vertex of $\mathcal{L}$ corresponding to the type $2$ lattice
$\Lambda\obot\mathrm{Span}\\{b({\bf x})\\}$. An edge corresponding to a type
zero lattice $\Lambda_{0}\subset\mathrm{Span}_{F}\\{b({\bf x})\\}^{\perp}$ is
broken into two pieces evenly and sent to the union of the two edges in
$\mathcal{L}$ joining the two vertices corresponding to
$\Lambda\obot\mathrm{Span}\\{b({\bf x})\\}$ and
$\Lambda^{\prime}\obot\mathrm{Span}\\{b({\bf x})\\}$ where $\Lambda$ and
$\Lambda^{\prime}$ are the two type $2$ lattices containing $\Lambda_{0}$.

####  9.2. Rank $1$ case

######  Lemma 9.2.

A point $z\in\mathcal{N}_{3}^{\mathrm{Pap}}(k)$ is in
$\mathcal{Z}^{\mathrm{Pap}}({\bf x})(k)$ if and only if $b_{\bf x}\in M(z)$.

  1. (1)

Assume $\Lambda_{0}\in\mathcal{V}^{0}$, then the superspecial point
$\mathcal{N}_{\Lambda_{0}}(k)$ is in $\mathcal{Z}^{\mathrm{Pap}}({\bf x})(k)$
if and only if $b_{\bf x}\in\Lambda_{0}$.

  2. (2)

Assume $\Lambda_{2}\in\mathcal{V}^{2}$, then

$\displaystyle\mathcal{Z}^{\mathrm{Pap}}({\bf
x})(k)\cap\mathcal{N}_{\Lambda_{2}}(k)=\begin{cases}\mathcal{N}_{\Lambda_{2}}(k)&\text{
if }b_{\bf x}\in\Lambda_{2}^{\sharp},\\\ \text{a superspecial point in
}\mathcal{N}_{\Lambda_{2}}(k)&\text{ if }b_{\bf
x}\in\Lambda_{2}\backslash\Lambda_{2}^{\sharp},\\\ \emptyset&\text{ if }b_{\bf
x}\notin\Lambda_{2}.\end{cases}$

###### Proof.

By Dieudonné theory, $z\in\mathcal{Z}^{\mathrm{Pap}}({\bf x})(k)$ if and only
if ${\bf x}(M(\mathbb{Y}))\subset M(z)$ if and only if $b_{\bf x}\in M(z)$
since $e$ is a generator of $M(\mathbb{Y})$. For
$z=\mathcal{N}_{\Lambda_{0}}(k)$ where $\Lambda_{0}\in\mathcal{V}^{0}$, we
have $M(z)=\Lambda_{0}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{\breve{F}}$. Hence
(1) immediately follows.

Now we proceed to prove (2). If $b_{\bf x}\in\Lambda^{\sharp}$, then (9.2)
tells us that $z\in\mathcal{Z}^{\mathrm{Pap}}({\bf x})(k)$ for any
$z\in\mathcal{N}^{\circ}_{\Lambda_{2}}(k)$. The fact that
$\Lambda_{2}^{\sharp}\subset\Lambda_{0}$ for any
$\Lambda_{0}\in\mathcal{L}^{0}$ contained in $\Lambda_{2}$ implies that
$\mathcal{N}_{\Lambda_{0}}(k)\in\mathcal{Z}^{\mathrm{Pap}}({\bf x})(k)$. So
$\mathcal{N}_{\Lambda_{2}}(k)\subset\mathcal{Z}^{\mathrm{Pap}}({\bf x})$.

If $b_{\bf x}\in\Lambda_{2}\backslash\Lambda_{2}^{\sharp}$, then
$\Lambda_{0}\coloneqq\Lambda^{\sharp}+\mathrm{Span}\\{b_{\bf x}\\}$ is a type
$0$ lattice contained in $\Lambda_{2}$ and
$\mathcal{N}_{\Lambda_{0}}(k)\in\mathcal{Z}^{\mathrm{Pap}}({\bf x})$. On the
other hand, since
$\tau(\Lambda_{0}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{\breve{F}})=\Lambda_{0}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{\breve{F}}$,
equation (9.2) tells us that $\mathcal{Z}^{\mathrm{Pap}}({\bf x})$ does not
contain any point in $\mathcal{N}^{\circ}_{\Lambda_{2}}(k)$.

If $b_{\bf x}\notin\Lambda_{2}$, then $b_{\bf x}\notin M(z)$ for any
$z\in\mathcal{N}_{\Lambda_{2}}(k)$, hence $\mathcal{Z}^{\mathrm{Pap}}({\bf
x})(k)\cap\mathcal{N}_{\Lambda_{2}}(k)=\emptyset$. ∎

######  Corollary 9.3.

Let $L\subset\mathbb{V}$. Assume $z\in\mathcal{Z}^{\mathrm{Pap}}(L)(k)$ and
$z\in\mathcal{N}_{\Lambda}(k)$ where $\Lambda\in\mathcal{V}^{2}$. Then
$\mathcal{N}_{\Lambda}\subset\mathcal{Z}^{\mathrm{Pap}}(\pi L).$

######  Corollary 9.4.

Assume ${\bf x}\in\mathbb{V}$ and $\mathrm{v}({\bf x})>0$. Assume
$\mathcal{N}_{\Lambda}\subset\mathcal{Z}^{\mathrm{Pap}}({\bf x})_{red}$ where
$\Lambda\in\mathcal{V}^{2}$, then either
$\mathcal{N}_{\Lambda}\subset\mathcal{Z}^{\mathrm{Pap}}(\frac{1}{\pi}{\bf
x})_{red}$ or
$\mathcal{N}_{\Lambda}\cap\mathcal{Z}^{\mathrm{Pap}}(\frac{1}{\pi}{\bf
x})_{red}$ is a unique superspecial point.

######  Lemma 9.5.

For $L\subset\mathbb{V}$ a lattice of arbitrary rank,
$\mathcal{Z}^{\mathrm{Pap}}(L)_{red}$ is connected.

###### Proof.

Suppose $\mathcal{Z}^{\mathrm{Pap}}(L)_{red}$ has two different connected
components $U_{1}$ and $U_{2}$. Since $\mathrm{SU}(\mathbb{V})$ acts
transitively on $\mathcal{L}$, we can find a ${\bf x}\in\mathbb{V}$ such that
$\mathcal{Z}^{\mathrm{Pap}}({\bf x})\cong\mathcal{N}^{\mathrm{Pap}}_{2,1}$
(i.e. $\\{{\bf x}\\}^{\bot}$ is split) and $\mathcal{Z}^{\mathrm{Pap}}({\bf
x})\cap U_{i}\neq\emptyset$ for $i=1,2$. Hence the reduced locus of

$\mathcal{Z}^{\mathrm{Pap}}(L\oplus\mathrm{Span}\\{{\bf
x}\\})\cong\mathcal{Z}^{\mathrm{Pap}}_{2,1}(L^{\prime})$

is not connected where $L^{\prime}$ is the orthogonal projection of $L$ onto
$\\{{\bf x}\\}^{\bot}$. This contradicts Corollaries 3.13, 3.15 and Lemma 3.16
of [HSY20]. ∎

Recall that for a lattice $L\subset\mathbb{V}$ (resp. ${\bf x}\in\mathbb{V}$),
we have defined $\mathcal{V}(L)$ and $\mathcal{L}(L)$ (resp. $\mathcal{V}({\bf
x})$ and $\mathcal{L}({\bf x})$) in Section 2.4.

######  Proposition 9.6.

Assume that ${\bf x}\in\mathbb{V}$ such that $h({\bf x},{\bf x})\neq 0$. Then
we have

$\mathcal{Z}^{\mathrm{Pap}}({\bf x})_{red}=\bigcup_{\Lambda\in\mathcal{V}({\bf
x})}\mathcal{N}_{\Lambda},$

where $\mathcal{V}({\bf x})$ is given as follows.

  1. (1)

When $\mathrm{v}({\bf x})=0$ and $\mathrm{Span}_{F}\\{{\bf x}\\}^{\perp}$ is
non-split, there is a unique vertex lattice $\Lambda_{\bf
x}\in\mathcal{V}^{0}$ containing $b_{\bf x}$. In this case $\mathcal{V}({\bf
x})=\\{\Lambda_{\bf x}\\}$.

  2. (2)

When $\mathrm{v}({\bf x})=d$ and $\mathrm{Span}_{F}\\{{\bf x}\\}^{\perp}$ is
non-split, we have

$\mathcal{V}({\bf x})=\\{\Lambda\in\mathcal{V}\mid d(\Lambda,\Lambda_{{\bf
x}/\pi^{d}})\leq d\\}$

where $\Lambda_{{\bf x}/\pi^{d}}$ is as in (1).

  3. (3)

When $\mathrm{v}({\bf x})=0$ and $\mathrm{Span}_{F}\\{{\bf x}\\}^{\perp}$ is
split, $\mathcal{L}({\bf x})$ is the tree $\mathcal{L}_{2,1}({\bf x})$
described after (9.3).

  4. (4)

When $\mathrm{v}({\bf x})=d$ and $\mathrm{Span}_{F}\\{{\bf x}\\}^{\perp}$ is
split, we have

$\mathcal{V}({\bf x})=\\{\Lambda\in\mathcal{V}\mid d(\Lambda,\mathcal{L}({\bf
x}/\pi^{d}))\leq d\\}$

where $\mathcal{L}({\bf x}/\pi^{d})$ is as in (3).

  5. (5)

When $h({\bf x},{\bf x})\notin\mathcal{O}_{F_{0}}$, $\mathcal{V}({\bf x})$ is
empty.

###### Proof.

Proof of (1): This is a direct consequence of Proposition 2.6 and the fact
that $\mathcal{N}_{2,-1}^{\mathrm{Pap}}$ has only one reduced point, see
[Shi20, Section 2] or [RSZ18, Section 8]. Alternatively since
$\mathrm{Span}_{F}\\{b_{\bf x}\\}^{\perp}$ is non-split of dimension $2$, it
contains a unique self dual lattice $\Lambda^{\prime}$, then $\Lambda_{\bf
x}\coloneqq\mathrm{Span}\\{b_{\bf x}\\}\oplus\Lambda^{\prime}$ is the unique
type $0$ lattice containing $b_{\bf x}$.

Proof of (3): Applying Proposition 2.6,we see that
$\mathcal{Z}^{\mathrm{Pap}}({\bf x})\cong\mathcal{N}_{2,1}^{\mathrm{Pap}}$ is
the Drinfeld $p$-adic half space, see [KR14a] and [HSY20]. The required
properties of $\mathcal{L}({\bf x})$ and $\mathcal{V}({\bf x})$ follow.

Proof of (2): We prove this by induction. The case $d=0$ is just (1). Now we
assume $d>0$ and that the statement holds for $d-1$, i.e.

$\mathcal{V}({\bf x}/\pi)=\\{\Lambda\in\mathcal{V}\mid d(\Lambda,\Lambda_{{\bf
x}/\pi^{d}})\leq d-1\\}.$

Then applying Corollary 9.3 to the lattice $L=\mathrm{Span}\\{{\bf x}/\pi\\}$
we have

$\bigcup_{\Lambda\in\mathcal{V}^{2},\ d(\Lambda,\Lambda_{{\bf x}/\pi^{d}})\leq
d}\mathcal{N}_{\Lambda}\subset\mathcal{Z}^{\mathrm{Pap}}({\bf x}).$

Corollary 9.4 and the induction hypothesis imply that every
$\Lambda_{2}\in\mathcal{V}^{2}({\bf x})$ satisfies $d(\Lambda,\Lambda_{{\bf
x}/\pi^{d}})\leq d$. By Lemma 9.5 there is no isolated
$\Lambda_{0}\in\mathcal{V}^{0}({\bf x})$, i.e. every
$\Lambda_{0}\in\mathcal{V}^{0}({\bf x})$ is contained in some
$\Lambda_{2}\in\mathcal{V}^{2}({\bf x})$ if $\mathrm{v}({\bf x})>0$. This
finishes the proof of (2).

Similarly we can prove (4) by an induction on $d$, the case $d=0$ is just (3).

(5) follows directly from Lemma 9.2. ∎

####  9.3. Rank $2$ case

######  Proposition 9.7.

Assume that $L^{\flat}=\mathrm{Span}\\{{\bf x}_{1},{\bf
x}_{2}\\}\subset\mathbb{V}$ is integral of rank $2$. Then

$\mathcal{Z}^{\mathrm{Pap}}(L^{\flat})_{red}=\bigcup_{\Lambda\in\mathcal{V}(L^{\flat})}\mathcal{N}_{\Lambda}$

is a finite union, where $\mathcal{V}(L^{\flat})$ is the set of vertices of
the tree $\mathcal{L}(L^{\flat})$ described as follows.

  1. (1)

Assume $L^{\flat}\approx\mathcal{H}_{2a+1}$ for some $a\in\mathbb{Z}_{\geq
0}$. Then $\mathcal{L}(L^{\flat})$ is a ball centered at a vertex lattice of
type $2$ with radius $\frac{2a+1}{2}$.

  2. (2)

Assume $L^{\flat}=\mathrm{Span}\\{\pi^{a}{\bf x}_{1},\pi^{a}{\bf x}_{2}\\}$
where $\mathrm{v}({\bf x}_{1})=0$, $\mathrm{v}({\bf x}_{2})\geq 0$ and
$\mathrm{Span}_{F}\\{{\bf x}_{1}\\}^{\bot}$ is nonsplit. Then
$\mathcal{L}(L^{\flat})$ is a ball centered at a vertex lattice of type $0$
with radius $a$.

  3. (3)

Assume $L^{\flat}=\mathrm{Span}\\{\pi^{a}{\bf x}_{1},\pi^{a+r}{\bf x}_{2}\\}$
where ${\bf x}_{1}\bot{\bf x}_{2}$, $\mathrm{v}({\bf x}_{1})=\mathrm{v}({\bf
x}_{2})=0$, $r\geq 0$ and $\mathrm{Span}_{F}\\{{\bf x}_{1}\\}^{\bot}$ is
split. Then

$\mathcal{L}(L^{\flat})=\\{\Lambda\in\mathcal{V}\mid
d(\Lambda,\mathcal{L}(\pi^{-a}L^{\flat}))\leq a\\},$

where

$\mathcal{L}(\pi^{-a}L^{\flat})=\\{\Lambda\in\mathcal{L}({\bf x}_{1})\mid
d(\Lambda,\Lambda_{0})\leq r\\},$

$\mathcal{L}({\bf x}_{1})$ is described in (3) of Proposition 9.6 and
$\Lambda_{0}$ is the unique type $0$ vertex lattice containing
$\\{x_{1},x_{2}\\}$.

###### Proof.

As in the proof of Proposition 9.2, for a $\Lambda\in\mathcal{V}$,
$\mathcal{N}_{\Lambda}\subset\mathcal{Z}^{\mathrm{Pap}}(L^{\flat})_{red}$ if
and only if $\Lambda^{\sharp}$ contains $b_{{\bf x}_{1}},b_{{\bf x}_{2}}$.

We first prove (1) when $a=0$. Suppose $\Lambda\in\mathcal{V}^{2}(L^{\flat})$.
Extend $\\{b_{{\bf x}_{1}},b_{{\bf x}_{2}}\\}$ to a basis $\\{b_{{\bf
x}_{1}},b_{{\bf x}_{2}},b_{3}\\}$ of $\mathbb{V}$ with Gram matrix
$\mathcal{H}_{1}\oplus\\{-\epsilon\\}$. Choose a basis
$\\{v_{1},v_{2},v_{3}\\}$ of $\Lambda^{\sharp}$ with the same Gram matrix
$\mathcal{H}_{1}\oplus\\{-\epsilon\\}$. Then $b_{{\bf
x}_{i}}\in\Lambda^{\sharp}$ ($i=1,2$) by Lemma 9.2 and

$b_{{\bf x}_{i}}=a_{i1}v_{1}+a_{i2}v_{2}+a_{i3}v_{3}$

where $a_{ij}\in\mathcal{O}_{F}$ ($j=1,2,3$). The fact that $(b_{{\bf
x}_{i}},b_{{\bf x}_{j}})_{1\leq i,j\leq 2}=T$ implies
$a_{i3}\in\pi\mathcal{O}_{F}$ for $i=1,2$ and $(a_{ij})_{1\leq i,j\leq 2}$ is
in $\mathrm{GL}_{2}(\mathcal{O}_{F})$. This guarantees that $L^{\flat}$ is a
direct summand of $\Lambda^{\sharp}$ by Gram-Schmit process. Hence
$\Lambda^{\sharp}$ is in fact the lattice
$\mathrm{Span}_{\mathcal{O}_{F}}\\{b_{{\bf x}_{1}},b_{{\bf x}_{2}},b_{3}\\}$.
The fact that all $\Lambda_{0}\in\mathcal{V}^{0}(L^{\flat})$ are in $\Lambda$
follows from Lemma 9.5.

When $a=0$, (2) follows from the fact that $\mathcal{Z}^{\mathrm{Pap}}({\bf
x}_{1})=\mathcal{N}^{\mathrm{Pap}}_{2,-1}$ (by Proposition 2.6) and
$\mathcal{Z}^{\mathrm{Pap}}(L^{\flat})_{red}=\mathcal{Z}^{\mathrm{Pap}}({\bf
x}_{1})_{red}$ is a unique superspecial point. Similarly, (3) follows from the
fact that $\mathcal{Z}^{\mathrm{Pap}}({\bf
x}_{1})=\mathcal{N}^{\mathrm{Pap}}_{2,1}$ and [HSY20, Corollary 3.13].

Now we prove (1), (2) and (3) for general $a$. First of all,
$\mathcal{L}(L^{\flat})=\mathcal{L}(\pi^{a}{\bf
x}_{1})\cap\mathcal{L}(\pi^{a}{\bf x}_{2})$ by definition. By Corollary 9.3 we
have

$\\{\Lambda\in\mathcal{V}\mid d(\Lambda,\mathcal{L}(\pi^{-a}L^{\flat}))\leq
a\\}\subset\mathcal{L}(\pi^{a}{\bf x}_{1})\cap\mathcal{L}(\pi^{a}{\bf
x}_{2}).$

Notice that for a sub-tree $\mathcal{L}^{\prime}$ of a tree $\mathcal{L}$ and
a vertex $x\in\mathcal{L}\setminus\mathcal{L}^{\prime}$, there is a unique
geodesic segment joining $x$ with $\mathcal{L}^{\prime}$. Given
$\Lambda\in\mathcal{L}(L^{\flat})=\mathcal{L}(\pi^{a}{\bf
x}_{1})\cap\mathcal{L}(\pi^{a}{\bf x}_{2})$, let $\gamma$ be the unique
geodesic segment joining $\Lambda$ with $\mathcal{L}(\pi^{-a}L^{\flat})$.
Assume that $\gamma$ intersects $\mathcal{L}(\pi^{-a}L^{\flat})$ at
$\Lambda(L^{\flat})$. Since $\mathcal{L}(\pi^{-a}L^{\flat})=\mathcal{L}({\bf
x}_{1})\cap\mathcal{L}({\bf x}_{2})$, $\gamma$ necessarily intersects both
$\mathcal{L}({\bf x}_{1})$ and $\mathcal{L}({\bf x}_{2})$. Without loss of
generality we assume that $\gamma$ intersects $\mathcal{L}({\bf x}_{1})$ at
$\Lambda({\bf x}_{1})$ first. Hence the intersection of $\gamma$ with
$\mathcal{L}({\bf x}_{2})$ is $\Lambda(L^{\flat})$ and

$d(\Lambda,\Lambda({\bf x}_{1}))=d(\Lambda,\mathcal{L}({\bf x}_{1}))\leq
d(\Lambda,\Lambda(L^{\flat}))=d(\Lambda,\mathcal{L}({\bf x}_{2})).$

Now by Proposition 9.6, we have

$d(\Lambda,\mathcal{L}({\bf x}_{1}))\leq a,\ d(\Lambda,\mathcal{L}({\bf
x}_{2}))\leq a.$

Hence $d(\Lambda,\mathcal{L}(\pi^{-a}L^{\flat}))\leq a$. This shows that

$\\{\Lambda\in\mathcal{V}\mid d(\Lambda,\mathcal{L}(\pi^{-a}L^{\flat}))\leq
a\\}=\mathcal{L}(\pi^{a}{\bf x}_{1})\cap\mathcal{L}(\pi^{a}{\bf x}_{2}).$

The general case of (1), (2) and (3) follows from the above equation and the
case $a=0$.

Notice that (1), (2) and (3) have covered all possibilities of $L^{\flat}$ due
to the classification of Hermitian lattices. Notice that in every case
$\mathcal{V}(L^{\flat})$ is finite. This finishes the proof of the
proposition. ∎

######  Definition 9.8.

Assume that $L^{\flat}$ is an integral lattice of rank $2$ in $\mathbb{V}$.
Define $\mathcal{S}(L^{\flat})$, the skeleton of $\mathcal{L}(L^{\flat})$, as
follows. If the fundamental invariant of $L^{\flat}$ is $(2a,b)$ ($b\geq 2a$),
define $\mathcal{S}(L^{\flat})\coloneqq\mathcal{L}(\pi^{-a}L^{\flat})$. If the
fundamental invariant of $L^{\flat}$ is $(2a+1,2a+1)$, define
$\mathcal{S}(L^{\flat})\coloneqq\emptyset$.

######  Remark 9.9.

The skeleton $\mathcal{S}(L^{\flat})$ is isomorphic to a ball in the Bruhat-
Tits tree of $\mathcal{N}^{\mathrm{Pap}}_{2,\pm 1}$.

######  Corollary 9.10.

For each $\Lambda_{2}\in\mathcal{V}^{2}(L^{\flat})$ not on the skeleton
$\mathcal{S}(L^{\flat})$, one can find
$\Lambda_{0}\in\mathcal{V}^{0}(L^{\flat})$ such that $\Lambda_{2}$ has the
largest distance to the boundary $\mathcal{B}(L^{\flat})$ of
$\mathcal{L}(L^{\flat})$ among all type $2$ lattices in
$\mathcal{V}^{2}(L^{\flat})$ containing $\Lambda_{0}$.

###### Proof.

Assmue the fundamental invariant of $L^{\flat}$ is $(2a,b)$ or $(2a+1,2a+1)$.
Define $M^{\flat}\coloneqq\pi^{-a}L^{\flat}$. Let $b$ be the unique integer
such that
$\Lambda_{2}\in\mathcal{L}(\pi^{b}M^{\flat})\setminus\mathcal{L}(\pi^{b-1}M^{\flat})$.
Choose any $\Lambda_{0}\in\mathcal{B}(\pi^{b}M^{\flat})$ such that
$\Lambda_{0}\subset\Lambda$. Then by Proposition 9.7, $\Lambda_{0}$ satisfies
the assumption of the corollary. ∎

####  9.4. The Krämer model

For $\Lambda\in\mathcal{V}^{2}$, let $\tilde{\mathcal{N}}_{\Lambda}$ be the
strict transform of $\mathcal{N}_{\Lambda}$ under the blow-up
$\mathcal{N}^{\mathrm{Kra}}\rightarrow\mathcal{N}^{\mathrm{Pap}}$. Since the
strict transform of a regular curve along any of its closed point is an
isomorphism, we know $\tilde{\mathcal{N}}_{\Lambda}\cong\mathbb{P}^{1}$.

######  Lemma 9.11.

For $\Lambda\neq\Lambda^{\prime}\in\mathcal{V}^{2}$,
$\tilde{\mathcal{N}}_{\Lambda}$ and $\tilde{\mathcal{N}}_{\Lambda^{\prime}}$
do not intersect.

###### Proof.

If $\mathcal{N}_{\Lambda}$ and $\mathcal{N}_{\Lambda^{\prime}}$ do not
intersect in $\mathcal{N}^{\mathrm{Pap}}$, then obviously
$\tilde{\mathcal{N}}_{\Lambda}$ and $\tilde{\mathcal{N}}_{\Lambda^{\prime}}$
do not intersect. Without loss of generality we can assume
$\Lambda=\mathrm{Span}\\{e_{1},e_{2},e_{3}\\}$ and
$\Lambda^{\prime}=\mathrm{Span}\\{\pi^{-1}e_{1},\pi e_{2},e_{3}\\}$ where the
Gram matrix of $\\{e_{1},e_{2},e_{3}\\}$ is
$\mathrm{Diag}(\mathcal{H},\epsilon)$. Take ${\bf x}_{0}=e_{3}$. Then by
Proposition 9.7, both $\tilde{\mathcal{N}}_{\Lambda}$ and
$\tilde{\mathcal{N}}_{\Lambda^{\prime}}$ are in $\tilde{\mathcal{Z}}({\bf
x}_{0})\cong\mathcal{N}^{\mathrm{Kra}}_{2,1}$. Now by [HSY20, Lemma 5.3],
$\tilde{\mathcal{N}}_{\Lambda}$ and $\tilde{\mathcal{N}}_{\Lambda^{\prime}}$
do not intersect. ∎

######  Lemma 9.12.

Let $\Lambda\in\mathcal{V}^{2}$ and $\Lambda_{0}\in\mathcal{V}^{0}$. When
$\Lambda_{0}\subset\Lambda$, $\tilde{\mathcal{N}}_{\Lambda}$ intersects
properly with $\mathrm{Exc}_{\Lambda_{0}}$ and

(9.4)
$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}})=1.$

When $\Lambda_{0}$ is not contained in $\Lambda$,
$\tilde{\mathcal{N}}_{\Lambda}$ does not intersect with
$\mathrm{Exc}_{\Lambda_{0}}$.

###### Proof.

First assume $\Lambda_{0}\subset\Lambda$. Since
$\tilde{\mathcal{N}}_{\Lambda}$ is a strict transformation of a curve, it
intersects the exceptional divisor properly. Let ${\bf x}_{0}$ be as in the
proof of Lemma 9.11. Then $\tilde{\mathcal{N}}_{\Lambda}$ is in
$\tilde{\mathcal{Z}}({\bf x}_{0})\cong\mathcal{N}^{\mathrm{Kra}}_{2,1}$.

$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}})$
$\displaystyle=\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes_{\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{0})}}\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{0})}\otimes\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}})$
$\displaystyle=\chi(\tilde{\mathcal{Z}}({\bf
x}_{0}),\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes_{\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{0})}}\mathcal{O}_{\mathrm{Exc}^{\prime}}).$

Here $\mathrm{Exc}^{\prime}\cong\mathbb{P}^{1}/k$ is the exceptional divisor
on $\tilde{\mathcal{Z}}({\bf x}_{0})$ corresponding to the rank $2$ self-dual
lattice

$\Lambda^{\prime}=\\{v\in\Lambda_{0}\mid v\bot{\bf x}_{0}\\}.$

By [HSY20, Lemma 5.2(a)], we know $\chi(\tilde{\mathcal{Z}}({\bf
x}_{0}),\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes_{\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{0})}}\mathcal{O}_{\mathrm{Exc}^{\prime}})=1$. When $\Lambda_{0}$ is not
contained in $\Lambda$, the superspecial point $\mathcal{N}_{\Lambda_{0}}(k)$
is not contained in $\mathcal{N}_{\Lambda}$, hence
$\tilde{\mathcal{N}}_{\Lambda}$ does not intersect with
$\mathrm{Exc}_{\Lambda_{0}}$. ∎

###  10\. Intersection of vertical components and special divisors

In this section we study the intersection of $\tilde{\mathcal{N}}_{\Lambda}$
and special divisors. The main result is Theorem 10.2. To proceed we first
study the decomposition of
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})$ when
$\mathrm{v}(L^{\flat})=0$. Since $n$ is odd, we can without loss of generality
assume that $\chi(\mathbb{V})=\chi(C)=1$. In the rest of the paper, we
identify $\mathbb{V}$ with $C$ by the isomorphism $b$ defined in (2.3).

####  10.1. Decomposition of
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})$

Let $L^{\flat}=\mathrm{Span}\\{{\bf x}_{1},{\bf x}_{2}\\}$ where ${\bf
x}_{1},{\bf x}_{2}\in\mathbb{V}$ are linearly independent and the Hermitian
form restricted to $L$ is non-degenerate.

######  Lemma 10.1.

${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L)=[\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}_{1})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}_{2})}]\in K_{0}(\mathcal{N}^{\mathrm{Kra}})$ is in fact in
$\mathrm{F}^{2}K_{0}(\mathcal{N}^{\mathrm{Kra}})$. Moreover we have the
decomposition

(10.1)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L)=\mathcal{Z}^{\mathrm{Kra}}(L)_{h}+{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L)_{v}.$

where $\mathcal{Z}^{\mathrm{Kra}}(L)_{h}$ is described in Theorem 4.1 and
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L)_{v}\in
F^{2}K_{0}^{(}\mathcal{N}^{\mathrm{Kra}})$.

###### Proof.

By Lemma 2.10, $\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})$ is Noetherian and has a
decomposition

$\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})=\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{h}\cup\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{v}.$

Expressing $\mathcal{Z}({\bf x}_{i})$ ($i=1,2$) as in (3.1) and applying
Propositions 3.2, 3.3 and Lemma 3.4,
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})$ equals

$[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{1})}\otimes^{\mathbb{L}}\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{2})}]+\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L^{\flat})}(2m_{\Lambda_{0}}({\bf
x}_{1})m_{\Lambda_{0}}({\bf x}_{2})+m_{\Lambda_{0}}({\bf
x}_{1})+m_{\Lambda_{0}}({\bf x}_{2}))H_{\Lambda_{0}}.$

$\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{h}$ is contained in
$\tilde{\mathcal{Z}}({\bf x}_{1})\cap\tilde{\mathcal{Z}}({\bf x}_{2})$ and has
dimension $1$ by Theorem 4.1. $\tilde{\mathcal{Z}}({\bf
x}_{1})\cap\tilde{\mathcal{Z}}({\bf x}_{2})_{v}$ also has dimension $1$ as it
is supported on the reduced locus of $\mathcal{N}^{\mathrm{Kra}}$ by Lemma
2.11 and does not contain any exceptional divisor
$\mathrm{Exc}_{\Lambda_{0}}$. Hence

(10.2) $[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{1})}\otimes^{\mathbb{L}}\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{2})}]=[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{1})\cap\tilde{\mathcal{Z}}({\bf
x}_{2})}]\in\mathrm{F}^{2}K_{0}(\mathcal{N}^{\mathrm{Kra}}),$

see for example [Zha21, Lemma B.2]. Hence we know that
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{v}\in\mathrm{F}^{2}K_{0}(\mathcal{N}^{\mathrm{Kra}})$
and we have the desired decomposition. ∎

By Lemma 10.1 and (2.9) we know that
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{v}\in
K^{\prime}_{0}(Y)$ where we can take $Y$ to be the reduced locus of
$\mathcal{N}^{\mathrm{Kra}}$. By the Bruhat-Tits stratification of
$\mathcal{N}^{\mathrm{Kra}}$ and the fact that
$\mathrm{Gr}^{1}K_{0}^{\mathrm{Exc}_{\Lambda_{0}}}(\mathcal{N}^{\mathrm{Kra}})\cong\mathrm{CH}^{1}(\mathrm{Exc}_{\Lambda_{0}})$
is generated by $H_{\Lambda_{0}}$, we have the following decomposition in
$\mathrm{Gr}^{2}K_{0}(\mathcal{N}^{\mathrm{Kra}})$:

(10.3)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{v}=\sum_{\Lambda_{2}\in\mathcal{V}^{2}(L^{\flat})}m(\Lambda_{2},L^{\flat})[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda_{2}}}]+\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L^{\flat})}m(\Lambda_{0},L^{\flat})H_{\Lambda_{0}}.$

We will determine the multiplicities $m(\Lambda_{2},L^{\flat})$ and
$m(\Lambda_{0},L^{\flat})$ when $\mathrm{v}(L^{\flat})=0$ in this section and
deal with the general case in Section 11.

Now assume $L^{\flat}=\mathrm{Span}\\{{\bf x}_{1},{\bf x}_{2}\\}$ with Gram
matrix $\mathrm{Diag}(u_{1},u_{2}(-\pi_{0})^{n})$ with
$u_{1},u_{2}\in\mathcal{O}_{F_{0}}^{\times}$. Applying Proposition 3.2 to
$\mathcal{Z}^{\mathrm{Kra}}({\bf x}_{1})$, we find

${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})=[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf
x}_{1})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}_{2})}]+\sum_{\Lambda_{0}\in\mathcal{V}^{0}({\bf
x}_{1})}[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}_{2})}].$

By Proposition 2.6, we know the intersection $\tilde{\mathcal{Z}}({\bf
x}_{1})\cap\mathcal{Z}^{\mathrm{Kra}}({\bf x}_{2})$ is proper and is
isomorphic to $\mathcal{Z}^{\mathrm{Kra}}_{2,\chi(u_{1})}({\bf x}_{2})$.
Combining this with Corollary 3.5 we obtain

(10.4)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})=i_{*}({}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}_{2,\chi(u_{1})}({\bf
x}_{2}))-\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L^{\flat})}H_{\Lambda_{0}}.$

where $i_{*}$ is the map
$\mathrm{Gr}^{1}K_{0}(\mathcal{N}^{\mathrm{Kra}}_{2,\chi(u_{1})})\rightarrow\mathrm{Gr}^{2}K_{0}(\mathcal{N}^{\mathrm{Kra}}_{3,1})$
induced by the closed immersion
$i:\mathcal{N}^{\mathrm{Kra}}_{2,\chi(u_{1})}\rightarrow\mathcal{N}^{\mathrm{Kra}}_{3,1}$.
Equation (10.4) reduces the problem of decomposing
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})$ in this case to [Shi20,
Theorem 4.5] and [HSY20, Theorem 4.1]. We do not make the effort to write the
complete result down, but instead look at two basic examples.

Let us begin by the case when $L^{\flat}$ is unimodular. By (10.4) and either
[Shi20, Theorem 4.5] or [HSY20, Theorem 4.1], we have

(10.5)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})=[\mathcal{O}_{\tilde{\mathcal{Z}}(L^{\flat})^{o}}]$

in the notation of Theorem 4.1.

Next consider $L^{\flat}=\mathrm{Span}\\{{\bf x}_{1},{\bf x}_{2}\\}$ with Gram
matrix $\mathrm{Diag}(1,-u\pi_{0})$ where $u\in\mathcal{O}_{F_{0}}^{\times}$.
Then $\mathrm{Span}\\{{\bf x}_{1}\\}^{\perp}$ is split and
$\tilde{\mathcal{Z}}({\bf x}_{1})\cong\mathcal{N}^{\mathrm{Kra}}_{2,1}$. By
Proposition 9.7 (3), $\mathcal{V}^{2}(L^{\flat})$ consists of two adjacent
lattices $\Lambda$ and $\Lambda^{\prime}$. Moreover by [HSY20, Theorem 4.1]
and (10.4), we have

(10.6)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})_{v}=[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}]+[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda^{\prime}}}]+H_{\Lambda\cap\Lambda^{\prime}}.$

####  10.2. The intersection number

Assume $\Lambda\in\mathcal{V}^{2}$. For ${\bf
x}\in\mathbb{V}\backslash\\{0\\}$, define

(10.7) $\mathrm{Int}_{\Lambda}({\bf
x})\coloneqq\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x})}).$

In this subsection we prove the following theorem.

######  Theorem 10.2.

Let $\Lambda\in\mathcal{V}^{2}$ and ${\bf x}\in\mathbb{V}\backslash\\{0\\}$.
Then

$\mathrm{Int}_{\Lambda}({\bf x})=1_{\Lambda}({\bf x})$

where $1_{\Lambda}$ is the characteristic function of
$\Lambda\subset\mathbb{V}$.

######  Corollary 10.3.

Assume that $\Lambda_{0}\in\mathcal{L}_{0}$ and $\Lambda\in\mathcal{L}_{2}$
such that $\Lambda_{0}\subset\Lambda$. Then for any $y_{0}\in\Lambda_{0}$ such
that $y_{0}^{\bot}$ is nonsplit, we have

$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}\otimes^{\mathbb{L}}\mathcal{O}_{\tilde{\mathcal{Z}}(y_{0})})=0.$

###### Proof.

By Proposition 3.2, we know

$\mathcal{Z}^{\mathrm{Kra}}(y_{0})=\tilde{\mathcal{Z}}(y_{0})+\mathrm{Exc}_{\Lambda_{0}}.$

Now the corollary follows immediately from Theorem 10.2 and Lemma 9.12. ∎

Proof of Theorem 10.2: We consider three different cases. First if
$x\notin\Lambda$ or $\mathrm{v}({\bf x})<0$, then by Lemma 9.2,
$\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cap\tilde{\mathcal{N}}_{\Lambda}=\emptyset$ hence
$\mathrm{Int}_{\Lambda}({\bf x})=0$. From now on we assume $x\in\Lambda$ and
$\mathrm{v}({\bf x})\geq 0$. Write ${\bf x}={\bf x}_{0}\pi^{n}$ with ${\bf
x}_{0}\in\Lambda\backslash\pi\Lambda$ and $n\geq 0$.

Case 1: First we assume ${\bf x}_{0}\in\Lambda\backslash\Lambda^{\sharp}$.
Choose a basis $\\{e^{\prime}_{1},e^{\prime}_{2},e^{\prime}_{3}\\}$ of
$\Lambda$ with Gram matrix $\mathcal{H}^{3,1}_{1}$ such that

${\bf x}_{0}=xe^{\prime}_{1}+ye^{\prime}_{2}+ze^{\prime}_{3}.$

Then one of $x$ and $y$ is in $\mathcal{O}_{F}^{\times}$ as
$\Lambda^{\sharp}=\mathrm{Span}\\{\pi e^{\prime}_{1},\pi
e^{\prime}_{2},e_{3}\\}$. Apparently the equation

$2u-v\bar{v}=h({\bf x}_{0},{\bf x}_{0})$

has a solution $(u,v)\in\mathcal{O}_{F_{0}}^{2}$ with
$u\in\mathcal{O}_{F_{0}}^{\times}$. Now according to Lemma A.2, we can find a
matrix $g\in\mathrm{U}(\mathcal{H}^{3,1}_{1})(\mathcal{O}_{F_{0}})$ such that

$g\left(\begin{array}[]{c}x\\\ y\\\
z\end{array}\right)=\left(\begin{array}[]{c}\pi u\\\ 1\\\
v\end{array}\right).$

Now replace the basis $\\{e^{\prime}_{1},e^{\prime}_{2},e^{\prime}_{3}\\}$ by
$\\{e_{1},e_{2},e_{3}\\}=\\{e^{\prime}_{1},e^{\prime}_{2},e^{\prime}_{3}\\}g^{-1}$,
we have

${\bf x}_{0}=\pi ue_{1}+e_{2}+ve_{3}$

where $u\in\mathcal{O}_{F_{0}}^{\times}$, $v\in\mathcal{O}_{F}$.

Define

(10.8) $f_{1}=\frac{1}{\pi}u^{-1}e_{2},\ f_{2}=\pi ue_{1},\ f_{3}=e_{3}.$

Then $\\{f_{1},f_{2},f_{3}\\}$ has also Gram matrix $\mathcal{H}^{3,1}_{1}$
and $\Lambda^{\prime}\coloneqq\mathrm{Span}\\{f_{1},f_{2},f_{3}\\}$ is a type
$2$ lattice adjacent to $\Lambda$ with
$\Lambda_{c}=\Lambda\cap\Lambda^{\prime}=\mathrm{Span}\\{\pi
e_{1},e_{2},e_{3}\\}$ is a type $0$ lattice. Now in terms of the basis
$\\{f_{1},f_{2},f_{3}\\}$ we have

${\bf x}=\pi^{n}(\pi uf_{1}+f_{2}+vf_{3}).$

Define $\theta\in\mathrm{U}(\mathbb{V})$ by taking the basis
$\\{e_{1},e_{2},e_{3}\\}$ to $\\{f_{1},f_{2},f_{3}\\}$. Then

$\theta({\bf x})={\bf x},\quad\theta(\Lambda)=\Lambda^{\prime}.$

In particular $\theta(\mathcal{Z}^{\mathrm{Kra}}({\bf
x}))=\mathcal{Z}^{\mathrm{Kra}}({\bf x})$ and

(10.9) $\mathrm{Int}_{\Lambda^{\prime}}({\bf x})=\mathrm{Int}_{\Lambda}({\bf
x}).$

Now let

$\mathbf{y}_{0}=e_{3},\quad\mathbf{y}_{1}=\pi(-\pi ue_{1}+e_{2}),$

$L^{\flat}=\mathrm{Span}\\{\mathbf{y}_{0},\mathbf{y}_{1}\\}$, and
$L=\mathrm{Span}\\{\mathbf{y}_{0},\mathbf{y}_{1},{\bf x}\\}$. Then by (10.6)
and Theorem 4.1 we have

(10.10)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})=[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}]+[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda^{\prime}}}]+H_{\Lambda_{c}}+[\mathcal{O}_{\tilde{\mathcal{Z}}(M^{\flat})}].$

where $\tilde{\mathcal{Z}}(M^{\flat})$ is the quasi canonical-lifting cycle of
the lattice

$M^{\flat}\coloneqq\mathrm{Span}\\{e_{3},-\pi ue_{1}+e_{2}\\}.$

Combining with (10.9), we have

(10.11) $\mathrm{Int}(L)=2\mathrm{Int}_{\Lambda}({\bf
x})+\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cdot
H_{\Lambda_{c}})+\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cdot[\mathcal{O}_{\tilde{\mathcal{Z}}(M^{\flat})}]).$

Let ${\bf x}^{\prime}=\pi^{n}(\pi ue_{1}+e_{2})={\bf x}-\pi^{n}ve_{3}$. Then
we have

$\displaystyle\mathrm{Int}(L)$
$\displaystyle=\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{0})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1}))$
$\displaystyle=\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{0})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1}))$
$\displaystyle=\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_{0})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})$
$\displaystyle\quad+\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L)}\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})$

where we have used linear invariance ([How19, Corollary D]) and Proposition
3.2. Notice that the Gram matrix of $\\{{\bf x}^{\prime},\mathbf{y}_{1}\\}$ is
$\mathrm{Diag}(2u(-\pi_{0})^{n},-2u\pi_{0})$. By Proposition 2.6 and [HSY20,
Theorem 1.1],

$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_{0})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})=\begin{cases}1&\text{
if }n=0,\\\ 1+n-2q&\text{ if }n\geq 1.\end{cases}$

By Corollary 3.6 and [HSY20, Lemma 3.15], we know that

$\displaystyle\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L)}\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})=|\mathcal{V}^{0}(L)|=\begin{cases}1&\text{
if }n=0,\\\ 2q+1&\text{ if }n\geq 1.\end{cases}$

Combining the above two equations we know that

$\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat}))=n+2.$

On the other hand, by Corollary 3.8,

$\chi(\mathcal{N}^{\mathrm{Kra}},H_{\Lambda_{0}}\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x}))=-1.$

By [Gro86, Proposition 3.3]

$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{Z}}(M^{\flat})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x})})=n+1.$

Hence we obtain by (10.11)

(10.12) $\mathrm{Int}_{\Lambda}({\bf x})=1.$

Case 2: Now we Assume ${\bf x}_{0}\in\Lambda^{\sharp}\backslash\pi\Lambda$. As
in the proof of the previous case, we can find a basis
$\\{e_{1},e_{2},e_{3}\\}$ of $\Lambda$ with Gram matrix
$\mathcal{H}^{3,1}_{1}$ by Lemma A.4 such that

${\bf x}=\pi^{n}(ue_{3}+\pi e_{1}).$

where $u\in\mathcal{O}_{F_{0}}^{\times}$. Define

$\Lambda^{\prime}=\mathrm{Span}\\{\pi e_{1},\frac{1}{\pi}e_{2},e_{3}\\},\
\Lambda_{c}=\Lambda\cap\Lambda^{\prime},$

then ${\bf x}_{0}\in\Lambda^{\prime}\backslash\Lambda^{\prime\sharp}$. Also
define

$\mathbf{y}_{0}=e_{3},\ \mathbf{y}_{1}=\pi(-\pi e_{1}+e_{2}),$

and $L^{\flat}\coloneqq\mathrm{Span}\\{\mathbf{y}_{0},\mathbf{y}_{1}\\}$. Then
by Theorem 4.1 and (10.6) we have

(10.13)
${}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})=[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}]+[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda^{\prime}}}]+H_{\Lambda_{c}}+[\mathcal{O}_{\tilde{\mathcal{Z}}(M^{\flat})}],$

where $\tilde{\mathcal{Z}}(M^{\flat})$ is the quasi-canonical lifting cycle of
the lattice

$M^{\flat}\coloneqq\mathrm{Span}\\{e_{3},-\pi e_{1}+e_{2}\\}.$

Let ${\bf x}^{\prime}\coloneqq\pi^{n+1}e_{1}={\bf x}-\pi^{n}ue_{3}$, then we
have

$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat}))=$
$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_{0})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})$
$\displaystyle+\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L)}\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})}).$

Notice that the Gram matrix of $\\{{\bf x}^{\prime},\mathbf{y}_{1}\\}$ is
equivalent to $\mathcal{H}_{1}$ when $n=0$, and to
$\mathrm{Diag}(u_{1}\pi_{0}^{n},u_{2}\pi_{0})$ for some
$u_{1},u_{2}\in\mathcal{O}_{F_{0}}^{\times}$ when $n\geq 1$. Hence by
Proposition 2.6 and [HSY20, Theorem 1.1], we know that

$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_{0})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})=\begin{cases}-(q-1)&\text{
if }n=0,\\\ 1+n-2q&\text{ if }n\geq 1.\end{cases}$

By Corollary 3.6 and Lemmas 3.15 and 3.16 of [HSY20], we know that

$\displaystyle\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L)}\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x}^{\prime})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}(\mathbf{y}_{1})})=|\mathcal{V}^{0}(L)|=\begin{cases}q+1&\text{
if }n=0,\\\ 2q+1&\text{ if }n\geq 1.\end{cases}$

Hence we know that

$\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat}))=n+2.$

On the other hand, by Corollary 3.8,

$\chi(\mathcal{N}^{\mathrm{Kra}},H_{\Lambda_{0}}\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x}))=-1.$

By [Gro86, Proposition 3.3]

$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{O}_{\tilde{\mathcal{Z}}(M^{\flat})}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^{\mathrm{Kra}}({\bf
x})})=n+1.$

Since ${\bf x}\in\Lambda^{\prime}\backslash\Lambda^{\prime\sharp}$, by the
previous case we also have

$\mathrm{Int}_{\Lambda^{\prime}}({\bf x})=1.$

Combining all above, we have by (10.13)

(10.14) $\mathrm{Int}_{\Lambda}({\bf x})=1.$

This finishes the proof of Theorem 10.2. ∎

###  11\. Proof of the modified Kudla-Rapoport conjecture: three dimension
case

In this section, we will prove Theorem 1.2. We need some preparation.

######  Proposition 11.1.

Assume that $L\subset\mathbb{V}$ has a Gram matrix

$T=\mathrm{Diag}(u_{1},u_{2}(-\pi_{0})^{b},u_{3}(-\pi_{0})^{c})$ with
$u_{i}\in\mathcal{O}_{F_{0}}^{\times}$ and $0\leq b\leq c$. Then

$\displaystyle\mathrm{Int}(L)=\partial\mathrm{Den}(L).$

Moreover, for every decomposition $L=L^{\flat}\oplus\mathrm{Span}\\{{\bf
x}\\}$, we have

$\mathrm{Int}(L)^{(2)}=\partial\mathrm{Den}(L)^{(2)}.$

###### Proof.

Fix a basis $\\{{\bf x}_{1},{\bf x}_{2},{\bf x}_{3}\\}$ of $L$ such that the
Gram matrix of $\\{{\bf x}_{1},{\bf x}_{2},{\bf x}_{3}\\}$ is
$T=\mathrm{Diag}(u_{1},T_{2})$ where $u_{1}\in\mathcal{O}_{F_{0}}^{\times}$
and $T_{2}\in\mathrm{Herm}_{2}(\mathcal{O}_{F_{0}})$. Let $u_{1}^{-1}\cdot L$
be a lattice represented by $u_{1}^{-1}\cdot T$. Since
$\mathrm{Int}(u_{1}^{-1}\cdot L)=\mathrm{Int}(L)$ and
$\partial\mathrm{Den}(u_{1}^{-1}\cdot L)=\partial\mathrm{Den}(L)$, we may
assume $u_{1}=1$. Let $L^{\flat}=\mathrm{Span}\\{{\bf x}_{2},{\bf x}_{3}\\}$.
According to Propositions 2.6, 3.2, and Corollary 3.6, we have

(11.1) $\displaystyle\mathrm{Int}(L)-\mathrm{Int}(L^{\flat})$
$\displaystyle=\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}({\bf
x}_{1})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat}))-\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\tilde{\mathcal{Z}}({\bf
x}_{1})\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat}))$
$\displaystyle=\sum_{\Lambda_{0}\in\mathcal{V}^{0}(L)}\chi(\mathcal{N}^{\mathrm{Kra}},[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}]\cdot{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat}))$
$\displaystyle=|\\{\mathcal{V}^{0}(L)\\}|.$

Now the result we want follows by comparing (11.1) with (8.5), and the
identity $\mathrm{Int}(L^{\flat})=\partial\mathrm{Den}(L^{\flat})$ proved in
[Shi20, Theorem 1.3] and [HSY20, Theorem 1.3]. (2) follows from (1) and
Theorem 5.6 (2). ∎

Proof of Theorem 1.4: Under the assumption $\mathrm{v}(L^{\flat})>0$, we can
decompose $\mathcal{D}(L^{\flat})$ in
$\mathrm{Gr}^{2}K_{0}(\mathcal{N}^{\mathrm{Kra}})$ as

(11.2)
$\displaystyle\mathcal{D}(L^{\flat})=\sum_{\Lambda_{2}\in\mathcal{V}(L^{\flat})}m(\mathcal{D}(L^{\flat}),\Lambda_{2})[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda_{2}}}]+\sum_{\Lambda_{0}\in\mathcal{V}(L^{\flat})}m(\mathcal{D}(L^{\flat}),\Lambda_{0})H_{\Lambda_{0}},$

by (10.3) and Proposition 4.5.

Claim 1: $m(\mathcal{D}(L^{\flat}),\Lambda_{0})=0$ unless
$L^{\flat}\subset\Lambda_{0}$. In such a case,

$\displaystyle m(\mathcal{D}(L^{\flat}),\Lambda_{0})=\begin{cases}q+1&\text{
if $\Lambda_{0}\in\mathcal{V}(L^{\flat})\setminus\mathcal{B}(L^{\flat})$},\\\
1&\text{ if $\Lambda_{0}\in\mathcal{B}(L^{\flat})$}.\end{cases}$

Indeed, since $\Lambda_{0}$ is of type $0$, we may choose a
$y_{0}\in\mathbb{V}\setminus L^{\flat}_{F}$ such that
$\mathrm{Span}\\{y_{0}\\}^{\bot}$ is non-split and
$y_{0}\in\Lambda_{0}\setminus\pi\Lambda_{0}$. In this case, Proposition 3.2,
Corollaries 3.7 and 3.8 imply that

$\chi(\mathcal{N}^{\mathrm{Kra}},H_{\Lambda_{0}}\cdot[\mathcal{O}_{\tilde{\mathcal{Z}}(y_{0})}])=1.$

So by (11.2), Corollaries 10.3 and 2.7, we have

$\displaystyle
m(\mathcal{D}(L^{\flat}),\Lambda_{0})=\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot[\mathcal{O}_{\tilde{\mathcal{Z}}(y_{0})}]).$

Let $(2a,2b)$ ($b>a$) be the fundamental invariant of the projection of
$L^{\flat}$ onto $\mathrm{Span}\\{y_{0}\\}^{\bot}$. Let $\varphi$ be the
natural quotient map $\Lambda_{0}\rightarrow\Lambda_{0}/\pi\Lambda_{0}$ and
define

$m\coloneqq\mathrm{dim}_{\mathbb{F}_{q}}\varphi(L^{\flat})\leq 2.$

Equation (9.1) implies that $m=0$ if and only if
$\Lambda_{0}\in\mathcal{L}(L^{\flat})\setminus\mathcal{B}(L^{\flat})$. First
assume $m=0$, in other words, $L^{\flat}\subset\pi\Lambda_{0}$ so $b\geq a\geq
1$. By the definition of $\mathcal{D}(L^{\flat})$ and [Shi20, Theorem 1.2], we
have

$m(\mathcal{D}(L^{\flat}),\Lambda_{0})=\mu(a,b)-q\mu(a-1,b)-\mu(a,b-1)+q\mu(a-1,b-1)=q+1,$

as claimed where

(11.3)
$\displaystyle\mu(a,b)=\chi(\mathcal{N}^{\mathrm{Kra}},{}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^{\flat})\cdot[\mathcal{O}_{\tilde{\mathcal{Z}}(y_{0})}])=\begin{cases}2\sum_{s=0}^{a}q^{s}(a+b+1-2s)-a-b-2&\text{
if }a\geq 0\\\ 0&\text{ if }a<0.\end{cases}$

Now assume $m=1$, then $\varphi(L^{\flat})$ is a line $\ell$ and $b\geq 1$. By
the assumption that $y_{0}\notin L^{\flat}_{F}$, we know $\ell$ is not in
$\mathrm{Span}\\{\varphi(y_{0})\\}$, hence the projection of $\ell$ onto
$\varphi(y_{0})^{\bot}$ is nonzero. Since $\varphi(y_{0})^{\bot}$ is nonsplit,
we must have $a=0$. Hence by the definition of $\mathcal{D}(L^{\flat})$ and
(11.3), we have

$\displaystyle m(\mathcal{D}(L^{\flat}),\Lambda_{0})$
$\displaystyle=\mu(0,b)-q\mu(-1,b)-\mu(0,b-1)+q\mu(-1,b-1)=1$

as claimed. Finally, $m=2$ is impossible since $v(L^{\flat})>0$. This finshes
the proof of Claim 1.

Claim 2: $m(\mathcal{D}(L^{\flat}),\Lambda_{2})=2$ for any
$\Lambda_{2}\in\mathcal{V}^{2}(L^{\flat})$.

Indeed, according to Lemma 3.9, we have
$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}])=0$.
On the other hand, Corollary 3.7 and Lemma 9.12 imply that

(11.4)
$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}])=\sum_{\Lambda_{0}\subset\Lambda_{2}}m(\mathcal{D}(L^{\flat}),\Lambda_{2})-2m(\mathcal{D}(L^{\flat}),\Lambda_{0}).$

Combining the above with Claim 1, we have

(11.5) $\displaystyle 0$
$\displaystyle=\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}])=\begin{cases}\sum_{\Lambda_{0}\subset\Lambda_{2}}m(\mathcal{D}(L^{\flat}),\Lambda_{2})-2(q+1)&\text{
if $\Lambda_{0}\in\mathcal{L}(L^{\flat})\setminus\mathcal{B}(L^{\flat})$},\\\
\sum_{\Lambda_{0}\subset\Lambda_{2}}m(\mathcal{D}(L^{\flat}),\Lambda_{2})-2&\text{
if $\Lambda_{0}\in\mathcal{B}(L^{\flat})$}.\end{cases}$

Recall $\mathcal{S}(L^{\flat})$ in Definition 9.8. First assume
$\Lambda_{2}\in\mathcal{L}(L^{\flat})\setminus\mathcal{S}(L^{\flat})$. If
$d(\Lambda_{2},\mathcal{B}(L^{\flat}))$ equals to $\frac{1}{2}$, choose
$\Lambda_{0}\in\mathcal{B}(L^{\flat})$ such that
$\Lambda_{0}\subset\Lambda_{2}$, then $\Lambda_{2}$ is the unique lattice in
$\mathcal{V}^{2}(L^{\flat})$ that contains $\Lambda_{0}$. Hence (11.5) implies
that $m(\mathcal{D}(L^{\flat}),\Lambda_{2})=2$. Now Corollary 9.10 allows us
to show $m(\mathcal{D}(L^{\flat}),\Lambda_{2})=2$ by induction on the distance
$d(\Lambda_{2},\mathcal{B}(L^{\flat}))$ for any
$\Lambda_{2}\in\mathcal{L}(L^{\flat})\setminus\mathcal{S}(L^{\flat})$.

Similarly for $\Lambda_{2}\in\mathcal{S}(L^{\flat})$, we can show
$m(\mathcal{D}(L^{\flat}),\Lambda_{2})=2$ by induction on its distance to
$\mathcal{S}(L^{\flat})\cap\mathcal{B}(L^{\flat})$. This finishes the proof of
Claim 2.

Now the proposition follows from the fact that for
$\Lambda_{0}\in\mathcal{V}(L^{\flat})$

$\sum_{\Lambda_{2}\in\mathcal{V}^{2}(L^{\flat})}\sum_{\Lambda_{0}\subset\Lambda_{2}}1=\begin{cases}q+1&\hbox{if
}\Lambda_{0}\in\mathcal{V}(L^{\flat})\setminus\mathcal{B}(L^{\flat}),\\\
1&\hbox{if }\Lambda_{0}\in\mathcal{B}(L^{\flat}).\end{cases}$

This finishes the proof of Theorem 1.4.

In the following discussion we freely use Theorem 10.2 and Corollary 3.8
without explicitly referring to them.

######  Proposition 11.2.

Assume $L=L^{\flat}\obot\mathrm{Span}\\{{\bf x}\\}$ with Gram matrix

$T=\mathrm{Diag}(\mathcal{H}_{a},u_{3}(-\pi_{0})^{c})$

where $a$ is a positive odd integer, and $c\geq 0$. Then

(11.6)
$\displaystyle\mathrm{Int}(L)^{(2)}=\partial\mathrm{Den}(L)^{(2)}=\begin{cases}1-q^{a}&\text{
if $a\leq 2c$},\\\ 1-q^{2c+1}&\text{ if $a>2c$}.\end{cases}$

###### Proof.

By Proposition 8.5, it suffices to prove the identity for
$\mathrm{Int}(L)^{(2)}$.

Now we compute $\mathrm{Int}(L)^{(2)}$. We may take
$L^{\flat}=\mathrm{Span}\\{\pi^{\frac{a+1}{2}}e_{1},\pi^{\frac{a+1}{2}}e_{2}\\}$,
where the Gram matrix of $\\{e_{1},e_{2}\\}$ is $\mathcal{H}$. Let
$e_{3}=\pi^{-c}{\bf x}$. Then $\mathcal{L}(L^{\flat})$ is centered at
$\mathrm{Span}\\{e_{1},e_{2},e_{3}\\}$ of radius $\frac{a}{2}$ by Proposition
9.7.

Assume $a\leq 2c$ first. In this case,
$\mathcal{L}(L^{\flat})\subset\mathcal{L}({\bf x})$. As a result, we have
$\mathrm{Int}_{\Lambda_{2}}({\bf x})=1$ and $\mathrm{Int}_{\Lambda_{0}}({\bf
x})=-1$ for any $\Lambda_{2}\in\mathcal{L}^{2}(L^{\flat})$ and
$\Lambda_{0}\in\mathcal{L}^{0}(L^{\flat})$. Hence by Theorem 1.4, we have

(11.7) $\displaystyle\mathrm{Int}(L)^{(2)}$
$\displaystyle=\sum_{\Lambda_{2}\in\mathcal{L}(L^{\flat})}\chi(\mathcal{N}^{\mathrm{Kra}},(2[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda_{2}}}]+\sum_{\Lambda_{0}\subset\Lambda_{2}}H_{\Lambda_{0}})\cdot\mathcal{Z}^{\mathrm{Kra}}({\bf
x}))$
$\displaystyle=(1-q)|\\{\Lambda_{2}\mid\Lambda_{2}\in\mathcal{L}(L^{\flat})\\}|$
$\displaystyle=(1-q)(1+(1+q)q+(1+q)q^{3}+\cdots+(1+q)q^{a-2})$
$\displaystyle=1-q^{a},$

as claimed.

Now we assume $a>2c$. We consider the case $c=0$ first. Recall that
$\tilde{\mathcal{Z}}(e_{3})\approx\mathcal{N}^{\mathrm{Kra}}_{2,1}$, hence
$\mathcal{L}(L^{\flat})\cap\mathcal{L}(e_{3})$ is a ball of radius
$\frac{a}{2}$ in the Bruhat-Tits tree $\mathcal{L}_{2,1}$ of
$\mathcal{N}^{\mathrm{Pap}}_{2,1}$, within which a vertex lattice
$\Lambda_{0}$ of type $0$ is contained in two vertex lattices of type $2$, and
a vertex lattice $\Lambda_{2}$ of type $2$ contains $q+1$ vertex lattice of
type $0$. Hence

$\displaystyle|\\{\Lambda_{0}\mid\Lambda_{0}\in(\mathcal{L}(L^{\flat})\setminus\mathcal{B}(L^{\flat}))\cap\mathcal{L}(e_{3})\\}|=1+q+(1+q)q+\cdots+(1+q)q^{\frac{a-3}{2}},$

and

$\displaystyle|\\{\Lambda_{0}\mid\Lambda_{0}\in\mathcal{B}(L^{\flat})\cap\mathcal{L}(e_{3})\\}|=(1+q)q^{\frac{a-1}{2}}.$

Moreover, notice that if $e_{3}\in\Lambda_{0}$, then
$\mathrm{Int}_{\Lambda_{2}}(e_{3})=1$ for any $\Lambda_{2}$ such that
$\Lambda_{0}\subset\Lambda_{2}$. However, if
$\Lambda_{0}\in\mathcal{B}(L^{\flat})\cap\mathcal{L}(e_{3})$, there is a
unique $\Lambda_{2}$ such that $m(\mathcal{D}(L^{\flat}),\Lambda_{2})\not=0$
among the $q+1$ vertex lattices of type $2$ that contain $\Lambda_{0}$. As a
result,

$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot\mathcal{Z}^{\mathrm{Kra}}(e_{3}))$
$\displaystyle=2(1+q\cdot|\\{\Lambda_{0}\mid\Lambda_{0}\in(\mathcal{L}(L^{\flat})\setminus\mathcal{B}(L^{\flat}))\cap\mathcal{L}(e_{3})\\}|)$
$\displaystyle\quad-(q+1)|\\{\Lambda_{0}\mid\Lambda_{0}\in(\mathcal{L}(L^{\flat})\setminus\mathcal{B}(L^{\flat}))\cap\mathcal{L}(e_{3})\\}|$
$\displaystyle\quad-|\\{\Lambda_{0}\mid\Lambda_{0}\in\mathcal{B}(L^{\flat}))\cap\mathcal{L}(e_{3})\\}|$
$\displaystyle=2+(q-1)(1+q+(1+q)q+\cdots+(1+q)q^{\frac{a-3}{2}})-(1+q)q^{\frac{a-1}{2}}$
$\displaystyle=1-q,$

which is compatible with (11.6).

Next we show

$\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot(\mathcal{Z}^{\mathrm{Kra}}(\pi
e_{3})-\mathcal{Z}^{\mathrm{Kra}}(e_{3})))=q-q^{3}.$

According to Proposition 9.6, $\mathcal{L}(\pi e_{3})=\\{\Lambda\mid
d(\Lambda,\mathcal{L}(e_{3}))\leq 1\\}$. Hence, around each
$\Lambda_{0}\in(\mathcal{L}(L^{\flat})\setminus\mathcal{B}(L^{\flat}))\cap\mathcal{L}(e_{3})$,
there will be $q(q-1)$ many new vertex lattices of type $0$ in
$\mathcal{L}(L^{\flat})\cap\mathcal{L}(\pi
e_{3})\setminus\mathcal{L}(L^{\flat})\cap\mathcal{L}(e_{3})$. Hence,

$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot(\mathcal{Z}^{\mathrm{Kra}}(\pi
e_{3})-\mathcal{Z}^{\mathrm{Kra}}(e_{3})))$ $\displaystyle=2q\cdot
q(q-1)(1+q+(1+q)q+(1+q)q^{2}+\cdots+(1+q)q^{\frac{a-1}{2}-2})$
$\displaystyle\quad-q(q-1)(q+1)(1+q+(1+q)q+(1+q)q^{2}+\cdots+(1+q)q^{\frac{a-1}{2}-2})$
$\displaystyle\quad-q(q-1)(1+q)q^{\frac{a-1}{2}-1}$ $\displaystyle=q-q^{3}.$

Continuing in this way, we can show

$\displaystyle\chi(\mathcal{N}^{\mathrm{Kra}},\mathcal{D}(L^{\flat})\cdot(\mathcal{Z}^{\mathrm{Kra}}(\pi^{i}e_{3})-\mathcal{Z}^{\mathrm{Kra}}(\pi^{i-1}e_{3})))=q^{2i-1}-q^{2i+1}$

for $2i<a$. So

$\mathrm{Int}(L)^{(2)}=\mathcal{D}(L^{\flat})\cdot\mathcal{Z}^{\mathrm{Kra}}(\pi^{c}e_{3})=1-q^{2c+1}=\partial\hbox{Den}^{(2)}(L)$

as claimed. ∎

######  Proposition 11.3.

Assume $L=L^{\flat}\obot\mathrm{Span}\\{{\bf x}\\}$ with Gram matrix

$T=\mathrm{Diag}(u_{1}(-\pi_{0})^{a},u_{2}(-\pi_{0})^{b},u_{3}(-\pi_{0})^{c})$

where $0<a\leq b\leq c$, then

$\mathrm{Int}(L)^{(2)}=\partial\mathrm{Den}(L)^{(2)}=1+\chi(-u_{2}u_{3})q^{a}(q^{a}-q^{b})-q^{a+b}.$

###### Proof.

By Proposition 8.4, it suffices to show

(11.8)
$\displaystyle\mathrm{Int}(L)^{(2)}=1+\chi(-u_{2}u_{3})q^{a}(q^{a}-q^{b})-q^{a+b}.$

If $\chi(-u_{2}u_{3})=-1$, then $\chi(S)\chi(u_{1})=1$, and (11.8) specializes
to

$\displaystyle\partial\mathrm{Den}(T)^{(2)}=1-q^{2a}.$

On the other hand, if $\chi(-u_{2}u_{3})=-1$, then $\mathcal{L}(L^{\flat})$ is
a ball centered at a vertex lattice of type $0$ with radius $a$. Since $a<c$,
$\mathcal{L}(L^{\flat})\subset\mathcal{L}({\bf x})$, and one can show
$\mathrm{Int}(L)^{(2)}=1-q^{2a}$ exactly as in (11.7).

Now we assume $\chi(-u_{2}u_{3})=1$, hence $\chi(S)\chi(u_{1})=-1$. In this
case, (11.8) specializes to

$\displaystyle\partial\mathrm{Den}(L)^{(2)}=1+q^{2a}-2q^{a+b}.$

Let $r=b-a$, and $L^{\flat}=\mathrm{Span}\\{x_{1},x_{2}\\}$. Then
$\mathcal{L}(\pi^{-a}L^{\flat})$ is a ball centered at a vertex lattice of
type $0$ with radius $r$ in the Bruhat-Tits tree $\mathcal{L}_{2,1}$. Hence,

$\mathcal{L}(L^{\flat})=\\{\Lambda\mid\Lambda\in\mathcal{L}_{3},\
d(\Lambda,\mathcal{L}(\pi^{-a}L^{\flat}))\leq a\\}.$

When $a=1$,
$\mathcal{V}^{0}(\pi^{-1}L^{\flat})=\mathcal{V}^{0}(L^{\flat})\setminus\mathcal{B}(L^{\flat})$.
Then combining with Theorem 1.4, it is not hard to see

$\displaystyle\mathrm{Int}(L)^{(2)}$ $\displaystyle=2(q+1+q\cdot
2(q+q^{2}+\cdots+q^{r}))-(q+1)|\mathcal{V}^{0}(L^{\flat})\setminus\mathcal{B}(L^{\flat})|-|\mathcal{B}(L^{\flat})|$
$\displaystyle=1+q^{2}-2q^{b+1},$

where we use the fact

$|\mathcal{V}^{0}(\pi^{-1}L^{\flat})|=1+2(q+q^{2}+\cdots+q^{r}),$

and

$|\mathcal{B}(L^{\flat})|=(q-1)q(1+2(q+q^{2}+\cdots+q^{r-1}))+2q^{r+2}.$

Now assume $a>1$. Let $T$ be the Hermitian matrix associated with
$L^{\flat}\obot\mathrm{Span}\\{{\bf x}\\}$, then

$\displaystyle\partial\mathrm{Den}(\pi L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}-\partial\mathrm{Den}(L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}$ $\displaystyle=1+q^{2a+2}-2q^{r+2a+2}-(1+q^{2a}-2q^{r+2a})$
$\displaystyle=q^{2a}(q^{2}-1)(1-q^{2r})$
$\displaystyle=q^{2}\left(\partial\mathrm{Den}(L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}-\partial\mathrm{Den}(\pi^{-1}L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}\right),$

and

$\displaystyle\mathrm{Int}(\pi L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}-\mathrm{Int}(L^{\flat}\obot\mathrm{Span}\\{{\bf x}\\})^{(2)}$
$\displaystyle=2q|\mathcal{B}(L^{\flat})|-q|\mathcal{B}(L^{\flat})|-|\mathcal{B}(\pi
L^{\flat})|$ $\displaystyle=(2q-q-q^{2})|\mathcal{B}(L^{\flat})|$
$\displaystyle=q^{2}\left(\mathrm{Int}(L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}-\mathrm{Int}(\pi^{-1}L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})^{(2)}\right),$

where we use the fact $|\mathcal{B}(\pi L^{\flat}\obot\mathrm{Span}\\{{\bf
x}\\})|=q^{2}|\mathcal{B}(L^{\flat})|$. Since $r$ is arbitrary, an induction
on $a$ gives the result we want. ∎

Proof of Theorem 1.2: The case $\mathrm{v}(L)<0$ follows from Proposition 8.1
and the fact that $\mathrm{Int}(L)=0$ under this condition. Assume $v(L)\geq
0$. There are three cases.

Case 1: When $L$ has a Gram matrix
$\mathrm{Diag}(u_{1},u_{2}(-\pi_{0})^{b},u_{3}(-\pi_{0})^{c})$ as in
Proposition 11.1, it is proved by Proposition 11.1.

Case 2: When $L$ has a basis $\\{{\bf x}_{1},{\bf x}_{2},{\bf x}_{3}\\}$ whose
Gram matrix is $T=\mathrm{Diag}(\mathcal{H}_{a},u_{3}(-\pi_{0})^{c})$, take
$L^{\flat}=\hbox{Span}({\bf x}_{1},{\bf x}_{2})$ and ${\bf x}={\bf x}_{3}$. By
Propositions 11.1, 11.2, and 11.3, we have

$\mathrm{Int}(L^{\flat,\prime}\oplus\mathrm{Span}\\{{\bf
x}\\})^{(2)}=\partial\mathrm{Den}(L^{\flat,\prime}\oplus\mathrm{Span}\\{{\bf
x}\\})^{(2)}$

for any $L^{\flat}\subset L^{\flat,\prime}\subset L^{\flat}_{F}$ (direct sums
in the above identity are actually orthogonal direct sums). Thus we have by
Theorem 5.6 (1)

$\mathrm{Int}(L)=\partial\mathrm{Den}(L).$

Case 3: When $L$ has a Gram matrix
$\mathrm{Diag}(u_{1}(-\pi_{0})^{a},u_{2}(-\pi_{0})^{b},u_{3}(-\pi_{0})^{c})$
with $0\leq a\leq b\leq c)$, the same argument as Case 2 gives
$\mathrm{Int}(L)=\partial\mathrm{Den}(L)$. This finishes the proof of the
theorem.

Theorem 1.2 and Theorem 5.6 imply the following corollary.

######  Corollary 11.4.

For any lattice $L=L^{\flat}\oplus\mathcal{O}_{F}{\bf x}\subset\mathbb{V}$ of
rank $3$, we have

$\mathrm{Int}(L)^{(2)}=\partial\mathrm{Den}(L)^{(2)}.$

##  Part IV Arithmetic Siegel-Weil formula

###  12\. Unitary Shimura varieties

In this section we briefly review the definition of an integral model of
unitary Shimura variety following [Shi18] and [RSZ21, Section 6] (see also
[RSZ20] and [Cho20]). Let $F$ be a CM field over $\mathbb{Q}$ with totally
real subfield $F_{0}$ of index $2$ in it. Let $d=[F_{0}:\mathbb{Q}]$. We
denote by $a\mapsto\bar{a}$ the nontrivial automorphism of $F/F_{0}$. Define

(12.1) $\mathcal{V}_{\mathrm{ram}}=\\{\text{finite places }v\text{ of
}F_{0}\mid v\text{ ramifies in }F\\}.$

In this paper we assume that $\mathcal{V}_{\mathrm{ram}}$ is non-empty. We
also make the assumption as in [RSZ21, Section 6] that every
$v\in\mathcal{V}_{\mathrm{ram}}$ is unramified over $\mathbb{Q}$ and does not
divide $2$.

Fix a totally imaginary element $\sqrt{\Delta}\in F$. Denote by $\Sigma_{0}$
(resp. $\Sigma_{F}$) the set of real (resp. complex) embeddings of $F_{0}$
(resp. $F$). Define a CM type of $F$ by

(12.2)
$\Sigma=\\{\sigma\in\Sigma_{F}\mid\sigma(\sqrt{\Delta})\in\sqrt{-1}\mathbb{R}_{>0}\\}.$

We fix a distinguished element $\tau_{0}\in\Sigma_{0}$ and let $\sigma_{0}$ be
the unique element in $\Sigma$ above $\tau_{0}$. For $\sigma\in\Sigma_{F}$,
denote its complex conjugate by $\bar{\sigma}$.

####  12.1. The Shimura datum

Define a function $r:\Sigma_{F}\rightarrow\mathbb{Z}_{\geq 0}$ by

$\sigma\mapsto r_{\sigma}\coloneqq\begin{cases}1&\text{ if
}\sigma=\sigma_{0};\\\ 0&\text{ if }\sigma\in\Sigma,\sigma\neq\sigma_{0};\\\
n-r_{\bar{\sigma}}&\text{ if }\sigma\notin\Sigma.\end{cases}.$

Assume that $W$ is a $n$ dimensional $F$-vector space with a Hermitian form
$(,)$ such that

$\mathrm{sig}W_{\sigma}=(r_{\sigma},r_{\bar{\sigma}}),\forall\sigma\in\Sigma$

where $W_{\sigma}\coloneqq W\otimes_{F,\sigma}\mathbb{C}$ and
$\mathrm{sig}W_{\sigma}$ is its signature with respect to $(,)$. Let
$\mathrm{U}(W)$ (resp. $\mathrm{GU}(W)$) be the unitary group (resp. general
unitary group) of $(W,(,))$. Recall that for an $F_{0}$-algebra $R$, we have

$\mathrm{GU}(W)(R)=\\{g\in\mathrm{GL}_{R}(W\otimes_{F_{0}}R)\mid(gv,gw)=c(g)(v,w),\forall
v,w\in W\otimes_{F_{0}}R\\}.$

Define the following groups.

$Z^{\mathbb{Q}}\coloneqq\\{z\in\mathrm{Res}_{F/\mathbb{Q}}\mathbb{G}_{m}\mid\mathrm{Nm}_{F/F_{0}}(z)\in\mathbb{G}_{m}\\},$
$G=\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(W),$ (12.3)
$G^{\mathbb{Q}}\coloneqq\\{g\in\mathrm{Res}_{F/\mathbb{Q}}\mathrm{GU}(W)\mid
c(g)\in\mathbb{G}_{m}\\}.$

Notice that

$Z^{\mathbb{Q}}(\mathbb{R})=\\{(z_{\sigma})\in(\mathbb{C}^{\times})^{d}\mid|z_{\sigma}|=|z_{\sigma_{0}}|,\forall\sigma\in\Sigma\\}.$

Define the Hodge map

$h_{Z^{\mathbb{Q}}}:\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{M}\rightarrow
Z^{\mathbb{Q}}(\mathbb{R}),z\mapsto(\bar{z},\ldots,\bar{z}).$

For each $\sigma\in\Sigma$ choose a $\mathbb{C}$-basis of $W_{\sigma}$ such
that $(,)$ is given by the matrix
$\mathrm{diag}(1_{r_{\sigma}},-1_{r_{\bar{\sigma}}})$. Define the Hodge map

$h_{\mathrm{GU}(W)}:\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{M}\rightarrow\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{GU}(W)(\mathbb{R})\cong\prod_{\tau\in\Sigma_{0}}\mathrm{GU}(W_{\tau})$

by sending $z$ to $\mathrm{diag}(z\cdot 1_{r_{\sigma}},\bar{z}\cdot
1_{r_{\bar{\sigma}}})$ for each $\sigma$ component. Then $h_{\mathrm{GU}(W)}$
factors as

$h_{\mathrm{GU}(W)}=i\circ h_{G^{\mathbb{Q}}}$

where
$i:G^{\mathbb{Q}}\rightarrow\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{GU}(W)$ is
the natural inclusion.

Define

$\tilde{G}\coloneqq Z^{\mathbb{Q}}\times_{\mathbb{G}_{m}}G^{\mathbb{Q}}$

where the maps from the factors on the right hand side to $\mathbb{G}_{m}$ are
$\mathrm{Nm}_{F/F_{0}}$ and the similitude character $c(g)$ respectively.
Notice that the map

(12.4) $\tilde{G}\rightarrow Z^{\mathbb{Q}}\times G,(z,g)\mapsto(z,z^{-1}g)$

is an isomorphism. We define the Hodge map $h_{\tilde{G}}$ by

$h_{\tilde{G}}:\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{M}\rightarrow\tilde{G}(\mathbb{R}),z\mapsto(h_{Z^{\mathbb{Q}}}(z),h_{G^{\mathbb{Q}}}(z)).$

Then $(\tilde{G},h_{\tilde{G}})$ is a Shimura datum whose reflex field
$E\subset\bar{\mathbb{Q}}$ is defined by

(12.5)
$\mathrm{Aut}(\bar{\mathbb{Q}}/E)=\\{\sigma\in\mathrm{Aut}(\bar{\mathbb{Q}})\mid\sigma\circ\Sigma=\Sigma,\sigma^{*}(r)=r\\}.$

######  Remark 12.1.

$F$ always embeds into $E$ via $\sigma_{0}$ ([RSZ20, Remark 3.1]). Furthermore
$E=F$ when $F$ is Galois over $\mathbb{Q}$ or when $F=F_{0}K$ where $K$ is an
imaginary quadratic field over $\mathbb{Q}$ and $\Sigma$ is induced from a CM
type of $K/\mathbb{Q}$.

For a compact group $K\in\tilde{G}(\mathbb{A}_{f})$, we can define a Shimura
variety $S(\tilde{G},h_{\tilde{G}})_{K}$ which has a canonical model over the
$\mathrm{Spec}\,E$. We refer to [RSZ21, Section 3] for the moduli problem
$S(\tilde{G},h_{\tilde{G}})_{K}$ represents.

####  12.2. Integral Model

In this subsection, we define the integral model for
$S(\tilde{G},h_{\tilde{G}})_{K}$ (as a Deligne-Mumford stack) in terms of a
moduli functor for a particular choice of $W$ and $K$.

For a lattice $\Lambda$ in $W$, we let $\Lambda^{\vee}$ denote its dual with
respect to the symplectic form
$\mathrm{tr}_{F/\mathbb{Q}}(\sqrt{\Delta}^{-1}(,))$ and $\Lambda^{\sharp}$
denote its dual with respect to the Hermitian form $(,)$. Then we have

(12.6) $\Lambda^{\vee}=\sqrt{\Delta}\partial^{-1}\Lambda^{\sharp}$

where $\partial$ is the different ideal of $F/\mathbb{Q}$. From now on we
assume that $W$ contains a lattice $\Lambda$ such that
$\Lambda^{\vee}=\Lambda$. Define the compact subgroup $K_{G}\subset
G(\mathbb{A}_{f})$ by

$K_{G}\coloneqq\\{g\in G(\mathbb{A}_{f})\mid
g(\Lambda\otimes\hat{\mathbb{Z}})=\Lambda\otimes\hat{\mathbb{Z}}\\}.$

Let $K_{Z^{\mathbb{Q}}}$ be the unique maximal compact subgroup of
$Z^{\mathbb{Q}}(\mathbb{A}_{f})$:

$K_{Z^{\mathbb{Q}}}\coloneqq\\{z\in(\mathcal{O}_{F}\otimes\hat{\mathbb{Z}})^{\times}\mid\mathrm{Nm}_{F/F_{0}}(z)\in\hat{\mathbb{Z}}\\}.$

Define the compact subgroup

(12.7) $K\coloneqq K_{Z^{\mathbb{Q}}}\times
K_{G}\subset\tilde{G}(\mathbb{A}_{f})$

under the isomorphism (12.4).

First we define an auxilary moduli functor $\mathcal{M}_{0}$ over
$\mathrm{Spec}\,\mathcal{O}_{E}$. For a locally Notherian
$\mathcal{O}_{E}$-scheme $S$, we define $\mathcal{M}_{0}(S)$ to be the
groupoid of triples $(A_{0},\iota_{0},\lambda_{0})$ where

  1. (1)

$A_{0}$ is an Abelian scheme over $S$.

  2. (2)

$\iota_{0}:\mathcal{O}_{F}\rightarrow\mathrm{End}(A_{0})$ is an
$\mathcal{O}_{F}$-action satisfying the Kottwitz condition of signature
$((0,1)_{\sigma\in\Sigma})$, namely

$\mathrm{charpol}(\iota_{0}(a)\mid\mathrm{Lie}\,A_{0})=\prod_{\sigma\in\Sigma}(T-\bar{\sigma}(a)),\forall
a\in\mathcal{O}_{F}.$

  3. (3)

$\lambda_{0}$ is a principal polarization of $A_{0}$ whose Rosati involution
induces on $\mathcal{O}_{F}$ via $\iota_{0}$ the nontrivial Galois
automorphism of $F/F_{0}$.

A morphism between two objects $(A_{0},\iota_{0},\lambda_{0})$ and
$(A^{\prime}_{0},\iota^{\prime}_{0},\lambda^{\prime}_{0})$ is a
$\mathcal{O}_{F}$-linear isomorphism $A_{0}\rightarrow A^{\prime}_{0}$ that
pulls $\lambda^{\prime}_{0}$ back to $\lambda$.

Since we assume $\mathcal{V}_{\mathrm{ram}}$ is nonempty, $\mathcal{M}_{0}$ is
nonempty ([RSZ21, Remark 3.7]). Then $\mathcal{M}_{0}$ is a Deligne-Mumford
stack, finite and étale over $\mathrm{Spec}\,\mathcal{O}_{E}$ ([How12,
Proposition 3.1.2]). Moreover, we choose a $1$ dimensional $F$ vector space
$W_{0}$ such that $W_{0}$ has a $\mathcal{O}_{F}$ lattice $\Lambda_{0}$ with a
nondegenerate alternating form $\langle,\rangle_{0}$ satisfying

  1. (1)

$\langle ax,y\rangle_{0}=\langle x,\bar{a}y\rangle_{0}$ for all
$a\in\mathcal{O}_{F}$ and $x,y\in\Lambda_{0}$.

  2. (2)

The quadratic form $x\mapsto\langle\sqrt{\Delta}x,x\rangle_{0}$ is negative
definite.

  3. (3)

The dual lattice $\Lambda_{0}^{\vee}$ of $\Lambda_{0}$ with respect to
$\langle,\rangle_{0}$ is $\Lambda_{0}$.

Let $(,)_{0}$ be the unique Hermitian form on $W_{0}$ such that
$\mathrm{tr}_{F/\mathbb{Q}}(\sqrt{\Delta}^{-1}(,)_{0})=\langle,\rangle_{0}$.
Then $(W_{0},(,)_{0})$ determines a certain similarity class $\xi$ of hermtian
forms which in turn give us an open and closed substack
$\mathcal{M}_{0}^{\xi}$ of $\mathcal{M}_{0}$ (see [RSZ20, Lemma 3.4]). Define
the $F$-vector space

(12.8) $V=\mathrm{Hom}_{F}(W_{0},W)$

with the Hermitian form $(,)_{V}$ determined by $(,)$ and $(,)_{0}$ via

(12.9) $(x(a),y(b))=(x,y)_{V}(a,b)_{0},\forall x,y\in V,\forall a,b\in W_{0}.$

The lattice

(12.10) $L\coloneqq\mathrm{Hom}_{\mathcal{O}_{F}}(\Lambda_{0},\Lambda)\subset
V$

is a self dual lattice with respect to the Hermitian form $(,)_{V}$.

We define the functor $\mathcal{M}$ on the category of locally Notherian
schemes over $\mathrm{Spec}\,\mathcal{O}_{E}$ as follows. For a scheme $S$ in
this category, $\mathcal{M}(S)$ is the groupoid of tuples
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F})$ where

  * •

$(A_{0},\iota_{0},\lambda_{0})$ is an object of $\mathcal{M}_{0}^{\xi}(S)$.

  * •

$A$ is an Abelian scheme over $S$.

  * •

$\iota:\mathcal{O}_{F}\rightarrow\mathrm{End}(A)$ is an
$\mathcal{O}_{F}$-action satisfying the Kottwitz signature condition

$\mathrm{charpol}(\iota(a)\mid\mathrm{Lie}\,A)=(T-\sigma_{0}(a))(T-\bar{\sigma}_{0}(a))^{n-1}\prod_{\sigma\in\Sigma\backslash\\{\sigma_{0}\\}}(T-\bar{\sigma}(a))^{n},\
\forall a\in\mathcal{O}_{F}.$

  * •

$\lambda:A\rightarrow A^{\vee}$ is a principal polarization whose associated
Rosati involution induces on $\mathcal{O}_{F}$ via $\iota$ the nontrivial
Galois automorphism of $F/F_{0}$.

  * •

$\mathcal{F}$ is a direct summand $\mathcal{O}_{S}$-submodule of
$\mathrm{Lie}\,A$ which is stable under the $\mathcal{O}_{F}$-action. Moreover
$\mathcal{O}_{F}$ acts on $\mathcal{F}$ by the structural morphism and on
$\mathrm{Lie}\,A/\mathcal{F}$ by the Galois conjugate of the structural
morphism.

We assume further that the tuple
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda)$ satisfies the sign condition,
the Krämer condition and the Eisenstein condition, all of which are defined
with respect to the signature
$((1,n-1)_{\\{\sigma_{0}\\}},(0,n)_{\Sigma\backslash\\{\sigma_{0}\\}})$ above.
(i) The sign condition. Let $s$ be a geometric point of $S$. Let
$(A_{0,s},\iota_{0,s},\lambda_{0,s},A_{s},\iota_{s},\lambda_{s},\mathcal{F}_{s})$
be the pull back of
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F})\in\mathcal{M}(S)$
to $s$. For every nonsplit place $v$ of $F_{0}$, we impose the condition

(12.11)
$\chi_{v}(\mathrm{inv}_{v}^{r}(A_{0,s},\iota_{0,s},\lambda_{0,s},A_{s},\iota_{s},\lambda_{s}))=\chi_{v}(V_{v}).$

We refer to [RSZ20, Appedix A] for the definition of

$\mathrm{inv}_{v}^{r}(A_{0,s},\iota_{0,s},\lambda_{0,s},A_{s},\iota_{s},\lambda_{s}).$

(ii) In order to introduce the Krämer condition and Eisenstein condition which
are only needed when $S$ has non-empty special fibers in certain
characteristics, we need more notations. We temporarily fix a finite prime $p$
of $\mathbb{Q}$. Fix an embedding
$\tilde{\nu}:\bar{\mathbb{Q}}\rightarrow\bar{\mathbb{Q}}_{p}$. This determines
a $p$-adic place $\nu$ of $E$. $\tilde{\nu}$ induces an identification

$\mathrm{Hom}_{\mathbb{Q}}(F,\bar{\mathbb{Q}})\xrightarrow{\sim}\mathrm{Hom}_{\mathbb{Q}}(F,\bar{\mathbb{Q}}_{p}):\sigma\mapsto\tilde{\nu}\circ\sigma.$

Let $\mathcal{V}_{p}(F)$ be the set of places of $F$ over $p$. For each
$w\in\mathcal{V}_{p}(F)$, define

(12.12)
$\mathrm{Hom}_{w}(F,\bar{\mathbb{Q}})\coloneqq\\{\sigma\in\mathrm{Hom}_{\mathbb{Q}}(F,\bar{\mathbb{Q}})\mid\tilde{\nu}\circ\sigma\text{
induces }w\\}.$

For each $w$, let $F^{t}_{w}$ be the maximal unramified extension of
$\mathbb{Q}_{p}$ in $F_{w}$. For
$\tau\in\mathrm{Hom}_{\mathbb{Q}_{p}}(F^{t}_{w},\bar{\mathbb{Q}}_{p})$, define

(12.13)
$\mathrm{Hom}_{w,\tau}(F,\bar{\mathbb{Q}})\coloneqq\\{\sigma\in\mathrm{Hom}_{w}(F,\bar{\mathbb{Q}})\mid\tilde{\nu}\circ\sigma|_{F^{t}_{w}}=\tau\\}.$

The definition of the above two sets depends on the choice of $\tilde{\nu}$ in
general but the partition of $\mathrm{Hom}_{\mathbb{Q}}(F,\bar{\mathbb{Q}})$
into unions of $\mathrm{Hom}_{w,\tau}(F,\bar{\mathbb{Q}})$ does not.

We make a base change and assume that $S$ is a scheme over
$\mathrm{Spec}\,\mathcal{O}_{E,\nu}$ where $\mathcal{O}_{E,\nu}$ is the
completion of $\mathcal{O}_{E}$ with respect to the $\nu$-adic topology. Then
the $\mathcal{O}_{F}$ action on $A$ induces an action of

$\mathcal{O}_{F}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}\cong\prod_{w\in\mathcal{V}_{p}(F)}\mathcal{O}_{F,w}$

on $\mathrm{Lie}\,A$. Hence we have decompositions

(12.14) $\mathrm{Lie}\,A=\bigoplus_{w}\mathrm{Lie}_{w}A,\ \hbox{ and
}\,\mathcal{F}=\bigoplus_{w}\mathcal{F}_{w}.$

For each $w$, the $\mathcal{O}_{F^{t}_{w}}$-action on $\mathrm{Lie}_{w}A$
induces decompositions

(12.15)
$\mathrm{Lie}_{w}A=\bigoplus_{\tau\in\mathrm{Hom}_{\mathbb{Q}_{p}}(F^{t}_{w},\bar{\mathbb{Q}}_{p})}\mathrm{Lie}_{w,\tau}A,\
\hbox{ and
}\,\mathcal{F}_{w}=\bigoplus_{\tau\in\mathrm{Hom}_{\mathbb{Q}_{p}}(F^{t}_{w},\bar{\mathbb{Q}}_{p})}\mathcal{F}_{w,\tau}.$

Here we make a further base change to
$\mathrm{Spec}\,\mathcal{O}_{\breve{E}_{\nu}}$ where $\breve{E}_{\nu}$ is the
completion of the maximal unramified extension of $E$ in
$\bar{\mathbb{Q}}_{p}$.

(iii) The Krämer condition. Assume that $w$ is a finite place of $F$ that is
ramified over $F_{0}$. We further assume that the underlying place of $w$ in
$\mathbb{Q}$ is $p$ and we make a base change so that $S$ is a
$\mathrm{Spec}\,\mathbb{Z}_{p}$-scheme. The Krämer condition is only needed
when $S$ has nonempty special fiber over $\mathrm{Spec}\,\mathbb{F}_{p}$. By
our assumption, the underlying place $v$ of $F_{0}$ is unramified over
$\mathbb{Q}$. Hence $F^{t}_{w}=F_{0,v}$ and
$\mathrm{Hom}_{w,\tau}(F,\bar{\mathbb{Q}})=\\{\sigma_{\tau},\bar{\sigma}_{\tau}\\}$
for all $\tau\in\mathrm{Hom}_{\mathbb{Q}_{p}}(F^{t}_{w},\bar{\mathbb{Q}}_{p})$
where we assume $\sigma_{\tau}\in\Sigma$. We impose the Krämer condition of
[Krä03]:

(12.16)
$\mathrm{rank}_{\mathcal{O}_{S}}\mathcal{F}_{w,\tau}=r_{\sigma_{\tau}},\
\forall\tau\in\mathrm{Hom}_{\mathbb{Q}_{p}}(F^{t}_{w},\bar{\mathbb{Q}}_{p}).$

(iv) The Eisenstein condition. Assume that $w$ is a finite place of $F$ whose
underlying place $v$ in $F_{0}$ is ramified over $\mathbb{Q}$. By our
assumption $w$ is unramified over $v$. Again assume that the underlying place
of $w$ in $\mathbb{Q}$ is $p$ and we make a base change so that $S$ is a
$\mathrm{Spec}\,\mathbb{Z}_{p}$-scheme. Decompose $\mathrm{Lie}\,A$ as in
(12.14) and (12.15). The Eisenstein condition is a set of conditions on
$\mathrm{Lie}_{w,\tau}A$ and is only needed when $S$ has nonempty special
fiber over $\mathrm{Spec}\,\mathbb{F}_{p}$. We do not describe the condition
in detail but instead refer to [RSZ21, Section 5.2, case (1) and (2)].

Finally a morphism between two objects
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F})$ and
$(A^{\prime}_{0},\iota^{\prime}_{0},\lambda^{\prime}_{0},A^{\prime},\iota^{\prime},\lambda^{\prime},\mathcal{F}^{\prime})$
is a morphism from $(A_{0},\iota_{0},\lambda_{0})$ to
$(A^{\prime}_{0},\iota^{\prime}_{0},\lambda^{\prime}_{0})$ in
$\mathcal{M}_{0}^{\xi}(S)$ together with an $\mathcal{O}_{F}$-linear
isomorphism
$(A,\iota,\lambda)\rightarrow(A^{\prime},\iota^{\prime},\lambda^{\prime})$
that pulls $\lambda^{\prime}$ back to $\lambda$ and pulls
$\mathcal{F}^{\prime}$ back to $\mathcal{F}$.

######  Proposition 12.2.

$\mathcal{M}$ is a Deligne-Mumford stack flat over $\mathcal{O}_{E}$. And

$\mathcal{M}\times_{\mathrm{Spec}\,\mathcal{O}_{E}}\mathbb{C}=S(\tilde{G},h_{\tilde{G}})_{K}.$

Let $\mathcal{V}_{\mathrm{ram}}(E)$ be the set of finite places of $E$ over
$\mathcal{V}_{\mathrm{ram}}$. Moreover we have:

(i) $\mathcal{M}$ is smooth of relative dimension $n-1$ over the open
subscheme of $\mathrm{Spec}\,\mathcal{O}_{E}$ obtained by removing the set
$\mathcal{V}_{\mathrm{ram}}(E)$. If $n=1$, then $\mathcal{M}$ is finite étale
over all of $\mathrm{Spec}\,\mathcal{O}_{E}$.

(ii) If $n\geq 2$ and $\nu\in\mathcal{V}_{\mathrm{ram}}(E)$ is unramified over
$F$, then $\mathcal{M}$ is flat and has semi-stable reduction over
$\mathrm{Spec}\,\mathcal{O}_{E,\nu}$.

###### Proof.

The proposition in fact summarizes [RSZ21, Theorem 3.5, 4.4 and 6.7]. The only
new input here is the fact that $\mathcal{M}$ is the blow-up of the integral
model considered by loc.cit. along its singularities. This follows from the
corresponding fact on local models, see for example [Shi20, Appendix]. ∎

From now on, in addition to the assumption that every
$v\in\mathcal{V}_{\mathrm{ram}}$ is unramified over $\mathbb{Q}$ and does not
divide $2$, we also assume that every $\sigma\in\mathcal{V}_{\mathrm{ram}}(E)$
is unramified over $F$ so that $\mathcal{M}$ is regular.

###  13\. Incoherent Eisenstein series

####  13.1. Eisenstein Series on unitary groups

In this section we assume that $W_{n}$ is the $F/F_{0}$ skew-Hermitian space
defined by the Hermitian matrix $w_{n}=\begin{pmatrix}0&I_{n}\\\
-I_{n}&0\end{pmatrix}$. Let $H=\mathrm{U}(W_{n})$. Let $P=MN$ be its standard
Siegel Parabolic subgroup where

$M(R)=\left\\{m(a)=\begin{pmatrix}a&0\\\ 0&{}^{t}\bar{a}^{-1}\end{pmatrix}\mid
a\in\mathrm{GL}_{n}(R\otimes_{\mathcal{O}_{F_{0}}}\mathcal{O}_{F})\right\\},$
$N(R)=\left\\{n(b)=\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}\mid
b\in\mathrm{Herm}_{n}(R)\right\\}$

for an $\mathcal{O}_{F_{0}}$-algebra $R$.

Let
$\chi:F_{0}^{\times}\backslash\mathbb{A}_{F_{0}}^{\times}\rightarrow\mathbb{C}^{\times}$
be the quadratic character associated to $F/F_{0}$. Fix a character
$\eta:\mathbb{A}_{F}^{\times}\rightarrow\mathbb{C}^{\times}$ such that
$\eta|_{\mathbb{A}_{F_{0}}^{\times}}=\chi^{n}$. Define the degenerate
principal series $I(s,\eta)$ to be the smooth induction

$I(s,\eta)=\mathrm{Ind}_{P(\mathbb{A}_{F_{0}})}^{H(\mathbb{A}_{F_{0}})}(\eta\cdot|\cdot|_{\mathbb{A}_{F}}^{s+\frac{n}{2}}).$

It consists of smooth functions $\Phi$ on $H(\mathbb{A}_{F_{0}})$ satisfying

$\Phi(n(b)m(a)g,s)=\eta(\det a)|\det
a|_{\mathbb{A}_{F}}^{s+\frac{n}{2}}\Phi(g,s).$

For such a section $\Phi$, define the Eisenstein series

$E(g,s,\Phi)=\sum_{\gamma\in P(F_{0})\backslash H(F_{0})}\Phi(\gamma g,s),g\in
H(\mathbb{A}_{F_{0}}).$

It converges when $\hbox{Re}(s)\gg 0$ and has a meromorphic continuation to
$s\in\mathbb{C}$. Choose an additive character
$\psi:F_{0}\backslash\mathbb{A}_{F_{0}}\rightarrow\mathbb{C}^{\times}$ such
that $\psi$ is unramified outside the set of finite places of $F_{0}$ split in
$F$. An argument like that of [LZ19, Footnote 4] shows the existence of such
$\psi$. Then we have a Fourier expansion

$E(g,s,\Phi)=\sum_{T\in\mathrm{Herm}_{n}(F_{0})}E_{T}(g,s,\Phi)$

where

$E_{T}(g,s,\Phi)=\int_{N(F_{0})\backslash
N(\mathbb{A}_{F_{0}})}E(n(b)g,s,\Phi)\psi(-\mathrm{tr}(Tb))db$

Here $db$ is the self-dual Haar measure on $N(\mathbb{A}_{F_{0}})$ with
respect to the pairing $\langle
n(b_{1}),n(b_{2})\rangle=\psi(\mathrm{tr}(b_{1},b_{2}))$. When $T$ is non-
singular and $\Phi=\otimes_{v}\Phi_{v}$ is factorizable, we have the product
expansion which is convergent when $\hbox{Re}(s)\gg 0$

(13.1) $E_{T}(g,s,\Phi)=\prod_{v}W_{T,v}(g_{v},s,\Phi_{v})$

where the local Whittaker function is defined by

(13.2)
$W_{T,v}(g_{v},s,\Phi_{v})=\int_{\mathrm{Herm}_{n}(F_{0,v})}\Phi_{v}(w_{n}^{-1}n(b)g,s)\psi_{v}(-\mathrm{tr}(Tb))db,$

where $db$ is the Haar measure on $\mathrm{Herm}_{n}(F_{0,v})$ that is self-
dual with respect to the pairing $\psi_{v}(\mathrm{tr}((\cdot,\cdot)))$.

####  13.2. Weil representation and degenerate principal series

Assume that $V_{v}$ is a $F_{v}/F_{0,v}$-Hermitian space of dimension $n$. The
fixed choice $\eta$ and $\psi$ gives a Weil representation $\omega_{v}$ of
$\mathrm{U}(V_{v})\times H(F_{0,v})$ on the Bruhat-Schwartz space
$\mathcal{S}(V_{v}^{n})$. Explicitly for $\varphi\in\mathcal{S}(V_{v}^{n})$
and ${\bf x}\in V_{v}^{n}$ we have

(13.3) $\displaystyle\omega_{v}(m(a))\varphi({\bf x})=$
$\displaystyle\eta_{v}(\det a)|\det a|_{F_{v}}^{\frac{n}{2}}\varphi({\bf
x}\cdot a),$ $\displaystyle m(a)\in M(F_{0,v})$ (13.4)
$\displaystyle\omega_{v}(n(b))\varphi({\bf x})=$
$\displaystyle\psi_{v}(\mathrm{tr}(b({\bf x},{\bf x})))\varphi({\bf x}),$
$\displaystyle n(b)\in N(F_{0,v})$ $\displaystyle\omega_{v}(w_{n})\varphi({\bf
x})=$ $\displaystyle\gamma(V_{v})^{n}\cdot\hat{\varphi}({\bf x}),$
$\displaystyle\omega_{v}(g)\varphi({\bf x})=$
$\displaystyle\varphi(g^{-1}\cdot{\bf x}),$ $\displaystyle g\in\mathrm{U}(V).$

Here $\hat{\varphi}$ is the Fourier transform of $\varphi$ with respect to
$\psi_{v}$ and $\gamma(V_{v})$ is an eighth root of unity called the Weil
index. By [KR14b, Equation (10.3)], we have

(13.5) $\gamma(V_{v})=(\Delta,\det
V_{v})_{v}\gamma_{v}(-\Delta,\psi_{v,\frac{1}{2}})^{n}\gamma_{v}(-1,\psi_{v,\frac{1}{2}})^{-n}.$

Here $(,)_{v}$ is the local Hilbert symbol and
$\gamma_{v}(a,\psi_{v,\frac{1}{2}})$ is the Weil index with respect to the
additive character $\psi_{v,\frac{1}{2}}$ given by
$\psi_{v,\frac{1}{2}}(x)=\psi_{v}(\frac{1}{2}x)$, see for example [Kud94] and
[Kud96, Chapter I.4]. Given a Schwartz function
$\phi\in\mathcal{S}(V_{v}^{n})$, we associated a “Siegel-Weil section”
(standard section) as follows:

(13.6) $\lambda(V_{v}):\mathcal{S}(V_{v}^{n})\rightarrow
I_{v}(s,\eta_{v}),\quad\varphi\mapsto\lambda(V_{v})(\varphi)(g,s)\coloneqq|\det
a|_{F_{v}}^{s}\cdot\omega(g)\varphi(0).$

The map is $H(F_{0,v})$-equivariant for $s=0$. Analogously for a
$\mathbb{A}_{F}/\mathbb{A}_{F_{0}}$-Hermitian space $\mathcal{C}$ of rank $n$,
$\mathrm{U}(\mathcal{C})\times H(\mathbb{A}_{F_{0}})$ acts on the Bruhat-
Schwartz space $\mathcal{S}(\mathcal{C}^{n})$ by the Weil representation
$\omega$ and we have a map

$\lambda(\mathcal{C}):\mathcal{S}(\mathcal{C}^{n})\rightarrow I(s,\eta),$

which is $H(\mathbb{A}_{F_{0}})$-equivariant for $s=0$. So (13.2) and (13.3)
give us

(13.7) $\displaystyle
W_{T,v}(g,0,\lambda(V_{v})(\varphi))=\gamma(V_{v})^{n}\int_{\mathrm{Herm}_{n}(F_{0,v})}\int_{V_{v}^{n}}\omega_{v}(g)\varphi({\bf
x})\psi_{v}(\mathrm{tr}(b({\bf x},{\bf
x})))\cdot\psi_{v}(-\mathrm{tr}(bT))d{\bf x}db,$

where $db$ is the measure as in (13.2) and $d{\bf x}$ is the Haar measure on
$V_{v}^{n}$ self-dual with respect to the pairing $\langle{\bf
x},\mathbf{y}\rangle=\psi_{v}(\mathrm{tr}_{F_{v}/F_{0,v}}(\mathrm{tr}({\bf
x},\mathbf{y})))$.

####  13.3. Equivalent form of Conjecture 1.1

Now assume $v\in\mathcal{V}_{\mathrm{ram}}$, let $q$ be the size of the
residue field of $F_{v}$. For $\epsilon=\pm 1$, let $V_{v,n,\epsilon}$ be the
(unique up to isomorphism) $F_{v}/F_{0,v}$-Hermitian space of dimension $n$
and sign $\epsilon$ and $I_{v,n,\epsilon}$ be a self-dual
$\mathcal{O}_{F,v}/\mathcal{O}_{F_{0},v}$-Hermitian lattice in it. We also use
the same notation to denote a Gram matrix of $I_{v,n,\epsilon}$. Let
$\mathcal{H}_{v}$ be the $\mathcal{O}_{F,v}/\mathcal{O}_{F_{0},v}$-Hermitian
lattice of rank $2$ with Gram matrix
$\begin{pmatrix}0&\frac{1}{\sqrt{\Delta}}\\\
-\frac{1}{\sqrt{\Delta}}&0\end{pmatrix}$. As in (1.7), define

(13.8) $\mathcal{H}_{v,\epsilon}^{n,i}=\mathcal{H}_{v}^{i}\obot
I_{v,n-2i,\epsilon}\subset V_{v,n,\epsilon}.$

When $n=2r$ is even we take $\mathcal{H}_{v,1}^{n,r}=\mathcal{H}_{v}^{r}$. Let
$\mathds{1}_{v,\epsilon}^{n,i}\in\mathcal{S}((V_{v,n,\epsilon})^{n})$ be the
characteristic function of $(\mathcal{H}_{v,\epsilon}^{n,i})^{n}$ and we
simply write $\mathds{1}_{v,\epsilon}^{n}$ for
$\mathds{1}_{v,\epsilon}^{n,0}$.

For $T\in\mathrm{Herm}_{n}(F_{0,v})$, choose a lattice $L_{T}$ in the
Hermitian space $V_{v,n,\epsilon}$ with Gram matrix $T$. By [Shi20,
Proposition 9.7], we have

(13.9) $\displaystyle
W_{T,v}(1,r,\lambda_{v}(V_{v,n,\epsilon})(\mathds{1}_{v,\epsilon}^{n,i}))=\gamma_{v}(V_{v,n,\epsilon})^{n}\cdot|\det(\mathcal{H}_{v,\epsilon}^{n,i})|_{F_{v}}^{n}\cdot|\Delta|_{F_{v}}^{e}\cdot\alpha_{v}(\mathcal{H}^{n,i}_{v,\epsilon},L_{T},X)|_{X=q^{-2r}},$

where $e=\frac{1}{2}n^{2}+\frac{1}{4}n(n-1)$ and $\alpha_{v}(\cdot,\cdot,X)$
is the local density polynomial defined in (5.1) at the place $v$. Notice that
by (13.5), we know

(13.10) $\gamma_{v}(V_{v,n,\epsilon})=-\gamma_{v}(V_{v,n,-\epsilon}).$

Choose a function $A_{v,\epsilon}^{n,i}(s)$ of $s$ such that

(13.11)
$A_{v,\epsilon}^{n,i}(0)=0,\quad\frac{d}{ds}A_{v,\epsilon}^{n,i}|_{s=0}=\frac{(-1)^{n}}{q^{2i}}\cdot
c_{\epsilon}^{n,i}\cdot\log q.$

where $c_{\epsilon}^{n,i}$ is as in (1.8) for the place $v$. In particular
$c_{\epsilon}^{n,i}$ is determined in Theorem 6.1. One such choice of
$A_{v,\epsilon}^{n,i}(s)$ is simply

$A_{v,\epsilon}^{n,i}(s)=\frac{(-1)^{n}}{2q^{2i}}\cdot
c_{\epsilon}^{n,i}\cdot(q^{s}-q^{-s}).$

We now define a section $\Phi_{v,n,\epsilon}\in I_{v}(s,\chi_{v})$ by

(13.12)
$\Phi_{v,n,\epsilon}=\lambda(V_{v,n,\epsilon})(\mathds{1}_{v,\epsilon}^{n})+\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}A_{v,-\epsilon}^{n,i}(s)\cdot\lambda(V_{v,n,-\epsilon})(\mathds{1}_{v,-\epsilon}^{n,i}).$

Then equation (13.9) tells us that Conjecture 1.1 at the place $v$ is
equivalent to the following conjecture.

######  Conjecture 13.1.

Let $\mathbb{V}_{v}$ be the space of special quasi-homomorphisms as in (1.2)
such that $\chi_{v}(\mathbb{V}_{v})=-\epsilon$. Let $\mathbb{L}_{v}$ be a
lattice of rank $n$ in $\mathbb{V}_{v}$ with Gram matrix $T$ and
$S=I_{v,n,\epsilon}$. Then

$\mathrm{Int}(\mathbb{L}_{v})\cdot\log
q=\frac{W^{\prime}_{T,v}(1,0,\Phi_{v,n,\epsilon})}{W_{S,v}(1,0,\Phi_{v,n,\epsilon})}.$

####  13.4. Incoherent Eisenstein series associated to the Shimura data

Now assume we are in the situation of Section 12. In particular $V$ is as in
(12.8) and $L$ is as in (12.10). We identify the archimedean places of $F_{0}$
with $\Sigma_{0}$. Let $\mathcal{C}$ be the
$\mathbb{A}_{F}/\mathbb{A}_{F_{0}}$-Hermitian space of rank $n$ such that
$\mathcal{C}_{v}=V_{v}$ for $v\neq\tau_{0}$ and $\mathcal{C}_{\tau_{0}}$ has
signature $(n,0)$. Then by the Hasse principal, $V$ is incoherent in the sense
that it does not come from a $F/F_{0}$-Hermitian space.

Define $\mathds{1}_{L}\in\mathcal{S}(\mathcal{C}^{n})$ to be the
characteristic function of $L^{n}\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}$. In
particular $\mathds{1}_{L,v}$ is the characteristic function of
$L^{n}\otimes_{\mathcal{O}_{F_{0}}}\mathcal{O}_{F_{0},v}$. We now define a
section $\Phi(g,s)=\otimes_{v}\Phi_{v}\in I(s,\chi)$ by

  1. (1)

If $v$ is finite and unramified in $F$, define
$\Phi_{v}=\lambda(\mathcal{C}_{v})(\mathds{1}_{L,v})$.

  2. (2)

If $v\in\mathcal{V}_{\mathrm{ram}}$ , let
$\epsilon=\chi(\mathcal{C}_{v})=\chi(V_{v})$. Define
$\Phi_{v}=\Phi_{v,n,\epsilon}$ as in (13.12).

  3. (3)

If $v$ is archimedean, take $\Phi_{v}=\lambda(\mathcal{C}_{v})(\varphi_{v})$
where $\varphi_{v}$ is the Gaussian function

$\varphi_{v}({\bf x})=\psi_{v}(i\cdot\mathrm{tr}(({\bf x},{\bf x}))).$

We then associate the Eisenstein series $E(g,s,\Phi)$.

Let $\mathbb{H}_{n}$ be the symmetric domain of $\mathrm{U}(W_{n})$. It is the
Hermitian upper half space

$\displaystyle\mathbb{H}_{n}=\\{z=x+iy\mid
x\in\mathrm{Herm}_{n}(F_{0,\infty}),y\in\mathrm{Herm}_{n}(F_{0,\infty})_{>0}\\}.$

where $F_{0,\infty}=\prod_{v\in\Sigma_{0}}F_{0,v}$ and
$\mathrm{Herm}_{n}(F_{0,\infty})_{>0}$ is the set of totally positive definite
matrices in $\mathrm{Herm}_{n}(F_{0,\infty})$. Define the classical Eisenstein
series $E(z,s,\Phi)$ by (see [KR14b, Equation (9.1)])

(13.13) $E(z,s,\Phi)=\eta_{\infty}(\det a)^{-1}(\det
y)^{-\frac{n}{2}}E(g_{z},s,\Phi),\ g_{z}\coloneqq n(x)m(a)\in
H(\mathbb{A}_{F_{0}})$

where $a\in\mathrm{GL}_{n}(F\otimes F_{0,\infty})$ such that
$y=a{}^{t}\bar{a}$. Notice that $E(z,s,\Phi)$ is independent of the choice of
$\chi$. Then we have a Fourier expansion

$E(z,s,\Phi)=\sum_{T\in\mathrm{Herm}_{n}(F_{0})}E_{T}(z,s,\Phi)$

as well. We define the following central derivatives

(13.14) $E^{\prime}(g,\Phi)\coloneqq\frac{d}{ds}E(g,s,\Phi)|_{s=0},\ \text{
and }\,E^{\prime}_{T}(z,\Phi)=\frac{d}{ds}E_{T}(z,s,\Phi)|_{s=0}.$

For an archimedean place $v$ and nonsingular $T\in\mathrm{Herm}_{n}(F_{0})$,
define

(13.15) $\displaystyle W_{T,v}(z_{v},s,\Phi)=\eta_{v}(\det a_{v})^{-1}(\det
y_{v})^{-\frac{n}{2}}W_{T,v}(n(x_{v})m(a_{v}),s,\Phi),\ n(u_{v})m(a_{v})\in
H(F_{0,v}).$

Then we have

(13.16)
$E_{T}(z,s,\Phi)=\prod_{v\nmid\infty}W_{T,v}(1,s,\Phi_{v})\prod_{v\mid\infty}W_{T,v}(z_{v},s,\Phi_{v}),$

and

$W_{T,\infty}(z,0,\Phi)=c_{\infty}q^{T},\
q^{T}\coloneqq\psi_{\infty}(\mathrm{tr}(Tz))$

for some constant $c_{\infty}$, see Lemma 15.1 below.

For a nonsingular $T\in\mathrm{Herm}_{n}(F_{0})$, let $V_{T}$ be the
$F/F_{0}$-Hermitian space represented by the Gram matrix $T$. Define

(13.17) $\mathrm{Diff}(T,\mathcal{C})=\\{v\text{ a place of
}F_{0}\mid(V_{T})_{v}\text{ is not isometric to }\mathcal{C}_{v}\\}.$

In particular any $v\in\mathrm{Diff}(T,\mathcal{C})$ must be nonsplit in $F$.
Since $\mathcal{C}$ is incoherent, by the Hasse principal
$|\mathrm{Diff}(T,\mathcal{C})|$ is an odd integer. The following lemma is
standard and well-known, c.f. [Kud03].

######  Lemma 13.2.

We have $E(z,0,\Phi)=0$. Moreover

  1. (1)

$E^{\prime}_{T}(z,\Phi)=0$ if $|\mathrm{Diff}(T,\mathcal{C})|>1$.

  2. (2)

If $\mathrm{Diff}(T,\mathcal{C})=\\{v\\}$ and $v$ is finite, then

$\displaystyle E^{\prime}_{T}(z,\Phi)=E^{\prime}_{T,v}(z,\Phi)\coloneqq
W^{\prime}_{T,v}(1,0,\Phi_{v})(\prod_{v^{\prime}\nmid\infty,v^{\prime}\neq
v}W_{T,v^{\prime}}(1,s,\Phi_{v^{\prime}})\prod_{v^{\prime}\mid\infty}W_{T,v^{\prime}}(z_{v^{\prime}},s,\Phi_{v^{\prime}}))|_{s=0}.$

  3. (3)

If $\mathrm{Diff}(T,\mathcal{C})=\\{v\\}$ and $v$ is archimedean, then

$\displaystyle E^{\prime}_{T}(z,\Phi)=E^{\prime}_{T,v}(z,\Phi)\coloneqq
W^{\prime}_{T,v}(z_{v},0,\Phi_{v})(\prod_{v^{\prime}\nmid\infty}W_{T,v^{\prime}}(1,s,\Phi_{v^{\prime}})\prod_{v^{\prime}\mid\infty,v^{\prime}\neq
v}W_{T,v^{\prime}}(z_{v^{\prime}},s,\Phi_{v^{\prime}}))|_{s=0}.$

###  14\. Special cycles

####  14.1. Special cycles in the Chow group

Let $\mathrm{Herm}_{m}(\mathcal{O}_{F_{0}})_{\geq 0}$ (resp.
$\mathrm{Herm}_{m}(\mathcal{O}_{F_{0}})_{>0}$) be the subset of totally (i.e.
for all archimedean places) positive semi-definite (resp. definite) matrices
in $\mathrm{Herm}_{m}(\mathcal{O}_{F_{0}})$. We define special cycles as in
[KR14b] and [RSZ21]. For a locally Notherian scheme $S$ over
$\mathrm{Spec}\,\mathcal{O}_{E}$ and
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F})\in\mathcal{M}(S)$,
we have the finite rank locally free $\mathcal{O}_{F}$-module

$\mathbb{L}(A_{0},A)\coloneqq\mathrm{Hom}_{\mathcal{O}_{F}}(A_{0},A).$

We can define a Hermitian form $h^{\prime}$ on $\mathbb{L}(A_{0},A)$ by

(14.1) $h^{\prime}(x,y)=\iota_{0}^{-1}(\lambda_{0}^{-1}\circ
y^{\vee}\circ\lambda\circ x)\in\mathcal{O}_{F}.$

######  Definition 14.1.

For $T\in\mathrm{Herm}_{m}(\mathcal{O}_{F})_{\geq 0}$, the special cycle
$\mathcal{Z}(T)$ is the stack of tuples
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F},{\bf x})$ such that
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F})\in\mathcal{M}(S)$
and ${\bf x}=(x_{1},\ldots,x_{m})\in\mathbb{L}(A_{0},A)^{m}$ such that

$h^{\prime}({\bf x},{\bf x})=(h^{\prime}(x_{i},x_{j}))=T.$

If $T\in\mathrm{Herm}_{m}(\mathcal{O}_{F})$ is not positive semi-definite, we
define $\mathcal{Z}(T)$ to be empty.

[KR14b, Proposition 2.9] generalizes to our case and shows that the natural
map $\mathcal{Z}(T)\rightarrow\mathcal{M}$ is finite and unramified. For $t\in
F_{0}$ totally positive, the same proof as [How15, Proposition 3.2.3] shows
that $\mathcal{Z}(t)$ is a Cartier divisor. In general $\mathcal{Z}(T)$ may
not have the expected dimension, hence we make the following definition as in
the local situation. Recall from [Gil09, Theorem 2.8] that
$\mathrm{Gr}^{m}(K_{0}(\mathcal{M}))_{\mathbb{Q}}\cong\mathrm{CH}^{m}(\mathcal{M})_{\mathbb{Q}}$.

######  Definition 14.2.

Assume $T\in\mathrm{Herm}_{m}(\mathcal{O}_{F})_{>0}$ and has diagonal entries
$\\{t_{1},\ldots,t_{m}\\}$. Define

$\mathcal{Z}^{\mathbb{L}}(T)=[(\mathcal{O}_{\mathcal{Z}(t_{1})}\otimes^{\mathbb{L}}\cdots\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(t_{m})})_{\mathcal{Z}(T)}]\in\mathrm{Gr}^{m}(K_{0}(\mathcal{M}))_{\mathbb{Q}}\cong\mathrm{CH}^{m}(\mathcal{M})_{\mathbb{Q}}.$

Here $\otimes^{\mathbb{L}}$ is the derived tensor product and we take the part
that is supported on $\mathcal{Z}(T)$.

####  14.2. Green currents

Under the isomorphism (12.4) we may identify the
$\tilde{G}(\mathbb{R})$-conjugacy class $h_{\tilde{G}}$ as

$\\{h_{Z^{\mathbb{Q}}}\\}\times\prod_{\tau\in\Sigma_{0}}h_{G,\tau}$

each of the factor above is a singleton except for $h_{G,\tau_{0}}$ which is
the Hermitian symmetric domain

$D=\mathrm{U}(n-1,1)/(\mathrm{U}(n-1)\times\mathrm{U}(1)).$

We may identify $D$ with the open subset of negative $\mathbb{C}$-lines in
$\mathbb{P}^{1}(V_{\tau_{0}})$. For a vector $x\in
V_{\tau_{0}}\setminus\\{0\\}$ and $z\in D$, write

$x=x_{z}+x_{z^{\bot}},\ x_{z}\in z,x_{z^{\bot}}\in z^{\bot},$

for the orthogonal decomposition with respect to $z$. Define

$R(x,z)=-(x_{z},x_{z})$

and

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

Notice that by the signature of $V_{\tau_{0}}$, $D(x)$ is nonempty if and only
if $(x,x)>0$, in which case $D(x)$ is a divisor. Define Kudla’s green function
(see [Kud97]) by

$g(x,z)=-\mathrm{Ei}(-2\pi R(x,z))$

where $\mathrm{Ei}$ is the exponential integral

$\mathrm{Ei}(u)=\int_{-\infty}^{u}\frac{e^{u}}{u}du$

where the path of integration lies in the complex plan cut along the line
$\mathrm{Re}(u)>0$. Then $g(x,\cdot)$ is a smooth function on $D\setminus
D(x)$ with logarithmic singularities along $D(x)$. By [Liu11, Proposition
4.9], it satisfies the Green current relation for $D(x)$

(14.3) $dd^{c}[g(x,z)]+\delta_{D(x)}=[\omega(x,z)]$

where $\omega(x,z)=\exp(2\pi(x,x))\varphi_{KM}(x,z)$ and
$\varphi_{KM}(x,z)\in\mathcal{S}(V_{\tau_{0}})\otimes\Omega^{(1,1)}(D)$ is the
Kudla-Millson Schwartz form defined in [KM90].

For ${\bf x}=(x_{1},\ldots.x_{m})\in V^{m}_{\tau_{0}}$ with nonsingular gram
matrix $T=({\bf x},{\bf x})$, define

$D({\bf x})=D(x_{1})\cap\cdots\cap D(x_{m}),$

which is nonempty if and only if $T>0$. Define the green current $g({\bf
x},z)$ using the star product

$g({\bf x},z)\coloneqq g(x_{1},z)*\cdots*g(x_{m},z).$

Then it satisfies the Green current relation for $D({\bf x})$

(14.4) $dd^{c}[g({\bf x},z)]+\delta_{D({\bf x})}=[\omega({\bf x},z)].$

[GS19] gives another way of defining a Green current for $D({\bf x})$ which
give rises to equivalent elements in the arithmetic Chow group (see below) as
$g({\bf x},z)$.

####  14.3. Complex uniformization of special cycles

For each $\tau$ in $\Sigma_{0}$, denote by $\Sigma(E,\tau)$ the set of complex
embeddings of $E$ such that $\nu\in\Sigma(E,\tau)$ if and only if
$\nu|_{F}\in\Sigma$ and $\nu|_{F_{0}}=\tau$. For each $\tau\in\Sigma_{0}$,
define the Hermitian space $V[\tau]$ such that

  1. (1)

$(V[\tau])_{\tau}$ has signature $(n-1,1)$

  2. (2)

$(V[\tau])_{\tau^{\prime}}$ has signature $(n,0)$ for $\tau^{\prime}\neq\tau$.

  3. (3)

$(V[\tau])_{v}\cong V_{v}$ for each finite place $v$ of $F_{0}$.

Hasse principal guarantees the existence of such $V[\tau]$ and
$V[\tau_{0}]\cong V$. We fix once and for all identifications

$V[\tau]\otimes_{\mathbb{Q}}\mathbb{A}_{f}\cong
V\otimes_{\mathbb{Q}}\mathbb{A}_{f},$

which induces isomorphisms

$\mathrm{U}(V[\tau])(\mathbb{A}_{f})\cong\mathrm{U}(V)(\mathbb{A}_{f})$

and

$\tilde{G}[\tau](\mathbb{A}_{f})\cong\tilde{G}(\mathbb{A}_{f})$

where $\tilde{G}[\tau]\cong
Z^{\mathbb{Q}}\times\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V[\tau])$ as in
(12.4). In particular we can view $K$ as a subgroup of
$\tilde{G}[\tau](\mathbb{A}_{f})$ and $K_{G}$ as a subgroup of
$\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V[\tau])(\mathbb{A}_{f})$. Then we
have

(14.5)
$\mathcal{M}(E_{\infty})=\bigsqcup_{\tau\in\Sigma_{0}}\bigsqcup_{\nu\in\Sigma(E,\tau)}\mathcal{M}(E_{\nu})$

where

(14.6) $\displaystyle\mathcal{M}(E_{\nu})=$
$\displaystyle\tilde{G}[\tau](\mathbb{Q})\backslash
D\times\tilde{G}[\tau](\mathbb{A}_{f})/K,\ \text{ if }\nu\in\Sigma(E,\tau).$
$\displaystyle=$ $\displaystyle\bigsqcup_{j\in J_{\tau}}\Gamma_{j}\backslash
D.$

Here $J_{\tau}$ is a finite index set such that

(14.7) $\tilde{G}[\tau](\mathbb{A}_{f})=\bigsqcup_{j\in
J_{\tau}}\tilde{G}[\tau](\mathbb{Q})\tilde{g}_{j}K$

and

$\Gamma_{j}=\tilde{G}[\tau](\mathbb{Q})\cap\tilde{g}_{j}K\tilde{g}_{j}^{-1}.$

Assume $T\in\mathrm{Herm}_{m}(F_{0})$ is nonsingular. On each
$\mathcal{M}(E_{\nu})$ we can define the special cycle

$Z(T)(E_{\nu})=\sum_{j\in J_{\tau}}\sum_{\begin{subarray}{c}{\bf
x}\in\tilde{G}[\tau](\mathbb{Q})\backslash(V[\tau])^{m}\\\ ({\bf x},{\bf
x})=T\end{subarray}}\ \mathds{1}_{\mathbb{L}}(\tilde{g}_{j}^{-1}x)C({\bf
x})_{\nu,j}$

where $C({\bf x})_{\nu,j}$ is the image of $D({\bf x})$ under the quotient map
$D\rightarrow\Gamma_{j}\backslash D$. Then we have (see [KR14b, Proposition
3.5])

(14.8)
$\mathcal{Z}(T)(E_{\infty})=\bigsqcup_{\tau\in\Sigma_{0}}\bigsqcup_{\nu\in\Sigma(E,\tau)}Z(T)(E_{\nu})$

where $\mathcal{Z}(T)$ is defined in Definition 14.1.

####  14.4. Special cycles in the arithmetic Chow group

For $T\in\mathrm{Herm}_{n}(F_{0})$ nonsingular,
$y_{\nu}\in\mathrm{Herm}_{m}(E_{\nu})_{>0}$, and $j\in J_{\tau}$ as in (14.7),
define a Green current $g(y_{\nu},T,z)$ whose restriction to the component
$\Gamma_{j}\backslash D$ is

(14.9) $g(y_{\nu},T,z)_{j}=\sum_{\begin{subarray}{c}{\bf x}\in(V[\tau])^{m}\\\
({\bf x},{\bf
x})=T\end{subarray}}\mathds{1}_{\mathbb{L}}(\tilde{g}_{j}^{-1}{\bf x})g({\bf
x}\cdot a,z)$

where $a\in\mathrm{GL}_{m}(\mathbb{C})$ such that $y_{\nu}=a{}^{t}\bar{a}$.
Then $g(y_{\nu},T,\cdot)$ is a Green current for $Z(T)(E_{\nu})$. For
$y=(y_{\nu})_{\nu}$, define $g(y,T,\cdot)$ to be the Green current whose
$\nu$-th component is $g(y_{\nu},T,\cdot)$. Then $g(y,T,\cdot)$ is a Green
current for $\mathcal{Z}(T)(E_{\infty})$.

Now we are ready to define the element
$\widehat{Z}(T,y)\in\widehat{\mathrm{CH}}^{m}(\mathcal{M})$ for nonsingular
$T\in\mathrm{Herm}_{m}(F_{0})$. Here we follow [SABK94, Section III.2]. Define
the group of cochains

$Z^{m}_{\mathrm{fin}}(\mathcal{M})\coloneqq\\{Z\in
Z^{m}(\mathcal{M})\mid|Z|\cap\mathcal{M}_{E}=\emptyset\\}$

where $Z^{m}(\mathcal{M})$ is the formal linear combination of codimensional
$m$ substack on $\mathcal{M}$, $|Z|$ is the support of $Z$ and
$\mathcal{M}_{E}$ is the generic fiber of $E$. Define

$\mathrm{CH}_{\mathrm{fin}}^{m}(\mathcal{M})\coloneqq
Z^{m}_{\mathrm{fin}}(\mathcal{M})/\langle\mathrm{div}f\rangle$

where $f$ is a rational function on some substack of
$\mathcal{M}\setminus\mathcal{M}_{E}$ of codimension $m-1$. Define

$\widehat{Z}^{m}(\mathcal{M}_{E})=\\{(Z,g)\mid Z\in
Z^{m}(\mathcal{M}_{E}),g\text{ is a Green current for }Z\\}.$

Then we have an isomorphism

$\widehat{\mathrm{CH}}^{m}(\mathcal{M})\cong(\mathrm{CH}_{\mathrm{fin}}^{m}(\mathcal{M})\oplus\widehat{Z}^{m}(\mathcal{M}_{E}))/\langle\widehat{\mathrm{div}}f;(0,0,\mathrm{im}\partial+\mathrm{im}\bar{\partial})\rangle$

where $f$ is a rational function on some substack of $\mathcal{M}$ of
codimension $m-1$, and

$\widehat{\mathrm{div}}f=(\mathrm{div}f,\mathrm{div}f,(-\log|f|_{\nu}^{2})_{\nu}).$

Since $\mathcal{Z}(T)_{E}$ has the expected codimension which is $m$, as in
the proof of Theorem 2 in loc.cit, there is a map

(14.10)
$\mathrm{CH}_{\mathcal{Z}(T)}^{m}(\mathcal{M})_{\mathbb{Q}}\rightarrow\mathrm{CH}_{\mathrm{fin}}^{m}(\mathcal{M})_{\mathbb{Q}}\oplus
Z_{\mathcal{Z}(T)_{E}}^{m}(\mathcal{M}_{E})_{\mathbb{Q}}.$

Denote still by $\mathcal{Z}^{\mathbb{L}}(T)$ the image of
$\mathcal{Z}^{\mathbb{L}}(T)$ (Definition 14.2) under the above map, then we
obtain an element

$\widehat{\mathcal{Z}}(T,y)=[(\mathcal{Z}^{\mathbb{L}}(T),g(y,T,\cdot))]\in\widehat{\mathrm{CH}}^{m}(\mathcal{M})_{\mathbb{Q}}.$

###  15\. Arithmetic Siegel-Weil formula

####  15.1. Local Siegel-Weil formula

In this section let $G_{1}=\mathrm{U}(V)$ which can be canonically identified
with $\mathrm{U}(W)$ where $W$ is as in (12.8). Let $h$ be the moment map

$h:V^{n}\rightarrow\mathrm{Herm}_{n}(F_{0}),\ {\bf x}\mapsto({\bf x},{\bf
x}).$

Define

$V^{n}_{\mathrm{reg}}=\\{{\bf x}\in V^{n}\mid\det(h({\bf x}))\neq 0\\}.$

For any ${\bf x}\in V^{n}_{\mathrm{reg}}$ with $h({\bf x})=T$, the map

$i_{x}:G_{1}(F_{0})\rightarrow\Omega(T)=h^{-1}(T),\ g\mapsto g\cdot{\bf x}$

is an isomorphism.

In the local situation, for fixed Haar measures $d{\bf x}$ on $V_{v}^{n}$ and
$dT$ on $\mathrm{Herm}_{n}(F_{0,v})$ respectively, there exists a unique Haar
measure $\mu$ on $G_{1}(F_{0,v})$ such that for any
$\varphi\in\mathcal{S}(V_{v}^{n})$, we have

(15.1) $\int_{V_{v}^{n}}\varphi({\bf x})d{\bf
x}=\int_{\mathrm{Herm}_{n}(F_{0,v})}O_{T,v}(\varphi)dT$

where

(15.2) $O_{T,v}(\varphi)=\int_{G_{1}(F_{0,v})}\varphi(g^{-1}{\bf x})d\mu(g).$

The following local Siegel-Weil formula was proved in [BY20, Section 2] in the
orthogonal case and the proof is similar in the unitary case and is left to
the reader.

######  Lemma 15.1.

(Local Siegel-Weil formula) Fix Haar measures $d{\bf x}$ on $V_{v}^{n}$ and
$dT$ on $\mathrm{Herm}_{n}(F_{0,v})$ respectively such that they are self-dual
with respect to pairings defined by $\psi_{v}$ as in (13.7). Let $\mu$ be the
Haar measure on $G_{1}(F_{0,v})$ such that (15.1) holds. For any
$\varphi\in\mathcal{S}(V_{v}^{n})$ and nonsingular
$T\in\mathrm{Herm}_{n}(F_{0,v})$, we have

(15.3) $W_{T,v}(1,0,\lambda(V_{v})(\varphi))=\gamma(V_{v})^{n}\cdot
O_{T,v}(\varphi).$

Moreover,

  1. (1)

When $v$ is finite and $L_{v}\subset V_{v}$ is a lattice, let
$\mathds{1}_{L_{v}}\in\mathcal{S}(V_{v}^{n})$ be the characteristic function
of $L_{v}^{n}$ and $K_{L_{v}}=\\{g\in G_{1}(F_{0,v})\mid gL_{v}=L_{v}\\}$. Let
$T$ be a Gram matrix of $L_{v}$, then

$W_{T,v}(1,0,\lambda(V_{v})(\mathds{1}_{L_{v}}))=\gamma(V_{v})^{n}\cdot\mathrm{Vol}(K_{L_{v}},d\mu).$

  2. (2)

When $v$ is archimedean and $V_{v}$ is positive definite, then
$G_{1}(F_{0,v})\cong\mathrm{U}(n)$ and

$\displaystyle
W_{T,v}(z_{v},0,\varphi)=\gamma(V_{v})^{n}\mathrm{Vol}(G_{1}(F_{0,v}),d\mu)q^{T},q^{T}=\psi_{v}(\mathrm{tr}(Tz_{v}))$

where $\varphi=\psi_{v}(i\mathrm{tr}(({\bf x},{\bf x})))$ is the Gaussian.

####  15.2. Main global result

We use the notation of Section 14 and specialize to the case $m=n$. Let
$\mathcal{C}$ be as in Subsection 13.4.

######  Lemma 15.2.

Let $T\in\mathrm{Herm}_{n}(\mathcal{O}_{F_{0}})$.

  1. (1)

If $|\mathrm{Diff}(T,\mathcal{C})|>1$, then $\mathcal{Z}(T)=\emptyset$ and
$\widehat{\mathcal{Z}}(T,y)=0$.

  2. (2)

If $\mathrm{Diff}(T,\mathcal{C})=\\{v\\}$ where $v$ is finite, then
$\mathcal{Z}(T)$ is supported on finite places of $E$ over $v$ and

$\widehat{\mathcal{Z}}(T,y)=[(\mathcal{Z}^{\mathbb{L}}(T),0)].$

  3. (3)

If $\mathrm{Diff}(T,\mathcal{C})=\\{\tau\\}$ where $\tau\in\Sigma_{0}$ is
archimedean, then $\widehat{\mathcal{Z}}(T,y)=0$ unless $\tau(T)$ has
signature $(n-1,1)$, in which case

$\widehat{\mathcal{Z}}(T,y)=[0,g(y,T,\cdot))]$

where $g(y,T,\cdot)$ is supported on
$\bigsqcup_{\nu\in\Sigma(E,\tau)}\mathcal{M}(E_{\nu})$.

###### Proof.

Notice that the summation in the definition of the green current
$g(y,T,\cdot)$ in (14.9) is over an empty set unless $\tau(T)$ has the same
signature as $V[\tau]$. The lemma now follows from this observation together
with the proof of [Shi18, Proposition 5.4], see also [KR14b, Proposition
2.22]. ∎

We can now define the arithmetic degree of $\widehat{\mathcal{Z}}(T,y)$ by

(15.4)
$\displaystyle\widehat{\mathrm{deg}}(\widehat{\mathcal{Z}}(T,y))=\sum_{\nu\in\Sigma(E,\mathrm{fin})}\chi(\mathcal{Z}(T)_{\nu},\mathcal{Z}^{\mathbb{L}}(T))\cdot\log
q_{\nu}+\frac{1}{2}\sum_{\nu\in\Sigma(E,\mathrm{arch})}\int_{\mathcal{M}(E_{\nu})}g(y_{\nu},T,z)_{\nu}.$

where $\Sigma(E,\mathrm{fin})$ (resp. $\Sigma(E,\mathrm{arch})$) is finite
(resp. archimedean) places of $E$, $q_{\nu}$ is the size of the residue field
of $E_{\nu}$ and $\mathcal{Z}(T)_{\nu}$ is the base change of $\mathcal{Z}(T)$
to $\mathrm{Spec}\,\mathcal{O}_{E,\nu}$. By Lemma 15.2, for a single $T$, the
above definition is reduced to calculation over $\Sigma(E,v)$ for a single
$v$.

We then have the following case of arithmetic Siegel-Weil formula.

######  Theorem 15.3.

Assume that Conjecture 13.1 or equivalently Conjecture 1.1 is true. For a
nonsingular $T\in\mathrm{Herm}_{n}(\mathcal{O}_{F_{0}})$ such that
$\mathrm{Diff}(T,\mathcal{C})$ is not a place of $F_{0}$ with residue
characteristic $2$, we know

(15.5) $\frac{1}{[E:F]}\widehat{\mathrm{deg}}(\widehat{\mathcal{Z}}(T,y))\cdot
q^{T}=\frac{h}{\mathrm{Vol}(K_{G_{1}},d\mu)\mathrm{Vol}(G_{1}(F_{0,\infty}),d\mu)}E^{\prime}_{T}(z,\Phi)$

where $\Phi$ is defined as in Subsection 13.4 and $h$ is the stack cardinality
of $Z^{\mathbb{Q}}(\mathbb{Q})\backslash
Z^{\mathbb{Q}}(\mathbb{A})/K_{Z^{\mathbb{Q}}}$. Finally, $K_{G_{1}}$ is the
compact open subgroup of $G_{1}(\mathbb{A}_{F_{0}})$ which preserves $L$.

###### Proof.

Both sides are zero when $|\mathrm{Diff}(T,\mathcal{C})|>1$. When
$\mathrm{Diff}(T,\mathcal{C})=\\{v\\}$ where $v$ is unramified in $F$, (15.5)
is proved in [LZ19, Theorem 13.6.1] . When
$\mathrm{Diff}(T,\mathcal{C})=\\{\tau\\}$ where $\tau\in\Sigma_{0}$ is
archimedean, (15.5) is proved in [Liu11, Theorem 4.17, 4.20] and independently
in [GS19, Theorem 1.1.2]. It remains to prove the case when
$\mathrm{Diff}(T,\mathcal{C})=\\{v\\}$ where $v$ is ramified in $F$ where we
need to assume Conjecture 13.1. The following proof is uniform for all $v$.

Let $\nu\in\Sigma(E,v)$ and $k_{\nu}$ be the residue field of $E_{\nu}$ which
we assume has characteristic $p$. By [Shi18, Proposition 5.4],
$\mathcal{Z}(T)_{\mathcal{O}_{E,\nu}}$ is supported on the basic locus
$\mathcal{M}^{ss}_{\nu}$ which is the locus of $\mathcal{M}^{ss}_{k_{\nu}}$
where the universal $p$-divisible group over $\mathcal{M}$ is isoclinic. Let
$\widehat{\mathcal{M}}^{ss}_{\nu}$ be the formal completion of $\mathcal{M}$
along $\mathcal{M}^{ss}_{\nu}$ By [Shi18, Theorem 1.3], we have the following
uniformization theorem

$\displaystyle\widehat{\mathcal{M}}^{ss}_{\nu}\cong(Z^{\mathbb{Q}}(\mathbb{Q})\backslash
Z^{\mathbb{Q}}(\mathbb{A})/K_{Z^{\mathbb{Q}}})\times(G_{1}(F_{0})\backslash\mathcal{N}^{\prime}\times
G_{1}(\mathbb{A}_{F_{0},f}^{p})/K_{G})$

where

$\mathcal{N}^{\prime}=\mathcal{N}^{\mathrm{Kra}}_{n}\times_{\mathrm{Spf}\,\mathcal{O}_{\breve{F},v}}\mathrm{Spf}\,\mathcal{O}_{\breve{E},\nu}\times\prod_{v^{\prime}\neq
v}G_{1}(F_{0,v^{\prime}})/K_{G,v^{\prime}}$

where $\mathcal{N}^{\mathrm{Kra}}_{n}$ is as in Definition 2.3 and the product
is taken over places of $F_{0}$ over $p$. Notice that $K_{G}=K_{G_{1}}$. Then
the base change of $\mathcal{Z}(T)$ to $\widehat{\mathcal{M}}^{ss}_{\nu}$ is
isomorphic to (see for example [KR14b, Proposition 6.3] or the proof of
[Shi18, Theorem 5.7])

$(Z^{\mathbb{Q}}(\mathbb{Q})\backslash
Z^{\mathbb{Q}}(\mathbb{A})/K_{Z^{\mathbb{Q}}})\times\bigsqcup_{g\in
G_{1}(F_{0})\backslash
G_{1}(\mathbb{A}_{F_{0},f}^{v})/K_{G}^{v}}\bigsqcup_{\begin{subarray}{c}{\bf
x}\in\Omega(T)(F_{0})\\\ g^{-1}x\in
L^{v}\end{subarray}}\mathcal{Z}^{\mathrm{Kra}}({\bf x})$

where $\mathcal{Z}^{\mathrm{Kra}}({\bf x})$ is as in Subsection 2.3 and
$L^{v}=L\otimes_{\mathcal{O}_{F_{0}}}\hat{\mathcal{O}}_{F_{0}}^{v}$(all places
away from $v$). By Conjecture 13.1 and the fact that $i_{\bf
x}:G_{1}(F_{0})\rightarrow\Omega(T)$ is an isomorphism, we have

$\displaystyle\chi(\mathcal{Z}(T)_{\nu},\mathcal{Z}^{\mathbb{L}}(T))\cdot\log
q_{\nu}$ $\displaystyle=h\log q_{\nu}\cdot\sum_{g\in G_{1}(F_{0})\backslash
G_{1}(\mathbb{A}_{F_{0},f}^{v})/K_{G}^{v}}\sum_{g_{1}\in
G_{1}(F_{0})}\mathds{1}_{L}^{v}(g^{-1}g_{1}^{-1}{\bf
x})\cdot\mathrm{Int}(L_{T})$ $\displaystyle=h[E_{\nu}:F_{v}]\cdot\sum_{g\in
G_{1}(\mathbb{A}_{F_{0},f}^{v})/K_{G}^{v}}\mathds{1}_{L}^{v}(g^{-1}{\bf
x})\cdot\frac{W^{\prime}_{T,v}(1,0,\Phi_{v})}{W_{S,v}(1,0,\Phi_{v})}.$

By Lemma 15.1, we have

$\displaystyle\sum_{g\in
G_{1}(\mathbb{A}_{F_{0},f}^{v})/K_{G}^{v}}\mathds{1}_{L}^{v}(g^{-1}{\bf x})=$
$\displaystyle\int_{G_{1}(\mathbb{A}_{F_{0},f}^{v})/K_{G}^{v}}\mathds{1}_{L}^{v}(g^{-1}{\bf
x})dg$ $\displaystyle=$
$\displaystyle\frac{\gamma(V(\mathbb{A}_{F_{0},f}^{v}))^{-n}}{\mathrm{Vol}(K_{G}^{v},d\mu)}\prod_{v\neq
v^{\prime}}W_{T,v^{\prime}}(1,0,\lambda(\mathbb{V}_{v^{\prime}})(\mathds{1}_{L_{v^{\prime}}})).$

Hence by Lemma 13.2, we have

$\displaystyle\chi(\mathcal{Z}(T)_{\nu},\mathcal{Z}^{\mathbb{L}}(T))\cdot\log
q_{\nu}$
$\displaystyle=h[E_{\nu}:F_{0,v}]\frac{\gamma(V(\mathbb{A}_{F_{0},f}^{v}))^{-n}}{\mathrm{Vol}(K_{G}^{v},d\mu)}\prod_{v\neq
v^{\prime}}W_{T,v^{\prime}}(1,0,\lambda(\mathbb{V}_{v^{\prime}})(\mathds{1}_{L_{v^{\prime}}}))\frac{W^{\prime}_{T,v}(1,0,\Phi_{v})}{W_{S,v}(1,0,\Phi_{v})}$
$\displaystyle=h[E_{\nu}:F_{0,v}]\frac{1}{\mathrm{Vol}(K_{G}^{v},d\mu)}\cdot
C_{1}\cdot E^{\prime}(\tau,0,\Phi)$

where

$C_{1}^{-1}=\gamma(V(\mathbb{A}_{F_{0},f}^{v}))^{n}W_{S,v}(1,0,\lambda(\mathbb{V}_{v}))W_{T,\infty}(\tau,0,\Phi_{\infty}).$

Finally, by Lemma 15.1 again, we have

$\displaystyle C_{1}^{-1}=$
$\displaystyle\prod_{v}\gamma(V_{v})^{n}\mathrm{Vol}(K_{G,v},d\mu)\mathrm{Vol}(G_{1}(F_{0,\infty}),d\mu)q^{T}$
$\displaystyle=$
$\displaystyle\mathrm{Vol}(K_{G,v},d\mu)\mathrm{Vol}(G_{1}(F_{0,\infty}),d\mu)q^{T}.$

Finally by the assumption that the place of $F$ over $v$ is unramified in $E$,
we know

$\sum_{\nu\in\Sigma(E,v)}[E_{\nu}:F_{v}]=[E:F].$

The theorem is proved. ∎

Theorem 1.2 and the main results of [Shi20] and [HSY20] then imply the
following.

######  Corollary 15.4.

Equation (15.5) is true when $n=2,3$ assuming
$T\in\mathrm{Herm}_{n}(\mathcal{O}_{F_{0}})$ such that $T$ is nonsingular and
$\mathrm{Diff}(T,\mathcal{C})$ is not a place of $F_{0}$ with residue
characteristic $2$.

###  Appendix A Calculation of primitive local density

In this appendix, we provide the proof of Propositions 5.9 and 5.10.
Throughout this section, $M$ is unimodular of rank $m\geq 2$ unless clearly
stated otherwise. Let $\\{v_{1},\cdots,v_{2k},v_{2k+1},\cdots,v_{2k+m}\\}$ be
a basis of $M_{k}=\mathcal{H}^{k}\obot M$ with Gram matrix
$\mathrm{Diag}(\mathcal{H}^{k},S)$ and $S=\mathrm{Diag}(I_{m-1},\nu)$ Let $L$
be a Hermitian lattice of rank $n$ with Gram matrix $T$. An isometric
embedding $\varphi:L\to M$ is called primitive if its image in $M/\pi M$ has
dimension $\mathrm{rank}_{\mathcal{O}_{F}}(L)$. We call a vector $v$ primitive
in $M$ if $\pi^{-1}v\not\in M$, or equivalently the natural embedding
$\varphi:\mathrm{Span_{\mathcal{O}_{F}}}\\{v\\}\hookrightarrow M$ is
primitive. For a $v\in M_{k}$, we let $\Pr_{\mathcal{H}^{k}}(w_{i})$ be the
projection of $w_{i}$ to $\mathcal{H}^{k}$.

####  A.1. Proof of Proposition 5.9

The main purpose of this subsection is to prove the first four parts of
Proposition 5.9. Part (5) of this proposition follows from Proposition 5.7 and
Corollaries A.10 and A.12.

###### Proof.

For (1), Choose $M(1)=\frac{t\pi}{2}v_{1}+v_{2}\in M_{k}$ with $q(M(1))=t$.
Then

(A.1) $\displaystyle M(1)^{\perp}$
$\displaystyle=\mathrm{Span_{\mathcal{O}_{F}}}\\{\frac{-t\pi}{2}v_{1}+v_{2},v_{3},...,v_{2k},v_{2k+1},\cdots,v_{2k+m}\\}$
$\displaystyle\cong\langle-t\rangle\obot\mathcal{H}^{k-1}\obot M,$

which is represented by $\mathrm{Diag}(-t,\mathcal{H}^{k-1},S)$. It is easy to
check

$|M_{k}:M(1)\obot
M(1)^{\perp}|^{-1}|M(1)^{\vee}:M(1)|=|t\pi|_{F}|t\pi|_{F}^{-1}=1.$

For (2) and (3), assume first that $M$ is isotropic (and unimodular). In this
case, we may choose a basis $\\{v_{2k+1}^{\prime},\dots,v_{2k+m}^{\prime}\\}$
of $M$ with Gram matrix $\mathrm{Diag}(\mathcal{H}_{0},1,\dots,1,-\nu)$.
Choose $M(0)=\frac{t}{2}v_{2k+1}^{\prime}+v_{2k+2}^{\prime}$ with $q(M(0))=t$.
Then

$\displaystyle M(0)^{\perp}$
$\displaystyle=\hbox{Span}\\{v_{1},\cdots,v_{2k},-\frac{t}{2}v_{2k+1}^{\prime}+v_{2k+2}^{\prime},v_{2k+3}^{\prime},\cdots,v_{2k+m}^{\prime}\\}$
$\displaystyle\cong\mathcal{H}^{k}\obot\hbox{Span}\\{v_{2k+3}^{\prime},\cdots,v_{2k+m}^{\prime}\\}\obot\langle-t\rangle.$

as claimed. Moreover

$|M_{k}:M(0)\perp
M(0)^{\perp}|^{-1}|M(0)^{\vee}:M(0)|=|t|_{F}|t\pi|_{F}^{-1}=q$

Next, assume that $M$ is anisotropic. In this case, $M$ has rank $2$ and has
Gram matrix $S=\mathrm{Diag}(1,\nu)$ with $\chi(M)=\chi(-\nu)=-1$. In this
case, $E=F_{0}(\sqrt{-\nu})$ is a unramified quadratic field extension of
$F_{0}$, and
$N_{E/F_{0}}\mathcal{O}_{E}^{\times}=\mathcal{O}_{F_{0}}^{\times}$. When
$\mathrm{v}(t)=0$, $t\in N_{E/F_{0}}\mathcal{O}_{E}^{\times}$, i.e.,
$t=a\bar{a}+b\bar{b}\nu$. Take $M(0)=av_{2k+1}+bv_{2k+2}$. Then $q(M(0))=t$,
and

$M(0)^{\perp}=\hbox{Span}\\{v_{1},\cdots,v_{2k},-\nu\bar{b}v_{2k+1}+\bar{a}v_{2k+2}\\}=\mathcal{H}^{k}\obot\langle
t\nu\rangle,$

and

$|M_{k}:M(0)\perp M(0)^{\perp}|^{-1}|M(0)^{\vee}:M(0)|=|\pi|_{F}^{-1}=q.$

When $\mathrm{v}(t)>0$, $t\not\in N_{E/F_{0}}\mathcal{O}_{E}^{\times}$. So
there is no primitive $M(0)\in M$ with $q(M(0))=t$. This proves (1)—(3) of
Proposition 5.9.

The proof of (4) follows from the following 4 lemmas.

######  Lemma A.1.

For primitive vectors $w_{1},\ w_{2}\in\mathcal{H}_{i}$ with
$q(w_{1})=q(w_{2})$, we can find an element $g\in\mathrm{U}(\mathcal{H}_{i})$
such that $g(w_{1})=w_{2}.$

###### Proof.

We treat the case $i$ is odd first. Assume $v=a_{1}v_{1}+a_{2}v_{2}$. $v$ is
primitive implies that $a_{1}$ or $a_{2}$ is a unit. Without loss of
generality, we assume $a_{2}$ is a unit and we can further assume $a_{2}=1$ by
the action of $\begin{pmatrix}\bar{a}_{2}&0\\\ 0&a_{2}^{-1}\end{pmatrix}$. Now
notice that $q(v)=(v,v)=(a_{1}-\bar{a}_{1})\pi^{i}$. Hence we can write
$a_{1}=\alpha+\frac{q(v)\pi^{-i}}{2}$, where $\alpha\in\mathcal{O}_{F_{0}}$.
Now let $g=\begin{pmatrix}1&-\alpha\\\ 0&1\end{pmatrix}$, and it is
straightforward to check that $g\in\mathrm{U}(\mathcal{H}_{i})$ and
$g(v)=\frac{q(v)\pi^{-i}}{2}v_{1}+v_{2}$.

Now we deal with the case $i$ is even. Again, we can assume
$v=a_{1}v_{1}+v_{2}$. Then $q(v)=(a_{1}+\bar{a}_{1})\pi^{i}$. Hence we can
write $a_{1}=\frac{q(v)}{2}\pi^{-i}+\beta\pi$, where
$\beta\in\mathcal{O}_{F_{0}}$. Now let $g=\begin{pmatrix}1&-\beta\pi\\\
0&1\end{pmatrix}$, and it is straightforward to check that
$g\in\mathrm{U}(\mathcal{H}_{i})$ and
$g(v)=\frac{q(v)\pi^{-i}}{2}v_{1}+v_{2}$. ∎

######  Lemma A.2.

Assume $M$ is any lattice such that $\mathrm{v}(M)\geq i$. For $w_{1},\
w_{2}\in\mathcal{H}_{i}^{k}\obot M$, if $\Pr_{\mathcal{H}_{i}^{k}}(w_{1})$ and
$\Pr_{\mathcal{H}_{i}^{k}}(w_{2})$ are primitive and $q(w_{1})=q(w_{2})$, then
there exists $g\in\mathrm{U}(\mathcal{H}_{i}^{k}\obot M)$ with
$g(w_{1})=w_{2}$.

###### Proof.

Choose a basis $\\{v_{1},...,v_{2k}\\}$ of $\mathcal{H}_{i}^{k}$ such that the
associated Gram matrix is $\mathcal{H}_{i}^{k}$. We also choose a basis
$\\{v_{2k+1},...,v_{2k+m}\\}$ of $M$. Write
$w_{1}=\sum_{i=1}^{2k+m}a_{i}v_{i}$. Since
$\mathrm{Pr}_{\mathcal{H}_{i}^{k}}(w_{1})$ is primitive, $a_{i}$ is a unit for
some $i\in\\{1,...,2k\\}.$ Without loss of generality, we may assume
$a_{1}=1$. Let $w^{\prime}=w_{1}+\frac{(-1)^{i+1}q(w_{1})\pi^{-i}}{2}v_{2}$,
then

$q(w^{\prime})=q(w_{1})+(w_{1},\frac{(-1)^{i+1}q(w_{1})\pi^{-i}}{2}v_{2})+(\frac{(-1)^{i+1}q(w_{1})\pi^{-i}}{2}v_{2},w_{1})\\\
=0,$

and $(w^{\prime},v_{2})=(v_{1},v_{2})$. As a result,
$M_{1}=\mathrm{Span}_{\mathcal{O}_{F}}\\{w_{1},v_{2}\\}=\mathrm{Span}_{\mathcal{O}_{F}}\\{w^{\prime},v_{2}\\}$
is isometric to $\mathcal{H}_{i}$. Notice that
$\mathrm{val}_{\pi}(q(w_{1}))\geq i$ is guaranteed by the assumption
$\mathrm{v}(M)\geq i$.

Similarly, we can show $w_{2}\in M_{2}$ for some $M_{2}$ that is isometric to
$\mathcal{H}_{i}$. However, the assumption $\mathrm{v}(M)\geq i$ and [Jac62,
Proposition $4.2$] imply that there exist
$g\in\mathrm{U}(\mathcal{H}_{i}^{k}\obot M)$ such that $g(M_{1})=M_{2}$. In
particular, $g(w_{1})\in M_{2}$. Since both $g(w_{1})$ and $w_{2}$ are in
$M_{2}$, the problem is reduced to Lemma A.1. ∎

######  Lemma A.3.

For primitive vectors $w_{1},\ w_{2}\in M$ with $q(w_{1})=q(w_{2})$, we can
find an element $g\in\mathrm{U}(M)$ such that $g(w_{1})=w_{2}.$

###### Proof.

Since $M$ is unimodular, we can decompose

$M=\mathcal{H}_{0}^{k}\obot M^{\prime},$

where $M^{\prime}=0$ or an anisotropic unimodular Hermiatian lattice of rank
$1$ or $2$. If $\Pr_{\mathcal{H}_{0}^{k}}(w_{1})$ and
$\Pr_{\mathcal{H}_{0}^{k}}(w_{2})$ are primitive, this is Lemma A.2. If
$\Pr_{\mathcal{H}_{0}^{k}}(w_{1})$ is not primitive, then
$\Pr_{M^{\prime}}(w_{1})$ is primitive and thus
$q(\Pr_{M^{\prime}}(w_{1}))\in\mathcal{O}_{F}^{\times}$. This implies that
$q(w_{2})=q(w_{1})$ is a unit, and
$M=\mathcal{O}_{F}w_{i}\obot(\mathcal{O}_{F}w_{i})^{\perp}$. Therefore there
is some $g\in\mathrm{U}(M)$ with $g(w_{1})=w_{2}$. ∎

######  Lemma A.4.

Assume that $w_{1},\ w_{2}\in M_{k}$ are primitive and that
$\Pr_{\mathcal{H}^{k}}(w_{1})$ and $\Pr_{\mathcal{H}^{k}}(w_{2})$ are not
primitive. Then we can find $g\in\mathrm{U}(M_{k})$ such that
$g(w_{1})=w_{2}$.

###### Proof.

Let $\\{v_{1},...,v_{2k+m}\\}$ be a basis of $\mathcal{H}^{k}\obot M$, whose
Gram matrix is $\mathcal{H}^{k}\obot\mathrm{Diag}(1,...,\nu)$ where $\nu$ is a
unit. Assume $v\in M_{k}$ is primitive and $\Pr_{\mathcal{H}^{k}}(v)$ is not
primitive, then we can write $v=\sum_{i=1}^{2k}\pi
a_{i}v_{i}+\sum_{j=2k+1}^{2k+m}a_{j}v_{j}$, where some $a_{j}$ is a unit for
$2k+1\leq j\leq 2k+m.$ Again, without loss of generality, we may assume
$a_{2k+m}=1.$ For $i\leq k$, we set

$\displaystyle
v_{2i-1}^{\prime}=v_{2i-1}+\frac{\bar{a}_{2i}}{\nu}v_{2k+m},\quad
v_{2i}^{\prime}=v_{2i}+\frac{-\bar{a}_{2i-1}}{\nu}v_{2k+m}$

Let
$M_{v}=\mathrm{Span}_{\mathcal{O}_{F}}\\{v_{1}^{\prime},...,v_{2k}^{\prime}\\}.$
Then it is easy to check that $M_{v}$ is perpendicular to $v$. Moreover,
$M_{v}$ is isometric to $\mathcal{H}^{k}$ since
$\mathrm{val}_{\pi}((v^{\prime}_{2i-1},v^{\prime}_{2i}))=-1$ and
$0\leq\mathrm{val}_{\pi}((v^{\prime}_{i},v^{\prime}_{j}))$ for other $1\leq
i,j\leq 2k$. Hence we can find $g_{v}\in\mathrm{U}(M_{k})$ such that
$g_{v}(M_{v})=\mathrm{Span_{\mathcal{O}_{F}}}\\{v_{1},...,v_{2k}\\},$ and
$g_{v}(v)\in\mathrm{Span_{\mathcal{O}_{F}}}\\{v_{2k+1},...,v_{2k+m}\\}=M$.

Applying the above to $w_{1}$ and $w_{2}$, we can find
$g_{w_{1}},g_{w_{2}}\in\mathrm{U}(M_{k})$ such that $g_{w_{1}}(w_{1}),\
g_{w_{2}}(w_{2})\in M$. Now the problem is reduced to Lemma A.3, and the lemma
is proved. ∎

According to Lemma A.2 and Lemma A.4, a primitive vector $v\in M_{k}$ is
either in the same orbit of a vector $M(1)\in\mathcal{H}^{k}$ or a vector
$M(0)\in M$. Lemma A.1 implies that primitive vectors
$M(1),M^{\prime}(1)\in\mathcal{H}^{k}$ with $q(M(1))=q(M^{\prime}(1))$ lie in
the same orbit. Lemma A.3 implies the similar result for primitive
$M(0),M^{\prime}(0)\in M$ with $q(M(0))=q(M^{\prime}(0))$. A combination of
the above proves Part $(4)$ of Proposition 5.9. ∎

####  A.2. Proof of Proposition 5.10

In this subsection, we prove the first part of Proposition 5.10, which we
restate as follows for the convenience of the reader.

######  Proposition A.5.

Let $L$ be Hermitian $\mathcal{O}_{F}$-lattice of rank $2$ and
$\mathrm{v}(L)>0$. Let $\varphi:L\rightarrow M_{k}$ be a primitive isometric
embedding. Let $d(\varphi)$ be the dimension of the image of the map

$\displaystyle\mathrm{Pr}_{\mathcal{H}^{k}}\circ\varphi:L\rightarrow\mathcal{H}^{k}$

in $\mathcal{H}^{k}/\pi\mathcal{H}^{k}$. Then

$\varphi(L)^{\bot}\cong(-L)\obot\mathcal{H}^{k-d(\varphi)}\obot
M^{(d(\varphi))}$

where $M^{(d(\varphi))}$ is unimodular of rank equal to
$(\mathrm{rank}(M)-2(2-d(\varphi)))$ and
$\mathrm{det}M^{(d(\varphi))}=(-1)^{d(\varphi)}\mathrm{det}M$. In particular,
if $d(\varphi)=1$ then $\mathrm{rank}(M)\geq 2$, and if $d(\varphi)=0$ then
$\mathrm{rank}(M)\geq 4$.

###### Proof.

This proposition follows from Lemmas A.6 and A.7 below. ∎

######  Lemma A.6.

Let the notation be as in Proposition A.5. If $\mathrm{rank}(M_{k})\leq 4$,
then

$\varphi(L)^{\bot}\approx-L.$

In particular, such an $\varphi$ does not exist if $\chi(M_{k})=-1$ or
$\mathrm{rank}(M_{k})<4$.

###### Proof.

First, assume $M_{k}=\mathcal{H}^{2}$ and $L\approx\mathcal{H}_{i}$ where
$i>0$. Let $\varphi(L)=\mathrm{Span_{\mathcal{O}_{F}}}\\{w_{1},w_{2}\\}$ such
that the Gram matrix of $w_{1},w_{2}$ is $\mathcal{H}_{i}$. By Lemma A.1, we
may assume $w_{1}=v_{1}$. Then we may write
$w_{2}=a_{1}v_{1}+\pi^{i+1}v_{2}+a_{3}v_{3}+a_{4}v_{4},$ and
$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{3}),\mathrm{v}_{\pi}(a_{4})\\}=0$ by
assumption. Without loss of generality, we may assume $a_{3}=1$. Now a direct
calculation shows that

$\displaystyle\varphi(L)^{\perp}=\mathrm{Span_{\mathcal{O}_{F}}}\\{v_{1}+(-\pi)^{i+1}v_{4},v_{3}+\bar{a}_{4}v_{4}\\}.$

Its Gram matrix is

$\left(\begin{array}[]{cc}0&(-\pi)^{i}\\\
\pi^{i}&(a_{4}-\bar{a}_{4})\pi^{-1}\end{array}\right)=\left(\begin{array}[]{cc}0&(-\pi)^{i}\\\
\pi^{i}&-a_{1}(-\pi)^{i}-\bar{a}_{1}\pi^{i}\end{array}\right)\approx\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&0\end{array}\right),$

hence

$\displaystyle\varphi(L)^{\perp}\approx-L.$

Now we treat the case $M_{k}=\mathcal{H}^{2}$ and
$L\approx\mathrm{Diag}(u_{1}(-\pi_{0}^{a}),u_{2}(-\pi_{0})^{b})$ where
$0<a\leq b$. Again, let
$\varphi(L)=\mathrm{Span_{\mathcal{O}_{F}}}\\{w_{1},w_{2}\\}$ such that the
Gram matrix of $w_{1},w_{2}$ is
$\mathrm{Diag}(u_{1}(-\pi_{0}^{a}),u_{2}(-\pi_{0})^{b})$, and we can assume
$w_{1}=v_{1}-\frac{q(w_{1})\pi}{2}v_{2}$ without loss of generality. Then we
may write
$w_{2}=a_{1}(v_{1}+\frac{q(w_{1})\pi}{2}v_{2})+a_{3}v_{3}+a_{4}v_{4}$, hence
$\mathrm{min}\\{v_{\pi}(a_{3}),v_{\pi}(a_{4})\\}=0$ by assumption again. We
may assume $a_{3}=1$ and a direct calculation shows that

$\displaystyle\varphi(L)^{\perp}=\mathrm{Span_{\mathcal{O}_{F}}}\\{v_{1}+\frac{q(w_{1})\pi}{2}v_{2}-\bar{a}_{1}q(w_{1})\pi
v_{4},v_{3}+\bar{a}_{4}v_{4}\\}.$

Set $v_{3}^{\prime}=v_{1}+\frac{q(w_{1})\pi}{2}v_{2}-\bar{a}_{1}q(w_{1})\pi
v_{4}$ and $v_{4}^{\prime}=a_{1}v_{3}^{\prime}+v_{3}+\bar{a}_{4}v_{4}$. Then
$\varphi(L)^{\perp}=\mathrm{Span_{\mathcal{O}_{F}}}\\{v_{3}^{\prime},v_{4}^{\prime}\\}$
and the Gram matrix of $\\{v_{3}^{\prime},v_{4}^{\prime}\\}$ is

$\displaystyle\begin{pmatrix}-q(w_{1})&0\\\
0&a_{1}\bar{a}_{1}q(w_{1})-(a_{3}\bar{a}_{4}-\bar{a}_{3}a_{4})\pi^{-1}\end{pmatrix}=\begin{pmatrix}-q(w_{1})&0\\\
0&-q(w_{2})\end{pmatrix}.$

Now let $M_{k}=\mathcal{H}\obot M$, where $M$ is unimodular of rank $2$. We
only treat the case $L\approx\mathcal{H}_{i}$ in detail, and the argument for
$L$ represented by a diagonal matrix is similar. We assume that $M_{k}$ has a
basis $\\{v_{1},\ldots,v_{4}\\}$ with Gram matrix
$\mathcal{H}\obot\mathrm{Diag}(1,\nu)$ where $\nu$ is a unit. Let
$\varphi(L)=\mathrm{Span_{\mathcal{O}_{F}}}\\{w_{1},w_{2}\\}$ where the Gram
matrix of $\\{w_{1},w_{2}\\}$ is $\mathcal{H}_{i}$. Then one can check that at
least one of $w_{1}$ and $w_{2}$ is primitive in $\mathcal{H}$. By Lemma A.1,
we can assume that

$w_{1}\coloneqq\varphi(m_{1})=v_{1},w_{2}\coloneqq\varphi(m_{2})=a_{1}v_{1}+\pi^{i+1}v_{2}+a_{3}v_{3}+a_{4}v_{4}$

and

(A.2)
$(w_{2},w_{2})=a_{1}\pi^{i}-\bar{a}_{1}\pi^{i}+a_{3}\bar{a}_{3}+a_{4}\bar{a}_{4}\nu=0.$

By our assumption we know that
$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{3}),\mathrm{v}_{\pi}(a_{4})\\}=0$. Since
we assume $i\geq 1$, (A.2) implies that both $a_{3}$ and $a_{4}$ are in
$\mathcal{O}_{F_{0}}^{\times}$. This in turn implies that
$-\nu\in\mathrm{Nm}_{F/F_{0}^{\times}}(\mathcal{O}_{F}^{\times})=\mathcal{O}_{F_{0}}^{2}$.
Hence $M_{k}\approx\mathcal{H}\obot\mathcal{H}_{0}$ and we can instead assume
that $\\{v_{1},v_{2},v_{3},v_{4}\\}$ has Gram matrix
$\mathcal{H}\obot\mathcal{H}_{0}$. We can furthur assume that

$w_{1}=v_{1},w_{2}=a_{1}v_{1}+\pi^{i+1}v_{2}+v_{3}+a_{4}v_{4}$

with

$(w_{2},w_{2})=a_{1}\pi^{i}-\bar{a}_{1}\pi^{i}+a_{4}+\bar{a}_{4}=0.$

By direct calculation, it is easy to see that

$\varphi(L)^{\bot}=\mathrm{Span}_{\mathcal{O}_{F}}\\{v_{1}-(-\pi)^{i}v_{4},v_{3}-\bar{a}_{4}v_{4}\\}.$

Its Gram matrix is

$\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&-a_{4}-\bar{a}_{4}\end{array}\right)=\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&a_{1}\pi^{i}-\bar{a}_{1}\pi^{i}\end{array}\right)\approx\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&0\end{array}\right).$

Finally, let $M_{k}$ be unimodular of rank $4$. We treat the case
$L\approx\mathcal{H}_{i}$ in detail, and the other cases follow from a similar
argument. Let $\varphi(L)=\mathrm{Span_{\mathcal{O}_{F}}}\\{w_{1},w_{2}\\}$
such that the Gram matrix of $\\{w_{1},w_{2}\\}$ is $\mathcal{H}_{i}$.
Apparently $M_{k}$ contains a $\mathcal{H}_{0}$. We can assume that $M_{k}$
has a basis $\\{v_{1},v_{2},v_{3},v_{4}\\}$ with Gram matrix
$\mathcal{H}_{0}\obot\mathrm{diag}\\{1,\epsilon\\}$ where
$\epsilon\in\mathcal{O}_{F_{0}}^{\times}$. By Lemma A.3 we can assume that
$w_{1}=v_{1}$. Then we have

$w_{2}=a_{1}v_{1}+\pi^{i}v_{2}+\sum_{j=3}^{4}a_{j}v_{j},$

and

(A.3)
$(w_{2},w_{2})=a_{1}(-\pi)^{i}+\bar{a}_{1}\pi^{i}+a_{3}\bar{a}_{3}+a_{4}\bar{a}_{4}\epsilon=0.$

By our assumption we know that
$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{3}),\mathrm{v}_{\pi}(a_{4})\\}=0$. Since
we assume $i\geq 1$, (A.3) implies that both $a_{3}$ and $a_{4}$ are in
$\mathcal{O}_{F_{0}}^{\times}$. This in turn implies that
$-\epsilon\in\mathrm{Nm}_{F/F_{0}^{\times}}(\mathcal{O}_{F}^{\times})=\mathcal{O}_{F_{0}}^{2}$.
Hence $M_{k}=\mathcal{H}_{0}^{2}$ and we can instead assume that
$\\{v_{1},v_{2},v_{3},v_{4}\\}$ has Gram matrix
$\mathcal{H}_{0}\obot\mathcal{H}_{0}$. We can furthur assume that

$w_{1}=v_{1},w_{2}=a_{1}v_{1}+\pi^{i}v_{2}+v_{3}+a_{4}v_{4}$

with

$(w_{2},w_{2})=a_{1}(-\pi)^{i}+\bar{a}_{1}\pi^{i}+a_{4}+\bar{a}_{4}=0.$

By a direct calculation, it is easy to see that

$\varphi(L)^{\bot}=\mathrm{Span}_{\mathcal{O}_{F}}\\{v_{1}-(-\pi)^{i}v_{4},v_{3}-\bar{a}_{4}v_{4}\\}.$

Its Gram matrix is

$\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&-a_{4}-\bar{a}_{4}\end{array}\right)=\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&a_{1}(-\pi)^{i}+\bar{a}_{1}\pi^{i}\end{array}\right)\approx\left(\begin{array}[]{cc}0&-(-\pi)^{i}\\\
-\pi^{i}&0\end{array}\right).$

Notice that, as a byproduct of the above argument, we actually also proved
that if $\mathrm{rank}(M_{k})<4$ or $M$ is not split, then no such $\varphi$
exists. The lemma is proved.

∎

######  Lemma A.7.

Assume $\mathrm{v}(L)\geq 0$. Let $\varphi:L\rightarrow M_{k}$ be a primitive
isometric embedding. Let $d(\varphi)$ be the dimension of
$\mathrm{Pr}_{\mathcal{H}^{k}}(\varphi(L))\otimes_{\mathcal{O}_{F}}\mathbb{F}_{q}$
in $\mathcal{H}^{k}/\pi\mathcal{H}^{k}$. Then there exist a
$g\in\mathrm{U}(M_{k})$ such that

$\displaystyle g(\varphi(L))\subset\mathcal{H}^{d(\varphi)}\obot
I_{4-2d(\varphi)}\subset M_{k},$

where $I_{4-2d(\varphi)}$ is a unimodular sublattice of $M_{k}$ with rank
$4-2d(\varphi)$.

###### Proof.

We prove the case for $L\approx\mathcal{H}_{i}$ in detail, and the other cases
are similar. Let $\\{v_{1},\ldots,v_{2k+m}\\}$ be a basis of $M_{k}$ whose
Gram matrix is $\mathcal{H}^{k}\obot\mathrm{diag}\\{1,\ldots,1,\nu\\}$ where
$\nu$ is a unit. Set
$\varphi(L)=\mathrm{Span_{\mathcal{O}_{F}}}\\{w_{1},w_{2}\\}$.

Assume $d(\varphi)=2$. If $i=-1$, then there is nothing to prove. Therefore,
we may assume $i>-1$. By Lemma A.2, without loss of generality, we can assume
that $w_{1}=v_{1}$. Then

$w_{2}=a_{1}v_{1}+\pi^{i+1}v_{2}+\sum_{j=3}^{2k+m}a_{j}v_{j}.$

By the assumption that $d(\varphi)=2$, we know that

$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{j})\mid 3\leq j\leq 2k\\}=0.$

Hence applying Lemma A.2 to $\mathcal{H}^{k-1}\obot M$, we can find a
$g\in\mathrm{U}(M_{k})$ such that

$gw_{1}=v_{1},\quad gw_{2}\in\mathcal{H}^{2}.$

where $\mathcal{H}^{2}$ refers to the first direct summand in the
decomposition $\mathcal{H}^{k}\obot
M=\mathcal{H}^{2}\obot\mathcal{H}^{k-2}\obot M$.

When $d(\varphi)=1$, without loss of generality, we can assume
$\Pr_{\mathcal{H}^{k}}(w_{1})$ is primitive. By Lemma A.2, we can assume that
$w_{1}=v_{1}$. Then

$w_{2}=a_{1}v_{1}+\pi^{i+1}v_{2}+\sum_{j=3}^{2k+m}a_{j}v_{j}.$

By the assumption that $d(\varphi)=1$, we know that

$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{j})\mid 3\leq j\leq 2k\\}\geq 1.$

Since we assume $\varphi$ is primitive, we know that

$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{j})\mid 2k+1\leq j\leq 2k+r\\}=0.$

Then we are done by applying Lemma A.4 to $\mathcal{H}^{k-1}\obot M$.

When $d(\varphi)=0$, without loss of generality, we may assume
$w_{1}=v_{2k+1}+v_{2k+2}$ by Lemma A.4. Here, we pick $v_{2k+i}$ so that the
corresponding Gram matrix is $\mathrm{Diag}(1,-1,1,\dots,-\nu)$ (this is
possible since we assume $m\geq 4$). Since $\varphi$ is primitive with
$d(\varphi)=0$, then

$\displaystyle w_{2}=\sum_{i=1}^{2k}\pi
a_{i}v_{i}+\sum_{i=2k+1}^{2k+m}a_{i}v_{i},$

and

$\mathrm{min}\\{\mathrm{v}_{\pi}(a_{j})\mid 2k+3\leq j\leq 2k+r\\}=0.$

We are done by applying Lemma A.4 to
$\mathcal{H}^{k}\obot\mathrm{Span_{\mathcal{O}_{F}}}\\{v_{2k+3},\dots,v_{2k+m}\\}$.
∎

####  A.3. Calculation of primitive local density

In this subsection, we compute primitive local density polynomials and prove
the formulas in Propositions 5.9 and 5.10. Assume $L$ is represented by a
nonsingular Hermitian matrix $T$ of rank $n\leq 2$. We let $\bar{v}$ denote
the image of $v$ in $M_{k}\otimes_{\mathcal{O}_{F}}\mathbb{F}_{q}$. Let

$\displaystyle M_{k}^{n}(i)\coloneqq\\{(v_{j})\in
M^{n,(n)}_{k}|\mathrm{Span}_{\mathbb{F}_{q}}\\{\mathrm{Pr}_{\mathcal{H}^{k}}(\bar{v}_{j}),\
1\leq j\leq n\\}\text{ has rank $i$}\\}$

where $M^{n,(n)}$ is as in (5.5), and

(A.4)
$\displaystyle\beta_{i}(M,L,X)\coloneqq\int_{\mathrm{Herm}_{n}(F)}dY\int_{M_{k}^{n}(i)}\psi(\langle
Y,T({\bf x})-T\rangle)d{\bf x}.$

Notice that

(A.5) $\sum_{i=0}^{n}\beta_{i}(M,L,X)=\beta(M,L,X)^{(n)}$

is the primitive local density defined earlier, and we will shorten it as
$\beta(M,L,X)$.

######  Lemma A.8.

Assume $L$ is of rank $n\leq 2$, then

$\displaystyle\beta_{n}(M,L,X)=\begin{cases}\beta(\mathcal{H}^{k},L)&\text{ if
$k\geq n$ or $L=\mathcal{H}$,}\\\ 0&\text{ otherwise}.\end{cases}$

###### Proof.

Recall that $T({\bf x})=({\bf x},{\bf x})$ is the moment matrix of ${\bf x}\in
M_{k}^{n}$. Notice that

$\displaystyle\beta_{n}(M,L,X)$
$\displaystyle=\int_{\mathrm{Herm}_{n}(F)}dY\int_{M^{n}}\int_{(\mathcal{H}^{k})^{n,(n)}}\psi(\langle
Y,T({}^{t}({\bf x}_{1},{\bf x}_{2}))-T\rangle)d{\bf x}_{1}d{\bf x}_{2}$
$\displaystyle=\int_{\mathrm{Herm}_{n}(F)}dY\int_{M^{n}}\int_{(\mathcal{H}^{k})^{n,(n)}}\psi(\langle
Y,T({\bf x}_{1})+T({\bf x}_{2})-T\rangle)d{\bf x}_{1}d{\bf x}_{2}.$

When $k\geq n$ or $T=\mathcal{H}$, there exists a $g\in\mathrm{U}(M_{k})$ by
Lemma A.2 or Lemma A.7 such that

$\displaystyle g({\bf x}_{2})=0,\text{ and }T({\bf x}_{1})+T({\bf
x}_{2})=T(g({\bf x}_{1})).$

So

$\displaystyle\int_{\mathrm{Herm}_{n}(F)}dY\int_{M^{n}}\int_{(\mathcal{H}^{k})^{n,(n)}}\psi(\langle
Y,T({\bf x}_{1})+T({\bf x}_{2})-T\rangle)d{\bf x}_{1}d{\bf x}_{2}$
$\displaystyle=\int_{\mathrm{Herm}_{n}(F)}dY\int_{M^{n}}\int_{(\mathcal{H}^{k})^{n,(n)}}\psi(\langle
Y,T(g({\bf x}_{1}))-T\rangle)d{\bf x}_{1}d{\bf x}_{2}$
$\displaystyle=\int_{\mathrm{Herm}_{n}(F)}dY\int_{M^{n}}\int_{(\mathcal{H}^{k})^{n,(n)}}\psi(\langle
Y,T({\bf x}_{1})-T\rangle)d{\bf x}_{1}d{\bf x}_{2}$
$\displaystyle=\int_{\mathrm{Herm}_{n}(F)}dY\int_{(\mathcal{H}^{k})^{n,(n)}}\psi(\langle
Y,T({\bf x}_{1})-T\rangle)d{\bf x}_{1}$
$\displaystyle=\beta(\mathcal{H}^{k},L).$

When $k<n$ and $L\neq\mathcal{H}$, we can check $\beta_{n}(M,L,X)=0$ by the
argument in Lemma A.7. ∎

By a variant of [CY20], Chao Li and Yifeng Liu obtained the following formula
of $\beta(\mathcal{H}^{k},L)$.

######  Lemma A.9 (Lemma $2.16$ of [LL21]).

Let $b_{1}\leq\cdots\leq b_{n}$ be the unique integers such that
$L^{\vee}/L\approx\mathcal{O}_{F}/(\pi^{b}_{1})\oplus\cdots\oplus\mathcal{O}_{F}/(\pi^{b_{n}})$.
Let $t_{o}(L)$ be the number of nonzero entries in $(b_{1},\cdots,b_{n})$.
Then

$\displaystyle\beta(\mathcal{H}^{k},L)=\prod_{k-\frac{n+t_{o}(L)}{2}<i\leq
k}(1-q^{-2i}).$

Combining the above two lemmas, we have the following.

######  Corollary A.10.

Assume $L$ is of rank $1$, we have

$\displaystyle\beta_{1}(M,L,X)=1-X.$

Assume $L$ is of rank $2$, we have

$\displaystyle\beta_{2}(M,L,X)=\begin{cases}(1-X)&\text{ if
$L=\mathcal{H}_{-1}$},\\\ (1-X)(1-q^{2}X)&\text{ otherwise}.\end{cases}$

######  Lemma A.11.

For an $\mathcal{O}_{F}$ Hermitian lattice, let $\bar{L}=L/\pi L$ be its
reduction modulo $\pi$ with resulting quadratic form. Let
$r(\overline{M},\overline{L})$ to be the number of isometries from
$\overline{L}$ to $\overline{M}$. Then

$\displaystyle\beta_{0}(M,L,X)=X^{n}\beta(M,L)=q^{-mn+n^{2}}r(\overline{M},\overline{L})X^{n}.$

###### Proof.

For the first identity, we may repeat the same calculation in the proof of
Lemma A.8 by replacing $M^{n}\obot(\mathcal{H}^{k})^{n,(n)}$ with
$M^{n,(n)}\obot(\pi\mathcal{H}^{k})^{n}$. Notice that the factor $X^{n}$ shows
up because $\mathrm{vol}((\pi\mathcal{H}^{k})^{n})=X^{n}$. The second identity
follows from the same proof of [CY20, Theorem 3.12] and we leave the detail to
the reader.

∎

Notice that [LZ21, Lemma 3.2.1] provides a uniform formula for
$|r(\overline{M},\overline{L})|$. As a result, we obtain the following
corollaries.

######  Corollary A.12.

Assume $L=\mathcal{O}_{F}{\bf x}$ is of rank $1$ (we allow $q({\bf x})=0$).

  1. (1)

If $\mathrm{v}(L)=0$, then

$\displaystyle\beta_{0}(M,L,X)=\begin{cases}(1+\chi(M)\chi(L)q^{-\frac{m-1}{2}})X&\text{
if $m$ is odd},\\\ (1-\chi(M)q^{-\frac{m}{2}})X&\text{ if $m$ is
even}.\end{cases}$

  2. (2)

If $\mathrm{v}(L)>0$, then

$\displaystyle\beta_{0}(M,L,X)=\begin{cases}(1-q^{1-m})X&\text{ if $m$ is
odd},\\\ \left(1-q^{1-m}+\chi(M)(q-1)q^{-\frac{m}{2}}\right)X&\text{ if $m$ is
even}.\end{cases}$

######  Corollary A.13.

Assume $L$ is of rank $2$. When $t(L)=1$, we assume that $L$ has gram matrix
$T=\mathrm{Diag}(u_{1},u_{2}(-\pi_{0})^{b})$ with $b>0$.

  1. (1)

If $m$ is odd, then

$\displaystyle\beta_{0}(M,L,X)=\begin{cases}q(1-q^{1-m})X^{2}&\text{ if
$t(L)=0$},\\\ q(1+\chi(M)\chi(u_{1})q^{\frac{3-m}{2}})(1-q^{1-m})X^{2}&\text{
if $t(L)=1$},\\\ q(1-q^{1-m})(1-q^{3-m})X^{2}&\text{ if $t(L)=2$}.\end{cases}$

  2. (2)

If $m$ is even, then

$\displaystyle\beta_{0}(M,L,X)=\begin{cases}q\left(1-\chi(L)q^{1-m}+\chi(L)\chi(M)(q-\chi(L))q^{-\frac{m}{2}}\right)X^{2}&\text{
if $t(L)=0$},\\\ q(1-\chi(M)q^{-\frac{m}{2}})(1-q^{2-m})X^{2}&\text{ if
$t(L)=1$},\\\
q\left((1-q^{2-m})+\chi(M)(q^{2}-1)q^{-\frac{m}{2}}\right)(1-q^{2-m})X^{2}&\text{
if $t(L)=2$}.\end{cases}$

Finally, we calculate $\beta_{1}(M,L,X)$.

######  Proposition A.14.

Assume $L$ is as in Corollary A.13. Let $\delta_{e}(m)=1$ or $0$ depending on
whether $m$ is even or odd.

  1. (1)

If $t(L)=2$, then

$\displaystyle\beta_{1}(M,L,X)=q(q+1)\left((1-q^{1-m})+\delta_{e}(m)\chi(M)(q-1)q^{-\frac{m}{2}}\right)X(1-X).$

  2. (2)

If $t(L)=1$, then

$\displaystyle\beta_{1}(M,L,X)=\begin{cases}q(1+q-q^{1-m}+\chi(M)\chi(u_{1})q^{\frac{3-m}{2}})X(1-X)&\text{
if $m$ is odd},\\\ q(1+q-q^{1-m}-\chi(M)q^{-\frac{m}{2}})X(1-X)&\text{ if $m$
is even}.\end{cases}$

  3. (3)

If $t(L)=0$ and $\chi(L)=1$, i.e. $L\cong\mathcal{H}_{0}$, then

$\displaystyle\beta_{1}(M,L,X)$
$\displaystyle=q\left(q+1-2q^{1-m}+\delta_{e}(m)\chi(M)(q-1)q^{-\frac{m}{2}}\right)X(1-X).$

  4. (4)

If $t(L)=0$ and $\chi(L)=-1$, then

$\displaystyle\beta_{1}(M,L,X)$
$\displaystyle=q(q+1)\left(1-\delta_{e}(m)\chi(M)q^{-\frac{m}{2}}\right)X(1-X).$

###### Proof.

We first assume $L=\mathcal{H}_{i}$. We claim that