Arithmetic volumes of unitary Shimura varieties]Arithmetic volumes of unitary Shimura varieties
Jan H. Bruinier and Benjamin Howard]Jan Hendrik Bruinier and Benjamin Howard
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7,
D–64289 Darmstadt, Germany
Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467, USA

J.B. was  supported in part by the LOEWE research unit USAG.  B.H.  was supported in part by NSF grant  DMS-1801905.

The integral model of a $\mathrm{GU}(n-1,1)$ Shimura variety  carries a universal abelian scheme over it,
and the  dual top exterior power of its Lie algebra carries a natural hermitian metric.  We express the arithmetic volume of this metrized line bundle, defined as  an iterated self-intersection in the Gillet-Soulé arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet $L$-functions.

§ INTRODUCTION

The explicit calculation of arithmetic volumes of Shimura varieties began with the work of Kühn [36] and Kramer (independently, in unpublished work).  Over the moduli stack $\mathcal{X} \to \Spec(\Z)$ of elliptic curves there is a  line bundle of weight one modular forms, characterized as the dual  Lie algebra of the universal elliptic curve.
This line bundle carries a natural hermitian metric, and so determines a class in the codimension one arithmetic Chow of $\mathcal{X}$, in the sense of Gillet-Soulé.  Working on a suitable compactification, Kühn and Kramer computed the self-intersection multiplicity of this hermitian line bundle, and gave a simple formula for it in terms of the logarithmic derivative of the Riemann zeta function at $s=-1$.

More generally, if $\mathcal{X}$ is an  integral model of a PEL type (or even Hodge type) Shimura variety, then $\mathcal{X}$ carries a Hodge bundle: the dual of the top exterior power of the Lie algebra of the universal abelian scheme over $\mathcal{X}$.  The Hodge bundle again carries a natural hermitian metric, and one can ask if its arithmetic volume, defined as the  $\mathrm{dim}(\mathcal{X})$-fold self-intersection in the Gillet-Soulé arithmetic Chow group, is again related to logarithmic derivatives of Dirichlet $L$-functions.

In some instances this is known.
If $\mathcal{X}$ is the integral model of a quaternionic Shimura curve, the volume was computed by Kudla-Rapoport-Yang [34].
Hilbert modular surfaces and the Siegel threefold were considered by Bruinier-Burgos-Kühn [6] and Jung-von Pippich [29], respectively.
The arithmetic volumes of $\mathrm{GSpin}(n,2)$   Shimura varieties, which include all of the above examples as special cases, were computed (up to a $\Q$-linear combination of logarithms of certain bad primes)
by Hörmann [21].

The present paper deals with the case of $\mathrm{GU}(n-1,1)$ Shimura varieties.
The overall strategy is similar to that used in [6] and [21], in that it uses the theory of Borcherds products to express the metrized Hodge bundle as a  linear combination (in the arithmetic Chow group) of smaller unitary Shimura varieties embedded as divisors.
This allows one to compute the arithmetic self-intersection of the Hodge bundle using induction on the dimension of the ambient Shimura variety.

The added complications in the $\mathrm{GU}(n-1,1)$ case come mainly from the primes of bad reduction of the Shimura variety; that is to say, from primes dividing the discriminant of the quadratic imaginary field used to define the unitary group.  At such primes, the divisors of Borcherds products  include quite complicated linear combinations of vertical divisors, which then appear as correction terms when one expresses the Hodge bundle as a linear combination of smaller Shimura varieties.
These vertical components in the divisors of Borcherds products, which have no analogue in [6] or [21],  were calculated explicitly in [13], making the present work possible.

§.§ Statement of the main result

Let $\kk\subset \C$ be an imaginary quadratic field of odd discriminant

\[
-D = \mathrm{disc}(\kk).
\]

Given an integer $n\ge 2$, there is a regular  Deligne-Mumford stack

\[
\mathcal{M}_{(n-1,1)} \to \Spec(\co_\kk),
\]

flat of relative dimension $n-1$,
parametrizing principally polarized abelian schemes $A$, endowed with an action  $\co_\kk\to \End(A)$,  and extra data encoding a signature $(n-1,1)$ condition on $\Lie(A)$.
See <ref> for details.

This stack admits a decomposition

\[
\mathcal{M}_{(n-1,1)}  = \bigsqcup_W \mathcal{M}_W
\]

into open and closed substacks indexed by strict similarity classes of relevant $\kk$-hermitian spaces $(W,h)$ of signature $(n-1,1)$,  in such a way that the generic fiber of $\mathcal{M}_W$ is a   Shimura variety  for the unitary similitude group $\mathrm{GU}(W)$.
Here relevant means  that $W$ contains an $\co_\kk$-lattice $\mathfrak{a} \subset W$ that is self-dual, in the sense of (<ref>).
The notion of strict similarity is defined in <ref>, but the important fact is that there is a dichotomy between the cases of $n$ odd and $n$ even.
If $n$ is odd the  relevant $W$ form a single strict similarity class, and the disjoint union has a single term.
If $n$ is even then strict similarity is equivalent to isometry, and the number of terms in the disjoint union is $2^{o(D) -1}$, where

\begin{equation*}%\label{o(D)}
o(D) = \# \{ \mbox{prime divisors of }D\}.
\end{equation*}

Fix a relevant $W$ as above, and consider the restriction
\pi: A \to \mathcal{M}_W
of the universal abelian scheme.
Its relative dimension is $n=\operatorname{dim}(A)$, and  its metrized Hodge bundle

\[
\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} \in \widehat{\Pic}(\mathcal{M}_W)
\]

\omega^\mathrm{Hdg}_{A / \mathcal{M}_W} =  \pi_* \Omega^{\operatorname{dim}(A)}_{A / \mathcal{M}_W}
endowed with the  hermitian metric

\begin{equation}\label{hodge metric}
\| s_z \|^2 =   \left|\frac{1}{ (2\pi i )^{\operatorname{dim}(A) }  }   \int_{A_z (\C)} s_z \wedge \overline{s}_z  \right|
\end{equation}

for any  $z\in \mathcal{M}_W(\C)$ and
s_z\in \omega^\mathrm{Hdg}_{A / \mathcal{M}_W , z} \iso H^0( A_z , \Omega^{\operatorname{dim}(A_z) }_{A_z / \C } ) .

The stack $\mathcal{M}_W$ has a  canonical toroidal compactification $\bar{\mathcal{M}}_W$,  with boundary a Cartier divisor smooth over $\co_\kk$.
Although the Hodge bundle has a distinguished extension to the  compactification, the metric on it does not extend smoothly.  Instead, the metric is pre-log singular along the boundary,  in the sense of  Definition 1.20 of [6].
This implies the integrability of its Chern form:   the smooth $(1,1)$ form on $\mathcal{M}_W(\C)$ defined locally by

\begin{equation}\label{chern form}
\chern(\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} )
=  \frac{1}{2\pi i} \partial \overline{\partial} \log \|s\|^2
= - d d^c  \log \|s\|^2
\end{equation}

for any nonzero  holomorphic  section $s$. Define the complex volume of the metrized Hodge bundle

\[
\vol_\C (  \widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} )  =
\int_{\mathcal{M}_W(\C)} \chern(  \widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{M}_W}  )^{n-1}
\]

as the (orbifold) integral of the top exterior power of its Chern form.

The pre-log singular conditions also allow us to view

\[
\widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} \in   \widehat{\mathrm{CH}}^1( \bar{\mathcal{M}}_W ,\mathscr{D}_\BKK)
\]

as a class in the codimension one arithmetic Chow group of  Burgos-Kramer-Kühn [11].

These arithmetic Chow groups come with intersection  pairings, and in codimension $n=\operatorname{dim}( \bar{\mathcal{M}}_W)$ there is an arithmetic degree

\[
\widehat{\deg} :   \widehat{\mathrm{CH}}^n( \bar{\mathcal{M}}_W ,\mathscr{D}_\BKK)  \to \R.
\]

This allows us to define the arithmetic volume

\[
\widehat{\vol} (  \widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} )
\widehat{\deg} \big(  \widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} \cdots \widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} \big)
\]

as the arithmetic degree of the $n$-fold iterated intersection.

Our main result is the calculation of these complex and arithmetic volumes in terms of Dirichlet $L$-functions.
To state it, for any place $\ell \le \infty$ denote by

\[
\inv_\ell(W)= (\det W,-D)_\ell
\]

the local invariant of  $W$.
Our assumption that $W$ contains a self-dual $\co_\kk$-lattice implies that  $\mathrm{inv}_\ell(W) =1$ for all finite primes $\ell\nmid D$.  As  $\inv_\infty(W)=-1$, and the product of all local invariants is $1$, we obtain

\begin{equation}\label{signs 1}
\prod_{\ell\mid D}  \mathrm{inv}_\ell(W)
%= \prod_{\ell<\infty} \mathrm{inv}_\ell(W) = \mathrm{inv}_\infty(W)
=-1 .
\end{equation}

The quadratic Dirichlet character determined by  $\kk/\Q$ is denoted

\[
\eps:(\Z/D\Z)^\times \to \{ \pm 1\}.
\]

For an integer $k \ge 1$ set

\begin{equation}\label{a_k}
\mathbf{a}_k(s) =   \frac { D^{k/2} \Gamma ( s + k )  L(2s+k,\eps^k) }{2^k \pi^{s+k} } ,
\end{equation}

where, if $k$ is even,  we understand $L(s,\eps^k)=\zeta(s)$.    Define

\begin{align}
\mathbf{A}_W (s)  & =  \mathbf{a}_1(s) \cdots \mathbf{a}_n(s)  \nonumber \\
& \quad \times
\begin{cases}
\prod_{\ell \mid D} \big(   1+ \leg{-1}{\ell}^{\frac{n}{2} }  \mathrm{inv}_\ell(W) \ell^{ -s- \frac{n}{2} }   \big) & \mbox{if $n$ is even} \\
1 & \mbox{if $n$ is odd.}
\end{cases}
\label{A_V}
\end{align}

If $n$ is odd there is no dependence on $W$ beyond its dimension.

All $\mathbf{a}_{k}(0)$  are positive rational numbers, and hence so is $\mathbf{A}_W(0)$.
The following is our main result.  It  appears in the text as Theorem <ref>.

The metrized Hodge bundle has  complex volume

\[
\vol_\C (  \widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} )
\mathbf{A}_W (0) \cdot
\begin{cases}
2^{n-1  }    & \mbox{if $n$ is odd } \\
2^{n-o(D)  }   & \mbox{if $n$ is even}
\end{cases}
\]

and arithmetic volume

\[
\widehat{\vol}  ( \widehat{\omega}^\mathrm{Hdg}_{A/ \mathcal{M}_W  } )
\left(  2 \frac{  \mathbf{A}_W'(0) }{ \mathbf{A}_W(0) }    - nC_0(n)
+ \log(D)  \right)  \cdot     \vol_\C (  \widehat{\omega}^\mathrm{Hdg}_{A / \mathcal{M}_W} ) ,
\]

where we have set

\[
C_0 (n) = 2  \log\left( \frac{4\pi e^\gamma}{ \sqrt{D} } \right)     +     (n-4)  \left(   \frac{L'(0,\eps)}{L(0,\eps)}  +  \frac{\log(D)}{2}    \right) .
\]

Theorem <ref> is closely related to a conjecture of Kudla-Rapoport [32] on the degrees of arithmetic  $0$-cycles on integral models of unitary Shimura varieties.
Briefly, to a $\kk$-hermitian space $V$ of signature $(n-1,1)$ one can attach a Shimura variety $\mathcal{S}_V$ (see the discussion in <ref> below), endowed with a family of arithmetic $0$-cycles $\widehat{\mathcal{Z}}_V(T)$  indexed by $n\times n$ hermitian matrices $T$.
The conjecture of Kudla-Rapoport predicts that the arithmetic degrees of these $0$-cycles should agree with the Fourier coefficients of the derivative of an Eisenstein series on the quasi-split unitary group $\Uni(n,n)$.

For those $0$-cycles  with $\det(T)\neq 0$, and supported at primes $p\nmid D$,  the Kudla-Rapoport conjecture is now a theorem of Li-Zhang [40].
The relevance of Theorem <ref> is to the degenerate  cases in which $\det(T)=0$, and especially to the most degenerate case $T=0$.  In this case the associated arithmetic $0$-cycle is, up to some correction factors at the primes $p\mid D$,  the $n$-fold iterated intersection
of a certain hermitian line bundle $\widehat{\mathcal{L}}_V$ on $\mathcal{S}_V$.
In other words, the Kudla-Rapoport conjecture predicts that the arithmetic volume of $\widehat{\mathcal{L}}_V$ is essentially the constant term of the derivative of an Eisenstein series.

The calculation of the arithmetic volume  of $\widehat{\mathcal{L}}_V$, which  is intertwined with the proof of Theorem <ref>, can be found in Corollary <ref>.
We have not attempted to compare this formula with the constant term of an Eisenstein series, in part because it is not  clear precisely which Eisenstein series one should work with.
The Eisenstein series should be constructed from local data, and it remains an open problem to determine precisely which  data one should choose at $p\mid D$ in order to make the Kudla-Rapoport conjecture true for all choices of $T$.  Similarly, because of subtleties at the primes dividing $D$, the precise relation between the arithmetic degree of $\widehat{\mathcal{Z}}_V(0)$ and the arithmetic volume of $\widehat{\mathcal{L}}_V$ is not completely clear.

We expect that the  proof of Theorem <ref> makes accessible more degenerate cases of the Kudla-Rapoport conjecture,  beyond the most degenerate case  $T=0$. For example,  it seems likely that our methods could be used to approach  cases in which  $T$ has small rank,  or in which

\[
T = \begin{pmatrix}
T' &  \\
& 0_{n-d}
\end{pmatrix}
\]

with $T'$ a nonsingular  $d\times d$ hermitian matrix that is not too complicated (e.g. squarefree determinant).
We hope to return to these questions in future work.

§.§ Outline of the paper

Fix a  $\kk$-hermitian space $V$ of signature $(n-1,1)$, and a self-dual $\co_\kk$-lattice $L\subset V$.

We construct in  <ref>  an Eisenstein series  $E_L(\tau,s,n)$ of weight $n$, valued in a finite dimensional representation of $\SL_2(\Z)$.
Its Fourier coefficients  can be expressed in terms of local representation densities of the lattice $L$, which we then compute in order to make the  coefficients completely explicit.

In <ref>  we introduce a complex Shimura variety $\mathrm{Sh}_K(H,\mathcal{D})$ associated to the unitary group $H=\Uni(V)$.  It carries a family of special divisors $Z(m)$ indexed by positive $m\in \Z$, and a metrized line bundle $\widehat{\mathcal{L}}$.
To a harmonic Maass form $f$ of weight $2-n$ we attach a divisor $Z(f)$, defined as a linear combination of $Z(m)$'s.
We then construct a Green function $\Phi(f)$ for $Z(f)$ using the machinery of regularized theta lifts, and show in  Theorem <ref> that the normalized integral

\begin{equation}\label{intro green integral}
\mathcal{I}(f) \define \mathrm{vol}_\C (\widehat{\mathcal{L}})^{-1} \int_{\mathrm{Sh}_K(H,\mathcal{D}) } \Phi(f) \chern(\widehat{\mathcal{L}})^{n-1}
\end{equation}

can be expressed in terms of the Fourier coefficients of the derivative of $E_L(\tau,s,n)$ at $s_0=(n-1)/2$.  The proof follows  ideas of Kudla [31], who computed similar integrals on orthogonal Shimura varieties.

The  complex Shimura variety $\mathrm{Sh}_K(H,\mathcal{D})$ is not a moduli space of abelian varieties, but it can be covered by another Shimura variety that does have a moduli interpretation.
In <ref> we work with $\kk$-hermitian spaces $W_0$ and $W$ of signatures $(1,0)$ and $(n-1,1)$.   The Shimura varieties associated to the unitary similitude groups $\mathrm{GU}(W_0)$ and $\mathrm{GU}(W)$ have moduli interpretations, which  can be used to construct regular integral models $\mathcal{M}_{W_0}$ and $\mathcal{M}_W$ over $\co_\kk$.
We recall  the definition of these integral models, their toroidal compactifications,  and the structure of their reductions at $p\mid D$.  Then we recall those aspects of the Gillet-Soulé  arithmetic intersection theory (as extended by Burgos-Kramer-Kuhn [11]) that are needed in the sequel.

In <ref> we choose $W_0$ and $W$ so that

\[
V \iso \Hom_\kk(W_0,W)
\]

as hermitian spaces.   This allows us to define an open and closed substack

\[
\mathcal{S}_V \subset \mathcal{M}_{W_0} \times_{\co_\kk} \mathcal{M}_W,
\]

which is an integral model of the Shimura variety associated to the subgroup $G \subset \mathrm{GU}(W_0) \times \mathrm{GU}(W)$ of pairs for which the similitude characters agree.
There is a natural surjection $G \to H$, which induces a finite étale cover

\begin{equation}\label{intro cover}
\mathcal{S}_V(\C) \to \mathrm{Sh}_K(H,\mathcal{D}).
\end{equation}

We next define  the family of Kudla-Rapoport divisors $\mathcal{Z}_V(m)$ on $\mathcal{S}_V$.
The complex fibers of these divisors are the pullbacks of  the divisors already defined on $\mathrm{Sh}_K(H,\mathcal{D})$.  Similarly, we define a metrized line bundle $\widehat{\mathcal{L}}_V$ on $\mathcal{S}_V$, compatible with the one already defined on $\mathrm{Sh}_K(H,\mathcal{D})$.
There is another  metrized line bundle on $\mathcal{S}_V$: the universal abelian scheme over $\mathcal{M}_W$ pulls back to an abelian scheme $A \to \mathcal{S}_V$, which determines a metrized Hodge bundle $\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}$.

The main result of <ref> is Theorem <ref>, which explains the precise connection between $\widehat{\mathcal{L}}_V$ and $\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}$.  What we prove is that, up to numerical equivalence (Definition <ref>), they differ by shifting the metrics by an explicit constant, and adding vertical divisors supported in characteristics $p\mid D$.

Going back and forth between these line bundles, in order to exploit the most desirable properties of each, is key to the proof of Theorem <ref>.  More precisely, we introduce in Theorem <ref> a third metrized line bundle $\widehat{\mathcal{K}}_V$,  constructed so that it enjoys the best properties of  $\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}$ and $\widehat{\mathcal{L}}_V$.  In the generic fiber, it agrees with the square of $\widehat{\mathcal{L}}_V$, so one can construct rational sections of it using the theory of Borcherds products.
On the other hand, like the metrized Hodge bundle, it has trivial arithmetic intersection with all components of the exceptional divisor $\mathrm{Exc}_V \subset \mathcal{S}_V$ of Definition <ref>. This implies that certain error terms that appear in later calculations disappear after taking arithmetic intersection with $\widehat{\mathcal{K}}_V$.

In <ref> we prove Theorems <ref> and <ref>, which are essential to the induction arguments used to prove  Theorem <ref>.
The idea is that if $p$ is a prime split in $\kk$, the Kudla-Rapoport divisor $\mathcal{Z}_V(p)$ should be closely related to a Shimura variety $\mathcal{S}_{V'}$ defined in the same way as $\mathcal{S}_V$, but with $\mathrm{dim}(V')= \mathrm{dim}(V)-1$.
This is true, up to some error terms coming from divisors supported in characteristics dividing $pD$.
Using such a relation, we are able to express  the height of the divisor $\mathcal{Z}_V(p)$ with respect to $\widehat{\mathcal{K}}_V$ in terms of the  arithmetic volume of the metrized line bundle $\widehat{\mathcal{K}}_{V'}$ on the lower-dimensional Shimura variety $\mathcal{S}_{V'}$.
A similar relation holds with $\widehat{\mathcal{K}}_V$ replaced by  $\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}$.
Because of the many subtleties in the moduli problem defining the integral models, this is perhaps the most technical part of the proof of Theorem <ref>.

In <ref>  we recall the  Borcherds products on $\mathcal{S}_V$, which were studied in detail in [13].  Given a meromorphic modular form $f (\tau)= \sum_{m\gg 0} c(m) q^m$ of weight $2-n$ and level $\Gamma_0(D)$, holomorphic outside the cusp $\infty$, one can construct a Borcherds product $\psi(f)$.  This is a rational section of a suitable power $\mathcal{L}_V^{\otimes k(f)}$ on the toroidal compactification $\bar{\mathcal{S}}_V$, whose divisor is

\begin{equation} \label{intro Bdivisor}
\mathrm{div}(\psi(f)) = \sum_{m>0} c(-m) \mathcal{Z}_V(m),
\end{equation}

up to an error term consisting of a linear combination of boundary components and vertical divisors in characteristics dividing $D$.

In Theorem <ref> we show that  $f$ may be chosen so that the Borcherds product has many desirable properties:
the weight $k(f)$ is nonzero and can be expressed in terms of the functions (<ref>), the only divisors $\mathcal{Z}_V(m)$ appearing in (<ref>) are those with $m$ a prime congruent to $1$ modulo $D$, no boundary components appear in $\mathrm{div}(\psi(f))$, and the vertical divisors appearing in the error term have a particularly simple form.   See especially Proposition <ref>.

In <ref>, we put everything together to prove Theorem <ref> via induction on $n\ge 2$ (the case $n=1$ can be deduced from  the Chowla-Selberg formula; see the proof of Theorem <ref>).  When $n=2$ the Shimura variety $\mathcal{S}_V$ is closely related to a modular curve or quaternionic Shimura curve, and  Theorem <ref> can be deduced from the results of Kühn, Kramer, and Kudla-Rapoport-Yang alluded to above; this is carried out in [25].

Because of the relations between  $\widehat{\omega}^\mathrm{Hdg}_{A/\mathcal{S}_V}$,  $\widehat{\mathcal{L}}_V$, and $\widehat{\mathcal{K}}_V$ worked out in <ref>, in order to prove Theorem <ref> in general, it suffices to compute the complex and arithmetic volumes of $\widehat{\mathcal{K}}_V$, which can be found in Theorems <ref> and <ref>.
To do this, we use the fact that $\widehat{\mathcal{K}}_V$ agrees with $\widehat{\mathcal{L}}_V^{\otimes 2}$ in the generic fiber to view the Borcherds product $\psi(f)^{\otimes 2}$ as a rational section of $\widehat{\mathcal{K}}_V^{\otimes k(f)}$.  In the codimension one arithmetic Chow group of $\mathcal{S}_V$,  this allows us to represent a multiple of $\widehat{\mathcal{K}}_V$  by  the divisor (<ref>) endowed with the Green function $\Phi_V(f)$ obtained by pulling back  via  (<ref>)  the Green function appearing in (<ref>),  up to some contributions from vertical divisors in characteristics dividing $D$ (known by the results of <ref>).

Thus  the arithmetic volume of $\widehat{\mathrm{vol}} (  \widehat{\mathcal{K}}_V  )$ can be computed by intersecting the codimension $n-1$ arithmetic cycle $ \widehat{\mathcal{K}}_V^{n-1}$ against the divisor (<ref>) endowed with $\Phi_V(f)$.
This intersection can then be expressed in terms of the arithmetic heights of the $\mathcal{Z}_V(m)$'s with respect to $\widehat{\mathcal{K}}_V^{n-1}$, and the integral $\mathcal{I}(f)$ of (<ref>).
Using the results of <ref>, the arithmetic heights are essentially  the arithmetic volumes of  Shimura varieties  in one dimension lower, which can be expressed in terms of the functions $\mathbf{a}_k(s)$ of (<ref>) by the induction hypothesis.
The integral $\mathcal{I}(f)$ was expressed in terms of the coefficients of $E_L(\tau,s,n)$ in <ref>, and the formulas from <ref> for these coefficients allow us to write $\mathcal{I}(f)$ in terms of the functions $\mathbf{a}_k(s)$.
Thus the arithmetic volume of $\widehat{\mathrm{vol}} (  \widehat{\mathcal{K}}_V  )$ can be expressed in terms of $\mathbf{a}_k(s)$,
and so the same is true of both the metrized Hodge bundle and $\widehat{\mathcal{L}}_V$.

§.§ Notation and conventions

Throughout the paper $\kk \subset \C$ is a quadratic imaginary field  of discriminant $-D$, and $\eps: (\Z/D\Z)^\times \to \{\pm 1\}$ is the associated quadratic character.

The symbol   $V$   always denotes a $\kk$-hermitian space of signature $(n-1,1)$ with $n\ge 1$, and the hermitian form is denoted $\langle - , - \rangle$.
Hermitian forms are always linear in the first variable and conjugate-linear in the second, and are assumed to be nondegenerate.

Beginning in <ref>, and continuing for the rest of the paper, we assume that $D$ is odd and  the hermitian space admits an $\co_\kk$-lattice $L\subset V$ that is self-dual, in the sense that

\begin{equation}\label{self-dual lattice}
L = \{ x \in V :  \langle  x , L \rangle \subset \co_\kk \}.
\end{equation}

These restrictions are not imposed in  <ref> and <ref> unless stated explicitly.

The term stack  means Deligne-Mumford stack.

§ EISENSTEIN SERIES AND THETA FUNCTIONS

Fix  a $\kk$-hermitian space $V$ of signature $(n-1,1)$, with $n \ge 1$.
After reviewing some basics of  Eisenstein series, theta functions, and the Siegel-Weil formula,
we attach a particular Eisenstein series to a lattice $L \subset V$, and  express  its Fourier coefficients in terms of representation densities.
We then  compute these representation densities in the case of a self-dual lattice, and so obtain explicit formulas for the Fourier coefficients; these formulas have a different shape depending on whether $n$ is even  or odd.

§.§ A seesaw dual reductive pair

Let $V_0$ be the unique symplectic space over $\Q$ of dimension $2$, and denote the symplectic form by $\langle x,y\rangle_0$.
Set $V_{0\kk}=V_0\otimes_\Q\kk$,  and extend the symplectic form $\kk$-linearly in the first argument and $\kk$-conjugate-linearly in the second argument. This defines a skew-hermitian form on $V_{0\kk}$.

The   $\Q$-vector space underlying $V$ carries a $\Q$-bilinear form

\begin{equation}\label{Q bilinear}
[  x,y ] =\tr_{\kk/\Q} \langle x, y\rangle
\end{equation}

of signature $(2n-2,2)$, with associated quadratic form

\begin{equation}\label{Q quadratic}
Q(x) =  \langle x, x\rangle .
\end{equation}

This  data determine  a seesaw dual reductive pair

\begin{align*}
\xymatrix{
G=\Uni(V_{0\kk}) \ar@{-}[d] \ar@{-}[dr]& H' =\Orth(V)  \ar@{-}[d]\\
G' =\operatorname{Sp}(V_0) \ar@{-}[ur]& H=\Uni(V).
\end{align*}

In particular, there are compatible Weil representations of $G(\A)\times H(\A)$ and $G'(\A)\times H'(\A)$ on the space $S(V(\A))$ of Schwartz-Bruhat functions on $V(\A)$.  This will be made explicit in <ref>.

Let $e,f$ be a basis of $V_0$ with Gram matrix
\kzxz{0}{1}{-1}{0},
and use this to identify

\[
G \iso \Uni(1,1) \define \left\{ g\in \operatorname{Res}_{\kk/\Q} \GL_2  : g\zxz{0}{1}{-1}{0} {}^t\bar g= \zxz{0}{1}{-1}{0}\right\}
\]

and $G' \iso \operatorname{SU}(1,1) \iso \SL_2$.
Let $P=NM$ be the parabolic subgroup of $G$ with Levi factor

\[
M=\left\{m(a)= \zxz{a}{0}{0}{\bar a^{-1}}  :  a\in\operatorname{Res}_{\kk/\Q} \mathbb{G}_m\right\}
\]

and unipotent radical

\[
N=\left\{ n(b)=\zxz{1}{b}{0}{1}  :  b\in \mathbb{G}_{a} \right\}.
\]

For each prime $p\leq \infty$ of $\Q$, let $K_p \subset G(\Q_p)$ be the maximal compact subgroup defined by

\[
K_p= \begin{cases} G(\Q_p) \cap \GL_2(\co_{\kk,p}),&\text{if $p<\infty$,}\\
G(\R) \cap \Uni(2,\R),&\text{if $p=\infty$,}
\end{cases}
\]

and put $K=\prod_{p\leq\infty} K_p$.
If we let  $\kk^1$ be the torus of norm one elements in  $\kk^\times=\mathrm{Res}_{\kk/\Q}\mathbb{G}_m$, there is a natural homomorphism

\[
\operatorname{SU}(1,1)\times\kk^1\map{ (g,a)\mapsto  g \cdot m(a)  }   \Uni(1,1) ,
\]

and the image of $\kk^1$ is the center of $\Uni(1,1)$.
This homomorphism    induces an isomorphism

\[
\SO(2,\R)\times \Uni(1,\R) /\{\pm 1\}\cong K_\infty.
\]

Let  $\H$ be the complex upper half-plane.
The first isomorphism  in
\[
\H\iso \SL_2(\R)/\SO(2,\R)\iso G(\R)/K_\infty .
\]
sends   $τ=u+iv∈$̋ to the element

\begin{equation}\label{gtau}
g_\tau =  n(u)m(v^{1/2})  \in \SL_2(\R)
\end{equation}

satisfying $g_\tau \cdot  i =\tau$.  The second is induced $\SL_2(\R) = G'(\R) \subset G(\R)$.

§.§ Eisenstein series for $\Uni(1,1)$

Given an $s\in \C$ and a character $\chi : \A_\kk^\times/\kk^\times \to \C^\times$,  let

\[
\operatorname{Ind}_{P(\A)}^{G(\A)}(\chi |\cdot|_{\A_\kk}^s)
\]

be the induced representation, realized on the space of smooth $K$-finite functions $\Phi$ on $G(\A)$  satisfying

\[
\Phi(n(b)m(a)g,s)= \chi(a) |a|_{\A_\kk}^{s+\frac{1}{2}}\Phi(g,s)
\]

for $b\in \A$ and  $a\in \A_\kk^\times$.
Here  $|a|_{\A_\kk}$ denotes the norm on $\A_\kk^\times$.
In particular, at the archimedian place we have the normalized absolute value $|a|_{\infty}=a\bar a$.

Recall that a section  of $I(s,\chi)$ is called standard if its restriction to $K$ is independent on $s$.
For a standard section $\Phi$ of $I(s,\chi)$, the associated Siegel Eisenstein series

\[
E(g,s,\Phi) = \sum_{\gamma\in P(\Q)\backslash G(\Q)} \Phi(\gamma g, s)
\]

converges absolutely for $\operatorname{Re}(s)> 1/2$, and  defines an automorphic form on $G(\A)$.
The Eisenstein series has  meromorphic continuation to all $s\in \C$.
We now describe this Eisenstein series in more classical terms, by restricting it to the subgroup $\SL_2\iso G'\subset G$.

Assume that $\Phi(s)=\Phi_\infty(s)\otimes \Phi_f(s)$  is a factorizable standard section  with $\Phi_f=\otimes_{p<\infty} \Phi_p(s)$, and that $\Phi_\infty$ is a normalized standard section of weight $\ell\in \Z$.
In other words, $\Phi_\infty(1,s)=1$, and for all

\[
k=\zxz{a}{b}{-b}{a} \in \SO(2,\R) \subset K_\infty
\]

we have $\Phi_\infty(gk,s)=\Phi_\infty(g,s)  \cdot \underline{k}^\ell$,  where

\begin{equation}\label{underline k}
\underline{k} = a+ib \in \C^\times .
\end{equation}

Abbreviate $\Gamma=\SL_2(\Z) \subset G(\A)$,  and let $\Gamma_\infty=P(\Q)\cap \Gamma$ be is subgroup of   upper triangular matrices.

If $\Phi_\infty$ is a normalized standard section of weight $\ell\in \Z$ as above, then

\begin{align*}
\label{eq:eis01}
&= j(g_\tau,i)^{-\ell}  \sum_{\gamma\in \Gamma_\infty\backslash \Gamma}
\operatorname{Im}(\gamma \tau)^{s+ \frac{1}{2}-\frac{\ell } {2 }  } \cdot j(\gamma,\tau)^{-\ell}\cdot \Phi_f(\gamma).
\end{align*}

Here $g_\tau \in \SL_2(\R)$ is as in (<ref>), and  $j(\gamma,\tau)=c\tau+d$  the usual automorphy factor
associated to   $\gamma=\kabcd\in \SL_2(\R)$.

For $\gamma=\kabcd\in \Gamma$ we write
$\gamma g_\tau = n(\beta)m(\alpha) k$
with $\beta\in \R$, $\alpha\in \R^+$, and $k\in \SO(2,\R)$. Then a computation shows that

\begin{align*}
\alpha = \frac{v^{1/2}}{|c\tau +d|}, \qquad
\underline k = \frac{c\bar \tau +d}{|c\tau +d|},
\end{align*}

and hence

\[
\Phi_\infty(\gamma g_\tau,s) =
v^{\ell/2}\cdot \frac{v^{s+ \frac{1}{2}-\frac{\ell}{2}  }  }  {|c\tau+d|^{2s+1-\ell}} \cdot (c\tau+d)^{-\ell}.
\]

The  inclusions $\Gamma\subset \SL_2(\Q)\subset G(\Q)$ induce  bijections

\[
\Gamma_\infty\backslash \Gamma \iso  (P(\Q)\cap \SL_2(\Q))\backslash \SL_2(\Q)   \iso P(\Q)\backslash G(\Q),
\]

and hence

\begin{align*}
& = \sum_{\gamma\in \Gamma_\infty\backslash \Gamma} \Phi_\infty(\gamma g_\tau,s)\Phi_f(\gamma) \\
v^{\ell/2} \sum_{\gamma\in \Gamma_\infty\backslash \Gamma}
\operatorname{Im}(\gamma \tau)^{s+  \frac{ 1-\ell }{ 2 } } \cdot (c\tau+d)^{-\ell}\cdot \Phi_f(\gamma)\\
j(g_\tau,i)^{-\ell}  \sum_{\gamma\in \Gamma_\infty\backslash \Gamma}
\operatorname{Im}(\gamma \tau)^{s+  \frac{ 1-\ell }{ 2 } }  \cdot j(\gamma,\tau)^{-\ell}\cdot \Phi_f(\gamma),
\end{align*}

as desired.

§.§ The Siegel-Weil formula

Write $\psi$ for the standard additive character of $\A/\Q$, satisfying  $\psi_\infty(x)=e^{2\pi i x}$.

\[
\eps_\A : \A^\times /\Q^\times \to \{\pm 1\}
\]

be the idele class character determined by the quadratic extension $\kk/\Q$, and fix a character

\[
\chi :   \A_\kk^\times/ \kk^\times \to \C^\times
\]

such that $\chi |_{\A^\times}=\eps_\A^n$.

As explained in [19] and [27], the choices of $\psi$ and $\chi$ determine a Weil representation
$\omega=\omega_{\psi,\chi}$ of the group $G(\A)\times H(\A)$ on the space of Bruhat-Schwartz functions $S(V(\A))$.
Recalling the $\Q$-bilinear form (<ref>) and associated quadratic form  (<ref>), the action is given  by the formulas

\begin{align*}
\omega(m(a))\varphi (x) &= \chi(a) |a|_{\A_\kk}^{n/2} \varphi(xa),\\
\omega(n(b))\varphi(x) & = \psi( bQ(x) )\varphi(x),\\
\omega(\zxz{0}{1}{-1}{0})\varphi(x) &= \int_{V(\A)} \varphi(y) \psi( [  x,y] )\,dy,\\
\omega(h)\varphi(x) & = \varphi(h^{-1} x)
\end{align*}

for $\varphi\in S(V(\A))$ and $x\in V(\A)$. Here $a\in \A_\kk^\times$, $b\in \A$, $h\in H(\A)$, and the Fourier transform is taken with respect to the self-dual measure $dy$ on $V(\A)$.
To a Schwartz function $\varphi$ there is an associated theta function

\begin{equation}\label{theta kernel}
\theta(g,h,\varphi)=  \sum_{x\in V(\Q)} \omega(g,h)\varphi(x).
\end{equation}

Abbreviating $s_0=(n-1)/2$, there is a $G(\A)$-intertwining operator

\begin{align*}
\lambda:S(V(\A))\to I(s_0,\chi)
\end{align*}

defined by $\lambda(\varphi)(g)= (\omega(g,1)\varphi)(0)$.
We extend $\lambda(\varphi)$ to a standard section of $I(s,\chi)$ by setting

\[
\lambda(\varphi)(g,s)= |a(g)|_{\A_\kk}^{s-s_0}(\omega(g,1)\varphi)(0).
\]

The following theorem is a case of the Siegel-Weil formula;  see [52] or  Theorem 1.1 of [27].

Assume that $n>2$,  or that $V$ is anisotropic. For $\varphi\in S(V(\A))$ and $g\in G(\A)$ we have

\[
E(g,\lambda(\varphi),s_0) =\kappa  \int_{H(\Q)\backslash H(\A)} \theta(g,h,\varphi)\, dh ,
\]

where  Haar measure  is normalized by  $\vol(H(\Q)\backslash H(\A))=1$, and

\begin{equation}\label{eq:kappa}
\kappa =
\begin{cases}
1 & \mbox{if }n>1 \\
2 & \mbox{if }n=1.
\end{cases}
\end{equation}

§.§ A special Eisenstein series

We consider Eisenstein series constructed from  particular Schwartz functions on $V(\A)$.
At the archimedian place we take the Gaussian associated to a majorant and at the non-archimedian places we take the characteristic function of a coset of an $\co_\kk$-lattice.

Let $L \subset V$ be an $\co_\kk$-lattice on which the hermitian form is $\co_\kk$-valued.
Its dual lattice  under the $\Q$-bilinear form (<ref>) is  denoted

\begin{equation}\label{dual lattice}
L'  \define    \{ x\in V  :  [  x,L]  \subset \Z\} .
\end{equation}

For $\mu\in L'/L$  let

\[
\varphi_\mu=\operatorname{char}(\mu+\hat L)  \in S(V(\A_f))
\]

be the characteristic function of $\mu+\hat L\subset V(\A_f)$, and denote
by $S_L$ the subspace of $S(V(\A_f))$ generated by all $\varphi_\mu$ with  $\mu\in L'/L$.
Hence we may identify

\[
S_L\iso  \C[L'/L].
\]

The restriction to $\Gamma=\SL_2(\Z)$ of the Weil representation  of $\SL_2(\A_f)\subset G(\A_f)$ takes the subspace
$S_L$ to itself, giving rise to a representation

\begin{align}
\label{eq:ol}
\omega_L:\Gamma\to \Aut(S_L).
\end{align}

The corresponding dual representation

\[
\omega_L^\vee:\Gamma\to \Aut(S_L^\vee)
\]

is given by $\omega_L^\vee(\gamma)(f)=f\circ \omega_L^{-1}(\gamma)$ for $f\in S_L^\vee$.
On the space $S_L$ we also have the conjugate representation $\bar\omega_L$ defined by

\[
\bar\omega_L(\gamma)(\varphi)= \overline{\omega_L(\gamma)(\bar\varphi)}
\]

for $\varphi\in S_L$.
If we use the standard $\C$-bilinear pairing

\begin{equation}\label{S_L pairing}
\Big\langle \sum_\mu a_\mu\varphi_\mu, \sum_\mu b_\mu\varphi_\mu \Big\rangle = \sum_\mu a_\mu b_\mu
\end{equation}

to identify $S_L\iso S_L^\vee$ as $\C$-vector spaces, then $\bar{\omega}_L = \omega_L^\vee$,
and this agrees with  the representation denoted $\rho_L$  in [2], [4], and [10].

For $\ell\in \Z$ and $\tau\in \H$, we define an $S_L$-valued Eisenstein series
\begin{align}
\label{eq:eisl}
E_L(\tau,s,\ell)= \sum_{\gamma\in \Gamma_\infty\backslash \Gamma}
\operatorname{Im}(\gamma \tau)^{s+\frac{1-\ell}{2}}  (c\tau+d)^{-\ell}\cdot \bar\omega_L(\gamma)^{-1}\varphi_0
\end{align}
of weight $ℓ$ and representation $ω̅_L$.
Note that this Eisenstein series has a slightly different normalization than the Eisenstein series in equation (2.17) of [10], since it is adapted to the unitary setting.

Recall  the Maass lowering and raising operators in weight $ℓ$  defined by
\[
L_\ell =-2iv^2\frac{\partial}{\partial \bar \tau}, \qquad
R_\ell =2i\frac{\partial}{\partial \tau}+\ell v^{-1}.
\]
They lower (respectively raise) the weight of an automorphic form by $2$.
It is easily seen that
\begin{align}
\label{eq:leis}
E_L(\tau,s, \ell) &= \left(s+ \frac{1-\ell}{2} \right) E_L(\tau,s, \ell-2) \\
R_\ell E_L(\tau,s, \ell) &= \left(s+\frac{1+\ell}{2}\right) E_L(\tau,s,\ell+2). \nonumber
\end{align}
%(see also \cite[ Lemma 2.7]{Ku:Integrals}). In particular, we see that
The Eisenstein series $E_L(τ,s,ℓ)$ is an eigenform of the hyperbolic Laplacian $Δ_ℓ$ of weight $ℓ$ (normalized as in (3.1) of [7]) with eigenvalue $(ℓ-1/2)^2-s^2$.
The following is a consequence of \eqref{eq:leis}.

\begin{lemma}
\label{lem:eisl}
Assume that $n>2$, or that $V$ is anisotropic.
The  Eisenstein series $E_L(\tau,s_0,n)$  of weight $n=\mathrm{dim}(V)$ at   $s_0=(n-1)/2$ is  holomorphic  in $\tau$.
Its derivative
\[
E_L'(\tau,s_0,n) \define \frac{\partial}{\partial s} E_L'(\tau,s,n)\big|_{s=s_0}
\]
\begin{align*}
L_n(E_L'(\tau,s_0,n)) & = E_L(\tau,s_0,n-2).
%\big(L_n E_L'(\tau,s_0,n)\big) \, d\mu(\tau) &= -\bar\partial E_L'(\tau,s_0,n)\, d\tau .
\end{align*}
\end{lemma}

\begin{remark}
The Eisenstein series $E_L(\tau,s,n-2)$ is coherent as it arises via the Siegel-Weil formula from the global hermitian space $V$. On the other hand, the Eisenstein series $E_L(\tau,s,n)$ is incoherent as it is associated to the incoherent collection of local hermitian spaces obtained by replacing $V_\infty$ by the positive definite hermitian space of dimension $n$ and keeping the non-archimedian part. In particular, at $s=0$, the center of symmetry of the functional equation, the Eisenstein series $E_L(\tau,s,n)$ vanishes identically.
\end{remark}

%\note{Can take this out later.}
%An alternative construction for the Eisenstein series is to lift the Eisenstein series for $\Gamma_0(D)$ in the cusp $\infty$ (which is vanishing at the other cusps) to $M_n(\bar \omega_L)$.
%We briefly sketch this in the case when $D=p\equiv 3 \pmod{4} $ is a prime.
%%and $n$ is even.
%Then $\inv_p(V)=-1$.
%Let $g\in M_n(\Gamma_0(p))$ and denote the Fourier expansion at the cusps $\infty$ and $0$ by
%g&=\sum_{m=0}^\infty a(m) q^m,\\
%\tilde g &=g|_n S =  \sum_{m=0}^\infty \tilde a(m) q^m,
%respectively. Let
%G=\sum_{\gamma\in \Gamma_0(p)\backslash \SL_2(\Z)} (g|_n \gamma) (\bar \omega_L(\gamma)^{-1}\varphi_0)
%be the lift of $g$ to $M_m(\bar \omega_L)$. Using the explicit coset decomposition
%\SL_2(\Z)=\Gamma_0(p)\cup \bigcup_{j=1}^{p} \Gamma_0(p) ST^j
%one computes that
%the Fourier coefficients $A(m,\mu)$ of $G$ are given by
%A(m,\mu)=\begin{cases}(-p)^{1-n/2}\tilde a(m),&\text{if $\mu\neq 0$,}\\
%(-p)^{1-n/2}\tilde a(m)+a(m),&\text{if $\mu= 0$,}
%for $\mu\in L'/L$ and $m\in \Z+Q(\mu)$.
%Let $E_n= 1-\frac{2n}{B_n}\sum_{m\geq 1} \sigma_{n-1}(m) q^m$ the Eisenstein series of weight $n$ for $\SL_2(\Z)$.
%%and abbreviate $C=-\frac{2n}{B_n}$.
%It is easily seen that the normalized Eisenstein series $g$ of weight $n$ for $\Gamma_0(p)$ in the cusp $\infty$ is equal to
%g(\tau)=\frac{1}{1-p^n} E_n(\tau)- \frac{p^n}{1-p^n} E_n(p\tau),
%and that
%\tilde g(\tau)=(g|_n S)(\tau)=\frac{1}{1-p^n} E_n(\tau)- \frac{1}{1-p^n} E_n(\tau/p).
%The lift of $g$ to $M_n(\bar\omega_L)$ is equal to the Eisenstein series $E_L(\tau,s_0,n)$.
%The Fourier coefficients $A(m,\mu)$ of the lift are now given by
%A(m,\mu)= \frac{C}{1-p^n}\cdot
%&\text{if $\mu\neq 0$,}\\
%+\sigma_{n-1}(m)-p^n \sigma_{n-1}(\frac{m}{p})
%,&\text{if $\mu= 0$,}
%where $C=-\frac{2n}{B_n}$. Here we have used the convention that $\sigma_s(x)=0$ if $x\notin \Z^+$. A computation shows that this matches with the formula for the $B(m,0,s_0)$ of Corollary~\ref{cor:eis3}.

\subsection{Coefficients of Eisenstein series}
\label{ss:coefficients}

When the weight of the  Eisenstein series \eqref{eq:eisl}  is $n=dim(V)$, we denote its Fourier expansion by
\begin{align}
\label{eq:fouriereis}
E_L(\tau,s,n)= \sum_{\mu\in L'/L}\sum_{m\in \Z+\langle \mu,\mu\rangle} B(m,\mu;s,v) q^m\varphi_\mu.
\end{align}
We  summarize some well-known facts about the coefficients;  details can be found in [9] and [35].

If $m≠0$, there is a factorization
\begin{align}
\label{eq:coeff}
B(m,\mu;s,v)= B(m,\mu,s)\cdot \mathcal{W}_m(s,v)
\end{align}
in which the non-archimedian contribution $B(m,μ,s)$ is independent of $v$.
The second factor is the   archimedian Whittaker function
\begin{align}
\label{eq:whittaker}
\mathcal{W}_m(s,v) = (4\pi |m| v)^{-n/2} e^{2\pi m v} \cdot W_{\sgn(m)\frac{n}{2}, s}(4\pi |m| v),
\end{align}
\[
W_{\kappa,\mu}(z)= e^{-z/2}z^{1/2+\mu} U(\frac{1}{2}+\mu-\kappa, 1+2\mu,z)
\]
denotes the classical confluent hypergeometric function as in \cite[Chapter 13]{AS}, and
\begin{align}
\label{eq:uint}
U(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt} t^{a-1} (1+t)^{b-a-1} \,dt
\end{align}
for $(a)>0$, $|(z)|<π/2$.

If $m=0$, the Fourier coefficient has the form
\begin{align}
B(0,\mu;s,v)= \delta_{\mu,0}\cdot  v^{s+\frac{1}{2}-\frac{n}{2}}+ B(0,\mu,s)\cdot
\end{align}
where $B(0,μ,s)$ is independent of $v$, and $δ_μ,0$ is the Kronecker delta.

\begin{remark}
Note that our normalization of the archimedian Whittaker function in \eqref{eq:whittaker} differs from the normalization in \cite[Proposition 2.3]{KY}. The normalizing factor $\Gamma(s+\frac{n}{2}+\frac{1}{2})^{-1}$ for $m>0$, and $\Gamma(s-\frac{n}{2}+\frac{1}{2})^{-1}$ for $m<0$, appears in loc.~cit. In our case this factor is included in the normalization of the $B(m,\mu,s)$, since this leads to slightly cleaner formulas in Theorem~\ref{thm:int}.
\end{remark}

The integral representation \eqref{eq:uint} can be used to
describe the asymptotic behavior of the Whittaker function as $v→∞$, which is given by
\begin{align*}
\mathcal{W}_m(s,v)= \begin{cases} 1+O(v^{-1}) &\mbox{if }m>0 \\
O(e^{-4\pi |m| v})  &\mbox{if }m<0,
\end{cases}
\end{align*}
locally uniformly in $s$ and $m$.

We will also require the following lemma on the value at $s_0=(n-1)/2$ of the Whittaker function and its derivative $𝒲_m'(s_0,v)=∂/∂s𝒲_m(s,v)|_s=s_0$.

\begin{lemma}
\label{lem:wasymptotic}
The special value at $s_0$ of the Whittaker function is
\begin{align*}
\mathcal{W}_m(s_0,v)= \begin{cases} 1 &\mbox{if } m>0 \\
\Gamma(1-n,4\pi |m|v) &\mbox{if }m<0 .
\end{cases}
\end{align*}
For $m>0$ we have
\[
\mathcal{W}_m'(s_0,v)=
\sum_{j=1}^{n-1} \binom{n-1}{j} \frac{\Gamma(j)}{(4\pi mv)^j}.
\]
In particular,
\[
\mathcal{W}_m'(s_0,v)= O(v^{-1}), \quad \text{for $v\to \infty$.}
\]
\end{lemma}

\begin{proof}
We only prove this for $m>0$ and leave the similar case when $m<0$ to the reader.
According to \eqref{eq:whittaker} and \eqref{eq:uint}, we have for $m>0$ that
\begin{align*}
\mathcal{W}_m(s,v) &=  \frac{(4\pi m v)^{s-s_0} }{\Gamma(s-s_0)} \cdot
\int_0^\infty e^{-4\pi m v t} t^{s-s_0-1} (1+t)^{s+s_0} \, dt\\
&=  1
+\frac{(4\pi m v)^{s-s_0} }{\Gamma(s-s_0)} \cdot
\int_0^\infty e^{-4\pi m v t} t^{s-s_0-1} ((1+t)^{s+s_0}-1) \,dt,
\end{align*}
where the latter integral converges absolutely for $\Re(s)>s_0-1$. We immediately see that $\mathcal{W}_m(s_0,v)=1$, and in addition that
\begin{align*}
\mathcal{W}'_m(s_0,v) &=
\int_0^\infty e^{-4\pi m v t} ((1+t)^{n-1}-1) \,\frac{dt}{t}\\
&= \sum_{j=1}^{n-1} \binom{n-1}{j}
\int_0^\infty e^{-4\pi m v t} t^j \,\frac{dt}{t}\\
&= \sum_{j=1}^{n-1} \binom{n-1}{j} \frac{\Gamma(j)}{(4\pi mv)^j}.
\end{align*}
This gives the claimed formula and the bound for $v\to \infty$.
\end{proof}

We now turn to the explicit calculation of the factor $B(m,μ,s)$ in \eqref{eq:coeff}, using  formulas from [9] and [35].
Fix $μ∈L'/L$,  and let \[d_\mu=\min\{b\in \Z^+ :  b\mu=0\}\]  be the order of $μ$.
For $m∈+Q(μ)$ and $a∈$, we denote by $N_m,μ(a)$ the modulo $a$ representation number
\begin{align*}
N_{m,\mu}(a)= \#\{ r\in L/aL:\; Q(r+\mu)\equiv m\pmod{a\Z}\}.
\end{align*}
For a prime $p$, set
\begin{align}
\label{eq:lp}
L^{(p)}_{m,\mu}(p^{-s})= (1-p^{-s+2n-1}) \sum_{\nu=0}^\infty N_{m,\mu}(p^\nu) p^{-\nu s}
\end{align}
so that
\[
\sum_{a=1}^\infty N_{m,\mu}(a) a^{- s} = \zeta(s-2n+1)\prod_p L^{(p)}_{m,\mu}(p^{-s}).
\]

\begin{proposition}
\label{prop:eis1}
For nonzero $m\in \Z+Q(\mu)$ we have
\[
B(m,\mu,s)=-\frac{2^{n} \pi^{s+\frac{n}{2}+\frac{1}{2}}}{ \sqrt{|L'/L|}} \cdot \begin{cases}
\displaystyle
\frac{|m|^{s+\frac{n}{2}-\frac{1}{2}}}{\Gamma(s+\frac{n}{2}+\frac{1}{2})}
\prod_{p} L^{(p)}_{m,\mu}(p^{-2s-n})
& \mbox{if } m>0 \\[3ex]
\displaystyle
\frac{|m|^{s+\frac{n}{2}-\frac{1}{2}}}{\Gamma(s-\frac{n}{2}+\frac{1}{2})}
\prod_{p} L^{(p)}_{m,\mu}(p^{-2s-n})
& \mbox{if }m<0.
\end{cases}
\]
\end{proposition}

\begin{proof}
This is obtained from Proposition 3.2 of  [9] by noticing that our  Eisenstein series $E_L(\tau,s,n)$ is $1/2$  the Eisenstein series $E_0(\tau,s-\frac{n}{2}+\frac{1}{2})$ of [9] with $\kappa=n$ and $r=2n$.
\end{proof}

According to equation (3.20) of [9], the Euler factors  $L^(p)_m,μ(p^-s)$ are polynomials in $p^-s$.
Moreover, by (3.23) of [9], if $p$ does not divide $d_μ^2 m |L'/L|$ then
\[
L^{(p)}_{m,\mu}(p^{-s})=1-\chi_F(p) p^{n-1-s},
\]
where $χ_F$ is the quadratic Dirichlet character associated to  the quadratic extension $(√((-1)^n |L'/L|))$.
This implies that the Euler product in Proposition  \ref{prop:eis1} converges absolutely for $Re(s)>0$ and that $B(m,μ,s_0)$ is rational if  $n>1$ (so that $s_0=n-1/2$ lies in the region of convergence).
In this case it is nonpositive for positive $m$ and vanishes for negative $m$.

Later we will require the following lemma on the behavior of the coefficients of the Eisenstein series in the limit $v→∞$.

\begin{lemma}
\label{lem:coeffasy}
Assume that $n>2$,  or that $V$ is anisotropic.
The coefficients $B'(m,\mu;s_0,v)$ of $E_L'(\tau,s_0,n)$ satisfy
\[
\lim_{v\to \infty}\big( B'(m,\mu;s_0,v ) -\kappa\delta_{m,0}\delta_{\mu,0}\log v\big)=\begin{cases}
B'(m,\mu,s_0) &\text{if $m>0$} \\
B'(0,\mu,s_0) &\text{if $m=0$ and $n=1$} \\
0 &\text{if $m=0$ and $n>1$} \\
0 &\text{if $m<0$.}
\end{cases}
\]
\end{lemma}

\begin{proof}
For $m\neq 0$ the assertion follows from Proposition \ref{prop:eis1} together with Lemma \ref{lem:wasymptotic} on the asymptotic behavior of the Whittaker function $\mathcal{W}_m(s,v)$ and its derivative at $s=s_0$.

For $m=0$, we consider the Laurent expansion of $B(0,\mu;s,v)$ at $s=s_0$, which is given by
\begin{align*}
&B(0,\mu;s,v)= \delta_{\mu,0}+ B(0,\mu,s_0)v^{-2s_0}\\
&\phantom{=}{}+\left( \delta_{\mu,0} \log(v) - B(0,\mu,s_0)\log(v) v^{-2s_0} + B'(0,\mu,s_0)v^{-2s_0}\right)(s-s_0) \\
\end{align*}
If $n>1$, then the holomorphicity of $E_L(\tau,s_0,n)$ implies that $B(0,\mu,s_0)=0$. Since $2s_0>0$, the term involving $v^{-2s_0}$ vanishes in the limit $v\to \infty$,  and we obtain the assertion.

If $n=1$ then $s_0=0$ is the center of symmetry of the functional equation of the incoherent Eisenstein series, and hence  $E_L(\tau,0,n)$ vanishes. The vanishing of the constant Fourier coefficient  implies
\[
\]
and consequently
\[
B'(0,\mu;0,v)= 2\delta_{\mu,0} \log(v) + B'(0,\mu,s_0).
\]
This concludes the proof of the lemma.
\end{proof}

\subsection{Self-dual lattices}
%\label{ss:self-dual coefficients}

In this subsection  we assume  the $_$-lattice $L⊂V$ is self-dual \eqref{self-dual lattice} under the hermitian form, and that $D=- disc()$ is odd.
The first of these assumptions implies that   \eqref{dual lattice} satisfies  $L'= 𝔡_^-1 L$, where   $𝔡_⊂_$ is the different.
We will determine the coefficient $B(m,μ,s)$ from   \eqref{eq:coeff} more explicitly.

To compute the Euler factors $L^(p)_m,μ(p^-s)$ in Proposition \ref{prop:eis1}, we need to determine the representation numbers $N_m,μ(p^ν)$.
We now  derive explicit formulas for these quantities  using finite Fourier transforms and formulas for certain lattice Gauss sums.   As we will only compute the coefficients for $μ=0$, we abbreviate

Fix an  $a∈$.  For $c∈^+$,  consider the Gauss sum
\[
G(a, c ) = \sum_{x\in \mathcal{O}_\kk/ c  \mathcal{O}_\kk} e\left(\frac{a\norm(x)}{c}\right).
\]
We are mainly interested in the case when $c=p^ν$ is a prime power.

\begin{lemma}
\label{lem:gauss1}
Let $p^\nu$ be a prime power.
If we write
$\gcd(a,p^\nu) =p^\alpha$ with $0\leq \alpha \leq \nu$ and $a=p^\alpha a'$ with $a'\in \Z$, then
%If we write $\gcd(a,p^\nu)=p^\alpha a'$ with $0\leq \alpha\leq \nu$ and $\gcd(a',p)=1$, then\note{Can't be what you mean.  Isn't $a'=1$?
%Jan: You are right. I corrected it.}
\[
G(a,p^\nu)= \begin{cases}
p^{2\nu} &\text{if $\alpha=\nu$}\\
p^{\alpha+\nu}\leg{-D}{p}^{\nu-\alpha} &\text{if $\alpha <\nu$ and $p\nmid D$}\\
\epsilon_p p^{\alpha+\nu+\frac{1}{2}}\leg{a'}{p}\leg{D'}{p}^{\nu-\alpha-1} &\text{if $\alpha <\nu$ and $p\mid D$.}
\end{cases}
\]
Here, in the final case, we have set  $D'=D/p$ and
\[
\epsilon_r = \begin{cases}
1 & \mbox{if } r\equiv 1\pmod{4} \\
i & \mbox{if } r\equiv 3 \pmod{4} .
\end{cases}
\]
for any odd positive integer $r$.
\end{lemma}

\begin{proof}
It is easily seen that
\[
G(a,p^\nu)= p^{2\alpha} G(a',p^{\nu-\alpha}).
\]
Hence we may assume that $a$ is coprime to $p$.
Now, if $p$ is odd, the assertion can be easily deduced from the classical Gauss sum
%\note{$\epsilon_c =?$}
\[
\sum_{x\in \Z/c \Z} e\left(\frac{a x^2}{c}\right)=\epsilon_{c}\sqrt{c} \leg{a}{c}
\]
for odd positive $c$ with $\gcd(c,a)=1$.

For $p=2$ and $a$ odd we have
\begin{align}
\label{eq:gauss1}
G(a,2^\nu)= \sum_{x,y\in \Z/2^\nu\Z} e\left( a\frac{x^2+xy+\frac{1+D}{4}y^2}{2^\nu}\right),
\end{align}
where we have used that $D$ is odd.
If $\nu=1$ the asserted formula is easily checked. Hence, assume that $\nu\geq 2$. Then we use the classical Gauss sum
\[
\sum_{x\in\Z/2^\nu\Z}  e\left( a\frac{x^2+xy}{2^\nu}\right)
\displaystyle
0,& 2\nmid y,\\[1ex]
\displaystyle
e\left(-a\frac{y^2}{2^{\nu+2}}\right)\sum_{x\in \Z/2^\nu\Z}  e\left( a\frac{x^2}{2^\nu}\right), & 2\mid y,
\end{cases}
\]
to rewrite \eqref{eq:gauss1} as follows:
\begin{align*}
G(a,2^\nu)= \frac{1}{2}\sum_{x,y\in \Z/2^{\nu}\Z} e\left( a\frac{x^2+Dy^2}{2^\nu}\right).
\end{align*}
Inserting the evaluation of the  Gauss sum
\[
\sum_{x\in\Z/2^\nu\Z}  e\left( a\frac{x^2}{2^\nu}\right)
= (1+i)\epsilon_a^{-1}2^{\nu/2} \leg{2^\nu}{a},
\]
we finally find that
\begin{align*}
G(a,2^\nu)= 2^\nu \leg{2^\nu}{D}=2^\nu \leg{-D}{2}^\nu,
\end{align*}
concluding the proof of the lemma.
\end{proof}

\begin{lemma}
\label{lem:gauss2}
Let $p^\nu$ be a prime power.
If we write
$\gcd(a,p^\nu) =p^\alpha$ with $0\leq \alpha \leq \nu$ and $a=p^\alpha a'$ with $a'\in \Z$, then
the Gauss sum
\begin{align*}
G_L(a,p^\nu)= \sum_{x\in L/p^\nu L} e\left(\frac{a\langle x,x\rangle }{p^\nu}\right)
\end{align*}
is given by
\[
G_L(a,p^\nu)= \begin{cases}
p^{2n\nu }&\text{if $\alpha=\nu$ }\\
p^{n\alpha+n\nu}\leg{-D}{p}^{n(\nu-\alpha)} &\text{if $\alpha <\nu$ and $p\nmid D$}\\
\epsilon_p^n \inv_p(V) p^{n\alpha+n\nu+\frac{n}{2}}\leg{a'}{p}^n\leg{D'}{p}^{n(\nu-\alpha-1)} &\text{if $\alpha <\nu$ and $p\mid D$,}
\end{cases}
\]
where $\epsilon_p$ and $D'$  have the same meaning as in Lemma \ref{lem:gauss1}.
\end{lemma}

\begin{proof}
The Gauss sum $G_L(a,p^\nu)$ only depends on $L_p=L\otimes_\Z \Z_p$.
If we let  $b_1,\dots,b_n$ be an orthogonal  $\mathcal{O}_{\kk,p}$-module basis of $L_p$ (which exists because of our hypothesis that $D$ is odd), then
\[
G_L(a,p^\nu) = G(a\langle b_1,b_1\rangle,p^\nu)\cdots G(a\langle b_n,b_n\rangle,p^\nu),
\]
and  the claim follows from Lemma \ref{lem:gauss1}.
Note that in the case $p\mid D$, standard formulas for the Hilbert symbol imply
\[
\inv_p(V) = \leg{\langle b_1,b_1\rangle}{p}\cdots \leg{\langle b_n,b_n\rangle}{p} . \qedhere
\]
\end{proof}

\begin{proposition}
\label{prop:lp1}
Assume that $n$ is even, and $m\in \Z$ is nonzero.
Fix a prime $p$, and  factor $m=p^\beta m'$ with  $\gcd(m',p)=1$.
%\note{need $p$ odd to invoke  Lemma \ref{lem:gauss2}?}
If $p\nmid D$ then
\[
(1-p^{n-s-1})\sum_{\gamma=0}^{\beta} p^{(n-s)\gamma}.
\]
If $p\mid D$ then
\begin{align*}
&= 1-\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}p^{n-1-s} \\
& \quad + \leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}\left(1 -  p^{n-1-s}\right)\sum_{\gamma=1}^{\beta} p^{(n-s)\gamma} .
\end{align*}
\end{proposition}

\begin{proof}
We compute the representation number $N_m(p^\nu)$ using the identity
\begin{align*}
N_m(p^\nu) &= \frac{1}{p^{\nu}} \sum_{a \in \Z/p^\nu\Z }  G_L(a,p^\nu) e\Big(-\frac{am}{p^\nu}\Big)\\
&=  \frac{1}{p^{\nu}} \sum_{\alpha=0}^\nu
\sum_{ a' \in (\Z/ p^{\nu-\alpha} \Z)^\times}  G_L(p^\alpha a',p^\nu) e \Big(-\frac{a'm}{p^{\nu-\alpha}}  \Big).
\end{align*}
As $n$ is even,  Lemma \ref{lem:gauss2} implies
\[
G_L(a,p^\nu)= \begin{cases}
p^{2n\nu} &\text{if $\alpha=\nu$}\\
p^{n\alpha+n\nu} &\text{if $\alpha <\nu$ and $p\nmid D$ }\\
\leg{-1}{p}^{n/2} \inv_p(V) p^{n\alpha+n\nu+\frac{n}{2}} &\text{if $\alpha <\nu$ and $p\mid D$.}
\end{cases}
\]
If $p\nmid D$ we obtain
\begin{align*}
&= p^{n\nu-\nu}\sum_{\alpha=0}^{\nu}  p^{n\alpha}
\sum_{a' \in  (\Z/ p^{\nu-\alpha} \Z)^\times } e \Big(-\frac{a'm}{p^{\nu-\alpha}} \Big)   \\
&= p^{n\nu-\nu}\sum_{\alpha=0}^{\nu}  p^{n\alpha}
\sum_{d\mid  \gcd(p^{\nu-\alpha},m)}   \mu\Big(\frac{p^{\nu-\alpha}}{d}\Big)  d   \\
&=p^{n\nu-\nu}\sum_{\alpha=0}^{\nu}  p^{n\alpha}
\sum_{\gamma=0}^{\min(\nu-\alpha,\beta)}     \mu\big(p^{\nu-\alpha-\gamma}\big) p^\gamma   .
%&=p^{n\nu-\nu}\sum_{\gamma=0}^{\beta} \sum_{\alpha=0}^{\nu-\gamma} p^{n\alpha+\gamma} \mu(p^{\nu-\alpha-\gamma}).
%&=p^{(2n-1)\nu}(1 -p^{-n})\sum_{\gamma=0}^{\beta}  p^{\gamma(1-n)}.
\end{align*}
Here we have used the evaluation of the Ramanujan sum.
Consequently, the Euler factor \eqref{eq:lp} is
\begin{align*}
L^{(p)}_{m}(p^{-s})&= (1-p^{2n-s-1})
\sum_{\nu=0}^\infty N_{m}(p^\nu) p^{-\nu s}\\
\sum_{\gamma=0}^{\beta} \sum_{\alpha=0}^{\infty} \sum_{\nu=\alpha+\gamma}^\infty p^{n\alpha+\gamma} \mu(p^{\nu-\alpha-\gamma})p^{(n-1-s)\nu}\\
%&=(1-p^{2n-s-1})\sum_{\gamma=0}^{\beta} \sum_{\alpha=0}^\infty \left(p^{n\alpha+\gamma}p^{(n-1-s)(\alpha+\gamma)}-p^{n\alpha+\gamma}p^{(n-1-s)(\alpha+\gamma+1)}\right)\\
&=(1-p^{n-s-1})\sum_{\gamma=0}^{\beta} p^{(n-s)\gamma}.
\end{align*}

In the case $p\mid D$  we find
\begin{align*}
&= p^{2n\nu-\nu}+\leg{-1}{p}^{n/2} \inv_p(V)p^{n\nu-\nu+\frac{n}{2}}\sum_{\alpha=0}^{\nu-1}  p^{n\alpha}
\sum_{a'\in (\Z / p^{\nu-\alpha} \Z)^\times } e \Big(-\frac{a'm}{p^{\nu-\alpha}} \Big) \\
p^{2n\nu-\nu}+\leg{-1}{p}^{n/2} \inv_p(V)p^{n\nu-\nu+\frac{n}{2}}\sum_{\alpha=0}^{\nu-1}  p^{n\alpha}
\sum_{d\mid \gcd(p^{\nu-\alpha},m)}   \mu \Big(\frac{p^{\nu-\alpha}}{d} \Big) d \\
p^{2n\nu-\nu}+\leg{-1}{p}^{n/2} \inv_p(V)p^{n\nu-\nu+\frac{n}{2}}\sum_{\alpha=0}^{\nu-1}  p^{n\alpha} \sum_{\gamma=0}^{\min(\nu-\alpha,\beta)} \mu \big(p^{\nu-\alpha-\gamma}\big) p^\gamma.
%&=p^{2n\nu-\nu}-\leg{-1}{p}^{n/2} \inv_p(V)p^{2n\nu-\nu+\frac{n}{2}}\\
%\leg{-1}{p}^{n/2} \inv_p(V)p^{n\nu-\nu+\frac{n}{2}}\sum_{\alpha=0}^{\nu}  p^{n\alpha} \sum_{\gamma=0}^{\min(\nu-\alpha,\beta)} \mu(p^{\nu-\alpha-\gamma}) p^\gamma
\end{align*}
Consequently,  the Euler factor $L^{(p)}_{m}(p^{-s})$ of \eqref{eq:lp} is
\begin{align*}\lefteqn{
(1-p^{2n-s-1})\sum_{\nu=0}^\infty N_{m}(p^\nu) p^{-\nu s} } \\
&=1+\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}(1-p^{2n-s-1})\sum_{\nu=1}^\infty p^{(n-1-s)\nu}
\sum_{\alpha=0}^{\nu-1}  \sum_{\gamma=0}^{\min(\nu-\alpha,\beta)}p^{n\alpha+\gamma}  \mu(p^{\nu-\alpha-\gamma}) \\
&=1+\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}(1-p^{2n-s-1})
\sum_{\gamma=0}^{\beta}\sum_{\alpha=0}^{\infty}\sum_{\substack{\nu\geq \alpha+\gamma\\ \nu\geq \alpha +1}}p^{(n-1-s)\nu} p^{n\alpha+\gamma}  \mu(p^{\nu-\alpha-\gamma}).
\end{align*}
The contribution of $\gamma=0$ to the latter sum is given by
\begin{align*}
%\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}
(1-p^{2n-s-1})\sum_{\alpha=0}^{\infty}\sum_{\substack{  \nu\geq \alpha +1}}p^{(n-1-s)\nu} p^{n\alpha}  \mu(p^{\nu-\alpha})=-p^{n-1-s}.
\end{align*}
The contribution of $\gamma\geq 1$ is equal to
\begin{align*}
%\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}
&(1-p^{2n-s-1})\sum_{\gamma=1}^{\beta}\sum_{\alpha=0}^{\infty}\sum_{\substack{  \nu\geq \alpha +\gamma}}p^{n\alpha+\gamma} p^{(n-1-s)\nu}  \mu(p^{\nu-\alpha-\gamma})\\
&=(1-p^{2n-s-1})\sum_{\gamma=1}^{\beta}\sum_{\alpha=0}^{\infty} p^{n\alpha+\gamma} \left(p^{(n-1-s)(\alpha+\gamma)} -  p^{(n-1-s)(\alpha+\gamma+1)}\right)\\
%&=(1-p^{2n-s-1})\sum_{\gamma=1}^{\beta}\sum_{\alpha=0}^{\infty} p^{(2n-1-s)\alpha}p^{(n-s)\gamma} \left(1 -  p^{n-1-s}\right)\\
&=\left(1 -  p^{n-1-s}\right)\sum_{\gamma=1}^{\beta} p^{(n-s)\gamma} .
\end{align*}
Putting the terms together, we find
\begin{align*}
L^{(p)}_{m}(p^{-s})&= 1-\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}p^{n-1-s}   \\
& \quad + \leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}}\left(1 -  p^{n-1-s}\right)\sum_{\gamma=1}^{\beta} p^{(n-s)\gamma} .
\end{align*}
This concludes the proof of the proposition.
\end{proof}

\begin{proposition}
\label{prop:lp2}
Assume $n$ is odd, and $m\in \Z$ is nonzero.
Fix a prime $p$, and factor $m=p^\beta m'$ with  $\gcd(m',p)=1$.
%\note{need $p$ odd to use Lemma \ref{lem:gauss2}?}
If $p\nmid D$ then
\[
\left(1-\leg{-D}{p}p^{n-s-1}\right)\sum_{\gamma=0}^{\beta} \leg{-D}{p}^\gamma p^{(n-s)\gamma}.
\]
If $p\mid D$ then
\[
L^{(p)}_{m}(p^{-s})= 1+\inv_p(V)\leg{-1}{p}^{\frac{n-1}{2}}\leg{D'}{p}^\beta\leg{m'}{p} p^{
\]
where we have set $D'=D/p$.
\end{proposition}

\begin{proof}
We argue in the same way as in the proof of Proposition \ref{prop:lp1}, using the identity
\begin{align*}
N_m(p^\nu) &= \frac{1}{p^{\nu}} \sum_{a \in \Z/ p^\nu\Z }  G_L(a,p^\nu) e \Big(-\frac{am}{p^\nu} \Big)
%&=  \frac{1}{p^{\nu}} \sum_{\alpha=0}^\nu \sum_{a'\,(p^{\nu-\alpha})^*}  G_L(p^\alpha a',p^\nu) e(-\frac{a'm}{p^{\nu-\alpha}})
\end{align*}
and the value of the Gauss sum $G_L(a,p^\nu)$ computed in Lemma \ref{lem:gauss2}.
If we write $a=p^\alpha a'$ with $a'$ coprime to $p$, we have
\[
G_L(a,p^\nu)= \begin{cases}
p^{2n\nu}  &\text{if $\alpha=\nu$ }\\
p^{n\alpha+n\nu}\leg{-D}{p}^{\nu-\alpha}  &\text{if $\alpha <\nu$ and $p\nmid D$ }\\
\epsilon_p^n \inv_p(V) p^{n\alpha+n\nu+\frac{n}{2}}\leg{a'}{p}\leg{D'}{p}^{\nu-\alpha-1} &\text{if $\alpha <\nu$ and $p\mid D$.}
\end{cases}
\]
%Here in the latter case we have put $D'=D/p$.

If $p\nmid D$ we obtain
\begin{align*}
N_m(p^\nu) &= p^{n\nu-\nu}\sum_{\alpha=0}^{\nu}  p^{n\alpha}\leg{-D}{p}^{\nu-\alpha}
\sum_{a'  \in  (\Z /  p^{\nu-\alpha} \Z)^\times } e \Big(-\frac{a'm}{p^{\nu-\alpha}} \Big)\\
&= p^{n\nu-\nu}\sum_{\alpha=0}^{\nu}  p^{n\alpha}\leg{-D}{p}^{\nu-\alpha}
\sum_{d\mid  \gcd(p^{\nu-\alpha},m)}\mu \Big(\frac{p^{\nu-\alpha}}{d} \Big) d  \\
&=p^{n\nu-\nu}\sum_{\alpha=0}^{\nu}  p^{n\alpha} \leg{-D}{p}^{\nu-\alpha}\sum_{\gamma=0}^{\min(\nu-\alpha,\beta)} \mu\big(p^{\nu-\alpha-\gamma}\big) p^\gamma .
%&=p^{n\nu-\nu}\sum_{\gamma=0}^{\beta} \sum_{\alpha=0}^{\nu-\gamma} p^{n\alpha+\gamma} \mu(p^{\nu-\alpha-\gamma}).
%&=p^{(2n-1)\nu}(1 -p^{-n})\sum_{\gamma=0}^{\beta}  p^{\gamma(1-n)}.
\end{align*}
Here we have used the evaluation of the Ramanujan sum.
Hence,  the Euler factor \eqref{eq:lp} is
\begin{align*}
L^{(p)}_{m}(p^{-s})&= (1-p^{2n-s-1})
\sum_{\nu=0}^\infty N_{m}(p^\nu) p^{-\nu s}\\
\sum_{\gamma=0}^{\beta} \sum_{\alpha=0}^{\infty} \sum_{\nu=\alpha+\gamma}^\infty p^{n\alpha+\gamma} \leg{-D}{p}^{\nu-\alpha}\mu(p^{\nu-\alpha-\gamma})p^{(n-1-s)\nu}\\
%&=(1-p^{2n-s-1})\sum_{\gamma=0}^{\beta} \sum_{\alpha=0}^\infty \left(p^{n\alpha+\gamma}p^{(n-1-s)(\alpha+\gamma)}-p^{n\alpha+\gamma}p^{(n-1-s)(\alpha+\gamma+1)}\right)\\
&=\left(1-\leg{-D}{p}p^{n-s-1}\right)\sum_{\gamma=0}^{\beta} \leg{-D}{p}^\gamma p^{(n-s)\gamma}.
\end{align*}

When  $p\mid D$ we find
\begin{align*}
N_m(p^\nu)   = p^{2n\nu-\nu}
&\phantom{+}{}\times\sum_{\alpha=0}^{\nu-1} \leg{D'}{p}^{\nu-\alpha-1}  p^{n\alpha} \sum_{a'  \in  (\Z /  p^{\nu-\alpha} \Z)^\times } \leg{a'}{p} e \Big(-\frac{a'm}{p^{\nu-\alpha}}  \Big).
\end{align*}
The latter Gauss sum is equal to $p^{\nu-\alpha-\frac{1}{2}}\epsilon_p \leg{-m'}{p}$ if $\beta=\nu-\alpha-1$ and is zero otherwise. Inserting this we find that $N_m(p^\nu)$ is equal to
\begin{align*}
& \cdot \begin{cases}
1+\inv_p(V)\leg{-1}{p}^{\frac{n+1}{2}}\leg{D'}{p}^\beta\leg{-m'}{p} p^{(1-n)(\beta+1/2)}  &\text{if $\nu>\beta$} \\[1ex]
1 &\text{if $\nu\leq \beta$,}
\end{cases}
\end{align*}
and hence $L^{(p)}_{m}(p^{-s})$ is equal to
\begin{align*} \lefteqn{
%L^{(p)}_{m}(p^{-s}) } \\
(1-p^{2n-s-1})\sum_{\nu=0}^\infty N_{m}(p^\nu) p^{-\nu s} }\\
&=1+\inv_p(V)\leg{-1}{p}^{\frac{n+1}{2}}\leg{D'}{p}^\beta\leg{-m'}{p} p^{(1-n)(\beta+1/2)} p^{(2n-1-s)(\beta+1)}\\
&=1+\inv_p(V)\leg{-1}{p}^{\frac{n-1}{2}}\leg{D'}{p}^\beta\leg{m'}{p} p^{
\end{align*}
This concludes the proof of the proposition.
\end{proof}

Recall the function $𝐚_n(s)$ of \eqref{a_k}, and abbreviate $s_0=(n-1)/2$.
We now express the coefficients $B(m,0,s)$ in terms of
\[
\mathbf{a}_n(s-s_0) =
\frac { D^{n/2} \Gamma \left( s + \frac{n}{2} + \frac{1}{2}  \right)  L\left(2s +1,\eps^n\right) }   {2^n \pi^{  s + \frac{n}{2} + \frac{1}{2}  } } .
\]

\begin{corollary}
\label{cor:eis2}
Assume that $n\ge 2$ is even.
If $m>0$ then
\[
B(m,0,s)= -
\frac{  m^{s+\frac{n}{2}-\frac{1}{2}}  }{ \mathbf{a}_n(s-s_0)    }
\cdot \prod_{p\nmid D}\sum_{\gamma=0}^{v_p(m)} p^{-2s\gamma}\cdot
\prod_{p\mid D} \frac{L^{(p)}_{m}(p^{-2s-n})}{1-p^{-2s-1}}.
\]
If  $m>0$ is prime to $D$ then
\[
B(m,0,s)=-\frac{ m^{s+\frac{n}{2}-\frac{1}{2}}   \sigma_{-2s}(m) }{  \mathbf{a}_n(s-s_0)    }
%\prod_{p\nmid D}\sum_{\gamma=0}^{v_p(m)} p^{-2s\gamma}
\cdot
\prod_{p\mid D} \frac{1-\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}-1-2s} }{1-p^{-2s-1}},
\]
where $\sigma_s(m)=\sum_{d\mid m} d^s$ is the usual divisor sum.
\end{corollary}

\begin{proof}
Combine Propositions \ref{prop:eis1} and \ref{prop:lp1}.
\end{proof}

%Assume that $m$ is positive and relatively prime to $D$.
%The coefficient $B(m,0,s)$ is given by
%B(m,0,s)=-\frac{2^{n} \pi^{s+\frac{n}{2}+\frac{1}{2}}m^{s+\frac{n}{2}-\frac{1}{2}}   \sigma_{-2s}(m) }{ D^{n/2}\Gamma(s+\frac{n}{2}+\frac{1}{2})\zeta(2s+1)}
%%\prod_{p\nmid D}\sum_{\gamma=0}^{v_p(m)} p^{-2s\gamma}
% \prod_{p\mid D} \frac{1-\leg{-1}{p}^{\frac{n}{2}} \inv_p(V)p^{\frac{n}{2}-1-2s} }{1-p^{-2s-1}},
%where $\sigma_s(m)=\sum_{d\mid m} d^s$ is the usual divisor sum.
%In particular, the value at $s_0=(n-1)/2$ is
%%B(m,0,s_0)=-\frac{2^{n} \pi^{n}}{ \Gamma(n)\zeta(n)}
%%\cdot \sigma_{n-1}(m)\cdot
%% \prod_{p\mid D} \frac{p^{\frac{n}{2}}-\leg{-1}{p}^{\frac{n}{2}} \inv_p(V) }{p^{n}-1}.
%B(m,0,s_0)=-\frac{(2\pi)^{n}  \sigma_{n-1}(m) }{ \Gamma(n)\zeta(n)}
% \prod_{p\mid D} \Big(p^{\frac{n}{2}}+\leg{-1}{p}^{\frac{n}{2}} \inv_p(V) \Big)^{-1},
%and the logarithmic derivative is
%\log(\frac{ m}{4\pi}) -2\frac{\sigma_{1-n}'(m)}{\sigma_{1-n}(m)}\\
%&\phantom{=}{}+2\sum_{p\mid D} \leg{-1}{p}^{\frac{n}{2}} \frac{\inv_p(V)p^{\frac{n}{2}}}{p^n-1}\log(p).
%The first assertions follow immediately from Corollary \ref{cor:eis2} and Proposition \ref{prop:lp1}.
%For the  logarithmic derivative in \eqref{eq:bp1} we obtain
%% To this end we employ Corollary~\ref{cor:eis3} to obtain
%\frac{B'(m,0,s_0)}{B(m,0,s_0)} & =
%\log(\pi m) -\frac{\Gamma'(n)}{\Gamma(n)}-2\frac{\zeta'(n)}{\zeta(n)}-2\frac{\sigma_{1-n}'(m)}{\sigma_{1-n}(m)}   \\
%& \quad +2\sum_{p\mid D} \leg{-1}{p}^{\frac{n}{2}} \frac{\inv_p(V)p^{\frac{n}{2}}}{p^n-1}\log(p).
%Using the functional equation
%\zeta(1-s)= 2(2\pi)^{-s}\cos\left(\frac{\pi s}{2}\right)\Gamma(s)\zeta(s)
%and our hypothesis that $n$ is even,  we find
%\begin{equation}\label{even functional}
%\frac{\zeta'(n)}{\zeta(n)}= -\frac{\zeta'(1-n)}{\zeta(1-n)}
%%Using the functional equation $\sigma_s(m)=m^s\sigma_{-s}(m)$ of the divisor sum, we obtain $\frac{\sigma'_{s}(m)}{\sigma'_{s}(m)}=\log(m)-\frac{\sigma'_{-s}(m)}{\sigma'_{-s}(m)}$.
%and the claim follows.

\begin{corollary}
\label{cor:eisodd2}
Assume that $n\ge 1$ is odd.
If $m>0$ then
\begin{align*}
B(m,0,s)  =  -\frac{ m^{s+\frac{n}{2}-\frac{1}{2}}}{  \mathbf{a}_n(s-s_0) }
\cdot
\prod_{p\nmid D}\sum_{\gamma=0}^{v_p(m)} \eps(p)^\gamma p^{-2s\gamma}\cdot
\prod_{p\mid D} L^{(p)}_{m}(p^{-2s-n}).
\end{align*}
If $m>0$ is  prime to $D$, then
\begin{align*}
B(m,0,s)  =-\frac{ m^{s+\frac{n}{2}-\frac{1}{2}}\sigma_{-2s,\eps}(m)}{ \mathbf{a}_n(s-s_0) }
\cdot \prod_{p\mid D}
\left( 1+   \leg{-1}{p}^{\frac{n-1}{2}} \!\leg{m}{p}   \inv_p(V)   p^{\frac{n-1}{2}-2s} \right),
\end{align*}
where $\sigma_{s,\eps}(m)=\sum_{d\mid m} \eps(d) d^s$ is  the usual divisor sum.
\end{corollary}

\begin{proof}
Combine  Propositions   \ref{prop:eis1} and  \ref{prop:lp2}.
\end{proof}

%If $m$ is positive and prime to $D$, then
%B(m,0,s)  =-\frac{2^{n} \pi^{s+\frac{n}{2}+\frac{1}{2}}m^{s+\frac{n}{2}-\frac{1}{2}}\sigma_{-2s,\eps}(m)}{ D^{n/2}\Gamma(s+\frac{n}{2}+\frac{1}{2})L( 2s+1 , \eps )}
%\cdot \prod_{p\mid D}
%\left( 1+\inv_p(V)\leg{-1}{p}^{\frac{n-1}{2}} \!\leg{m}{p}p^{\frac{n-1}{2}-2s} \right),
%where $\sigma_{s,\eps}(m)=\sum_{d\mid m} \eps(d) d^s$.
%% denotes the usual divisor sum.
%In particular, the value at $s_0=(n-1)/2$ is equal to
%%B(m,0,s_0)=-\frac{2^{n} \pi^{n}}{ \Gamma(n)\zeta(n)}
%%\cdot \sigma_{n-1}(m)\cdot
%% \prod_{p\mid D} \frac{p^{\frac{n}{2}}-\leg{-1}{p}^{\frac{n}{2}} \inv_p(V) }{p^{n}-1}.
%B(m,0,s_0)   =-\frac{(2\pi)^{n} m^{n-1} \sigma_{1-n,\eps}(m)}{ D^{n/2}\Gamma(n)L( n , \eps)}
%\cdot  \prod_{p\mid D}
%\left( 1+\inv_p(V)\leg{-1}{p}^{\frac{n-1}{2}} \!\leg{m}{p}p^{\frac{1-n}{2}} \right),
%%\Big(p^{\frac{n}{2}}+\leg{-1}{p}^{\frac{n}{2}} \inv_p(V) \Big)^{-1}.
%and, if $n>1$,  the logarithmic derivative is
%\frac{B'(m,0,s_0)}{B(m,0,s_0)}&=2\frac{L'( 1-n , \eps )}{L( 1-n, \eps )}+\frac{\Gamma'(n)}{\Gamma(n)}+
%\log\left(\frac{ mD^2}{4\pi}\right) -2\frac{\sigma_{1-n,\eps}'(m)}{\sigma_{1-n,\eps}(m)}\\
%&\phantom{=}{}-2\sum_{p\mid D} \left(1+\inv_p(V)\leg{-1}{p}^{\frac{n-1}{2}}\leg{m}{p}p^{\frac{n-1}{2}}\right)^{-1}\log(p).
%The first assertions follow directly from Corollary \ref{cor:eisodd2}.
%For the  logarithmic derivative we compute
%%To prove  \eqref{eq:bp2} for odd $n$, we use Corollary~\ref{cor:eisodd3} to derive
%\log(\pi m) -\frac{\Gamma'(n)}{\Gamma(n)}-2\frac{L'( n , \eps)}{L( n , \eps)}-2\frac{\sigma_{1-n,\eps}'(m)}{\sigma_{1-n,\eps}(m)}\\
%&\phantom{=}{}-2\sum_{p\mid D} \left(1+\inv_p(V)\leg{-1}{p}^{\frac{n-1}{2}}\leg{m}{p}p^{\frac{n-1}{2}}\right)^{-1}\log(p).
%Using the assumption that $n$ is odd and the  functional equation
%L( 1-s , \eps )= 2(2\pi)^{-s}D^{s-1/2}\sin \left(\frac{\pi s}{2} \right)\Gamma(s)L( s , \eps )
%of the Dirichlet $L$-function function, we see that
%\begin{equation}\label{odd functional}
%\frac{L'( n , \eps)}{L(  n  , \eps)}= -\frac{L'(1-n , \eps )}{L( 1-n  ,  \eps )}
%Inserting this, we obtain the claimed formula.

\begin{remark}
If $n=1$, then
%the coeffients $B'(m,0,s_0)$ that appear in Theorem \ref{thm:int2} can be computed as follows. In this case
$s_0=0$ is the center of symmetry of the functional equation of the incoherent Eisenstein series $E_L(\tau,s,1)$. Hence $E_L(\tau,s,1)$ vanishes identically (to odd order) at $s=0$. For all  positive $m$ this implies  that the coefficients $B(m,\mu,s_0)$ also vanish. Consequently, in view of
Proposition \ref{prop:eis1}, there must be at least one prime $p<\infty$ for which the local factor $L^{(p)}_m(p^{-1})$ is zero. According to Proposition \ref{prop:lp2}, this happens exactly when
\[
\inv_p(V)= -\begin{cases}\leg{-D}{p}^\beta,&\text{if $p\nmid D$,}\\[1ex]
\leg{D'}{p}^\beta\leg{m'}{p},&\text{if $p\mid D$,}
\end{cases}
\]
where we have used the notation of the cited proposition. In terms of the local Hilbert symbol this is equivalent to the condition $\inv_p(V)= -(-D,m)_p$, which means that $V_p$ does not represent $m$.

The derivatives $B'(m,0,s_0)$ of the coefficients can be computed as follows.
Let $\mathbb{V}$ be the incoherent hermitian space over $\A_\kk$ whose non-archimedian contribution is given by $V\otimes_\Q \A_f$ and whose archimedian contribution is $V\otimes_\Q \R$ with the positive definite hermitian form $-\langle \cdot,\cdot \rangle$. Following [30] we define the `Diff set' associated with $\mathbb{V}$ and $m\in \Q^\times$ by
\[
\operatorname{Diff}(\mathbb{V},m)=\{ p\leq \infty \mid \;\text{$\mathbb{V}_p$ does not represent $m$}\}.
\]
This is a finite set of places of $\Q$ of odd cardinality. When $m>0$ it only consists of finite primes. The above argument shows that $L^{(p)}_m(p^{-1})=0$ if and only if $p\in \operatorname{Diff}(\mathbb{V},m)$. Hence $B'(m,0,s_0)\neq 0$ if and only if $ \operatorname{Diff}(\mathbb{V},m)$ consists of a single prime $q$. By Proposition \ref{prop:eis1} we find for $m\in \Z^+$
in this case that
\[
B'(m,0,0)= -\frac{2\pi}{\sqrt{D}}\cdot \prod_{\substack{\text{$p$ prime}\\ p\neq q}} L_m^{(p)} (p^{-1}) \cdot \frac{d}{ds} L_m^{(q)} (q^{-2s-1}) \big|_{s=0} .
\]
This expression  can be evaluated explicitly by means of Proposition \ref{prop:lp2}. For instance, when $(m,D)=1$ and $m=q^\beta m'$ with $m'$ coprime to $q$, we see that $\inv_q(V)=1$, the prime $q$ must be inert in $\kk$, and $\beta$ must be odd. Under these conditions we find that
\[
B'(m,0,0)= -\frac{2^{o(D)+1}\pi\,\sigma_{0,\eps}(m')}{\sqrt{D}L(\eps,1)}  \left(\beta+1\right)\cdot \log(q),
\]
where $o(D)$ denotes the number of prime factors of $D$.
%\note{Jan: Add the formula for $B'(0,0,0)$ for completeness?}
\end{remark}

Regardless of whether $n$ is even or odd, we have the  following lower bound for the coefficients of the Eisenstein series $E_L(τ,s_0,n)$.

\begin{corollary}
\label{cor:eisgrowth}
Assume that $n \geq 2$.
There is a constant $C>0$, depending  only  on $L$,  such that
\[
-B(m,0,s_0)> C\cdot m^{n-1}
\]
for all $m\in \Z^+$.
\end{corollary}

\begin{proof}
This follows  easily from   Corollary \ref{cor:eis2} and Corollary \ref{cor:eisodd2}.
\end{proof}

\section{Automorphic Green functions}
\label{s:green integrals}

Keep $V$ as in \S \ref{s:eisenstein}.
We recall the family of special divisors on the complex Shimura variety associated to the unitary group  $(V)$, and derive a geometric variant of the Siegel-Weil formula.
Using this, we compute the integrals of  automorphic Green functions for the special divisors.

\subsection{A complex Shimura variety}
\label{ss:u(v) shimura}

The  fixed embedding $⊂$  identifies $⊗_$, and  allows us to distinguish between the two  orthogonal idempotents $ϵ, ϵ ∈⊗_$, which we label in such a way that
\[
( \alpha \otimes 1) \epsilon = (1\otimes \alpha) \epsilon ,\qquad
(\alpha \otimes 1) \overline{\epsilon} =  (1\otimes \overline{\alpha}  )  \overline{\epsilon}
\]
for all $α∈$.

Viewing $V()=V⊗_$  as $$-hermitian space of signature $(n-1,1)$,  define a hermitian symmetric domain
\begin{equation}\label{hermitian model}
\mathcal{D} = \{  \mbox{negative definite complex lines } z \subset V(\R) \}.
\end{equation}
The natural map
\[
V(\R) \hookrightarrow V(\C) = \epsilon V(\C) \oplus \overline{\epsilon} V(\C) \map{\mathrm{proj}}  \epsilon V(\C)
\]
is a $$-linear isomorphism, and identifies
\begin{equation}\label{bilinear model}
\mathcal{D} = \{   z \in   \mathbb{P}( \epsilon V(\C))   : [ z , \overline{z} ]  <0 \}.
\end{equation}
Here we have endowed  $V() =V⊗_$ with the $$-bilinear extension of the $$-bilinear form $[-,-]$ of \eqref{Q bilinear}, and $z↦z$ is the complex conjugation on the second factor in the tensor product.
%One can also identify $\mathcal{D}$ with the complex $n-1$ ball, but we have no use for this interpretation.
Under either interpretation, there is an evident action of the real points of
\[
\]
on $𝒟$ by holomorphic automorphisms.

Fix  an $_$-lattice $L⊂V$ on which the hermitian form is $_$-valued.
Choose a compact open subgroup   $K⊂H(_f)$  that stabilizes $L$, and hence acts on the finite dimensional vector space $S_L = [ L' / L]$ of \S \ref{ss:lattice eisenstein}.
This choice determines a complex Shimura variety
\begin{align}
\label{eq:X}
\mathrm{Sh}_K(H,\mathcal{D})=H(\Q)\backslash \mathcal{D}\times H(\A_f)/K ,
\end{align}
which we view as a complex orbifold of dimension $n-1$.
If we write $H(_f)= _j H()h_j K$ as a finite disjoint union, then
\begin{equation}\label{X components}
\mathrm{Sh}_K(H,\mathcal{D}) =\bigsqcup_j  ( H(\Q)\cap K_j) \backslash \mathcal{D},
\end{equation}
where $K_j = h_j K  h_j^-1$ stabilizes the lattice $L_j = h_j L$.

The Shimura variety  carries a family of special divisors
\[
Z(m) \in \mathrm{Div}(  \mathrm{Sh}_K(H,\mathcal{D}) )
\]
indexed by positive integers $m$, which
we describe in terms of the decomposition \eqref{X components}.
For each $x∈V$ with $⟨x,x⟩>0$, define an analytic divisor
\[
\mathcal{D}(x) = \{ z\in \mathcal{D} :  x \perp z  \}
\]
where $⊥$ means orthogonal with respect to the hermitian or bilinear form, depending on whether the reader prefers the model \eqref{hermitian model} or \eqref{bilinear model}.
The pullback of  $Z(m)$ via  the uniformization
\[
\mathcal{D} \to ( H(\Q)\cap K_j) \backslash \mathcal{D} \subset \mathrm{Sh}_K(H,\mathcal{D})
\]
is the locally finite analytic divisor
\[
\sum_{ \substack{ x \in  L_j  \\  \langle x, x\rangle =m  } } \mathcal{D}(x) \in \mathrm{Div}(\mathcal{D}).
\]

More generally,  given a positive $m∈$ and a $K$-fixed function $φ∈S_L$,  there is a special divisor
\[
Z(m,\varphi) \in \mathrm{Div}_\C(  \mathrm{Sh}_K(H,\mathcal{D}) )
\]
whose pullback via the above uniformization is
\[
\sum_{ \mu \in L '/L  }  \varphi(\mu)
\sum_{ \substack{ x \in \mu_j+L_j   \\  \langle x, x\rangle =m  } }
\mathcal{D}(x) \in \mathrm{Div}_\C (\mathcal{D}),
\]
where  $μ_j$ is the image of $μ$ under $h_j : L'/L →L_j'/L_j$.
Taking $φ=φ_0$ to be the characteristic function of $0∈L'/L$ recovers $Z(m)$.

Let $$ be the tautological line bundle over $𝒟$ whose fiber over a point $z∈𝒟$ is the complex line $_z=z$.
Denote by  $$ the tautological  line bundle endowed with the hermitian metric  $·$ defined as follows:  if we use the interpretation \eqref{hermitian model} to view $z∈𝒟$ as a line in $V()$, then
\[
\| s\|^2 = - \frac{ \langle s,s\rangle}{4\pi e^\gamma}
\]
for every $s∈_z$.  If we instead use \eqref{bilinear model} to view $z$ as a line in $ϵV()$, the metric is the same, but with $⟨-,-⟩$ replaced by $[-,-]$.
The Chern form  $()$ is an $H()$-invariant K\"ahler form on $𝒟$.
The metrized tautological bundle descends to $Sh_K(H,𝒟)$, and its holomorphic sections are hermitian modular forms of weight $1$.  As in the introduction, we abbreviate
\[
\vol_\C( \widehat{\taut}) = \int_{\mathrm{Sh}_K(H,\mathcal{D})}  \chern(\widehat{\taut})^{n-1}.
\]

In the subsections below we will recall the construction of Green functions for certain linear combinations of special  divisors,  and  compute their integrals with respect to the volume form $()^n-1$.

\begin{remark}
Although we will not need it, we note that
\[
\vol_\C( \widehat{\taut} )^{-1}    \int_{Z(m,\varphi)}  \chern(\widehat{\taut})^{n-2} =
-  \sum_{ \mu \in L'/L } \varphi(\mu) B(m,\mu,s_0)
\]
exactly as in the case of orthogonal Shimura varieties [31].
On the right hand side the notation is that of \S \ref{ss:coefficients}.
\end{remark}

\subsection{A geometric variant of the Siegel-Weil formula}

For $z∈𝒟$ and $x∈V()$, let
\[
\varphi_\infty(z,x)= \exp\big(-2\pi \langle x_{z^\perp}, x_{z^\perp}\rangle +2\pi   \langle x_{z}, x_{z}\rangle \big)
\]
be the Gaussian.  Here we are using the model \eqref{hermitian model}, so that $z$ determines an orthogonal decomposition
V() = z^⊥⊕z,
and the vectors $x_z^⊥$ and $x_z$ are the projections of $x$ to the two summands.

As in \S \ref{ss:seesaw}, let $G= (1,1)$ be the quasi-split unitary group over $$, so that $_2 ⊂G$.
Fix a base point $z_0∈𝒟$ and consider the  Schwartz function
\[
\varphi_\infty(x) \define \varphi_\infty(z_0,x) \in S(V(\R)).
\]
It has weight $n-2$ under the action of the Weil representation (\S \ref{ss:siegel-weil}) of $_2()$.  In other words, using the notation of \eqref{underline k},
\begin{align*}
\omega(k)(\varphi_\infty)=\underline k^{n-2} \varphi_\infty
\end{align*}
for all $k∈(2,)$.

A choice of   $φ_f ∈S(V(_f))$  determines a Schwartz function $φ_∞⊗φ_f ∈S(V())$.
Applying the construction \eqref{theta kernel} yields a  corresponding theta function $θ( g,h, φ_∞⊗φ_f)$ of the variables $(g,h) ∈G() ×H()$.  Restricting the first variable to
$g∈_2()  ⊂G()$, we obtain the  Siegel theta function
\[
\theta(g  ,h,\varphi_\infty\otimes \varphi_f)
= \sum_{x\in V(\Q)} \omega(g,1) \varphi_\infty(z_0,h_\infty^{-1} x )\cdot \varphi_f(h_f^{-1}x),
\]
where $h=h_∞h_f∈H()$.

We may view this theta function as a function of the variables
\[
(\tau, z , h_f) \in \H \times \mathcal{D}  \times H(\A_f)
\]
as follows. Let $g_τ∈_2()$ be as in \eqref{gtau}, and  choose an  $h_∞∈H()$ with $h_∞z_0= z$,
so that $φ_∞(z_0,h_∞^-1 x)= φ_∞(z, x)$.  Now set
\begin{align}
\label{eq:siegel2}
\theta(\tau,z,h_f,\varphi_f)&=v^{1- \frac{n}{2} }
\theta(g_\tau,h_\infty h_f, \varphi_\infty\otimes\varphi_f)\\
%&= v^{1-n/2}\sum_{x\in V(\Q)} \omega(g_\tau,1) \varphi_\infty(z_0,h_\infty^{-1} x )\cdot \varphi_f(h_f^{-1}x)\\
\nonumber
&= v \sum_{x\in V(\Q)} e\big(\langle x_{z^\perp}, x_{z^\perp}\rangle \tau-  \langle x_{z}, x_{z}\rangle\bar\tau\big)\cdot \varphi_f(h_f^{-1}x).
\end{align}
In the variable $τ$, we have the transformation law
\begin{align}
\label{eq:thetatr}
\theta(\gamma\tau, z,h_f ,\varphi_f)=(c\tau +d)^{n-2}\theta(\tau,z,h_f, \omega(\gamma)^{-1}\varphi_f)
\end{align}
for $γ∈Γ$.
In the variable $z$, we have the transformation law
\begin{align}
\label{eq:thetatr2}
\theta(\tau,\delta z,\delta h_f ,\varphi_f)=\theta(\tau,z,h_f, \varphi_f).
\end{align}
for  $δ∈H()$,

Now suppose that $φ_f ∈S_L = [L'/L]$ is fixed by the action of $K$.
Using \eqref{eq:siegel2} and \eqref{eq:thetatr2}, we see that   the  Siegel theta function \eqref{eq:siegel2}
descends to a function on  $ ℋ×Sh_K(H,𝒟)$.    We are interested in the geometric theta integral
\begin{align}
\label{eq:intgeo}
\mathcal{I}(\tau,\varphi_f)
&=  \vol_\C( \widehat{\taut}  )^{-1} \int_{\mathrm{Sh}_K(H,\mathcal{D})}
\theta(\tau,z,h_f, \varphi_f)\,\chern(\widehat{\taut})^{n-1}
\end{align}
as a function of $τ∈$̋.

Assume that $n>2$ or that $V$ is anisotropic.
Recalling the notation of (<ref>) and (<ref>),  we have

\[
\kappa\cdot \mathcal{I}(\tau,\varphi_f) =
\left\langle E_L(\tau, s_0,n-2),\varphi_f\right\rangle,
\]

where the pairing on the right is (<ref>).

We rewrite the integral in (<ref>) as an integral over the quotient $H(\Q)\backslash H(\A)$.
To this end,  we normalize the Haar measure on the compact subgroup $H(\R)_{z_0}\subset H(\R)$ so that the volume is $1$.
Then we normalize the Haar measure $dh_\infty$ on $H(\R)$ so that the induced measure on the quotient
$H(\R)/H(\R)_{z_0}\cong \mathcal{D}$ agrees with the measure defined by $\chern(\widehat{\taut})^{n-1}$.
Finally, we normalize the Haar measure $dh_f$ on $H(\A_f)$ so that $\vol(H(\Q)\backslash H(\A))=1$.

\[
\vol(K) = \int_K dh_f,
\]

we then  have the identities

\[
\vol_\C(  \widehat{\taut} )  = \int_{   H(\Q) \backslash H(\A) / H(\R)_{z_0} K  }  dh_\infty dh_f = \vol(K)^{-1}.
\]

Using (<ref>) we obtain

\begin{align*}
%\kappa\cdot \mathcal{I}(\tau,\varphi_f) %&= \kappa
\mathcal{I}(\tau,\varphi) &= \vol(K)\int_{\mathrm{Sh}_K(H,\mathcal{D})} \theta(\tau,z, h_f,\varphi_f)\, \chern(\widehat{\taut})^{n-1}  \\
v^{1-\frac{n}{2}}\int_{H(\Q)\backslash H(\A)} \theta(g_\tau,h_\infty h_f,\varphi_\infty\otimes\varphi_f)\, dh_\infty dh_f,
\end{align*}

and applying the Siegel-Weil formula of  Theorem <ref> shows

\begin{align*}
\kappa\cdot \mathcal{I}(\tau,\varphi) &=v^{1-\frac{n}{2}}
E(g_\tau,\lambda(\varphi_\infty\otimes \varphi_f),s_0).
\end{align*}

As $\varphi_\infty$ has weight $n-2$,  Lemma <ref> implies

\begin{align*}\lefteqn{
E(g_\tau,\lambda(\varphi_\infty\otimes \varphi_f),s_0)  } \\
&=v^{\frac{n}{2}-1} \sum_{\gamma\in \Gamma_\infty\backslash \Gamma}
\operatorname{Im}(\gamma \tau)^{s_0+\frac{1}{2}-\frac{n-2}{2}}  (c\tau+d)^{2-n}\cdot (\omega_L(\gamma)\varphi_f)(0) \\
\nonumber
&=v^{\frac{n}{2}-1} \sum_{\gamma\in \Gamma_\infty\backslash \Gamma}
\operatorname{Im}(\gamma \tau)^{s_0+\frac{1}{2}-\frac{n-2}{2}}  (c\tau+d)^{2-n}\cdot (\omega_L^\vee(\gamma)^{-1}\delta_0)(\varphi_f) .
\end{align*}

Here $\delta_0\in S_L^\vee$ denotes the functional  $\varphi_f\mapsto \varphi_f(0)$.
Since the Weil representation is unitary,  we have

\[
(\omega_L^\vee(\gamma)^{-1}\delta_0)(\varphi_f) = \langle \bar \omega_L(\gamma)^{-1}\varphi_0, \varphi_f\rangle.
\]

To complete the proof, substitute this equality into the final expression above, and recall the definition of $E_L(\tau,s,n-2)$ from (<ref>).

§.§ Automorphic Green functions

Let $\sigma$ be a finite dimensional representation of $\Gamma=\SL_2(\Z)$
on a complex vector space $V_\sigma$, and assume that $\sigma$  factors through a finite quotient.
In our applications, $\sigma$ will be  the Weil representation

\[
\omega_L:\Gamma\to \Aut(S_L)
\]

of (<ref>),  or its complex conjugate, or its dual.

For $k\in \Z$,  denote by $H_k(\sigma)$ the vector space of (weak) harmonic Maass forms of weight $k$ for the group $\Gamma$ with representation $\sigma$ as in [10].
We write

\[
S_k(\sigma) \subset  M_k(\sigma)   \subset M_k^!(\sigma)
\]

for the subspaces of cusp forms,  holomorphic modular forms, and weakly holomorphic modular forms, respectively.

A harmonic Maass form $f\in H_k(\sigma)$ has a Fourier expansion

\begin{align}
\label{eq:fourierf}
f(\tau)=\sum_{\substack{m\in \Q\\ m\gg -\infty }} c^+(m) q^m
+\sum_{\substack{m\in \Q \\ m<0 } } c^-(m) \Gamma(1-k, 4\pi |m| v) q^m
\end{align}

with coefficients $c^\pm(m)\in V_\sigma$.
Here $\tau=i+i v \in \H$,  $q=e^2πiτ$, and
\[
\Gamma(s,x)=\int_x^\infty e^{-t}t^{s-1}dt
\]
is the incomplete gamma function.
The coefficients are supported on rational numbers with uniformly bounded denominators.
The first summand on the right hand side of \eqref{eq:fourierf} is denoted by $f^+$ and is called the holomorphic part of $f$, the second summand is denoted by $f^-$ and is called the non-holomorphic part.
%The $V_\sigma$-valued Fourier  polynomial
%P_f=\sum_{\substack{m\in \Q\\ m\leq 0}} c^+(m) q^m
%is called the principal part of $f$.

As in [7], there is a  surjective conjugate-linear differential operator
\begin{equation}\label{xi}
\xi_k: H_k(\omega_L)\to S_{2-k}(\bar\omega_L)
\end{equation}
defined by
\[
\xi_k(f)(\tau)=2iv^{k} \overline{\frac{\partial f}{\partial \bar\tau}}.
\]
Its kernel is   $M^!_k(ω_L)$.

Suppose  $f∈H_2-n(ω_L)$ is $K$-fixed  with Fourier coefficients  $c^±(m)∈S_L$.
Define the special divisor associated to $f$ by
\begin{align}
\label{eq:zf}
Z(f)= \sum_{m>0}\sum_{\mu \in L'/L} c^+(-m,\mu) Z(m, \varphi_\mu)\in \Div_\C(\mathrm{Sh}_K(H,\mathcal{D})).
\end{align}
Here $c^±(m,μ)∈$ is the value of $c^±(m)$ at  $μ∈L'/L$, and
$φ_μ$ is  the characteristic function of the coset $μ+L̂$.

We recall from \S 3  of  [8] the construction of an automorphic Green function for $Z(f)$.
Using \eqref{eq:siegel2}, define an $S_L$-valued Siegel theta function
\begin{equation}\label{lattice theta}
\theta_L(\tau,z,h)=\sum_{\mu \in L'/L} \theta(\tau,z,h,\varphi_\mu)\varphi_\mu
\end{equation}
for $τ∈$̋, $z\in \mathcal{D}$, and $h\in H(\A_f)$.
The transformation law (<ref>) implies that it defines a non-holomorphic modular form for $\Gamma$ of weight $n-2$ with representation $\bar \omega_L$,
and  so, using the notation (<ref>),  the pairing $\langle f, \theta_L(\tau,z,h)\rangle $ is  $\Gamma$-invariant as a  function of $\tau \in \H$.

Following [2] and [7], we consider the regularized theta lift
\begin{align}
\label{eq:AutoGreen}
\Phi(z,h,f)= \int_{\Gamma\backslash \H}^{\reg}
\langle f,\theta_L(\tau,z,h)\rangle \, d\mu(\tau)
\end{align}
of $f$, where $dμ(τ)= du dv/v^2$ is the invariant measure.
%The integral is regularized by taking the constant term in the Laurent expansion at $s=0$ of the meromorphic continuation of
%\Phi(z,f,s)= \lim_{T\to \infty}\int_{\mathcal{F}_T}
%\{ f,\Theta_L(\tau,z)\} v^{-s}d\mu(\tau).
%Here $\mathcal{F}_T$ denotes the standard fundamental domain for $\SL_2(\Z)$ truncated at height $T$.
Here the integral is regularized as in [2] and [8].
The main properties of
\[
\Phi(f) = \Phi(z,h,f) ,
\]
viewed as a function on $Sh_K(H,𝒟)$,  are summarized in the following theorem, see Theorem~3.3.1 and Proposition~3.3.4 of [8].

\begin{theorem}
The Green function $\Phi(z,h,f)$ is smooth on $\mathrm{Sh}_K(H,\mathcal{D})\smallsetminus Z(f)$. It has logarithmic singularities along $Z(f)$, and it is integrable over $\mathrm{Sh}_K(H,\mathcal{D})$ if $n>2$ or if $V$ is anisotropic. The corresponding current on smooth compactly supported $(n-1,n-1)$-forms on $\mathrm{Sh}_K(H,\mathcal{D})$ satisfies the Green equation
\[
dd^c [\Phi(f)] +\delta_{Z(f)} = [dd^c \Phi(f)].
\]
\end{theorem}

%For the behavior of $\Phi(f)$ on the canonical toroidal compactification of $\mathrm{Sh}_K(H,\mathcal{D})$ we refer to \cite[Corollary 4.3.5]{BHY} and to \S \ref{sect:4} of the present note.

\subsection{Integrals of Green functions}

In this subsection we assume that either $n>2$, or that $n≥1$ and $V$ is anisotropic.
Choose  an $S_L$-valued harmonic Maass form
\[
f\in H_{2-n}(\omega_L)
\]
fixed by the action of the compact open subgroup $K ⊂H(_f)$ on $S_L$.

The main result of this subsection expresses the integral
\begin{align}
\label{eq:greenint}
\mathcal{I}(f) =  \vol_\C(\widehat{\taut})^{-1} \int_{\mathrm{Sh}_K(H,\mathcal{D})} \Phi(f)  \chern( \widehat{\taut})^{n-1}
\end{align}
in terms of the derivatives  $B'(m,μ,s_0)$ at  $s_0=(n-1)/2$  of the coefficients   \eqref{eq:coeff} of $E_L (τ,s,n)$.
The proof follows the argument of [31] in the orthogonal case, which is based on the Siegel-Weil formula.  In fact, we slightly generalize this argument by allowing harmonic Maass forms for the inputs of the regularized theta lift.
Let  $κ∈{ 1,2}$ be as in  \eqref{eq:kappa}.

\begin{lemma}
\label{lem:int}
We have
\begin{align*}
\mathcal{I}(f) &=\lim_{T\to \infty}\left(\frac{1}{\kappa}
\int_{\mathcal{F}_T} \left\langle f , \,E_L(\tau,s_0,n-2)\right\rangle \, d\mu(\tau) -c^+(0,0) \log(T)\right).
\end{align*}
Here $\mathcal{F}_T=\{\tau\in \mathcal{F} :  \operatorname{Im}(\tau) \leq T\}$ is the truncation at height $T$ of the standard fundamental domain $\mathcal{F}$ for $\Gamma$.
\end{lemma}

\begin{proof}
Exactly as in  Proposition 2.5 of [31],  on $\mathrm{Sh}_K(H,\mathcal{D})\smallsetminus Z(f)$ we have the equality
\[
\Phi(z,h,f)= \lim_{T\to\infty}\left(\int_{\mathcal{F}_T} \langle f, \theta_L(\tau,z,h)\rangle \, d\mu(\tau) -c^+(0,0) \log (T) \right).
\]
The  integral $\mathcal{I}(f)$ is therefore given by the limit as $T\to \infty$ of
\begin{align*}
\vol_\C(\widehat{\taut})^{-1}
\int_{\mathrm{Sh}_K(H,\mathcal{D})}
\int_{\mathcal{F}_T} \langle f, \theta_L(\tau,z,h)\rangle \, d\mu(\tau) \chern(\widehat{\taut})^{n-1} -c^+(0,0) \log(T) .
\end{align*}
As $\mathcal{F}_T$ is compact, we may interchange the order of integration.
Inserting \eqref{lattice theta} and  \eqref{eq:intgeo}, we find that $\mathcal{I}(f)$ is  the limit as $T\to \infty$ of
\[
\vol_\C(\widehat{\taut})^{-1}
\int_{\mathcal{F}_T} \left\langle f, \int_{\mathrm{Sh}_K(H,\mathcal{D})}
\theta_L(\tau,z,h) \chern(\widehat{\taut})^{n-1}  \right\rangle \, d\mu(\tau) -c^+(0,0) \log(T)  ,
\]
which we rewrite as
\[
\int_{\mathcal{F}_T} \sum_{ \mu \in L' / L }  f(\tau)(\mu) \cdot \mathcal{I}(\tau,\varphi_\mu)  \, d\mu(\tau) -c^+(0,0) \log(T)  .
\]
Applying Proposition~\ref{lem:swgeo} completes the proof.
\end{proof}

\begin{theorem}
\label{thm:int}
If $n>2$, or if $n=2$ and $V$ is anisotropic,
the integral \eqref{eq:greenint} satisfies
\[
\mathcal{I}(f)=\sum_{\mu\in L'/L}\sum_{\substack{m\in \Q \\ m>0}} c^+(-m,\mu)B'(m,\mu,s_0).
\]
If $n=1$,  the integral \eqref{eq:greenint} satisfies
\[
\mathcal{I}(f)=\frac{1}{2}\sum_{\mu\in L'/L}\sum_{\substack{m\in \Q  \\ m\geq 0}} c^+(-m,\mu)B'(m,\mu,s_0).
\]
\end{theorem}

\begin{proof}
\[
A_T(f) =  \int_{\mathcal{F}_T} \left\langle f , \,E_L(\tau,s_0,n-2)\right\rangle \, d\mu(\tau),
\]
so that Lemma \ref{lem:int}  becomes
\begin{align}
\label{eq:if1}
\mathcal{I}(f) &=\lim_{T\to \infty}\left(\frac{A_T(f)}{\kappa}  -c^+(0,0) \log(T) \right).
\end{align}

Using Lemma \ref{lem:eisl} we see that
\[
E_L(\tau,s_0,n-2)\,d\mu(\tau)=\big(L_n E_L'(\tau,s_0,n)\big) \, d\mu(\tau) = -\bar\partial E_L'(\tau,s_0,n)\, d\tau ,
\]
and so
\begin{align*}
&=- \int_{\mathcal{F}_T} \left\langle f , \,\bar\partial E_L'(\tau,s_0,n)  \right\rangle d\tau\\
&=- \int_{\mathcal{F}_T} d\left(\left\langle f , \, E_L'(\tau,s_0,n)\,  \right\rangle  d\tau\right)+\int_{\mathcal{F}_T} \left\langle (\bar\partial f ), \, E_L'(\tau,s_0,n)  \right\rangle d\tau .
\end{align*}
Applying  Stokes' theorem to the first term,  and
\[
(\bar \partial f) d\tau = -(L_{2-n}f) \,d\mu(\tau)
\]
to the second,  we obtain
\begin{align*}
A_T(f)&=- \int_{\partial \mathcal{F}_T} \left\langle f , \, E_L'(\tau,s_0,n)\,  \right\rangle  d\tau - \int_{\mathcal{F}_T} \left\langle  L_{2-n} f , \, E_L'(\tau,s_0,n)  \right\rangle d\mu(\tau)\\
&=\int_{u=0}^1\left\langle f(u+iT ), \, E_L'(u+iT,s_0,n)\,  \right\rangle  du  \\
& \quad - \int_{\mathcal{F}_T} \left\langle  \overline{\xi_{2-n}(f)} , \, E_L'(\tau,s_0,n)  \right\rangle v^n d\mu(\tau),
\end{align*}
where $\xi_{2-n}$ is the differential operator \eqref{xi}.
Inserting the Fourier expansions \eqref{eq:fourierf} and \eqref{eq:fouriereis} of $f$ and the Eisenstein series yields
\begin{align*}
&=   \sum_{\substack{\mu\in L'/L\\ m\in \Q}} c^+(-m,\mu)B'(m,\mu;s_0,T)  \\
& \quad +\sum_{\substack{\mu\in L'/L\\ m\in \Q^+}} c^-(-m,\mu)\Gamma(n-1,4\pi |m| T) B'(m,\mu;s_0,T)  \\
&\quad - \int_{\mathcal{F}_T} \left\langle  \overline{\xi_{2-n}(f)} , \, E_L'(\tau,s_0,n)  \right\rangle v^n d\mu(\tau).
\end{align*}
The exponential decay of the incomplete gamma function and the polynomial growth of the coefficients $c^-(m,\mu)$ imply that the second term goes to zero in the limit $T\to \infty$. The third term converges to the Petersson scalar product of $\xi_{2-n}(f)$ and $E_L'(\tau,s_0,n)$. But since $ \xi_{2-n}(f)$ is cuspidal and hence orthogonal to Eisenstein series, this Petersson scalar product vanishes.
Inserting this into \eqref{eq:if1}, we find
\begin{align*}
\label{eq:if2}
\mathcal{I}(f) &=\lim_{T\to \infty}\bigg(\frac{1}{\kappa}
\sum_{\substack{\mu\in L'/L\\ m\in \Q  }} c^+(-m,\mu)B'(m,\mu;s_0,T)
-c^+(0,0) \log (T) \bigg).
\end{align*}

The exponential decay of the coefficients $B'(m,\mu;s_0,T)$ in \eqref{eq:coeff} for $m<0$ and the subexponential growth of the coefficients $c^+(-m,\mu)$ imply that the contribution of all $m<0$ in the above term vanishes.
If $n>1$ we obtain by virtue of Lemma~\ref{lem:coeffasy} that
\begin{align*}
\mathcal{I}(f)
\sum_{\substack{\mu\in L'/L\\ m> 0}} c^+(-m,\mu)B'(m,\mu,s_0),
\end{align*}
as desired.
When $n=1$, again by Lemma~\ref{lem:coeffasy}, there is an additional  contribution to the sum from $m=0$.
\end{proof}

\section{Integral models and arithmetic intersection theory}
\label{s:integral models}

We recall the integral models of $GU(n-1,1)$ Shimura varieties constructed by Pappas and Kr\"amer,  and the arithmetic intersection theory of Gillet-Soul\'e and Burgos-Kramer-K\"uhn on their toroidal compactifications.

From here until the end of the paper,  we assume $D=-disc()$ is odd.

\subsection{Moduli problems}
\label{ss:basic moduli}

For any $n≥1$,  work of Pappas [45] and Kr\"amer [37], as summarized in \S 2.3  of [13],   provides us with a regular and flat  $_$-stack
\[
\mathcal{M}_{(n-1,1)} \to \Spec(\co_\kk)
\]
with reduced fibers.  For an  $_$-scheme $S$, the objects of the groupoid $ℳ_(n-1,1)(S)$  are quadruples $(A,ι,ψ,ℱ)$ in which
\begin{itemize}
\item
$A \to S$ is an abelian scheme of dimension $n$,
\item
$\iota : \co_\kk \to \End(A)$ is an $\co_\kk$-action,
\item
$\psi : A \to A^\vee$ is a principal polarization whose induceded Rosati involution   restricts to complex conjugation on the image of $\iota$,
\item
$\mathcal{F} \subset \Lie(A)$ is  an $\co_\kk$-stable hyperplane\footnote{That is to say, an $\co_S$-module local direct summand     of rank $n-1$}
satisfying  the signature $(n-1,1)$ condition of Kr\"amer [37]: the actions of $\co_\kk$  on $\mathcal{F}$  and on $\Lie(A)/\mathcal{F}$ are through the structure morphism $i_S:\co_\kk \to \co_S$ and its complex conjugate, respectively.
\end{itemize}

\begin{remark}
The stack $\mathcal{M}_{(n-1,1)}$ is denoted $\mathcal{M}^\Kra_{(n-1,1)}$ in [13].
\end{remark}

\begin{remark}
One does not know a good theory of integral models $\mathcal{M}_{(n-1,1)}$ if $2$ is ramified in $\kk$, and
this lack of knowledge is the main reason for restricting to quadratic imaginary fields of odd discriminant.
\end{remark}

\begin{definition}\label{def:relevant}
A finite dimensional $\kk$-hermitian space $W$ is \emph{relevant} if it admits a self-dual $\co_\kk$-lattice $\mathfrak{a} \subset W$.
Two relevant hermitian spaces $W$ and $W'$ are \emph{strictly similar} if there is a $\kk$-linear isomorphism $W\iso W'$ identifying the hermitian forms up to scaling by a positive rational number.
\end{definition}

\begin{proposition}\label{prop:jacobowitz}
Let $o(D)$ denote the number of prime divisors of $D$.
\begin{enumerate}
\item
Any two  self-dual $\co_\kk$-lattices in a relevant $\kk$-hermitian space  are isometric everywhere locally.

\item
There are $2^{o(D) -1}$ isomorphism classes of relevant hermitian spaces of any fixed signature $(r,s)$.
If $r+s$ is odd all  lie in the same  strict similarity class, but if $r+s$ is even all lie in different strict similarity classes.
\end{enumerate}
\end{proposition}

\begin{proof}
The first claim is a theorem of Jacobowitz [28], which uses our assumption that $D$ is odd.  The second claim is Lemma 2.11 of  [32].
\end{proof}

As in Proposition 2.12(i) of [32],  there is a decomposition
\begin{equation}\label{relevant decomp}
\mathcal{M}_{(n-1,1)} = \bigsqcup_W \mathcal{M}_W,
\end{equation}
where the disjoint union is over the strict similarity classes of relevant hermitian spaces $W$ of signature $(n-1,1)$,
and the generic fiber of each $ℳ_W$ is a Shimura variety of type $GU(W)$.
When $n$ is odd,    Proposition \ref{prop:jacobowitz} implies that the disjoint union has a single term.
When $n$ is even, the disjoint union is over the $2^o(D)-1$ isomorphism classes of relevant $$-hermitian spaces.

Denote by
\[
\mathcal{M}_{(1,0)} \to \Spec(\co_\kk)
\]
the moduli stack of elliptic curves with CM by $_$.  More precisely, the functor of points assigns to an $_$-scheme $S$ the groupoid $ℳ_(1,0)(S)$ whose objects are pairs $(A_0,ι_0 )$ in which
\begin{itemize}
\item
$A_0 \to S$ is an elliptic curve,
\item
$\iota_0 : \co_\kk \to \End(A_0)$ is an $\co_\kk$-action satisfying the signature $(1,0)$ condition: the induced action of $\co_\kk$ on $\Lie(A_0)$  is through the  structure morphism $i_S: \co_\kk \to \co_S$.
\end{itemize}
As $A_0$ is an elliptic curve, there is a  unique principal polarization
\begin{itemize}
\item
$\psi_0 : A_0 \to A_0^\vee$,
\end{itemize}
and the induced Rosati involution on $(A_0)$ restricts to complex conjugation on the image of $ι_0$.  By Proposition 5.1 of [33] or  Proposition 2.1.2 of  [23],  $ℳ_(1,0)$ is  finite \'etale over $_$.

\begin{remark}
For ease of notation, we usually write objects of $\mathcal{M}_{(1,0)}$ simply as $A_0$,  suppressing the remaining data from the notation.  Similarly, objects of $\mathcal{M}_{(n-1,1)}$ will usually be written simply as  $A$.
\end{remark}

When $n=1$, the data of $ψ$ and $ℱ$ in the moduli problem  $ℳ_ (n-1,1) $ are uniquely determined, and  $ℳ_ (0,1) $  classifies pairs $(A,ι)$ over $_$-schemes $S$ in which $A→S$ is an elliptic curve and $ι: _→(A)$ is an action such that the induced action of  $_$ on the   $_S$-module  $(A)$ is through the conjugate of the structure morphism $_→_S$.  Replacing $ι$ by its conjugate therefore defines a canonical isomorphism
\[
\mathcal{M}_{ (0,1) } \iso \mathcal{M}_{(1,0)}.
\]

In particular, there is a decomposition of $ℳ_(1,0)$ analogous to \eqref{relevant decomp}, but this statement is vacuous, as the decomposition has a single term by Proposition \ref{prop:jacobowitz}.
If  $W_0$ is the unique, up to strict similarly, relevant $$-hermitian space of signature $(1,0)$,  it will be convenient to set
\[
\mathcal{M}_{W_0} = \mathcal{M}_{(1,0)} .
\]

Fix relevant $$-hermitian spaces $(W_0,h_0)$ and $(W,h)$ of signatures $(1,0)$ and $(n-1,1)$, and
self-dual $_$-lattices $𝔞_0 ⊂W_0$ and $𝔞 ⊂W$.
The hermitian forms induce alternating $$-bilinear forms
\begin{equation}\label{symplectic}
e_0 ( a, b ) = \mathrm{Tr}_{\kk/\Q} \left(  \frac{h_0(a,b) }{ \sqrt{-D} }  \right) ,\qquad e ( a , b ) = \mathrm{Tr}_{\kk/\Q} \left(  \frac{h(a,b) }{ \sqrt{-D} }  \right)
\end{equation}
on $W_0$ and $W$, and $𝔞_0$ and $𝔞$ are self-dual with respect to these forms.

\begin{definition}\label{def:kramer-pappas level}
Suppose we are given an integer $N\ge 1$ and an $\co_\kk$-scheme $S$ with $N\in \co_S^\times$.
A  \emph{level $N$-structure} on $A_0 \in \mathcal{M}_{W_0}(S)$ is a pair $(\eta_0, \zeta_0)$ consisting of isomorphisms
\[
\eta_0: A_0[N] \iso  \underline{\mathfrak{a}_0 / N \mathfrak{a}_0} ,\qquad \xi_0 : \mu_N \iso \underline{\Z/N\Z}
\]
of \'etale sheaves on $S$ that identify the Weil pairing on $A_0[N]$ induced by the principal polarization with the pairing on $\mathfrak{a}_0/N\mathfrak{a}_0$ induced by $e_0$.
Similarly, a  \emph{level $N$-structure} on $A \in \mathcal{M}_{W}(S)$ is a pair $(\eta, \zeta)$ consisting of isomorphisms
\[
\eta: A[N] \iso  \underline{\mathfrak{a} / N \mathfrak{a}} ,\qquad \xi : \mu_N \iso \underline{\Z/N\Z}
\]
that identify the Weil pairing on $A[N]$ with the alternating pairing on $\mathfrak{a}/N\mathfrak{a}$.
\end{definition}

Over $_[1/N]$, we obtain finite  \'etale covers
\begin{equation}\label{level covers}
\mathcal{M}_{W_0}(N) \to \mathcal{M}_{W_0/\co_\kk[1/N]} , \qquad
\mathcal{M}_{W}(N) \to \mathcal{M}_{W/\co_\kk[1/N]}
\end{equation}
by adding level structures to the moduli problems.   By the first claim of Proposition \ref{prop:jacobowitz}, these are independent of the choices of $𝔞_0$ and $𝔞$.

\subsection{The exceptional divisor}

Now assume $n≥2$.
We recall some more constructions from \S 2.3  of [13], following work of Pappas [45] and Kr\"amer [37].

There is a normal and flat $_$-stack
\[
\mathcal{M}^\Pap_{(n-1,1)} \to \Spec(\co_\kk)
\]
defined as the moduli space of triples  $(A,ι,ψ)$ whose components are  as in the definition of $ℳ_(n-1,1)$.  However,   instead of including a hyperplane $ℱ ⊂(A)$ as part of the moduli problem, we demand that the action of $_$ on $(A)$ satisfy a Kottwitz-style signature $(n-1,1)$ condition, and the wedge conditions of Pappas. We refer the reader to  \S 2.3  of [13] for the precise definitions.

%The first  is the Kottwitz determinant condition:  for all $\alpha\in \co_\kk$ we have
%\det( T- \iota(\alpha)  \mid \Lie(A) ) = (T- i_S(\alpha))^{n-1} (T- i_S(\overline{\alpha}) ) \in \co_S[T] ,
% where $i_S : \co_\kk \to \co_S$ is the structure map.
%  The second is the wedge condition of Pappas:  if we fix, as in \cite[\S 1.7]{BHKRY-1},  any $\pi\in \co_\kk$ satisfying $\co_\kk = \Z + \Z\pi$ and set
%\epsilon_S   & = \pi \otimes 1 - 1 \otimes i_S(\overline{\pi}) \in  \co_\kk \otimes_\Z \co_S  \label{idempotents} \\
%\overline{\epsilon}_S   & =  \overline{\pi} \otimes 1 - 1\otimes i_S( \overline{\pi} ) \in  \co_\kk \otimes_\Z \co_S , \nonumber
%the induced operators
%\bigwedge\nolimits^n  \epsilon_S : \bigwedge\nolimits^n \Lie(A) \to \bigwedge\nolimits^n \Lie(A) \\
%\bigwedge\nolimits^2 \overline{\epsilon}_S : \bigwedge\nolimits^2 \Lie(A) \to \bigwedge\nolimits^2 \Lie(A)
%are trivial.  This condition does not depend on the choice of $\pi$.

Forgetting the hyperplane $ℱ ⊂(A)$ defines a  morphism
\begin{equation}\label{kramer-pappas}
\mathcal{M}_{(n-1,1)} \to \mathcal{M}^\Pap_{(n-1,1)},
\end{equation}
which can be realized as a blow-up.  To interpret it as such, define the  \emph{singular locus}
\begin{equation}\label{singular}
\mathrm{Sing}_{(n-1,1)} \subset \mathcal{M}^\Pap_{(n-1,1)},
\end{equation}
as the reduced locus of nonsmooth points.
It  is a  proper  $_$-stack of  dimension $0$, supported in characteristics dividing $D$.

The morphism \eqref{kramer-pappas} can be identified with the blow-up of the singular locus, and so there is a canonical cartesian diagram
\begin{equation}\label{basic exceptional}
\xymatrix{
{\mathrm{Exc}_{(n-1,1)} }  \ar[r] \ar[d]  &  { \mathcal{M}_{(n-1,1)} } \ar[d] \\
{\mathrm{Sing}_{(n-1,1)} } \ar[r]  & { \mathcal{M}^\Pap_{(n-1,1)} }
\end{equation}
in which the upper left corner is the  \emph{exceptional divisor}.
In particular,  \eqref{kramer-pappas} is relatively representable and projective, and restricts to an isomorphism
\[
\mathcal{M}_{(n-1,1)} \smallsetminus  \mathrm{Exc}_{(n-1,1)}
\iso \mathcal{M}^\Pap_{(n-1,1)} \smallsetminus \mathrm{Sing}_{(n-1,1)} .
\]

\begin{remark}\label{rem:singular description}
If $\mathfrak{p} \subset \co_\kk$ is a prime  above $p\mid D$ with residue field $\F_\mathfrak{p}$,   the $\F_\mathfrak{p}^\alg$-points of the singular locus are those
\[
A\in \mathcal{M}^\Pap_{(n-1,1)} ( \F_\mathfrak{p}^\alg  )
\]
for which the action of $\co_\kk$ on the $\F_\mathfrak{p}^\alg$-vector space $\Lie(A)$ is through the reduction map  $\co_\kk \to \F_\mathfrak{p}^\alg$.
At such a point,   \emph{any} hyperplane $\mathcal{F} \subset \Lie(A)$ is $\co_\kk$-stable and satisfies Kr\"amer's signature condition, and the fiber over $A$ of the left vertical morphism in \eqref{basic exceptional} is the projective space parametrizing all such $\mathcal{F}$.
\end{remark}

\subsection{Toroidal compactification}
\label{ss:toroidal}

We turn to the study of compactifications of $ℳ_(n-1,1)$.
When $n=1$ this stack is finite (hence proper)  over $_$, and so we set
\[
\bar{\mathcal{M}}_{(0,1)} = \mathcal{M}_{(0,1)}.
\]
For the rest of this subsection, assume $n≥2$.

One finds in \S 2 of  [23] the construction of a toroidal compactification
\[
\mathcal{M}_{(n-1,1)}  \subset  \bar{\mathcal{M}}_{(n-1,1) },
\]
obtained by imitating the constructions of Faltings and Chai [15].  If one works over $_[1/D]$, these are special cases of the constructions of  Lan [38].
The compactification is regular, and is smooth outside  the exceptional divisor of \eqref{basic exceptional}.

The universal abelian scheme  $A →ℳ_(n-1,1)$ extends uniquely to a semi-abelian scheme
\[
\bar{A} \to \bar{\mathcal{M}}_{(n-1,1)}
\]
with $_$-action.
At a geometric point of the boundary, this semi-abelian scheme is an extension of an abelian variety  by a torus of rank two.
The universal hyperplane $ℱ ⊂(A)$ over $ℳ_(n-1,1)$ extends uniquely to an $_$-stable hyperplane
\begin{equation}\label{hyperplane extension}
\bar{\mathcal{F}} \subset \Lie(\bar{A}),
\end{equation}
which again  satisfies Kr\"amer's signature $(n-1,1)$ condition.

There is a similar toroidal compactification of $ℳ_(n-1,1)^$ (now only normal instead of regular), and  a Cartesian diagram
\begin{equation}\label{basic exceptional compact}
\xymatrix{
{\mathrm{Exc}_{(n-1,1)} }  \ar[r] \ar[d]  &  { \bar{\mathcal{M}}_{(n-1,1)} } \ar[d] \\
{\mathrm{Sing}_{(n-1,1)} } \ar[r]  & { \bar{\mathcal{M}}^\Pap_{(n-1,1)} }
\end{equation}
extending  \eqref{basic exceptional}.
The vertical arrow on the right is the blow-up along the closed immersion in the bottom row.

\begin{remark}\label{rem:no cone}
For general Shimura varieties, a toroidal compactification depends on a choice of rational polyhedral cone decomposition of a convex cone sitting inside the real points of a $\Q$-vector space.
For our $\mathrm{GU}(n-1,1)$ Shimura varieties, the vector space in question is one-dimensional.
Hence the rational polyhedral cone decomposition is uniquely determined, and the toroidal compactifications are canonical.
\end{remark}

Let $𝒜_n$ be the moduli stack of principally polarized abelian schemes of dimension $n$.
There are canonical morphisms
\begin{equation}\label{to siegel}
\mathcal{M}_{(n-1,1)} \map{\eqref{kramer-pappas}} \mathcal{M}_{(n-1,1)}^\Pap \to \mathcal{A}_{n/\co_\kk},
\end{equation}
in which the first arrow is projective, and the second is  finite.

\begin{lemma}\label{lem:normalized compactification}
If $\bar{\mathcal{A}}_n$ is any choice of smooth Faltings-Chai toroidal compactification,
$\bar{\mathcal{M}}^\Pap_{(n-1,1)}$ is canonically identified with the normalization of
\begin{equation}\label{pap-siegel}
\mathcal{M}_{(n-1,1)}^\Pap \to \bar{\mathcal{A}}_{n/\co_\kk}.
\end{equation}
See \S 29.53 of [51] for normalization.
\end{lemma}

\begin{proof}
By  [23],  the universal abelian scheme $A \to \mathcal{M}^\Pap_{(n-1,1)}$  extends (necessarily uniquely) to a semi-abelian scheme $\bar{A}$ over the compactification.
This extension satisfies a universal  property: Suppose $S$ is an irreducible normal scheme with generic point $\eta \to  S$, and we are given a morphism
\[
\eta \to  \mathcal{M}^\Pap_{(n-1,1)}.
\]
If the pullback $A_\eta$  extends to a semi-abelian scheme over $S$,  then this extension is the pullback of
$\bar{A}$ via  a (necessarily unique) morphism
\[
S \to \bar{\mathcal{M}}^\Pap_{(n-1,1)}
\]
restricting to  the given morphism at the generic point.

This extension  property is analogous to Theorem IV.5.7(5) of [15], but the statement is  simplified by  the uniqueness of the rational polyhedral cone decomposition of Remark \ref{rem:no cone}.
If one works over  $\co_\kk[1/D]$, the extension property is  a special case of  Theorem 6.4.1.1(6) of [38].  The proof over $\co_\kk$ is exactly the same.

On the other hand, Theorem 11.4 of [39] shows that the compactification of $\mathcal{M}^\mathrm{Pap}_{(n-1,1)}$ defined by normalization  also carries a semi-abelian scheme satisfying the same universal  property, and the lemma follows.
\end{proof}

Fix an $ℳ_W$ as in \eqref{relevant decomp}.
We   need compactifications of the \'etale covers
\[
\mathcal{M}_{W}(N) \to \mathcal{M}_{W/\co_\kk[1/N]}
\]
of \eqref{level covers}.
Rather than repeat the constructions of [23] with level structure,
we will  appeal to the results of [39] and [41], in which compactifications of quite general Shimura varieties are constructed  as normalizations of Faltings-Chai compactifications of Siegel moduli spaces.
These results cannot be applied directly to the covers above, as the failure of the first arrow in \eqref{to siegel} to be finite (it contracts the entire exceptional divisor to a $0$-dimensional substack) implies the failure of  Lemma \ref{lem:normalized compactification} if $ℳ^_(n-1,1)$ is replaced by  $ℳ_(n-1,1)$.  Thus the proof of Proposition \ref{prop:full compactification} below requires the roundabout step of  first compactifying covers of  $ℳ^_(n-1,1)$.

For any $N≥1$,  define  $ ℳ̅_W(N)$ as the  normalization of
\[
\mathcal{M}_W(N)  \to \bar{\mathcal{M}}_{(n-1,1)/\co_\kk[1/N]}.
\]

\begin{proposition}\label{prop:full compactification}
The stack $\bar{\mathcal{M}}_W(N)$ is regular, flat, and proper over $\co_\kk[1/N]$, and is a projective scheme if $N\ge 3$.
It is smooth away from its exceptional divisor
\[
\mathrm{Exc}_{(n-1,1)} \times_{ \bar{\mathcal{M}}_{(n-1,1)} }  \bar{\mathcal{M}}_W(N).
\]
In particular, it is smooth  in a neighborhood of its boundary, which is a Cartier divisor  smooth over $\co_\kk[1/N]$.
\end{proposition}

\begin{proof}

There is a decomposition
\[
\mathcal{M}^\Pap_{(n-1,1)} = \bigsqcup_W \mathcal{M}^\Pap_W,
\]
exactly as in \eqref{relevant decomp}.  We may add $N\ge 1$ level structure, exactly as in Definition \ref{def:kramer-pappas level},  to obtain a finite \'etale cover
\[
\mathcal{M}^\Pap_W (N) \to \mathcal{M}^\Pap_{W/\co_\kk[1/N]}.
\]
This cover has a  compactification $\bar{\mathcal{M}}^\Pap_W (N)$ defined as the normalization of
\[
\mathcal{M}^\Pap_W (N) \to  \bar{\mathcal{M}}^\Pap_{(n-1,1) / \co_\kk[1/N] },
\]
which, by Lemma \ref{lem:normalized compactification},  is the same   as the normalization of
\begin{equation}\label{pap-siegel level}
\mathcal{M}^\Pap_W (N) \to   \bar{\mathcal{A}}_{n/\co_\kk[1/N]}.
\end{equation}

The properness,  flatness, and normality of $\bar{\mathcal{M}}^\Pap_W (N)$  follow  from its construction as the  normalization of a proper, flat, and smooth $\co_\kk[1/N]$-stack.
It is not regular (as this is not true even of its interior).  However, its singular locus
\[
\mathrm{Sing}_W(N) = \mathrm{Sing}_{(n-1,1)} \times_{ \mathcal{M}^\Pap_{(n-1,1) }} \mathcal{M}^\Pap_W(N)
\]
is proper over $\co_\kk[1/N]$, and the smoothness of
\[
\mathcal{M}^\Pap_W (N) \smallsetminus \mathrm{Sing}_W(N)
\]
implies the smoothness of
\[
\bar{\mathcal{M}}^\Pap_W (N) \smallsetminus \mathrm{Sing}_W(N)  .
\]
This is a consequence of the following principle, found  in \S 14 of [39] and Theorem 1 of [41]:  if $U  \subset \bar{\mathcal{M}}^\Pap_W (N)$ is an  open neighborhood of the boundary $\partial\bar{\mathcal{M}}^\Pap_W (N)$, then the singularities of  $U$ are no worse than the singularities of $U \smallsetminus \partial\bar{\mathcal{M}}^\Pap_W (N)$.

By the same principle (and  its proof), the smoothness of $\bar{\mathcal{M}}^\Pap_W (N)$ near the boundary implies the smoothness of the boundary  divisor itself.  This is actually a particular feature of our $\mathrm{GU}(W)$ Shimura varieties.  For more  general  Shimura varieties the boundary has a canonical stratification by locally closed substacks (see Theorem 4.1.5 of [41] or \S 9 of [39]), and the above cited principle tells us that each \emph{stratum} is smooth.  In our special case each boundary stratum is a divisor, and is simply  a union of connected components of $\partial\bar{\mathcal{M}}^\Pap_W (N)$.  Hence the entire boundary  is  a Cartier divisor smooth over $\co_\kk[1/N]$.

Using the characterization of \eqref{basic exceptional compact} as a blow-up,  one may identify $\bar{\mathcal{M}}_W(N)$  with the blow-up of $\bar{\mathcal{M}}^\Pap_W (N)$ along its singular locus.  As the singular locus does not meet the boundary, it follows that $\bar{\mathcal{M}}_W(N)$ is itself proper and flat with boundary a smooth Cartier divisor.  Moreover, this shows that $\bar{\mathcal{M}}_W(N)$ is smooth away from the exceptional divisor of the blow-up, and in particular in an open neighborhood of its boundary.  As the interior is regular, being  \'etale over the regular stack $\mathcal{M}_W$,  we find that $\bar{\mathcal{M}}_W(N)$ is regular.

% Lemma 29.53.4 of [51] provides us with  the diagonal arrow in the commutative diagram
%{  \mathcal{S}_V(N)  }  \ar[d] \ar[r] &  {  \mathcal{M}_{W_0}(N) \times \bar{\mathcal{M}}_W(N)   }  \ar[d]   \\
%{  \bar{\mathcal{S}}_V(N)   }   \ar[r] \ar[ur]  &  { \mathcal{M}_{W_0/\co_\kk[1/N] } } \times \bar{\mathcal{M}}_{W/\co_\kk[1/N] },
%which realizes $\bar{\mathcal{S}}_V(N)$ as an open and closed substack of the upper right corner.
%All claims about $\bar{\mathcal{S}}_V(N)$ therefore follow from the corresponding properties of $  \bar{\mathcal{M}}_W(N)$, and the fact that $\mathcal{M}_{W_0}(N)$ is finite \'etale over $\co_\kk[1/N]$.

Finally, when $N\ge 3$ we claim that both the source and target of
\[
\bar{\mathcal{M}}_W(N) \to \bar{\mathcal{M}}^\Pap_W(N)
\]
are  projective schemes.  As this morphism is a blow-up, it suffices to prove this for the target.
For this, we use the results of Chapter V.5 of [15].  Adding principal level structure yields a finite \'etale cover
\[
\mathcal{A}_n(N) \to \mathcal{A}_{n/\Z[1/N]},
\]
and the  compactification  $\bar{\mathcal{A}}_n$ may be chosen so that \eqref{pap-siegel level} factors as
\[
\mathcal{M}^\Pap_W (N) \to   \bar{\mathcal{A}}_n(N)  \to  \bar{\mathcal{A}}_{n/\co_\kk[1/N]},
\]
where   $\bar{\mathcal{A}}_n(N)$ is the smooth projective scheme obtained by normalizing
\[
\mathcal{A}_n(N)  \to  \bar{\mathcal{A}}_{n/\co_\kk[1/N]}.
\]
This realizes $\bar{\mathcal{M}}^\Pap_W (N)$ as the normalization of the first arrow in the composition, and hence provides us with a finite and relatively representable morphism
\[
\bar{\mathcal{M}}^\Pap_W (N) \to  \bar{\mathcal{A}}_n(N).
\]
As the target is a projective scheme, so is the source.
\end{proof}

\begin{remark}
One could avoid the constructions of [23] and [38] altogether, and define
$\bar{\mathcal{M}}_{(n-1,1)}$ from the start as a blow-up of the normalization of \eqref{pap-siegel}.
All properties that we need to know about this compactification could then be deduced from [39] and [41], except for the extension \eqref{hyperplane extension} of the universal hyperplane across the boundary (which is needed in the proof of Proposition \ref{prop:taut-hodge1}).
For this one  needs  detailed information about the local  charts near the boundary,  as found  in [23].
\end{remark}

%\begin{proposition}\label{prop:too compact}
%Let $\mathcal{A}_n$ be the smooth $\Z$-stack of finite type parametrizing  principally polarized abelian schemes of relative dimension $n$.   For any choice of smooth toroidal compactification  $ \mathcal{A}_n \subset \bar{\mathcal{A}}_n$ as in [15],  the $\co_\kk$-stack $\bar{ \mathcal{M}}^\Pap_{(n-1,1)}$ is isomorphic to the  normalization  of the natural morphism
%\begin{equation}\label{pappas to FC}
%\mathcal{M}^\Pap_{(n-1,1)} \to  \bar{\mathcal{A}}_{n/\co_\kk}
% in the sense of  \cite[\href{https://stacks.math.columbia.edu/tag/0BAK}{Tag 0BAK}]{stacks-project}.
% \begin{proof}
% This is not difficult, but giving all the details would require first recalling most of the machinery of arithmetic compactifications of Shimura varieties.  Hence we only sketch the ideas.
%By applying the criterion of \cite[Theorem IV.5.7(5)]{faltings-chai} to the semi-abelian extension $\bar{A}$ of Proposition \ref{prop:kramer semi-abelian}, one finds that \eqref{pappas to FC} extends uniquely to\note{details?}
%\begin{equation}\label{pappas to siegel extension}
%\bar{\mathcal{M}}_{(n-1,1)}^\Pap \to \bar{\mathcal{A}}_{n/\co_\kk}.
%One can again check that the fiber of \eqref{pappas to siegel extension} over any geometric point  is finite. \note{finiteness details?}
%The morphism \eqref{pappas to siegel extension} is obviously proper  (the domain is proper over $\co_\kk$)  and so  is finite  by \cite[\href{https://stacks.math.columbia.edu/tag/02OG}{Tag 02OG}]{stacks-project}, and is therefore integral.
% Combining this with  \cite[\href{https://stacks.math.columbia.edu/tag/035I}{Tag 035I}]{stacks-project} shows that the normalization of \eqref{pappas to FC} is the same as the normalization of the open immersion
% \mathcal{M}^\Pap_{(n-1,1)}   \to   \bar{\mathcal{M}}^\Pap_{(n-1,1)} ,
%which is just $\bar{\mathcal{M}}^\Pap_{(n-1,1)}$ again.
% \end{proof}

%\begin{proposition}\label{prop:too compact}
%The compactifications $\bar{\mathcal{S}}_V$ and $\mathcal{M}_{(n-1,1)}^*$ agree with those constructed in [23], and hence also with those of  [13] and [14].
%Let $\mathcal{M}_{(n-1,1)}'$ be the compactification constructed in  [23].  The essential point is that the universal abelian scheme $A \to \mathcal{M}_{(n-1,1)}$ extends to semi-abelian scheme over $\mathcal{M}_{(n-1,1)}'$, and one can use the criterion of \cite[Theorem IV.5.7(5)]{faltings-chai} to extend \ref{to FC} to an integral  morphism\note{this requires more explanation}
%\begin{equation}\label{FC extension property}
%\mathcal{M}'_{ (n-1,1) } \to \mathcal{A}^*_{n/\co_\kk}.
%By  \cite[\href{https://stacks.math.columbia.edu/tag/035I}{Tag 035I}]{stacks-project}, the canonical  morphism
%$\bar{\mathcal{M}}_{(n-1,1)} \to \mathcal{A}^*_n$ factors as
%\bar{\mathcal{M}}_{ (n-1,1) } \to \mathcal{M}'_{ (n-1,1) } \map{\eqref{FC extension property}} \mathcal{A}^*_n,
%in such a way that  the first arrow  identifies $\bar{\mathcal{M}}_{ (n-1,1) }$ with the normalization of the open immersion $\mathcal{M}_{ (n-1,1) } \to \mathcal{M}'_{ (n-1,1) }$. As the codomain of this open immersion is already normal (in fact, regular), we deduce that
%\bar{\mathcal{M}}_{ (n-1,1) } \iso \mathcal{M}'_{ (n-1,1) }.
%The analogous claim $\bar{\mathcal{S}}_V \iso \mathcal{S}_V'$ follows  from this,  as both sides are identified with the Zariski closure of $\mathcal{S}_V$ in
%\mathcal{M}_{(1,0)} \times \bar{\mathcal{M}}_{ (n-1,1) } \iso\mathcal{M}_{(1,0)} \times \mathcal{M}'_{ (n-1,1) }.
%Denote by $\partial \bar{\mathcal{S}}_V = \bar{\mathcal{S}}_V \smallsetminus \mathcal{S}_V$ the boundary of our toroidal compactification.  By Proposition \ref{prop:unitary compactification} it is a smooth divisor.

\subsection{Arithmetic Chow groups}
\label{ss:chow}

We summarize the arithmetic intersection theory of
Gillet-Soul\'e [17, 49] and  Burgos-Kramer-K\"uhn [6, 11]
for the  Shimura varieties  $ℳ_W$ of \eqref{relevant decomp},
mostly for the purposes of fixing normalizations of arithmetic degrees, heights, and volumes.

As usual,  denote by $Pic(ℳ_W)$ the group of isomorphism classes of hermitian line bundles on $ℳ_W$, and similarly for the toroidal compactification $ℳ̅_W$.
In order to account for metrics that do not extend smoothly across the boundary, when we work on the compactification $ℳ̅_W$ we must relax the notion of a hermitian metric.

Accordingly, denote by $( ℳ̅_W, 𝒟_)$ the group of isomorphism classes of pre-log singular hermitian line bundles on $ℳ̅_W$, in the sense of  Definition 1.20 of [6].
Exactly as in Proposition 5.2.1 of [25], the natural restriction maps
\begin{equation}\label{pre-log injection}
\widehat{\Pic}( \bar{\mathcal{M}}_W ) \to \widehat{\Pic}( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK ) \to \widehat{\Pic}( \mathcal{M}_W)
\end{equation}
are injective.

Fix an integer $N≥3$, so that the toroidal compactification  $ℳ̅_W(N)$ of Proposition \ref{prop:full compactification} is a regular scheme of dimension $n$, projective and flat over $_[1/N]$.
For $0≤d ≤n$,  we have a homomorphism of   arithmetic Chow groups
\[
\widehat{\mathrm{CH}}^d_\GS ( \bar{\mathcal{M}}_W(N) )
\to \widehat{\CH}^d (\bar{\mathcal{M}}_W(N) , \mathscr{D}_\BKK )
\]
from Theorem 6.23 and  (7.49) of [11].
The domain is the arithmetic Chow group of Gillet-Soul\'e [17, 49], while
the codomain is the  arithmetic Chow group of Burgos-Kramer-K\"uhn [6, 11].
In both groups  elements are rational equivalence classes of pairs $(𝒵, g )$ in which $𝒵$ is a codimension $d$ cycle on $ℳ̅_W(N)$.
In the Gillet-Soul\'e construction  $g$ is a $(d-1,d-1)$ current on $ℳ̅_W(N)()$ satisfying the Green equation
\[
dd^c g + \delta_\mathcal{Z} = [\omega ]
\]
of currents  for some smooth $(d,d)$ form $ω$.
In the Burgos-Kramer-K\"uhn construction $g$ is a more elaborate \emph{Green object},  allowed to have pre-log-log singularities along the boundary of the compactification.

Whenever $N |N'$ the  map
ℳ̅_W(N') →ℳ̅_W(N)
induces a pullback
\[
\widehat{\CH}^d (\bar{\mathcal{M}}_W(N) , \mathscr{D}_\BKK )
\to \widehat{\CH}^d (\bar{\mathcal{M}}_W(N') , \mathscr{D}_\BKK ),
\]
allowing us to define, exactly as in \S 6.3 of [11],
\[
\widehat{\CH}^d (\bar{\mathcal{M}}_W , \mathscr{D}_\BKK )
= \mil_{N\ge 3}  \widehat{\CH}^d (\bar{\mathcal{M}}_W(N) , \mathscr{D}_\BKK ).
\]
There is an \emph{arithmetic intersection pairing}
\[
\widehat{\CH}^{d_1} (\bar{\mathcal{M}}_W , \mathscr{D}_\BKK ) \otimes \widehat{\CH}^{d_2} (\bar{\mathcal{M}}_W , \mathscr{D}_\BKK )
\to \widehat{\CH}^{d_1+d_2} (\bar{\mathcal{M}}_W , \mathscr{D}_\BKK )_\Q,
\]
an  \emph{arithmetic Chern class map}
\[
\widehat{\Pic}( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK )  \to  \widehat{\CH}^1 (\bar{\mathcal{M}}_W , \mathscr{D}_\BKK ),
\]
and an  \emph{arithmetic degree}
\[
\widehat{\deg} : \widehat{\CH}^n(\bar{\mathcal{M}}_W , \mathscr{D}_\BKK ) \to \R
\]
induced by the analogous structures on the arithmetic Chow group of each scheme $ℳ̅_W(N)$ with $N≥3$.

\begin{remark}
The conventions for delta currents, currents associated to  smooth forms,  and Green forms used  in  [11] and [6]   differ from those of Gillet-Soul\'e [17, 49] by powers of $2$ and  $2\pi i$.
We will always use the conventions of Gillet-Soul\'e.
For example, the arithmetic Chern class map sends a pre-log singular hermitian line bundle $\widehat{\mathcal{L}}$ to the arithmetic divisor
\[
\widehat{\mathrm{div}}(s) = ( \mathrm{div}(s) ,  - \log\| s\|^2  )
\]
for any nonzero rational section $s$.
\end{remark}

To make explicit the normalization of the arithmetic degree, first note that for each $N ≥3$,
the  finite \'etale cover $ℳ_W(N) →ℳ_W/_[1/N]$ has constant fiber degree, in the sense that there is a $d_N∈$ satisfying
\begin{equation}\label{level degree}
\frac{d_N}{ \# \Aut(x)    }
=  \#  \{  \mbox{geometric points}\ y \to \mathcal{M}_W(N)\ \mbox{above}\ x\}
\end{equation}
for every geometric point    $x→ℳ_W/_[1/N]$.
Now suppose we have a class
\[
\widehat{\mathcal{Z}}=( \mathcal{Z}_N , g_N )_{N\ge 3} \in
\widehat{\CH}^n (\bar{\mathcal{M}}_W  , \mathscr{D}_\BKK ) .
\]
If we write  each
𝒵_N = ∑_i m_N,i 𝒵_N,i
as a $$-linear combination of $0$-dimensional irreducible closed subschemes on $ℳ̅_W(N)$,
then the real number
\[
\widehat{\deg}_N ( \mathcal{Z}_N, g_N ) =
\sum_i  m_{N,i}   \cdot   \# \mathcal{Z}_{N,i}( \F^\alg_{i})   \cdot   \log (  \#  \F_{i} )
+   \int_{ \mathcal{M}_W(N)(\C) } g_N,
\]
where  $_i$   is the residue field of the unique prime of $_[1/N]$ below  $𝒵_N,i$,
is well-defined up to adding a $$-linear combination of ${ log(p) : p |N}$, and
\begin{equation}\label{degree normalization}
\widehat{\deg} (\widehat{\mathcal{Z}}) = \frac{1}{d_N}   \cdot \widehat{\deg}_N ( \mathcal{Z}_N, g_N )
\end{equation}
up to the same ambiguity.

\begin{remark}\label{rem:no half}
There is no $1/2$ in front of the integral, in apparent disagreement with \S 3.4.3 of [17].
In fact there is no disagreement.  For us  $\mathcal{M}_W(N)(\C)$ means the set of all
morphisms $\Spec(\C) \to \mathcal{M}_W(N)$ as $\kk$-schemes, whereas Gillet-Soul\'e would take morphisms as $\Q$-schemes.
Thus for Gillet-Soul\'e the domain of integration would be our $\mathcal{M}_W(N)(\C)$  \emph{together with its complex conjugate}, yielding an integral  twice as big as  ours.
\end{remark}

\subsection{Volumes and heights}
\label{ss:volumes}

For a pre-log singular hermitian line bundle
\[
\widehat{\mathcal{L} } \in  \widehat{\Pic}(\bar{\mathcal{M}}_W , \mathscr{D}_\BKK ),
\]
define the  \emph{arithmetic volume}  by
\[
\widehat{\vol}( \widehat{\mathcal{L}} )
=  \widehat{\deg}(   \widehat{\mathcal{L}}^n )  .
\]
Here $ ℒ^n $ is the $n$-fold  iterated intersection of the arithmetic Chern class
\[
\widehat{\mathcal{L}} \in  \widehat{\CH}^1 (\bar{\mathcal{M}}_W , \mathscr{D}_\BKK ) .
\]

Let $𝒵$ be a  divisor on $ℳ̅_W$, and assume that $𝒵$  intersects the boundary
$∂ℳ̅_W$  properly in the generic fiber.
As in the discussion following Theorem 7.58  of  [11], one can then define the \emph{arithmetic height}    (or \emph{Faltings height})   \emph{of $\mathcal{Z}$ with respect to $\widehat{\mathcal{L}}$}, denoted
\[
\mathrm{ht}_{\widehat{\mathcal{L}}}(\mathcal{Z})   \in \R.
\]
To make it explicit,  fix $N≥3$.  Choose a pair $(𝒴,  g_𝒴)$ representing the iterated intersection
\[
\widehat{\mathcal{L}}^{n-1} \in  \widehat{\CH}^{n-1}(\bar{\mathcal{M}}_W(N), \mathscr{D}_\BKK ),
\]
and  do this in such a way  that  the supports of $𝒵_N$ and  $𝒴$ do not intersect in the generic fiber, where $𝒵_N$ is the pullback of $𝒵$ via
\[
\bar{\mathcal{M}}_W(N) \to \bar{\mathcal{M}}_{W/\co_\kk[1/N]}.
\]
Now  form the intersection
\[
\mathcal{Z}_N \cdot \mathcal{Y} \in \CH^n_{  \mathcal{Z}_N  \cap  \mathcal{Y} } (\bar{\mathcal{M}}_W(N))_\Q
\]
in the Chow group with support along  the closed subset $Sppt(𝒵_N) ∩Sppt(𝒴)$.
As this closed subset  is supported in finitely many nonzero characteristics, there is a natural change-of-support map
\[
\CH^n_{  \mathcal{Z}  \cap  \mathcal{Y} } (\bar{\mathcal{M}}_W(N) )  \to
\bigoplus_{  \substack{ \mathfrak{p} \subset \co_\kk   \\   \mathfrak{p} \nmid N\co_\kk }   }
\CH^n_\mathfrak{p}  (\bar{\mathcal{M}}_W (N) )
\]
where $^n_𝔭  (ℳ̅_W(N) )$ is the Chow group with support in the mod $𝔭$ fiber of $ℳ̅_W (N)$.
For each $𝔭$ there is a degree map
\[
\deg_\mathfrak{p} : \CH^n_\mathfrak{p}  (\bar{\mathcal{M}}_W (N) ) \to \Z
\]
sending a reduced and irreducible codimension $n$ subscheme (a.k.a., a closed point)  $𝒞  ⊂ℳ̅_W (N)$ supported at $𝔭$ to
\[
\deg_\mathfrak{p} (\mathcal{C}) = \# \mathcal{C}( \F_\mathfrak{p}^\alg)
\]
where $_𝔭 = _/𝔭$.
The arithmetic height  satisfies
\[
d_N\cdot \mathrm{ht}_{\widehat{\mathcal{L}}}(\mathcal{Z})  =
\sum_{  \substack{ \mathfrak{p} \subset \co_\kk   \\   \mathfrak{p} \nmid N\co_\kk }   }
\deg_\mathfrak{p} ( \mathcal{Z}_N \cdot \mathcal{Y} ) \cdot \log(\#\F_\mathfrak{p})
+  \int_{ \mathcal{Z}_N(\C) } g_\mathcal{Y}
\]
up to a $$-linear combination of ${ log(p) : p|N}$, where $d_N$ is as in  \eqref{level degree}.

A choice of pre-log singular hermitian metric on the line bundle $(𝒵)$ determines a class
\[
\widehat{\mathcal{Z}} \in \widehat{\mathrm{Pic}}(\bar{\mathcal{M}}_W, \mathscr{D}_\BKK).
\]
If we view  the constant function $1$ as a global section of  $(𝒵)$  and set $g_𝒵= - log1^2$,
the arithmetic Chern class of this hermitian line bundle is
\[
( \mathcal{Z} , g_\mathcal{Z} ) \in \widehat{\mathrm{CH}}^1( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK) ,
\]
and Proposition 7.56 of [11] gives the relation
\begin{equation}\label{height degree}
\widehat{\deg}(   \widehat{\mathcal{Z}}  \cdot  \widehat{\mathcal{L}}^{n-1}   )
= \mathrm{ht}_{\widehat{\mathcal{L}}}(\mathcal{Z}) + \int_{ \mathcal{M}_W(\C) }  g_\mathcal{Z} \cdot \chern(\widehat{\mathcal{L}})^{n-1} .
\end{equation}

From the point of view of arithmetic volumes and heights, some hermitian line bundles are essentially indistinguishable.

\begin{definition}\label{def:numerical}
We say that
\[
\widehat{\mathcal{L}}_1 , \widehat{\mathcal{L}}_2   \in   \widehat{\Pic}(\mathcal{M}_W  )
\]
are \emph{numerically equivalent} if the difference
$\widehat{\mathcal{E}} =   \widehat{\mathcal{L}}_1 - \widehat{\mathcal{L}}_2$ lies in the subgroup
\[
\widehat{\Pic}( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK )  \subset \widehat{\Pic}(\mathcal{M}_W  )
\]
of \eqref{pre-log injection}  and satisfies
\widehat{\deg}( \widehat{\mathcal{Z}} \cdot \widehat{\mathcal{E}} ) =0
for all $\widehat{\mathcal{Z}} \in \widehat{\CH}^{n-1}(\bar{\mathcal{M}}_W,\mathscr{D}_\BKK ).$
\end{definition}

\begin{lemma}\label{lem:numerical basics}
If $\widehat{\mathcal{L}}_1$ and  $\widehat{\mathcal{L}}_2$ are numerically equivalent, then their Chern forms are equal.  Moreover,
\begin{equation}\label{trivial shift}
\widehat{\vol}( \widehat{\mathcal{L}_1} ) = \widehat{\vol}( \widehat{\mathcal{L}}_2 )
\quad\mbox{and}\quad
\mathrm{ht}_{ \widehat{\mathcal{L}}_1  } (\mathcal{Z})
= \mathrm{ht}_{ \widehat{\mathcal{L}}_2 }  (\mathcal{Z})
\end{equation}
for any divisor $\mathcal{Z}$ as above.
\end{lemma}

\begin{proof}
Set  $\widehat{\mathcal{E}} =  \widehat{\mathcal{L}}_1 - \widehat{\mathcal{L}}_2$.
If  $g$ is any smooth $(n-2,n-2)$  form on $\bar{\mathcal{M}}_W(\C)$, the arithmetic cycle class
\[
\widehat{\mathcal{Z}} = ( 0, g) \in\widehat{\CH}^{n-1}(\bar{\mathcal{M}}_W , \mathscr{D}_\BKK)
\]
\[
0 = \widehat{\deg}( \widehat{\mathcal{Z}} \cdot \widehat{\mathcal{E}} )
=  \int_{\mathcal{M}_W(\C)}    \chern( \widehat{\mathcal{E}}  )  \wedge g.
\]
As this holds for all choices of $g$, we must have $ \chern( \widehat{\mathcal{E}}  )=0$.
This proves the desired equality of Chern forms.
The first equality of \eqref{trivial shift} is clear from
\[
\widehat{\vol}( \widehat{\mathcal{L}_1} )
= \sum_{i=0}^n \widehat{\deg} ( \widehat{\mathcal{L}}_2^{n-i} \cdot \widehat{\mathcal{E}}^i ),
\]
as every term with $i>0$ vanishes. The second  equality follows from \eqref{height degree} and the equality of Chern forms already proved.
\end{proof}

If $𝒵$ is a  Cartier divisor on $ℳ̅_W$ supported in nonzero characteristics,  the line bundle
$( 𝒵 )$   is canonically trivialized in the generic fiber by the constant function $1$.
For any real number $c$, denote by
\begin{equation}\label{constant metrics}
(\mathcal{Z} ,c)  \in    \widehat{\Pic}( \bar{\mathcal{M}}_W )
\end{equation}
the line bundle $(𝒵)$  endowed with the constant metric characterized by  $-log 1^2 =   c$.
In particular, adding $(0,c)$ to a hermitian line bundle just rescales $ ·^2$  by $e^-c$.  The following lemma describes the effect of this on arithmetic volumes.

\begin{lemma}\label{lem:trivial volume shift}
For  any $\widehat{\mathcal{L}} \in \widehat{\Pic}( \bar{\mathcal{M}}_W, \mathscr{D}_\BKK )$ and  any constant $c\in \R$,  we have
\[
\widehat{\vol}\big( \widehat{\mathcal{L}} + (0 ,c ) \big) =
\widehat{\vol}(  \widehat{\mathcal{L}} )  + n c   \int_{\mathcal{M}_W(\C) }  \chern(\widehat{\mathcal{L}})^{n-1}  ,
\]
where $ \chern(\widehat{\mathcal{L}})$ is the Chern form \eqref{chern form}.
\end{lemma}

\begin{proof}
A trivial induction argument shows that
\[
\widehat{\mathcal{L}}^{k-1} \cdot (0 , c )   =  ( 0  , c  \cdot  \chern(\widehat{\mathcal{L}})^{k-1} )
\in \widehat{\mathrm{CH}}^k( \bar{\mathcal{M}}_W ,\mathscr{D}_\BKK)
\]
for all $k>0$.  As $(0,c) \cdot (0,c) =0$  we  deduce
\begin{align*}
\big( \widehat{\mathcal{L}} + (0 ,c ) \big)^k
& = \widehat{\mathcal{L}} ^k +  k \widehat{\mathcal{L}}^{k-1} \cdot (0 , c )   \\
& = \widehat{\mathcal{L}} ^k + k ( 0  ,  c  \cdot  \chern(\widehat{\mathcal{L}})^{k-1} ).
\end{align*}
Setting $k=n$ and taking the arithmetic degree  completes the proof.
\end{proof}

\section{A Shimura variety and its line bundles}
\label{s:special shimura bundles}

For the rest of this paper we fix a $$-hermitian space $V$ of signature $(n-1,1)$, with $n≥1$, and assume that $V$ admits a self-dual $_$-lattice \eqref{self-dual lattice}.
We will associate to $V$ an $n$-dimensional open and closed substack
\[
\mathcal{S}_V \subset \mathcal{M}_{(1,0)} \times_{\co_\kk} \mathcal{M}_{(n-1,1)},
\]