values in the open substack $\mathcal{U}_0^\dagger$,  which is inverse to the map $\mathcal{U}_0^\dagger \to \mathcal{T}_{V'}$ constructed above.
\end{proof}

Proposition \ref{prop:i_0 construction} gives us morphisms
\begin{equation}\label{i_0 pullback}
\xymatrix{
{  \mathcal{S}_{V'/\co_\kk[1/p] } }
& & { \mathcal{T}_{V'} }  \ar[rr]       \ar[ll]_{\eqref{tau cover}}
& & { \mathcal{S}_{V/\co_\kk[1/p] } }
\end{equation}
in which the  arrow on the right is the (finite and unramified) composition
\[
{\mathcal{T}_{V'} } \map{i_0 } \mathcal{U}^\dagger_0 \hookrightarrow   \mathcal{Z}_V(p)_{/\co_\kk[1/p]}  \to \mathcal{S}_{V/\co_\kk[1/p]} ,
\]
sending  $(A_0' , A' , t ) ↦(A_0,A)$, where $A_0=A_0'$, and $A$ is related to $A'$ by a diagram \eqref{abelian correspondence}.

\begin{proposition}\label{prop:i_0 pullbacks}
The hermitian line bundles
\[
\widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}
\in   \widehat{\Pic}(  \mathcal{S}_{V' } )
\]
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{S}_{V}}     \in   \widehat{\Pic}(  \mathcal{S}_{V } )
have the same images under the homomorphisms
\[
\xymatrix{
{ \widehat{\Pic}( \mathcal{S}_{V'}   ) }  \ar[r] &  {  \widehat{\Pic} (    \mathcal{T}_{V'} ) _\Q }&  {  \widehat{\Pic}(  \mathcal{S}_{V } ) }  \ar[l]
\]
induced by \eqref{i_0 pullback}.
The same is true of
$(\mathrm{Exc}_{V'} ,0 )$ and $(\mathrm{Exc}_V,0)$, and of $\widehat{\taut}_{V'}$ and $\widehat{\taut}_V$.
\end{proposition}

\begin{proof}
If $(A'_0,A')$ denotes the pullback of the universal object via the left arrow in \eqref{i_0 pullback}, and $(A_0,A)$ denotes the pullback of the universal object via the right arrow in \eqref{i_0 pullback},
examination of the proof of Proposition \ref{prop:i_0 construction} shows that there is a quasi-isogeny
\[
f \in \Hom( A'_0 \times A' , A)[1/p]
\]
of degree $\deg(f)=p$.
As in the  proof of  Proposition \ref{prop:i_p pullbacks}, this implies that
\[
\widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{T}_{V'}}
\widehat{\omega}^\mathrm{Hdg}_{ A'_0 / \mathcal{T}_{V'}}
\widehat{\omega}^\mathrm{Hdg}_{ A' / \mathcal{T}_{V'}}   + (0 , -\log(p))
\]
as elements of $\widehat{\Pic}(\mathcal{T}_{V'} )$, and  the term $(0,-\log(p))$ is torsion.
The first claim follows  from this.
The remaining claims are essentially the same as in Proposition \ref{prop:i_p pullbacks}, and we leave the details to the reader.
\end{proof}

\begin{proposition}\label{prop:i_0 nonexceptional}
The nonexceptional locus $ \mathcal{U}_0^\nonexc \subset  \mathcal{U}_0$ satisfies
\[
\mathcal{U}_0^\nonexc \subset \mathcal{U}_0^\dagger.
\]
Moreover, the  morphism of Proposition \ref{prop:i_0 construction}  restricts to an isomorphism
\[
\mathcal{S}^\nonexc_{V' / \co_\kk[1/p] }    \iso
\mathcal{U}^\nonexc_0  .
\]
\end{proposition}

\begin{proof}
The proof is essentially the same as Proposition \ref{prop:i_p nonexceptional}, and the details are left to the reader.
\end{proof}

\begin{proposition}\label{prop:i_0 toroidal}
Denote by $\bar{\mathcal{T}}_{V'}$  the normalization of  $ \mathcal{T}_{V'} \to \bar{\mathcal{S}}_{V'/\co_\kk[1/p] }$.
The closed immersion $i_0:\mathcal{T}_{V'} \to \mathcal{U}_0$ of Proposition \ref{prop:i_0 construction} extends to a proper morphism
\[
\bar{\mathcal{T}}_{V'} \to  \bar{\mathcal{U}}_0,
\]
where the codomain is defined as the Zariski closure of
\[
\mathcal{U}_0 \subset \bar{\mathcal{Z}}_V(p)_{/\co_\kk[1/p]},
\]
or, equivalently, as the normalization of
$ \mathcal{U}_0 \to \bar{\mathcal{S}}_{V/\co_\kk[1/p]}$.
\end{proposition}

\begin{proof}
Consider the universal triple $(A_0,A',t)$ over $\mathcal{T}_{V'}$, and let  $(A_0, A)$ be the pullback of the universal pair  over $\mathcal{S}_V$ via the composition
\[
\mathcal{T}_{V'} \map{i_0} \mathcal{U}_0  \to  \mathcal{S}_{V} .
\]
We  know that  $A'$ extends to a semi-abelian scheme over $\bar{\mathcal{T}}_{V'}$, obtained as a pullback via
\[
\bar{\mathcal{T}}_{V'} \to \bar{\mathcal{S}}_{V'} \to \bar{\mathcal{M}}_{(n-2,1)},
\]
and that $A_0$ extends to an elliptic curve over $\bar{\mathcal{T}}_{V'}$, obtained as a pullback via
\[
\bar{\mathcal{T}}_{V'} \to \bar{\mathcal{S}}_{V'} \to \mathcal{M}_{(1,0)},
\]
Using Lemma \ref{lem:semiextension} and the isogenies \eqref{abelian correspondence},   we deduce that $A$  also extends to  a semi-abelian scheme  over $\bar{\mathcal{T}}_{V'}$.  With this extension in hand, the proof is essentially identical to that of Proposition \ref{prop:i_p toroidal}.
\end{proof}

\subsection{Proof of Theorems \ref{thm:height descent 1} and  \ref{thm:height descent 2}}

Define an $_[1/p]$-stack
\[
\mathcal{Z}^\dagger_V(p)  = \mathcal{T}_{V'} \sqcup  \mathcal{S}_{V' / \co_\kk[1/p] } \sqcup  \mathcal{S}_{V' /\co_\kk[1/p]}  .
\]
Propositions \ref{prop:i_p construction} and \ref{prop:i_0 construction} provide us with a canonical  closed immersion
\[
\mathcal{Z}^\dagger_V(p)
\map{i = i_0 \sqcup i_\mathfrak{p}  \sqcup i_{\overline{\mathfrak{p}} }}    \mathcal{U}_0 \sqcup \mathcal{U}_\mathfrak{p} \sqcup \mathcal{U}_{\overline{\mathfrak{p}}} = \mathcal{Z}_V(p)_{ / \co_\kk[1/p] }
\]
with image the closed substack $ 𝒰^†_0 ⊔𝒰^†_𝔭 ⊔𝒰^†_𝔭$.  Hence we have a diagram
\begin{equation}\label{divisor correspondence}
\xymatrix{
{ \mathcal{Z}^\dagger_V(p) } \ar[rr]^{i}   \ar[d]_\alpha  & & {      \mathcal{Z}_V(p)_{ / \co_\kk[1/p] } }  \ar[d]^\beta  \\
{ \mathcal{S}_{V'/\co_\kk[1/p]} }   & & { \mathcal{S}_{V/\co_\kk[1/p] } }
\end{equation}
in  which $α$ is a finite \'etale surjection of degree $p^n-1+1$, and $β$ is finite.

\begin{lemma}\label{lem:exceptional KR error}
As divisors on $ \mathcal{S}_{V/\co_\kk[1/p] }$, we have
\[
\mathcal{Z}_V(p)_{ / \co_\kk[1/p] }  = \mathcal{Z}^\dagger_V(p)  + E,
\]
where $\mathcal{Z}^\dagger_V(p)$ has no irreducible components contained in the exceptional divisor $\mathrm{Exc}_{V/\co_\kk[1/p]} \subset \mathcal{S}_{V/\co_\kk[1/p]}$ of Definition \ref{def:special exceptional}, and   $E$ is supported entirely on the exceptional divisor.
\end{lemma}

\begin{proof}
The \emph{exceptional locus} of $  \mathcal{Z}^\dagger_V(p)$ is the preimage of the exceptional divisor under $\alpha$, and  an  irreducible component of it is \emph{exceptional}  if it is contained in the exceptional locus.
Similarly, the \emph{exceptional locus} of $  \mathcal{Z}_V(p)_{ / \co_\kk[1/p] }$ is the preimage of the exceptional divisor under $\beta$, and  an irreducible component of it  is \emph{exceptional} if it is contained in the exceptional locus.

The final claims of Propositions \ref{prop:i_p nonexceptional} and \ref{prop:i_0 nonexceptional} show that $i$  restricts to an isomorphism between the non-exceptional loci of the source and target.  In particular, it establishes a bijection between the  generic points of non-exceptional irreducible components, and corresponding non-exceptional generic have local rings of the same (finite) length.

However, the morphism $\alpha$ is finite \'etale, so no irreducible component of the source can map into the exceptional divisor of the target.
Thus the  closed immersion $i$ actually establishes a bijection between  irreducible components of the source and  non-exceptional irreducible components of the target, in a way that matches up their multiplicities.  The claim follows immediately.
\end{proof}

Propositions \ref{prop:i_p toroidal} and \ref{prop:i_0 toroidal} imply that the diagram above extends  to
\begin{equation}\label{compact divisor correspondence}
\xymatrix{
{  \bar{\mathcal{Z}}^\dagger_V(p) } \ar[rr]^i  \ar[d]_\alpha   & & {  \bar{\mathcal{Z}}_V(p)_{ / \co_\kk[1/p] } }  \ar[d]^\beta   \\
{  \bar{\mathcal{S}}_{V'/\co_\kk[1/p]} }  &  & { \bar{\mathcal{S}}_{V/\co_\kk[1/p] } },
\end{equation}
where we have defined
\begin{equation}\label{dagger KR}
\bar{\mathcal{Z}}^\dagger_V(p) =  \bar{\mathcal{T}}_{V'} \sqcup  \bar{\mathcal{S}}_{V' / \co_\kk[1/p] } \sqcup  \bar{\mathcal{S}}_{V' /\co_\kk[1/p]}.
\end{equation}
Equivalently, this is the  normalization of
\[
\alpha: \mathcal{Z}^\dagger_V(p) \to \bar{\mathcal{S}}_{V'/\co_\kk[1/p] }.
\]

For any $N ≥1$ prime to $p$,  we can add level structure to  $𝒯̅_V'$  by  defining
\[
\mathcal{T}_{V'}(N) = \mathcal{T}_{V'} \times_{\mathcal{S}_{V'}} \mathcal{S}_{V'}(N),
\]
and letting   $𝒯̅_V'(N)$ be  the normalization of
\[
\mathcal{T}_{V'}(N) \to  \bar{\mathcal{T}}_{V'  / \co_\kk[1/N]}.
\]
Equivalently, this is the normalization of
\[
\mathcal{T}_{V'}(N) \to  \bar{\mathcal{S}}_{V'}(N)_{ / \co_\kk[ \frac{1}{Np} ]}.
\]

\begin{lemma}\label{lem:T compact}
The stack $\bar{\mathcal{T}}_{V'}(N)$ is regular,  flat, and proper over $\co_\kk[\frac{1}{Np} ]$, and is a projective scheme if $N\ge 3$.  It is smooth in a neighborhood of its boundary
\[
\partial \bar{\mathcal{T}}_{V'}(N) = \bar{\mathcal{T}}_{V'}(N) \smallsetminus \mathcal{T}_{V'}(N) ,
\]
which is a Cartier divisor smooth over $\co_\kk[\frac{1}{Np}]$.
\end{lemma}

\begin{proof}
The proof is similar to that of Proposition \ref{prop:full compactification}, and is again based on the results of [39] and [41].

Recall from Remark \ref{rem:honest shimura} that the generic fiber of $\mathcal{S}_{V'}$ is the Shimura variety associated to a reductive group $G'$ and a certain compact open subgroup of $G'(\A_f)$, and one can easily check that the generic fiber of the finite \'etale cover
\begin{equation}\label{alt level}
\mathcal{T}_{V'}(N) \to \mathcal{S}_{V'}(N)_{ /\co_\kk[ \frac{1}{Np} ]}
\end{equation}
is obtained by shrinking  compact open subgroup.

Recall from \S \ref{ss:special shimura} that $\mathcal{S}_{V'}(N)$ is constructed as an open and closed substack
\[
\mathcal{S}_{V'}(N) \subset \mathcal{M}_{W_0}(N) \times_{\co_\kk[1/N]}  \mathcal{M}_{W'}(N).
\]
In this construction, we may replace $  \mathcal{M}_{W'}(N)$ with the stack $\mathcal{M}^\Pap_{W'}(N)$ appearing in the proof of Proposition \ref{prop:full compactification} to obtain a blow-down
\[
\mathcal{S}^\Pap_{V'}(N) \subset \mathcal{M}_{W_0}(N) \times_{\co_\kk[1/N]}  \mathcal{M}^\Pap_{W'}(N).
\]
Repeating the construction of \eqref{alt level} with $\mathcal{S}_{V'}(N)$ replaced by $\mathcal{S}^\Pap_{V'}(N)$ yields a  finite \'etale cover
\[
\mathcal{T}^\Pap_{V'}(N) \to \mathcal{S}^\Pap_{V'}(N)_{ /\co_\kk[ \frac{1}{Np} ]},
\]
and $\mathcal{T}_{V'}(N)$ is the blow-up of the normal stack $\mathcal{T}^\Pap_{V'}(N)$  along its  proper  and $0$-dimensional locus of nonsmooth points.

As in the proof of Proposition \ref{prop:full compactification}, we now have morphisms
\[
\mathcal{T}^\Pap_{V'}(N) \to  \mathcal{S}^\Pap_{V' }(N)_{/\co_\kk[ \frac{1}{Np} ]}
\to \mathcal{M}^\Pap_{W'}(N)_{/\co_\kk[ \frac{1}{Np} ]}
\to \bar{\mathcal{A}}_{/\co_\kk[ \frac{1}{Np} ]},
\]
and we may define $\bar{\mathcal{T}}^\Pap_{V'}(N)$ as the normalization  of the composition.
We may now apply the results of [39] and [41] to deduce properties of  $\bar{\mathcal{T}}^\Pap_{V'}(N)$  from properties of its interior.  Arguing exactly as  in the proof of Proposition \ref{prop:full compactification}, this compactification is normal and flat, and is a projective scheme if $N\ge 3$.
Its boundary is a smooth Cartier divisor.  The nonsmooth locus has dimension $0$, and does not meet the boundary.  As $\bar{\mathcal{T}}_{V'}(N)$ can be identified with the blow-up of $\bar{\mathcal{T}}^\Pap_{V'}(N)$ along its nonsmooth locus, it has all the same properties, and is also regular (being smooth near the boundary and regular in the interior).
\end{proof}

Lemma \ref{lem:T compact} and Remark \ref{rem:special compact level} provide us with an  $_[ 1/Np ]$-stack
\[
\bar{\mathcal{Z}}^\dagger_V(p,N)
\define  \bar{\mathcal{T}}_{V'}(N) \sqcup
\bar{\mathcal{S}}_{V'}(N)_{ / \co_\kk[\frac{1}{Np}] } \sqcup  \bar{\mathcal{S}}_{V'}(N)_{/\co_\kk[ \frac{1}{Np} ] } ,
\]
with all the nice properties enumerated in Proposition \ref{prop:full compactification}.  Thus  \eqref{dagger KR} has its own theory of pre-log singular hermitian line bundles and Burgos-Kramer-K\"uhn arithmetic Chow groups
\[
\widehat{\CH}^d (  \bar{\mathcal{Z}}^\dagger_V(p) , \mathscr{D}_\BKK )
= \mil_{ \substack{ N\ge 3 \\ p \nmid N} }  \widehat{\CH}^d (  \bar{\mathcal{Z}}^\dagger_V(p,N) , \mathscr{D}_\BKK ) ,
\]
exactly as in \S \ref{ss:chow}.   These Chow groups include   notions of arithmetic heights and volumes as in  \S \ref{ss:volumes},  taking  values in the abelian group $/ log(p)$.

One can repeat the construction of the diagram \eqref{compact divisor correspondence} with level structures, and so obtain pullbacks
\[
\xymatrix{
{  \widehat{\CH}^d(  \bar{\mathcal{S}}_{V'} ,\mathscr{D}_\BKK  )   }   \ar[rr]^{\alpha^*} & &
{   \widehat{\CH}^d(  \bar{\mathcal{Z}}^\dagger_V(p) ,\mathscr{D}_\BKK  )   }    & &
{  \widehat{\CH}^d(  \bar{\mathcal{S}}_V ,\mathscr{D}_\BKK  )   }  \ar[ll]_{(\beta \circ i)^*}
\]
\[
\xymatrix{
{  \widehat{\Pic}(  \bar{\mathcal{S}}_{V'} ,\mathscr{D}_\BKK  )_\Q   }   \ar[rr]^{\alpha^*} &  &
{   \widehat{\Pic}(  \bar{\mathcal{Z}}^\dagger_V(p) ,\mathscr{D}_\BKK  )   }_\Q  & &
{  \widehat{\Pic}(  \bar{\mathcal{S}}_V ,\mathscr{D}_\BKK  )_\Q   }  \ar[ll]_{(\beta \circ i)^*}
\]

\begin{lemma}\label{lem:formal height induction}
If two pre-log singular hermitian line bundles
\[
\widehat{\mathcal{P}}' \in \widehat{\Pic}( \bar{\mathcal{S}}_{V'} , \mathscr{D}_\BKK)
\quad \mbox{and}\quad
\widehat{\mathcal{P}} \in \widehat{\Pic}( \bar{\mathcal{S}}_{V} , \mathscr{D}_\BKK)
\]
have the same pullback to $ \widehat{\Pic}(  \bar{\mathcal{Z}}^\dagger_V(p) ,\mathscr{D}_\BKK  ) _\Q$   then
\[
\int_{ \mathcal{Z}_V(p) (\C)  }  \chern( \widehat{\mathcal{P}})^{n-2}
=  ( p^{n-1}+1)  \int_{  \mathcal{S}_{V'} (\C)  }   \chern( \widehat{\mathcal{P}}'  )^{n-2}.
\]
Moreover, there is an $a(p) \in \Q$ such that
\[
\mathrm{ht}_{\widehat{\mathcal{P}}}  (\bar{\mathcal{Z}}_V(p))
(p^{n-1}+1 ) \cdot  \widehat{\vol}(   \widehat{\mathcal{P}}'  )  +  \mathrm{ht}_{\widehat{\mathcal{P}}}  (E) + a(p) \log(p) ,
\]
where $E$ is the divisor of Lemma \ref{lem:exceptional KR error}.
\end{lemma}

\begin{proof}
As in the proof of Lemma \ref{lem:exceptional KR error}, the closed immersion $i$ in \eqref{divisor correspondence}
restricts to an isomorphism of non-exceptional loci, and hence induces an isomorphism in the generic fiber.
\begin{align*}
\int_{ \mathcal{Z}_V(p) (\C)  } \beta^* \chern( \widehat{\mathcal{P}})^{n-2}
& =
\int_{ \mathcal{Z}^\dagger_V(p) (\C)  }  (\beta \circ i)^*  \chern( \widehat{\mathcal{P}})^{n-2}  \\
&   =   \int_{ \mathcal{Z}^\dagger_V(p) (\C)  }   \alpha^* \chern( \widehat{\mathcal{P}}'  )^{n-2} \\
& =  ( p^{n-1}+1)  \int_{  \mathcal{S}_{V'} (\C)  }   \chern( \widehat{\mathcal{P}}'  )^{n-2} .
\end{align*}
The second claim follows from Lemma \ref{lem:exceptional KR error} and the equalities
\[
\mathrm{ht}_{ \widehat{\mathcal{P}}}  (\bar{\mathcal{Z}}^\dagger_V(p))
\widehat{\vol}(   (\beta \circ i)^*\widehat{\mathcal{P}}  )
\widehat{\vol}(   \alpha^*\widehat{\mathcal{P}}'  )
(p^{n-1}+1 ) \cdot  \widehat{\vol}(   \widehat{\mathcal{P}}'  )
\]
in $\R/\Q\log(p)$, where the  first and last equalities are obtained directly by unpacking the definitions of pullbacks, heights, and arithmetic intersections in [11], and using the fact that $\alpha$ is (away from the boundary) a finite \'etale surjection of degree $p^{n-1}+1$.
%By Theorem 7.47 of [11], any arithmetic cycle class
%\widehat{\mathcal{D}} \in \widehat{\CH}^{n-1} (  \bar{\mathcal{S}}_{V'} , \mathscr{D}_\BKK )
%has a pullback
%f^* \widehat{\mathcal{D}} \in \widehat{\CH}^{n-1} (  \bar{\mathcal{Z}}^\dagger_V(p) , \mathscr{D}_\BKK ),
%which satisfies
%\begin{equation}\label{pullback degrees}
%\widehat{\deg} ( f^*   \widehat{\mathcal{D}} )= (p^{n-1}+1 ) \cdot   \widehat{\deg}( \widehat{\mathcal{D}}  ).
%To justify \eqref{pullback degrees}, fix  $N \ge 3$  prime to $p$, and represent the pullback of $\widehat{\mathcal{D}}$ to
%$\bar{\mathcal{S}}_{V'}(N)$ by an arithmetic $0$-cycle $(\mathcal{D} , g_\mathcal{D})$ with $\mathcal{D}$ supported in the interior $\mathcal{S}_{V'}(N)$.   This can be done, for example, by applying Chow's moving lemma  for projective varieties  to the reductions of $ \bar{\mathcal{S}}_{V'}(N)$.
%The equality \eqref{pullback degrees} then follows immediately from the fact that
%f: \mathcal{Z}^\dagger_V(p,N) \to \mathcal{S}_{V'}(N)_{/\co_\kk[ \frac{1}{Np} ]}
% is a finite \'etale surjection of degree $p^{n+1}+1$.
% The final equality of \eqref{correspondence swap} follows by applying \eqref{pullback degrees}  with $\widehat{\mathcal{D}}$ equal to the iterated self-intersection of   $\widehat{\mathcal{P}}'$, and using the compatibility of arithmetic intersection with $f^*$.
\end{proof}

\begin{proof}[Proof of Theorems \ref{thm:height descent 1} and \ref{thm:height descent 2}]
Propositions \ref{prop:i_p pullbacks} and \ref{prop:i_0 pullbacks} imply that the hypotheses of
Lemma \ref{lem:formal height induction} are satisfied by
\[
\widehat{\mathcal{P}}' = \widehat{\tautmod}_{V'}
\quad \mbox{and} \quad
\widehat{\mathcal{P}} = \widehat{\tautmod}_{V}
\]
which therefore gives the equality
\[
\int_{ \mathcal{Z}_V(p) (\C)  }  \chern( \widehat{\tautmod}_V)^{n-2}
=  ( p^{n-1}+1)  \int_{  \mathcal{S}_{V'} (\C)  }   \chern( \widehat{\tautmod}_{V'}  )^{n-2},
\]
and the equality
\[
\mathrm{ht}_{ \widehat{\tautmod}_V }(\bar{\mathcal{Z}}_V(p))
( p^{n-1}+1 )  \widehat{\vol} (\widehat{\tautmod}_{V'} )
+   \mathrm{ht}_{ \widehat{\tautmod}_V }(E)
\]
up to a rational multiple of $\log(p)$.    The second term on the right vanishes by claim (1) of Theorem \ref{thm:taut-hodge compare}, completing the proof of Theorem  \ref{thm:height descent 1}.

Similarly,  Propositions \ref{prop:i_p pullbacks} and \ref{prop:i_0 pullbacks} show  that  the hypotheses of Lemma \ref{lem:formal height induction} are satisfied by
\[
\widehat{\mathcal{P}}' = \widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}
\quad \mbox{and} \quad
\widehat{\mathcal{P}} = \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V} ,
\]
and so
\[
\mathrm{ht}_{ \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}  }  (\bar{\mathcal{Z}}_V(p))
(p^{n-1}+1 ) \cdot  \widehat{\vol}(   \widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}    )  +  \mathrm{ht}_{ \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}  }  (E)
\]
up to a rational multiple of $\log(p)$.  One again,  the second term on the right vanishes by claim (1) of Theorem \ref{thm:taut-hodge compare}.
For the first term on the right,  Proposition \ref{prop:easy numerical} and  Lemmas  \ref{lem:numerical basics} and \ref{lem:trivial volume shift} imply
\begin{align*}
\widehat{\vol}(  \widehat{\omega}^\mathrm{Hdg}_{A_0'/ \mathcal{S}_{V'}}
+ \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}    )
& =
\widehat{\vol}(   (0 , C_1)  +
\widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}   ) \\
&=   \widehat{\vol}(    \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}  )
+ (n-1) C_1 \int_{\mathcal{S}_{V'}(\C) } \chern(  \widehat{\omega}^\mathrm{Hdg}_{A'/ \mathcal{S}_{V'}}  )^{n-2} ,
\end{align*}
\[
= \log(2\pi) + 2h^\mathrm{Falt}_\kk
=  -  \frac{L'(0,\eps)}{L(0,\eps)}   - \frac{\log(D)}{2}   .
\]
This proves Theorem \ref{thm:height descent 2}.
\end{proof}

\section{Borcherds products}
\label{s:borcherds}

We continue to work with the Shimura variety $𝒮_V$ of \eqref{moduli inclusion}
associated to a $$-hermitian space $V$ of signature $(n-1,1)$, and now assume $n≥2$.

After explaining the connection between the complex orbifold $𝒮_V()$ and the Shimura variety \eqref{eq:X} associated to the unitary group $(V)$, we will use the results of \S \ref{ss:green functions}  to construct Green functions for certain linear combinations of the Kudla-Rapoport divisors  $𝒵_V(m) →𝒮_V$.

We then recall the arithmetic theory of Borcherds products on $𝒮_V$ from [13],
and show that one can produce Borcherds products whose divisors are linear combinations of only those $𝒵_V(p)$ with $p$ a prime congruent to $1$ modulo $D$, up to a linear combination of vertical divisors which can be computed explicitly.

\subsection{Green functions and Borcherds products}
\label{ss:general borcherds}

%We now combine Theorem~\ref{thm:int} with the formulas of \S~\ref{s:eisenstein} to derive an explicit formula for the integrals of automorphic Green functions as  in \eqref{eq:greenint}.
%For simplicity we restrict ourselves to the case when the principal part of the input harmonic Maass form is supported on the trivial coset (up to the constant term). Such input forms can be obtained as lifts of scalar valued harmonic Maass forms for $\Gamma_0(D)$.

\[
V\iso \Hom_\kk(W_0,W)
\]
as in \S \ref{ss:special shimura},  fix self-dual $_$-lattices
$𝔞_0 ⊂W_0$ and $𝔞 ⊂W$ as in \S \ref{ss:basic moduli}, and define a self-dual $_$-lattice
\begin{equation}\label{lattice choice}
L = \Hom_{\co_\kk}(\mathfrak{a}_0 , \mathfrak{a}) \subset V.
\end{equation}
The subgroup  $G⊂GU(W_0) ×GU(W)$ of  Remark \ref{rem:honest shimura} acts on both $W_0$ and $W$ via unitary similitudes, and we denote by $K ⊂G(_f)$ the largest compact open subgroup fixing the lattices $𝔞_0$ and $𝔞$.
Recalling the hermitian symmetric domain $𝒟$ of \S \ref{ss:u(v) shimura}, we identify
\[
\mathcal{S}_V(\C) \iso  G(\Q) \backslash \mathcal{D} \times G(\A_f) / K
\]
as in \S 2 of [13].

The group $G$ also acts on $V$, defining a surjective homomorphism
\[
G \to H=\Uni(V)
\]
with kernel the diagonally embedded $Res_/ 𝔾_m$.
Denoting again by $K$ the image of the above compact open subgroup under $G(_f) →H(_f)$,
and recalling the complex Shimura variety \eqref{eq:X}, we obtain a  finite cover
\begin{equation}\label{cover}
\mathcal{S}_V(\C)  \to \mathrm{Sh}_K(H,\mathcal{D}).
\end{equation}

Let $H^∞_2-n(D,^n)$ denote the space of $$-valued harmonic
Maass forms $f$ of weight $2-n$, level  $Γ_0(D)$,  and character $^n$ such that
\begin{itemize}
\item
$f$ is bounded at all cusps of $\Gamma_0(D)$ different from the cusp $\infty$,
\item
$f$ has polynomial growth at $\infty$, in sense that there is a
\[
P_f = \sum_{m\leq 0} c^+(m)q^m \in \C[q^{-1}]
\]
such that $f-P_f=o(1)$ as $q$ goes to $0$.
\end{itemize}
Such a harmonic Maass form has a Fourier expansion analogous to \eqref{eq:fourierf} with Fourier coefficients $c^±(m)∈$.

Fix an  $f∈H_2-n^∞(D,^n)$ with Fourier coefficients $c^±(m)$.
This form  can be lifted to a vector valued harmonic Maass form, in the sense of \S \ref{ss:green functions},   by setting
\begin{align}
\label{eq:lift}
\tilde f=\sum_{\gamma\in \Gamma_0(D)\backslash \SL_2(\Z)} (f|_{2-n} \gamma) (\omega_L(\gamma)^{-1}\varphi_0)
\in H_{2-n}(\omega_L) ,
\end{align}
where $φ_0 ∈S_L = [L'/L]$ is the characteristic function of $0 ∈L'/L$.
We denote the Fourier coefficients of $f̃$ by $c̃^±(m,μ)$ for $μ∈L'/L$ and $m∈$.
The coefficients of $f̃$ can be computed in terms of the coefficients of $f$, and for $m<0$ we have
\begin{align}
\label{eq:coeffrel}
\tilde c^{+}(m,\mu) =\begin{cases}
c^+(m) & \text{if $\mu=0$}\\
0&\text{if $\mu\neq 0$}
\end{cases}
\end{align}
as in  Proposition 6.1.2 of  [13] or  \S 5 of [47].

Under the covering map \eqref{cover},  the divisors $Z(m)$ and the hermitian line bundle $ℒ$ of \S \ref{ss:u(v) shimura} pull pack to the  divisors $𝒵_V(m)$ and $ℒ_V$ of \S \ref{ss:special shimura}.   This allows us to  apply the construction \eqref{eq:AutoGreen} to the vector valued form \eqref{eq:lift} to obtain a Green function
$Φ(z,h,f̃)$ for the analytic divisor
\[
Z(f) = \sum_{m>0} c^+(-m) Z(m)  \in \mathrm{Div}_\C(  \mathrm{Sh}_K(H,\mathcal{D})  )
\]
of \eqref{eq:zf}, which we pull back  to  a Green function $Φ_V(f)$ for the divisor
\[
\mathcal{Z}_V(f)= \sum_{m>0} c^+(-m) \mathcal{Z}_V(m) \in \mathrm{Div}_\C (  \mathcal{S}_V ) .
\]
If  $n>2$, or if $n=2$ and $V$ is anisotropic, then
\begin{equation}\label{S_V green integral}
\vol_\C(\widehat{\taut}_V )^{-1} \int_{  \mathcal{S}_V(\C)  } \Phi_V (f)   \chern( \widehat{\taut}_V )^{n-1}
=  \sum_{  m>0  } c^+(-m)B'(m, 0 ,s_0)
\end{equation}
by Theorem \ref{thm:int} and \eqref{eq:coeffrel}.  Here, as always, $s_0=(n-1)/2$.

Now consider the subspace of weakly holomorphic forms
\[
M_{2-n}^{!,\infty}(D,\eps^n) \subset  H^\infty_{2-n}(D,\eps^n) .
\]
These are meromorphic modular forms of the indicated weight, level, and character that are holomorphic outside the cusp $∞$ of $Γ_0(D)$.  If we fix an
\begin{equation}\label{input form}
f(\tau) = \sum_{m \gg -\infty} c(m) q^m \in M_{2-n}^{ ! , \infty}( D , \eps^n )
\end{equation}
in such a way that $c(m) ∈$ for all $m$ then,
after possibly replacing $f$ by a nonzero integer multiple, Theorem 5.3.1 of [13] provides us with a   Borcherds product $ψ(f)$.
This is a rational section of the hermitian line bundle $_V^⊗k(f)$ on $𝒮̅_V$
\begin{equation}\label{borcherds norm}
\Phi_V(f) = - \log\| \psi(f) \|^2,
\end{equation}
whose divisor (at least if $n>2$)  is
\begin{equation}\label{naive Bdiv}
\bar{\mathcal{Z}}_V(f) = \sum_{m>0} c(-m) \bar{\mathcal{Z}}_V(m)
\end{equation}
plus an explicit (but complicated) linear combination of boundary components and vertical divisors in characteristics $p|D$.

\begin{remark}\label{rem:lifted weight}
The integer $k(f)$, called the \emph{weight} of the Borcherds product, is given as follows:
Applying the construction \eqref{eq:lift} yields a weakly holomorphic vector valued form
\[
\tilde{f} = \sum_{ \substack{ m\in \Q \\ m \gg -\infty } } \tilde{c}(m) q^m  \in M_{2-n}^!(\omega_L)
\]
as in \eqref{eq:fourierf}, with coefficients $\tilde{c}(m) \in S_L=\C [L'/L]$.
The weight
\begin{equation}\label{wt def}
k(f) =  \tilde{c}(0,0)
\end{equation}
is the value of $\tilde{c}(0)$ at the trivial coset in $L'/L$.
\end{remark}

The space of all forms \eqref{input form} is an infinite dimensional $$-vector space.
As evidenced by the following theorem,   this gives us a  great deal of freedom to choose a Borcherds product $ψ(f)$ with prescribed properties.

\begin{theorem}\label{thm:good borcherds}
Assume $n>2$.
Given any infinite subset $\mathscr{A} \subset \Z^+$, there is a weakly holomorphic form    \eqref{input form} satisfying
\begin{enumerate}
\item  $c(m) \in \Z$ for all $m\in \Z$,
\item $c(-m) =0$ for all positive integers $m \not\in \mathscr{A}$,
\item   $k(f) \neq 0$,
\item
up to a vertical divisor supported  in characteristics  $p\mid D$,
the divisor of  $\psi(f)$  is equal to \eqref{naive Bdiv}.
\end{enumerate}
In particular, the divisor of $\psi(f)$ contains no irreducible components supported on the boundary.
\end{theorem}

The proof will occupy the entirety of the next subsection.

\subsection{Borcherds products with prescribed properties}

As $L' ⊂V$ is the dual lattice of $L$ with respect to the quadratic form  \ref{Q quadratic},    any $μ∈L'/L$ determines a coset $ -Q(μ)   ⊂$.
Denote by  $P(ω_L)$  the space of finite Fourier polynomials
\[
\sum_{\mu\in L'/L}\sum_{\substack{m \in \Z-Q(\mu)\\  m\leq  0}} c(m ,\mu )  \varphi_\mu \cdot q^m
\]
valued in  $S_L=[L'/L]$, whose coefficients satisfy  $c(m,μ)=c(m,-μ)$.
As $n≥2$,    we may  view
H_2-n(ω_L) ⊂P(ω_L)
by sending a harmonic Maass form  \eqref{eq:fourierf} to its principal part
\[
\sum_{m\le 0} c^+(m)  \cdot  q^m
\sum_{\mu\in L'/L} \sum_{m\le 0} c^+(m,\mu) \varphi_\mu \cdot  q^m .
\]
In particular, this allows us to view
M^!_2-n(ω_L) ⊂P(ω_L).

Denote by $S_L[[q^1/D]]$   the space of  $S_L$-valued formal power series in  the variable $q^1/D$, and denote by
\[
\vartheta \define q\frac{d}{dq} : S_L[[q^{1/D}]]  \to S_L[[q^{1/D}]]
\]
the Ramanujan theta operator.
For any $k∈$, taking $q$-expansions allows us to view  $M_k(ω̅_L) ⊂S_L[[q^1/D]]$.

Following [7] we consider the  $$-bilinear pairing
\[
\{ - , - \} : P(\omega_L)\times S_L[[q^{1/D}]] \to \C
\]
defined by
\[
\{p,g\} =  \sum_{ \substack{ m\ge 0 \\ \mu\in L'/L   } } c(-m,\mu) b(m,\mu) ,
\]
where $c(m,μ)$ and $b(m,μ)$ denote the coefficients of $p$ and $g$, respectively.

\begin{proposition}
\label{prop:crit}
For any
\[
p=\sum_{\mu\in L'/L}\sum_{m<0} c(m,\mu)\varphi_\mu\, q^{m}\in P(\omega_L),
\]
the following are equivalent:
\begin{enumerate}
\item
There exists a weakly holomorphic modular form $f\in M_{2-n}^!(\omega_L)$
whose principal part  agrees with $p$ up to a constant in $S_L$.
\item
We have $\{p, g\}=0$ for every $g\in S_{n}(\bar\omega_L)$.
\end{enumerate}
When these conditions holds,  the constant term
$c(0,0)$ of  $f$ is related to the value of the Eisenstein series
\begin{equation}\label{E_L}
E_L=E_L(\tau,s_0,n)\in M_n(\bar\omega_L)
\end{equation}
of \ref{eq:fouriereis} at $s_0=(n-1)/2$ by
\begin{align}
\label{eq:ctcond}
-c(0,0)= \{p,E_L\}=\sum_{\mu\in L'/L}\sum_{m>0}  c(-m,\mu)\, B(m,\mu, s_0) .
\end{align}
\end{proposition}

\begin{proof}
See  [3], or Corollary 3.9 of [7].
\end{proof}

\begin{proposition}
\label{prop:keyprop}
Let $\mathscr{A} \subset  \Z^+$ be any infinite subset.
If $n>2$,  there exists a weakly holomorphic   form $f\in M_{2-n}^!(\omega_L)$  whose  Fourier coefficients $c(m,\mu)$  are integers  satisfying the following properties:
\begin{itemize}
\item[(i)]
if  $c(-m,\mu)\neq 0$ with $m >0$,  then $\mu=0$ and $m\in \mathscr{A}$,
\item[(ii)]
$c(0,0)\neq 0$,
\item[(iii)] $\{f,\vartheta(g)\}= 0$ for all $g\in M_{n-2}(\bar \omega_L)$.
\end{itemize}
\end{proposition}

%The weakly holomorphic form $f\in M^!_{2-n}(\bar\omega_L)$ of the proposition can be obtained as the lift of an element of $M_{2-n}^{!,\infty}(D,\chi)$ as in \eqref{eq:lift}.

\begin{proof}
We generalize the argument of \cite[Proposition~3.1]{Br2}.
It follows from the main result of  [42] that
the space $M_{2-n}^!(\omega_L)$ has a basis of weakly holomorphic modular forms with integral coefficients. Hence, it suffices to show the existence of an $f\in M_{2-n}^!(\omega_L)$ with {\em rational} coefficients satisfying  the stated properties.

Write $M_n(\bar \omega_L,\Q)$ for the $\Q$-vector space of modular forms in $M_\kappa(\bar\omega_L)$ with rational coefficients, and $S_n(\bar \omega_L,\Q)$ for the subspace of cusp forms with rational coefficients.
To lighten  notation, throughout the proof we denote by $M$ the finite dimensional $\Q$-vector space
\[
M=M_n(\bar \omega_L,\Q)\oplus \vartheta M_{n-2}(\bar \omega_L,\Q)\subset
\]
and by $S$ the subspace
\[
S= S_n(\bar \omega_L,\Q)\oplus \vartheta M_{n-2}(\bar \omega_L,\Q)\subset M.
\]
The $\Q$-duals are denoted  $M^\vee$ and $S^\vee$, and we denote by
\[
\pr: M^\vee\to S^\vee .
\]
the surjection induced by  $S\subset M$.

For $\mu\in L'/L$ and $m\in \Z+Q(\mu)$,  denote by
$a_{m,\mu} \in M^\vee$  the linear functional  sending
\[
g=\sum_{\nu \in L'/L} \sum_{\ell \ge 0}  b(\ell,\nu) \varphi_\nu  \cdot q^\ell   \in M
\]
to the Fourier coefficient $a_{m,\mu}(g)= b(m,\mu).$
Let  $M_{\mathscr{A}}^\vee\subset M^\vee$  be the subspace generated by all functionals $a_{m,0}$ with $m\in \mathscr{A}$, and fix   $e_1,\dots,e_d\in M_{\mathscr{A}}^\vee$ such that
$\pr(e_1),\dots,\pr(e_d)$ is a basis of the subspace $\pr(M_{\mathscr{A}}^\vee)\subset S^\vee$.

For every $m\in \mathscr{A}$ there is a unique tuple
\[
r(m) = ( r_1(m),\dots,r_d(m) ) \in \Q^d
\]
such that
\[
\pr(a_{m,0}) = r_1(m)\cdot \pr(e_1)+\ldots+r_d(m)\cdot \pr(e_d).
\]
The linear combination
\[
\tilde a_{m,0} \define a_{m,0} -  ( r_1(m)\cdot e_1+\cdots+r_d(m)\cdot e_d)  \in M_{\mathscr{A}}^\vee
\]
clearly  lies in the kernel of $\pr$.

\begin{lemma}
There is an $m\in \mathscr{A}$ such that  the Eisenstein series  \eqref{E_L} satisfies   $\tilde a_{m,0}(E_{L}) \neq 0$.
\end{lemma}

\begin{proof}
We assume on the contrary that $\tilde a_{m,0}(E_{L})=0$ for all $m\in \mathscr{A}$.
In other words, that the coefficients \eqref{eq:coeff} satisfy
\begin{align}
\label{eq:eisa}
B(m,0,s_0) =  r_1(m)\cdot e_1( E_{L})+\ldots+r_d(m)\cdot e_d(E_{L})
\end{align}
for all such $m$.

Let $\|r\|$ be the euclidian norm of a vector $r\in \R^d$, and denote by $\|\cdot\|$ a norm on $S^\vee\otimes_\Q \R$, say the operator norm with respect to a fixed norm on the finite dimensional vector space $S\otimes_\Q \R$.
Since $\pr(e_1),\dots,\pr(e_d)$ are linearly independent, there exists an $\eps>0$ such that
\[
\eps \|r\|\leq \| r_1 \pr(e_1)+\ldots+r_d  \pr(e_d)\|
\]
for all $r=(r_1,\dots,r_d)\in \R^d$.
Taking $r=r(m)$,   we obtain
\[
\eps \cdot \|r(m)\| \leq \| \pr (a_{m,0}) \| .
\]
On the other hand, \eqref{eq:eisa} implies that there is a constant $c>0$ such that
\[
|B(m,0,s_0)|\leq c\cdot \|r(m)\|.
\]
Combining these last two inequalities, we find
\begin{align}
\label{eq:3}
|B(m,0,s_0)| \leq \frac{c}{\eps}\cdot \| \pr (a_{m,0}) \|
\end{align}
for all $m\in \mathscr{A}$.

The Hecke bound for the coefficients of (scalar valued) cusp forms of weight $n$ for $\Gamma(D)$ implies that
\[
|\pr (a_{m,0})(g)|= O(m^{n/2}),
\]
as $m\to \infty$ for any $g\in S_n(\bar \omega_L)$.
On the other hand, an elementary estimate shows that
\[
|\pr (a_{m,0})(g)|= O(m^{n-2+\delta}),
\]
as $m\to \infty$ for any $\delta>0$ and any $g\in\vartheta ( M_{n-2}(\bar \omega_L))$.
As $n>2$, these bounds imply
$\| \pr (a_{m,0}) \| = O(m^{n-3/2})$.
Combining this with \eqref{eq:3} shows that
\[
|B(m,0,s_0)| = O(m^{n-3/2})
\]
for $m\in \mathscr{A}$ and $m\to \infty$,  contradicting Corollary \ref{cor:eisgrowth}.
\end{proof}

We now complete the proof of Proposition \ref{prop:keyprop}.
The lemma provides us with an $a=\tilde a_{m,0}\in M_{\mathscr{A}}^\vee$ satisfying $\pr(a)=0$ and $a(E_L)\neq 0$.    By definition of $M_{\mathscr{A}}^\vee$, we may expand $a$ as a finite linear combination
\[
a = \sum_{m \in \mathscr{A}} c(m,0)  a_{m,0}
\]
with $c(m,0) \in \Q$,
and then form the Fourier polynomial
\[
p = \sum_{m \in \mathscr{A}} c(m,0) \varphi_0 \cdot q^m \in P(\omega_L).
\]

The condition $\pr(a)=0$ implies that $\{ p,g\}=0$ for all $g\in S_{n}(\bar\omega_L)$, and
$\{ p, \vartheta(g)\}=0$ for all $g\in S_{n-2}(\bar\omega_L)$.
In particular,   Proposition \ref{prop:crit}  provides us with a form $f\in M^!_{2-n}(\omega_L)$
whose principal part agrees with $p$ up to a constant, satisfies
\[
\{ f, \vartheta(g) \} = \{ p,\vartheta(g)\} =0
\]
for all $g\in S_{n-2}(\bar\omega_L)$, and has constant term   $ \{ p,E_L\} =a(E_L) \neq 0$.
\end{proof}

As a special case of the following proposition,  the form $f$ of Proposition \ref{prop:keyprop} lies in the image of the lifting map
\begin{equation}\label{holomorphic lift}
M^{!,\infty}_{2-n}(D,\varepsilon^n) \map{h\mapsto \tilde{h}} M_{2-n}^!(\omega_L)
\end{equation}
defined by \eqref{eq:lift}.

\begin{proposition}
\label{prop:keypropscal}
Assume $n>2$, and let $f\in  M_{2-n}^!(\omega_L)$ be a form whose Fourier coefficients $c(m,\mu)$ satisfy $c(m,\mu)=0$ for all $(m,\mu)$ with $m<0$ and $\mu\neq 0$.
In the notation of \eqref{eq:lift}, there exists an  $h\in M^{!,\infty}_{2-n}(D,\varepsilon^n)$
with principal part
\begin{align}
\label{eq:prinh}
\sum_{m<0} c(m,0) q^m + \mathrm{constant},
\end{align}
and  such that $\tilde h=f$.
\end{proposition}

\begin{proof}
Using the basis $\{ \varphi_\mu\}_{\mu\in L'/L}$ of $S_L$, any form $g(\tau)  \in S_n(\bar\omega_L)$ can be written as $g(\tau) =\sum_{\mu\in L'/L} g_\mu(\tau)   \varphi_\mu$.
Taking the component corresponding to $\mu=0$ defines a linear map
\[
S_n(\bar\omega_L)\map{ g\mapsto g_0}   S_{n}(D,\varepsilon^n).
\]

We claim that the map $g\mapsto g_0$  is surjective.  Indeed,  this is equivalent to the injectivity of the adjoint map $ S_{n}(D,\varepsilon^n)\to S_n(\bar\omega_L)$, which is just the map $g\mapsto \tilde{g}$ of \eqref{eq:lift}.
This injectivity follows from the explicit formula for the Fourier expansion of $\tilde g $ found in Proposition 3.3.2 of [14], along with the fact that the $\Q/\Z$-valued quadratic form on $L'/L$ represents all elements of $\frac{1}{D}\Z/\Z$ primitively.

Now suppose we are given a cusp form
\begin{equation}\label{dual cusp}
\sum_{m>0} b(m) q^m \in S_{n}(D,\varepsilon^n).
\end{equation}
If we choose a  $g\in S_n(\bar\omega_L)$ such that $g_0 = \sum_m b(m) q^m$, then
\[
\sum_{m>0} c(-m,0)b(m)= \{ f,g\}=0,
\]
where the first equality follows from our hypotheses on the coefficients of $f$, and the second follows from the residue theorem.
As this vanishing holds for all forms \eqref{dual cusp}, it follows from
Serre duality on the modular curve $X_0(D)$ that there exists an $h\in M^{!,\infty}_{2-n}(D,\varepsilon^n)$ with principal part  \eqref{eq:prinh}.
In particular, it follows from  \eqref{eq:coeffrel} that $\tilde{h} -f$ is holomorphic at $\infty$.  As $\tilde{h} -f$ has  negative weight, it is identically $0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:good borcherds}]
First pick a vector-valued form  $\tilde{f}\in M_{2-n}^!(\omega_L)$ as in  Proposition \ref{prop:keyprop}, and then apply Proposition \ref{prop:keypropscal} to pick a scalar-valued form
\[
f(\tau) = \sum_{ m \gg -\infty } c(m) q^m  \in M^{!,\infty}_{2-n}(D,\varepsilon^n)
\]
that maps to it under \eqref{holomorphic lift}.
It follows from the relation \eqref{eq:coeffrel} between the coefficients of $f$ and $\tilde{f}$ that
for all $m>0$ the coefficient  $c(-m)$ is an integer, and vanishes unless $m\in \mathscr{A}$.

Now we use the fact that $M^{!,\infty}_{2-n}(D,\varepsilon^n)$ has a basis of forms with integer coefficients\footnote{One can deduce this from the corresponding statement for holomorphic modular forms, by multiplying weakly holomorphic modular forms by powers of Ramanujan's discriminant to kill the poles at $\infty$}.    For any  $\sigma \in \Aut(\C/\Q)$, it follows that the formal $q$-expansion $f^\sigma$ again lies in $M^{!,\infty}_{2-n}(D,\varepsilon^n)$.  The difference  $f(\tau)-f^\sigma(\tau)$ is then  holomorphic at every cusp with  weight $n-2<0$, and so vanishes identically.
Thus all coefficients of $f(\tau)$ are rational, and we may replace $f(\tau)$  by a positive integer multiple to assume that $c(m)\in \Z$ for all $m\in \Z$.

As we chose $\tilde{f}$ so that its constant term $\tilde{c}(0,0)$ is nonzero, the associated Borcherds product $\psi(f)$ has nonzero weight by Remark \ref{rem:lifted weight}.

It only remains to verify  property (4) in Theorem \ref{thm:good borcherds}.
For this we appeal to Theorem 5.3.3 of [13], which tells us that
\begin{equation}\label{divisor boundary}
\mathrm{div}(\psi(f)) = \bar{\mathcal{Z}} _V(f)
+ \sum_{m>0} c(-m) \mathcal{B}(m) ,
\end{equation}
up to a linear combination of divisors supported in characteristics $p\mid D$.
Here, as in (5.3.3) of [13],
\begin{equation}\label{boundary mult}
\mathcal{B}(m) = \frac{m}{n-2} \sum_\Phi \rho_{L_0}(m)  \cdot \mathcal{S}_V(\Phi),
\end{equation}
where the sum is over a finite set $\{ \Phi \}$ indexing the irreducible boundary components $\mathcal{S}_V(\Phi) \subset \partial \bar{\mathcal{S}}_V$, each of which is connected and smooth over $\co_\kk$.  Inside the sum, each $L_0$ is a self-dual hermitian $\co_\kk$-module (which depends on $\Phi$)  of signature $(n-2,0)$, and
\[
\rho_{L_0}(m) = \# \{ x\in L_0 : \langle x,x\rangle =m \}
\]
is the number of times that lattice represents $m$.

As explained  in \S 3.1 of [13],  each $\Phi$ is an equivalence class of pairs $(I , g)$ in which $I \subset V$ is an isotropic $\kk$-line, and $g\in G(\A_f)$.  This data determines a filtration
\[
\mathfrak{a} \subset \mathfrak{a}^\perp \subset gL
\]
by $\co_\kk$-module direct summands, with  $\mathfrak{a} = I \cap gL$  isotropic of rank one and  $\mathfrak{a}^\perp = \{ x\in gL  :   \langle x, \mathfrak{a} \rangle =0 \}$.
The quotient $L_0 = \mathfrak{a}^\perp / \mathfrak{a}$  inherits a self-dual hermitian form from that  on $gL \subset V$, and the filtration admits a (non-canonical) splitting
\[
gL = \mathfrak{a} \oplus L_0 \oplus \mathfrak{b}
\]
in which  $\mathfrak{b}$ is rank one and isotropic,  and $\mathfrak{a} \oplus \mathfrak{b}$  is orthogonal to $L_0$.

As $L_0$ and $gL$ are themselves self-dual hermitian $\co_\kk$-modules, we may form modular forms valued in the finite dimensional vector spaces $S_{L_0}=\C [L_0'/L_0]$ and $S_{gL} = \C[(gL)'/(gL)]$, exactly we did for $L$.  The action of   $g\in G(\A_f)$ defines a canonical  bijection
L' / L \iso (gL)'/(gL),
which induces an isomorphism $S_L \iso S_{gL}$  respecting the Weil representations on source and target.

To each $\Phi$,  we may  attach the $S_{L_0}$-valued theta series
\[
\Theta_\Phi(\tau) = \sum_{ \mu \in L_0' / L_0  }  \Theta_{\Phi,\mu} (\tau)   \varphi_\mu   \in M_{n-2}(\bar{\omega}_{L_0} )
\]
\Theta_{\Phi,\mu} (\tau) =  \sum_{ x\in \mu+L_0  } q^{\langle x,x\rangle} .
As in Theorem 4.1 of [48], there is an induced $S_{gL}$-valued modular form
\[
\ind_L(\Theta_\Phi) = \sum_{\substack{\mu\in L_0'/L_0\\
\beta \in \mathfrak{d}_\kk^{-1} \mathfrak{a}/\mathfrak{a}}} \Theta_{E,\mu}(\tau)\varphi_{\mu+\beta}
\in  M_{n-2}(\bar\omega_{gL}) ,
\]
and we  use  $S_L \iso S_{gL}$ to view  $\ind_L( \Theta_\Phi )\in M_{n-2}(\bar\omega_{L})$.

Using the relation \eqref{eq:coeffrel} between the coefficients of $\tilde{f}$ and $f$, we see that
\[
\sum_{m>0} m  c(-m) \rho_{L_0}(m) =  \{ \tilde{f} , \vartheta(\ind_L( \Theta_\Phi )) \} =0,
\]
where the second equality follows from condition (iii) in Proposition \ref{prop:keyprop}.  Comparing with \eqref{boundary mult} shows that
\sum_{m>0}   c(-m) \mathcal{B}(m) = 0,
and property (4) of Theorem  \ref{thm:good borcherds} follows by comparison with  \eqref{divisor boundary}.
\end{proof}

%We now prove a second existence result for Borcherds products. It can be used to determine the arithmetic volumes of the special divisors $Z(p,0)$ for almost all primes $p\equiv 1\pmod{D}$.  \note{I don't think we need this.  There is a cleaner way to get at volume of $Z(p,0)$}
%Assume that $n\geq 3$ and let $\kk$ and $L$ be as above. Let $\mathscr{A}$ be an infinite set of positive integers.
%There exists a positive  integer $t_0$ such that for all $m_0\in \mathscr{A}$ with $m_0\geq t_0$ there is a Borcherds product $\Psi$ of weight $0$ on $X^*$ with
%\dv_{X^*}(\Psi) = \sum_{\substack{m\in \mathscr{A}\\m\geq t_0}} c(m) Z^*(m,0),
%where $c(m_0)\neq 0$ and only finitely many of the other coefficients $c(m)$ are non-zero.
%In particular, the divisor of $\Psi$ does not include any boundary components.
%In view of   \cite[Theorem 8.1]{Hof} (see also \cite[Theorem 13.3]{Bo1}) and Corollary \ref{cor:boundary},  Theorem \ref{thm:bpdiv2} is a consequence of the following proposition.
%%Assume that $n\geq 3$.
%There exists a positive  integer $t_0$ such that for all $m_0\in \mathscr{A}$ with $m_0\geq t_0$ there is a
%weakly holomorphic modular form $f\in M_{2-n}^!(\omega_L)$ with integral Fourier coefficients $c(m,\mu)$ with the properties:
%if $(m,\mu)\in \Q^+\times L'/L$ with $c(-m,\mu)\neq 0$, then $\mu=0$ and $m\in \mathscr{A}$,
%\item[(ii)] $c(-m_0,0)\neq 0$ and
%$c(0,0)= 0$,
%\item[(iii)] $\{f,\vartheta(g)\}= 0$ for all $g\in M_{n-2}(\bar \omega_L)$.
%We use the notation of the proof of
%Proposition \ref{prop:keyprop}. For $l\in \Z^+$ we let
%$A_l\subset M^\vee$ be the linear span of all  functionals
%a_{m,0}\in M^\vee \quad \text{with $m\in \mathscr{A}$ and $m\geq l$.}
%We obtain a descending chain of subspaces
%M^\vee \supset A_1 \supset A_2 \supset  A_3 \supset\dots .
%Since $M^\vee$ has finite dimension this chain becomes stationary, i.e., there exists a positive integer $t_0$ such that $A_l=A_{t_0}$ for all $l\geq t_0$.
%Hence for every $m_0\in \mathscr{A}$ with $m_0\geq t_0$ and for every $l\geq m_0$ we have
%a_{m_0,0}\in A_l.
%Consequently, there exist $m_1,\dots, m_d\in A_l$ and coefficients $c(m_1),\dots ,c(m_d)\in \Q$ such that
%-a_{m_0,0}= \sum_{j=1}^d c(m_j) a_{m_j,0}.
%According to Proposition~\ref{prop:crit}, there  exists an $f\in M_{2-n}^!(\omega_L)$ with principal part
%q^{-m_0}\varphi_0+\sum_{j=1}^d c(m_j)q^{-m_j}\varphi_0
%that satisfies (i)--(iii).

\subsection{A carefully chosen Borcherds product}

Recalling the functions $𝐚_k(s)$ of  \eqref{a_k},  set
𝐛_V,1(s)  =𝐚_1(s).
For  $k ≥2$  even,  set
\[
\mathbf{b}_{V,k}(s)
= \mathbf{a}_k(s)
\prod_{\ell \mid D} \left(   1+ \leg{-1}{\ell}^{ \frac{k}{2} } \mathrm{inv}_\ell(V) \ell^{ -s- \frac{k}{2} }   \right).
\]
For  $k ≥3$ odd,  set
\[
\mathbf{b}_{V,k}(s)
= \mathbf{a}_k(s)
\prod_{\ell \mid D} \left(  1+ \leg{-1}{\ell}^{ \frac{k-1}{2} } \mathrm{inv}_\ell(V) \ell^{ -s +\frac{1-k}{2}}  \right)^{-1}.
\]
When  $k>1$ we have
$𝐛_V,k(s)  𝐛_V,k+1(s) =  𝐚_k(s)  𝐚_k+1 (s)$,
which implies  that the function \eqref{A_V} factors as
\begin{equation}\label{A_V factorization}
\mathbf{A}_V(s) = \mathbf{b}_{V,1}(s) \cdots \mathbf{b}_{V,n}(s)  .
\end{equation}

Now  suppose $n>2$,  abbreviate
\[
\mathscr{A} = \{ \mbox{primes } p \equiv 1\pmod{D} \},
\]
and assume that  the weakly modular form
\[
f(\tau) = \sum_{m \gg -\infty} c(m) q^m \in M_{2-n}^{ ! , \infty}( D , \eps^n )
\]
of \eqref{input form} is chosen as in Theorem \ref{thm:good borcherds}.
In particular, the divisor of the Borcherds product $ψ(f)$ contains no components of the boundary, and so
\begin{equation}\label{borcherds divisor}
\mathrm{div} ( \psi(f)  ) =  \bar{\mathcal{Z}}_V(f)  +
\vertical(f)
\end{equation}
in which
\[
\bar{\mathcal{Z}}_V(f)
=  \sum_{ p \in \mathscr{A} }  c(-p) \bar{\mathcal{Z}}_V(p),
\]
and $ (f)$  is  supported in characteristics dividing $D$.

\begin{remark}
The notation $\vertical(f)$ is slightly misleading, as $ \bar{\mathcal{Z}}_V(f)$ may itself have vertical components.
Any such components are supported on the exceptional divisor $\mathrm{Exc}_V \subset \bar{\mathcal{S}}_V$, by
Corollary 3.7.3 of    [13].
\end{remark}

\begin{proposition}\label{prop:awesome B}
The  Borcherds product $\psi(f)$ has weight
\[
k(f) =  \frac{1}{\mathbf{b}_{V,n}(0)}  \sum_{p \in \mathscr{A} } c(-p) (   p^{n-1} + 1  ) ,
\]
and, recalling the notation \eqref{constant metrics}, satisfies
\[
( \vertical (f)  , 0 )  =  (E,0) -   k(f)   \sum_{\ell \mid D} \left(  0  , \frac{ \log(\ell) }{1+ \beta_\ell } \right)  \in \widehat{\Pic}(\mathcal{S}_V)_\Q,
\]
where $E$ is a divisor (which may depend on  $f$)  supported on the exceptional divisor
of Definition \ref{def:special exceptional}, and
\begin{equation}\label{bang}
\beta_\ell  = (-1)^{n+1}  \cdot \begin{cases}
\leg{-1}{\ell}^{\frac{n}{2} }    \inv_\ell(V)     \ell^{ \frac{n}{2} }  &  \mbox{if $n$ is even} \\[2ex]
\leg{-1}{\ell}^{\frac{n-1}{2} }   \inv_\ell(V)    \ell ^{ \frac{n-1}{2}  } &  \mbox{if $n$ is odd.}
\end{cases}.
\end{equation}
\end{proposition}

\begin{proof}
The proof requires a short digression on Eisenstein series.

For any divisor $r\mid D$ set $r'=D/r$.  Our assumption that $D$ is odd implies that the quadratic character $\eps : (\Z/D\Z)^\times \to \{ \pm 1\}$ determined by $\kk$ is
\[
\eps(a) =  \leg{a}{D}.
\]
Hence we may  factor  $\eps = \eps_r \cdot \eps_{r'}$ with
\[
\eps_r(a) = \left( \frac{a}{r} \right) \quad \mbox{and} \quad \eps_{r'} (a) = \left( \frac{a}{r'} \right).
\]
Define the quadratic Gauss sum
\[
\tau(\eps_r) =  \sum_{  a \in (\Z/r\Z)^\times } \eps_r(a) e^{2\pi i a/r} = \begin{cases}
\sqrt{r}  & \mbox{if } r\equiv 1\pmod{4} \\
i\sqrt{r} & \mbox{if } r \equiv 3\pmod{4},
\end{cases}
\]
and similarly with $r$ replaced by $r'$.

\begin{lemma}\label{lem:eisenstein formulas}
For every divisor  $r\mid D$ there is an Eisenstein series
\[
E_r =  \sum_{m \ge 0} e_r(m) \cdot q^m \in M_n(\Gamma_0(D),\eps^n)
\]
whose Fourier coefficients are as  follows.
The constant term  is
\[
e_r(0)= \begin{cases}
1 & \mbox{if }r=1 \\
0 &  \mbox{otherwise.}
\end{cases}
\]
If $n$ is even the coefficients indexed by $m>0$ are
\[
e_r(m)   =
\frac{  r^{n/2} (-2\pi i )^n    }{ D^n  \Gamma(n)  L_D( n ,\eps^n ) }
\sum_{ \substack{   c\mid m \\ c>0  \\ \gcd( m /c, r) =1   } }
c^{n-1}   \sum_{ d \mid \gcd( c , r'  ) } d \mu( r' / d) .
\]
If $n$ is odd the coefficients indexed by $m>0$ are
\[
e_r(m)   =
\eps_r(r')   \frac{ r^{n/2} (-2\pi i )^n  \tau(\eps_{r'})    }{  D^n   \Gamma(n) L_D( n ,\eps^n ) }
\sum_{ \substack{  c\mid m \\ c>0  \\ \gcd( m /c, r) =1   } }
\eps_r( m/c)   \eps_{r'}(c) \cdot c^{n-1}.
\]
In both formulas $L_D(s,\eps^n)$ is the Dirichlet $L$-function with Euler factors at all $\ell\mid D$ removed.
\end{lemma}

\begin{proof}
% Choose a matrix
% \[
% R_r = \left(\begin{matrix}    \alpha & \beta \\  \gamma r' & \delta r  \end{matrix}\right)  \in \Gamma_0(D/r)
% \]
% with $\alpha,\beta,\gamma,\delta\in \Z$,  set
% $
% W_r = R_r   \cdot  \left( \begin{smallmatrix}   r \\ & 1 \end{smallmatrix}\right),
% $
%and  define
%E (z)
%  =
% \sum_{ \left( \begin{smallmatrix}  a & b \\ c & d \end{smallmatrix} \right)    \in  \Gamma_\infty \backslash \Gamma_0(D) }   \frac{ \eps^n (d) } {    (c z + d)^n  }  \in M_n( \Gamma_0(D) , \eps^n).
% \]
% Here $\Gamma_\infty \subset \mathrm{SL}_2(\Z)$ is the subgroup of upper triangular matrices.
%The claim is that
%  \[
%E_r = \eps_r(-\beta) \eps_{r'}(\alpha r) \cdot ( E \mid_n W_r ) \in M_n(\Gamma_0(D),\eps^n).
% has the stated Fourier expansion.
For each $s,t\in \Z/D\Z$ define an Eisenstein series
\[
G_{(s,t)}(z) = \sum_{  \substack{   c,d \in \Z  \\  (c,d) \neq (0,0) \\ (c,d)  \equiv (s,t)  \pmod{D}  } }    (   cz + d)^{-n}  .
\]
Theorem 7.1.3 of [43] shows that
\[
E_r(z)   =
\frac{   \eps^n_r(r')  }{  2 r^{n/2}  L_D( n ,\eps^n ) }
\sum_{ s,t  \in \Z/D\Z  }   \eps_r^n(  s )  \eps_{r'}^n(  t)   G_{ (  s,t   )  }  ( r' z )
\]
has the desired Fourier expansion.
% Indeed, one first checks that [warning:there was a bad search and replace in the formulas below. Some $\alpha$ should be $\beta$]
% \[
%L_D( n ,\eps^n )   \cdot  E (z)
% = \frac{1}{2}  \sum_{ s , t  \in \Z/D\Z  }  \eps^n(t) G_{(s,t)} (Dz) .
%Applying $|_nW_r$ to both sides and using
% \[
% W_r
% =
%  \left( \begin{matrix} D^{-1} & \\ & 1  \end{matrix} \right)
%  \left(\begin{matrix}     \alpha  r &  \alpha r' \\ \gamma  & \delta  \end{matrix}\right)
%   \left( \begin{matrix} D & \\ & r  \end{matrix} \right),
% \]
%one then finds
% \begin{align*}
% L_D( n ,\eps^n )  \cdot   E_r(z)
% &  =  \frac{ \eps^n_r(-\alpha) \eps^n_{r'}(\alpha r) }{ 2 D^{n/2} }  \sum_{ s ,t\in \Z/D\Z   }  \eps^n(t) G_{(s,t)} \big|_n \left( \begin{matrix} D \\ & 1 \end{matrix} \right) W_r \\
%& =
% \frac{\eps^n_r(-\alpha) \eps^n_{r'}(\alpha r)}{ 2 D^{n/2}}  \sum_{  s , t  \in \Z/D\Z   }  \eps^n(t)
% G_{ (    \alpha rs + \gamma t  ,  \alpha  r' s  + \delta t    )  }  \big|_n
%   \left( \begin{matrix} D & \\ & r  \end{matrix} \right) \\
%   & =
% \frac{\eps^n_r(-\alpha) \eps^n_{r'}(\alpha r) }{ 2r^{n/2}   }  \sum_{ s,t  \in \Z/D\Z  }  \eps^n(t)
% G_{ (    \alpha rs + \gamma t  ,  \alpha  r' s  + \delta t    )  }  ( r' z )  \\
%  & =
% \frac{\eps^n_r(-\alpha) \eps^n_{r'}(\alpha r) }{ 2 r^{n/2}  }  \sum_{ s,t  \in \Z/D\Z  }  \eps^n( -\alpha r' s + \alpha rt)
% G_{ (  s,t    )  }  ( r' z ) \\
%  & =
% \frac{\eps^n_r(r')  }{ 2 r^{n/2}  }
% \sum_{ s,t  \in \Z/D\Z  }   \eps_r^n(  s )  \eps_{r'}^n(  t)   G_{ (  s,t   )  }  ( r' z )
%   \end{align*}
%where the second to last formula  is obtained by
%(s,t) \mapsto ( \delta s - \gamma t , -\alpha r' s + \alpha rt)
%= (s,t) \cdot \left( \begin{matrix}     \delta   &   -\alpha r'  \\  -\gamma & r   \end{matrix} \right)  .
%First suppose that $n$ is even, so that $\eps_r^n$ and $\eps_{r'}^n$ are  the trivial Dirichlet characters modulo $r$ and $r'$, respectively.
%The constant term is
%g_r(0) = \begin{cases}
%2 \zeta_D(n ) & \mbox{if } r=1  \\
%0 & \mbox{otherwise,}
%and when $m>0$ we have
%g_r (m) =
%  \frac{2 r^n (-2\pi i )^n   }{ D^n \Gamma(n) }  \sum_{ \substack{  c\mid m \\ c>0  \\ \gcd( m/c, r) =1   } }
%   c^{n-1}   \sum_{ d \mid \gcd( c , r'  ) } d \mu( r' / d) .
%Now suppose that $n$ is odd, so that   $\eps_r^n=\eps_r$ and $\eps_{r'}^n=\eps_{r'}$ are  primitive Dirichlet characters modulo $r$ and $r'$.  The constant term is
%g_r(0) = \begin{cases}
%2 L (n, \eps ) & \mbox{if } r=1  \\
%0 & \mbox{otherwise,}
%and when  $m>0$ we have
%g_r (m) =
%  \frac{2 r^n (-2\pi i )^n  \tau(\eps_{r'})   }{ D^n  \Gamma(n) }
%   \sum_{ \substack{  c\mid m \\ c>0  \\ \gcd( m/c, r) =1   } }
% \eps_r( m/c)   \eps_{r'}(c) \cdot c^{n-1} .
\end{proof}

%The above formulas can be rewritten using the relations
% \frac{  2 \zeta(n) } {  \zeta(1-n)  }  =    \frac{ (-2\pi i )^n }{  \Gamma(n)  }
%   \frac{ 2 L(n,\eps_{r'}) }{  L(1-n,\eps_{r'}) }   =   \frac{ (-2\pi i  )^n  \tau (\eps_{r'})   r^n   } {      D^n \Gamma(n) }
%for $n$ even and odd (respectively).

Suppose  $r\mid D$, and set $r'=D/r$.
If $n$ is even, set
\[
a   =  \frac{   (-2\pi i )^n \mu(D)   }{ D^n \Gamma(n) L_D(n , \eps^n) }  \quad\mbox{and}  \quad
b_r   = r^{n/2}  \mu( r )  .
\]
If $n$ is odd, set
\[
a =
\frac{   (-2\pi i )^n   \tau(\eps) }{ D^n \Gamma(n) L_D(n , \eps^n)  }   \quad\mbox{and}  \quad
b_r = \frac{ r^{n/2} \eps_r( r')     \tau(\eps_{r'}) }{  \tau(\eps)  }.
\]
One may check directly that
b_r=\prod_{\ell\mid r} b_\ell,
where the product is over all primes $\ell\mid r$, and that when $p \in \mathscr{A}$ the formulas of Lemma \ref{lem:eisenstein formulas} simplify to
\begin{equation}\label{simple miyake}
e_r(p) = a b_r \cdot  ( p^{n-1}+1 ) .
\end{equation}