1. Introduction
In their seminal work [Reference Kudla and RapoportKR11] and [Reference Kudla and RapoportKR14b], Kudla and Rapoport made a conjectural local arithmetic Siegel–Weil formula (the Kudla–Rapoport conjecture) relating the intersection numbers of special divisors on unitary Rapoport–Zink (RZ) spaces to the central derivative of certain local density polynomials. A unitary RZ space is a local version of a unitary Shimura variety associated to a general unitary group 
$\mathrm {GU}(1,n-1)$. The Kudla–Rapoport conjecture plays a central role in the arithmetic Siegel–Weil formula for unitary Shimura varieties, which was first proposed by Kudla in [Reference KudlaKud97] for orthogonal Shimura varieties. When 
$n=1$ or 
$2$, the Kudla–Rapoport conjecture was proved in [Reference Kudla and RapoportKR11]. The case when 
$n=3$ was proved in [Reference TerstiegeTer13]. The general case was proved recently in [Reference Li and ZhangLZ22a] by an ingenious induction. The Archimedean analogue of the Kudla–Rapoport conjecture was proved in [Reference LiuLiu11] and [Reference Garcia and SankaranGS19]. The analogue of the Kudla–Rapoport conjecture for GSpin RZ space is formulated and proved in [Reference Li and ZhangLZ22b].
Originally the Kudla–Rapoport conjecture was proposed only for good primes, namely inert primes over which the RZ space has hyperspecial level structure. A modified Kudla–Rapoport conjecture for RZ space with minuscule parahoric level structure over inert primes has been proposed in [Reference ChoCho22]. For ramified primes, there are two kinds of well-understood arithmetic models of RZ spaces. One is the exotic smooth model which has good reduction, the other is the Krämer model proposed in [Reference KrämerKrä03] which only has semi-stable reduction. The analogue of the Kudla–Rapoport conjecture for the even-dimensional exotic smooth model was studied in [Reference Li and LiuLL22], in which case the conjecture can be proved by the same strategy as in [Reference Li and ZhangLZ22a]. For the Krämer model, however, it was expected that serious modification of the original Kudla–Rapoport conjecture is needed. A precise formulation has not previously been given. One of the main goals of this paper is to formulate a precise conjecture (Conjecture 1.1) based on earlier work of [Reference ShiShi22] and [Reference He, Shi and YangHSY23] for the case 
$n=2$. We then prove Conjecture 1.1 for 
$n = 3$.
In a very recent joint work with Chao Li [Reference He, Li, Shi and YangHLSY22], we proved the conjecture completely. One of the major innovations of [Reference He, Li, Shi and YangHLSY22] is a decomposition formula of primitive local density polynomials, which is inspired by the results in the appendix of this work. The geometric side of the ‘horizontal’ part in [Reference He, Li, Shi and YangHLSY22] essentially follows from the current work. To deal with the vertical part in general, [Reference He, Li, Shi and YangHLSY22] uses partial Fourier transform inspired by [Reference Li and ZhangLZ22b]. The current work uses explicit computation instead. Finally, it was discovered in [Reference He, Li, Shi and YangHLSY22] that the ‘central’ derivative of the primitive local density polynomials has a surprisingly simple formula.
1.1 The naive conjecture
Let 
$p$ be an odd prime and 
$F$ be a ramified quadratic field extension of a 
$p$-adic number field 
$F_0$ with residue field 
$\mathbb {F}_q$. Fix an algebraic closure 
$k$ of 
$\mathbb {F}_q$. Fix a uniformizer 
$\pi$ of 
$F$ such that 
$\pi _0=\pi ^2$ is a uniformizer of 
$F_0$ and let 
$\mathrm {v}_\pi$ be the valuation on 
$F$ such that 
$\mathrm {v}_\pi (\pi )=1$. Let 
$\breve {F}_0$ be the completion of a maximal unramified extension of 
$F_0$ and 
$\breve {F} := F\otimes _{F_0} \breve {F}_0$. Let 
$\mathcal {O}_{\breve F}$ and 
$\mathcal {O}_{\breve {F}_0}$ be the rings of integers of 
$\breve {F}$ and 
$\breve {F}_0$, respectively. For a Hermitian lattice or space 
$M$ of rank 
$n$, we define its sign as
\begin{equation} \chi(M) = \chi((-1)^{{n(n-1)}/{2}}\mathrm{det}(M)) =\pm 1, \end{equation}
where 
$\chi$ is the quadratic character of 
$F_0^\times$ associated to 
$F/F_0$. We call 
$M$ split or non-split depending on whether 
$\chi (M)=1$ or 
$-1$. For a Hermitian matrix 
$T$, define 
$\chi (T)$ to be the sign of its associated Hermitian lattice.
Let 
$\mathbb {Y}$ and 
$\mathbb {X}$ be pre-fixed framing Hermitian formal 
$\mathcal {O}_F$-modules of signature 
$(0,1)$ and 
${(1,n-1)}$, respectively, over 
$\mathrm {Spec}\, k$. Recall that Hermitian formal 
$\mathcal {O}_F$-modules are a particular kind of formal 
$p$-divisible groups with 
$\mathcal {O}_F$-action, see § 2.1. The space of special quasi-homomorphisms
\begin{equation} \mathbb{V}=\mathrm{Hom}_{\mathcal{O}_F}(\mathbb{Y},\mathbb{X})\otimes_\mathbb{Z} \mathbb{Q} \end{equation}
is equipped with a Hermitian form 
$h(\,{,}\,)$, see (2.2). Let 
$\epsilon =\chi (\mathbb {V})$. The RZ space 
$\mathcal {N}^\mathrm {Kra}_{n,\epsilon }$ parameterizes certain classes of supersingular Hermitian formal 
$\mathcal {O}_F$-modules of signature 
$(1,n-1)$ over 
${\mathrm {Spf}\,\mathcal {O}_{\breve {F}} }$, see § 2.1. It is a formal scheme over 
${\mathrm {Spf}\,\mathcal {O}_{\breve {F}} }$ with semi-stable reduction and can be viewed as a regular model of the formal completion of the corresponding global unitary Shimura variety along its basic locus over 
$p$. When 
$n$ is odd, 
$\mathcal {N}^\mathrm {Kra}_{n,1}$ is isomorphic to 
$\mathcal {N}^\mathrm {Kra}_{n,-1}$. We often write 
$\mathcal {N}^\mathrm {Kra}$ instead of 
$\mathcal {N}^\mathrm {Kra}_{n,\epsilon }$ for simplicity.
For each subset 
$L \subset \mathbb {V}$, define 
$\mathcal {Z}^\mathrm {Kra}(L)$ to be the formal subscheme of 
$\mathcal {N}^\mathrm {Kra}$ where 
${\bf x}$ deforms to a homomorphism for any 
${\bf x}\in L$. Let 
$L\subset \mathbb {V}$ be an 
$\mathcal {O}_F$-lattice of rank 
$r$. We say 
$L$ is integral if 
$h(\,{,}\,)|_L$ is non-degenerate and takes values in 
$\mathcal {O}_F$. Let 
${\bf x}_1,\ldots,{\bf x}_r$ be a basis of 
$L$. We define
\begin{equation} {}^\mathbb{L} \mathcal{Z}^\mathrm{Kra}(L)=[\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_1)}\otimes^{\mathbb{L}}\cdots \otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_r)}]\in K_0(\mathcal{N}^\mathrm{Kra}), \end{equation}
where 
$\otimes ^\mathbb {L}$ is the derived tensor product of complex of coherent sheaves on 
$\mathcal {N}^\mathrm {Kra}$ and 
$K_0(\mathcal {N}^\mathrm {Kra})$ is the Grothendieck groups of finite complexes of coherent locally free sheaves on 
$\mathcal {N}^\mathrm {Kra}$. By [Reference HowardHow19, Corollary C], 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}(L)$ is independent of the choice of basis of 
$L$. When 
$L$ has rank 
$n$, we define the intersection number
\begin{equation} \mathrm{Int}(L)=\chi(\mathcal{N}^\mathrm{Kra},{}^\mathbb{L} \mathcal{Z}^\mathrm{Kra}(L)), \end{equation}
where 
$\chi$ is the Euler characteristic. One can show that 
$\mathrm {Int}(L)$ is finite, see Lemma 2.14.
Let 
$L$ and 
$M$ be Hermitian lattices of rank 
$n$ and 
$m$ respectively. Moreover, we assume 
$\mathrm {v}(M):= \min \{\mathrm {v}_\pi ( h(v,v'))\mid v,v'\in M\} \ge -1.$ We use 
$\mathrm {Herm}_{L,M}$ to denote the scheme of Hermitian 
$\mathcal {O}_F$-module homomorphisms from 
$L$ to 
$M$, which is a scheme of finite type over 
$\mathcal {O}_{F_0}$. More specifically, for an 
$\mathcal {O}_{F_0}$-algebra 
$R$, we define
\[ L_{R}:= L\otimes_{\mathcal{O}_{F_0}}R, \quad (x\otimes a,y\otimes b)_{R}:= \pi (x ,y )\otimes_{\mathcal{O}_{F_0}} ab\in \mathcal{O}_{F}\otimes_{\mathcal{O}_{F_0}} R \text{ where }x,y \in L, a,b\in R. \]
Then
\[ \mathrm{Herm}_{L, M}(R)=\{ \phi \in \mathrm{Hom}_{\mathcal{O}_F}(L_{R}, M_{R}) \mid (\phi(x),\phi(y))_{R}\equiv (x,y)_{R} \text{ for all } x,y \in L_{R}\}. \]
To simplify the notation, let 
$I(M,L,d)$ denote 
$\mathrm {Herm}_{L, M}(\mathcal {O}_{F_0}/(\pi _0^d))$. Then direct calculation shows that
\begin{equation} \alpha(M, L) = q^{-d n (2m -n)} |I(M,L,d)| \end{equation}
becomes constant for sufficiently large integers 
$d >0$. We call it the local density (of 
$M$ representing 
$L$).
Let 
$\mathcal {H}$ be the (Hermitian) hyperbolic plane with Gram matrix 
$\mathcal {H}=\bigl (\begin {smallmatrix} 0 & \pi ^{-1}\\ -\pi ^{-1} & 0 \end {smallmatrix}\bigr )$. One can show that there is a (local density) polynomial 
$\alpha (M,L,X)\in \mathbb {Q}[X]$ such that
\[ \alpha(M{\unicode{x29BA}}\mathcal{H}^k,L)=\alpha(M,L,q^{-2k}). \]
Define its derivative by
\[ \alpha'(M,L):= -\frac{\partial}{\partial X} \alpha(M,L,X)|_{X=1}. \]
As 
$\alpha (M,L, X)$ (respectively, 
$\alpha '(M,L)$) only depends on their Gram matrices 
$S$ and 
$T$, we will also denote it by 
$\alpha (S, T, X)$ (respectively, 
$\alpha '(S,T)$). Let 
$M$ be the unique unimodular Hermitian 
$\mathcal {O}_F$-lattice of rank 
$n$ with 
$\chi (M)=-\chi (L)$. The naive analogue of the local Kudla–Rapoport conjecture is
\begin{equation} \mathrm{Int}(L)=2 \frac{\alpha'(M,L)}{\alpha(M,M)}. \end{equation}
However, this conjectural formula is not even true for 
$n=2$ according to the main theorem of [Reference He, Shi and YangHSY23]. The analytic side of the conjecture needs to be modified.
1.2 The precise conjecture
By [Reference ShiShi18, Theorem 1.2], 
$\mathcal {Z}^\mathrm {Kra}(L)$ is empty when 
$L$ is not integral, so we have
\[ \mathrm{Int}(L)=0. \]
On the analytic side, the right-hand side of (1.6) is automatically zero only when 
$\mathrm {v}(L) \leq -2$, and is sometimes non-zero when 
$\mathrm {v}(L) =-1$. Thus, there should be correction terms involving Hermitian lattices 
$M$ with 
$\mathrm {v}(M) =-1$. By [Reference JacobowitzJac62], there are 
$n-1$ equivalent classes of Hermitian lattices which are direct sum of copies of 
$\mathcal {H}$ and unimodular lattices:
\begin{equation} \mathcal{H}_{n,i}^{\epsilon}:= \mathcal{H}^i{\unicode{x29BA}} I_{n-2i}^{\epsilon}, \quad \text{for } 1\leq i \leq \frac{n}{2}, \ \epsilon =\pm 1, \end{equation}
where we use 
$I_{n-2i}^{\epsilon }$ to denote the unimodular Hermitian lattice of rank 
$n-2i$ with 
$\chi (I_{n-2i}^{\epsilon })= \chi (\mathcal {H}_{n,i}^{\epsilon } )=\epsilon$. When 
$n=2r$ is even, we take 
$I_{0, \epsilon } =0$ and 
$\mathcal {H}_{n, r}^1 = \mathcal {H}^{r}$. Then the local arithmetic Siegel–Weil formula (also known as the Kudla–Rapoport conjecture at a ramified prime) should be of the following form:
\begin{equation} \mathrm{Int}(L) = 2 \frac{\alpha'(I_{n}^{-\epsilon}, L)}{\alpha(I_{n}^{-\epsilon}, I_{n}^{-\epsilon})} + \sum_{ i} c_{n,i}^\epsilon \frac{\alpha(\mathcal{H}_{n,i}^{\epsilon}, L)}{\alpha(I_{n}^{-\epsilon}, I_{n}^{-\epsilon})}, \end{equation}
where 
$\epsilon =\chi ( L)$. Since 
$\mathrm {Int}(\mathcal {H}_{n, j}^{\epsilon })=0$, we should have
\begin{equation} 2 \frac{\alpha'(I_{n}^{-\epsilon},\mathcal{H}_{n, j}^{\epsilon} )}{\alpha(I_{n}^{-\epsilon}, I_{n}^{-\epsilon})} + \sum_{ i} c_{n,i}^\epsilon \frac{\alpha(\mathcal{H}_{n,i}^{\epsilon},\mathcal{H}_{n,j}^{\epsilon})}{\alpha( I_{n}^{-\epsilon}, I_{n}^{-\epsilon})} =0. \end{equation}
This system of equations turns out to determine the coefficients 
$c_{n,i}^\epsilon$ uniquely by Theorem 6.1. We propose the following Kudla–Rapoport conjecture at a ramified prime.
Conjecture 1.1 The identity (1.8) always holds with the coefficients 
$c_{n,i}^\epsilon$ uniquely determined by (1.9).
For convenience, we set
\begin{equation} \partial \mathrm{Den}(L) =2 \frac{\alpha'(I_{n}^{-\epsilon}, L)}{\alpha(I_{n}^{-\epsilon},I_{n}^{-\epsilon} )} + \sum_{ i} c_{n,i}^\epsilon \frac{\alpha(\mathcal{H}_{n,i}^{\epsilon}, L)}{\alpha(I_{n}^{-\epsilon},I_{n}^{-\epsilon})}. \end{equation}
We remark that because 
$\mathrm {Int}(L)$ is always an integer, Conjecture 1.1 suggests that 
$\partial \mathrm {Den}(L)$ should be an integer, which is already not obvious.
The conjecture holds for 
$\mathcal {N}^\mathrm {Kra}_{2,\pm 1}$ by results in [Reference ShiShi22] and [Reference He, Shi and YangHSY23]. In this paper, we prove the conjecture for 
$n=3$ and provide some partial results in the general case.
Theorem 1.2 Conjecture 1.1 is true when 
$n=3$.
1.3 Special difference cycles
One of the novelties of this paper is the concept of special difference cycles. Let 
$L_1$ be an 
$\mathcal {O}_F$-lattice of 
$\mathbb {V}$ of rank 
$n_1 \le n$. Define the special difference cycle 
$\mathcal {D}(L_1)\in K_0 (\mathcal {N}^\mathrm {Kra})$ by
\begin{equation} \mathcal{D}(L_1)={}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L_1)+\sum_{i=1}^{n_1} (-1)^i q^{i(i-1)/2} \sum_{\substack{L_1 \subset L'\subset \frac{1}{\pi} L_1\\ \mathrm{dim}_{\mathbb{F}_q}(L'/L_1)=i}} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L')\in K_0(\mathcal{N}^\mathrm{Kra}). \end{equation}
Here 
$\mathcal {D}(L_1)$ can be seen as a higher codimensional analogue of the difference divisor first introduced in [Reference TerstiegeTer13, Definition 2.10]. By the definition and a 
$q$-adic linear-algebraic inclusion–exclusion principle, we have (see Lemma 2.16)
\begin{equation} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L_1) =\sum_{\substack{L'\, \mathrm{integral}\\ L_1 \subset L' \subset L_{1,F}}} \mathcal{D}(L'). \end{equation}
Here 
$L_F =L\otimes _{\mathcal {O}_F} F$ for an 
$\mathcal {O}_F$-lattice 
$L$. The above summation is, in fact, finite. Assume that we have a decomposition 
$L=L_1\oplus L_2$ of 
$\mathcal {O}_F$-lattices such that 
$L_i$ has rank 
$n_i$ and 
$n_1+n_2=n$. Define
\begin{equation} \mathrm{Int}(L)^{(n_1)}=\chi(\mathcal{N}^\mathrm{Kra}, \mathcal{D}(L_1)\cdot \mathcal{Z}^\mathrm{Kra}(L_2)), \end{equation}
where 
$\cdot$ is the product on 
$K_0(\mathcal {N}^\mathrm {Kra})$ induced by tensor product of complexes. Note that this definition, in fact, depends on the decomposition of 
$L$.
On analytic side, we define
\begin{equation} \partial \mathrm{Den}(L)^{(n_1)}:= \partial \mathrm{Den}(L)-\sum_{i=1}^{n_1} (-1)^{i-1} q^{i(i-1)/2} \sum_{\substack{L_1 \subset L_1' \subset L_{1, F} \\ \dim {L_1'/L_1}=i}} \partial \mathrm{Den}(L'_1\oplus L_2). \end{equation}
This definition again depends on the decomposition of 
$L$. What motivates the definition of 
$\partial \mathrm {Den}(L)^{(n_1)}$ and 
$\mathcal {D}(L_1)$ is the fact that 
$\partial \mathrm {Den}(L)^{(n_1)}$ is equal to the derivative of certain primitive local density polynomials, see [Reference KatsuradaKat99, Proposition 2.1] or Theorem 5.2 below. The analogue of (1.12) holds for 
$\partial \mathrm {Den}(L)^{(n_1)}$. As a consequence, we have the following theorem (see Theorem 5.6 for a refinement).
Theorem 1.3 Conjecture 1.1 is true if and only if for every lattice 
$L=L_1\oplus L_2$ such that 
$L_i$ has rank 
$n_i$, we have
\begin{equation} \mathrm{Int}(L)^{(n_1)}=\partial \mathrm{Den}(L)^{(n_1)}. \end{equation} We speculate that 
$\mathcal {D}(L_1)$ is of a simple form when 
$n_1=n-1$. One strong piece of evidence for this is that the ‘horizontal’ part of 
$\mathcal {D}(L_1)$ is either empty or isomorphic to one or two copies of 
$\mathrm {Spf}\, W_s$ where 
$W_s$ is the integer ring of an extension of 
$\breve {F}$ of degree 
$q^s$, see Proposition 4.6. Another evidence is that the intersection of 
$\mathcal {D}(L_1)$ with an exceptional divisor in 
$\mathcal {N}^\mathrm {Kra}$ is 
$\pm 1$ or 
$0$, see Lemma 3.9. When 
$n=3$, we show that 
$\mathcal {D}(L_1)$ has a simple decomposition, see Theorem 1.4.
1.4 The case 
$n=3$
The proof of Theorem 1.2 is divided into three cases, see § 11. For 
$\mathrm {v}(L)<0$, we show directly 
$\partial \mathrm {Den}(L)=\mathrm {Int}(L)=0$. The case 
$\mathrm {v}(L)=0$ is reduced to the case 
$n=2$, which was proved in [Reference ShiShi22] and [Reference He, Shi and YangHSY23]. For 
$\mathrm {v}(L) >0$, we prove that 
$\mathrm {Int}(L)^{(2)} = \partial \mathrm {Den}(L)^{(2)}$ for a decomposition 
$L=L^\flat {\unicode{x29BA}} \mathrm {Span}\{{\bf x}\}$, and then apply Theorem 1.3 (more precisely Theorem 5.6).
In order to prove 
$\mathrm {Int}(L)^{(2)} = \partial \mathrm {Den}(L)^{(2)}$, we need to understand the decomposition of 
$\mathcal {D}(L^\flat )$. We say a lattice 
$\Lambda \subset \mathbb {V}$ is a vertex lattice if 
$\pi \Lambda \subseteq \Lambda ^\sharp \subseteq \Lambda$ where 
$\Lambda ^\sharp$ is dual lattice of 
$\Lambda$ with respect to 
$h(\,{,}\,)$ and we call 
$t=\mathrm {dim}_{\mathbb {F}_q}(\Lambda /\Lambda ^\sharp )$ the type of 
$\Lambda$. This has to be an even integer between 
$0$ and 
$n$. We denote the set of vertex lattices of type 
$t$ by 
$\mathcal {V}^t$. When 
$n=3$, a type 
$2$ lattice 
$\Lambda _2$ corresponds to a line 
$\tilde {\mathcal {N}}_{\Lambda _2}\cong \mathbb {P}^1_k$ in 
$\mathcal {N}^\mathrm {Kra}_3$ and a type 
$0$ lattice 
$\Lambda _0$ corresponds to a divisor 
$\mathrm {Exc}_{\Lambda _0}\cong \mathbb {P}^2_k$. Let 
$H_{\Lambda _0}$ be the hyperplane class of 
$\mathrm {Exc}_{\Lambda _0}$. We have the following theorem.
Theorem 1.4 If 
$\mathrm {v}(L^\flat )>0$, we have the following decomposition of cycles in 
$\mathrm {Gr}^2 K_0(\mathcal {N}^\mathrm {Kra}_3)$
\[ \mathcal{D}(L^\flat)=\sum_{\substack{
\Lambda_2\in \mathcal{V}^2 \\ L^\flat \subset
\Lambda_2^\sharp}}
\biggl(2[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda_2}}]+\sum_{\substack{\Lambda_0\in\mathcal{V}^0\\\Lambda_0\subset
\Lambda_2}}H_{\Lambda_0}\biggr), \]
where 
$\mathrm {Gr}^\bullet K_0(\mathcal {N}^\mathrm {Kra}_3)$ is the associated graded ring of 
$K_0(\mathcal {N}^\mathrm {Kra}_3)$ with respect to the codimension filtration.
Theorem 1.4 is proved by intersecting 
$\mathcal {D}(L^\flat )$ with special divisors that are isomorphic to 
$\mathcal {N}^\mathrm {Kra}_{2,-1}$ and computing the intersection numbers in two different ways. One way relates the intersection numbers to the main result of [Reference ShiShi22]. The other way uses the decomposition in Theorem 1.4 and detects the multiplicity of each component that shows up on the right-hand side.
1.5 Global application
In the last part of the paper, we apply the local results above to the global intersection problem proposed by [Reference Kudla and RapoportKR14b]. For brevity and clarity of exposition we restrict our attention to the case when 
$F$ is an imaginary quadratic field. We remark here that our results can be applied to the case when 
$F$ is a general CM field given correct local assumptions. Let 
$\mathcal {M}^\mathrm {Kra}_{(1,n-1)}$ be the moduli functor over 
$\mathrm {Spec}\, \mathcal {O}_F$ which parametrizes principally polarized abelian varieties 
$A$ with an action 
$\iota :\mathcal {O}_F \rightarrow \mathrm {End}(A)$, a compatible polarization 
$\lambda : A\rightarrow A^\vee$ and a filtration 
$\mathcal {F}_A\subset \mathrm {Lie}\, A$ which satisfies the signature 
$(1,n-1)$ condition (see § 12.1). Let 
$V$ be a Hermitian vector space over 
$F$ of signature 
$(n-1,1)$ containing a self-dual lattice 
$L$. The lattice 
$L$ determines an open and closed substack
\[ \mathcal{M}\subset\mathcal{M}_{(0,1)}\times_{\mathrm{Spec}\, \mathcal{O}_F} \mathcal{M}^\mathrm{Kra}_{(1,n-1)} \]
which is an integral model of a unitary Shimura variety. For a point in 
$\mathcal {M}(S)$ (
$S$ an 
$\mathcal {O}_F$-scheme), i.e. a pair 
$(E,\iota _0,\lambda _0)\in \mathcal {M}_{(0,1)}(S)$, 
$(A,\iota,\lambda,\mathcal {F}_A)\in \mathcal {M}^\mathrm {Kra}_{(1,n-1)}(S)$, define the locally free 
$\mathcal {O}_F$-module
\[ V'(E,A)=\mathrm{Hom}_{\mathcal{O}_F}(E,A). \]
It is equipped with the Hermitian form 
$h'(x,y)=\iota _0^{-1}(\lambda _0^{-1}\circ y^\vee \circ \lambda \circ x)$. For a 
$m \times m$ non-singular Hermitian matrix 
$T$ with values in 
$\mathcal {O}_F$, let 
$\mathcal {Z}(T)$ be the stack of collections 
$(E,\iota _0,\lambda _0, A,\iota, \lambda,\mathcal {F}_A,{\bf x})$ such that 
$(E,\iota _0,\lambda _0, A,\iota, \lambda,\mathcal {F}_A)\in \mathcal {M}(S)$, 
${\bf x} \in V'(E,A)^m$ with 
$h'({\bf x},{\bf x})=T$. Then 
$\mathcal {Z}(T)$ is representable by a Deligne–Mumford stack which is finite and unramified over 
$\mathcal {M}$ (see [Reference Kudla and RapoportKR14b, Proposition 2.9]). When 
$t\in \mathbb {Z}_{>0}$, each component of 
$\mathcal {Z}(t)$ can be viewed as a divisor by [Reference HowardHow15, Proposition 3.2.3]. In general, 
$\mathcal {Z}(T)$ does not necessarily have the expected codimension which is the rank of 
$T$.
Let 
$\mathcal {C} =\{ \mathcal {C}_p\}$ be an incoherent collection of local Hermitian spaces of rank 
$n$ associated to 
$V$ such that 
$\mathcal {C}_\ell \cong V_\ell$ for all finite 
$\ell$ and 
$\mathcal {C}_\infty$ is positive definite. It is ‘incoherent’ in the sense that it does not come from a global Hermitian space. For a non-singular Hermitian matrix 
$T$ of rank 
$n$ with values in 
$\mathcal {O}_F$, let 
$V_T$ be the Hermitian space with Gram matrix 
$T$. Define
\begin{equation} \mathrm{Diff}(T,\mathcal{C}):= \{p \text{ a place of } \mathbb{Q} \mid \mathcal{C}_p \text{ is not isomorphic to } (V_T)_p\}. \end{equation}
Then 
$\mathcal {Z}(T)$ is empty if 
$|\mathrm {Diff}(T,\mathcal {C})|>1$. If 
$\mathrm {Diff}(T,\mathcal {C})=\{p\}$ for a finite prime 
$p$ inert or ramified in 
$F$, then the support of 
$\mathcal {Z}(T)$ is on the supersingular locus of 
$\mathcal {M}$ over 
$\mathrm {Spec}\, \mathbb {F}_p$. Let 
$e$ be the ramification index of 
$F_p/\mathbb {Q}_p$. Define the arithmetic degree
\begin{equation}
\widehat{\mathrm{deg}}_T=\chi(\mathcal{Z}(T),\mathcal{O}_{\mathcal{Z}(t_1)}\otimes^{\mathbb{L}}
\cdots \otimes^{\mathbb{L}}
\mathcal{O}_{\mathcal{Z}(t_n)})\cdot \log p^{2/e},
\end{equation}
where 
$\otimes ^\mathbb {L}$ stands for derived tensor product on the category of coherent sheaves on 
$\mathcal {M}$, 
$\chi$ is the Euler characteristic and 
$t_i$ (
$1 \le i \le n$) are the diagonal entries of 
$T$. When 
$\mathrm {Diff}(T,\mathcal {C})=\{\infty \}$, then 
$\mathcal {Z}(T)$ is, in fact, empty and one can use integration of a green current to define the arithmetic degree 
$\widehat {\mathrm {deg}}_{T}(v)$ with the parameter 
$v$ being a positive-definite Hermitian matrix 
$v$ of order 
$n$ (which will be the imaginary part of 
$\tau$); see, for example, [Reference Li and ZhangLZ22a, § 15.3].
On the analytic side, we consider an incoherent Eisenstein series 
$E(\tau,s,\Phi )$ for a non-standard section 
$\Phi$ in a degenerate principal series representation of 
$\mathrm {U}(n,n)(\mathbb {A})$, see § 12.2. Here 
$\tau$ is in the Hermitian Siegel upper half space
\begin{equation} \mathbb{H}_n=\{\tau =u+i v\mid u\in \mathrm{Herm}_n, v\in \mathrm{Herm}_{n,>0}\}, \end{equation}
where 
$\mathrm {Herm}_n$ (respectively, 
$\mathrm {Herm}_{n,>0}$) is the set of 
$n\times n$ (positive-definite) Hermitian matrices with values in 
$\mathbb {C}$ and 
$s\in \mathbb {C}$. Our local conjecture and result imply the following theorem, which extends [Reference Li and ZhangLZ22a, Theorem 1.3.1] to include ramified primes.
Theorem 1.5 (Arithmetic Siegel–Weil formula for non-singular coefficients)
Assume that the fundamental discriminant of 
$F$ is 
$d_F \equiv 1 \pmod 8$ and that Conjecture 1.1 holds for every 
$F_p$ with 
$p|d_F$. For any non-singular Hermitian matrix 
$T$ with values in 
$\mathcal {O}_F$ of size 
$n$, we have
\[ E'_T(\tau, 0,\Phi)=C \cdot \widehat{\mathrm{deg}}_T(v) \cdot q^T, \quad q^T=\exp(2\pi i \mathrm{tr}(T\tau)), \]
where 
$E'_T(\tau, 0,\Phi )$ is the 
$T$th Fourier coefficient of 
$E'(\tau, 0,\Phi )$ and 
$C$ is a constant that only depends on 
$F$ and 
$L$. In particular, the arithmetic Siegel–Weil formula holds for 
$n=2, 3$ for non-singular 
$T$.
In a very recent joint work with Chao Li [Reference He, Li, Shi and YangHLSY22], we proved Conjecture 1.1, and so this theorem is now unconditional.
1.6 Notation
For 
$\mathcal {O}_F$-lattices (respectively, 
$\mathcal {O}_{\breve F}$-lattices) 
$L$ and 
$L'$, we write 
$L \stackrel {t}{\subset } L'$ if 
$L\subset L' \subset ({1}/{\pi }) L$ and 
$\dim _{\mathbb {F}_q}(L'/L)=t$ (respectively, 
$\dim _k (L'/L)=t$). We say a vector 
$v\in L$ is primitive if 
$({1}/{\pi })v\notin L$.
Throughout the paper, we always assume a Hermitian lattice is non-degenerate. For Hermitian lattices 
$L$ and 
$L'$, we use 
$L{\unicode{x29BA}} L'$ to denote orthogonal direct sum, and 
$L \oplus L'$ as direct sum of lattices. Given a Hermitian lattice 
$L$ with Hermitian form 
$(\, ,\, )$, we consider two different dual lattices of 
$L$. We use 
$L^{\sharp }$ (respectively, 
$L^{\vee })$ to denote the dual lattice of 
$L$ with respect to 
$(\,{,}\,)$ (respectively, 
$\mathrm {tr}_{F/F_0}(\,{,}\,))$. Recall that 
$\mathrm {v}(L)$ is defined to be 
$\min \{\mathrm {v}_\pi ( h(v,v'))\mid v,v'\in L\}$. For each Hermitian lattice 
$L$, there exists a Jordan decomposition 
$L={\unicode{x29BA}} _{i\ge s} L_i$ such that 
$L_i^{\sharp }=\pi ^{-i} L_i$. We call 
$L$ integral if 
$s\ge 0$. For an integral lattice 
$L$, we define
\[ t(L):= \sum_{i\ge 1}\mathrm{rank}_{\mathcal{O}_F}(L_i). \]
Following [Reference Li and LiuLL22, Definition 2.11], for a lattice 
$L$ with Hermitian form 
$(\, ,\, )$, we may find a basis of 
$L$ whose Gram matrix is
\[ \big(\beta_{1} \pi^{2 b_{1}}\big)
\oplus \cdots \oplus\big(\beta_{s} \pi^{2 b_{s}}\big)
\oplus\left(\begin{array}{@{}cc@{}} 0 & \pi^{2 c_{1}+1} \\
-\pi^{2 c_{1}+1} & 0 \end{array}\right) \oplus \cdots
\oplus\left(\begin{array}{@{}cc@{}} 0 & \pi^{2 c_{t}+1} \\
-\pi^{2 c_{t}+1} & 0 \end{array}\right) \]
for some 
$\beta _{1}, \ldots, \beta _{s} \in \mathcal {O}_{F_0}^{\times }$ and 
$b_{1}, \ldots, b_{s}, c_{1}, \ldots, c_{t} \in \mathbb {Z}$. Moreover, we define its (unitary) fundamental invariants 
$(a_1,\ldots,a_n)$ to be the unique non-decreasing rearrangement of 
$(2 b_1,\ldots,2b_s,2c_1+1,\ldots,2c_t+1)$. The partial order of 
$\mathbb {Z}^n$ induces a partial order on the set of fundamental invariants.
Let 
$\mathcal {H}_i=\bigl (\!\begin {smallmatrix}0 & \pi ^{i}\\ (-\pi )^{i} & 0 \end {smallmatrix}\!\bigr )$ and 
$\mathcal {H}=\mathcal {H}_{-1}$. We also use it to denote a Hermitian lattice with Gram matrix 
$\mathcal {H}_i$. Given a Hermitian lattice 
$M$, we use 
$M^{[k]}$ to denote 
$M{\unicode{x29BA}} \mathcal {H}^k$. We use 
$I_m^\epsilon$ to denote a unimodular Hermitian lattice of rank 
$m$ and 
$\chi (I_m^\epsilon )=\epsilon$. For a Hermitian matrix 
$T$, we define 
$\mathrm {v}(T)=\mathrm {v}(L)$ where 
$L$ is a lattice whose Gram matrix is 
$T$. We use 
$\mathrm {Herm}_n(F)$ to denote the set of Hermitian matrices over 
$F$ of size 
$n$. When there is no confusion, we also simply denote it as 
$\mathrm {Herm}_n$. For 
$T,T'\in \mathrm {Herm}_n(F)$, we say 
$T$ is equivalent to 
$T'$ if there is a 
$U\in \mathrm {GL}_n(\mathcal {O}_F)$ such that 
$U^*TU=T'$, where 
$U^*= {}^t \bar U$. In this case, we denote it as 
$T\approx T'$.
For 
$t\in \mathcal {O}_{F_0}$, let 
$\mathrm {v}(t):= \mathrm {val}_{\pi _0}(t)$ and write 
$t=t_0(-\pi _0)^{\mathrm {v}(t)}$. For 
$x\in \mathbb {V}$, we set 
$q(x)=(x,x)$ and 
$\mathrm {v}(x)=\mathrm {v}(q(x))$. We use 
$\langle t \rangle$ to denote a lattice 
$\mathcal {O}_F x$ of rank one with 
$q(x)=t$.
The notation in each section that is not mentioned here will be explained at the very beginning of the section.
1.7 The structure of the paper
The paper is divided into three parts. In Part 1, we prove some facts about special cycles for arbitrary 
$n$. More specifically, in § 2 we recall some basic facts about 
$\mathcal {N}^\mathrm {Kra}$ and define special cycles and special difference cycles on it. In § 3, we compute the intersection number between special cycles and the exceptional divisors. In § 4, we prove a decomposition theorem for the horizontal component of 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}(L^\flat )$ when 
$L^\flat$ has rank 
$n-1$.
Part 2 is about Hermitian local densities. In § 5, we study induction formulas of local density polynomials and relate the local density polynomials with primitive local densities. In § 6, we show that the coefficients 
$c_{n,i}^\epsilon$ in (1.9) are uniquely determined and give an algorithm to compute them. In §§ 7 and 8, we compute the local density polynomials when 
$n\leq 3$.
In Part 3 we prove Theorem 1.2, i.e. Conjecture 1.1 for 
$n=3$. In § 9, we study the reduced locus of the special cycles for 
$n=3$. In § 10, we decompose 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}(L^\flat )$ for 
$L^\flat$ of rank 
$2$ and of valuation 0, and compute the intersection number of 
$\tilde {\mathcal {N}}_{\Lambda _2}$ with 
$\mathcal {Z}^\mathrm {Kra}({\bf x})$. Finally, we prove Theorem 1.4, finish the proof of Theorem 1.2 in § 11, and explain its global applications in § 12.
In Appendix A, we compute the primitive local densities that are used in Part 2 of the paper.
Part 1. The geometric side
2. RZ space and special cycle
We denote 
$\bar {a}$ the Galois conjugate of 
$a\in F$ over 
$F_0$. Let 
$\mathrm {Nilp}\, \mathcal {O}_{\breve F}$ be the category of 
$\mathcal {O}_{\breve F}$-schemes 
$S$ such that 
$\pi$ is locally nilpotent on 
$S$. For such an 
$S$, denote its special fiber 
$S\times _{\mathrm {Spf}\, \mathcal {O}_{\breve F}} \mathrm {Spec}\, k$ by 
$\bar S$. Let 
$\sigma$ be the Frobenius element of 
$\breve {F}_0/F_0$.
2.1 RZ spaces
Let 
$S\in \mathrm {Nilp}\, \mathcal {O}_{\breve F}$. A 
$p$-divisible strict 
$\mathcal {O}_{F_0}$-module over 
$S$ is a 
$p$-divisible group over 
$S$ with an 
$\mathcal {O}_{F_0}$ action whose induced action on its Lie algebra is via the structural morphism 
$\mathcal {O}_{F_0}\rightarrow \mathcal {O}_S$.
Definition 2.1 A formal Hermitian 
$\mathcal {O}_F$-module of dimension 
$n$ over 
$S$ is a triple 
$(X,\iota,\lambda )$ where 
$X$ is a supersingular 
$p$-divisible strict 
$\mathcal {O}_{F_0}$-module over 
$S$ of dimension 
$n$ and 
$F_0$-height 
$2n$ (supersingular means the relative Dieudonné module of 
$X$ at each geometric point of 
$S$ has slope 
$\tfrac {1}{2}$), 
$\iota :\mathcal {O}_F\rightarrow \mathrm {End}(X)$ is an 
$\mathcal {O}_F$-action and 
$\lambda :X\rightarrow X^\vee$ is a principal polarization in the category of strict 
$\mathcal {O}_{F_0}$-modules such that the Rosati involution induced by 
$\lambda$ is the Galois conjugation of 
$F/F_0$ when restricted on 
$\mathcal {O}_F$. We say 
$(X,\iota,\lambda )$ satisfies the signature condition 
$(1,n-1)$ if for all 
$a\in \mathcal {O}_F$ we have:
(i)
$\mathrm {char}(\iota (a)\mid \mathrm {Lie}\, X)=(T-s(a))\cdot (T-s(\bar {a}))^{n-1}$ where $s:\mathcal {O}_F\rightarrow \mathcal {O}_S$ is the structure morphism;(ii) the wedge condition proposed in [Reference PappasPap00],
\[ \wedge^n(\iota(a)-s(a)\mid \mathrm{Lie}\, X)=0,\quad \wedge^2(\iota(a)-s(\bar{a})\mid \mathrm{Lie}\, X)=0. \]
Let 
$(\mathbb {X},\iota _\mathbb {X},\lambda _\mathbb {X})$ be a formal Hermitian 
$\mathcal {O}_F$-module of dimension 
$n$ over 
$k$, and 
$N$ be its rational relative Dieudonne module. Then 
$N$ is an 
$2n$-dimensional 
$\breve {F}_0$-vector space equipped with a 
$\sigma$-linear operator 
$\mathbf {F}$ and a 
$\sigma ^{-1}$-linear operator 
$\mathbf {V}$. The 
$\mathcal {O}_F$-action 
$\iota _\mathbb {X}:\mathcal {O}_F\rightarrow \mathrm {End}(\mathbb {X})$ induces on 
$N$ an 
$\mathcal {O}_F$-action commuting with 
$\mathbf {F}$ and 
$\mathbf {V}$. We still denote this induced action by 
$\iota _\mathbb {X}$ and denote 
$\iota _\mathbb {X}(\pi )$ by 
$\pi$. Let 
$\tau := \pi \mathbf {V}^{-1}$ and 
$C:= N^\tau$. Then 
$C$ is an 
$n$-dimensional 
$F$-vector space equipped with a Hermitian form 
$(\,{,}\,)_\mathbb {X}$ defined using the polarization 
$\lambda _\mathbb {X}$, see [Reference ShiShi18, Equation (2.7)]. When 
$n$ is odd, there is a unique choice of 
$(\mathbb {X},\iota _\mathbb {X},\lambda _\mathbb {X})$ up to quasi-isogenies that preserves the polarization by a factor in 
$\mathcal {O}_{F_0}^\times$. When 
$n$ is even, there are two such choices according to the sign 
$\epsilon =\chi (C)$ (see (1.1)) of 
$C$. See [Reference ShiShi18, Remark 2.16] and [Reference Rapoport, Terstiege and WilsonRTW14, Remark 4.2]. Fix a formal Hermitian 
$\mathcal {O}_F$-module 
$(\mathbb {Y},\iota _\mathbb {Y},\lambda _\mathbb {Y})$ of signature 
$(0,1)$ over 
$\mathrm {Spec}\, k$. It is unique up to 
$\mathcal {O}_F$-linear isomorphisms. Define
\begin{equation} \mathbb{V}=\mathrm{Hom}_{\mathcal{O}_F}(\mathbb{Y},\mathbb{X})\otimes \mathbb{Q}, \end{equation}which is equipped with a Hermitian form
\begin{equation} h(x,y)=\lambda_\mathbb{Y}^{-1}\circ y^\vee \circ \lambda_\mathbb{X} \circ x\in \mathrm{End}^0_F(\mathbb{Y})\overset{\sim}{\rightarrow}F, \end{equation}
where 
$y^\vee$ is the dual quasi-homomorphism of 
$y$ and 
$\mathrm {End}^0_F(\mathbb {Y})$ is the ring of 
$F$-linear quasi-endomorphisms of 
$\mathbb {Y}$. The Hermitian spaces 
$(\mathbb {V},h(\,{,}\,))$ and 
$(C,(, )_\mathbb {X})$ are related by the 
$F$-linear isomorphism
\begin{equation} b: \mathbb{V}\rightarrow C, \quad {\bf x}\mapsto {\bf x}(e), \end{equation}
where 
$e$ is a generator of the relative covariant Dieudonné module 
$M(\mathbb {Y})$ of 
$\mathbb {Y}$. Let 
$(\,{,}\,)_\mathbb {Y}$ be the analogue of 
$(\,{,}\,)_\mathbb {X}$ for 
$\mathbb {Y}$, namely the Hermitian form on the rational relative Dieudonné module of 
$\mathbb {Y}$ defined by 
$\lambda _\mathbb {Y}$. By [Reference ShiShi18, Lemma 3.6], we have
\begin{equation} h({\bf x},{\bf x})(e,e)_\mathbb{Y}=(b({\bf x}),b({\bf x}))_\mathbb{X}. \end{equation}
By scaling the Hermitian form 
$(\,{,}\,)_\mathbb {Y}$ we can assume that
\[ (e,e)_\mathbb{Y}=1, \]
so 
$\mathbb {V}$ and 
$C$ are isomorphic as Hermitian spaces. We will sometimes identify 
$\mathbb {V}$ and 
$C$.
Definition 2.2 Fix a formal Hermitian 
$\mathcal {O}_F$-module 
$(\mathbb {X},\iota _\mathbb {X},\lambda _\mathbb {X})$ of dimension 
$n$ over 
$k$ with the sign 
$\epsilon =\chi (C)$. The moduli space 
$\mathcal {N}_{n,\epsilon }^\mathrm {Pap}$ is the functor sending each 
$S\in \mathrm {Nilp}\, \mathcal {O}_{\breve F}$ to the groupoid of isomorphism classes of quadruples 
$(X,\iota,\lambda,\rho )$ where 
$(X,\iota,\lambda )$ is a formal Hermitian 
$\mathcal {O}_F$-module over 
$S$ of signature 
$(1,n-1)$ and 
$\rho :X\times _{S}\bar {S} \rightarrow \mathbb {X} \times _{\mathrm {Spec}\, k} \bar {S}$ is a quasi-morphism of formal 
$\mathcal {O}_F$-modules of height 
$0$. An isomorphism between two such quadruples 
$(X,\iota,\lambda,\rho )$ and 
$(X',\iota ',\lambda ',\rho ')$ is given by an 
$\mathcal {O}_F$-linear isomorphism 
$\alpha :X\rightarrow X'$ such that 
$\rho '\circ (\alpha \times _S \bar {S})=\rho$ and 
$\alpha ^*(\lambda ')$ is a 
$\mathcal {O}_{F_0}^\times$ multiple of 
$\lambda$. We drop the subscript 
$\epsilon$ in 
$\mathcal {N}_{n,\epsilon }^\mathrm {Pap}$ when we do not emphasize on the sign.
By the discussion before (2.1), when 
$n$ is odd, two different choices of 
$\epsilon$ give us isomorphic moduli spaces. When 
$n$ is even, two different choices of 
$\epsilon$ give us two sets of non-isomorphic moduli spaces. By [Reference Rapoport, Terstiege and WilsonRTW14], 
$\mathcal {N}_n^\mathrm {Pap}$ is representable by a formal scheme flat and of relative dimension 
$n-1$ over 
$\mathrm {Spf}\, \mathcal {O}_{\breve F}$. We remark here that although [Reference Rapoport, Terstiege and WilsonRTW14] works on the category of 
$p$-divisible groups (namely when 
$F_0=\mathbb {Q}_p$), their arguments and results easily extend to the category of strict formal 
$\mathcal {O}_{F_0}$-modules using relative Dieudonné theory or more generally the relative display theory developed in [Reference Ahsendorf, Cheng and ZinkACZ16]. When 
$n=1$, we have 
$\mathcal {N}_1^\mathrm {Pap}\cong \mathrm {Spf}\, \mathcal {O}_{\breve F}$. The universal Hermitian 
$\mathcal {O}_F$-module over 
$\mathcal {N}_1^\mathrm {Pap}$ is the canonical lifting 
$(\mathcal {G},\iota _\mathcal {G},\lambda _\mathcal {G},\rho _\mathcal {G})$ of 
$(\mathbb {Y},\iota _\mathbb {Y},\lambda _\mathbb {Y})$ to 
$\mathrm {Spf}\, \mathcal {O}_{\breve F}$ in the sense of [Reference GrossGro86]. When 
$n>1$, 
$\mathcal {N}_n^\mathrm {Pap}$ is regular outside the set of superspecial points over 
$\mathrm {Spec}\, k$, which are the points characterized by the condition 
$\iota (\pi )|_{\mathrm {Lie}\, X}=0$. The set of superspecial points is in fact the set of type 
$0$ lattices (see § 2.3), hence is isolated and we denote it by Sing.
Definition 2.3 Fix 
$(\mathbb {X},\iota _\mathbb {X},\lambda _\mathbb {X})$ be as in Definition 2.2. The moduli space 
$\mathcal {N}_{n,\epsilon }^\mathrm {Kra}$ is the functor sending each 
$S\in \mathrm {Nilp}\, \mathcal {O}_{\breve F}$ to the groupoid of isomorphism classes of quintuples 
$(X,\iota,\lambda,\rho,\mathcal {F})$ where 
$(X,\iota,\lambda,\rho )\in \mathcal {N}_{n,\epsilon }^\mathrm {Pap}(S)$ and 
$\mathcal {F}$ is a locally free direct summand of 
$\mathrm {Lie}\, X$ of rank 
$n-1$ as an 
$\mathcal {O}_S$-module such that 
$\mathcal {O}_F$ acts on 
$\mathrm {Lie}\, X/\mathcal {F}$ by the structural morphism and acts on 
$\mathcal {F}$ by the Galois conjugate of the structural morphism. An isomorphism between two such quintuples 
$(X,\iota,\lambda,\rho,\mathcal {F})$ and 
$(X',\iota ',\lambda ',\rho ',\mathcal {F}')$ is an isomorphism 
$\alpha :(X,\iota,\lambda,\rho )\rightarrow (X',\iota ',\lambda ',\rho ')$ in 
$\mathcal {N}_{n,\epsilon }^\mathrm {Pap}(S)$ such that 
$\alpha ^*(\mathcal {F}')=\mathcal {F}$. Again we drop the subscript 
$\epsilon$ in 
$\mathcal {N}_{n,\epsilon }^\mathrm {Kra}$ when we do not emphasize on the sign.
By [Reference KrämerKrä03] (see also [Reference ShiShi22, Proposition 2.7]), the natural forgetful functor 
$\Phi :\mathcal {N}_n^\mathrm {Kra}\rightarrow \mathcal {N}_n^\mathrm {Pap}$ forgetting 
$\mathcal {F}$ is the blow up of 
$\mathcal {N}_n^\mathrm {Pap}$ along its singular locus Sing. For each point 
$\Lambda \in \mathrm {Sing}$, its inverse image 
$\Phi ^{-1}(\Lambda )$ is an exceptional divisor 
$\mathrm {Exc}_\Lambda$ isomorphic to 
$\mathbb {P}^{n-1}_k$.
2.2 Special cycles
Definition 2.4 For an 
$\mathcal {O}_F$-lattice 
$L$ of 
$\mathbb {V}$, define 
$\mathcal {Z}^\mathrm {Pap}(L)$ to be the subfunctor of 
$\mathcal {N}_n^\mathrm {Pap}$ sending each 
$S\in \mathrm {Nilp}\, \mathcal {O}_{\breve F}$ to the isomorphism classes of tuples 
$(X,\iota,\lambda,\rho )\in \mathcal {N}_n^\mathrm {Pap}(S)$ such that for any 
$x\in L$ the quasi-homomorphism
\[ \rho^{-1}\circ x\circ \rho_\mathcal{G}: \mathcal{G}\times_{S} \bar{S} \rightarrow X\times_{S} \bar{S} \]
extends to a homomorphism 
$\mathcal {G}_S\rightarrow X$. For 
${\bf x}\in \mathbb {V}^m$, we let 
$\mathcal {Z}^\mathrm {Pap}({\bf x}):= \mathcal {Z}^\mathrm {Pap} (L)$ where 
$L=\mathrm {Span} \{{\bf x}\}$. Let
\[ \mathcal{Z}^\mathrm{Kra}({\bf x})=\mathcal{Z}^\mathrm{Kra}(L):= \mathcal{Z}^\mathrm{Pap}(L)\times_{\mathcal{N}_n^\mathrm{Pap}} \mathcal{N}_n^\mathrm{Kra}. \]
By Grothendieck–Messing theory 
$\mathcal {Z}^\mathrm {Pap}(L)$ (hence, 
$\mathcal {Z}^\mathrm {Kra}(L)$) is a closed formal subscheme of 
$\mathcal {N}_n^\mathrm {Pap}$ 
$(\mathcal{N}_n^{\mathrm{Kra}})$. We sometimes add the subscript 
$_{n,\epsilon }$ to 
$\mathcal {Z}^\mathrm {Pap}(L)$, 
$\mathcal {Z}^\mathrm {Pap}({\bf x})$, 
$\mathcal {Z}^\mathrm {Kra}(L)$ and 
$\mathcal {Z}^\mathrm {Kra}({\bf x})$ to indicate their ambient moduli spaces.
Definition 2.5 For an 
$\mathcal {O}_F$-lattice 
$L\subset \mathbb {V}$, define 
$\tilde {\mathcal {Z}}(L)$ to be the strict transform (see the definition after [Reference HartshorneHar13, Chapter II and Corrollary 7.15]) of 
$\mathcal {Z}^\mathrm {Pap}(L)$ under the blow up 
$\mathcal {N}_n^\mathrm {Kra}\rightarrow \mathcal {N}_n^\mathrm {Pap}$.
Proposition 2.6 Suppose 
$\chi (\mathbb {V})=\epsilon$. Let 
$L$ be a self-dual lattice of rank 
$m$ in 
$\mathbb {V}$ with 
$\eta =\chi (L)$. We have
\[
\mathcal{Z}_{n,\epsilon}^\mathrm{Pap}(L)\cong
\mathcal{N}_{n-m,\epsilon \eta}^\mathrm{Pap}\quad \text{and}\quad
\tilde{\mathcal{Z}}_{n,\epsilon}(L)\cong \mathcal{N}_{n-m,\epsilon \eta}^\mathrm{Kra}.
\]
Proof. Let us start with the case 
$L=\mathrm {Span}\{{\bf x}_0\}$ where 
${\bf x}_0\in \mathbb {V}$. Assume that 
$u=h({\bf x}_0,{\bf x}_0)$. Multiplying the Hermitian form 
$(\,{,}\,)_\mathbb {X}$ on 
$C$ by 
$u^{-1}$ does not affect the various moduli spaces involved. Thus, we can perform this and assume that 
$h({\bf x}_0,{\bf x}_0)=1$. Moreover, the sign of its orthogonal complement in 
$\mathbb {V}$ becomes
\[ \epsilon_1=\epsilon \cdot \chi(u^{-1})\cdot \chi(u^{-(n-1)})\cdot\chi(-1)^{n-1}=\epsilon\chi(u)^{n}\chi(-1)^{n-1}. \]
Then for 
$(X,\iota,\lambda,\rho )\in \mathcal {Z}^\mathrm {Pap}_{n,\epsilon }({\bf x}_0)(S)$, we define
\[ {\bf x}_0^*:= \lambda_\mathcal{G}^{-1}\circ {\bf x}_0^\vee\circ \lambda, \ e:= {\bf x}_0\circ {\bf x}_0^* \in \mathrm{End}(X). \]
By the fact that 
$h({\bf x}_0,{\bf x}_0)=1$ we know that 
$e$ is an idempotent. It is routine to check that
\[ ((1-e)X,(1-e)\iota, (1-e^\vee)\lambda (1-e),\rho (1-e)) \]
is an object in 
$\mathcal {N}_{n-1,\epsilon _1}^\mathrm {Pap}(S)$. Conversely given 
$(Y,\iota _Y,\lambda _Y,\rho _Y)\in \mathcal {N}_{n-1,\epsilon _1}^\mathrm {Pap}(S)$, the object
\[ (Y\times \mathcal{G}_S, \iota_Y\times \iota_{\mathcal{G}_S}, \lambda_Y\times \lambda_{\mathcal{G}_S},g\circ(\rho_Y\times \rho_{\mathcal{G}_S})) \]
is in 
$\mathcal {Z}_{n,\epsilon }^\mathrm {Pap}({\bf x}_0)(S)$ where 
$g\in \mathrm {U}(\mathbb {V})$ such that 
$g^{-1}{\bf x}_0$ is the inclusion 
$0\times \mathrm {id}:\mathbb {Y} \rightarrow \mathbb {X}_{n-1} \times \mathbb {Y}$ where 
$\mathbb {X}_{n-1}$ is the framing object of 
$\mathcal {N}_{n-1,\epsilon _1}^\mathrm {Pap}$. The above two functors are inverse to each other. This shows that 
$\mathcal {Z}_{n,\epsilon }^\mathrm {Pap}({\bf x}_0)\cong \mathcal {N}_{n-1,\epsilon _1}^\mathrm {Pap}$. For general 
$L$ of rank 
$m$ and determinant 
$u$, find a basis with Gram matrix 
$\{1,\ldots,1,u\}$ and apply the above result repeatedly. Thus, we have 
$\mathcal {Z}_{n,\epsilon }^\mathrm {Pap}(L)\cong \mathcal {N}_{n-m,\epsilon _m}^\mathrm {Pap}$ where
\[ \epsilon_m=\epsilon \chi(u)^{n-m+1}\chi(-1)^{(n-m)m+{m(m-1)}/{2}}. \]
Note that by scaling the Hermitian form by 
$(-1)^m u$ again we have a 
$\mathcal {N}_{n-m,\epsilon _m}^\mathrm {Pap}=\mathcal {N}_{n-m,\epsilon \eta }^\mathrm {Pap}$. It then follows from [Reference HartshorneHar13, Chapter II and Corollary 7.15] that 
$\tilde {\mathcal {Z}}_{n,\epsilon }(L)$ is the blow up of 
$\mathcal {Z}_{n,\epsilon }^\mathrm {Pap}(L)$ along its superspecial points, which is 
$\mathcal {N}_{n-m,\epsilon \eta }^\mathrm {Kra}$.
Corollary 2.7 Let 
$L$ be as in Proposition 2.6 and 
$\mathbf {y}\in \mathbb {V}$ such that 
$\mathbf {y}\bot L$. Then
\[ \mathcal{Z}_{n,\epsilon}^\mathrm{Kra}(\mathbf{y})\cap\tilde{\mathcal{Z}}_{n,\epsilon}(L)\cong\mathcal{Z}_{n-m,\epsilon \eta}^\mathrm{Kra}(\mathbf{y}). \]
Remark 2.8 It follows directly from the definition that 
$\tilde {\mathcal {Z}}(L)$ is a closed sub-formal scheme of 
$\tilde {\mathcal {Z}}({\bf x}_1)\cap \cdots \cap \tilde {\mathcal {Z}}({\bf x}_r)$ if 
$\{{\bf x}_1,\ldots,{\bf x}_r\}$ is a basis of 
$L$. However, in general, these two cannot be identified.
2.3 Bruhat–Tits stratification
For an 
$\mathcal {O}_{\breve F}$-lattice 
$M$ of 
$N$, define 
$M^\sharp$ to be the dual lattice of 
$M$ with respect to the form 
$(\,{,}\,)_\mathbb {X}$. Recall the following results.
Proposition 2.9 [Reference Rapoport, Terstiege and WilsonRTW14, Proposition 2.2 and 2.4]
Let 
$\mathcal {N}(k)$ be the set of 
$\mathcal {O}_{\breve F}$-lattices
\[ \mathcal{N}(k)=\{M\subset C\otimes_F \breve{F}\mid M^{\sharp}=M,\, \pi \tau(M) \subset M \subset \pi^{-1} \tau(M),\, \dim_{k} (M + \tau(M))/M \le 1 \}. \]
Then the map
$$ {\mathcal N}^{\mathrm Pap}(k) \rightarrow {N}(k), \quad x = (X, \iota, \lambda, \rho) \mapsto M(x)= \rho (M(X)) \subset N $$
is a bijection.
We say a lattice 
$\Lambda \subset C$ is a vertex lattice if 
$\pi \Lambda \subseteq \Lambda ^\sharp \subseteq \Lambda$ where 
$\Lambda ^\sharp$ is dual lattice of 
$\Lambda$ with respect to 
$(\,{,}\,)_\mathbb {X}$, and we call 
$t=\mathrm {dim}_{\mathbb {F}_q}(\Lambda /\Lambda ^\sharp )$ the type of 
$\Lambda$. We denote the set of vertex lattices (respectively, of type 
$t$) by 
$\mathcal {V}$ (respectively, 
$\mathcal {V}^t$). We say two vertex lattices 
$\Lambda _1$ and 
$\Lambda _2$ are neighbours if 
$\Lambda _1\subset \Lambda _2$ or 
$\Lambda _2\subset \Lambda _1$. Then we can define a simplicial complex 
$\mathcal {L}$ as follows. When 
$n$ is odd or when 
$n$ is even and 
$C$ is non-split, then an 
$r$-simplex is formed by 
$\Lambda _0,\ldots,\Lambda _r$ if any two members of this set are neighbours. When 
$n$ is even and 
$C$ is split, we refer to discussion before [Reference Rapoport, Terstiege and WilsonRTW14, 3.4] for the definition of 
$\mathcal {L}$. We also use 
$\mathcal {L}_{n,\epsilon }$ to denote 
$\mathcal {L}$ if 
$C$ has dimension 
$n$ and 
$\chi (C)=\epsilon$. Again when 
$n$ is odd, 
$\mathcal {L}_{n,1}=\mathcal {L}_{n,-1}$, hence we use 
$\mathcal {L}_n$ to denote it.
By results in §§ 4 and 6 of [Reference Rapoport, Terstiege and WilsonRTW14], to each 
$\Lambda \in \mathcal {V}^t$ we can associate Deligne–Lusztig varieties 
$\mathcal {N}_\Lambda$ and 
$\mathcal {N}_\Lambda ^\circ$ of dimension 
$t/2$, such that
\[ \mathcal{N}_{\Lambda}(k)=\{M\in \mathcal{N}(k)\mid M \subset \Lambda \otimes_{\mathcal{O}_F}\mathcal{O}_{\breve{F}}\}, \]
and
\[ \mathcal{N}_{\Lambda}^{\circ}(k)=\{M\in \mathcal{N}(k)\mid \Lambda(M)=\Lambda\}. \]
Here 
$\Lambda (M)$ is the minimal vertex lattice such that 
$\Lambda (M)\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F}$ contains 
$M$ which always exists by [Reference Rapoport, Terstiege and WilsonRTW14, Proposition 4.1]. By Theorem 1.1 of [Reference Rapoport, Terstiege and WilsonRTW14], we know that
\[ \mathcal{N}_{\Lambda}:= \bigsqcup_{\Lambda'\in\mathcal{L},\Lambda'\subseteq \Lambda}\mathcal{N}_{\Lambda'}^\circ, \]
and
\[ \mathcal{N}^\mathrm{Pap}_{\mathrm{red}}=\bigsqcup_{\Lambda\in\mathcal{L}}\mathcal{N}_\Lambda^\circ, \]
where each 
$\mathcal {N}_\Lambda$ is a closed subvariety of 
$\mathcal {N}^\mathrm {Pap}_{\mathrm {red}}$. By [Reference Rapoport, Terstiege and WilsonRTW14], we also know that
\[ \mathcal{N}_\Lambda\cap\mathcal{N}_{\Lambda'}=\begin{cases} \mathcal{N}_{\Lambda\cap\Lambda'}, & \text{if } \Lambda\cap\Lambda'\in \mathcal{V},\\ \emptyset, & \text{otherwise}. \end{cases} \]
For a lattice 
$L\subset \mathbb {V}$, define
\begin{equation} \mathcal{V}(L):= \{\Lambda\in \mathcal{V}\mid L\subseteq \Lambda^\sharp\} \quad\text{and}\quad \mathcal{V}^t(L):= \{\Lambda\in \mathcal{V}^t\mid L\subseteq \Lambda^\sharp\}. \end{equation}
When 
$L=\mathrm {Span} \{{\bf x}\}$ we also denote 
$\mathcal {V}(L)$ (respectively, 
$\mathcal {V}^t(L)$) by 
$\mathcal {V}({\bf x})$ (respectively, 
$\mathcal {V}^t({\bf x})$). For any subset 
$S$ of 
$\mathcal {V}$, we define 
$\mathcal {L}(S)$ to be the subcomplex of 
$\mathcal {L}$ such that a simplex is in 
$\mathcal {L}(S)$ if and only if every vertex in it is in 
$S$. For a lattice 
$L$ of 
$\mathcal {V}$ and 
${\bf x}\in C$, define
\begin{equation} \mathcal{L}(L)=\mathcal{L}(\mathcal{V}(L)). \end{equation}
When 
$L=\mathrm {Span} \{{\bf x}\}$ we also denote 
$\mathcal {L}(L)$ by 
$\mathcal {L}({\bf x})$.
2.4 Horizontal and vertical part
A formal scheme 
$X$ over 
$\mathrm {Spf}\, \mathcal {O}_{\breve F}$ is called horizontal (respectively, vertical) if it is flat over 
${\mathrm {Spf}\,\mathcal {O}_{\breve {F}} }$ (respectively, 
$\pi$ is locally nilpotent on 
$\mathcal {O}_X$). For a formal scheme 
$X$ over 
$\mathrm {Spf}\, O_{\breve F}$, its horizontal part 
$X_{h}$ is canonically defined by the ideal sheaf 
$\mathcal {O}_{X,\mathrm {tor}}$ of torsion sections on 
$\mathcal {O}_X$. If 
$X$ is Noetherian, there exists a 
$m\in \mathbb {Z}_{>0}$ such that 
$\pi ^m \mathcal {O}_{X,\mathrm {tor}}=0$. We define the vertical part 
$X_{v}\subset X$ to be the closed formal subscheme defined by the ideal sheaf 
$\pi ^m \mathcal {O}_X$. Since 
$\mathcal {O}_{X,\mathrm {tor}}\cap \pi ^m \mathcal {O}_X=\{0\}$, we have the following decomposition by primary decomposition:
\begin{equation} X=X_h\cup X_v \end{equation}
as a union of horizontal and vertical formal subschemes. Note that the horizontal part 
$X_{h}$ is canonically defined whereas the vertical part 
$X_{v}$ depends on the choice of 
$m$.
Lemma 2.10 For a lattice 
$L^\flat \subset \mathbb {V}$ of rank greater or equal to 
$n-1$ with non-degenerate Hermitian form, 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )$ is Noetherian.
Proof. The lemma can be proved as in [Reference Li and ZhangLZ22a, Lemma 2.9.2].
Lemma 2.11 For a rank 
$n-1$ lattice 
$L^\flat \subset \mathbb {V}$ with non-degenerate Hermitian form, 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )_v$ is supported on the reduced locus 
$\mathcal {N}^\mathrm {Kra}_{\mathrm {red}}$ of 
$\mathcal {N}^\mathrm {Kra}$, i.e. 
$\mathcal {O}_{\mathcal {Z}^\mathrm {Kra}(L^\flat )_v}$ is annihilated by a power of the ideal sheaf of 
$\mathcal {N}^\mathrm {Kra}_{\mathrm {red}}$.
Proof. The proof is the same as that of [Reference Li and ZhangLZ22a, Lemma 5.1.1].
2.5 Derived special cycles
For a locally Noetherian formal scheme 
$X$ together with a formal subscheme 
$Y$, denote by 
$K_0^Y(X)$ the Grothendieck group of finite complexes of coherent locally free 
$\mathcal {O}_X$-modules acyclic outside 
$Y$. For such a complex 
$A^\bullet$, denote by 
$[A^\bullet ]$ the element in 
$K_0^Y(X)$ represented by it. We use 
$K_0(X)$ to denote 
$K_0^X(X)$. Denote by 
$\mathrm {F}^i K_0^Y(X)$ the codimension 
$i$ filtration on 
$K_0^Y(X)$ and 
$\mathrm {Gr}^i K_0^Y(X)$ its 
$i$th graded piece. We have a cup product 
$\cdot$ on 
$K_0^Y(X)$ defined by tensor product of complexes:
\[ [A_1^\bullet]\cdot [A_2^\bullet]=[A_1^\bullet \otimes A_2^\bullet]. \]
When 
$X$ is a scheme, the cup product satisfies [Reference Soulé, Abramovich, Burnol and KramerSABK94, § I.3 and Theorem 1.3]
\begin{equation} \mathrm{F}^i K_0^Y(X)_\mathbb{Q} \cdot \mathrm{F}^j K_0^Y(X)_\mathbb{Q}\subset \mathrm{F}^{i+j} K_0^Y(X)_\mathbb{Q}. \end{equation}
It is expected that (2.8) is also true when 
$X$ is a formal scheme. We only need special cases of this fact which can be checked directly; see, for example, Lemmas 3.4 and 10.1.
Let 
$K'_0(Y)$ be the Grothendieck group of coherent sheaves of 
$\mathcal {O}_Y$-modules on 
$Y$. When 
$X$ is regular we have the following isomorphism:
\begin{equation} K^Y_0(X)\cong K'_0(Y). \end{equation}
In particular, 
$K_0(X)\cong K'_0(X)$.
When 
$X$ is a regular scheme of dimension 
$d$, there is an isomorphism of graded rings defined by the Chern character:
\[ \mathrm{ch}:K_0(X)_\mathbb{Q}\cong \bigoplus_{i=1}^d \mathrm{CH}^i(X)_\mathbb{Q}. \]
In particular, we have
\[ \mathrm{Gr}^i K_0(X)_\mathbb{Q}\cong \mathrm{CH}^i(X)_\mathbb{Q}. \]
Recall that for 
${\bf x}\in \mathbb {V}$, 
$\mathcal {Z}^\mathrm {Kra}({\bf x})$ is a divisor, see [Reference HowardHow19, Proposition 4.3].
Definition 2.12 For 
${\bf x}=({\bf x}_1,\ldots,{\bf x}_r)\in \mathbb {V}^r$, define 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}({\bf x})$ to be
\begin{equation} \big[\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_1)}\otimes^\mathbb{L}\cdots \otimes^\mathbb{L} \mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_r)}\big] \in K_0^{\mathcal{Z}^\mathrm{Kra}({\bf x})} (\mathcal{N}^\mathrm{Kra}), \end{equation}
where 
$\otimes ^\mathbb {L}$ is the derived tensor product of complexes of coherent locally free sheaves on 
$\mathcal {N}^\mathrm {Kra}$. By [Reference HowardHow19, Theorem B], 
${}^\mathbb {L}\mathcal {Z}^\mathrm {Kra}({\bf x})$ only depends on 
$L:= \mathrm {Span}\{{\bf x}\}$, hence can be denoted as 
${}^\mathbb {L}\mathcal {Z}^\mathrm {Kra}(L)$.
Definition 2.13 When 
$L$ has rank 
$n$, we define the intersection number
\begin{equation} \mathrm{Int}(L)=\chi(\mathcal{N}^\mathrm{Kra},{}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L)), \end{equation}
where 
$\chi$ is the Euler characteristic.
Lemma 2.14 The subscheme 
$\mathcal {Z}^\mathrm {Kra}(L)$ is properly supported on 
$\mathcal {N}_{\mathrm {red}}^\mathrm {Kra}$. In particular, 
$\mathrm {Int}(L)$ is finite.
Proof. This can be proved exactly the same way as [Reference Li and ZhangLZ22a, Lemma 2.10.1].
2.6 Special difference cycles
Conjecture 1.1 and Theorem 5.2 motivate us to make the following definition.
Definition 2.15 For 
$L\subset \mathbb {V}$ a rank 
$\ell$ lattice, define the special difference cycle 
$\mathcal {D}(L)\in K_0^{\mathcal {Z}^\mathrm {Kra}(L)} (\mathcal {N}^\mathrm {Kra})$ by
\begin{equation} \mathcal{D}(L)={}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L)+\sum_{i=1}^\ell (-1)^i q^{i(i-1)/2} \sum_{\substack{L\subset L'\subset \frac{1}{\pi} L\\ \mathrm{dim}_{\mathbb{F}_q}(L'/L)=i}} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L'). \end{equation} One interesting observation is the following decomposition of 
${}^\mathbb {L}\mathcal {Z}^\mathrm {Kra}(L)$.
Lemma 2.16 For 
$L\subset \mathbb {V}$ a lattice of rank 
$\ell$, we have the following identity in 
$K_0^{\mathcal {Z}^\mathrm {Kra}(L)} (\mathcal {N}^\mathrm {Kra})$ where the summation is finite:
\[ {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L) =\sum_{\substack{L'\,\mathrm{integral} \\L \subset L' \subset L_F}} \mathcal{D}(L'). \]
Proof. First, if 
$L$ is not integral, neither is 
$L'$ if 
$L\subset L'$. In this case 
${}^\mathbb {L}\mathcal {Z}^\mathrm {Kra}(L)=0$ and the summation index on the right-hand side of the identity in the lemma is empty. This proves the lemma when 
$\mathrm {v}(L)<0$. We can now prove the identity by induction on the fundamental invariant of 
$L$. Assume that the lemma is proved for all 
$L'\subset L_F$ with 
$L\subsetneq L'$.
For 
$L'$ with 
$L\subsetneq L'\subset ({1}/{\pi }) L$, we have
\[ {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L') =\displaystyle\sum_{L' \subset L'' \subset L'_F} \mathcal{D}(L'') \]
by the induction hypothesis. Combining this with (2.12), we can write
\[ {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L) =\displaystyle\sum_{L \subset L'' \subset L_F} m(L'') \mathcal{D}(L'') \]
where 
$m(L'')\in \mathbb {Z}$. Now it suffices to show 
$m(L'')=1$ for any 
$L''$ such that 
$L\subset L'' \subset L_F$.
First, note that 
$m(L)=1$. For any 
$L''$ such that 
$L\subset L''\subsetneq L_F$, let 
$M'=({1}/{\pi })L\cap L''$ and 
$m=\dim _{\mathbb {F}_q} (M'/L)$. We have
\begin{equation} m(L'')=-\sum_{i=1}^m (-1)^i q^{i(i-1)/2} \sum_{\substack{L\subset L'\subset M'\\ \mathrm{dim}_{\mathbb{F}_q}(L'/L)=i}} 1=1 \end{equation}
by evaluating the identity in the corollary to [Reference TamagawaTam63, Lemma 12] at 
$t=1$.
Remark 2.17 When 
$\ell =1$ and 
$L=\mathrm {Span}\{{\bf x}\}$, the Cartier divisor
\[ \mathcal{D}(L)=\mathcal{Z}({\bf x})-\mathcal{Z}\biggl(\frac{1}{\pi}{\bf x}\biggr) \]
is the difference divisor 
$\mathcal {D}({\bf x})$ defined in [Reference TerstiegeTer13, Definition 2.10].
Definition 2.18 Assume 
$L=L_1\oplus L_2$, where 
$L_i$ is of rank 
$n_i$ and 
$n_1+n_2=n$. We define
\begin{equation} \mathrm{Int}(L)^{(n_1)}=\chi(\mathcal{N}^\mathrm{Kra}, \mathcal{D}(L_1)\cdot {}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L_2)). \end{equation}
Notice that 
$\mathrm {Int}(L)^{(n_1)}$ depends on the decomposition 
$L=L_1\oplus L_2$.
3. Special cycles and exceptional divisors
For a formal subscheme 
$\mathcal {Z}$ of 
$\mathcal {N}^\mathrm {Kra}$, we use the notation 
$\otimes _{\mathcal {Z}}$ (respectively, 
$\otimes _{\mathcal {Z}}^\mathbb {L}$) instead of 
$\otimes _{\mathcal {O}_\mathcal {Z}}$ (respectively, 
$\otimes _{\mathcal {O}_\mathcal {Z}}^\mathbb {L}$). We also simply write 
$\otimes$ (respectively, 
$\otimes ^\mathbb {L}$) instead of 
$\otimes _{\mathcal {N}^\mathrm {Kra}}$ (respectively, 
$\otimes _{\mathcal {N}^\mathrm {Kra}}^\mathbb {L}$). Let us first recall the following distribution law of derived tensor product. In this section, we identify 
$\mathbb {V}$ with 
$C$ by the isomorphism 
$b$ defined in (2.3).
Lemma 3.1 Assume that 
$\mathcal {A}_i$ (
$1\leq i \leq k$) is in the derived category of bounded coherent sheaves on 
$\mathcal {N}^\mathrm {Kra}$ and 
$i:\mathcal {Z}\rightarrow \mathcal {N}^\mathrm {Kra}$ is a closed embedding of formal subscheme. Then the following identity holds in the derived category of bounded coherent sheaves on 
$\mathcal {Z}$:
\[ i^*(\mathcal{A}_1\otimes^\mathbb{L}\cdots \otimes^\mathbb{L}\mathcal{A}_k\otimes^\mathbb{L} \mathcal{O}_{\mathcal{Z}}) =i^*(\mathcal{A}_1\otimes^\mathbb{L} \mathcal{O}_{\mathcal{Z}}) \otimes_\mathcal{Z}^\mathbb{L}\cdots \otimes_\mathcal{Z}^\mathbb{L} i^*(\mathcal{A}_k\otimes^\mathbb{L} \mathcal{O}_{\mathcal{Z}}). \]
Proof. We can take locally free representatives of 
$A_i^\bullet$ of 
$\mathcal { A}_i$. Then 
$A_1^\bullet \otimes \cdots \otimes A_k^\bullet$ is again a complex of locally free sheaves on 
$\mathcal {N}^\mathrm {Kra}$, hence a locally free representatives of 
$\mathcal {A}_1\otimes ^\mathbb {L}\cdots \otimes ^\mathbb {L}\mathcal {A}_k$. Hence, 
$i^*(\mathcal {A}_1\otimes ^\mathbb {L}\cdots \otimes ^\mathbb {L}\mathcal {A}_k\otimes ^\mathbb {L} \mathcal {O}_{\mathcal {Z}})$ can be represented by 
$A_1^\bullet \otimes \cdots \otimes A_k^\bullet \otimes \mathcal {O}_\mathcal {Z}$. Meanwhile 
$A_i^\bullet \otimes \mathcal {O}_\mathcal {Z}$ is a representative of 
$\mathcal {A}_i\otimes ^\mathbb {L}\mathcal {O}_\mathcal {Z}$ in the derived category of bounded coherent sheaves on 
$\mathcal {N}^\mathrm {Kra}$ and is also a complex of locally free sheaves on 
$\mathcal {Z}$. Hence 
$i^*(\mathcal {A}_1\otimes ^\mathbb {L} \mathcal {O}_{\mathcal {Z}}) \otimes _\mathcal {Z}^\mathbb {L}\cdots \otimes _\mathcal {Z}^\mathbb {L} i^*(\mathcal {A}_k\otimes ^\mathbb {L} \mathcal {O}_{\mathcal {Z}})$ can be represented by 
$(A_1^\bullet \otimes \mathcal {O}_\mathcal {Z}) \otimes _{\mathcal {Z}}\cdots \otimes _{\mathcal {Z}} (A_k^\bullet \otimes \mathcal {O}_\mathcal {Z})$. Now by the distribution law of tensor products we have
\[ A_1^\bullet \otimes\cdots \otimes A_k^\bullet \otimes\mathcal{O}_\mathcal{Z}=(A_1^\bullet \otimes\mathcal{O}_\mathcal{Z}) \otimes_{\mathcal{Z}}\cdots \otimes_{\mathcal{Z}} (A_k^\bullet \otimes\mathcal{O}_\mathcal{Z}). \]
This finishes the proof of the lemma.
Proposition 3.2 Assume that the dimension of 
$\mathbb {V}$ is 
$n\geq 2$. Then for each 
${\bf x}\in \mathbb {V}$, 
$\mathcal {Z}^\mathrm {Kra}({\bf x})$ is a divisor. Moreover, we have the following decomposition of Cartier divisors:
\begin{equation} \mathcal{Z}^\mathrm{Kra}({\bf x})=\tilde{\mathcal{Z}}({\bf x})+\sum_{\Lambda \in \mathcal{V}^0, {\bf x} \in \Lambda} (m_\Lambda({\bf x})+1) \mathrm{Exc}_\Lambda, \end{equation}
where 
$m_\Lambda ({\bf x})$ is the largest integer 
$m$ such that 
$\pi ^{-m} \cdot {\bf x}\in \Lambda$.
Proof. The fact that 
$\mathcal {Z}^\mathrm {Kra}({\bf x})$ is a divisor is due to [Reference HowardHow19, Proposition 4.3]. By [Reference ShiShi18, Proposition 3.7], the superspecial point corresponding to a type 
$0$ lattice 
$\Lambda$ is in 
$\mathcal {Z}^\mathrm {Pap}({\bf x})$ if and only if 
${\bf x} \in \Lambda$. Hence, 
$\mathrm {Exc}_\Lambda \subset \mathcal {Z}^\mathrm {Kra}({\bf x})$ if and only if 
${\bf x} \in \Lambda$. Since 
$\mathcal {N}_{n,\epsilon }^\mathrm {Kra}$ is regular, we must have a decomposition as in (3.1) and the only job left is to determine the multiplicity of each 
$\mathrm {Exc}_\Lambda$.
Fix a type 
$0$ lattice 
$\Lambda$ and let 
$m:= m_\Lambda ({\bf x})$. Then 
$\pi ^{-m}\cdot {\bf x}$ is a primitive vector in 
$\Lambda$. By Lemma A.3, there exists a decomposition
\[ \Lambda=\Lambda_2{\unicode{x29BA}}\Lambda', \]
where 
$\Lambda _2$ and 
$\Lambda '$ are unimodular lattices of rank 
$2$ and 
$n-2$, respectively, and 
$\pi ^{-m}\cdot {\bf x}\in \Lambda _2$. Let 
$\eta =\chi (\Lambda ')$. By applying Proposition 2.6, we see that 
$\tilde {\mathcal {Z}}_{n,\epsilon }(\Lambda ')\cong \mathcal {N}_{2,\epsilon \eta }^\mathrm {Kra}$. Moreover, we have the following proper intersections
\[ \mathcal{Z}^\mathrm{Kra}_{n,\epsilon}({\bf x}) \cap \tilde{\mathcal{Z}}_{n,\epsilon}(\Lambda')=\mathcal{Z}^\mathrm{Kra}_{2,\epsilon \eta}({\bf x}),\quad \tilde{\mathcal{Z}}_{n,\epsilon}({\bf x}) \cap \tilde{\mathcal{Z}}_{n,\epsilon}(\Lambda')=\tilde{\mathcal{Z}}_{2,\epsilon \eta}({\bf x}), \]
and
\[ \mathrm{Exc}_\Lambda \cap \tilde{\mathcal{Z}}_{n,\epsilon}(\Lambda')=\mathrm{Exc}_{\Lambda_2}, \]
where 
$\mathrm {Exc}_{\Lambda _2}$ is the exceptional divisor in 
$\mathcal {N}_{2,\epsilon \eta }^\mathrm {Kra}$ corresponding to the vertex lattice 
$\Lambda _2$. Hence, the multiplicity of 
$\mathrm {Exc}_{\Lambda }$ in 
$\mathcal {Z}^\mathrm {Kra}_{n,\epsilon }({\bf x})$ is the same as the multiplicity of 
$\mathrm {Exc}_{\Lambda _2}$ in 
$\mathcal {Z}^\mathrm {Kra}_{2,\epsilon \eta }({\bf x})$. Now the proposition follows from [Reference ShiShi22, Theorem 4.6] and [Reference He, Shi and YangHSY23, Theorem 4.1].
The Chow ring 
$\mathrm {CH}^\bullet (\mathrm {Exc}_\Lambda )\cong \mathrm {Gr}^\bullet K_0(\mathrm {Exc}_\Lambda )$ is isomorphic to 
$\mathbb {Z}[H_\Lambda ]/(H_\Lambda ^{n-1}-1)$ where 
$H_\Lambda$ is the hyperplane class of 
$\mathrm {Exc}_\Lambda$ represented by any 
$\mathbb {P}^{n-2}_k$ in 
$\mathrm {Exc}_\Lambda$.
Proposition 3.3 Assume 
$\dim \mathbb {V} =n \geq 2$. Assume 
${\bf x}\in \mathbb {V}$ such that 
$h({\bf x},{\bf x})\neq 0$ and 
$\Lambda$ is a type 
$0$ vertex lattice containing 
${\bf x}$. Let 
$m:= m_\Lambda ({\bf x})$ as in Proposition 3.2. Then 
$\tilde {\mathcal {Z}}({\bf x})$ and 
$\mathrm {Exc}_\Lambda$ intersect properly and
\[ \big[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x})\cap \mathrm{Exc}_\Lambda}\big]=(2m+1)H_\Lambda \in \mathrm{CH}^1(\mathrm{Exc}_\Lambda). \]
Proof. First 
$\tilde {\mathcal {Z}}({\bf x})$ and 
$\mathrm {Exc}_\Lambda$ are Cartier divisors with no common component, so they intersect properly. Let 
$m=m_\Lambda ({\bf x})$ and 
${\bf x}':= \pi ^{-m}\cdot {\bf x}$. By assumption 
$m\geq 0$. By Proposition 5.9, we have
\[ \{v\in \Lambda\mid h({\bf x}',v)=0\}=\mathrm{Span}\{\mathbf{y}\}{\unicode{x29BA}} \Lambda', \]
where 
$\mathrm {v}(\mathbf {y})=\mathrm {v}({\bf x}')$ and 
$\Lambda '$ is unimodular. Let 
$\eta =\chi (\Lambda ')$ and
\[ \Lambda_2:= \{v\in \Lambda\mid v\bot \Lambda'\}. \]
$\Lambda _2$ is rank 
$2$ unimodular and contains 
${\bf x}'$.
By Proposition 2.6, we have 
$\tilde {\mathcal {Z}}(\Lambda ')\cong \mathcal {N}_{2,\epsilon \eta }^\mathrm {Kra}$. In particular, 
$\tilde {\mathcal {Z}}(\Lambda ')$ is regular. By Corollary 2.7, we know that 
$\tilde {\mathcal {Z}}(\Lambda ')\cap \tilde {\mathcal {Z}}({\bf x})=\tilde {\mathcal {Z}}_{2,\epsilon \eta }({\bf x})$. In particular, 
$\tilde {\mathcal {Z}}(\Lambda ')$ and 
$\tilde {\mathcal {Z}}({\bf x})$ intersect properly as 
$\tilde {\mathcal {Z}}_{2,\epsilon \eta }({\bf x})$ is a divisor in 
$\mathcal {N}^\mathrm {Kra}_{2,\epsilon \eta }$. On the other hand, 
$\tilde {\mathcal {Z}}(\Lambda ')\cap \mathrm {Exc}_\Lambda$ is the exceptional divisor 
$\mathrm {Exc}_{\Lambda _2}$ in 
$\mathcal {N}^\mathrm {Kra}_{2,\epsilon \eta }$. Since 
$\mathrm {Exc}_\Lambda \cong \mathbb {P}^{n-1}_k$, it is also regular. Our strategy is to compute the intersection number
\[ \chi\bigl(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x})}\otimes^{\mathbb{L}} \mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^{\mathbb{L}} \mathcal{O}_{\tilde{\mathcal{Z}}(\Lambda')}\bigr) \]
in two different ways. By Lemma 3.1, one way is
\begin{equation} \chi\big(\tilde{\mathcal{Z}}(\Lambda'), \mathcal{O}_{\tilde{\mathcal{Z}}(\Lambda')\cap \tilde{\mathcal{Z}}({\bf x})}\otimes^{\mathbb{L}}_{\tilde{\mathcal{Z}}(\Lambda')} \mathcal{O}_{\tilde{\mathcal{Z}}(\Lambda')\cap \mathrm{Exc}_\Lambda}\big), \end{equation}
where we use the fact that the intersections 
$\tilde {\mathcal {Z}}(\Lambda ')\cap \tilde {\mathcal {Z}}({\bf x})$ and 
$\tilde {\mathcal {Z}}(\Lambda ')\cap \mathrm {Exc}_\Lambda$ are proper (see, for example, [Reference ZhangZha21, Lemma B.2]). The other way is, by Lemma 3.1,
\begin{equation} \chi\big(\mathrm{Exc}_\Lambda, \mathcal{O}_{\tilde{\mathcal{Z}}({\bf x})\cap \mathrm{Exc}_\Lambda}\otimes^\mathbb{L}_{\mathrm{Exc}_\Lambda} \mathcal{O}_{\tilde{\mathcal{Z}}(\Lambda')\cap \mathrm{Exc}_\Lambda}\big). \end{equation}
When 
$\epsilon \eta =-1$, by Proposition 3.11 and Theorem 4.5 of [Reference ShiShi22], we know that (3.2) is equal to 
$2m+1$. When 
$\epsilon \eta =1$, by Lemma 3.10, Theorem 4.1 and Lemma 5.2 of [Reference He, Shi and YangHSY23], we know that (3.2) is equal to 
$2m+1$ as well. Since the intersection number of 
$H_\Lambda$ with 
$\mathrm {Exc}_{\Lambda _2}\cong \mathbb {P}^1_k$ in 
$\mathrm {Exc}_\Lambda$ is 
$1$, the proposition follows.
3.1 Intersection numbers involving the exceptional divisors
Lemma 3.4 The class of 
$\underbrace {\mathcal {O}_{\mathrm {Exc}_\Lambda }\otimes ^\mathbb {L} \cdots \otimes ^\mathbb {L}\mathcal {O}_{\mathrm {Exc}_\Lambda }}_{m}$ in 
$\mathrm {CH}^{m-1}(\mathrm {Exc}_\Lambda )$ is 
$(-2H_\Lambda )^{m-1}$.
Proof. To study this intersection, it suffices to consider the local model 
$N^\mathrm {Kra}$ constructed in [Reference KrämerKrä03]. Let 
$N^\mathrm {Kra}_s$ be its special fiber. Recall by [Reference KrämerKrä03, (4.11)], we have
\[ N^\mathrm{Kra}_s =\mathrm{Exc} + Z_2 \]
as Cartier divisors where 
$\mathrm {Exc}$ is the exceptional divisor of 
$N^\mathrm {Kra}$ and 
$Z_2$ is a divisor in 
$N^\mathrm {Kra}$ which intersect properly with 
$\mathrm {Exc}$. Their intersection is 
$2H$ where 
$H$ is the hyperplane class of 
$\mathrm {Exc}$. Since 
$\mathrm {Exc}$ is properly supported on 
$N^\mathrm {Kra}$, we have
\[ [\mathcal{O}_{\mathrm{Exc}}\otimes^\mathbb{L} \mathcal{O}_{N^\mathrm{Kra}_s}]=0. \]
Hence,
\begin{align*} 0&=[\mathcal{O}_\mathrm{Exc}\otimes^\mathbb{L}_{N^\mathrm{Kra}} \mathcal{O}_{N^\mathrm{Kra}_s}]\\ &=[\mathcal{O}_\mathrm{Exc}\otimes^\mathbb{L}_{N^\mathrm{Kra}} \mathcal{O}_\mathrm{Exc}]+[\mathcal{O}_\mathrm{Exc} \otimes^\mathbb{L}_{N^\mathrm{Kra}} \mathcal{O}_{Z_2}]\\ &= [\mathcal{O}_\mathrm{Exc}\otimes^\mathbb{L}_{N^\mathrm{Kra}} \mathcal{O}_\mathrm{Exc}]+2H. \end{align*}
This proves the lemma when 
$m=2$. The general case now follows from Lemma 3.1.
Corollary 3.5 Let 
$\Lambda \in \mathcal {V}^0$ and 
${\bf x}\in \Lambda$. Then we have the following identity in 
$\mathrm {CH}^1(\mathrm {Exc}_\Lambda )$:
\[ [\mathcal{O}_{\mathrm{Exc}_{\Lambda}}\otimes^\mathbb{L} \mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x})}]=-H_{\Lambda}. \]
Proof. By Propositions 3.2, 3.3 and Lemma 3.4, we have the following identity in 
$\mathrm {CH}^1(\mathrm {Exc}_\Lambda )$:
\[ \big[\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x})}\otimes^\mathbb{L}\mathcal{O}_{\mathrm{Exc}_\Lambda}\big]=\big[(2m_\Lambda({\bf x})+1)-2(m_\Lambda({\bf x})+1)\big] H_\Lambda=-H_\Lambda. \]
This finishes the proof of the corollary.
Corollary 3.6 Assume that 
$n-m\geq 1$ and 
$\mathrm {Exc}_\Lambda \subset \mathcal {Z}^\mathrm {Kra}({\bf x}_1)\cap \ldots \cap \mathcal {Z}^\mathrm {Kra}({\bf x}_m)$, then
\[ \chi(\mathcal{N}_n^\mathrm{Kra},\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_1)}\otimes^{\mathbb{L}}\ldots\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_m)}\otimes^\mathbb{L}\underbrace{\mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^{\mathbb{L}} \cdots \otimes^{\mathbb{L}} \mathcal{O}_{\mathrm{Exc}_\Lambda}}_{n-m})=(-1)^{n-1}\cdot 2^{n-m-1}. \]
Proof. By Corollary 3.5, Lemmas 3.1 and 3.4, we have
\begin{align*} &\chi(\mathcal{N}_n^\mathrm{Kra},\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_1)}\otimes^{\mathbb{L}}\ldots\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_m)}\otimes^\mathbb{L}\underbrace{\mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^{\mathbb{L}} \cdots \otimes^{\mathbb{L}} \mathcal{O}_{\mathrm{Exc}_\Lambda}}_{n-m})\\ &\qquad=\chi(\mathrm{Exc}_\Lambda,\underbrace{(-H_\Lambda)\otimes_{\mathrm{Exc}_\Lambda}^{\mathbb{L}} \cdots \otimes_{\mathrm{Exc}_\Lambda}^{\mathbb{L}} (-H_\Lambda)}_{m}\otimes_{\mathrm{Exc}_\Lambda}^\mathbb{L}\underbrace{(-2H_\Lambda)\otimes_{\mathrm{Exc}_\Lambda}^{\mathbb{L}} \cdots \otimes_{\mathrm{Exc}_\Lambda}^{\mathbb{L}} (-2H_\Lambda)}_{n-m-1})\\ &\qquad=(-1)^m\cdot(-2)^{n-m-1}.\end{align*}
□
For 
$\Lambda \in \mathcal {V}^0$, let 
$\mathbb {P}^1_\Lambda$ be any 
$\mathbb {P}^1_k$ in 
$\mathrm {Exc}_\Lambda$, and
\begin{equation} \mathrm{Int}_{\Lambda}({\bf x})=\chi(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x})}\otimes^\mathbb{L}\mathcal{O}_{\mathbb{P}_\Lambda^1}). \end{equation}Corollary 3.7 For 
$\Lambda \in \mathcal {V}^0$, we have
\begin{equation} \chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^\mathbb{L} \mathcal{O}_{\mathbb{P}^1_{\Lambda}})=-2. \end{equation}Proof. By Lemma 3.4, we have
\begin{align*} &\chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^\mathbb{L} \mathcal{O}_{\mathbb{P}^1_{\Lambda}})\\ &\qquad=\chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^\mathbb{L} (\mathcal{O}_{\mathrm{Exc}_\Lambda} \otimes_{\mathcal{O}_{\mathrm{Exc}_\Lambda}} \mathcal{O}_{\mathbb{P}^1_{\Lambda}}))\\ &\qquad=\chi(\mathrm{Exc}_\Lambda,(\mathcal{O}_{\mathrm{Exc}_\Lambda}\otimes^\mathbb{L} \mathcal{O}_{\mathrm{Exc}_\Lambda}) \otimes_{\mathcal{O}_{\mathrm{Exc}_\Lambda}} \mathcal{O}_{\mathbb{P}^1_{\Lambda}})\\ &\qquad=-2\chi(\mathrm{Exc}_\Lambda,H_\Lambda\cdot [\mathcal{O}_{\mathbb{P}^1_{\Lambda}}])\\ &\qquad=-2. \end{align*}
□
Corollary 3.8 For 
$\Lambda \in \mathcal {V}^0$, we have
\[ \mathrm{Int}_{\Lambda}({\bf x})=-1_{\Lambda}({\bf x}). \]
Proof. If 
${\bf x}\notin \Lambda$, then the intersection number is apparently 
$0$. Otherwise, by Corollary 3.5 we have
\begin{align*} &\chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x})}\otimes^\mathbb{L} \mathcal{O}_{\mathbb{P}^1_{\Lambda}})\\ &\qquad=\chi(\mathrm{Exc}_\Lambda,(\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x})}\otimes^\mathbb{L} \mathcal{O}_{\mathrm{Exc}_\Lambda}) \otimes_{\mathcal{O}_{\mathrm{Exc}_\Lambda}} \mathcal{O}_{\mathbb{P}^1_{\Lambda}})\\ &\qquad=-\chi(\mathrm{Exc}_\Lambda,H_\Lambda\cdot [\mathcal{O}_{\mathbb{P}^1_{\Lambda}}])\\ &\qquad=-1.\end{align*}
□
The above results suggest that the difficulty in computing 
$\mathrm {Int}(L)$ mainly lies in computing
\[ \chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{Z}}(x_1)}\otimes^{\mathbb{L}}\cdots \otimes^{\mathbb{L}}\mathcal{O}_{\tilde{\mathcal{Z}}(x_n)}). \]
We end this section by studying the intersection number of difference cycle with exceptional divisors.
Lemma 3.9 If 
$L^\flat$ has rank 
$n-1$, then for any 
$\Lambda \in \mathcal {V}^0(L^\flat )$, we have
\[ \chi(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat)\cdot [\mathcal{O}_{\mathrm{Exc}_{\Lambda}}])=\begin{cases} (-1)^{n-1}, & \text{if } L^\flat=\Lambda\cap L^\flat_F,\\ 0, & \text{otherwise}. \end{cases} \]
Remark 3.10 We have 
$L^\flat =\Lambda \cap L ^\flat _F$ if and only if 
$L^\flat$ is of type (see (4.2) and Lemma 4.3) 
$1$ or 
$0$ and 
$\Lambda$ is at the boundary of the 
$\mathcal {L}(L^\flat )$.
Proof. Define
\[ M':= \frac{1}{\pi}L^\flat \cap \Lambda \quad\text{and}\quad m:= \dim_{\mathbb{F}_q}(M'/L^\flat). \]
Then for 
$L'$ such that 
$L^\flat \subset L'\subset ({1}/{\pi }) L^\flat$, we know that 
$\mathcal {Z}^\mathrm {Kra}(L')$ intersects 
$\mathrm {Exc}_{\Lambda }$ if and only if 
$L'\subset M'$. For such 
$L'$, by Corollary 3.6, we have
\begin{equation} \chi(\mathcal{N}^\mathrm{Kra},{}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L')\cdot [\mathcal{O}_{\mathrm{Exc}_{\Lambda}}])=(-1)^{n-1}. \end{equation}Hence,
\[
\chi\big(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat)\cdot
[\mathcal{O}_{\mathrm{Exc}_{\Lambda}}]\big)=(-1)^{n-1}\biggl[1+\sum_{i=1}^m (-1)^i q^{i(i-1)/2}
\sum_{\substack{L^\flat\subset L'\subset M'\\
\mathrm{dim}_{\mathbb{F}_q}(L'/L^\flat)=i}} 1\biggr].
\]
Note that 
$m=0$ if and only if 
$M'=L^\flat$ which is equivalent to the condition 
$L^\flat =\Lambda \cap L^\flat _F$. In this case the summation in (4.8) is over an empty set, hence (4.8) is equal to 
$1$. If 
$m>0$ we know (4.8) is equal to 
$0$ by (2.13).
4. Horizontal components of special cycles
Given an integral Hermitian lattice 
$L$, we can have its Jordan decomposition:
\begin{equation} L={\unicode{x29BA}}_{t\geq 0} L_t, \end{equation}
where 
$L_t$ is 
$\pi ^t$-modular, see [Reference JacobowitzJac62]. Define the type of 
$L$ to be
\begin{equation} t(L)=\sum_{t\geq 1} \mathrm{rank}_{\mathcal{O}_F}(L_t). \end{equation}4.1 Quasi-canonical lifting cycles
Assume that 
$\mathrm {dim}(\mathbb {V})=2$. When 
$\chi (\mathbb {V})=-1$, for 
$\mathbf {y}\in \mathbb {V}$, by [Reference ShiShi22, Theorem 4.5], we have the following equality of Cartier divisors on 
$\mathcal {N}_{2,-1}^\mathrm {Kra}$:
\[ \tilde{\mathcal{Z}}_{2,-1}(\mathbf{y})=\mathcal{Z}_0+\sum_{s=1}^{\mathrm{v}(\mathbf{y})} (\mathcal{Z}_{s}^+ +\mathcal{Z}_{s}^-). \]
Here 
$\mathcal {Z}_0$ (respectively, 
$\mathcal {Z}_{s}^{\pm }$) is a canonical (respectively, quasi-canonical) lifting cycle of level 
$0$ (respectively, 
$s$), see [Reference ShiShi22, § 3]. Moreover by [Reference ShiShi22, Proposition 3.12], 
$\mathcal {Z}_s^+$ and 
$\mathcal {Z}_s^-$ do not intersect when 
$s\geq 1$. Let 
$\mathcal {O}_s:=\mathcal {O}_{F_0}+\mathcal {O}_{F}\cdot \pi _0^s$ and 
$M_s$ be the finite abelian extension of 
${\breve F}$ corresponding to the subgroup 
$\mathcal {O}_s^\times$ under local class field theory. Let 
$W_s$ be the integral closure of 
$\mathcal {O}_{\breve F}$ in 
$M_s$. Then we have 
$\mathcal {Z}_0\cong {\mathrm {Spf}\,\mathcal {O}_{\breve {F}} }$ and 
$\mathcal {Z}_{s}^{\pm }\cong \mathrm {Spf}\, W_s$. Define the primitive part of 
$\tilde {\mathcal {Z}}_{2,-1}(\mathbf {y})$ to be
\begin{align*}
\tilde{\mathcal{Z}}_{2,-1}(\mathbf{y})^\circ:=
\left\{\begin{array}{@{}ll}
\mathcal{Z}_{\mathrm{v}(\mathbf{y})}^+
+\mathcal{Z}_{\mathrm{v}(\mathbf{y})}^-, & \text{if }
\mathrm{v}(\mathbf{y})>0, \\ \mathcal{Z}_0, & \text{if }
\mathrm{v}(\mathbf{y})=0. \end{array}\right.
\end{align*}
When 
$\chi (\mathbb {V})=1$, for 
$\mathbf {y}\in \mathbb {V}$ such that 
$\mathrm {v}(\mathbf {y})\geq 0$, by [Reference He, Shi and YangHSY23, Theorem 4.1], we have the following equality of Cartier divisors on 
$\mathcal {N}_{2,1}^\mathrm {Kra}$:
\[ \tilde{\mathcal{Z}}_{2,1}(\mathbf{y})=\mathcal{Z}_0+\mathcal{Z}_v(\mathbf{y}), \]
where 
$\mathcal {Z}_0\cong {\mathrm {Spf}\,\mathcal {O}_{\breve {F}} }$ is a canonical lifting cycle and 
$\mathcal {Z}_v(\mathbf {y})$ is a Cartier divisor whose structure sheaf is annihilated by 
$\pi ^N$ for some 
$N> 0$. Define the primitive horizontal part of 
$\tilde {\mathcal {Z}}_{2,1}(\mathbf {y})$ as
\[
\tilde{\mathcal{Z}}_{2,1}(\mathbf{y})^\circ:=
\left\{\begin{array}{@{}ll} 0, & \text{if }
\mathrm{v}(\mathbf{y})>0, \\ \mathcal{Z}_0, & \text{if }
\mathrm{v}(\mathbf{y})=0. \end{array}\right.
\]
4.2 Horizontal cycles
Definition 4.1 Let 
$M^\flat$ be a rank 
$n-1$ integral lattice in 
$\mathbb {V}$. We say that 
$M^\flat$ is horizontal if one of the following conditions is satisfied:
(1)
$M^\flat$ is unimodular;(2)
$M^\flat$ is of the form $M^\flat =M{\unicode{x29BA}} \mathrm {Span}\{\mathbf {y}\}$ where $M$ is a unimodular sublattice of rank $n-2$ such that $(M_F)^\bot$ (the perpendicular complement of $M_F$ in $\mathbb {V}$) is non-split.
Note that condition (2) is independent of the choice of 
$M$. We denote the set of horizontal lattices by 
$\mathrm {Hor}$.
For a rank 
$n-1$ integral lattice 
$L^\flat$, define
\begin{equation} \mathrm{Hor}(L^\flat):= \{M^\flat \in \mathrm{Hor} \mid L^\flat\subseteq M^\flat\}. \end{equation}
Let 
$M^\flat \subset \mathbb {V}$ be a lattice of rank 
$n-1$ and type 
$1$ or 
$0$. We can decompose 
$M^\flat$ as
\begin{equation} M^\flat= M{\unicode{x29BA}} \mathrm{Span}\{\mathbf{y}\}, \end{equation}
for some unimodular lattice 
$M$ of rank 
$n-2$. Then Proposition 2.6 and its corollary imply that
\[ \tilde{\mathcal{Z}}(M^\flat)\cong \tilde{\mathcal{Z}}_{2,\chi((M_F)^\bot)}(\mathbf{y}). \]
Under this isomorphism, define 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ to be the formal subscheme of 
$\tilde {\mathcal {Z}}(M^\flat )$ isomorphic to 
$\tilde {\mathcal {Z}}_{2,\chi ((M_F)^\bot )}(\mathbf {y})^\circ$. By the discussion in § 4.1, 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ is nonempty if and only if 
$M^\flat \in \mathrm {Hor}$, in which case it consists of the union of irreducible components of 
$\tilde {\mathcal {Z}}(M^\flat )$ isomorphic to 
$\mathrm {Spf}\, W_s$. In particular, 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ is independent of the choice of 
$M$.
Theorem 4.2 Let 
$L^\flat$ be a rank 
$n-1$ integral lattice in 
$\mathbb {V}$, then
\begin{equation} \mathcal{Z}^\mathrm{Kra}(L^\flat)_h=\bigcup_{M^\flat \in \mathrm{Hor}(L^\flat)} \tilde{\mathcal{Z}}(M^\flat)^\circ. \end{equation}
In particular, 
$\mathcal {Z}^{\mathrm {Kra}}(L^\flat )_h$ is of pure dimension 
$1$. Moreover, we have the following identity in 
$\mathrm {Gr}^{n-1}K_0(\mathcal {N}^\mathrm {Kra})$:
\[ \big[\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(L^\flat)_h}\big]=\sum_{M^\flat \in \mathrm{Hor}(L^\flat)} \big[\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)^\circ}\big]. \]
Proof. The proof largely follows [Reference Li and ZhangLZ22a, § 4.4]. Let 
$K$ be a finite extension of 
$\breve F$. Assume that 
$z$ is an irreducible component of 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )(\mathcal {O}_K)=\mathcal {Z}^\mathrm {Pap}(L^\flat )(\mathcal {O}_K)$, and let 
$G$ be the corresponding formal 
$\mathcal {O}_F$-module over 
$\mathcal {O}_K$. Define
\[ L:= \mathrm{Hom}_{\mathcal{O}_F}(T_p\mathcal{G},T_p G), \]
where 
$\mathcal {G}$ is the canonical lifting and 
$T_p$ is the integral 
$p$-adic Tate module. Here 
$L$ is an 
$\mathcal {O}_F$-module of rank 
$n$ equipped with the Hermitian form
\[ \{x,y\}=\lambda_\mathcal{G}^\vee \circ y^\vee \circ \lambda_G\circ x, \]
under which it is self-dual. We have two inclusions (preserving Hermitian forms)
\[ i_K:\mathrm{Hom}_{\mathcal{O}_F}(\mathcal{G},G)_F \rightarrow L_F, \]
and
\[ i_k:\mathrm{Hom}_{\mathcal{O}_F}(\mathcal{G},G)_F \rightarrow \mathbb{V}. \]
By Lemma 4.4.1 of [Reference Li and ZhangLZ22a], we have
\begin{equation} \mathrm{Hom}_{\mathcal{O}_F}(\mathcal{G},G)=i^{-1}_K(L). \end{equation}Let
\[ M^\flat:= (L^\flat_F)\cap i_{k}(i_K^{-1}(L)) \cong \mathrm{Hom}_{\mathcal{O}_F}(\mathcal{G},G). \]
Then 
$z \subset \mathcal {Z}(M^\flat )(\mathcal {O}_K)$. Lemma 4.3 below implies that 
$t(M^\flat )\leq 1$. Hence, we know that 
$z$ is one of the irreducible component of 
$\tilde {\mathcal {Z}}(M^\flat )^\circ \cong \tilde {\mathcal {Z}}_{2,\chi ((M_F)^\bot )}(\mathbf {y})$ assuming the decomposition of 
$M^\flat$ as in (4.4). The non-emptiness of 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ implies that 
$M^\flat \in \mathrm {Hor}$. It remains to prove that 
$z$ has multiplicity 
$1$ in 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )$. Consider 
$R$-points of both sides of (4.5), where 
$R:= \mathcal {O}_K[x]/(x^2)$. As in [Reference KrämerKrä03] (see [Reference Rapoport and ZinkRZ96, Appendix of Chapter 3]) we know
\[ \mathbb{D}(\mathcal{G})(R)\cong \mathcal{O}_F\otimes_{\mathcal{O}_{F_0}}R, \quad\text{and}\quad \mathbb{D}(G)(R)\cong (\mathcal{O}_F\otimes_{\mathcal{O}_{F_0}}R)^n, \]
where 
$\mathbb {D}$ is the 
$\mathcal {O}_{F_0}$-relative Dieudonné crystal. Define
\[ \tilde{e}_0=1\otimes 1\in \mathbb{D}(\mathcal{G})(R),\quad \tilde{f}_0=\pi\otimes 1 \in \mathbb{D}(\mathcal{G})(R). \]
Then the Hodge submodule 
$\mathcal {F}_0$ of 
$\mathbb {D}(\mathcal {G})(R)$ is spanned by
\[ (1\otimes \pi )\tilde{e}_0+\tilde{f}_0. \]
Here 
$\mathbb {D}(G)(R)$ is equipped with an 
$\mathcal {O}_F$-invariant symplectic form 
$\langle,\rangle$ and we can assume that 
$\mathbb {D}(G)(R)$ has a basis 
$\{\tilde {e}_1,\ldots,\tilde {e}_n,\tilde {f}_1,\ldots,\tilde {f}_n\}$ such that
\[ (\pi\otimes 1)\tilde{e}_i=\tilde{f}_i, \quad \langle \tilde{e}_i,\tilde{f}_j \rangle=\delta_{ij}. \]
Since any element in 
$L^\flat$ is 
$\mathcal {O}_F$-linear, we can arrange a change of basis if necessary and assume that
\[ L^\flat((1\otimes \pi )\tilde{e}_0+\tilde{f}_0)=\mathrm{Span}_{R}\big\{(1\otimes\pi^{a_1})((1\otimes \pi )\tilde{e}_1+\tilde{f}_1), \ldots,(1\otimes\pi^{a_{n-1}})((1\otimes \pi )\tilde{e}_{n-1}+\tilde{f}_{n-1})\big\}. \]
Now 
$\mathbb {D}(G)(\mathcal {O}_K)=\mathbb {D}(G)(R)\otimes _R \mathcal {O}_K$. Let 
$e_i=\tilde {e}_i\otimes 1$ and 
$f_i=\tilde {f}_i\otimes 1$ respectively. There is an exact sequence of free 
$\mathcal {O}_{F}\otimes _{\mathcal {O}_{F_0}}\mathcal {O}_K$-modules (the Hodge filtration)
\[ 0\rightarrow \mathrm{Fil}\rightarrow \mathbb{D}(G)(\mathcal{O}_K)\rightarrow \mathrm{Lie}\, G\rightarrow 0, \]
where 
$\mathrm {Fil}$ is isotropic with respect to 
$\langle,\rangle$. We must have 
$L^\flat ((1\otimes \pi )e_0+f_0)\subset \mathrm {Fil}$. Hence, we have
\[ (1\otimes \pi )e_1+f_1, \ldots,(1\otimes \pi )e_{n-1}+f_{n-1}\subset \mathrm{Fil}. \]
Since 
$\mathrm {Fil}$ is isotropic and by the signature condition, we have
\[ \mathrm{Fil}=\mathrm{Span}_{\mathcal{O}_K}\{(1\otimes \pi )e_1+f_1, \ldots,(1\otimes \pi )e_{n-1}+f_{n-1},(1\otimes \pi )e_n-f_n\}. \]
Since 
$(x)\subset R$ has a nilpotent positive-definite structure, by Grothendieck–Messing theory, a lift 
$\tilde {z}$ of 
$z$ to 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )(R)$ corresponds to a lift of 
$\mathrm {Fil}$ to an isotropic 
$\mathcal {O}_F\otimes _{\mathcal {O}_{F_0}}R$-module 
$\widetilde {\mathrm {Fil}}$ in 
$\mathbb {D}(G)(R)$ containing the image of 
$L^\flat$. By the same reasoning as above, we must have
\[ \widetilde{\mathrm{Fil}}=\mathrm{Span}_{R}\big\{(1\otimes \pi )\tilde{e}_1+\tilde{f}_1, \ldots,(1\otimes \pi )\tilde{e}_{n-1}+\tilde{f}_{n-1},(1\otimes \pi )\tilde{e}_n-\tilde{f}_n\big\}. \]
Hence, such lift is unique. This implies that the multiplicity of 
$z$ in 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )$ is one.
Lemma 4.3 Let 
$L$ be a self-dual Hermitian lattice of rank 
$n$ and 
$W$ be a 
$n-1$ dimensional subspace of 
$L_F$. Then 
$t(M^\flat )\le 1$ for 
$M^\flat =L\cap W$.
Proof. This is exactly the same as the proof of [Reference Li and ZhangLZ22a, Lemma 4.5.1]. Note that in our case we may need some blocks 
$\bigl (\begin {smallmatrix} 0 & \pi ^a \\ (-\pi )^a & 0\end {smallmatrix}\bigr )$ in the upper left 
$(n-1)\times (n-1)$ block of 
$T$ as in [Reference Li and ZhangLZ22a]. Alternatively, see [Reference Li and LiuLL22, Lemma 2.24(2)].
We end this subsection with the following lemma.
Lemma 4.4 Assume 
$M^\flat \in \mathrm {Hor}$. Then 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ intersects the special fiber of 
$\mathcal {N}_{n,\epsilon }^\mathrm {Kra}$ at a unique 
$\mathrm {Exc}_\Lambda$ for some 
$\Lambda \in \mathcal {V}^0$. Moreover,
\[ \chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)^\circ}\otimes^{\mathbb{L}} \mathcal{O}_{\mathrm{Exc}_\Lambda})=\begin{cases} 1, & \text{if } M^\flat \text{ is unimodular}, \\ 2, & \text{otherwise}. \end{cases} \]
Proof. By the definition of 
$\mathrm {Hor}$, we can find a decomposition of 
$M^\flat$
\[ M^\flat=M{\unicode{x29BA}} \{{\bf x}\} \]
such that 
$M$ is self-dual. Let 
$\Lambda$ be any vertex lattice containing 
$M^\flat$. If 
$M^\flat$ is unimodular, then 
$\Lambda$ has to be of the form 
$M^\flat {\unicode{x29BA}} L'$ where 
$L'$ is the unique unimodular lattice in 
$(M^\flat _F)^\bot$. If 
$M^\flat$ is of the form 
$M{\unicode{x29BA}} L'$ such that 
$M$ is of rank 
$n-2$ and 
$(M_F)^\bot$ is non-split, then the proof of [Reference ShiShi18, Theorem 3.10] implies that there is a unique vertex lattice 
$\Lambda '$ in 
$(M_F)^\bot$ which is of unimodular (this fact is the same as the fact that the Bruhat–Tits building of 
$(M_F)^\bot$ has only one point). Then 
$\Lambda$ must be of the form 
$M{\unicode{x29BA}} \Lambda '$. In both cases, 
$\Lambda$ is unique and is of type 
$0$.
Assume 
$\chi (M)=\eta$. By Proposition 2.6, 
$\tilde {\mathcal {Z}}(M)\cong \mathcal {N}^\mathrm {Kra}_{2,\epsilon \eta }$. Moreover, 
$\tilde {\mathcal {Z}}(M)\cap \mathrm {Exc}_{\Lambda }=\mathbb {P}^1_k$ is an exceptional divisor in 
$\mathcal {N}^\mathrm {Kra}_{2,\epsilon \eta }$. Hence, by Lemma 3.1, we have
\[ \chi(\mathcal{N}_{n,\epsilon}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)^\circ}\otimes^{\mathbb{L}} \mathcal{O}_{\mathrm{Exc}_\Lambda})= \chi(\mathcal{N}_{2,\epsilon\eta}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)^\circ}\otimes_{\mathcal{N}^\mathrm{Kra}_{2,\epsilon\eta}}^{\mathbb{L}} \mathcal{O}_{\mathbb{P}^1_k}). \]
Now the lemma follows from [Reference He, Shi and YangHSY23, Lemma 5.2] when 
$\epsilon \eta =1$, and from [Reference ShiShi22, Proposition 3.11] when 
$\epsilon \eta =-1$.
4.3 The horizontal part of special difference cycles
Definition 2.15 motivates us to make the following definition.
Definition 4.5 When 
$L^\flat$ is a rank 
$n-1$ integral lattice, define 
$\mathcal {D}(L^\flat )_h\in \mathrm {Gr}^{n-1} K_0(\mathcal {N}^\mathrm {Kra})$ by
\begin{equation} \mathcal{D}(L^\flat)_h=[\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(L^\flat)_h}]+\sum_{i=1}^{n-1} (-1)^i q^{i(i-1)/2} \sum_{\substack{L^\flat\subset L'\subset \frac{1}{\pi} L^\flat\\ \mathrm{dim}_{\mathbb{F}_q}(L'/L^\flat)=i}} [\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(L')_h}]. \end{equation}Proposition 4.6 Assume 
$L^\flat$ is a rank 
$n-1$ integral lattice, then
\[ \mathcal{D}(L^\flat)_h=\begin{cases} \tilde{\mathcal{Z}}(L^\flat)^\circ, & \text{if } L^\flat\in \mathrm{Hor},\\ 0 , & \text{if } L^\flat\notin \mathrm{Hor}. \end{cases} \]
Proof. By Theorem 4.2, it suffices to compute the multiplicity of an irreducible component in 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ in 
$\mathcal {D}(L)_h$ for all 
$M^\flat \in \mathrm {Hor}(L^\flat )$ (see (4.3)). For such a 
$M^\flat$, define
\[ M':= \frac{1}{\pi}L^\flat \cap M^\flat \quad\text{and}\quad m:= \mathrm{dim}_{\mathbb{F}_q}(M'/L^\flat). \]
Then for a lattice 
$L'$ with 
$L^\flat \subset L'\subset ({1}/{\pi }) L^\flat$, we know that 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ is in 
$\mathcal {Z}^\mathrm {Kra}(L')_h$ if and only if 
$L'\subset M'$. Hence, the multiplicity of an irreducible components in 
$\tilde {\mathcal {Z}}(M^\flat )^\circ$ in 
$\mathcal {D}(L)_h$ is
\begin{equation} 1+\sum_{i=1}^m (-1)^i q^{i(i-1)/2} \sum_{\substack{L^\flat\subset L'\subset M'\\ \mathrm{dim}_{\mathbb{F}_q}(L'/L^\flat)=i}} 1. \end{equation}
Note that 
$m=0$ if and only if 
$M'=M^\flat =L^\flat$, in this case the summation in (4.8) is over an empty set, hence (4.8) is equal to 
$1$. If 
$m>0$, (4.8) is equal to 
$0$ by (2.13).
Part 2. The analytic side
5. Induction formula and primitive local density
In this section, we study various induction formulas of local density polynomials. Let 
$M$ be a Hermitian 
$\mathcal {O}_F$-lattice of rank 
$m$ with 
$\mathrm {v}(M):= \min \{\mathrm {v}_\pi ( h(v,v'))\mid v,v'\in M\} \ge -1$. and let 
$M^{[k]}= \mathcal {H}^k{\unicode{x29BA}} M$ for an integer 
$k \ge 0$. Let 
$L$ be a Hermitian 
$\mathcal {O}_F$-lattice of rank 
$n$.
There is a polynomial 
$\alpha (M, L, X)$ of 
$X$, the local density polynomial, such that
\begin{equation} \alpha(M, L, q^{-2k}) =\int_{\mathrm{Herm}_n(F)} \int_{(M^{[k]})^n} \psi(\langle Y,T({\bf x})-T\rangle)\, d{\bf x} \, dY, \end{equation}
where 
$T({\bf x})$ is the moment matrix of 
${\bf x}$, 
$d{\bf x}$ is the Haar measure on 
$(M^{[k]})^n$ with total volume 
$1$, 
$dY$ is the Haar measures on 
$\mathrm {Herm}_n(F)$ such that 
$\mathrm {Herm}_n(\mathcal {O}_{F})$ has total volume 
$1$ and 
$\psi$ is an additive character of 
$F_0$ with conductor 
$\mathcal {O}_{F_0}$. Finally, we define 
$\langle X,Y\rangle =\mathrm {Tr}(XY)$ on 
$\mathrm {Herm}_n$. We also use the notation 
$\alpha (M, L)=\alpha (M, L, 1)$ and
\begin{equation} \alpha'(M, L) =-\frac{\partial}{\partial X} \alpha(M,L, X)|_{X=1}. \end{equation} There is another way to define 
$\alpha (M,L,X)$ as follows. We use 
$\mathrm {Herm}_{L,M}$ to denote the scheme of Hermitian 
$\mathcal {O}_F$-module homomorphisms from 
$L$ to 
$M$, which is a scheme of finite type over 
$\mathcal {O}_{F_0}$. More specifically, for an 
$\mathcal {O}_{F_0}$-algebra 
$R$, we define
\[ L_{R}:= L\otimes_{\mathcal{O}_{F_0}}R, \quad (x\otimes a,y\otimes b)_{R}:= \pi (x ,y )\otimes_{\mathcal{O}_{F_0}} ab\in \mathcal{O}_{F}\otimes_{\mathcal{O}_{F_0}} R,\quad \text{where }x,y \in L, a,b\in R. \]
Then
\[ \mathrm{Herm}_{L, M}(R)=\{ \phi \in \mathrm{Hom}_{\mathcal{O}_F}(L_{R}, M_{R}) \mid (\phi(x),\phi(y))_{R}\equiv (x,y)_{R} \text{ for all } x,y \in L_{R}\}. \]
To simplify the notation, we let
\begin{equation} I(M,L,d) := \mathrm{Herm}_{L, M}(\mathcal{O}_{F_0}/(\pi_0^d)). \end{equation}Then a direct calculation as in [Reference ShiShi22, Lemma 6.1] shows that
\begin{equation} \alpha(M, L) = q^{-d n (2m -n)} |I(M,L,d)| \end{equation}
for sufficiently large integers 
$d >0$. Since 
$\alpha (M,L, X)$ only depends on the Gram matrices of 
$M$ and 
$L$, we may also denote it by 
$\alpha (S, T, X)$ if 
$S$ and 
$T$ are the Gram matrices of 
$M$ and 
$L$.
Now we define primitive local density polynomials. For 
$1 \le \ell \le n$, let
\begin{equation} (M^{[k]})^{n,(\ell)}=\big\{ {\bf x}=(x_1, \ldots, x_n) \in (M^{[k]})^n \mid \dim \text{Span}\{x_1, \ldots, x_\ell\} =\ell \text{ in } M^{[k]}/\pi M^{[k]}\big\}. \end{equation}
For 
$L=L_1\oplus L_2$, where 
$L_1=\mathrm {Span}\{l_1,\ldots,l_{\ell }\}$ and 
$L_2=\mathrm {Span}\{l_{\ell +1},\ldots,l_n\}$, we define the local 
$\ell$-primitive density to be
\begin{equation} \beta(M^{[k]}, L_1\oplus L_2)^{(\ell)}= \int_{\mathrm{Herm}_n(F)} \int_{(M^{[k]})^{n,(\ell)}} \psi( \langle Y, T({\bf x}) -T \rangle) \,d{\bf x} \, dY. \end{equation}
When 
$\ell \not = n$, the above definition depends on a choice of 
$L=L_1\oplus L_2$. Hence, we always fix such a decomposition 
$L=L_1\oplus L_2$ in this case. When 
$L=L_1{\unicode{x29BA}} L_2$, and 
$L_i$ is represented by 
$T_i$, we also denote 
$\beta (M, L_1{\unicode{x29BA}} L_2)^{(\ell )}$ as 
$\beta (S, \mathrm {Diag}(T_1,T_2))^{(\ell )}$. When 
$\ell =n$, we simply denote 
$\beta (M, L_1\oplus L_2)^{(\ell )}$ as 
$\beta (M,L)$.
Lemma 5.1 Assume 
$L=L_1 \oplus L_2$ where 
$\mathrm {rank}(L_1)=n_1$. Then
\[ \alpha(M,L,X)=\sum_{L_1\subset L_1'\subset L_{1,F}}(q^{n-m}X)^{\ell(L_1'/L_1)}\beta(M,L_1'\oplus L_2,X)^{(n_1)}, \]
where 
$\ell (L_1'/L_1)=\mathrm {length}_{\mathcal {O}_F}L_1'/L_1$.
Proof. This is the analogue of [Reference KitaokaKit83, Lemma 3]. Let 
$G=\mathrm {GL}_{n_1}(F)\cap \mathrm {M}_{n_1}(\mathcal {O}_F)$ and 
$U=\mathrm {GL}_{n_1}(\mathcal {O}_F)$. By choosing a basis 
$\{l_1,\ldots,l_{n_1}\}$ of 
$L_1$, we may identify 
$U\backslash G$ with 
$\{L_1'\mid L_1\subset L_1'\subset L_{1,F}\}$ by sending 
$g$ to 
$L_1\cdot g^{-1}$. Then the identity we want to prove is equivalent to
\[ \alpha(M,L,X)=\sum_{g\in U\backslash G}|{\det g}|^{2k+m-n} \beta(M,L_1\cdot g^{-1}{\unicode{x29BA}} L_2,X)^{(n_1)}, \]
where 
$|\pi |=q^{-1}.$ By a partition of 
$M^n_{k}$, we have
\begin{align*} \alpha(M,L,X)&=\int_{\mathrm{Herm}_n(F)} \,dY \int_{(M^{[k]})^n} \psi(\langle Y,T({\bf x})-T\rangle) \,d{\bf x}\\ &= \sum_{g\in U\backslash G } \int_{\mathrm{Herm}_n(F)} \,dY \int_{(M^{[k]})^{n,(n_1)}\cdot g_1} \psi(\langle Y,T({\bf x})-T\rangle) \,d{\bf x}, \end{align*}
where 
$g_1=\mathrm {Diag}(g, I_{n-n_1})$, and the action of 
$g_1$ is simply matrix multiplication on the 
$n$ components of 
$M^{n,(n_1)}$. Now
\begin{align*}
&\int_{\mathrm{Herm}_n(F)} \,dY
\int_{(M^{[k]})^{n,(n_1)}\cdot g_1} \psi(\langle Y,T({\bf
x})-T\rangle) \,d{\bf x}\\ &\qquad= |{\det g_1}|^{2k+m}
\int_{\mathrm{Herm}_n(F)} \,dY \int_{(M^{[k]})^{n,(n_1)}}
\psi(\langle Y,T({\bf x} g_1)-T\rangle) \,d{\bf x}\\
&\qquad= |{\det g_1}|^{2k+m} \int_{\mathrm{Herm}_n(F)} \,dY
\int_{(M^{[k]})^{n,(n_1)}} \psi(\langle Y,(T({\bf
x})-T[g_1^{-1}])[g_1]\rangle) \,d{\bf x}\\ &\qquad= |{\det g_1}|^{2k+m} \int_{\mathrm{Herm}_n(F)} \,dY
\int_{(M^{[k]})^{n,(n_1)}} \psi(\langle Y[g_1^*],T({\bf
x})-T[g_1^{-1}]\rangle) \,d{\bf x}\\ &\qquad= |{\det g_1}|^{2k+m-n} \int_{\mathrm{Herm}_n(F)} \,dY
\int_{(M^{[k]})^{n,(n_1)}} \psi(\langle Y,T({\bf
x})-T[g_1^{-1}]\rangle) \,d{\bf x}\\ &\qquad= |{\det g_1}|^{2k+m-n} \beta(M^{[k]},L\cdot g_1^{-1})^{(n_1)}.
\end{align*}
Here 
$T[g]:= g^{*}Tg$. Now the lemma is clear.
Theorem 5.2 Let 
$L$ be as in Lemma 5.1. Then
\[ \alpha(M, L, X) = \sum_{i=1}^{n_1} (-1)^{i-1} q^{i(i-1)/2+i(n-m)}X^{i} \cdot \sum_{\substack{L_1 \subset L_1' \subset \pi^{-1}L_{1} \\ \dim {L_1'/L_1}=i}} \alpha(M, L_1'\oplus L_2,X) + \beta(M, L,X)^{(n_1)}. \]
Proof. This is an analogue of [Reference KatsuradaKat99, Proposition 2.1]. The proof follows from a combination of the argument (in a reverse order) in Lemmas 2.16 and 5.1.
Motivated by Theorem 5.2, we give the following definition.
Definition 5.3 Let 
$L =L_1 \oplus L_2$ be as in Lemma 5.1. We define
\begin{equation} \partial \mathrm{Den}(L)^{(n_1)}:= \partial \mathrm{Den}(L)-\sum_{i=1}^{n_1} (-1)^{i-1} q^{i(i-1)/2} \sum_{\substack{L_1 \subset L_1' \subset L_{1, F} \\ \dim {L_1'/L_1}=i}} \partial \mathrm{Den}(L'_1\oplus L_2). \end{equation}Corollary 5.4 Let 
$L =L_1 \oplus L_2$ be as in Lemma 5.1, and 
$\epsilon =\chi (L)$. Then
\[ \partial
\mathrm{Den}(L)^{(n_1)}=\frac{1}{\alpha(I_{n}^{-\epsilon},I_{n}^{-\epsilon})}
\biggl(2\beta'(I_{n}^{-\epsilon},L)^{(n_1)}+\sum_{i}c^{n,i}_{\epsilon}\beta(\mathcal{H}_{n,i}^{\epsilon},L)^{(n_1)}\biggr).
\]
As a corollary of Lemma 5.1, we have the following.
Corollary 5.5 Let 
$L =L_1 \oplus L_2$ be as in Lemma 5.1. Then we have the following identity where the summation is finite:
\[ \partial \mathrm{Den}(L)=\sum_{L_1 \subset L_1' \subset L_{1,F}} \partial \mathrm{Den}(L_1'\oplus L_2)^{(n_1)}. \]
We may reduce the identity 
$\mathrm {Int}(L)=\partial \mathrm {Den}(L)$ to a primitive version as the following theorem shows.
Theorem 5.6 Let 
$L=L_1 \oplus L_2 \subset \mathbb {V}$ be as in Lemma 5.1.
(1) Conjecture 1.1 is true for
$L$ if for every $L_1 \subset L_1' \subset L_{1, F}$, we have
\[ \mathrm{Int}(L_1' \oplus L_2)^{(n_1)}= \partial \mathrm{Den}(L_1' \oplus L_2)^{(n_1)}. \]
(2) If Conjecture 1.1 holds for all lattices
$L'=L_1'\oplus L_2$ of $\mathbb {V}$ of rank $n$ with $L_1 \subset L_1' \subset L_{1, F}$, then
\[ \mathrm{Int}(L_1 \oplus L_2)^{(n_1)} =\partial \mathrm{Den}(L_1 \oplus L_2)^{(n_1)}. \]
(3) For
$1 \le n_1 \le n$, Conjecture 1.1 is true if and only if for every lattice $L =L_1 \oplus L_2 \subset \mathbb {V}$ with $\mathrm {rank}(L_1)=n_1$, one has
\[ \mathrm{Int}(L_1 \oplus L_2)^{(n_1)} =\partial \mathrm{Den}(L_1 \oplus L_2)^{(n_1)}. \]
Proof. Part (1) follows from Lemma 2.16 and Corollary 5.5. Part (2) follows from Definitions 2.15 and 5.3. Part (3) follows from parts (1) and (2).
For the rest of this section, we assume that 
$M$ is unimodular of rank 
$m$ with a Gram matrix 
$\mathrm {Diag}(I_{m-1}, \nu )$. To go further with the calculation of 
$\alpha (M,L,X)$, we need an induction formula for 
$\beta (M,L,X)^{(\ell )}$ as follows. The proof is essentially the same as that of Corollary 9.11 of [Reference Kudla and RapoportKR11], and is left to the reader.
Proposition 5.7 Let 
$L=L_1 {\unicode{x29BA}} L_2$, where 
$L_j$ is of rank 
$n_j$. Let 
$C(M^{[k]}, L_1)$ be the 
$\mathrm {U}(M^{[k]})$-orbits of sublattices 
$M(i) \subset M^{[k]}$ such that 
$M(i)$ is isometric to 
$L_1$, and write 
$C(M^{[k]}, L_1)=\sqcup _{i\in J}\{ M(i) \}$. Then
\[ \beta(M, L, X)^{(n_1)} =\sum_{i\in J} |M:M(i) {\unicode{x29BA}} M(i)^{\perp}|^{-n_2} |M(i)^{\vee}:M(i)|^{n_2} \beta_i(M, L_1, X) \alpha(M(i)^{\perp},L_2), \]
where
\begin{align*} &\beta_i(M, L_1, X)\\ &\quad=\lim_{d\to
\infty}q^{-d n_1 (2m+4k -n_1)} \#\big\{ \phi \in I(M^{[k]},
L_1,d)^{(n_1)} \mid \exists\, \ \Phi \in \mathrm{U}(M)
\text{ with } \phi(L_1)=\Phi(M(i))\big\}, \end{align*}
and
\[ I(M^{[k]},L_1,d)^{(n_1)}:= \big\{\phi\in I(M^{[k]},L_1,d)\mid \mathrm{rank}_{\mathbb{F}_q}\phi(L_1)\otimes_{\mathcal{O}_F} \mathbb{F}_q =n_1\big\}. \]
Recall that 
$I(M^{[k]},L_1,d)$ is defined in (5.3).
One special case is that 
$L=\mathcal {H}^i{\unicode{x29BA}} L_2$. Since any sublattice of 
$M^{[k]}=M{\unicode{x29BA}} \mathcal {H}^k$ isometric to 
$\mathcal {H}^i$ is always a direct summand of 
$M^{[k]}$ and 
$\alpha (M,\mathcal {H}^i,X)=\beta (M,\mathcal {H}^i,X)^{(2i)}=\beta (M,\mathcal {H}^i,X)$, the above proposition specializes as follows.
Corollary 5.8 Assume 
$L=\mathcal {H}^i{\unicode{x29BA}} L_2$, then
\begin{equation} \alpha(M,L,X)=\beta(M,\mathcal{H}^i,X)\alpha(M,L_2,q^{2i}X)=\alpha(M,\mathcal{H}^i,X)\alpha(M,L_2,q^{2i}X). \end{equation}We end this section with two more special cases of Proposition 5.7. Proofs are given in Appendix A.
Proposition 5.9 Let the notation be as in Proposition 5.7. Assume 
$n_1=1$ and 
$L_1 =\langle t \rangle$ where 
$t\in \mathcal {O}_{F_0}.$
(1) There always exists a primitive vector
$M(1)\in \mathcal {H}^k$ with $q(M(1))=t$, and
Here\[ M(1)^{\perp} \cong \mathcal{H}^{k-1} {\unicode{x29BA}} I_{m}^{\chi(M)} {\unicode{x29BA}} \langle -t \rangle. \]
$\langle t \rangle$ denotes a lattice $\mathcal {O}_F v$ of rank one with $(v, v)=t$.(2) If
$\mathrm {v}(t)=0$, then there exist a primitive vector $M(0)\in M$ with $q(M(0))=t$, and
Here\[ M(0)^{\perp}\cong \mathcal{H}^{k}{\unicode{x29BA}} I_{m-2}^{\epsilon_{m-2}} {\unicode{x29BA}} \langle \nu t \rangle. \]
$\epsilon _{m-2}=\chi ((-1)^{(m-2)(m-3)/2})$.(3) If
$\mathrm {v}(t)>0$, then there exist a primitive vector $M(0)\in M$ with $q(M(0))=t$ only when $M$ is isotropic (i.e. $\exists \, v\in M$ with $q(v)=0$). In this case,
Assuming the existence of\[ M(0)^{\perp}\cong \mathcal{H}^{k}{\unicode{x29BA}} I_{m-2}^{\chi(M)} {\unicode{x29BA}} \langle -t \rangle. \]
$M(1)$ and $M(0)$, we have
\[ |M^{[k]}:M(i){\unicode{x29BA}} M(i)^{\perp}|^{-1}|M(i)^{\vee}:M(i)|=\begin{cases} 1, & \text{if }i=1,\\ q, & \text{if }i=0. \end{cases} \]
(4) Under the action of
$\mathrm {U}(M^{[k]})$, $v$ is either in the same orbit of a fixed vector $M(1)\in \mathcal {H}^k$ or a fixed vector $M(0)\in M$.(5) We have the following induction formula:
Moreover:\[ \beta(M, L,X)^{(1)} =\beta_1(M,L_1,X) \alpha(M(1)^{\perp},L_2)+q^{n-1}\beta_0(M, L_1,X) \alpha(M(0)^{\perp},L_2). \]
(a) for any
$L_1$,
\begin{align*} \beta_1(M, L_1,X)=1-X; \end{align*}
(b) assume
$\mathrm {v}(t)=0$, then
\[ \beta_0(M, L_1 ,X)=\begin{cases}
\big(1+\chi(M) \chi(L)q^{-({m-1})/{2}})X, & \text{if
}m\text{ is odd},\\ \big(1-\chi(M)q^{-{m}/{2}})X, &
\text{if }m\text{ is even}; \end{cases}
\]
(c) assume
$\mathrm {v}(t)>0$, then
\[ \beta_0(M, L_1 ,X)=\begin{cases}
\big(1-q^{1-m}\big)X, & \text{if }m\text{ is odd},\\
\big(1-q^{1-m}+\chi(M)(q-1)q^{-{m}/{2}}\big)X, & \text{if
}m\text{ is even}. \end{cases} \]
Proof. Parts (1)–(4) are proved in § A.1. The induction formula for 
$\beta (M, L,X)^{(1)}$ follows from Proposition 5.7. For the formula of 
$\beta _i(M, L_1 ,X)$, see Corollaries A.10 and A.12.
Proposition 5.10 Let the notation be as in Proposition 5.7. Assume 
$\mathrm {v}(L_1)>0$ and 
${n_1=2}$. Then we have a partition of 
$C(M^{[k]}, L_1)=\bigsqcup _{i=0}^{2}C_i(M^{[k]},L_1)$ such that for any 
$M(i)\in C_i(M^{[k]},L_1)$, 
$M(i)^{\bot }$ is isometric to
\[ (-L_1) {\unicode{x29BA}} \mathcal{H}^{k-i} {\unicode{x29BA}} M^{(i)}. \]
Here 
$M^{(i)}$ is a unimodular 
$\mathcal {O}_F$-lattice of rank 
$m-2(2-i)$ and has determinant 
$(-1)^i \det L$.
Moreover, we have
\begin{equation} \beta(M, L,X)^{(2)}=\sum_{i=0}^{2}q^{(2-i)(n-2)}\beta_i(M,L_1,X)\alpha(M(i)^{\perp}, L_2,X), \end{equation}where
\begin{align*}
\beta_2(M,L_1,X)&=(1-X)(1-q^2 X),\\
\beta_1(M,L_1,X)&= q(q+1)\big( (1-q^{1-m})+\delta_{e}(m)\chi(M)(q-1)q^{-{m}/{2}}\big)X(1-X),\\
\beta_0(M,L_1,X) &=\begin{cases}
q(1-q^{1-m})(1-q^{3-m})X^2, & \text{if }m\text{ is odd},\\
q\big( (1-q^{2-m})+\chi(M)(q^2-1)q^{-{m}/{2}}\big)(1-q^{2-m})X^2, & \text{if }m\text{ is even}. \end{cases}
\end{align*}
Here 
$\delta _e(m)=1$ or 
$0$ depending on whether 
$m$ is even or odd.
6. The modified Kudla–Rapoport conjecture
Recall that the Hermitian lattices used to define the correction terms are of the following forms:
\begin{equation} \mathcal{H}_{n,i}^{\epsilon}:= \mathcal{H}^i{\unicode{x29BA}} I_{n-2i}^{\epsilon}, \quad \text{for } 1\leq i \leq \frac{n}{2}, \quad \epsilon =\pm 1, \end{equation}
where 
$I_{n-2i}^{\epsilon }$ is the unimodular Hermitian lattice of rank 
$n-2i$ with 
$\chi (I_{n-2i}^{\epsilon })= \chi (\mathcal {H}_{n,i}^{\epsilon } )=\epsilon$. When 
$n=2r$ is even, we take 
$I_{0}^{\epsilon } =0$ and 
$\mathcal {H}_{n, r}^1 = \mathcal {H}^{r}$.
Theorem 6.1 Let 
$r_\epsilon =({n-1})/2$ when 
$n$ is odd, and 
$r_\epsilon =\lfloor ({n+\epsilon })/2\rfloor$ when 
$n$ is even. In the following, we just write 
$r_{\epsilon }$ as 
$r$:
\begin{align*} A^\epsilon =(A_{i, j}^\epsilon) &= \begin{pmatrix} \alpha(\mathcal{H}_{n,1}^{\epsilon}, \mathcal{H}_{n,1}^{\epsilon}) & \alpha(\mathcal{H}_{n,2}^{\epsilon}, \mathcal{H}_{n,1}^{\epsilon}) & \cdots & \alpha(\mathcal{H}_{n,r}^{\epsilon}, \mathcal{H}_{n,1}^{\epsilon}) \\ 0 & \alpha(\mathcal{H}_{n,2}^{\epsilon}, \mathcal{H}_{n,2}^{\epsilon}) & \cdots & \alpha(\mathcal{H}_{n,r}^{\epsilon}, \mathcal{H}_{n,2}^{\epsilon}) \\ \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \alpha(\mathcal{H}_{n,r}^{\epsilon}, \mathcal{H}_{n,r}^{\epsilon}) \end{pmatrix},\\ B^\epsilon &= {}^t(\alpha'(I_{n}^{-\epsilon}, \mathcal{H}_{\epsilon}^{n, 1}), \ldots, \alpha'(I_{n}^{-\epsilon}, \mathcal{H}_{n,r}^{\epsilon})), \end{align*}
and
\begin{align*} C^\epsilon={}^t(c_\epsilon^{n, 1}, \ldots, c_\epsilon^{n, r}),\end{align*}
where 
$c_{n,i}^\epsilon$ is as in Conjecture 1.1.
Then 
$C^\epsilon$ is the solution of the equation
\begin{align} A^\epsilon C^\epsilon=-2 B^{\epsilon}. \end{align}Moreover,
\begin{align}
A_{j,j}^\epsilon&=2q^{{(n-2j)(n-2j-1)}/{2}} \prod_{0<
s\le j}(1-q^{-2s})\prod_{1 \le s \le
\lfloor({n-2j-1})/{2}\rfloor}(1-q^{-2s})\nonumber\\ &\quad
\times \begin{cases} 1, & \text{if }n\text{ is odd},\\
1, -\epsilon q^{-({n-2j})/{2}}, & \text{if }n\text{ is
even}. \end{cases} \end{align}
Finally, for 
$i< j$,
\begin{equation} A_{i,j}^\epsilon=
A_{j,j}^\epsilon\cdot \begin{cases}
I\left(n-2i,\dfrac{n-2i-1}{2},j-i\right), & \text{if
}n\text{ is odd},\\
I\left(n-2i,\dfrac{n-2i-1+\epsilon}{2},j-i\right), &
\text{if }n\text{ is even}, \end{cases}
\end{equation}
where
\[ I(n,d,k):= \prod_{s=1}^{k}\frac{(q^{d-s+1}-1)(q^{n-d-s}+1)}{q^s-1}. \]
Proof. First note that 
$\alpha (\mathcal {H}_{n, i}^\epsilon, \mathcal {H}_{n,j}^{\epsilon })=0$ if 
$i< j$. Thus, (1.9) is indeed equivalent to (6.2), and there exists a unique solution 
$C_{\epsilon }$.
Now we compute 
$A_{j,j}^\epsilon$ explicitly. Corollary 5.8 and Lemma A.9 imply that
\[ \alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}_{n,j}^{\epsilon})=\alpha(\mathcal{H}^j,\mathcal{H}^j)\alpha(I_{n-2j}^{\epsilon},I_{n-2j}^{\epsilon}). \]
According to Lemma A.8,
\begin{align*} \alpha(\mathcal{H}^j,\mathcal{H}^j)=\prod_{0< s\le j}(1-q^{-2s}). \end{align*}
By Lemma A.11,
\[ \alpha(I_{n-2j}^{\epsilon},I_{n-2j}^{\epsilon})=|\mathrm{O}(\overline{I}_{n-2j}^{\epsilon})(\mathbb{F}_q)|, \]
where 
$\overline {I}_{n-2j}^{\epsilon }=I_{n-2j}^\epsilon \otimes _{\mathcal {O}_F} \mathcal {O}_{F}/(\pi )$ is the space over 
$\mathbb {F}_q$ with the naturally induced quadratic form. Now (6.3) follows from the well-known formula:
\[
|\mathrm{O}(\overline{I}_{n-2j}^{\epsilon})(\mathbb{F}_q)|
=\begin{cases}
2q^{{(n-2j)(n-2j-1)}/{2}}\displaystyle\prod _{s=1}^{({n-2j-1})/{2}}(1-q^{-2s}), & \text{if }n\text{ is odd},\\
2q^{{(n-2j)(n-2j-1)}/{2}}(1-\epsilon q^{-({n-2j})/{2}})\displaystyle\prod _{s=1}^{(({n-2j})/{2})-1}(1-q^{-2s}), & \text{if }n\text{
is even}. \end{cases} \]
To obtain (6.4), note that (Corollary 5.8)
\[ \alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}_{n,i}^{\epsilon})=\alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}^{i}) \alpha(\mathcal{H}_{n-2i,j-i}^{\epsilon},I_{n-2i}^{\epsilon}), \]
and
\[ \alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}_{n,j}^{\epsilon})=\alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}^{i}) \alpha(\mathcal{H}_{n-2i,j-i}^{\epsilon},\mathcal{H}_{n-2i,j-i}^{\epsilon}). \]
Hence,
\[ \frac{A_{i, j}^\epsilon}{A_{j, j}^\epsilon}= \frac{\alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}_{n,i}^{\epsilon})}{\alpha(\mathcal{H}_{n,j}^{\epsilon},\mathcal{H}_{n,j}^{\epsilon})}=\frac{\alpha(\mathcal{H}_{n-2i,j-i}^{\epsilon},I_{n-2i}^{\epsilon})}{\alpha(\mathcal{H}_{n-2i,j-i}^{\epsilon},\mathcal{H}_{n-2i,j-i}^{\epsilon})}. \]
Fix an 
$\mathcal {O}_F$-lattice 
$L$ that is represented by 
$I_{n-2i}^{\epsilon }$. According to Lemma 6.2, to compute 
${A_{i, j}^\epsilon }/{A_{j, j}^\epsilon }$, we need to count the number of lattices 
$L'$ in 
$L_{F}$ such that contain 
$L\subset L'$ and 
$L' \cong \mathcal {H}_{n-2i,j-i}^{\epsilon }$, which is equivalent to the following condition:
\[ \pi L \stackrel{j-i}{\subset} \pi L' \stackrel{n-2j}{\subset} (L')^\sharp\stackrel{j-i}{\subset} L \stackrel{j-i}{\subset}{L'}. \]
Since 
$L'$ and 
$\pi L'$ determine each other, we just need to count 
$\pi L'$ satisfying the above condition. We regard 
$\pi L'/\pi L$ as a 
$(j-i)$-dimensional subspace of 
$L/\pi L$, where 
$L/\pi L$ is equipped with quadratic form 
$(x,y)/\pi$.
Claim The condition
\[ \pi L' \stackrel{}{\subset} (L')^\sharp \]
is equivalent to the condition that 
$\pi L'/\pi L$ is an isotropic subspace of 
$L/\pi L.$
Indeed, assume 
$\pi L'/\pi L$ is an isotropic subspace of 
$L/\pi L.$ Then 
$(\pi x, \pi y)\in \pi \mathcal {O}_F$ for any 
$x,y\in L'$, which is equivalent to 
$(x,\pi y)\in \mathcal {O}_F$ for any 
$x,y\in L'$. The latter condition is the same as 
$L' \stackrel {}{\subset } (L')^\sharp$. The other direction is clear.
Therefore, 
${A_{i, j}^\epsilon }/{A_{j, j}^\epsilon }$ is the number of 
$(j-i)$-dimensional isotropic subspaces of 
$L/\pi L$. According to [Reference Li and ZhangLZ22b, Lemma 3.2.2], it equals to
\[ \begin{cases}
I\left(n-2i,\dfrac{n-2i-1}{2},j-i\right)\!, & \text{if
}n\text{ is odd},\\
I\left(n-2i,\dfrac{n-2i-1+\epsilon}{2},j-i\right)\!, &
\text{if }n\text{ is even}. \end{cases}
\]
□
According to Theorem 6.1, in order to solve 
$C^\epsilon$, we need to know 
$B^\epsilon$ and 
$A^\epsilon$. Here, 
$B^\epsilon$ can be calculated by applying Corollary 5.8 and Proposition 5.9 inductively. The following lemma can be used to compute 
$A^\epsilon$.
Lemma 6.2 Let 
$F/F_0$ be a quadratic 
$p$-adic field extension, and let 
$L$ and 
$M$ be two 
$\mathcal {O}_F$-Hermitian lattices of rank 
$n$. Then 
${\alpha (M, L)}/{\alpha (M, M)}$ is equal to the number of lattices 
$L'$ in 
$L_F$ containing 
$L$ and isometric to 
$M$.
Proof. The proof is a generalization of that of Proposition 10.2 of [Reference Kudla and RapoportKR14b] and works for both inert and ramified primes.
Let us assume that there is an isometric embedding from 
$L$ into 
$M$, otherwise both sides of the identity in the lemma are zero. In this case, we have a fixed 
$L_F \cong M_F$. Let 
$\alpha$ (respectively, 
$\beta$) be a top degree translation invariant form on 
$L_F^n$ (respectively, 
$\mathrm {Herm}_n(F)$). Let 
$\nu _p=\alpha /h^*(\beta )$ where
\[ h: \quad L_F^n\rightarrow \mathrm{Herm}_n(F),\quad x\mapsto (x,x). \]
Define 
$X$ to be the set of 
$F$-linear isometric embeddings from 
$L$ into 
$M$. By fixing a basis of 
$L_F$ and regarding 
$\phi \in X$ as a linear isometry from 
$L_F$ to itself, we identify 
$X$ as a subset of 
$L_F^n$. By the argument in § 3 of [Reference Gan and YuGY00] (in particular, Lemma 3.4), we know that
\begin{equation} \alpha(M,L)=\mathrm{vol}(X,d\nu_p) \frac{\mathrm{vol}(\mathrm{Herm}_n(\mathcal{O}_{F}),d\beta)}{\mathrm{vol}((M)^n,d\alpha)}. \end{equation}
For any 
$\phi \in X$ regarded as a linear isometry from 
$L_F$ to itself, the lattice 
$L_\phi := \phi ^{-1}(M)$ is a lattice containing 
$L$. Conversely, for any 
$L'$ containing 
$L$ and isometric to 
$M$, there is a 
$\phi \in X$ such that 
$L_\phi =L'$. Hence, we have a partition
\[ X=\bigsqcup_{L \subset L'} X_{L'}, \quad X_{L'}:= \{\phi\in X\mid L_\phi=L'\}. \]
Since each 
$L'$ is isomorphic to 
$M$, all the 
$X_{L'}$ have the same volume as that of 
$X_M$. Specializing (6.5) to 
$L=M$, we see
\begin{equation} \alpha(M,M)=\mathrm{vol}(X_{M},d\nu_p) \frac{\mathrm{vol}(\mathrm{Herm}_n(\mathcal{O}_{F}),d\beta)}{\mathrm{vol}((M)^n,d\alpha)}. \end{equation}Remark 6.3 When 
$F/F_0$ is unramified and 
$M$ is unimodular, the lemma was proved by (3.6.1.1) of [Reference Li and ZhangLZ22a].
Now we specialize Theorem 6.1 to the case 
$n=3$.
Lemma 6.4 Assume 
$n=3$ and 
$\epsilon =\chi (L)$. Then 
$c_{3,1}^\epsilon ={q^2}/({1+q})$, hence
\[ \partial \mathrm{Den}(L)=2\frac{\alpha'(I_{3}^{-\epsilon},L)}{\alpha(I_{3}^{-\epsilon},I_{3}^{-\epsilon})}+\frac{q^2}{1+q}\frac{\alpha(\mathcal{H}_{3,1}^{\epsilon},L)}{\alpha(I_{3}^{-\epsilon},I_{3}^{-\epsilon})}. \]
Proof. First, according to Theorem 6.1,
\begin{equation} \alpha(\mathcal{H}_{3,1}^{\epsilon},\mathcal{H}_{3,1}^{\epsilon})=2(1-q^{-2}). \end{equation}By Corollary 5.8, we have
\[ \alpha (I_{3}^{-\epsilon}, \mathcal{H}_{3,1}^{\epsilon},X)=\alpha(I_{3}^{-\epsilon}, \mathcal{H},X)\alpha(I_{3}^{-\epsilon},I_{1}^{\epsilon},q^2 X). \]
According to Lemmas A.8 and A.9,
\[ \alpha(I_{3}^{-\epsilon}, \mathcal{H},X)= \beta(\mathcal{H}^k,\mathcal{H})=1-X. \]
Lemma 7.1 gives that
\[ \alpha(I_{3}^{-\epsilon},I_{1}^{\epsilon},q^2 X)=1-qX . \]
Hence,
\[ \alpha (I_{3}^{-\epsilon}, \mathcal{H}_{3,1}^{\epsilon},X)=(1-X)(1-qX), \]
and
\[ \alpha' (I_{3}^{-\epsilon}, \mathcal{H}_{3,1}^{\epsilon})=1-q. \]
Combining this with (6.7), we solve (6.2) and obtain
\[ c_{3,1}^\epsilon=\frac{q^2}{1+q}. \]
Now the lemma follows from (1.10).
7. Local density formula when $\mathrm {rank}(T)\le 2$
The main purpose of this section is to give an explicit formula for 
$\alpha (I, T, X)$ where 
$I=\mathrm {Diag}(I_{m-1}, \nu )$ with 
$\nu \in \mathcal {O}_{F_0}^\times$ and 
$\mathrm {rank}(T)\le 2$.
7.1 The case $T=(t)$
In order to apply induction formulas to calculate 
$\alpha (I, T, X)$ for 
$T$ with 
$\text {rank}(T) =2$, we need to consider the case 
$T=(t)$ first. Write 
$t=t_0(-\pi _0)^{\mathrm {v}(t)}$ for 
$t_0\in \mathcal {O}_{F_0}^\times$, and
\begin{equation} I_{a, b}=\mathrm{Diag}(I,\nu_1(-\pi_0)^a,\nu_2(-\pi_0)^{b}) =\mathrm{Diag}(s_1,\ldots,s_{m+2}) \end{equation}
for integers 
$0\le a \le b$.
Lemma 7.1 Assume 
$0\le a\le b\le \mathrm {v}(t)$.
(1) If
$m$ is odd, then
\begin{align*} \alpha(I_{a,b},(t),X)&=1+\chi(I)\chi(-\nu_1)(q-1)\sum_{s=a+1}^{b}q^{-ms+a+({m-1})/{2}}X^s\\ &\quad +\chi(I_{a,b})\chi( t_0)q^{-(m+1)\mathrm{v}(t)+a+b-({m+1})/{2}}X^{\mathrm{v}(t)+1}. \end{align*}
(2) If
$m$ is even, then
\begin{align*} \alpha(I_{a,b},(t),X)
&=1+\chi(I) (q-1) \sum_{s=1}^{a}q^{-(m-1)s+{m}/{2}-1}X^s\\
&\quad+\chi(I_{a,b}) q^{a+b}\biggl((q-1)
\sum_{s=b+1}^{\mathrm{v}(t)}q^{-(m+1)s+{m}/{2}} X^s -
q^{-(m+1)\mathrm{v}(t)-1-{m}/{2}}X^{\mathrm{v}(t)+1}\biggr).
\end{align*}
Proof. Direct calculation gives
\begin{align*} \alpha(I_{a,b},(t),X)
&=\int_{F_0}\, dY \int_{\mathcal{O}_F^{2k+m+2}} \psi
(\langle Y, \mathrm{Diag}(\mathcal{H}^k,I_{a,b})[{\bf x}]-t
\rangle)\, d{\bf x} \nonumber\\ &=\int_{F_0} \psi(-tY)\, dY\\
&\quad \times \int_{\mathcal{O}_F^{2k} \times \mathcal{O}_F^{m+2}}\psi
\biggl( Y \sum_{i=1}^{k}
\mathrm{tr}\biggl(\frac{1}{\pi}x_i\bar{y}_i\biggr) +Y
\sum_{l=1}^{m+2}s_lz_l\bar{z}_{l}\biggr)
\prod_{i}\,dx_i \,dy_i \prod_{l}\,dz_l\nonumber\\
&=1+\sum_{s=1}^{\infty}\int_{\mathrm{v}(Y)=-s} I_k(Y)
I_{I_{a,b}}(Y)\psi(-tY) \, dY. \end{align*}
Here, according to [Reference ShiShi22, Lemma 7.6],
\begin{align*} I_k(Y)=\int_{\mathcal{O}_F^{2k}}\psi
\biggl( Y \sum_{i=1}^{k} \mathrm{tr}\biggl(
\frac{1}{\pi}x_i\bar{y}_i\biggr)\biggr) \prod
\,dx_i \,dy_i= q^{-2ks} , \end{align*}
and
\begin{align*} I_{I_{a,b}}(Y)
=\int_{\mathcal{O}_F^{m+2}}\psi \biggl( Y
\sum_{l=1}^{m+2}s_lz_l\bar{z}_{l}\biggr) \prod
\,dz_l=\prod_{l=1}^{m+2} J(s_l Y), \end{align*}
where
\begin{equation} J(t) =\int_{\mathcal{O}_F} \psi(t z \bar z) \,dz = \begin{cases} 1, & \text{if } \mathrm{v}(t) \ge 0,\\ q^{\mathrm{v}(t)} \chi(-t_0) g\bigl(\chi,\psi_{{1}/{\pi_0}}\bigr), & \text{if } \mathrm{v}(t) <0, \end{cases} \end{equation}and
\begin{align*} g(\chi,\psi_{{1}/{\pi_0}}) =\sum_{x \in \mathcal{O}_{F_0}/\pi_0} \chi(x) \psi\biggl(\frac{x}{\pi_0}\biggr) \end{align*}
is the Gauss sum. Write 
$\psi '=\psi _{{1}/{\pi _0}}$. Then
\begin{align*}
&\alpha(I_{a,b},(t),X)\nonumber\\
&\quad=1+\sum_{s=1}^{a}q^{s}\int_{\mathcal{O}_{F_0}^{\times}}q^{-2ks}\cdot
q^{-ms} \chi(\nu (-Y)^m)g(\chi,\psi')^{m}\psi(-(-\pi_0)^s
Yt) \,dY\nonumber\\ &\qquad
+\sum_{s=a+1}^{b}\int_{\mathcal{O}_{F_0}^{\times}}q^{-2ks}\cdot
q^{-m s+a} \chi(\nu_1\nu
(-Y)^{m+1})g(\chi,\psi')^{m+1}\psi(-(-\pi_0)^{-s} Yt)\,
dY\nonumber\\
&\qquad+\sum_{s=b+1}^{\infty}\int_{\mathcal{O}_{F_0}^{\times}}q^{-2ks}\cdot
q^{-(m+1)s+a+b} \chi(\nu_1\nu_2\nu
(-Y)^{m+2})g(\chi,\psi')^{m+2}\psi(-(-\pi_0)^{-s} Yt)\, dY.
\end{align*}
Recall the well-known facts that
\begin{equation}
\begin{aligned} g(\chi,\psi')^2 & =\chi(-1)\cdot q, \\
\int_{\mathcal{O}_{F_0}^\times} \psi((-\pi_0)^{-s} Y t)
\,dY & = \operatorname{Char}(\pi_0^s\mathcal{O}_{F_0})(t) -
q^{-1}\operatorname{Char}(\pi_0^{s-1}\mathcal{O}_{F_0})(t),\\
\int_{\mathcal{O}_{F_0}^\times }\chi(Y)\psi((-\pi_0)^{-s} Y
t) \,dY & =\chi(-t_0) q^{-1} g(\chi,
\psi')\operatorname{Char}(\pi_0^{s-1}\mathcal{O}_{F_0}^\times)(t).
\end{aligned} \end{equation}
When 
$m$ is odd, we have
\begin{align*} \alpha(I_{a,b}, (t), X)&=1+\chi((-1)^{({m+1})/{2}}\nu_1\nu )(q-1)
\sum_{s=a+1}^{b}q^{-ms+a+({m-1})/{2}}X^s\\ &\quad +\chi((-1)^{({m+1})/{2}}\nu_1\nu_2\nu
t_0)q^{-(m+1)(\mathrm{v}(t)+1)+a+b+({m+1})/{2}}X^{\mathrm{v}(t)+1}.
\end{align*}
When 
$m$ is even, we have
\begin{align*} &\alpha(I_{a,b}, (t),
X)\\ &\quad=1+\chi((-1)^{{m}/{2})}\nu) (q-1)
\sum_{s=1}^{a}q^{-(m-1)s+{m}/{2}-1}X^s +\chi((-1)
^{({m+2})/{2}}\nu_1\nu_2\nu)\\ &\qquad \cdot
\biggl((q-1)
\sum_{s=b+1}^{\mathrm{v}(t)}q^{-(m+1)s+a+b+{m}/{2}} X^s -
q^{-(m+1)(\mathrm{v}(t)+1)+a+b+{m}/{2}}X^{\mathrm{v}(t)+1}\biggr).
\end{align*}
Finally, note that for 
$I$ of rank 
$m$ we have
\[ \chi(I)=\begin{cases} \chi((-1)^{({m-1})/{2}}\nu), & \text{if }m\text{ is odd},\\ \chi((-1)^{{m}/{2}}\nu), & \text{if }m\text{ is even}. \end{cases} \]
Now the lemma is clear.
Similarly, we have the following lemma.
Lemma 7.2 Let 
$I$ be unimodular with odd rank 
$m$. Then
\[ \alpha( I{\unicode{x29BA}} \mathcal{H}_{i},(t), X)=\begin{cases} 1+\chi(I)\chi( t_0)q^{-(\mathrm{v}(t)+1)(m+1)+({m+1})/{2}+i} X^{\mathrm{v}(t)+1}, & \text{if }i\le 2\mathrm{v}(t),\\ 1+\chi(I)\chi( t_0)q^{-(\mathrm{v}(t)+1)(m-1)+({m-1})/{2}}X^{\mathrm{v}(t)+1}, & \text{if }i> 2\mathrm{v}(t). \end{cases} \]
7.2 The case $T=\mathrm {Diag}(u_1(-\pi _0)^a,u_2(-\pi _0)^b)$
In this subsection, we compute 
$\alpha (I,T,X)$ for 
$I$ unimodular of rank 
$m\ge 2$ and 
$T=\mathrm {Diag}(u_1(-\pi _0)^a,u_2(-\pi _0)^b)$ with 
$0\le a \le b$. Notice that 
$\alpha (I, T, X)=0$ when 
$a<0$.
Proposition 7.3 Assume 
$T=\mathrm {Diag}(u_1(-\pi _0)^a,u_2(-\pi _0)^b)$ and that 
$I$ is isotropic of even rank 
$m \ge 2$, then
\begin{align*} \alpha(I,T,X)&=(1-X)\biggl(\sum_{i=0}^{a}(q^{2-m}X)^{i}+\gamma_e(I,T,X)\biggr)\\ &\quad+q X(q^{2-m}X)^{a}(1-\chi(I)q^{-({m}/{2})})( 1+ \chi(I)\chi(T)q^{({m-2})/{2}}(q^{2-m} X)^{b+1})\\ &\quad+ \big(1-q^{-(m-1)}+(q-1)\chi(I)q^{-({m}/{2})} \big) X\\ &\quad\cdot \biggl(q \sum_{i=0}^{a-1}(q^{2-m}X)^i+\gamma_e(I,T,X)-\chi(I)\chi(T)q^{{m}/{2}}(q^{2-m}X)^{a+b+1} \biggr), \end{align*}
where
\[ \gamma_{e}(I,T,X) = \chi(I)q^{{m}/{2}}\biggl(\sum_{d=1}^a(q^d-1)(q^{2-m}X)^{d} +\chi(T)q^a (q^{2-m}X)^{b+1}\sum_{i=0}^{a}(q^{1-m}X)^{i}\biggr). \]
Proof. Since 
$I$ is of even rank, 
$u_1\cdot I^{[k]}\approx I^{[k]}$, and we may assume 
$T$ is of the form 
$\mathrm {Diag}( (-\pi _0)^a,u(-\pi _0)^b)$ without loss of generality.
According to Theorem 5.2 and Proposition 5.9, we have
\begin{align*} \alpha(I,T, X)&=\beta_1(I, (-\pi_0)^a,X) \alpha(M(1)^{\perp},u(-\pi_0)^b)\\ &\quad +q\beta_0(I, (-\pi_0)^a,X) \alpha(M(0)^{\perp},u(-\pi_0)^b) \\ &\quad +q^{2-m}X\alpha(I, \mathrm{Diag}((-\pi_0)^{a-1},u(-\pi_0)^b),X) \end{align*}
where 
$M(1)^{\perp }=\mathrm {Diag}(\mathcal {H}^{k-1},-(-\pi _0)^i,I)$ and
\[ M(0)^{\perp}=\mathrm{Diag}(\mathcal{H}^k,\underbrace{-(-\pi_0)^i,1,\ldots,1,-\nu}_{m-1}). \]
Continuing this process, we obtain
\begin{align*}
\alpha(I,T,X)&=\sum_{i=0}^{a}(q^{2-m}X)^{a-i}\\
&\quad\cdot \big(\beta_1(I,(-\pi_0)^i,X)\alpha(M(1)^{\perp},u(-\pi_0)^b) +q\beta_0(I,(-\pi_0)^i,X)
\alpha(M(0)^{\perp},u(-\pi_0)^b)\big).
\end{align*}
By the formulas in Proposition 5.9 and Lemma 7.1, the above is equal to
\begin{align*} &\sum_{i=0}^{a}(q^{2-m}X)^{a-i}(1-X)\\ &\quad\cdot \biggl(1+(q-1)\chi(I)q^{({m-2})/{2}}\sum_{s=-i}^{-1}(q^{(m-1)}(q^2 X)^{-1})^s+\chi(I)\chi(T)q^{i+{m}/{2}}(q^{2-m} X)^{b+1}\biggr)\\ &\quad+q (q^{2-m}X)^{a}X(1-q^{-({m}/{2})}\chi(I))( 1+ \chi(I)\chi(T)q^{-(b+1)(m-2)+({m-2})/{2}}X^{b+1})\\ &\quad+ q\sum_{i=1}^{a}(q^{2-m}X)^{a-i}X\big((1-q^{-(m-1)})+(q-1)\chi(I) q^{-(m-1)+({m-2})/{2}}\big)\\ &\quad\cdot \biggl(1+(q-1)\chi(I)q^{({m-4})/{2}}\sum_{s=-i}^{-1}(q^{(m-3)}X^{-1})^s+\chi(I)\chi(T) q^{i+({m-2})/{2}}(q^{2-m}X)^{b+1}\biggr). \end{align*}
Now the transformation
\[ \sum_{i=0}^a \sum_{s=1}^{i}q^s (q^{2-m}X)^{a-i+s} =\sum_{d=1}^a\sum_{s=1}^{d}q^s (q^{2-m}X)^d \]
and some calculation gives us the result we want.
The case that 
$I$ is anisotropic (i.e. when 
$m=2$ and 
$\chi (I)=-1$) can be computed similarly and is simpler. We omit the detail here. In particular, we may recover the following formula.
Proposition 7.4 [Reference ShiShi22, Theorem 6.2(1)]
Assume 
$I=\mathrm {Diag}(1,\nu )$, then
\begin{align*} &\alpha(I, T, X)\\ &\quad= (1-X) (1+ \chi(I) + q \chi(I)) \sum_{e=0}^\alpha (q X)^e -\chi(T) q^{\alpha+1} X^{\beta+1} (1-X) \sum_{e=0}^{\alpha} (q^{-1}X)^e\\ &\qquad -\chi(T) (1+q) ( X^{\alpha+ \beta +2}+\chi(I)\chi(T)) + (1+\chi(I)) q^{\alpha+1} X^{\alpha+1} (1+\chi(T) X^{\beta-\alpha}). \end{align*}
Moreover, a similar computation yields the following, and we leave the detail to reader.
Proposition 7.5 Assume that 
$I$ is unimodular of odd rank 
$m\ge 3$. Then
\begin{align*} &\alpha(I,T,X)\\
&\quad=(1\!-\!X)\biggl(\sum_{i=0}^{a}(q^{2-m} X)^{i} \!+\!\gamma_{o,1}(I,T,X)\biggr) \!+\!(1\!-\!q^{-(m-1)})X \biggl(q \sum_{i=0}^{a-1}(q^{2-m}
X)^{i}+\gamma_{o,0}(S,T,X)\biggr)\\
&\qquad +q X(q^{2-m} X)^{a} \big(1+\chi(I)\chi(u_1)q^{-({m-1})/{2}}\big)\big(1-\chi(I)\chi(u_1)
q^{(2-m)b-({m-1})/{2}}X^{b+1}\big),
\end{align*}
where 
$\gamma _{o,1}(I,T,X)$ equals
\[ \chi(I)\chi(u_1) q^{({m-1})/{2}}\biggl(\sum_{d=a+1}^{a+b}(q^{a+b+1-d}-1)(q^{2-m}X)^d-\sum_{i=b+1}^{a+b+1} (q^{2-m} X)^{i}\biggr), \]
and 
$\gamma _{o,0}(I,T,X)$ equals
\[ \chi(I)\chi(u_1)q^{({m-1})/{2}}\biggl(\sum_{d=a+1}^{a+b}(q^{a+b+1-d}-q)(q^{2-m}X)^d-\sum_{i=b+1}^{a+b} (q^{2-m} X)^{i}\biggr). \]
8. Local density formula when $\mathrm {rank}(T)=3$
In this section, we always assume 
$\mathrm {rank}(T)=3$ and 
$S=I_3^{-\chi (T)}$. The aim of this section is to compute 
$\partial \mathrm {Den}(T)$ explicitly. We treat the case 
$\mathrm {v}(T)\le -1$ in the first subsection. In the second subsection, we deal with the case when 
$T=\mathrm {Diag}(1,T_2)$ for 
$T_2$ diagonal. In the last subsection, instead of 
$\partial \mathrm {Den}(T)$, we compute 
$\partial \mathrm {Den}(T)^{(2)}$ for 
$T$ of the form not covered by previous subsections.
8.1 $\partial \mathrm {Den}(T)$ for $T$ with $\mathrm {v}(T)\le -1$
Proposition 8.1 If 
$\mathrm {v}(T)\le -1$, then 
$\mathrm {Int}(T)=\partial \mathrm {Den} (T)=0$.
Proof. If 
$\mathrm {v}(T)< -1$, then 
$\partial \mathrm {Den} (T)=0$ since 
$\mathrm {v}(S^{[k]})\ge -1$. If 
$\mathrm {v}(T)= -1$, then 
$T$ is of the form 
$\mathrm {Diag}(\mathcal {H},(u_3(-\pi _0)^c))$ with 
$\chi (T)=\chi (u_3)$. In this case, according to Corollary 5.8, Lemmas A.8 and A.9, we have
\[ \alpha(S,T,X)=(1-X)\alpha(S,(u_3(-\pi_0)^c),q^2 X). \]
Similarly, we have
\begin{align*} \alpha(\mathcal{H}^{3,1}_{\chi(T)},T)&=\beta(\mathcal{H}^{3,1}_{\chi(T)},\mathcal{H})\alpha(u_3,(u_3(-\pi_0)^c))\\ &=(1-q^{-2})\alpha(u_3,(u_3(-\pi_0)^c)). \end{align*}
Hence, applying Lemma 7.1 to 
$I_{0,0}=S$ where 
$I$ is of rank 1, we have
\begin{align*} \partial \mathrm{Den}(T)&=2\alpha(S,(u_3(-\pi_0)^c),q^2)+\frac{q^2}{1+q}(1-q^{-2})\alpha((u_3),(u_3(-\pi_0)^c))\\ &=2(1+\chi(S)\chi(u_3)q)+2(q-1)\\ &=2(1-q)+2(q-1)\\ &=0. \end{align*}
Here we are using the fact 
$\chi (S)\chi (T)=\chi (S)\chi (u_3)=-1$.
8.2 $\partial \mathrm {Den}(T)$ for $T=\mathrm {Diag}(1,T_2)$ with $T_2$ diagonal
In this subsection, we assume 
$T=\mathrm {Diag}(1,T_2)$, where 
$T_2=\mathrm {Diag}(u_1(-\pi _0)^a,u_2(-\pi _0)^b)$ with 
$0\le a\le b$. Let 
$u=u_1u_2$. In addition, let 
$S=\mathrm {Diag}(1,1,\nu )$ and 
$S_2=\mathrm {Diag}(1,\nu )$. We compare 
$\partial \mathrm {Den}(T)$ and 
$\partial \mathrm {Den}(T_2)$ in this subsection.
Recall that
\[ \partial \mathrm{Den}(T)=2\frac{\alpha'(S,T)}{\alpha(S,S)}+\frac{q^2}{1+q}\frac{\alpha(\mathcal{H}^{3,1}_{\chi(T)},T)}{\alpha(S,S)}. \]
Moreover, according to [Reference ShiShi22, Theorem 1.3] and [Reference He, Shi and YangHSY23, Theorem 1.1], the analytic side in the case 
$n=2$ is
\[ \partial \mathrm{Den}(T_2)=2\frac{\alpha'(S_2,T_2)}{\alpha(S_2,S_2)}-\frac{2q^2}{q^2-1}\frac{\alpha(\mathcal{H},T_2)}{\alpha(S_2,S_2)}. \]
Proposition 8.2
\[ \partial \mathrm{Den}(T)-\partial \mathrm{Den}(T_2)=\begin{cases} 1+2\displaystyle\sum_{i=1}^a q^i, & \text{if }\chi(T)=1,\\ 1, & \text{if }\chi(T)=-1. \end{cases} \]
Proof. Proposition 5.9 implies that 
$\alpha (S,T,X)$ equals
\[ (1-X)\alpha( \mathrm{Diag}(-1,S),T_2,q^2 X)+q^{2} (1+q^{-1}\chi(S))X\alpha( S_2,T_2,X). \]
Hence,
\begin{equation} \alpha'(S,T) =\alpha( \mathrm{Diag}(-1,S),T_2,q^2 )+q^{2} (1+q^{-1}\chi(S))\alpha'( S_2,T_2). \end{equation}
According to Lemma A.11, one can check that 
$\alpha (S,S)=\beta (S,S)=2q(q^2-1)$, and 
$\alpha (S_2,S_2)=2(q-\chi (S_2))$. Then
\begin{equation} \frac{\alpha'(S,T)}{\alpha(S,S)}-\frac{\alpha'(S_2,T_2)}{\alpha(S_2,S_2)} =\frac{\alpha( \mathrm{Diag}(-1,S),T_2,q^2 )}{\alpha(S,S)}. \end{equation}Hence, we just need to check that
\begin{align*} &2\frac{\alpha( \mathrm{Diag}(-1,S),T_2,q^2 )}{\alpha(S,S)}+ \frac{q^2}{1+q^2}\frac{\alpha( \mathcal{H}^{3,1}_{\chi(T)},T)}{\alpha(S,S)}+\frac{2q^2}{q^2-1}\frac{\alpha(\mathcal{H},T_2)}{\alpha(S_2,S_2)}\\ &\quad=\begin{cases} 1+2\displaystyle\sum_{i=1}^a q^i, & \text{if }\chi(T)=1,\\ 1, & \text{if }\chi(T)=-1. \end{cases} \end{align*}
By Proposition 7.3, we may check that
\begin{equation} 2\alpha(\mathrm{Diag}(-1,S),T_2,q^2)= \begin{cases} 2(2q^{a+2}-(q+1)^2)(q-1), & \text{if }\chi(T)=1,\\ 2(q-1)(q^2-1), & \text{if }\chi(T)=-1. \end{cases} \end{equation}
To compute 
$({q^2}/({1+q}))\alpha (\mathcal {H}^{3,1}_{\chi (T)},T)$, we may choose 
$\mathcal {H}^{3,1}_{\chi (T)}=\mathrm {Diag}(\mathcal {H}, 1)$ when 
$\chi (T)=1$. By Corollary 5.8, Proposition 7.4, and a direct calculation, we have
\[ \frac{q^2}{1+q}\frac{\alpha(\mathcal{H}^{3,1}_{\chi(T)},T)}{\alpha(S,S)} =\frac{1}{2q(q^2-1)}\cdot\begin{cases} (q-1) \alpha(\mathrm{Diag}(-1,1),T_2)+\dfrac{2q^2}{q-1} \alpha(\mathcal{H},T_2), & \text{if }\chi(T)=1,\\ (q-1) \alpha(\mathrm{Diag}(-1,-u),T_2), & \text{if }\chi(T)=-1. \end{cases} \]
Combining this with the formulas in [Reference He, Shi and YangHSY23, Theorem 6.1], we have
\begin{equation} \frac{q^2}{1+q^2}\frac{\alpha( \mathcal{H}^{3,1}_{\chi(T)},T)}{\alpha(S,S)}+\frac{2q^2}{q^2-1}\frac{\alpha(\mathcal{H},T_2)}{\alpha(S_2,S_2)} =\frac{1}{q(q^2-1)}\cdot\begin{cases} 4q^{a+2}-q^2-2q-1, & \text{if }\chi(T)=1,\\ (q^2-1), & \text{if }\chi(T)=-1. \end{cases} \end{equation}Now a direct computation combined with (8.3) and (8.4) proves the proposition.
Corollary 8.3 Assume 
$L$ is a Hermitian lattice with Gram matrix 
$T$, then
\begin{equation} \partial \mathrm{Den}(T) -\partial \mathrm{Den}(T_2)=|\{\mathcal{V}^0(L)\}|. \end{equation}Proof. We can write 
$L=L^\flat {\unicode{x29BA}} \mathcal {O}_F {\bf x}$ where 
$q({\bf x})=1$. If 
$L^\flat$ is non-split, then 
$|\{\mathcal {V}^0(L)\}|=1$.
If 
$L^\flat$ is split, then 
$|\{\mathcal {V}^0(L)\}|= 1+2\sum _{i=1}^a q^i$ since 
$\mathcal {L}_3(L)$ can be identified with 
$\mathcal {L}_{2,1}(L^\flat )$, which is a ball in 
$\mathcal {L}_{2,1}$ centered at a vertex lattice of type 
$0$ with radius 
$a$ (see [Reference He, Shi and YangHSY23] for more detail). Here 
$\mathcal {L}_{2,1}$ is the Bruhat–Tits tree associated with 
$\mathcal {N}^{\mathrm {Kra}}_{2,1}$ and 
$\mathcal {L}_{2,1}(L^\flat )$ is the subtree of 
$\mathcal {L}_{2,1}$ associated with 
$L^\flat$.
8.3 $\partial \mathrm {Den}(T)^{(2)}$
In this subsection, we assume 
$T=\mathrm {Diag}(T_2,u_3(-\pi _0)^c)$ with 
$\mathrm {v}(T_2)>0$, and compute 
$\partial \mathrm {Den}(T)^{(2)}$. Recall that 
$\partial \mathrm {Den}(T)^{(2)}=\partial \mathrm {Den}(L^\flat {\unicode{x29BA}} \mathcal {O}_F {\bf x})^{(2)}$ where the Gram matrix of 
$L=L^\flat {\unicode{x29BA}} \mathcal {O}_F {\bf x}$ is 
$T$. We consider two cases separately in Propositions 8.4 and 8.5.
Proposition 8.4 Let 
$T=\mathrm {Diag}(u_1(-\pi _0)^a,u_2(-\pi _0)^b, u_3(-\pi _0)^c)$ where 
$0< a\le b \le c$. Then
\[ \partial \mathrm{Den}(T)^{(2)}= 1+\chi(-u_2u_3)q^{a}(q^a-q^b)-q^{a+b}. \]
Proof. Recall that
\[ \partial \mathrm{Den}(T)^{(2)}=\frac{1}{2q(q^2-1)}\bigl(2\beta'(S,T)^{(2)}+\frac{q^2}{1+q}\beta(\mathcal{H}^{3,1}_{\chi(T)},T)^{(2)}\bigr). \]
We compute 
$\beta '(S,T)^{(2)}$ first. According to Proposition 5.10, 
$\beta _0(S,T_2,X)=0$ and
\begin{align*} \beta(S,T,X)^{(2)}& =\beta_2(S,T_2,X)\alpha(\mathrm{Diag}(S,-T_2),u_3(-\pi_0)^c,q^4 X)\\ &\quad +q \beta_1(S,T_2,X)\alpha(\mathrm{Diag}(-\nu,-T_2),u_3(-\pi_0)^c,q^2 X)\\ &=(1-X)(1-q^2 X)\alpha(\mathrm{Diag}(S,-T_2),u_3(-\pi_0)^c,q^4 X) \\ &\quad+(q+1)(q^2-1)X(1-X) \alpha(\mathrm{Diag}(-\nu,-T_2),u_3(-\pi_0)^c,q^2 X). \end{align*}
According to Lemma 7.1,
\begin{align*}
\alpha(\mathrm{Diag}(S, -T_2),u_3(-\pi_0)^c,q^4 X) &=1+\chi(S)\chi(u_1)(q-1)\\
&\quad\times\sum_{s=a+1}^bq^{a+1}(q X)^s +\chi(u_1u_2u_3\nu)q^{a+b+2} X^{c+1},
\end{align*}
and
\begin{align*}
\alpha(\mathrm{Diag}(-\nu,-T_2),u_3(-\pi_0)^c,q^2 X) &=1+\chi(S)\chi(u_1)(q-1)q^a\\
&\quad\times\sum_{s=a+1}^b(q X)^s+\chi(u_1u_2u_3\nu)q^{a+b+1}X^{c+1}.
\end{align*}
The relation 
$\chi (u_1u_2u_3\nu )=\chi (S)\chi (T)=-1$ and a direct calculation show that
\[ \beta'(S,T_2)^{(2)}= 1+\chi(-u_2u_3)q^{a}(q^a-q^b)-q^{a+b}. \]
Finally, 
$\beta (\mathcal {H}^{3,1}_{\chi (T)},T)^{(2)}=0$ by Proposition 5.10. The proposition is proved.
Proposition 8.5 Recall that 
$\mathcal {H}_a=\bigl (\begin {smallmatrix} 0 & \pi ^a\\ (-\pi )^a & 0\end {smallmatrix}\bigr )$. Let 
$T= \mathrm {Diag}(\mathcal {H}_a, u_3(-\pi _0)^c)$ where 
$a$ is a positive odd integer and 
$c\ge 0$. Then
\[ \partial \mathrm{Den}(T)^{(2)}= \begin{cases} (1-q^{a}), & \text{if }a\le 2c,\\ (1-q^{2c+1}), & \text{if }a> 2c. \end{cases} \]
Proof. Recall that
\[ \partial \mathrm{Den}(T)^{(2)}=\frac{1}{2q(q^2-1)}\biggl(2\beta'(S,T)^{(2)}+\frac{q^2}{1+q}\beta(\mathcal{H}^{3,1}_{\chi(T)},T)^{(2)}\biggr). \]
We need to compute 
$\beta '(S,T)^{(2)}$ and 
$\beta (\mathcal {H}^{3,1}_{\chi (T)},T)^{(2)}$.
According to Proposition 5.10, 
$\beta _0(S,T_2,X)=0$ and
\begin{align*} \beta(S,T,X)^{(2)} &=\beta_2(S,\mathcal{H}_a,X)\alpha(\mathrm{Diag}(S, \mathcal{H}_a),u_3(-\pi_0)^c,q^4 X)\\ &\quad +q \beta_1(S,\mathcal{H}_a,X) \alpha(\mathrm{Diag}( -\nu,\mathcal{H}_a),u_3(-\pi_0)^c,q^2 X)\\ &=(1-X)\bigl((1-q^2 X)\alpha(\mathrm{Diag}(S, \mathcal{H}_a),u_3(-\pi_0)^c,q^4 X)\\ &\quad +(q+1)(q^2-1)X \alpha(\mathrm{Diag}( -\nu,\mathcal{H}_a),u_3(-\pi_0)^c,q^2 X)\bigr). \end{align*}
According to Lemma 7.2,
\[ \alpha(\mathrm{Diag}(S, \mathcal{H}_a),u_3(-\pi_0)^c,q^4 X)= \begin{cases} 1+\chi(S)\chi(u_3)q^{2+a} X^{c+1}, & \text{if }a\le 2c,\\ 1+ \chi(S)\chi(u_3)q^{2c+3} X^{c+1}, & \text{if }a> 2c, \end{cases} \]
and
\[ \alpha(\mathrm{Diag}( -\nu,\mathcal{H}_a),u_3(-\pi_0)^c,q^2 X)=\begin{cases} 1+\chi(S)\chi(u_3)q^{1+a} X^{c+1}, & \text{if }a\le 2c,\\ 1+ \chi(S)\chi(u_3)q^{2c+2} X^{c+1}, & \text{if }a> 2c. \end{cases} \]
A short computation shows that
\[ \beta'(S,T)^{(2)}=q(q^2-1) \cdot \begin{cases} 1+\chi(S)\chi(u_3)q^{a}, & \text{if }a\le 2c,\\ 1+\chi(S)\chi(u_3)q^{2c+1}, & \text{if }a> 2c. \end{cases} \]
Note that 
$\chi (S)\chi (u_3)=\chi (S)\chi (T)=-1$. Finally, 
$\beta (\mathcal {H}^{3,1}_{\chi (T)},T)^{(2)}=0$ by Proposition 5.10. The proposition is proved.
Part 3. Proof of the main theorem
9. Reduced locus of special cycle
As remarked in § 2, results of [Reference Rapoport, Terstiege and WilsonRTW14] extend to the category of strict formal 
$\mathcal {O}_{F_0}$-modules using relative Dieudonné theory.
9.1 The Bruhat–Tits building for $n=3$
From now on we assume 
$n=3$ and 
$\mathcal {L}=\mathcal {L}_3$ as in § 2.3.
Lemma 9.1
(1) For every
$\Lambda _2\in \mathcal {V}^2$, $\mathcal {N}_{\Lambda _2}$ is isomorphic to the projective line $\mathbb {P}^1$ over $k$. Its $q+1$ rational points correspond to all $\Lambda _0\in \mathcal {V}^0$ contained in $\Lambda _2$.(2) Every
$\Lambda _0\in \mathcal {V}^0$ is contained in $q+1$ type $2$ lattices. In other words, there are $q+1$ projective lines in $(\mathcal {N}_3^\mathrm {Pap})_{\mathrm {red}}$ passing through the superspecial point $\mathcal {N}_{\Lambda _0}(k)$. Moreover,
(9.1)\begin{equation} \bigcap_{\Lambda_2\in \mathcal{V}_2, \Lambda_0 \subset \Lambda_2} \Lambda_2^\sharp=\pi \Lambda_0. \end{equation}
Proof. Suppose 
$z\in \mathcal {N}(k)$ and 
$M:= M(z)\subset N$ is defined as in Proposition 2.9. Since 
$n=3$, by [Reference Rapoport, Terstiege and WilsonRTW14, Proposition 4.1] we have 
$\Lambda (M)\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F}=M+\tau (M)$.
Proof of part (1). Suppose 
$z\in \mathcal {N}_{\Lambda _2}(k)$, i.e. 
$M\subset \Lambda _2$.
If 
$M=\tau (M)$, then 
$M=\Lambda _0\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F}$ for some 
$\Lambda _0\in \mathcal {V}^0$ contained in 
$\Lambda _2$.
If 
$M\neq \tau (M)$, then by taking the dual of 
$M\subset \Lambda _2\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F}$ we have the following sequence of inclusions
\begin{equation} (\Lambda_2\otimes_{\mathcal{O}_F} \mathcal{O}_{\breve F})^\sharp \stackrel{1}{\subset} M \stackrel{1}{\subset} M+\tau(M)= \Lambda_2\otimes_{\mathcal{O}_F} \mathcal{O}_{\breve F}. \end{equation}
In both cases the class of 
$M$ in 
$\Lambda _2\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F}/(\Lambda _2\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F})^\sharp \cong k^2$ is a line. This finishes the proof of part (1).
Proof of part (2). For each 
$\Lambda _0\in \mathcal {V}^0$ we just need to count the number of lattices 
$\Lambda _2\in \mathcal {V}^2$ that contains 
$\Lambda _0$. We have the following sequence of inclusions
\[ \pi \Lambda_0 \stackrel{2}{\subset} \Lambda_2^\sharp \stackrel{1}{\subset} \Lambda_0\stackrel{1}{\subset}\Lambda_2. \]
With respect to the quadratic form 
$(\,{,}\,)\pmod {\pi }$ on 
$\Lambda _0/\pi \Lambda _0$, the dual lattice 
$\Lambda _2^\sharp$ corresponds to the 
$2$-dimensional subspaces 
$U:= \Lambda _2^\sharp /\pi \Lambda _0$ in 
$\Lambda _0/\pi \Lambda _0$ such that 
$U^\bot \stackrel {1}{\subset } U$. Thus, we just need to count the number of isotropic lines 
$U^\bot$. Assume that 
$\{e_1,e_2,e_3\}$ is a basis of 
$\Lambda _0/\pi \Lambda _0$ whose Gram matrix with respect to the quadratic form 
$(\,{,}\,)_\mathbb {X} \pmod \pi$ is
\[ \left(\begin{array}{ccc} & 1 & \\ 1 & & \\ & & \epsilon \end{array}\right). \]
It is easy to see that the isotropic lines are 
$\mathrm {Span}\{e_1\}$, 
$\mathrm {Span}\{e_2\}$ and 
$\mathrm {Span}\{e_1-({\epsilon a^2}/{2})e_2+a e_3\}$ (
$a\in \mathbb {F}_q^\times$). Finally, (9.1) can be checked directly using this basis.
It is well known that 
$\mathcal {L}$ is a tree; see, for example, [Reference BrownBro89, § 3 of Chapter VI]. More specifically, the vertices of 
$\mathcal {L}$ correspond to vertex lattices of type 
$2$ or 
$0$. There is an edge between 
$\Lambda \in \mathcal {V}^2$ and 
$\Lambda _0\in \mathcal {V}^0$ if 
$\Lambda _0\subset \Lambda$. We give each edge length 
$\tfrac {1}{2}$. This defines a metric 
$d(\,{,}\,)$ on 
$\mathcal {L}$. Recall that we have defined 
$\mathcal {L}(L)$ in (2.6). Then the boundary of 
$\mathcal {L}(L)$ is the set
\begin{equation} \mathcal{B}(L)=\{\Lambda\in \mathcal{V}^0(L)\mid \exists\, \Lambda_2\in\mathcal{V}^2\text{ such that }\Lambda\subset\Lambda_2,\Lambda_2\notin\mathcal{L}(L)\}. \end{equation} Recall we have the isomorphism 
$b:\mathbb {V}\rightarrow C$ defined in (2.3). Recall from [Reference Kudla and RapoportKR14a] or [Reference He, Shi and YangHSY23] that the vertices of 
$\mathcal {L}_{2,1}$ correspond to vertex lattices of type 
$2$, and an edge corresponds to a vertex lattice of type 
$0$. Each vertex of 
$\mathcal {L}_{2,1}$ is contained in 
$q+1$ edges and each edge connects exactly two vertices. For 
${\bf x}\in \mathbb {V}$ with 
$\mathrm {v}({\bf x})=0$ and 
$\mathrm {Span}_F\{{\bf x}\}^{\perp }$ split, recall that 
$\mathcal {L}_{2,1}$ is the Bruhat–Tits tree of 
$\mathcal {Z}^\mathrm {Pap}({\bf x})\cong \mathcal {N}^\mathrm {Pap}_{2,1}$. Then 
${\bf x}$ determines an embedding 
$\mathcal {L}_{2,1}\hookrightarrow \mathcal {L}$ defined as follows. First we send each vertex of 
$\mathcal {L}_{2,1}$ corresponding to a vertex lattice 
$\Lambda \subset \mathrm {Span}_F\{b({\bf x})\}^{\perp }$ of type 
$2$ to the vertex of 
$\mathcal {L}$ corresponding to the type 
$2$ lattice 
$\Lambda {\unicode{x29BA}} \mathrm {Span} \{b({\bf x})\}$. An edge of 
$\mathcal {L}_{2,1}$ corresponding to a type 
$0$ 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 
$\Lambda {\unicode{x29BA}} \mathrm {Span} \{b({\bf x})\}$ and 
$\Lambda '{\unicode{x29BA}} \mathrm {Span} \{b({\bf x})\}$ where 
$\Lambda$ and 
$\Lambda '$ 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) 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) Assume
$\Lambda _2\in \mathcal {V}^2$, then
\[ \mathcal{Z}^\mathrm{Pap}({\bf x})(k) \cap \mathcal{N}_{\Lambda_2}(k)=\begin{cases} \mathcal{N}_{\Lambda_2}(k), & \text{if } b({\bf x}) \in \Lambda_2^\sharp, \\ \text{a superspecial point in } \mathcal{N}_{\Lambda_2}(k), & \text{if } b({\bf x})\in \Lambda_2 \backslash \Lambda_2^\sharp,\\ \emptyset, & \text{if } b({\bf x})\notin \Lambda_2. \end{cases} \]
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 part (1) immediately follows.
Now we proceed to prove part (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 
$\mathcal {N}_{\Lambda _0}(k)\in \mathcal {Z}^\mathrm {Pap}({\bf x})(k)$. Thus, 
$\mathcal {N}_{\Lambda _2}(k)\subset \mathcal {Z}^\mathrm {Pap}({\bf x})(k)$.
If 
$b({\bf x})\in \Lambda _2\backslash \Lambda _2^\sharp$, then 
$\Lambda _0:= \Lambda ^\sharp +\mathrm {Span}\{b({\bf x})\}$ is a type 
$0$ lattice contained in 
$\Lambda _2$ and 
$\mathcal {N}_{\Lambda _0}(k)\in \mathcal {Z}^\mathrm {Pap}({\bf x})(k)$. On the other hand, since 
$\tau (\Lambda _0\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F})=\Lambda _0\otimes _{\mathcal {O}_F} \mathcal {O}_{\breve F}$, (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 
$\mathcal {N}_{\Lambda }\subset \mathcal {Z}^\mathrm {Pap}({\bf x})_{\mathrm {red}}$ where 
$\Lambda \in \mathcal {V}^2$, then either 
$\mathcal {N}_{\Lambda }\subset \mathcal {Z}^\mathrm {Pap}(({1}/{\pi }){\bf x})_{\mathrm {red}}$ or 
$\mathcal {N}_{\Lambda }\cap \mathcal {Z}^\mathrm {Pap}(({1}/{\pi }){\bf x})_{\mathrm {red}}$ is a unique superspecial point.
Lemma 9.5 For 
$L\subset \mathbb {V}$ a lattice of arbitrary rank, 
$\mathcal {Z}^\mathrm {Pap}(L)_{\mathrm {red}}$ is connected.
Proof. Suppose 
$\mathcal {Z}^\mathrm {Pap}(L)_{\mathrm {red}}$ 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 
$\mathcal {Z}^\mathrm {Pap}({\bf x})_{\mathrm {red}}\cap U_i\neq \emptyset$ for 
$i=1,2$. Hence, the reduced locus of
\[ \mathcal{Z}^\mathrm{Pap}(L\oplus\mathrm{Span}\{{\bf x}\})\cong \mathcal{Z}^\mathrm{Pap}_{2,1}(L') \]
is not connected where 
$L'$ is the orthogonal projection of 
$L$ onto 
$\{{\bf x}\}^\bot$. This contradicts Corollaries 3.13 and 3.15 and Lemma 3.16 of [Reference He, Shi and YangHSY23].
Recall that for a lattice 
$L\subset \mathbb {V}$ (respectively, 
${\bf x}\in \mathbb {V}$), we have defined 
$\mathcal {V}(L)$ and 
$\mathcal {L}(L)$ (respectively, 
$\mathcal {V}({\bf x})$ and 
$\mathcal {L}({\bf x})$) in § 2.3.
Proposition 9.6 Assume that 
${\bf x}\in \mathbb {V}$ such that 
$h({\bf x},{\bf x})\neq 0$. Then we have
\[ \mathcal{Z}^\mathrm{Pap}({\bf x})_{\mathrm{red}}=\bigcup_{\Lambda\in \mathcal{V}({\bf x})} \mathcal{N}_\Lambda, \]
where 
$\mathcal {V}({\bf x})$ is given as follows.
(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) When
$\mathrm {v}({\bf x})=d$ and $\mathrm {Span}_F\{{\bf x}\}^{\perp }$ is non-split, we have
where\[ \mathcal{V}({\bf x})=\{\Lambda\in \mathcal{V}\mid d(\Lambda, \Lambda_{{\bf x}/\pi^d})\le d \} \]
$\Lambda _{{\bf x}/\pi ^d}$ is as in part (1).(3) When
$\mathrm {v}({\bf x})=0$ and $\mathrm {Span}_F\{{\bf x}\}^{\perp }$ is split, $\mathcal {L}({\bf x})$ is the tree $\mathcal {L}_{2,1}$.(4) When
$\mathrm {v}({\bf x})=d$ and $\mathrm {Span}_F\{{\bf x}\}^{\perp }$ is split, we have
where\[ \mathcal{V}({\bf x})=\{\Lambda\in \mathcal{V}\mid d(\Lambda, \mathcal{L}({\bf x}/\pi^d))\le d \} \]
$\mathcal {L}({\bf x}/\pi ^d)$ is as in part (3).(5) When
$h({\bf x}, {\bf x}) \notin \mathcal {O}_{F_0}$, $\mathcal {V}({\bf x})$ is empty.
Proof. Proof of part (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 [Reference ShiShi22, § 2] or [Reference Rapoport, Smithling and ZhangRSZ18, § 8]. Alternatively since 
$\mathrm {Span}_F\{b({\bf x})\}^{\perp }$ is non-split of dimension 
$2$, it contains a unique self-dual lattice 
$\Lambda '$, then 
$\Lambda _{\bf x}:= \mathrm {Span}\{b({\bf x})\}\oplus \Lambda '$ is the unique type 
$0$ lattice containing 
$b({\bf x})$.
Proof of part (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 [Reference Kudla and RapoportKR14a] and [Reference He, Shi and YangHSY23]. The required properties of 
$\mathcal {L}({\bf x})$ and 
$\mathcal {V}({\bf x})$ follow.
Proof of part (2). We prove this by induction. The case 
$d=0$ is just part (1). Now we assume 
$d>0$ and that the statement holds for 
$d-1$, i.e.
\[ \mathcal{V}({\bf x}/\pi)=\{\Lambda\in \mathcal{V}\mid d(\Lambda, \Lambda_{{\bf x}/\pi^d})\le d-1 \}. \]
Then applying Corollary 9.3 to the lattice 
$L=\mathrm {Span}\{{\bf x}/\pi \}$ we have
\[ \bigcup_{\Lambda\in \mathcal{V}^2,\ d(\Lambda, \Lambda_{{\bf x}/\pi^d})\le d} \mathcal{N}_{\Lambda}\subset \mathcal{Z}^\mathrm{Pap}({\bf x})_{\mathrm{red}}. \]
Corollary 9.4 and the induction hypothesis imply that every 
$\Lambda _2\in \mathcal {V}^2({\bf x})$ satisfies 
$d(\Lambda, \Lambda _{{\bf x}/\pi ^d}) \le d$. 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 part (2).
Similarly we can prove part (4) by an induction on 
$d$, the case 
$d=0$ is just part (3).
Part (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
\[ \mathcal{Z}^\mathrm{Pap}(L^\flat)_{\mathrm{red}}=\bigcup_{\Lambda\in \mathcal{V}(L^\flat)} \mathcal{N}_\Lambda \]
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) Assume
$L^\flat \approx \mathcal {H}_{2a+1}$ for some $a\in \mathbb {Z}_{\ge 0}$. Then $\mathcal {L}(L^\flat )$ is a ball centered at a vertex lattice of type $2$ with radius $({2a+1})/{2}$.(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 non-split. Then $\mathcal {L}(L^\flat )$ is a ball centered at a vertex lattice of type $0$ with radius $a$.(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
where\[ \mathcal{L}(L^\flat)=\{\Lambda\in \mathcal{V}\mid d(\Lambda, \mathcal{L}(\pi^{-a}L^\flat))\le a\}, \]
\[ \mathcal{L}(\pi^{-a}L^\flat)=\{\Lambda\in \mathcal{L}({\bf x}_1)\mid d(\Lambda, \Lambda_0)\le r\}, \]
$\mathcal {L}({\bf x}_1)$ is described in part (3) of Proposition 9.6 and $\Lambda _0$ is the unique type $0$ vertex lattice containing $\{{\bf x}_1,{\bf x}_2\}$.
Proof. As in the proof of Proposition 9.2, for a 
$\Lambda \in \mathcal {V}$, 
$\mathcal {N}_\Lambda \subset \mathcal {Z}^\mathrm {Pap}(L^\flat )_{\mathrm {red}}$ if and only if 
$\Lambda ^\sharp$ contains 
$b({\bf x}_1),b({\bf x}_2)$.
We first prove part (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 the Gram–Schmidt 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 
$a=0$, part (2) follows from the fact that 
$\mathcal {Z}^\mathrm {Pap}({\bf x}_1)=\mathcal {N}^\mathrm {Pap}_{2,-1}$ (by Proposition 2.6) and 
$\mathcal {Z}^\mathrm {Pap}(L^\flat )_{\mathrm {red}}=\mathcal {Z}^\mathrm {Pap}({\bf x}_1)_{\mathrm {red}}$ is a unique superspecial point. Similarly when 
$a=0$, (3) follows from the fact that 
$\mathcal {Z}^\mathrm {Pap}({\bf x}_1)=\mathcal {N}^\mathrm {Pap}_{2,1}$ and [Reference He, Shi and YangHSY23, Corollary 3.13].
Now we prove parts (1), (2) and (3) for general 
$a$. First, 
$\mathcal {L}(L^\flat )=\mathcal {L}(\pi ^a{\bf x}_1)\cap \mathcal {L}(\pi ^a{\bf x}_2)$ by definition. By Corollary 9.3 we have
\[ \{\Lambda\in \mathcal{V} \mid d(\Lambda,\mathcal{L}(\pi^{-a}L^\flat))\le a\}\subset \mathcal{L}(\pi^a {\bf x}_1)\cap \mathcal{L}(\pi^a {\bf x}_2). \]
Note that for a sub-tree 
$\mathcal {L}'$ of a tree 
$\mathcal {L}$ and a vertex 
$x\in \mathcal {L}\setminus \mathcal {L}'$, there is a unique geodesic segment joining 
$x$ with 
$\mathcal {L}'$. 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
\[ \{\Lambda\in \mathcal{V}\mid d(\Lambda,\mathcal{L}(\pi^{-a}L^\flat))\le a\}= \mathcal{L}(\pi^a {\bf x}_1)\cap \mathcal{L}(\pi^a {\bf x}_2). \]
The general case of parts (1), (2) and (3) follows from the above equation and the case 
$a=0$.
Note that parts (1), (2) and (3) have covered all possibilities of 
$L^\flat$ due to the classification of Hermitian lattices. Note 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 
$\mathcal {S}(L^\flat ):= \mathcal {L}(\pi ^{-a}L^\flat )$. If the fundamental invariant of 
$L^\flat$ is 
$(2a+1,2a+1)$, define 
$\mathcal {S}(L^\flat ):= \emptyset$.
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. Assume the fundamental invariant of 
$L^\flat$ is 
$(2a,b)$ or 
$(2a+1,2a+1)$. Define 
$M^\flat := \pi ^{-a}L^\flat$. 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 
$\Lambda \ne \Lambda '\in \mathcal {V}^2$, 
$\tilde {\mathcal {N}}_\Lambda$ and 
$\tilde {\mathcal {N}}_{\Lambda '}$ do not intersect.
Proof. If 
$\mathcal {N}_{\Lambda }$ and 
$\mathcal {N}_{\Lambda '}$ do not intersect in 
$\mathcal {N}^\mathrm {Pap}$, then obviously 
$\tilde {\mathcal {N}}_\Lambda$ and 
$\tilde {\mathcal {N}}_{\Lambda '}$ do not intersect. Without loss of generality we can assume 
$\Lambda =\mathrm {Span}\{e_1,e_2,e_3\}$ and 
$\Lambda '=\mathrm {Span}\{\pi ^{-1}e_1,\pi e_2,e_3\}$ 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 
$\tilde {\mathcal {N}}_{\Lambda '}$ are in 
$\tilde {\mathcal {Z}}({\bf x}_0)\cong \mathcal {N}^\mathrm {Kra}_{2,1}$. Now by [Reference He, Shi and YangHSY23, Lemma 5.3], 
$\tilde {\mathcal {N}}_{\Lambda }$ and 
$\tilde {\mathcal {N}}_{\Lambda '}$ 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
\begin{equation} \chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{N}}_\Lambda}\otimes \mathcal{O}_{\mathrm{Exc}_{\Lambda_0}})=1. \end{equation}
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 
$\tilde {\mathcal {Z}}({\bf x}_0)\cong \mathcal {N}^\mathrm {Kra}_{2,1}$:
\begin{align*} \chi(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{N}}_\Lambda}\otimes \mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}) &=\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}})\\ &=\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}'}). \end{align*}
Here 
$\mathrm {Exc}'\cong \mathbb {P}^1_k$ is the exceptional divisor on 
$\tilde {\mathcal {Z}}({\bf x}_0)$ corresponding to the rank 
$2$ self-dual lattice
\[ \Lambda'=\{v\in \Lambda_0\mid v\bot {\bf x}_0\}. \]
By [Reference He, Shi and YangHSY23, Lemma 5.2(a)], we know 
$\chi (\tilde {\mathcal {Z}}({\bf x}_0),\mathcal {O}_{\tilde {\mathcal {N}}_\Lambda }\otimes _{\mathcal {O}_{\tilde {\mathcal {Z}}({\bf x}_0)}} \mathcal {O}_{\mathrm {Exc}'})=1$. 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 We have that 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}(L^\flat )=[\mathcal {O}_{\mathcal {Z}^\mathrm {Kra}({\bf x}_1)}\otimes ^\mathbb {L} \mathcal {O}_{\mathcal {Z}^\mathrm {Kra}({\bf x}_2)}]\in K_0(\mathcal {N}^\mathrm {Kra})$ is, in fact, in 
$\mathrm {F}^{2} K_0(\mathcal {N}^\mathrm {Kra})$. Moreover, we have the decomposition in 
$\mathrm {Gr}^2 K_0^{\mathcal {Z}^\mathrm {Kra}(L^\flat )}(\mathcal {N}^\mathrm {Kra})$:
\begin{equation} {}^\mathbb{L} \mathcal{Z}^\mathrm{Kra}(L^\flat)=\mathcal{Z}^\mathrm{Kra}(L^\flat)_h+{}^\mathbb{L} \mathcal{Z}^\mathrm{Kra}(L^\flat)_v, \end{equation}
where 
$\mathcal {Z}^\mathrm {Kra}(L^\flat )_h$ is described in Theorem 4.2 and 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}(L)_v\in \mathrm {Gr}^2 K_0^{\mathcal {Z}^\mathrm {Kra}(L^\flat )_v}(\mathcal {N}^\mathrm {Kra})$.
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
\[ \big[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x}_1)}\otimes^\mathbb{L}\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x}_2)}\big]+\sum_{\Lambda_0\in \mathcal{V}^0(L^\flat)} (2m_{\Lambda_0}({\bf x}_1)m_{\Lambda_0}({\bf x}_2)+m_{\Lambda_0}({\bf x}_1)+m_{\Lambda_0}({\bf x}_2)) H_{\Lambda_0}. \]
$\mathcal {Z}^\mathrm {Kra}(L^\flat )_h$ is contained in 
$\tilde {\mathcal {Z}}({\bf x}_1)\cap \tilde {\mathcal {Z}}({\bf x}_2)$ and has dimension 
$1$ by Theorem 4.2; 
$\tilde {\mathcal {Z}}({\bf x}_1)\cap \tilde {\mathcal {Z}}({\bf x}_2)_v$ 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,
\begin{equation} \big[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x}_1)}\otimes^\mathbb{L}\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x}_2)}\big]=\big[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x}_1)\cap\tilde{\mathcal{Z}}({\bf x}_2)}\big] \in \mathrm{F}^{2} K_0(\mathcal{N}^\mathrm{Kra}), \end{equation}
see, for example, [Reference ZhangZha21, Lemma B.2]. Hence, we know that 
${}^\mathbb {L}\mathcal {Z}^\mathrm {Kra}(L^\flat )\in \mathrm {F}^{2} K_0(\mathcal {N}^\mathrm {Kra})$. The desired decomposition then follows from Theorem 4.2.
By Lemma 10.1 and (2.9) we know that 
${}^\mathbb {L} \mathcal {Z}^\mathrm {Kra}(L^\flat )_v\in K'_0(Y)$ 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 
$\mathrm {Gr}^2 K_0(\mathcal {N}^\mathrm {Kra})$:
\begin{equation} {}^\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}. \end{equation}
We 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 § 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
\[ {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L^\flat)=\big[\mathcal{O}_{\tilde{\mathcal{Z}}({\bf x}_1)}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_2)}\big]+\sum_{\Lambda_0\in \mathcal{V}^0({\bf x}_1)} \big[\mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}_2)}\big]. \]
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
\begin{equation} {}^\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}, \end{equation}
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 [Reference ShiShi22, Theorem 4.5] and [Reference He, Shi and YangHSY23, 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 [Reference ShiShi22, Theorem 4.5] or [Reference He, Shi and YangHSY23, Theorem 4.1], we have
\begin{equation} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L^\flat)=[\mathcal{O}_{\tilde{\mathcal{Z}}(L^\flat)^\circ}] \end{equation}in the notation of Theorem 4.2.
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 
$\Lambda '$. Moreover, by [Reference He, Shi and YangHSY23, Theorem 4.1] and (10.4), we have
\begin{equation} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L^\flat)_v=[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}]+[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda'}}]+H_{\Lambda\cap \Lambda'}. \end{equation}10.2 The intersection number
Assume 
$\Lambda \in \mathcal {V}^2$. For 
${\bf x}\in \mathbb {V}\backslash \{0\}$, define
\begin{equation} \mathrm{Int}_\Lambda({\bf x}):= \chi\big(\mathcal{N}^\mathrm{Kra},\mathcal{O}_{\tilde{\mathcal{N}}_\Lambda}\otimes^\mathbb{L} \mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x})}\big). \end{equation}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 
$y_0\in \Lambda _0\setminus \pi \Lambda _0$ such that 
$y_0^\bot$ is non-split, 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 
$n \ge 0$.
Case 1. First we assume 
${\bf x}_0\in \Lambda \backslash \Lambda ^\sharp$. Choose a basis 
$\{e'_1,e'_2,e'_3\}$ of 
$\Lambda$ with Gram matrix 
$\mathcal {H}_{3,1}^1$ such that
\[ {\bf x}_0=x e'_1+y e'_2+z e'_3. \]
Then one of 
$x$ and 
$y$ is in 
$\mathcal {O}_{F}^\times$ as 
$\Lambda ^\sharp =\mathrm {Span}\{\pi e'_1, \pi e'_2, e_3\}$. 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 
$\{e'_1,e'_2,e'_3\}$ by 
$\{e_1,e_2,e_3\}=\{e'_1,e'_2,e'_3\}g^{-1}$, we have
\[ {\bf x}_0=\pi u e_1+e_2+v e_3, \]
where 
$u\in \mathcal {O}_{F_0}^\times$, 
$v\in \mathcal {O}_F$.
Define
\begin{equation} f_1=\frac{1}{\pi} u^{-1} e_2,\quad f_2=\pi u e_1,\quad f_3=e_3. \end{equation}
Then 
$\{f_1,f_2,f_3\}$ has also Gram matrix 
$\mathcal {H}_{3,1}^1$ and 
$\Lambda ':= \mathrm {Span}\{f_1,f_2,f_3\}$ is a type 
$2$ lattice adjacent to 
$\Lambda$ with 
$\Lambda _c=\Lambda \cap \Lambda '=\mathrm {Span}\{\pi e_1,e_2,e_3\}$ is a type 
$0$ lattice. Now in terms of the basis 
$\{f_1,f_2,f_3\}$ we have
\[ {\bf x}=\pi^n(\pi u f_1+f_2+v f_3). \]
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
\[ \theta({\bf x})={\bf x}, \quad \theta(\Lambda)=\Lambda'. \]
In particular, 
$\theta (\mathcal {Z}^\mathrm {Kra}({\bf x}))=\mathcal {Z}^\mathrm {Kra}({\bf x})$ and
\begin{equation} \mathrm{Int}_{\Lambda'}({\bf x})=\mathrm{Int}_\Lambda({\bf x}). \end{equation}Now let
\[ \mathbf{y}_0=e_3,\quad \mathbf{y}_1=\pi(-\pi u e_1+e_2), \]
$L^\flat =\mathrm {Span}\{\mathbf {y}_0,\mathbf {y}_1\}$, and 
$L=\mathrm {Span}\{\mathbf {y}_0,\mathbf {y}_1,{\bf x}\}$. Then by (10.6) and Theorem 4.2, we have
\begin{equation} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L^\flat)=[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}]+[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda'}}]+H_{\Lambda_c}+[\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)}], \end{equation}
where 
$\tilde {\mathcal {Z}}(M^\flat )$ is the quasi-canonical lifting cycle of the lattice
\[ M^\flat:= \mathrm{Span}\{e_3,-\pi u e_1+e_2\}. \]
Combining with (10.9), we have
\begin{equation} \mathrm{Int}(L) =2 \cdot \mathrm{Int}_\Lambda({\bf x}) + \chi\big(\mathcal{N}^{\mathrm{Kra}},{}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}({\bf x}) \cdot H_{\Lambda_c}\big)+\chi\big(\mathcal{N}^{\mathrm{Kra}},{}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}({\bf x}) \cdot [\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)}]\big). \end{equation}
Let 
${\bf x}'=\pi ^n(\pi u e_1+e_2)= {\bf x} - \pi ^n v e_3$. Then we have
\begin{align*} \mathrm{Int}(L) &=\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))\\ &=\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))\\ &=\chi(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_0)}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)})\\ &\quad +\sum_{\Lambda_0\in \mathcal{V}^0(L)} \chi\big(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)}\big), \end{align*}
where we have used linear invariance [Reference HowardHow19, Corollary D] and Proposition 3.2. Note that the Gram matrix of 
$\{{\bf x}',\mathbf {y}_1\}$ is 
$\mathrm {Diag}(2u(-\pi _0)^n,-2u\pi _0)$. By Proposition 2.6 and [Reference He, Shi and YangHSY23, Theorem 1.1],
\[ \chi(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_0)}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)})=\begin{cases} 1, & \text{if } n=0,\\ 1+n-2q, & \text{if } n\geq 1. \end{cases} \]
By Corollary 3.6 and [Reference He, Shi and YangHSY23, Lemma 3.15], we know that
\[ \sum_{\Lambda_0\in \mathcal{V}^0(L)} \chi(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)})=|\mathcal{V}^0(L)| =\begin{cases} 1, & \text{if } n=0,\\ 2q+1, & \text{if } n\geq 1.\end{cases} \]
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 [Reference GrossGro86, 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)
\begin{equation} \mathrm{Int}_\Lambda({\bf x})=1. \end{equation} 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
\[ \Lambda'=\mathrm{Span}\biggl\{\pi e_1,\frac{1}{\pi}e_2,e_3\biggr\},\ \Lambda_c=\Lambda\cap\Lambda', \]
then 
${\bf x}_0\in \Lambda '\backslash \Lambda '^\sharp$. Also define
\[ \mathbf{y}_0=e_3,\ \mathbf{y}_1=\pi(-\pi e_1+e_2), \]
and 
$L^\flat := \mathrm {Span}\{\mathbf {y}_0,\mathbf {y}_1\}$. Then by Theorem 4.2 and (10.6) we have
\begin{equation} {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L^\flat)=[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda}}]+[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda'}}]+H_{\Lambda_c}+[\mathcal{O}_{\tilde{\mathcal{Z}}(M^\flat)}], \end{equation}
where 
$\tilde {\mathcal {Z}}(M^\flat )$ is the quasi-canonical lifting cycle of the lattice
\[ M^\flat:= \mathrm{Span}\{e_3,-\pi e_1+e_2\}. \]
Let 
${\bf x}':= \pi ^{n+1} e_1={\bf x} -\pi ^n u e_3$, then we have
\begin{align*} \chi(\mathcal{N}^\mathrm{Kra}, {}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}({\bf x})\cdot{}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}(L^\flat)) &=\chi\big(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_0)}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)}\big)\\ &\quad +\sum_{\Lambda_0\in \mathcal{V}^0(L)} \chi\big(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)}\big). \end{align*}
Note that the Gram matrix of 
$\{{\bf x}',\mathbf {y}_1\}$ 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 [Reference He, Shi and YangHSY23, Theorem 1.1], we know that
\[ \chi(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\tilde{\mathcal{Z}}(\mathbf{y}_0)}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)})=\begin{cases} -(q-1), & \text{if } n=0,\\ 1+n-2q, & \text{if } n\geq 1. \end{cases} \]
By Corollary 3.6 and Lemmas 3.15 and 3.16 of [Reference He, Shi and YangHSY23], we know that
\[ \sum_{\Lambda_0\in \mathcal{V}^0(L)} \chi\big(\mathcal{N}^\mathrm{Kra}, \mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}({\bf x}')}\otimes^\mathbb{L}\mathcal{O}_{\mathcal{Z}^\mathrm{Kra}(\mathbf{y}_1)}\big) =|\mathcal{V}^0(L)|=\begin{cases} q+1, & \text{if } n=0,\\ 2q+1, & \text{if } n\geq 1. \end{cases} \]
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 [Reference GrossGro86, 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 
${\bf x}\in \Lambda '\backslash \Lambda '^\sharp$, by the previous case we also have
\[ \mathrm{Int}_{\Lambda'}({\bf x})=1. \]
Combining all of the above, we have by (10.13)
\begin{equation} \mathrm{Int}_\Lambda({\bf x})=1. \end{equation}This finishes the proof of Theorem 10.2.
11. Proof of the modified Kudla–Rapoport conjecture: three-dimensional case
In this section, we 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 
$0 \le b \le c$. Then
\[ \mathrm{Int}(L)=\partial \mathrm{Den}(L). \]
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 
$T_2\in \mathrm {Herm}_2(\mathcal {O}_{F})$. 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 and 3.2 and Corollary 3.6, we have
\begin{align} \mathrm{Int}(L)-\mathrm{Int}(L^\flat) & =\chi\big(\mathcal{N}^\mathrm{Kra},{}^\mathbb{L}\mathcal{Z}^\mathrm{Kra}({\bf x}_1)\cdot {}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^\flat)\big)-\chi\big(\mathcal{N}^\mathrm{Kra},{}^\mathbb{L}\tilde{\mathcal{Z}}({\bf x}_1)\cdot {}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^\flat)\big)\nonumber\\ &=\sum_{\Lambda_0\in \mathcal{V}^0(L)} \chi\big(\mathcal{N}^\mathrm{Kra},[\mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}]\cdot {}^{\mathbb{L}}\mathcal{Z}^{\mathrm{Kra}}(L^\flat)\big)\nonumber\\ &=|\{\mathcal{V}^0(L)\}|. \end{align}
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 [Reference ShiShi22, Theorem 1.3] and [Reference He, Shi and YangHSY23, Theorem 1.3]. Part (2) follows from part (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
\begin{equation} \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}, \end{equation}by (10.3) and Proposition 4.6.
Claim 1 We claim that 
$m(\mathcal {D}(L^\flat ),\Lambda _0)=0$ unless 
$L^\flat \subset \Lambda _0$. In such a case,
\begin{equation} 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} \end{equation}
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 and Corollaries 3.7 and 3.8 imply that
\[ \chi\big(\mathcal{N}^\mathrm{Kra},H_{\Lambda_0} \cdot [\mathcal{O}_{\tilde{\mathcal{Z}}(y_0)}]\big)=1. \]
Thus, by (11.2) and Corollaries 10.3 and 2.7, we have
\[ m(\mathcal{D}(L^\flat),\Lambda_0) =\chi\big(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat) \cdot [\mathcal{O}_{\tilde{\mathcal{Z}}(y_0)}]\big). \]
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
\[ m:= \mathrm{dim}_{\mathbb{F}_q} \varphi(L^\flat) \le 2. \]
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 [Reference ShiShi22, 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
\begin{equation} \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\displaystyle\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}\end{equation} 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 non-zero. Since 
$\varphi (y_0)^\bot$ is non-split, we must have 
$a=0$. Hence, by the definition of 
$\mathcal {D}(L^\flat )$ and (11.4), we have
\[ m(\mathcal{D}(L^\flat),\Lambda_0) =\mu(0,b)-q\mu(-1,b)-\mu(0,b-1)+q \mu(-1,b-1)=1 \]
as claimed. Finally, 
$m=2$ is impossible since 
$v(L^\flat ) >0$. This finishes the proof of Claim 1.
Claim 2 We claim that 
$m(\mathcal {D}(L^\flat ),\Lambda _2)=2$ 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
\begin{equation} \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). \end{equation}Combining the above with Claim 1, we have
\begin{equation} 0=\chi(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat) \cdot [\mathcal{O}_{\mathrm{Exc}_{\Lambda_0}}])=\begin{cases} \displaystyle\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),\\ \displaystyle\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} \end{equation}
Recall 
$\mathcal {S}(L^\flat )$ in Definition 9.8. First assume 
$\Lambda _2\in \mathcal {L}(L^\flat )\setminus \mathcal {S}(L^\flat )$. If 
$d(\Lambda _2,\mathcal {B}(L^\flat ))$ is equal to 
$\tfrac {1}{2}$, 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 (11.6) implies that 
$m(\mathcal {D}(L^\flat ),\Lambda _2)=2$ in this case. 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.
Note that for 
$\Lambda _0 \in \mathcal {V}(L^\flat )$
\[ \sum_{\Lambda_2 \in \mathcal{V}^2(L^\flat)} \sum_{\Lambda_0 \subset \Lambda_2} 1 = \begin{cases} q+1, & \text{if }\Lambda_0 \in \mathcal{V}(L^\flat)\setminus \mathcal{B}(L^\flat),\\ 1, & \text{if } \Lambda_0 \in \mathcal{B}(L^\flat). \end{cases} \]
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 
$L=L^\flat {\unicode{x29BA}} \mathrm {Span}\{{\bf x}\}$ with Gram matrix
\[ T= \mathrm{Diag}(\mathcal{H}_a, u_3(-\pi_0)^c), \]
where 
$a$ is a positive odd integer, and 
$c\ge 0$. Then
\begin{equation} \mathrm{Int}(L)^{(2)}=\partial \mathrm{Den} (L)^{(2)}= \begin{cases} 1-q^{a}, & \text{if }a\le 2c,\\ 1-q^{2c+1}, & \text{if }a> 2c. \end{cases} \end{equation}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 
$L^\flat =\mathrm {Span}\{\pi ^{({a+1})/2} e_1,\pi ^{({a+1})/2} e_2\}$, 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 
${a}/{2}$ by Proposition 9.7.
Assume 
$a\le 2c$ 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 
$\Lambda _2\in \mathcal {V}^2(L^\flat )$ and 
$\Lambda _0\in \mathcal {V}^0(L^\flat )$. Hence, by Theorem 1.4, we have
\begin{align} \mathrm{Int}(L)^{(2)} &=\sum_{\Lambda_2\in \mathcal{L}(L^{\flat})}\chi\biggl(\mathcal{N}^\mathrm{Kra}, \biggl(2\big[\mathcal{O}_{\tilde{\mathcal{N}}_{\Lambda_2}}\big]+\sum_{\Lambda_0\subset \Lambda_2}H_{\Lambda_0}\biggr)\cdot \mathcal{Z}^{\mathrm{Kra}}({\bf x})\biggr)\nonumber\\ &=(1-q)|\{\Lambda_2\mid \Lambda_2\in \mathcal{L}(L^\flat)\}| \nonumber\\ &=(1-q)(1+(1+q)q+(1+q)q^3+\cdots+(1+q)q^{a-2})\nonumber\\ &=1-q^a, \end{align}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 
${a}/{2}$ in the Bruhat–Tits tree 
$\mathcal {L}_{2,1}$ of 
$\mathcal {N}^{\mathrm {Pap}}_{2,1}$ centered at the vertex lattice corresponding to 
$\pi ^{-({a+1})/{2}}\,{\cdot}\, L^\flat$, 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,
\[ |\{\Lambda_0\mid \Lambda_0\in (\mathcal{L}(L^\flat)\setminus \mathcal{B}(L^\flat))\cap \mathcal{L}(e_3)\}|=1+q+(1+q)q+\cdots+(1+q) q^{({a-3})/{2}}, \]
and
\[ |\{\Lambda_0\mid \Lambda_0\in \mathcal{B}(L^\flat)\cap \mathcal{L}(e_3)\}|=(1+q) q^{({a-1})/{2}}. \]
Moreover, note that if 
$e_3\in \Lambda _0$, then 
$\mathrm {Int}_{\Lambda _2}(e_3)=1$ for any 
$\Lambda _2$ such that 
$\Lambda _0\subset \Lambda _2$. As a result,
\begin{align*}
\chi\big(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat)\cdot
\mathcal{Z}^{\mathrm{Kra}}(e_3)\big) &=2(1+
q\cdot|\big\{\Lambda_0\mid \Lambda_0\in
\big(\mathcal{L}(L^\flat)\setminus
\mathcal{B}(L^\flat)\big)\cap \mathcal{L}(e_3)\big\}|)\\
&\quad -(q+1) |\{\Lambda_0\mid \Lambda_0\in
(\mathcal{L}(L^\flat)\setminus \mathcal{B}(L^\flat))\cap
\mathcal{L}(e_3)\}|\\ &\quad-|\big\{\Lambda_0\mid
\Lambda_0\in \mathcal{B}(L^\flat)\cap
\mathcal{L}(e_3)\big\}|\\
&=2+(q-1)(1+q+(1+q)q+\cdots+(1+q)q^{({a-3})/{2}})\\
&\quad-(1+q)q^{({a-1})/{2}}=1-q, \end{align*}
which is compatible with (11.7).
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, 
$\mathcal {V}(\pi e_3)=\{\Lambda \mid d(\Lambda,\mathcal {L}(e_3))\le 1\}$. 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,
\begin{align*} &\chi(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat)\cdot (\mathcal{Z}^{\mathrm{Kra}}(\pi e_3)-\mathcal{Z}^{\mathrm{Kra}}(e_3)))\\ &\quad =2q\cdot q(q-1)(1+q+(1+q)q+(1+q)q^2+\cdots+(1+q)q^{({a-1})/{2}-2})\\ &\qquad -q(q-1)(q+1)(1+q+(1+q)q+(1+q)q^2+\cdots+(1+q)q^{({a-1})/{2}-2})\\ &\qquad -q(q-1) (1+q)q^{({a-1})/{2}-1}\\ &\quad=q-q^{3}. \end{align*}
Continuing in this way, we can show
\[ \chi\big(\mathcal{N}^\mathrm{Kra},\mathcal{D}(L^\flat)\cdot (\mathcal{Z}^{\mathrm{Kra}}(\pi^i e_3)-\mathcal{Z}^{\mathrm{Kra}}(\pi^{i-1}e_3))\big) =q^{2i-1}-q^{2i+1} \]
for 
$2i< a$. Thus,
\[ \mathrm{Int}(L)^{(2)}=\mathcal{D}(L^\flat)\cdot \mathcal{Z}^{\mathrm{Kra}}( \pi^c e_3) =1- q^{2c+1} =\partial \text{Den}^{(2)}(L) \]
as claimed.
Proposition 11.3 Assume 
$L=L^\flat {\unicode{x29BA}} \mathrm {Span}\{{\bf x}\}$ with Gram matrix
\[ T=\mathrm{Diag}(u_1(-\pi_0)^a,u_2(-\pi_0)^b, u_3(-\pi_0)^c) \]
where 
$0< a\le b \le c$, 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
\begin{equation} \mathrm{Int}(L)^{(2)}=1+\chi(-u_2u_3)q^{a}(q^a-q^b) -q^{a+b}. \end{equation}
Note that since 
$a\le b\le c$, we have 
$\mathcal {L}(L^\flat )\subset \mathcal {L}({\bf x})$ by Propositions 9.6 and 9.7.
First, we assume 
$\chi (-u_2u_3)=-1$, then (11.9) specializes to
\[ \partial \mathrm{Den} (T)^{(2)}= 1-q^{2a}. \]
On the other hand, 
$\mathcal {L}(L^\flat )$ is a ball centered at a vertex lattice of type 
$0$ with radius 
$a$ in this case. One can show 
$\mathrm {Int}(L)^{(2)}=1-q^{2a}$ exactly as in (11.8).
Now we assume 
$\chi (-u_2u_3)=1$. In this case, (11.9) specializes to
\[ \partial \mathrm{Den} (L)^{(2)}= 1+q^{2a}-2q^{a+b}. \]
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,
\[ \mathcal{L}(L^\flat)=\big\{\Lambda\mid \Lambda\in \mathcal{L}_3,\ d(\Lambda,\mathcal{L}(\pi^{-a}L^\flat))\le a\big\}. \]
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
\begin{align*} \mathrm{Int}(L)^{(2)} &=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)|\\ &=1+q^2-2q^{b+1}, \end{align*}
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 
$L^\flat {\unicode{x29BA}} \mathrm {Span}\{{\bf x}\}$, then
\begin{align*} &\partial \mathrm{Den}
(\pi L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf x}\})^{(2)}- \partial
\mathrm{Den} (L^\flat{\unicode{x29BA}} \mathrm{Span}\{{\bf
x}\})^{(2)}\\ &\quad= 1+q^{2a+2}-2q^{r+2a+2}-(
1+q^{2a}-2q^{r+2a})\\ &\quad=q^{2a}(q^2-1)(1-q^{2r})\\
&\quad=q^2\big(\partial \mathrm{Den}
(L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf x}\})^{(2)}- \partial
\mathrm{Den} (\pi^{-1}L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf
x}\})^{(2)}\big), \end{align*}
and
\begin{align*} &\mathrm{Int} (\pi
L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf x}\})^{(2)}- \mathrm{Int}
(L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf x}\})^{(2)}\\
&\quad=2q|\mathcal{B}(L^\flat)|-q|\mathcal{B}(L^\flat)|-|\mathcal{B}(\pi
L^\flat)|\\ &\quad=(2q-q-q^2)|\mathcal{B}(L^\flat)|\\
&\quad=q^2\bigl(\mathrm{Int}
(L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf x}\})^{(2)}- \mathrm{Int}
(\pi^{-1} L^\flat{\unicode{x29BA}}\mathrm{Span}\{{\bf
x}\})^{(2)}\bigr), \end{align*}
where we use the fact 
$|\mathcal {B}(\pi L^\flat {\unicode{x29BA}} \mathrm {Span}\{{\bf x}\})|=q^2|\mathcal {B}(L^\flat )|$. 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 
$v(L) \ge 0$. 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 
$L^\flat = \text {Span}({\bf x}_1, {\bf x}_2)$ 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, by Theorem 5.6(1) we have
\[ \mathrm{Int}(L)=\partial \mathrm{Den}(L). \]
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 
$0\le a \le b \le c$, the same argument as Case 2 gives 
$\mathrm {Int}(L)=\partial \mathrm {Den}(L)$. This finishes the proof of the theorem.
Theorems 1.2 and 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
\[ \mathrm{Int}(L)^{(2)}=\partial \mathrm{Den} (L)^{(2)}. \]
12. Global applications
In this section we assume that 
$F$ is an imaginary quadratic field with discriminant 
$d_F$. Denote by 
$a\mapsto \bar a$ the complex conjugation on 
$F$. The result in this section can be easily extended to CM number fields (totally imaginary quadratic extension of a totally real number field) and more general level structures at split places, see [Reference Li and ZhangLZ22a] and [Reference He, Li, Shi and YangHLSY22]. We restrict to the imaginary quadratic fields to make the exposition as simple as possible.
12.1 Unitary Shimura varieties and special cycles
In this subsection, we briefly review the definition of an integral model of Shimura variety defined in [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20] over 
$\mathrm {Spec}\, \mathcal {O}_F$ . Let
\[ \mathcal{M}^{\mathrm{Kra}}_{(1,n-1)} \rightarrow \mathrm{Spec}\, \mathcal{O}_F \]
be the algebraic stack which assigns to each 
$\mathcal {O}_F$-scheme 
$S$ the groupoid of isomorphism classes of quadruples 
$(A,\iota,\lambda,\mathcal {F}_A)$ where:
(1)
$A\rightarrow S$ is an abelian scheme of relative dimension $n$;(2)
$\iota :\mathcal {O}_F\rightarrow \mathrm {End}(A)$ is an action satisfying the following determinant condition (the Kottwitz condition of signature $(1,n-1)$)
for all\[ \mathrm{char}(T-\iota(\alpha)\mid \mathrm{Lie}\, A)=(T-s(\alpha)) (T-s(\bar{\alpha}))^{n-1} \in \mathcal{O}_S[T], \]
$\alpha \in \mathcal {O}_F$ where $s:\mathcal {O}_F\rightarrow \mathcal {O}_S$ is the structure morphism;(3)
$\lambda :A\rightarrow A^\vee$ is a principal polarization whose Rosati involution satisfies $\iota (\alpha )^*=\iota (\bar {\alpha })$ for all $\alpha \in \mathcal {O}_F$;(4)
$\mathcal {F}_A \subset \mathrm {Lie}\, A$ is an $\mathcal {O}_F$-stable $\mathcal {O}_S$-module local direct summand of rank $n-1$ satisfying the Krämer condition; $\mathcal {O}_F$ acts on $\mathrm {Lie}\, A/\mathcal {F}_A$ by the structure map $s:\mathcal {O}_F \rightarrow \mathcal {O}_S$ and acts on $\mathcal {F}_A$ by the complex conjugate of the structure map.
Two objects 
$(A,\iota,\lambda,\mathcal {F}_A)$ and 
$(A',\iota ',\lambda ',\mathcal {F}_{A'})$ in 
$\mathcal {M}^{\mathrm {Kra}}_{(1,n-1)}(S)$ are isomorphic if there is an 
$\mathcal {O}_F$-linear isomorphism 
$f:A\rightarrow A'$ of abelian schemes such that 
$f^*(\lambda ')=\lambda$ and 
$f_*(\mathcal {F}_A)=\mathcal {F}_{A'}$. The stack 
$\mathcal {M}^{\mathrm {Kra}}_{(1,n-1)}$ is flat of dimension 
$n-1$ over 
$\mathrm {Spec}\, \mathcal {O}_F$. It is smooth over 
$\mathrm {Spec}\, \mathcal {O}_F[{1}/{{d_F}}]$ and has semi-stable reduction over primes of 
$F$ dividing 
${d_F}$. This is indicated by the corresponding behaviour of its local model studied in [Reference KrämerKrä03]. Analogously one can define 
$\mathcal {M}_{(0,1)}\rightarrow \mathcal {O}_F$ be the algebraic stack which assigns to each 
$\mathcal {O}_F$-scheme 
$S$ the groupoid of isomorphism classes of triples 
$(E,\iota _0,\lambda _0)$ where:
(1)
$E\rightarrow S$ is an abelian scheme of relative dimension $1$;(2)
$\iota _0:\mathcal {O}_F\rightarrow \mathrm {End}(E)$ is an action such that its induced action on $\mathrm {Lie}\, E$ agrees with the complex conjugate of the structural map $s:\mathcal {O}_F\rightarrow \mathcal {O}_S$;(3)
$\lambda _0:E\rightarrow E^\vee$ is a principal polarization whose Rosati involution satisfies $\iota _0(\alpha )^*=\iota _0(\bar {\alpha })$ for all $\alpha \in \mathcal {O}_F$.
The stack 
$\mathcal {M}_{(0,1)}$ is smooth of relative dimension 
$0$ over 
$\mathrm {Spec}\, \mathcal {O}_F$, see, for example, [Reference HowardHow15, Proposition 2.1.2].
Assume that 
$\mathbb {F}$ is an algebraically closed field of characteristic 
$p$ over 
$\mathcal {O}_F$. Let
\[ \big(E_0,\iota_0,\lambda_0,A,\iota,\lambda,\mathcal{F}_A\big)\in \big(\mathcal{M}_{(0,1)}\times \mathcal{M}^{\mathrm{Kra}}_{(1,n-1)}\big)(\mathrm{Spec}\, \mathbb{F}). \]
For any prime number 
$\ell \neq p$, we can define a Hermitian form 
$h(x,y)$ on the Tate module
\begin{equation} T_\ell(E_0,A):=\mathrm{Hom}_{\mathcal{O}_F}(T_\ell(E_0),T_\ell(A)) \end{equation}
as in [Reference Kudla and RapoportKR14b, § 2.3] using the polarizations 
$\lambda _0,\lambda$ and Weil pairings on 
$E_0,A$.
Fix a Hermitian space 
$W$ over 
$F$ of signature 
$(1,n-1)$ that contains a self-dual lattice 
$\mathfrak {a}$ and a Hermitian space 
$W_0$ over 
$F$ of signature 
$(0,1)$ that contains a self-dual lattice 
$\mathfrak {a}_0$. Define
\begin{equation} V:=\mathrm{Hom}_F (W_0,W), \quad L:=\mathrm{Hom}_{\mathcal{O}_F}(\mathfrak{a}_0,\mathfrak{a}). \end{equation}
Here 
$V$ and 
$L$ are equipped with Hermitian forms coming from those on 
$W_0$ and 
$W$. Define 
$G:=\mathrm {U}(W)$. Also define the group scheme 
$\mathrm {GU}(W)$ over 
$\mathbb {Q}$ by
\[ \mathrm{GU}(W)(R)=\{g\in \mathrm{GL}_R(W\otimes R)\mid (gv,gw)=c(g)(v,w),\forall v,w\in W\otimes R\}, \]
where 
$R$ is any 
$\mathbb {Q}$-algebra. Also define 
$Z:=\mathrm {Res}_{F/\mathbb {Q}} \mathbb {G}_m=\mathrm {GU}(W_0)$ and
\begin{equation} \tilde{G}:=Z\times_{\mathbb{G}_m} \mathrm{GU}(W), \end{equation}
where the maps from the factors on the right-hand side to 
$\mathbb {G}_m$ are 
$\mathrm {Nm}_{F/\mathbb {Q}}$ and the similitude character 
$c(g)$, respectively. We have an isomorphism of group schemes
\begin{equation} \tilde{G}\rightarrow Z\times \mathrm{U}(W), (z,g)\mapsto (z,z^{-1}g). \end{equation}
Let 
$K_G$ be the compact subgroup of 
$G(\mathbb {A}_f)$ that stabilizes the lattice 
$\mathfrak {a}\otimes \hat {\mathbb {Z}}$ and 
$K_Z=\hat {\mathbb {Z}}^\times \subset Z(\mathbb {A}_f)$. Under the isomorphism (12.4), define
\begin{equation} K:=K_Z\times K_G. \end{equation} Now define 
$\mathcal {M}\subset \mathcal {M}_{(0,1)}\times \mathcal {M}^{\mathrm {Kra}}_{(1,n-1)}$ to be the open and closed substack such that
\[ (E_0,\iota_0,\lambda_0,A,\iota,\lambda,\mathcal{F}_A)\in \mathcal{M}(S) \]
if and only if there is an isomorphism of Hermitian 
$\mathcal {O}_F\otimes \mathbb {Z}_\ell$-modules
\begin{equation} T_\ell(E_{0,s},A_s)\cong L\otimes \mathbb{Z}_\ell \end{equation}
for any geometric point 
$s\in S$ and prime 
$\ell$ that is not the same as the characteristic of 
$s$. Then 
$\mathcal {M}$ is an integral model of the Shimura variety associated to the group 
$\tilde {G}$ with level structure defined by 
$K$.
Now we review the definition of special cycles. For 
$(E,\iota _0,\lambda _0, A,\iota, \lambda,\mathcal {F}_A)\in \mathcal {M}(S)$ where 
$S$ is an 
$\mathcal {O}_F$-scheme, consider the projective 
$\mathcal {O}_F$-module of finite rank
\[ V'(E,A)=\mathrm{Hom}_{\mathcal{O}_F}(E,A). \]
On this module there is a Hermitian form 
$h'(x,y)$ defined by
\begin{equation} h'(x,y)=\iota_0^{-1}(\lambda_0^{-1}\circ y^\vee \circ \lambda \circ x), \end{equation}
where 
$y^\vee$ is the dual homomorphism of 
$y$. It is proved in [Reference Kudla and RapoportKR14b, Lemma 2.7] that 
$h'(x,y)$ is positive semi-definite. The following is [Reference Kudla and RapoportKR14b, Definition 2.8].
Definition 12.1 For 
$T\in \mathrm {Herm}_m(\mathbb {Z})_{>0}$, the special cycle 
$\mathcal {Z}(T)$ is the stack of collections 
$(E,\iota _0,\lambda _0,A,\iota,\lambda,\mathcal {F}_A,{\bf x})$ where
\[ (E,\iota_0,\lambda_0,A,\iota ,\lambda,\mathcal{F}_A)\in \mathcal{M}(S) \]
and 
${\bf x}=(x_1,\ldots,x_m)\in \mathrm {Hom}_{\mathcal {O}_F}(E,A)^m$ such that
\[ h'({\bf x},{\bf x})=(h'(x_i,x_j))=T. \]
When 
$t\in \mathbb {Z}_{>0}$, each component of 
$\mathcal {Z}(t)$ is a divisor by [Reference HowardHow15, Proposition 3.2.3]. In general, 
$\mathcal {Z}(T)$ does not necessarily have the expected codimension which is the rank of 
$T$.
Let 
$\mathcal {C} =\{ \mathcal {C}_p\}$ be a incoherent collection of local Hermitian spaces of rank 
$n$ such that 
$\mathcal {C}_\ell \cong V_\ell$ for all finite 
$\ell$ and 
$\mathcal {C}_\infty$ is positive definite. For a non-singular Hermitian matrix 
$T$ of rank 
$n$ with values in 
$\mathcal {O}_F$, Let 
$V_T$ be the Hermitian space with Gram matrix 
$T$. Define
\begin{equation} \mathrm{Diff}(T,\mathcal{C}):= \big\{p \text{ a place of } \mathbb{Q} \mid \mathcal{C}_p \text{ is not isomorphic to } (V_T)_p\big\}. \end{equation}
Then 
$\mathrm {Diff}(T,\mathcal {C})$ is a finite set consisting of places of 
$\mathbb {Q}$ inert or ramified in 
$F$. By [Reference Kudla and RapoportKR14b, Proposition 2.22], 
$\mathcal {Z}(T)$ is empty if 
$|\mathrm {Diff}(T,\mathcal {C})|>1$. If 
$\mathrm {Diff}(T,\mathcal {C})=\{p\}$ for a finite prime 
$p$ inert or ramified in 
$F$, it is proved in [Reference Kudla and RapoportKR14b] that the support of 
$\mathcal {Z}(T)$ is on the supersingular locus of 
$\mathcal {M}$ over 
$\mathrm {Spec}\, \bar {\mathbb {F}}_p$. Let 
$e$ be the ramification index of 
$F_p/\mathbb {Q}_p$. Define the arithmetic degree
\begin{equation} \widehat{\mathrm{deg}}_T=\chi(\mathcal{Z}(T),\mathcal{O}_{\mathcal{Z}(t_1)}\otimes^{\mathbb{L}} \mathcal{O}_{\mathcal{Z}(t_2)} \otimes^{\mathbb{L}} \mathcal{O}_{\mathcal{Z}(t_n)})\cdot \log p^{2/e}, \end{equation}
where 
$\otimes ^\mathbb {L}$ denotes the derived tensor product on the category of coherent sheaves on 
$\mathcal {M}$, 
$\chi$ is Euler characteristic and 
$t_i$ (
$1 \le i \le n$) are the diagonal entries of 
$T$. When 
$\mathrm {Diff}(T,\mathcal {C})=\{\infty \}$, 
$\mathcal {Z}(T)$ is empty [Reference Kudla and RapoportKR14b, Lemma 2.7] and the arithmetic degree 
$\widehat {\mathrm {deg}}_T(v)$ is the integration of a green current 
$G(T, v)$ (
$v >0$ is a positive-definite Hermitian matrix of order 
$n$) defined by Liu [Reference LiuLiu11] and Garcia and Sankaran [Reference Garcia and SankaranGS19]; see, for example, [Reference Li and ZhangLZ22a, Equation (15.3.0.2)].
12.2 Eisenstein series
On the analytic side, let 
$\chi :\mathbb {A}/\mathbb {Q}^\times \rightarrow \mathbb {C}^\times$ be the quadratic character associated to the extension 
$F/\mathbb {Q}$. Fix a character 
$\eta :\mathbb {A}_F^\times \rightarrow \mathbb {C}^\times$ such that 
$\eta |_{\mathbb {A}^\times }=\chi ^n$. We consider an incoherent Eisenstein series 
$E(z,s,\Phi )$ associated to a section 
$\Phi =\otimes \Phi _p$ in a degenerate principal series representation 
$I(s, \eta )$ of 
$\mathrm {U}(n,n)(\mathbb {A})$ (see [Reference Li and ZhangLZ22a, § 12] or [Reference Kudla and RapoportKR14b]), where 
$\tau \in \mathbb {H}_n$ (see (1.18)), 
$s\in \mathbb {C}$ and 
$\Phi _p$ is given as follows. The section 
$\Phi _\infty$ is the standard weight 
$n$ section. When 
$p < \infty$ is unramified in 
$F$, 
$\Phi _p$ is the standard section associated to the characteristic function of 
$L_p^n$ via the map 
$\lambda : S(\mathcal {C}_p^n) \rightarrow I(0, \eta _p)$:
\[ \lambda(\varphi)(g)=\omega(g)\varphi(0), \]
where 
$\omega$ is the Weil representation of 
$\mathrm {U}(n,n)$ associated to the character 
$\chi$. When 
$p$ is ramified in 
$F$, define
\begin{equation} \Phi_{p}=\Phi_p^0+\sum_{i=1}^{\lfloor {n}/{2}\rfloor} A_{p,\epsilon}^i(s) \cdot \Phi_p^i. \end{equation}
Here 
$\Phi _p^0$ is the standard section associated to the characteristic function 
$L_p^n$, 
$\Phi _p^i$ is the standard section associated to the characteristic function of 
$(\mathcal {H}_{n,i}^{\epsilon })^n$ at 
$p$ with 
$\epsilon =-\chi _p(L_p)$, and
\begin{equation} A_{p, \epsilon}^{i}(0)=0,\quad \frac{d}{ds}A_{p}^{i}|_{s=0}=\frac{(-1)^n}{p^{2i}}\cdot c_{n,i}^\epsilon\cdot \log p, \end{equation}
where 
$c_{n,i}^\epsilon$ are as in (1.9). Let 
$\psi$ be the standard additive character of 
$\mathbb {A}/\mathbb {Q}$, i.e.
\begin{equation} \psi_\infty(x)=\exp(2\pi i x), \quad \psi_\ell(x)=\exp(-2 \pi i \lambda(x)), \end{equation}
where 
$\lambda$ is the canonical map 
$\mathbb {Q}_\ell \rightarrow \mathbb {Q}_\ell /\mathbb {Z}_\ell \hookrightarrow \mathbb {Q}/\mathbb {Z}$. Let 
$E_T(\tau,s,\Phi )$ (respectively, 
$E'_T(\tau,\Phi )$) be the 
$T$th Fourier coefficient of 
$E(\tau,s,\Phi )$ (respectively, 
$E'(\tau, 0, \Phi )$) with respect to 
$\psi$. Then, for 
$s\gg 0$, we have the following product formula (see, for example, [Reference Li and ZhangLZ22a, § 12.4]):
\begin{equation} E_T(\tau,s,\Phi)=c_\infty \cdot \prod_{p<\infty} W_{T,p}(1,s,\Phi_p) \cdot q^T, \end{equation}
where 
$c_\infty$ is a constant independent of 
$T$ calculated in [Reference LiuLiu11, Proposition 4.5], and 
$W_{T,p}(1,s,\Phi _p)$ is the local Whittaker integral defined in [Reference Kudla and RapoportKR14b, Equation (10.2)].
12.3 An equivalent form of Conjecture 1.1
In this subsection we assume 
$p$ is a prime of 
$\mathbb {Q}$ ramified in 
$F$. Let 
$|\cdot |_p$ be the non-Archimedean valuation on 
$F_p$ normalized so that 
$|\sqrt {d_F}|_p={1}/{p}$. For 
$\epsilon =\pm 1$, let 
$V_{p}^{\epsilon }$ be the (unique up to isomorphism) 
$F_p/\mathbb {Q}_p$-Hermitian space of dimension 
$n$ and sign 
$\epsilon$. For any lattice 
$M_p$ of rank 
$n$ in 
$V_p^\epsilon$, let 
$\Phi _{M_p}\in I_p(s,\eta _p)$ be the standard section associated to the characteristic function of 
$M_p^n$. By [Reference ShiShi22, Proposition 9.7], we have
\begin{equation} W_{T,p}(1,r,\Phi_{M_p}) =\gamma_p (V_{p}^{\epsilon})^n \cdot |\!\det (M_p)|_p^n \cdot |d_F|_p^f \cdot \alpha_v(M_p,L_T,X)|_{X=p^{-2r}}, \end{equation}
where 
$f=\tfrac {1}{2}n^2+\tfrac {1}{4}n(n-1)$, 
$\alpha _p(\cdot,\cdot,X)$ is the local density polynomial defined in (5.1) at the place 
$p$ and 
$\gamma _p(V_{p}^{\epsilon })$ is an 
$8$th root of unity defined in [Reference Kudla and RapoportKR14b, Equation (10.3)]. By [Reference Kudla and RapoportKR14b], we know
\begin{equation} \gamma_p (V_{p}^{\epsilon})=-\gamma_p (V_{p}^{-\epsilon}). \end{equation}
For 
$T\in \mathrm {Herm}_n(F)$, choose a lattice 
$L_T$ in the Hermitian space 
$V_{p}^{\chi _p(T)}$ with Gram matrix 
$T$. Then (12.14) and (12.15) imply that Conjecture 1.1 at the place 
$p$ is equivalent to the following conjecture.
Conjecture 12.2 Let 
$\mathbb {V}$ be the space of special quasi-homomorphisms as in (1.2) such that 
$\chi _p(\mathbb {V}) =\epsilon$. Let 
$T\in \mathrm {Herm}_n(F)$ such that 
$\chi _p(T)=\epsilon$ and 
$L_T$ be a lattice of rank 
$n$ in 
$\mathbb {V}$ with Gram matrix 
$T$. Then
\[ \mathrm{Int}(L_T)\cdot \log p= \frac{W'_{T,p}(1,0,\Phi_p)}{W_{I_n^{-\epsilon},p}(1,0,\Phi_p)}, \]
where 
$\Phi _p$ is defined in (12.10).
12.4 (Global) arithmetic Siegel–Weil formula
Similar to [Reference Li and ZhangLZ22a, Theorem 1.3.1], we have the following theorem.
Theorem 12.3 (Arithmetic Siegel–Weil formula for non-singular coefficients)
Assume that the fundamental discriminant of 
$F$ is 
$d_F \equiv 1 \pmod 8$ and that Conjecture 1.1 holds for every 
$F_p$ with 
$p|d_F$. For any non-singular Hermitian matrix 
$T$ with values in 
$\mathcal {O}_F$ of size 
$n$, we have
\[ E'_T(\tau, 0, \Phi)=C \cdot \widehat{\mathrm{deg}}_{T}(v) \cdot q^T, \quad q^T=\exp(2\pi i \mathrm{tr}(T\tau)), \]
where 
$C$ is an explicit constant that only depends on 
$F$ and 
$L$, 
$\widehat {\mathrm {deg}}_{T}(v)=\widehat {\mathrm {deg}}_{T}$ for positive-definite 
$T$, and 
$\tau =u + i v$. In particular, the arithmetic Siegel–Weil formula holds for 
$n=2, 3$ for non-singular 
$T$.
Proof. We sketch the main idea of the proof. When 
$|\text {Diff}(T, \mathcal {C})| >1$, both sides are zero. When 
$\text {Diff}(T, \mathcal {C}) =\{ p\}$ for a finite prime 
$p\ne 2$ (as 
$d_F \equiv 1 \pmod 8$), then 
$T$ is positive definite and the support of 
$\mathcal {Z}(T)$ is on the supersingular locus of 
$\mathcal {M}$ over 
$\mathrm {Spec}\, \bar {\mathbb {F}}_p$ although it has higher than expected dimension and needs ‘derivation’ to make it correct dimensional cycle (we skip it here and just define its degree in the following). When 
$p$ is inert in 
$F$, the theorem is proved in [Reference Li and ZhangLZ22a, Theorem 1.3.1]. When 
$p$ is ramified in 
$F$, the theorem can be proved in a similar fashion assuming Conjecture 1.1. The key is that by the 
$p$-adic uniformization theorem [Reference Rapoport and ZinkRZ96, Chapter 6], for each component 
$\mathcal {Z}$ of 
${Z}(T) (\bar {\mathbb {F}}_p)$, the arithmetic degree of 
$\mathcal {Z}(T)$ supported on 
$\mathcal {Z}$
\begin{equation} \chi(\mathcal{Z},\mathcal{O}_{\mathcal{Z}(t_1)}\otimes^{\mathbb{L}} \mathcal{O}_{\mathcal{Z}(t_2)} \otimes^{\mathbb{L}} \mathcal{O}_{\mathcal{Z}(t_n)})\cdot \log p, \end{equation}
is the same as 
$\mathrm {Int}(L) \log p$ (
$L$ has Gram matrix 
$T$) in Conjecture 1.1. In particular, this number is independent of the choice of 
$\mathcal {Z}$ and depends only on 
$T$. Assuming that Conjecture 1.1 (or rather its equivalent form Conjecture 12.2) holds, this is then equal to 
$c_{p, 1} W_{T, p}'(1, 0, \Phi _p)$ for some constant 
$c_{p, 1}\ne 0$. Thus, 
$\widehat {\mathrm {deg}}_T$ is this number times the number of components of 
${Z}(T)(\bar {\mathbb {F}}_p)$, which can be counted via the Siegel–Weil formula. Combining these results together with (12.13), we can prove
\[ C_p\cdot \widehat{\mathrm{deg}}_T\cdot q^T = E_T'(\tau, 0, \Phi) \]
for come explicit constant 
$C_p$ independent of 
$T$. Similar argument holds when 
$\text {Diff}(T, \mathcal {C}) =\{ \infty \}$ in which case the theorem is proved in [Reference LiuLiu11] and [Reference Garcia and SankaranGS19]. Finally, one checks that 
$C_p$ is independent of the choice of 
$p$.
Acknowledgements
We thank C. Li for his help during the preparation of this paper. We thank the referees for their careful reading of the paper and their comments which make the paper more readable.
Conflicts of Interest
None.
Appendix A Calculation of primitive local density
In this appendix, we provide the proofs of Propositions 5.9 and 5.10. Throughout this section, 
$M$ is unimodular of rank 
$m\ge 2$ unless clearly stated otherwise. Let 
$\{ v_1, \ldots, v_{2k}, v_{2k+1}, \ldots, v_{2k+m}\}$ be a basis of 
$M^{[k]}=\mathcal {H}^k{\unicode{x29BA}} M$ with Gram matrix 
$\mathcal {H}^k{\unicode{x29BA}} \mathrm {Diag}(I_{m-1}, \nu )$. 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 part (1), choose 
$M(1)=({t\pi }/{2})v_1+v_2 \in M^{[k]}$ with 
$q(M(1))=t$. Then
\begin{align} M(1)^{\perp}&=\mathrm{Span_{\mathcal{O}_F}}\biggl\{\frac{-t\pi }{2}v_1+v_2,v_3,\ldots,v_{2k}, v_{2k+1}, \ldots, v_{2k+m}\biggr\}\nonumber\\ &\cong \langle -t \rangle {\unicode{x29BA}} \mathcal{H}^{k-1} {\unicode{x29BA}} M , \end{align}
which is represented by 
$\mathrm {Diag}(-t, \mathcal {H}^{k-1}, S)$. It is easy to check
\[ |M^{[k]}:M(1){\unicode{x29BA}} M(1)^{\perp}|^{-1}|M(1)^{\vee}:M(1)|=|t \pi|_F |t \pi|_F^{-1}=1. \]
For parts (2) and (3), assume first that 
$M$ is isotropic (and unimodular). In this case, we may choose a basis 
$\{v_{2k+1}',\ldots,v_{2k+m}'\}$ of 
$M$ with Gram matrix 
$\mathrm {Diag}(\mathcal {H}_0,1,\ldots,1,-\nu )$. Choose 
$M(0)= ({t}/2) v_{2k+1}'+ v_{2k+2}'$ with 
$q(M(0))=t$. Then
\begin{align*} M(0)^\perp&=\text{Span}\biggl\{v_1, \ldots, v_{2k}, -\frac{t}2 v_{2k+1}'+ v_{2k+2}', v_{2k+3}', \ldots, v_{2k+m}'\biggr\}\\ & \cong \mathcal{H}^k {\unicode{x29BA}} \text{Span}\{v_{2k+3}', \ldots, v_{2k+m}'\} {\unicode{x29BA}} \langle -t \rangle. \end{align*}
as claimed. Moreover,
\[ |M^{[k]}:M(0){\unicode{x29BA}} M(0)^{\perp}|^{-1}|M(0)^{\vee}:M(0)|=|t|_F |t \pi|_F^{-1}=q. \]
Next, assume that 
$M$ is anisotropic. In this case, 
$M$ has rank 
$2$ and has Gram matrix 
$\mathrm {Diag}(1, \nu )$ 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 
$M(0) = a v_{2k+1} + b v_{2k+2}$. Then 
$q(M(0))=t$, and
\[ M(0)^\perp= \text{Span}\big\{v_1, \ldots, v_{2k}, -\nu \bar b v_{2k+1} +\bar a v_{2k+2}\big\}= \mathcal{H}^k{\unicode{x29BA}} \langle t\nu \rangle, \]
and
\[ |M^{[k]}:M(0){\unicode{x29BA}} M(0)^{\perp}|^{-1}|M(0)^{\vee}:M(0)|=|\pi|_F^{-1} =q. \]
When 
$\mathrm {v}(t) >0$, 
$t \not \in N_{E/F_0}\mathcal {O}_E^\times$. Thus, there is no primitive 
$M(0) \in M$ with 
$q(M(0))=t$. This proves Proposition 5.9(1)–(3).
The proof of part (4) follows from the following four 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 
$v=a_1v_1+a_2v_2$. Then 
$v$ 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 
$\bigl (\begin {smallmatrix}\bar {a}_2 & 0\\0 & a_2^{-1}\end {smallmatrix}\bigr )$. Now note that 
$q(v)=(v,v)=(a_1-\bar {a}_1)\pi ^{i}$. Hence, we can write 
$a_1=\alpha +{q(v)\pi ^{-i}}/{2}$, where 
$\alpha \in \mathcal {O}_{F_0}$. Now let 
$g=\bigl (\begin {smallmatrix}1 & -\alpha \\0 & 1\end {smallmatrix}\bigr )$, and it is straightforward to check that 
$g\in \mathrm {U}(\mathcal {H}_i)$ and 
$g(v)=({q(v)\pi ^{-i}}/{2})v_1 + v_2$.
Now we deal with the case 
$i$ is even. Again, we can assume 
$v=a_1v_1+v_2$. Then 
$q(v)=(a_1+\bar {a}_1)\pi ^i$. Hence we can write 
$a_1=({q(v)}/{2})\pi ^{-i}+\beta \pi$, where 
$\beta \in \mathcal {O}_{F_0}$. Now let 
$g=\bigl (\begin {smallmatrix}1 & -\beta \pi \\0 & 1\end {smallmatrix}\bigr )$, and it is straightforward to check that 
$g\in \mathrm {U}(\mathcal {H}_i)$ and 
$g(v)=({q(v)\pi ^{-i}}/{2})v_1+v_2$.
Lemma A.2 Assume 
$M$ is any lattice such that 
$\mathrm {v}(M)\geq i$. For 
$w_1,\ w_2 \in \mathcal {H}_i^k{\unicode{x29BA}} M$, 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 
$g\in \mathrm {U}(\mathcal {H}_i^k{\unicode{x29BA}} M)$ with 
$g(w_1)=w_2$.
Proof. Choose a basis 
$\{v_1,\ldots,v_{2k}\}$ of 
$\mathcal {H}_i^{k}$ such that the associated Gram matrix is 
$\mathcal {H}_i^k$. We also choose a basis 
$\{v_{2k+1},\ldots,v_{2k+m}\}$ of 
$M$. Write 
$w_1=\sum _{i=1}^{2k+m}a_iv_i$. Since 
$\mathrm {Pr}_{\mathcal {H}_i^k}(w_1)$ is primitive, 
$a_i$ is a unit for some 
$i\in \{1,\ldots,2k\}$. Without loss of generality, we may assume 
$a_1=1$. Let 
$w'=w_1+({(-1)^{i+1}q(w_1)\pi ^{-i}}/{2}) v_2$, then
\[ q(w')=q(w_1)+\left(w_1,\frac{(-1)^{i+1}q(w_1)\pi^{-i}}{2} v_2\right) +\left(\frac{(-1)^{i+1}q(w_1)\pi^{-i}}{2} v_2,w_1\right)=0, \]
and 
$(w', v_2) =(v_1, v_2)$. As a result, 
$M_1=\mathrm {Span}_{\mathcal {O}_F}\{w_1,v_2\}=\mathrm {Span}_{\mathcal {O}_F}\{w',v_2\}$ is isometric to 
$\mathcal {H}_i$. Note that 
$\mathrm {val}_{\pi }(q(w_1))\ge i$ is guaranteed by the assumption 
$\mathrm {v}(M)\ge i$.
Similarly, we can show 
$w_2\in M_2$ for some 
$M_2$ that is isometric to 
$\mathcal {H}_i$. However, the assumption 
$\mathrm {v}(M)\ge i$ and [Reference JacobowitzJac62, Proposition 
$4.2$] imply that there exist 
$g\in \mathrm {U}(\mathcal {H}_i^k{\unicode{x29BA}} M)$ 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
\[ M=\mathcal{H}_0^k {\unicode{x29BA}} M', \]
where 
$M'=0$ 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 
$\Pr _{M'}(w_1)$ is primitive and, thus, 
$q(\Pr _{M'}(w_1)) \in \mathcal {O}_F^\times$. This implies that 
$q(w_2)=q(w_1)$ is a unit, and 
$M= \mathcal {O}_F w_i {\unicode{x29BA}} (\mathcal {O}_F w_i)^\perp$. 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 
$\{v_1,\ldots,v_{2k+m}\}$ be a basis of 
$\mathcal {H}^k{\unicode{x29BA}} M$, whose Gram matrix is 
$\mathcal {H}^k{\unicode{x29BA}} \mathrm {Diag}(1,\ldots,\nu )$ 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 
$v=\sum _{i=1}^{2k}\pi a_i v_i + \sum _{j=2k+1}^{2k+m}a_jv_j$, 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
\[ v_{2i-1}'=v_{2i-1}+\frac{\bar{a}_{2i}}{\nu} v_{2k+m},\quad v_{2i}'=v_{2i}+\frac{-\bar{a}_{2i-1}}{\nu} v_{2k+m}. \]
Let 
$M_v=\mathrm {Span}_{\mathcal {O}_F}\{v_1',\ldots,v_{2k}'\}.$ Then it is easy to check that 
$M_v$ is perpendicular to 
$v$. Moreover, 
$M_v$ is isometric to 
$\mathcal {H}^{k}$ since 
$\mathrm {val}_{\pi }((v'_{2i-1},v'_{2i}))=-1$ and 
$0\leq \mathrm {val}_{\pi }((v'_i,v'_j))$ for other 
$1\leq i,j\leq 2k$. Hence, we can find 
$g_v\in \mathrm {U}(M^{[k]})$ such that 
$g_v(M_v)=\mathrm {Span_{\mathcal {O}_F}}\{v_1,\ldots,v_{2k}\}$ and 
$g_v(v)\in \mathrm {Span_{\mathcal {O}_F}}\{v_{2k+1},\ldots,v_{2k+m}\}=M$.
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 Lemmas A.2 and 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 
$M(1),M'(1)\in \mathcal {H}^k$ with 
$q(M(1))=q(M'(1))$ lie in the same orbit. Lemma A.3 implies the similar result for primitive 
$M(0),M'(0)\in M$ with 
$q(M(0))=q(M'(0))$. 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 a 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
\[ \mathrm{Pr}_{\mathcal{H}^k}\circ \varphi:L\rightarrow \mathcal{H}^k \]
in 
$\mathcal {H}^k/\pi \mathcal {H}^k$. Then
\[ \varphi(L)^\bot\cong (-L){\unicode{x29BA}} \mathcal{H}^{k-d(\varphi)}{\unicode{x29BA}} M^{(d(\varphi))}, \]
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 
$\mathrm {rank}(M)\geq 4$.
Lemma A.6 Let the notation be as in Proposition A.5. If 
$\mathrm {rank}(M^{[k]})\le 4$, 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 
$w_2=a_1v_1+\pi ^{i+1}v_2+a_3v_3+a_4v_4,$ and 
$\min \{\mathrm {v}_{\pi }(a_3),\mathrm {v}_{\pi }(a_4)\}=0$ by assumption. Without loss of generality, we may assume 
$a_3=1$. Now a direct calculation shows that
\[ \varphi(L)^{\perp}=\mathrm{Span_{\mathcal{O}_F}}\{v_1+(-\pi)^{i+1}v_4,v_3+\bar{a}_4v_4\}. \]
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
\[ \varphi(L)^{\perp}\approx -L. \]
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 
$0< a\le b$. 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 
$w_1=v_1-({q(w_1)\pi }/{2})v_2$ without loss of generality. Then we may write 
$w_2=a_1(v_1+({q(w_1)\pi }/{2})v_2)+a_3v_3+a_4v_4$, hence 
$\min \{v_{\pi }(a_3),v_{\pi }(a_4)\}=0$ by assumption again. We may assume 
$a_3=1$ and a direct calculation shows that
\[ \varphi(L)^{\perp}=\mathrm{Span_{\mathcal{O}_F}}\biggl\{v_1+\frac{q(w_1)\pi}{2}v_2-\bar{a}_1q(w_1)\pi v_4, v_3+\bar{a}_4v_4\biggr\}. \]
Set
\[
v_3'=v_1+\frac{q(w_1)\pi}{2}v_2-\bar{a}_1q(w_1)\pi v_4\quad\text{and}\quad v_4'=a_1v_3'+v_3+\bar{a}_4v_4.
\]
Then 
$\varphi (L)^\perp =\mathrm {Span_{\mathcal {O}_F}}\{v_3',v_4'\}$ and the Gram matrix of 
$\{v_3',v_4'\}$ is
\[ \begin{pmatrix} -q(w_1) & 0\\0 & a_1\bar{a}_1q(w_1)-(a_3\bar{a}_4-\bar{a}_3a_4)\pi^{-1} \end{pmatrix}= \begin{pmatrix} -q(w_1) & 0\\0 & -q(w_2) \end{pmatrix}. \]
Now assume 
$M^{[k]}=\mathcal {H}{\unicode{x29BA}} M$, 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 
$\mathcal {H}{\unicode{x29BA}} \mathrm {Diag}(1,\nu )$ 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
\[ w_1:= \varphi(m_1)=v_1, w_2:= \varphi(m_2)=a_1 v_1 +\pi^{i+1}v_2 +a_3v_3 +a_4 v_4 \]
and
\begin{equation} (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. \end{equation}
By our assumption we know that 
$\min \{\mathrm {v}_\pi (a_3),\mathrm {v}_\pi (a_4)\}=0$. 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, 
$M^{[k]}\approx \mathcal {H}{\unicode{x29BA}} \mathcal {H}_0$ and we can instead assume that 
$\{v_1,v_2,v_3,v_4\}$ has Gram matrix 
$\mathcal {H}{\unicode{x29BA}} \mathcal {H}_0$. We can further assume that
\[ w_1=v_1, \quad w_2=a_1 v_1+ \pi^{i+1} v_2+v_3 +a_4 v_4 \]
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
\[ \varphi(L)^\bot=\mathrm{Span}_{\mathcal{O}_F}\big\{v_1-(-\pi)^i v_4,v_3-\bar{a}_4 v_4\big\}. \]
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, assume 
$M^{[k]}$ is 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 
$\mathcal {H}_0{\unicode{x29BA}} \mathrm {diag}\{1,\epsilon \}$ 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
\begin{equation} (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. \end{equation}
By our assumption we know that 
$\min \{\mathrm {v}_\pi (a_3),\mathrm {v}_\pi (a_4)\}=0$. 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 
$\mathcal {H}_0{\unicode{x29BA}} \mathcal {H}_0$. We can further assume that
\[ w_1=v_1, \quad w_2=a_1 v_1+ \pi^i v_2+v_3 +a_4 v_4 \]
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
\[ \varphi(L)^\bot=\mathrm{Span}_{\mathcal{O}_F}\big\{v_1-(-\pi)^i v_4,v_3-\bar{a}_4 v_4\big\}. \]
Its Gram matrix is
\[ \left(\begin{array}{@{}cc@{}} 0 & -(-\pi)^i
\\ -\pi^i & -a_4-\bar{a}_4
\end{array}\right)=\left(\begin{array}{@{}cc@{}} 0 & -(-\pi)^i \\
-\pi^i & a_1 (-\pi)^i+\bar{a}_1 \pi^i
\end{array}\right)\approx \left(\begin{array}{@{}cc@{}} 0 &
-(-\pi)^i \\ -\pi^i & 0 \end{array}\right).
\]
Note 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 
$\mathrm {v}(L)\ge 0$. 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
\[ g(\varphi(L))\subset \mathcal{H}^{d(\varphi)}{\unicode{x29BA}} I_{4-2d(\varphi)}\subset M^{[k]}, \]
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 
$\mathcal {H}^k{\unicode{x29BA}} \mathrm {diag}\{1,\ldots,1,\nu \}$ 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
\[ w_2=a_1v_1+\pi^{i+1} v_2+\sum_{j=3}^{2k+m} a_j v_j. \]
By the assumption that 
$d(\varphi )=2$, we know that
\[ \min\{\mathrm{v}_\pi(a_j)\mid 3\leq j \leq 2k\}=0. \]
Hence, applying Lemma A.2 to 
$\mathcal {H}^{k-1}{\unicode{x29BA}} M$, we can find a 
$g\in \mathrm {U}(M^{[k]})$ such that
\[ g w_1=v_1, \quad g w_2\in \mathcal{H}^2, \]
where 
$\mathcal {H}^2$ refers to the first direct summand in the decomposition 
$\mathcal {H}^k{\unicode{x29BA}} M=\mathcal {H}^2{\unicode{x29BA}} \mathcal {H}^{k-2}{\unicode{x29BA}} M$.
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
\[ w_2=a_1v_1+\pi^{i+1} v_2+\sum_{j=3}^{2k+m} a_j v_j. \]
By the assumption that 
$d(\varphi )=1$, we know that
\[ \min\{\mathrm{v}_\pi(a_j)\mid 3\leq j \leq 2k\}\geq 1. \]
Since we assume 
$\varphi$ is primitive, we know that
\[ \min\{\mathrm{v}_\pi(a_j)\mid 2k+1\leq j \leq 2k+r\}=0. \]
Then we are done by applying Lemma A.4 to 
$\mathcal {H}^{k-1} {\unicode{x29BA}} M$.
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 
$\mathrm {Diag}(1,-1,1,\ldots,-\nu )$ (this is possible since we assume 
$m\ge 4$). Since 
$\varphi$ is primitive with 
$d(\varphi )=0$, then
\[ w_2=\sum_{i=1}^{2k}\pi a_iv_i+\sum_{i=2k+1}^{2k+m}a_iv_i, \]
and
\[ \min\{\mathrm{v}_\pi(a_j)\mid 2k+3\leq j \leq 2k+r\}=0. \]
We are done by applying Lemma A.4 to 
$\mathcal {H}^k{\unicode{x29BA}} \mathrm {Span_{\mathcal {O}_F}}\{v_{2k+3},\ldots,v_{2k+m}\}$.
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 non-singular Hermitian matrix 
$T$ of rank 
$n\le 2$. We let 
$\bar {v}$ denote the image of 
$v$ in 
$M^{[k]}\otimes _{\mathcal {O}_F}\mathbb {F}_q$. Let
\[ (M^{[k]})^n(i):= \big\{(v_j)\in M^{n,(n)}_{k}\mid \mathrm{Span}_{\mathbb{F}_q}\big\{\mathrm{Pr}_{\mathcal{H}^k}(\bar{v}_j),\ 1\le j\le n \big\} \text{ has rank }i \big\}, \]
where 
$M^{n,(n)}$ is as in (5.5), and
\begin{equation} \beta_{i}(M,L,X) :=\int_{\mathrm{Herm}_{n}(F)} \,dY \int_{(M^{[k]})^n(i)} \psi (\langle Y, T({\bf x} )-T \rangle)\,d{\bf x}. \end{equation}Note that
\begin{equation} \sum_{i=0}^{n}\beta_i(M, L,X)=\beta(M, L,X)^{(n)} \end{equation}
is the primitive local density defined earlier, and we will shorten it as 
$\beta (M, L,X)$. Note that if 
$L$ is of the form 
$\mathcal {H}^{j}$, then 
$\beta (M, L,X)=\beta _{n}(M, L,X)$.
First, by a variant of [Reference Cho and YamauchiCY20], Chao Li and Yifeng Liu obtained the following formula of 
$\beta (\mathcal {H}^k,L)$.
Lemma A.8 [Reference Li and LiuLL22, Lemma 
$2.16$]
Let 
$b_1\le \cdots \le b_n$ 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 non-zero entries in 
$(b_1,\ldots,b_n)$. Then
\[ \beta(\mathcal{H}^k,L)= \prod_{k-({n+t_o(L)})/{2}< i\le k} (1-q^{-2i}). \]
Lemma A.9 Assume 
$L$ is of rank 
$n$, then
\[ \beta_{n}(M,L,q^{-2k})=\beta(\mathcal{H}^k,L). \]
In particular, if 
$L$ is of the form 
$\mathcal {H}^j$, then
\[ \beta(M,L,q^{-2k})= \beta_{n}(M,L,q^{-2k})=\beta(\mathcal{H}^k,L). \]
Proof. Recall that 
$T({\bf x})=({\bf x}, {\bf x})$ is the moment matrix of 
${\bf x} \in (M^{[k]})^n$. For a 
${\bf x}_2 \in M^n$, let 
$T'({\bf x}_2)=T-T({\bf x}_2)$. Then
\begin{align*} \beta_n( M,L,q^{-2k}) &=\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}\\ &= \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\\ &= \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)\rangle)\,d{\bf x}_1 \,d{\bf x}_2. \end{align*}
Note that if 
$L$ and 
$L'$ are two Hermitian 
$\mathcal {O}_F$-lattices with moment matrix 
$T$ and 
$T'$ such that 
$T-T'\in \mathrm {Herm}_n(\mathcal {O}_{F_0})$, then 
$t_o(L)=t_o(L')$. Hence, for any 
${\bf x}_2 \in M^n$, we have by Lemma A.8
\begin{align*} \beta(\mathcal{H}^k,T'({\bf x}_2))&= \int_{\mathrm{Herm}_{n}(F)} \,dY \int_{(\mathcal{H}^k)^{n,(n)}} \psi (\langle Y, T({\bf x}_1)-T'({\bf x}_2)\rangle)\,d{\bf x}_1 \\ &=\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\\ &=\beta(\mathcal{H}^k,T). \end{align*}
Therefore,
\begin{align*} \beta_n(M,L,q^{-2k}) &=\mathrm{vol}(M^n,d{\bf x}_2) \cdot \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\\ &= \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\\ &= \beta(\mathcal{H}^k,L). \end{align*}
□
Combining the above two lemmas, we have the following.
Corollary A.10
(1) If
$L$ is of rank $1$, then we have
\[ \beta_{1}(M, L,X)=1-X. \]
(2) If
$L$ is of rank $2$, then we have
\[ \beta_{2}(M,L,X)=\begin{cases} (1-X), & \text{if }L=\mathcal{H},\\ (1-X)(1-q^2X), & \text{otherwise}. \end{cases} \]
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
\[ \beta_{0}(M,L,X)=X^n\beta(M,L) =q^{-mn+n^2}r(\overline{M},\overline{L}) X^n. \]
Proof. The second identity follows from the same proof of [Reference Cho and YamauchiCY20, Theorem 3.12]. Then a similar argument as in the proof of Lemma A.9 gives the first identity. In this case, we need to replace 
$M^n{\unicode{x29BA}} (\mathcal {H}^{k})^{n,(n)}$ in the proof of Lemma A.9 with 
$M^{n,(n)}{\unicode{x29BA}} (\pi \mathcal {H}^{k})^n$. The factor 
$X^n$ shows up because 
$\mathrm {vol}((\pi \mathcal {H}^k)^n)=(q^{-2k})^n$. We leave the details to the reader.
Note that [Reference Li and ZhangLZ22b, 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) If
$\mathrm {v}(L)=0$, then
\[ \beta_0(M,L,X)=\begin{cases}
(1+\chi(M) \chi(L)q^{-({m-1})/{2}})X, & \text{if }m\text{
is odd},\\ (1-\chi(M)q^{-{m}/{2}})X, & \text{if }m\text{
is even}. \end{cases} \]
(2) If
$\mathrm {v}(L)>0$, then
\[ \beta_0(M,L,X)=\begin{cases}
(1-q^{1-m})X , & \text{if }m\text{ is odd},\\
(1-q^{1-m}+\chi(M)(q-1)q^{-{m}/{2}})X, & \text{if
}m\text{ is even}. \end{cases} \]
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) If
$m$ is odd, then
\[ \beta_0(M,L,X)=\begin{cases} q(1-q^{1-m})X^2, & \text{if }t(L)=0,\\ q(1+\chi(M)\chi( u_1)q^{({3-m})/{2}})(1-q^{1-m})X^2, & \text{if }t(L)=1,\\ q(1-q^{1-m})(1-q^{3-m})X^2, & \text{if }t(L)=2. \end{cases} \]
(2) If
$m$ is even, then
\[ \beta_0(M,L,X)=\begin{cases} q (
1-\chi(L)q^{1-m} +\chi(L)\chi(M)(q-\chi(L))q^{-{m}/{2}} )
X^2, & \text{if }t(L)=0,\\
q(1-\chi(M)q^{-{m}/{2}})(1-q^{2-m})X^2, & \text{if
}t(L)=1,\\ q ( (1-q^{2-m})+\chi(M)(q^2-1)q^{-{m}/{2}}
)(1-q^{2-m})X^2, & \text{if }t(L)=2. \end{cases}
\]
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) If
$t(L)=2$, then
\[ \beta_1(M,L,X)= q(q+1) (
(1-q^{1-m})+\delta_{e}(m)\chi(M)(q-1)q^{-{m}/{2}}
)X(1-X). \]
(2) If
$t(L)=1$, then
\[ \beta_1(M,L,X) =\begin{cases}
q(1+q-q^{1-m}+\chi(M)\chi( u_1)q^{({3-m})/{2}})X(1-X), &
\text{if }m\text{ is odd},\\
q(1+q-q^{1-m}-\chi(M)q^{-{m}/{2}})X(1-X), & \text{if
}m\text{ is even}. \end{cases} \]
(3) If
$t(L)=0$ and $\chi (L)=1$, i.e. $L\cong \mathcal {H}_0$, then
\[ \beta_1(M,L,X)= q (
q+1-2q^{1-m}+\delta_{e}(m)\chi(M)(q-1)q^{-{m}/{2}}
)X(1-X). \]
(4) If
$t(L)=0$ and $\chi (L)=-1$, then
\[ \beta_1(M,L,X)= q (q+1) (
1-\delta_{e}(m)\chi(M)q^{-{m}/{2}} )X(1-X).
\]
Proof. First we assume 
$L=\mathcal {H}_i$. We claim that
\[ \beta_1(M, \mathcal{H}_i, X) =\begin{cases} q(1-X)\biggl(2\beta_0(M,0,X)+\displaystyle\sum_{\alpha \in (\mathcal{O}_{F_0}/(\pi_0))^{\times}}\beta_0(M, \langle -2\alpha \rangle,X) \biggr), & \text{if }i=0,\\ q(q+1) (1-X)\beta_0(M,0,X), & \text{if }i\ge 1. \end{cases} \]
Here 
$\alpha (M, 0, X)=\alpha (M, \mathcal {O}_F {\bf x}, X)$ with 
$q({\bf x}) =0$ and 
${\bf x} \ne 0$. Assuming the claim, the proposition for 
$L=\mathcal {H}_i$ follows from Corollary A.12.
To prove the claim, it suffices to show the identity for 
$X=q^{-2k}$ for sufficiently many 
$k\ge 0$. Recall
\begin{align*} I(M^{[k]},L,d)&=\big\{
\phi \in \mathrm{Hom}_{\mathcal{O}_F}(L/\pi_0^d L, M^{[k]}/
\pi_0^d M^{[k]}) \mid\\ &\qquad (\phi(x),\phi(y))\equiv(x,y) \text{ mod } \pi^{2d-1},\
\forall x,y \in L\big\}. \end{align*}
Let
\[ J(M^{[k]},L,d):= \{\phi\in I(M^{[k]},L,d)\mid \mathrm{dim}_{\mathbb{F}_{q}}\overline{\mathrm{Pr}_{\mathcal{H}^k}(\phi(L))}= \mathrm{dim}_{\mathbb{F}_{q}}\overline{\mathrm{Pr}_{M}(\phi(L))}=1\}. \]
Then
\[ \beta_1(M,L,q^{-2k})=\lim_{d\to \infty}q^{-(4(2k+m)-4)d}|J(M^{[k]},L,d)|. \]
Let 
$\{l_1,l_2\}$ be a basis of 
$L$ with Gram matrix 
$\mathcal {H}_i$. For 
$\phi \in J(M^{[k]},L,d)$, it will be determined by 
$w_i=\phi (l_i)$. Let 
$w_{i,\mathcal {H}}=\mathrm {Pr}_{\mathcal {H}^k}(w_i)$, and 
$w_{i,M}=\mathrm {Pr}_{M}(w_i)$. Since 
$\mathrm {rank}_{\mathbb {F}_{q}}\overline {\mathrm {Pr}_{\mathcal {H}^k}(\phi (L))}=1$, 
$\mathrm {rank}_{\mathbb {F}_{q}}\overline {\mathrm {Pr}_{\mathcal {H}^k}(w_i)}=1$ for 
$i=1$ or 
$2$.
Now we define a partition of 
$J(M^{[k]},L,d)$. Assume 
$\alpha \in \mathcal {O}_{F_0}$. Let
\begin{align*}
J_{\alpha}(M^{[k]},L,d)&:= \{\phi\in
I(M^{[k]},L,d)\mid
\mathrm{rank}_{\mathbb{F}_{q}}\overline{w}_{1,\mathcal{H}}
=1,\ \overline{w}_{2,\mathcal{H}}=\alpha
\overline{w}_{1,\mathcal{H}}\},\\
J_{\infty}(M^{[k]},L,d)&:= \{\phi\in
I(M^{[k]},L,d)\mid
\mathrm{rank}_{\mathbb{F}_{q}}\overline{w}_{2,\mathcal{H}}=1,\
\overline{w}_{1,\mathcal{H}}=0 \}.
\end{align*}
Then it is easy to verify
\[ J(M^{[k]},L,d)=\bigcup_{\alpha\in \mathcal{O}_{F_0}/(\pi_0)} J_{\alpha}(M^{[k]},L,d)\cup J_{\infty}(M^{[k]},L,d). \]
Now we compute 
$|J_{\alpha }(M^{[k]},L,d)|$. To determine a 
$\phi \in J_{\alpha }(M^{[k]},L,d)$, we choose 
$w_1=\phi (l_1)$ first. By definition, we have
\begin{align} & \lim_{d\to \infty}q^{(2(2k+m)-1)d}\#\big\{w_1\in M^{[k]}/\pi_0^d M^{[k]} \mid w_{1,\mathcal{H}}\text{ is primitive, and } q(w_1)\equiv 0 \text{ mod } \pi_0^d \big\}\nonumber\\ &\qquad=\beta_1(M^{[k]},0)=1-q^{-2k}. \end{align} Given such a 
$w_1$, now we find the number of 
$w_2=\phi (l_2)$ such that 
$\phi$ lies in 
$J_{\alpha }(M^{[k]},L,d)$. By Lemma A.2, we may assume 
$w_{1,S}=0$. Let 
$w_2=w_{2,M}+\alpha w_1+\pi w_\mathcal {H}$, where 
$w_\mathcal {H}\in \mathcal {H}^k$. Then the corresponding 
$\phi$ lies in 
$J_{\alpha }(M^{[k]},L,d)$ if and only if
\[ \pi^i\equiv (w_1,w_2)\equiv(w_1,\pi w_\mathcal{H}) \mod \pi^{2d-1} \]
and
\begin{align*} 0\equiv q(w_2)&\equiv\mathrm{tr}((\alpha w_1,\pi w_\mathcal{H}))-\pi_0 q(w_\mathcal{H})+q(w_{2,M}) \\ &\equiv\alpha\mathrm{tr}(\pi^i)-\pi_0 q(w_\mathcal{H})+q(w_{2,M}) \mod \pi^{2d-1}. \end{align*}
First,
\begin{equation} \lim_{d\to
\infty}q^{-2d(2k-1)}\#\big\{\pi w_\mathcal{H} \in
\mathcal{H}^k/\pi_0^d \mathcal{H}^k\mid(w_1,\pi
w_{\mathcal{H}})\equiv\pi^i\ \text{mod}\ \pi^{2d-1}\big\}=q^{1-2k}.
\end{equation}
Second, for each fixed 
$\pi w_\mathcal {H}$ we have
\begin{align} &\lim_{d\to
\infty}q^{(-2m+1)d} \#\{ w_{2,M} \in M/\pi_0^d M\mid
w_{2,M} \text{ primitive, } q(w_{2,M})\equiv -\alpha
\mathrm{tr}(\pi^i)+\pi_0 q(w_\mathcal{H})\ \text{mod } \pi_0^d\} \nonumber\\ &\qquad =\beta(M,\langle
-\alpha \mathrm{tr}(\pi^i)+\pi_0
q(w_\mathcal{H})\rangle)\nonumber\\ &\qquad =\begin{cases}
\beta(M, \langle -2 \alpha \rangle), & \text{if } i=0,\\
\beta(M, 0), & \text{if } i > 0. \end{cases}
\end{align}
By symmetry, 
$| J_{\infty }(M^{[k]},L,d)|=| J_{0}(M^{[k]},L,d)|$. Now a combination of (A.6), (A.7) and (A.8) implies that
\begin{align*} \beta_1(M, \mathcal{H}_i, q^{-2k}) &=\lim_{d\to \infty}q^{(-4(2k+m)+4)d} \biggl(\sum_{\alpha\in \mathcal{O}_{F_0}/(\pi_0)}| J_{\alpha}(M^{[k]},L,d)|+| J_{\infty}(M^{[k]},L,d)|\biggr)\\ &=\begin{cases} q(1-X)\biggl(2\beta_0(M,0,q^{-2k})+\displaystyle\sum_{\alpha \in \mathcal{O}_{F_0}^{\times}/(\pi_0)}\beta_0(M,-2\alpha,q^{-2k}) \biggr), & \text{if }i=0,\\ q(q+1) (1-X)\beta_0(M,0,q^{-2k}), & \text{if }i\ge 1, \end{cases} \end{align*}
as claimed.
Next, we assume 
$L$ has a basis 
$\{ l_1, l_2\}$ whose Gram matrix is 
$\mathrm {Diag}(u_1(-\pi _0)^a,u_2(-\pi _0)^b)$ with 
$0 \le a \le b$. Let 
$w_i=\phi (l_i)$ as before. Then the number of possible choices for 
$w_1$ is given by
\[ q^{(2(2k+m)-1)d}\beta_1(M,\langle u_1(-\pi_0)^a \rangle , q^{-2k}) \]
for sufficiently large 
$d$. We may assume 
$w_1=w_{1,\mathcal {H}}$ without loss of generality. Let 
$w_2=w_{2,M}+\alpha w_1+\pi w_\mathcal {H}$ as before. Then 
$\phi$ lies in 
$J_{\alpha }(M^{[k]},L,d)$ if and only if
\[ 0 \equiv (w_1,w_2)\equiv(w_1,\alpha w_1)+(w_1,\pi w_\mathcal{H}) \mod \pi^{2d-1} \]
and
\begin{align*} u_2(-\pi_0)^b&\equiv q(w_2)\equiv(w_{2,M}+\alpha w_{1}+\pi w_\mathcal{H} , w_2)\\ &\equiv q(w_{2,M})-\alpha^2 q( w_1)-\pi_0 q(w_\mathcal{H})\mod \pi^{2d-1}. \end{align*}
Now
\[ \lim_{d\to \infty}q^{(-4k+2)d}\#\{\pi w_\mathcal{H} \in \mathcal{H}^k/\pi_0^d \mathcal{H}^k\mid (w_1,\pi w_{\mathcal{H}})\equiv -(w_1,\alpha w_1)\ \text{mod}\ \pi^{2d-1}\}=q^{1-2k}, \]
and for a fixed 
$\pi w_\mathcal {H}$ we have
\begin{align*}
&\lim_{d\to \infty}q^{(-2m+1)d} \#\bigl\{ w_{2,M} \in L_{S}/\pi_0^d
L_{S}\mid w_{2,M} \text{ primitive},\\ & q(w_{2,M})\equiv u_2(-\pi_0)^b+\alpha^2 q( w_1)+\pi_0
q(w_\mathcal{H})\ \text{mod}\ \pi^{2d-1}\bigr\}\\
&\quad=\beta(M, \langle u_2(-\pi_0)^b+\alpha^2 q(
w_1)+\pi_0 q(w_\mathcal{H})\rangle).
\end{align*}
Now this proposition follows from a similar argument as before, and we leave the details to the reader.
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