# A proof of the Kudla–Rapoport conjecture for Krämer models

Qiao He  Department of Mathematics, University of Wisconsin Madison, Van Vleck
Hall, Madison, WI 53706, USA<EMAIL_ADDRESS>,  Chao Li  Columbia University,
Department of Mathematics, 2990 Broadway, New York, NY 10027, USA
<EMAIL_ADDRESS>,  Yousheng Shi  Department of Mathematics,
University of Wisconsin Madison, Van Vleck Hall, Madison, WI 53706, USA
<EMAIL_ADDRESS>and  Tonghai Yang  Department of Mathematics, University of
Wisconsin Madison, Van Vleck Hall, Madison, WI 53706, USA
<EMAIL_ADDRESS>
###### Abstract.

We prove the Kudla–Rapoport conjecture for Krämer models of unitary
Rapoport–Zink spaces at ramified places. It is a precise identity between
arithmetic intersection numbers of special cycles on Krämer models and
modified derived local densities of hermitian forms. As an application, we
relax the local assumptions at ramified places in the arithmetic Siegel–Weil
formula for unitary Shimura varieties, which is in particular applicable to
unitary Shimura vartieties associated to unimodular hermitian lattices over
imaginary quadratic fields.

###### Contents

  1. 1 Introduction
  2. 2 Krämer models of Rapoport-Zink spaces and special cycles
  3. 3 Bruhat–Tits Stratification of Krämer models
  4. 4 Fourier transform: the geometric side
  5. 5 Review of local densities and primitive local densities
  6. 6 Decomposition of primitive local densities
  7. 7 Explicit formulas for $\mathrm{Pden}^{\prime}(L)$
  8. 8 Fourier transform: the analytic side
  9. 9 Proof of the main theorem
  10. 10 Global applications

##  1\. Introduction

###  1.1. Background

The classical _Siegel–Weil formula_ ([Sie35, Sie51, Wei65]) relates certain
Siegel Eisenstein series to the arithmetic of quadratic forms, namely it
expresses special _values_ of these series as theta functions — generating
series of representation numbers of quadratic forms. Kudla ([Kud97, Kud04])
initiated an influential program to establish the _arithmetic Siegel–Weil
formula_ relating certain Siegel Eisenstein series to objects in arithmetic
geometry, which among others, aims to express the _central derivative_ of
these series as the arithmetic analogue of theta functions — generating series
of arithmetic intersection numbers of $n$ special divisors on Shimura
varieties associated to $\mathrm{SO}(n-1,2)$ or $\mathrm{U}(n-1,1)$.

For $\mathrm{U}(n-1,1)$–Shimura varieties with hyperspecial level at an
_unramified_ place, Kudla–Rapoport [KR11] conjectured a local arithmetic
Siegel–Weil formula, now known as the (local) _Kudla–Rapoport conjecture_. It
is a precise identity between the _central derivative_ of local representation
densities of hermitian forms (the analytic side) and the arithmetic
intersection number of special cycles on unitary Rapoport–Zink spaces (the
geometric side). This conjecture was recently proved by Zhang and one of us
[LZ22a], and we refer to the introduction of [LZ22a] for more background and
related results.

It is a natural question, which is also important for global applications, to
formulate and prove an analogue of the Kudla–Rapoport conjecture at a
_ramified_ place. At a ramified place, there are two well-studied level
structures for unitary Rapoport–Zink spaces, one gives rise to the _exotic
smooth model_ which has good reduction, and the other one gives rise to the
_Krämer model_ which has bad (semistable) reduction. For the even dimensional
exotic smooth model, the analogue of Kudla–Rapoport conjecture was formulated
and proved by Liu and one of us [LL22] using a strategy similar to [LZ22a].

For the Krämer model, however, the situation is more complicated — it is
expected that the analytic side of the conjecture requires nontrivial
modification, by a certain linear combination of _central values_ of local
representation densities. The necessity of such modification in the presence
of bad reduction was first discovered by Kudla–Rapoport [KR00] via explicit
computation in the context of the Drinfeld $p$-adic half plane. In [HSY21],
three of us formulated _the Kudla–Rapoport conjecture for Krämer models_
(recalled in §1.2) by providing a conceptual recipe for the precise
modification needed for the analytic side. Moreover, this conjecture was
proved for $n=2$ (based on the previous works [Shi20, HSY20]) and $n=3$ in
[HSY21].

The main theorem of the present paper settles this conjecture for any $n$ (and
the proof is new even for $n=2,3$). As a first application, we relax the local
assumptions in the arithmetic Siegel–Weil formula for
$\mathrm{U}(n-1,1)$–Shimura varieties by allowing Krämer models at ramified
places. The main theorem should also be useful to relax the local assumptions
at ramified places in the arithmetic inner product formula [LL21, LL22] and
its $p$-adic avatar by Disegni–Liu [DL22].

###  1.2. Kudla–Rapoport conjecture for Krämer models

Let $p$ be an odd prime. Let $F_{0}$ be a finite extension of $\mathbb{Q}_{p}$
with residue field $\kappa=\mathbb{F}_{q}$. Let $F$ be a ramified quadratic
extension of $F_{0}$. Let $\pi$ be a uniformizer of $F$ such that
$\operatorname{Tr}_{F/F_{0}}(\pi)=0$. Then $\pi_{0}:=\pi^{2}$ is a uniformizer
of $F_{0}$. Let $\breve{F}$ be the completion of the maximal unramified
extension of $F$. Let $O_{F},{O_{\breve{F}}}$ be the ring of integers of
$F,{\breve{F}}$ respectively.

Let $n\geq 2$ be an integer. To define the Krämer model of the unitary
Rapoport–Zink space, we fix a (principally polarized) supersingular hermitian
$O_{F}$-modules $\mathbb{X}$ of signature $(1,n-1)$ over ${\bar{\kappa}}$
(Definition 2.1). The _Krämer model_ $\mathcal{N}=\mathcal{N}_{n}$ is the
formal scheme over $\mathrm{Spf}\,O_{\breve{F}}$ parameterizing hermitian
formal $O_{F}$-modules $X$ of signature $(1,n-1)$ within the quasi-isogeny
class of $\mathbb{X}$, together with a rank 1 filtration
$\mathcal{F}\subseteq\operatorname{Lie}X$ satisfying the Krämer condition
(Definition 2.2). The space $\mathcal{N}$ is locally of finite type, and
semistable of relative dimension $n-1$ over $\mathrm{Spf}\,{O_{\breve{F}}}$.
There are two choices of the framing object $\mathbb{X}$ (up to quasi-
isogeny), giving rise to two non-isomorphic (resp. isomorphic) spaces
$\mathcal{N}$ when $n$ is even (resp. odd) (§2.2).

Let $\mathbb{Y}$ be the framing hermitian $O_{F}$-modules of signature $(1,0)$
over ${\bar{\kappa}}$ defined as in Definition 2.1. The _space of quasi-
homomorphisms_
$\mathbb{V}=\mathbb{V}_{n}\coloneqq\operatorname{Hom}_{O_{F}}(\mathbb{Y},\mathbb{X})\otimes_{O_{F}}F$
carries a natural $F/F_{0}$-hermitian form, which makes $\mathbb{V}$ a non-
degenerate $F/F_{0}$-hermitian space of dimension $n$ (§2.2). The two choices
of the framing object $\mathbb{X}$ exactly correspond to the two isomorphism
classes of $\mathbb{V}$, classified by
$\chi(\mathbb{V}):=\chi((-1)^{\frac{n(n-1)}{2}}\det(\mathbb{V}))\in\\{\pm
1\\}$, where $\chi:F_{0}^{\times}\rightarrow\\{\pm 1\\}$ is the quadratic
character associated to $F/F_{0}$. For any subset $L\subseteq\mathbb{V}$, the
_special cycle_ $\mathcal{Z}(L)$ (§2.3) is a closed formal subscheme of
$\mathcal{N}$, over which each quasi-homomorphism $x\in L$ deforms to
homomorphisms.

Let $L\subseteq\mathbb{V}$ be an $O_{F}$-lattice (of full rank $n$). We will
associate to $L$ two integers: the _arithmetic intersection number_
$\operatorname{\mathrm{Int}}(L)$ and the _modified derived local density_
$\partial\mathrm{Den}(L)$.

######  Definition 1.1.

Let $L\subseteq\mathbb{V}$ be an $O_{F}$-lattice. Let $x_{1},\ldots,x_{n}$ be
an $O_{F}$-basis of $L$. Define the _arithmetic intersection number_

(1.1)
$\operatorname{\mathrm{Int}}(L)\coloneqq\chi(\mathcal{N},\mathcal{O}_{\mathcal{Z}(x_{1})}\otimes^{\mathbb{L}}\cdots\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x_{n})})\in\mathbb{Z},$

where $\mathcal{O}_{\mathcal{Z}(x_{i})}$ denotes the structure sheaf of the
special divisor $\mathcal{Z}(x_{i})$, $\otimes^{\mathbb{L}}$ denotes the
derived tensor product of coherent sheaves on $\mathcal{N}$, and $\chi$
denotes the Euler–Poincaré characteristic (Definition 2.10). By Howard [How19,
Corollary D]), we know that $\operatorname{\mathrm{Int}}(L)$ is independent of
the choice of the basis $x_{1},\ldots,x_{n}$ and hence is a well-defined
invariant of $L$ itself.

For $M$ another hermitian $O_{F}$-lattice (of arbitrary rank), denote by
$\mathrm{Herm}_{L,M}$ the $O_{F_{0}}$-scheme of hermitian $O_{F}$-module
homomorphisms from $L$ to $M$ (Definition 5.1) and define its _local density_
to be

$\mathrm{Den}(M,L)\coloneqq\lim_{d\rightarrow+\infty}\frac{|\mathrm{Herm}_{L,M}(O_{F_{0}}/\pi_{0}^{d})|}{q^{N\cdot
d_{L,M}}},$

where $d_{L,M}$ is the dimension of
$\mathrm{Herm}_{L,M}\otimes_{O_{F_{0}}}F_{0}$. Let $H$ be the standard
hyperbolic hermitian $O_{F}$-lattice of rank 2 (given by the hermitian matrix
$\left(\begin{smallmatrix}0&\pi^{-1}\\\ -\pi^{-1}&0\end{smallmatrix}\right)$).
It is well-known that there exists a _local density polynomial_
$\mathrm{Den}(M,L,X)\in\mathbb{Q}[X]$ such that for any integer $k\geq 0$,

(1.2) $\mathrm{Den}(M,L,q^{-2k})=\mathrm{Den}(H^{k}\obot M,L).$

When $M$ has also rank $n$ and $\chi(M)=-\chi(L)$, we have
$\mathrm{Den}(M,L)=0$ (Lemma 5.7) and in this case we write

$\mathrm{Den}^{\prime}(M,L):=-2\cdot\frac{\mathrm{d}}{\mathrm{d}X}\bigg{|}_{X=1}\mathrm{Den}(M,L,X).$

Define the (normalized) _derived local density_

(1.3)
$\mathrm{Den}^{\prime}(L):=\frac{\mathrm{Den}^{\prime}(I_{n},L)}{\mathrm{Den}(I_{n},I_{n})}\in\mathbb{Q}.$

Here $I_{n}$ is the unimodular lattice of rank $n$ with
$\chi(I_{n})=-\chi(L)$. Recall that a hermtian $O_{F}$-lattice $L$ is
_unimodular_ 111We refrain from using the terminology _self-dual_ in the
ramified case to avoid possible confusion with a lattice $L$ such that
$L=L^{\vee}$, where $L^{\vee}$ is the dual lattice with respect to the
underlying quadratic form, see §4.2. if $L=L^{\sharp}$, where $L^{\sharp}$ is
the dual lattice of $L$ with respect to the hermitian form (see §1.5 for
notation).

The naive analogue of the Kudla–Rapoport conjecture for Krämer model states
that

$\operatorname{\mathrm{Int}}(L)\stackrel{{\scriptstyle?}}{{=}}\mathrm{Den}^{\prime}(L).$

However, as explained in [HSY21] this naive analogue does not hold for trivial
reasons. In fact, by definition $\operatorname{\mathrm{Int}}(L)$ vanishes
unless $L$ is integral (i.e., $L\subseteq L^{\sharp}$), while
$\mathrm{Den}^{\prime}(L)$ does not vanish for non-integral lattices $L$ which
are dual to vertex lattices. More precisely, recall that an integral
$O_{F}$-lattice $\Lambda\subseteq\mathbb{V}$ is called a _vertex lattice (of
type $t$)_ if $\Lambda^{\sharp}/\Lambda$ is a $\kappa$-vector space (of
dimension $t$). For a vertex lattice $\Lambda\subseteq\mathbb{V}$ of type
$t>0$, we have $\Lambda^{\sharp}$ is non-integral so
$\operatorname{\mathrm{Int}}(\Lambda^{\sharp})=0$, while
$\mathrm{Den}^{\prime}(\Lambda^{\sharp})\neq 0$ in general (see e.g. (5.7)).
In general, we define the type $t(L)$ of $L$ to be the number of positive
fundamental invariants of $L$ (see §1.5).

To account for these discrepancies, we will define $\partial\mathrm{Den}(L)$
by modifying $\mathrm{Den}^{\prime}(L)$ with a linear combination of the
(normalized) _local densities_ (Corollary 5.8)

(1.4)
$\mathrm{Den}_{t}(L):=\frac{\mathrm{Den}(\Lambda_{t}^{\sharp},L)}{\mathrm{Den}(\Lambda_{t}^{\sharp},\Lambda_{t}^{\sharp})}\in\mathbb{Z}.$

Here $\Lambda_{t}\subseteq\mathbb{V}$ is a vertex lattice of type $t$ (in
particular $\chi(\Lambda_{t}^{\sharp})=\chi(L)$). Recall that the possible
vertex type $t$ is given by any even integer such that $0\leq t\leq
t_{\mathrm{max}}$, where

$t_{\mathrm{max}}=\begin{cases}n,&\mathrm{if}\,n\text{ even,
}\chi(\mathbb{V})=+1,\\\ n-1,&\mathrm{if}\,n\text{ odd},\\\
n-2,&\mathrm{if}\,n\text{ even, }\chi(\mathbb{V})=-1.\end{cases}$

######  Definition 1.2.

Let $L\subseteq\mathbb{V}$ be an $O_{F}$-lattice. Define the _modified derived
local density_ (Corollary 7.2)

(1.5)
$\partial\mathrm{Den}(L):=\mathrm{Den}^{\prime}(L)+\sum_{j=1}^{t_{\mathrm{max}}/2}c_{2j}\cdot\mathrm{Den}_{2j}(L)\in\mathbb{Z}.$

Here the coefficients $c_{2j}\in\mathbb{Q}$ are chosen to satisfy

(1.6) $\partial\mathrm{Den}(\Lambda_{2i}^{\sharp})=0,\quad 1\leq i\leq
t_{\mathrm{max}}/2,$

which turns out to be a linear system in
$(c_{2},c_{4},\ldots,c_{t_{\mathrm{max}}})$ with a unique solution ([HSY21,
Theorem 6.1].

The main purpose of this paper is to prove the following local arithmetic
Siegel-Weil formula, settling the main conjecture of [HSY21]. We will prove
this theorem in §9.

######  Theorem 1.3 (Kudla–Rapoport conjecture for Krämer models).

Let $L\subseteq\mathbb{V}$ be an $O_{F}$-lattice. Then

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

###  1.3. The arithmetic Siegel–Weil formula

Next let us describe some global applications of our main theorem, following
the setting of [LZ22a, §1.3]. We now switch to global notations. Let $F$ be a
CM number field with maximal total real subfield $F_{0}$. Fix an embedding
$\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and fix a CM type
$\Phi\subseteq\operatorname{Hom}(F,\overline{\mathbb{Q}})=\operatorname{Hom}(F,\mathbb{C})$
of $F$. We aslo identify the CM type $\Phi$ with the set of archimedean places
of $F_{0}$. Let $V$ be an $F/F_{0}$-hermitian space of dimension $n$ and
$G=\operatorname{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V)$. Assume the signatures
of $V$ are $\\{(n-1,1)_{\phi_{0}},(n,0)_{\phi\in\Phi-\\{\phi_{0}\\}}\\}$ for
some distinguished element $\phi_{0}\in\Phi$. Define a torus
$Z^{\mathbb{Q}}=\\{z\in\operatorname{Res}_{F/\mathbb{Q}}\mathbb{G}_{m}:\mathrm{Nm}_{F/F_{0}}(z)\in\mathbb{G}_{m}\\}$.
Associated to $\widetilde{G}\coloneqq Z^{\mathbb{Q}}\times G$ there is a
natural Shimura datum $(\widetilde{G},\\{h_{\widetilde{G}}\\})$ of PEL type
([LZ22a, §11.1]). Let $K=K_{Z^{\mathbb{Q}}}\times
K_{G}\subseteq\widetilde{G}(\mathbb{A}_{f})$ be a compact open subgroup. Then
the associated Shimura variety
$\mathrm{Sh}_{K}=\mathrm{Sh}_{K}(\widetilde{G},\\{h_{\widetilde{G}}\\})$ is of
dimension $n-1$ and has a canonical model over its reflex field $E$.

Assume that $K_{Z^{\mathbb{Q}}}\subseteq Z^{\mathbb{Q}}(\mathbb{A}_{f})$ is
the unique maximal open compact subgroup. Assume that
$K_{G}=\prod_{v}K_{G,v}$, where $v$ runs over the finite places of $F_{0}$
such that $K_{G,v}\subseteq\mathrm{U}(V)(F_{0,v})$ is given by

  * •

the stabilizer of a self-dual or almost self-dual lattice
$\Lambda_{v}\subseteq V_{v}$ if $v$ is inert in $F$,

  * •

the stabilizer of a unimodular lattice $\Lambda_{v}\subseteq V_{v}$ if $v$ is
ramified in $F$,

  * •

a principal congruence subgroup of
$\mathrm{U}(V)(F_{0,v})\simeq\mathrm{GL}_{n}(F_{0,v})$ if $v$ is split in $F$.

Let $\mathcal{V}_{\mathrm{ram}}$ (resp. $\mathcal{V}_{\mathrm{asd}}$) be the
set of finite places $v$ of $F_{0}$ such that $v$ is ramified in $F$ (resp.
$v$ is inert in $F$ and $\Lambda_{v}$ is almost self-dual). Further assume
that all places of $E$ above
$\mathcal{V}_{\mathrm{ram}}\cup\mathcal{V}_{\mathrm{asd}}$ are unramified over
$F$. Then we obtain a global regular integral model $\mathcal{M}_{K}$ of
$\mathrm{Sh}_{K}$ over $O_{E}$ as in [LZ22a, §14.1-14.2], which is semistable
at all places of $E$ above
$\mathcal{V}_{\mathrm{ram}}\cup\mathcal{V}_{\mathrm{asd}}$ (for more precise
technical conditions required, see (G0)-(G4)). When $K_{G}$ is the stabilizer
of a global unimodular lattice, the regular integral model $\mathcal{M}_{K}$
recovers that in [HSY21], and that in [BHK+20] if $F_{0}=\mathbb{Q}$.

Let $\mathbb{V}$ be the _incoherent_ hermitian space over $\mathbb{A}_{F}$
associated to $V$, namely $\mathbb{V}$ is totally positive definite and
$\mathbb{V}_{v}\cong V_{v}$ for all finite places $v$. Let
$\varphi_{K}\in\mathscr{S}(\mathbb{V}^{n}_{f})$ be a $K$-invariant (where $K$
acts on $\mathbb{V}_{f}$ via the second factor $K_{G}$) factorizable Schwartz
function such that $\varphi_{K,v}=\mathbf{1}_{(\Lambda_{v})^{n}}$ at all $v$
nonsplit in $F$. Let $T\in\mathrm{Herm}_{n}(F_{0})$ be a nonsingular
$F/F_{0}$-hermitian matrix of size $n$. Associated to $(T,\varphi_{K})$ we
have _arithmetic special cycles_ $\mathcal{Z}(T,\varphi_{K})$ over
$\mathcal{M}_{K}$ ([LZ22a, §14.3]) generalizing the $\mathcal{Z}(T)$ in
[KR14]. Analogous to the local situation (1.1), we can define its local
arithmetic intersection numbers
$\operatorname{\mathrm{Int}}_{T,v}(\varphi_{K})$ at finite places $v$. Using
the star product of Kudla’s Green functions, we can also define its local
arithmetic intersection number
$\operatorname{\mathrm{Int}}_{T,v}(\mathsf{y},\varphi_{K})$ at infinite places
([LZ22a, §15.3]), which depends on a parameter
$\mathsf{y}\in\mathrm{Herm}_{n}(F_{0,\infty})_{>0}$ where
$F_{0,\infty}=F_{0}\otimes_{\mathbb{Q}}\mathbb{R}$. Combining all the local
arithmetic numbers together, define the _global arithmetic intersection
number_ , or the _arithmetic degree_ of the special cycle
$\mathcal{Z}(T,\varphi_{K})$ in the arithmetic Chow group of
$\mathcal{M}_{K}$,

$\widehat{\deg}_{T}(\mathsf{y},\varphi_{K})\coloneqq\sum_{v\nmid\infty}\operatorname{\mathrm{Int}}_{T,v}(\varphi_{K})+\sum_{v\mid\infty}\operatorname{\mathrm{Int}}_{T,v}(\mathsf{y},\varphi_{K}).$

On the other hand, associated to
$\varphi:=\varphi_{K}\otimes\varphi_{\infty}\in\mathscr{S}(\mathbb{V}^{n})$,
where $\varphi_{\infty}$ is the Gaussian function, there is a classical
_incoherent Eisenstein series_ $E(\mathsf{z},s,\varphi)$ ([LZ22a, §12.4]) on
the hermitian upper half space

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

This is essentially the Siegel Eisenstein series associated to a standard
Siegel–Weil section of the degenerate principal series ([LZ22a, §12.1]). The
Eisenstein series here has a meromorphic continuation and a functional
equation relating $s\leftrightarrow-s$. The central value
$E(\mathsf{z},0,\varphi)=0$ by the incoherence. We thus consider its _central
derivative_

$\mathrm{Eis}^{\prime}(\mathsf{z},\varphi_{K})\coloneqq\frac{\mathrm{d}}{\mathrm{d}s}\bigg{|}_{s=0}E(\mathsf{z},s,\varphi).$

Analogous to the local situation, we need to modify
$\mathrm{Eis}^{\prime}(\mathsf{z},\varphi_{K})$ by central values of coherent
Eisenstein series. For
$v\in\mathcal{V}_{\mathrm{ram}}\cup\mathcal{V}_{\mathrm{asd}}$, let
${}^{v}\mathbb{V}$ be the _coherent_ hermitian space over $\mathbb{A}_{F}$
nearby $\mathbb{V}$ at $v$, namely $(^{v}\mathbb{V})_{w}\simeq\mathbb{V}_{w}$
exactly for all places $w\neq v$. For any vertex lattice
$\Lambda_{t,v}\subseteq(^{v}\mathbb{V})_{v}$ of type $t$, the Schwartz
function
$\varphi^{v}\otimes\mathbf{1}_{(\Lambda_{t,v}^{\sharp})^{n}}\in\mathscr{S}((^{v}\mathbb{V})^{n})$
gives a classical _coherent Eisenstein series_
$E(\mathsf{z},s,\varphi^{v}\otimes\mathbf{1}_{(\Lambda_{t,v}^{\sharp})^{n}})$.
Analogous to (1.4), define the (normalized) _central values_

(1.7)
$^{v}\mathrm{Eis}_{t}(\mathsf{z},\varphi_{K}):=\frac{\operatorname{vol}(K_{G,v})}{\operatorname{vol}(K_{\Lambda_{t,v}^{\sharp}})}\cdot
E(\mathsf{z},0,\varphi^{v}\otimes\mathbf{1}_{(\Lambda_{t,v}^{\sharp})^{n}}).$

Here $K_{\Lambda_{t,v}^{\sharp}}\subseteq\mathrm{U}(^{v}\mathbb{V})(F_{0,v})$
is the stabilizer of $\Lambda_{t,v}^{\sharp}$, and the volumes are taken with
respect to the Haar measures on $\mathrm{U}(V)(F_{0,v})$ and
$\mathrm{U}(^{v}\mathbb{V})(F_{0,v})$ as defined in [LL21, Definition 3.8].
When $v\in\mathcal{V}_{\mathrm{ram}}$, analogous to (1.5), define the linear
combination

(1.8)
${}^{v}\mathrm{Eis}(\mathsf{z},\varphi_{K}):=\sum_{j=1}^{t_{\mathrm{max},v}/2}c_{2j,v}\cdot{}^{v}\mathrm{Eis}_{2j}(\mathsf{z},\varphi_{K}),$

where $t_{\mathrm{max},v}$ and $c_{2j,v}$ are the numbers $t_{\mathrm{max}}$
and $c_{2j}$ respectively in (1.5) for the local hermitian space
$(^{v}\mathbb{V})_{v}$ over the ramified extension $F_{v}/F_{0,v}$. When
$v\in\mathcal{V}_{\mathrm{asd}}$, define

(1.9)
$^{v}\mathrm{Eis}(\mathsf{z},\varphi_{K}):=c_{0,v}\cdot{}^{v}\mathrm{Eis}_{0}(\mathsf{z},\varphi_{K}),$

where $c_{0,v}=-\frac{1}{1+q_{v}}$ and $q_{v}$ is the size of the residue
field of $F_{0,v}$. Define the _modified central derivative_

(1.10)
$\partial\mathrm{Eis}(\mathsf{z},\varphi_{K}):=\mathrm{Eis}^{\prime}(\mathsf{z},\varphi_{K})+(-1)^{n}\sum_{v\in\mathcal{V}_{\mathrm{ram}}\cup\mathcal{V}_{\mathrm{asd}}}{}^{v}\mathrm{Eis}(\mathsf{z},\varphi_{K}).$

Associated to an additive character
$\psi:\mathbb{A}_{F_{0}}/F_{0}\rightarrow\mathbb{C}^{\times}$ (as explained in
[LZ22a, §12.2] we assume that $\psi$ is unramified outside the set of finite
places of $F_{0}$ split in $F$), it has a decomposition into Fourier
coefficients

(1.11)
$\partial\mathrm{Eis}(\mathsf{z},\varphi_{K})=\sum_{T\in\mathrm{Herm}_{n}(F_{0})}\partial\mathrm{Eis}_{T}(\mathsf{z},\varphi_{K}).$

The following result asserts an identity between the arithmetic degrees of
special cycles and the nonsingular Fourier coefficients of the modified
central derivative of the incoherent Eisenstein series, which generalizes
[LZ22a, Theorem 1.3.1] from inert places to all nonsplit places. In
particular, when $F$ is an imaginary quadratic field of discriminant $d\equiv
1\pmod{8}$, we have an unconditional arithmetic Siegel-Weil formula for all
unimodular lattices of signature $(n-1,1)$ at non-singular coefficients, i.e.,
[HSY21, Theorem 1.5] holds without conditions.

######  Theorem 1.4 (Arithmetic Siegel–Weil formula: nonsingular terms).

Let $\mathrm{Diff}(T,\mathbb{V})$ be the set of places $v$ such that
$\mathbb{V}_{v}$ does not represent $T$ ([LZ22a, §12.3]). Let
$T\in\mathrm{Herm}_{n}(F_{0})$ be nonsingular such that
$\mathrm{Diff}(T,\mathbb{V})=\\{v\\}$ where $v$ is nonsplit in $F$ and not
above 2. Then

$\widehat{\deg}_{T}(\mathsf{y},\varphi_{K})q^{T}=c_{K}\cdot\partial\mathrm{Eis}_{T}(\mathsf{z},\varphi_{K}),$

where $q^{T}\coloneqq\psi_{\infty}(\operatorname{Tr}T\mathsf{z})$, $c_{K}$ is
a nonzero constant independent of $T$ and $\varphi_{K}$ (to be specified in
Theorem 10.1).

We form the _generating series of arithmetic degrees_

(1.12)
$\widehat{\deg}(\mathsf{z},\varphi_{K})\coloneqq\sum_{T\in\mathrm{Herm}_{n}(F_{0})\atop\det
T\neq 0}\widehat{\deg}_{T}(\mathsf{y},\varphi_{K})q^{T}.$

The following result relates this generating series to the modified central
derivative of the incoherent Eisenstein series, which removes the assumption
that $F/F_{0}$ is unramified at all finite places from [LZ22a, Theorem 1.3.2].

######  Theorem 1.5 (Arithmetic Siegel–Weil formula).

Assume that $F/F_{0}$ is split at all places above 2. Further assume that
$\varphi_{K}$ is nonsingular ([LZ22a, §12.3]) at two places split in $F$. Then

$\widehat{\deg}(\mathsf{z},\varphi_{K})=c_{K}\cdot\partial\mathrm{Eis}(\mathsf{z},\varphi_{K}).$

In particular, $\widehat{\deg}(\mathsf{z},\varphi_{K})$ is a nonholomorphic
hermitian modular form of genus $n$.

###  1.4. Strategy and novelty of the proof of the Main Theorem 1.3

Our general strategy is closest to the unramified orthogonal case proved in
[LZ22b]. More precisely, fix an $O_{F}$-lattice $L^{\flat}\subseteq\mathbb{V}$
of rank $n-1$ and denote by
$\mathbb{W}=(L^{\flat}_{F})^{\perp}\subseteq\mathbb{V}$. Consider functions on
$\mathbb{V}\setminus L^{\flat}_{F}$,

$\operatorname{\mathrm{Int}}_{L^{\flat}}(x)\coloneqq\operatorname{\mathrm{Int}}(L^{\flat}+\langle
x\rangle),\quad\partial\mathrm{Den}_{L^{\flat}}(x)\coloneqq\partial\mathrm{Den}(L^{\flat}+\langle
x\rangle).$

Then it remains to show the equality of the two functions
$\operatorname{\mathrm{Int}}_{L^{\flat}}=\partial\mathrm{Den}_{L^{\flat}}$. To
show this equality, we find a decomposition

$\operatorname{\mathrm{Int}}_{L^{\flat}}=\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{H}}+\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}},\quad\partial\mathrm{Den}_{L^{\flat}}=\partial\mathrm{Den}_{L^{\flat},\mathscr{H}}+\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}$

into “horizontal” and “vertical” parts such that the horizontal identity
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{H}}=\partial\mathrm{Den}_{L^{\flat},\mathscr{H}}$
holds and that the vertical parts
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}$ and
$\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}$ behaves well under Fourier
transform along $L^{\flat}_{F}$.

The horizontal identity essentially reduces to the horizontal computation for
$n=2$ in [Shi20, HSY20]. For the vertical identity, we perform a _partial
Fourier transform_ along $L^{\flat}_{F}$ and consider new functions on
$\mathbb{W}\setminus\\{0\\}$,

$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\perp}(x):=\int_{L^{\flat}_{F}}\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}(y+x)\mathrm{d}y,\quad\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}^{\perp}(x):=\int_{L^{\flat}_{F}}\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}(y+x)\mathrm{d}y.$

The key is to show that
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\perp}$ and
$\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}^{\perp}$ are both _constant_ on
$\mathbb{W}^{\geq
0}\setminus\\{0\\}:=\\{x\in\mathbb{W}\setminus\\{0\\}:{\mathrm{val}}(x)\geq
0\\}$ of $\mathbb{W}$ (see §1.5 for notation) as in Theorem 4.16 and Theorem
8.2. Using an induction on the valuation of $L^{\flat}$, we show that the
difference
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\perp}-\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}^{\perp}$
vanishes on $\mathbb{W}^{\leq 0}:=\\{x\in\mathbb{W}:{\mathrm{val}}(x)\leq
0\\}$, and hence it vanishes identically and allows us to conclude that
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}=\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}$.

On the geometric side, we prove a _Bruhat–Tits stratification_ for the Krämer
model (Theorem 3.19), analogous to the case of the Pappas model treated in
Rapoport–Terstiege–Wilson [RTW14]. We make use of the linear invariance of
special cycles [How19] to express
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}$ as a linear combination
of functions on $\mathbb{V}$ which are _translation invariant_ under vertex
lattices. A new observation in our ramified case is that the translation
invariance already allows us to control the support of its Fourier transform
well enough (Lemma 4.13) to conclude the desired key constancy of
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\perp}$ on
$\mathbb{W}^{\geq 0}$. Compared to the unramified case, we _completely avoid_
the Tate conjecture of generalized Deligne-Lusztig varieties and explicit
computation of their intersections with special divisors. §3 studies the
structure of $\mathcal{N}_{\mathrm{red}}$ and special cycles, and should be of
independent interest (in addition to preparation for §4) .

On the analytic side, we make use of the _primitive decomposition_ of the
local density polynomial into primitive local density polynomials and obtain a
decomposition.

(1.13) $\partial\mathrm{Den}(L)=\sum_{L\subseteq
L^{\prime}}\partial\mathrm{Pden}(L^{\prime}),$

where $L^{\prime}$ runs over $O_{F}$-lattices in $L_{F}$ containing $L$, and
the symbol $\mathrm{Pden}$ stands for the primitive version of $\mathrm{Den}$
(Corollary 5.4). Unlike the unramified or exotic smooth case, the primitive
local density _polynomial_ itself seems rather complicated (see e.g. Corollary
6.2). Nevertheless we manage to prove a simple formula for its _modified
central derivative_ $\partial\mathrm{Pden}(L)$, which we find quite striking.

######  Theorem 1.6 (Theorem 7.1).

Let $L\subset\mathbb{V}$ be an $O_{F}$-lattice (of full rank $n$).

  1. (1)

If $L$ is not integral, then $\partial\mathrm{Pden}(L)=0.$

  2. (2)

If $L$ is unimodular, then

$\partial\mathrm{Pden}(L)=\begin{cases}1,&\text{if $n$ is odd},\\\ 0,&\text{if
$n$ is even}.\end{cases}$

  3. (3)

If $L$ is integral and of type $t>0$, then

$\partial\mathrm{Pden}(L)=\begin{cases}\displaystyle\prod_{\ell=1}^{\frac{t-1}{2}}(1-q^{2\ell}),&\text{
if $t$ is odd},\\\
\displaystyle(1-\chi(L^{\prime})q^{\frac{t}{2}})\prod_{\ell=1}^{\frac{t}{2}-1}(1-q^{2\ell}),&\text{
if $t$ is even}.\end{cases}$

Here we write $L\simeq I_{n-t}\obot L^{\prime}$ with $I_{n-t}$ unimodular of
rank $n-t$.

The proof of this theorem occupies the entire §6 and §7, and is our major
technical innovation. One key difference between our case and the unramified
or exotic smooth case is that in our case $I_{n}$ and $H$ (see (1.2) and
(1.3)) have different fundamental invariants, hence it is not clear how to
reduce the calculation of $\partial\mathrm{Pden}$ into the embedding-counting
problems over finite fields in the style of [CY20, §3]. To deal with this
difficulty, we first decompose $\partial\mathrm{Pden}(L)$ according to orbits
of Hermitian embeddings (Theorem 6.1). Now a new observation is that the
primitive local density polynomial becomes simpler when $L$ is “very integral”
(i.e., when its fundamental invariants are all $\geq 1$, see Proposition 7.6)
in which case the decomposition in Theorem 6.1 is simple. The primitive local
density polynomial vanishes when $L$ is “very non-integral” (e.g., when one of
its fundamental invariants is $\leq-2$, see the proof of Lemma 5.2). When $L$
is the dual of a vertex lattice of positive type, this is just our assumption
(1.6). The remaining cases (in particular the unimodular lattice case) are
much trickier to handle, whose proof occupies most of §7 and is summarized in
§7.2. The proof relies on a series of non-trivial polynomial identities (e.g.,
Lemma 7.15 and Lemma 7.16) involving algebraic combinatorics of quadratic
spaces over finite fields, which should be of independent interest.

With the simple formula for $\partial\mathrm{Pden}(L)$ at hand, we finally
prove the desired key constancy of
$\partial\mathrm{Den}_{L^{\flat},\mathscr{V}}^{\perp}$ on $\mathbb{W}^{\geq
0}\setminus\\{0\\}$ via involved lattice-theoretic computation in §8, in a
fashion similar to [LZ22b]. The techniques developed here on the analytic side
seem to have wide applicability and we hope that they may shed new light on
the Kudla–Rapoport conjecture in the context of more general level structures,
e.g., for minuscule parahoric levels at unramified places formulated by Cho
[Cho22].

###  1.5. Notation and terminology

In this paper, a lattice means a hermitian $O_{F}$-lattice without explicit
mentioning. Unless otherwise stated, $L$ always means a non-degenerate lattice
of rank $n$ with a hermitian form $(\,,\,)$.

  * •

We say a sublattice $L^{\flat}$ of a hermitian space is non-degenerate if the
restriction of the hermitian form to it is non-degenerate.

  * •

We define $L^{\sharp}$ to be the dual lattice of $L$ with respect to the
hermitian form $(\,,\,)$. If $L\subset L^{\sharp}$, we say $L$ is integral.

  * •

Following [LL22, Definition 2.11], for a lattice $L$ with hermitian form
$(\,,\,)$, we say that a basis $\left\\{\ell_{1},\ldots,\ell_{n}\right\\}$ of
$L$ is a normal basis (which always exists by [LL22, Lemma 2.12]) if its
moment matrix $T=\left(\left(\ell_{i},\ell_{j}\right)\right)_{1\leq i,j\leq
n}$ is conjugate to

$\displaystyle\left(\beta_{1}\pi^{2b_{1}}\right)\oplus\cdots\oplus\left(\beta_{s}\pi^{2b_{s}}\right)\oplus\left(\begin{array}[]{cc}0&\pi^{2c_{1}+1}\\\
-\pi^{2c_{1}+1}&0\end{array}\right)\oplus\cdots\oplus\left(\begin{array}[]{cc}0&\pi^{2c_{t}+1}\\\
-\pi^{2c_{t}+1}&0\end{array}\right)$

by a permutation matrix, for some $\beta_{1},\ldots,\beta_{s}\in
O_{F_{0}}^{\times}$ and $b_{1},\ldots,b_{s},c_{1},\ldots,c_{t}\in\mathbb{Z}$.
Moreover, we define its (unitary) fundamental invariants
$(a_{1},\cdots,a_{n})$ to be the unique nondecreasing rearrangement of
$(2b_{1},\cdots,2b_{s},2c_{1}+1,\cdots,2c_{t}+1)$.

  * •

We define the type $t(L)$ of $L$ to be the number of positive fundamental
invariants of $L$. We use $r(L)$ to denote the rank of $L$ and call $L$ a full
type lattice if $r(L)=t(L)$.

  * •

We define the valuation of $L$ to be
$\mathrm{val}(L)\coloneqq\sum_{i=1}^{n}a_{i}$, where $(a_{1},\cdots,a_{n})$
are the fundamental invariants of $L$. For $x\in L$, we define
${\mathrm{val}}(x)={\mathrm{val}}((x,x))$, where ${\mathrm{val}}(\pi_{0})=1$.

  * •

For a hermitian space $\mathbb{V}$, we let
$\mathbb{V}^{?i}\coloneqq\\{x\in\mathbb{V}\mid{\mathrm{val}}(x)?i\\}$ where
$?$ can be $\geq$, $\leq$ or $=$.

  * •

For a ring $R$, we use $\langle\ell_{1},\cdots,\ell_{n}\rangle_{R}$ to denote
$\mathrm{Span}_{R}\\{\ell_{1},\cdots,\ell_{n}\\}$. When $R=O_{F}$, we simply
write $\langle\ell_{1},\cdots,\ell_{n}\rangle$. We use $L_{F}$ to denote
$L\otimes_{O_{F}}F$.

  * •

For a hermitian lattice of rank $n$, we define its sign as

$\chi(L)\coloneqq\chi((-1)^{\frac{n(n-1)}{2}}\mathrm{det}(L))=\pm 1$

where $\chi$ is the quadratic character of $F_{0}^{\times}$ associated to
$F/F_{0}$. Without explicit mentioning, we always use $\epsilon$ to denote
$\chi(L)$.

  * •

Let $I_{m}^{\epsilon}$ denote a unimodular lattice of rank $m$ with
$\chi(I_{m}^{\epsilon})=\epsilon$. We also simply denote a unimodular lattice
of rank $m$ by $I_{m}$ if we do not need to remember its sign or its sign is
clear in the context. In particular, when we consider
$\mathrm{Den}^{\prime}(I_{n},L)$, we mean $I_{n}=I_{n}^{-\epsilon}$.

  * •

We call a sublattice $N\subset M$ primitive in $M$ if
$\mathrm{dim}_{\mathbb{F}_{q}}\overline{N}=r(N)$, where $\overline{N}=(N+\pi
M)/\pi M$. We also use $\overline{L}$ to denote $L\otimes_{O_{F}}O_{F}/(\pi).$

  * •

For two lattices $L,L^{\prime}$ of same rank, let
$n(L^{\prime},L)=\\#\\{L^{\prime\prime}\subset L_{F}\mid L\subset
L^{\prime\prime},L^{\prime\prime}\cong L^{\prime}\\}.$

  * •

We let $\delta_{\mathrm{odd}}(n)=1$ if $n$ is an odd integer and
$\delta_{\mathrm{odd}}(n)=0$ if $n$ is an even integer.

Acknowledgments: C. L. was partially supported by the NSF grant DMS-2101157.
T.Y. was partially supported by by the Dorothy Gollmar Chair’s Fund and Van
Vleck research fund. Q.H. and T.Y. are partially supported by a graduate
school grant of UW-Madison.

##  2\. Krämer models of Rapoport-Zink spaces and special cycles

We denote $\bar{a}$ the Galois conjugate of $a\in F$ over $F_{0}$. Denote by
$\mathrm{Nilp}\,O_{\breve{F}}$ be the category of $O_{\breve{F}}$-schemes $S$
such that $\pi$ is locally nilpotent on $S$. For such an $S$, denote its
special fiber
$S\times_{{\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}}\mathrm{Spec}\,\bar{\kappa}$
by $\bar{S}$. Let $\sigma\in\mathrm{Gal}(\breve{F}_{0}/F_{0})$ be the
Frobenius element. We fix an injection of rings
$i_{0}:O_{F_{0}}\rightarrow\mathcal{O}_{\breve{F}_{0}}$ and an injection
$i:O_{F}\rightarrow O_{\breve{F}}$ extending $i_{0}$. Denote by
$\bar{i}:O_{F}\rightarrow O_{\breve{F}}$ the map $a\mapsto i(\bar{a})$.

###  2.1. RZ spaces

Let $S\in\mathrm{Nilp}\,O_{\breve{F}}$. A $p$-divisible strict
$O_{F_{0}}$-module over $S$ is a $p$-divisible group over $S$ with an
$O_{F_{0}}$-action whose induced action on its Lie algebra is via
$O_{F_{0}}\xrightarrow[]{i_{0}}O_{\breve{F_{0}}}\rightarrow\mathcal{O}_{S}$.

######  Definition 2.1.

A formal hermitian $O_{F}$-module of dimension $n$ over $S$ is a triple
$(X,\iota,\lambda)$ where $X$ is a supersingular $p$-divisible strict
$O_{F_{0}}$-module over $S$ of dimension $n$ and $F_{0}$-height $2n$
(supersingular means the $O_{F_{0}}$-relative Dieudonné module of $X$ at each
geometric point of $S$ has slope $\frac{1}{2}$),
$\iota:O_{F}\rightarrow\operatorname{End}(X)$ is an $O_{F}$-action and
$\lambda:X\rightarrow X^{\vee}$ is a principal polarization in the category of
strict $O_{F_{0}}$-modules such that the Rosati involution induced by
$\lambda$ is the Galois conjugation of $F/F_{0}$ when restricted on $O_{F}$.

######  Definition 2.2.

Fix a formal hermitian $O_{F}$-module
$(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$ of dimension $n$ over
$\bar{\kappa}$. The moduli space $\mathcal{N}_{n}$ is the functor such that
$\mathcal{N}_{n}(S)$ for any $S\in\mathrm{Nilp}\,O_{\breve{F}}$ is the set of
isomorphism classes of quintuples $(X,\iota,\lambda,\rho,\mathcal{F})$ such
that

  1. (1)

$(X,\iota,\lambda)$ is a formal hermitian $O_{F}$-module over $S$.

  2. (2)

$\rho:X\times_{S}\bar{S}\rightarrow\mathbb{X}\times_{\mathrm{Spec}\,\bar{\kappa}}\bar{S}$
is a morphism of formal $O_{F}$-modules of height $0$.

  3. (3)

$\mathcal{F}$ satisfies Krämer’s ([Krä03]) signature condition: it is a local
direct summand of $\operatorname{Lie}X$ of rank $n-1$ as an
$\mathcal{O}_{S}$-module such that $O_{F}$ acts on $\mathcal{F}$ by
$O_{F}\xrightarrow[]{\bar{i}}O_{\breve{F}}\rightarrow\mathcal{O}_{S}$ and acts
on $\operatorname{Lie}X/\mathcal{F}$ by
$O_{F}\xrightarrow[]{i}O_{\breve{F}}\rightarrow\mathcal{O}_{S}$.

An isomorphism between two such quintuples
$(X,\iota,\lambda,\rho,\mathcal{F})$ and
$(X^{\prime},\iota^{\prime},\lambda^{\prime},\rho^{\prime},\mathcal{F}^{\prime})$
is an isomorphism $\alpha:X\rightarrow X^{\prime}$ such that
$\rho^{\prime}\circ(\alpha\times_{S}\bar{S})=\rho$,
$\alpha^{*}(\lambda^{\prime})$ is a $O_{F_{0}}^{\times}$-multiple of $\lambda$
and $\alpha_{*}(\mathcal{F})=\mathcal{F}^{\prime}$.

Notice that $\mathcal{N}_{n}$ is a relative Rapoport-Zink space in the sense
of [Mih22]. By [How19, Proposition 2.2], $\mathcal{N}_{n}$ is representable by
a flat formal scheme of relative dimension $n-1$ over
$\mathrm{Spf}\,O_{\breve{F}}$. We drop the subscript $n$ in $\mathcal{N}_{n}$
when there is no ambiguity.

###  2.2. Associated hermitian spaces

For a strict $O_{F_{0}}$-module $X$ over $\bar{\kappa}$, let $M(X)$ be the
$O_{F_{0}}$-relative Dieudonné module of $X$. Let
$(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$ be the framing object
as in Definition 2.2, and $N=M(\mathbb{X})\otimes_{O_{F_{0}}}F_{0}$ be its
rational relative Dieudonne module. Then $N$ is a $2n$-dimensional
$\breve{F}_{0}$-vector space equipped with a $\sigma$-linear operator
$\mathbf{F}$ and a $\sigma^{-1}$-linear operator $\mathbf{V}$. The
$O_{F}$-action
$\iota_{\mathbb{X}}:O_{F}\rightarrow\operatorname{End}(\mathbb{X})$ induces on
$N$ an $O_{F}$-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 $\Pi$. The principal polarization of $\mathbb{X}$
induces a skew-symmetric $\breve{F}_{0}$-bilinear form $\langle\,,\,\rangle$
on $N$ satisfying

$\langle\mathbf{F}x,y\rangle=\langle
x,\mathbf{V}y\rangle^{\sigma},\quad\langle\iota(a)x,y\rangle=\langle
x,\iota(\bar{a})y\rangle,$

for any $x,y\in N$, $a\in O_{F}$. Then $N$ is an $n$-dimensional
$\breve{F}$-vector space equipped with a $\breve{F}/\breve{F}_{0}$-hermitian
form $(\,,\,)$ defined by (see [Shi18, (2.6)])

(2.1) $(x,y):=\delta(\langle\Pi x,y\rangle+\pi\langle x,y\rangle),$

where $\delta$ is a fixed element in $\mathcal{O}_{\breve{F}_{0}}^{\times}$
such that $\sigma(\delta)=-\delta$. We can use the relation

(2.2) $\langle
x,y\rangle=\frac{1}{2\delta}\operatorname{Tr}_{\breve{F}/\breve{F}_{0}}(\pi^{-1}(x,y)).$

to recover $\langle\,,\,\rangle$. Let $\tau\coloneqq\Pi\mathbf{V}^{-1}$ and
$C\coloneqq N^{\tau}$. Then $C$ is an $F$-vector space of dimension $n$ and
$N=C\otimes_{F_{0}}\breve{F}_{0}$. The restriction of $(\,,\,)$ to $C$ is a
$F/F_{0}$-hermitian form which we still denote by $(\,,\,)$. There are two
choices of $(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$ up to quasi-
isogenies preserving the polarization on the nose, according to the sign
$\epsilon=\chi(C)$ of $C$. Here $\chi:F_{0}^{\times}\rightarrow\\{\pm 1\\}$ is
the character associated to the quadratic extension $F/F_{0}$ and we define
the sign of $C$ as

$\chi(C):=\chi((-1)^{n(n-1)/2}\det(C)).$

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. See [Shi18, Remark 2.16] and [RTW14, Remark
4.2].

Fix a formal hermitian $O_{F}$-module
$(\mathbb{Y},\iota_{\mathbb{Y}},\lambda_{\mathbb{Y}})$ of dimension $1$ over
$\mathrm{Spec}\,\bar{\kappa}$. Define

(2.3)
$\mathbb{V}_{n}=\operatorname{Hom}_{O_{F}}(\mathbb{Y},\mathbb{X})\otimes\mathbb{Q}.$

We drop the subscript $n$ of $\mathbb{V}_{n}$ unless we need to specify the
dimension. The vector space $\mathbb{V}$ is equipped with a hermitian form
$(\,,\,)_{\mathbb{V}}$ such that for any $x,y\in\mathbb{V}$

(2.4) $(x,y)_{\mathbb{V}}=\lambda_{\mathbb{Y}}^{-1}\circ
y^{\vee}\circ\lambda_{\mathbb{X}}\circ
x\in\operatorname{End}(\mathbb{Y})\otimes_{\mathbb{Z}}\mathbb{Q}\overset{\sim}{\rightarrow}F$

where $y^{\vee}$ is the dual quasi-homomorphism of $y$. The hermitian spaces
$(\mathbb{V},(\,,\,)_{\mathbb{V}})$ and $(C,(,))$ are related by the
$F$-linear isomorphism

(2.5) $b:\mathbb{V}\rightarrow C,\quad x\mapsto x(e)$

where $e$ is a generator of $\tau$-fixed points of the $O_{F_{0}}$-relative
Dieudonné module $M(\mathbb{Y})$. The relative Diedonné module $M(\mathbb{Y})$
is equipped with a hermitian form $(\,,\,)_{\mathbb{Y}}$ such that
$(e,e)_{\mathbb{Y}}\in O_{F_{0}}^{\times}$. By [Shi18, Lemma 3.6], we have

(2.6) $(x,x)_{\mathbb{V}}\cdot(e,e)_{\mathbb{Y}}=(b(x),b(x)).$

By scaling the polarization $\lambda_{\mathbb{Y}}$ by a factor in
$O_{F_{0}}^{\times}$ we can assume that

$(e,e)_{\mathbb{Y}}=1,$

so $\mathbb{V}$ and $C$ are isomorphic as hermitian spaces. When the context
is clear we often drop the subscript $\mathbb{V}$ in $(\,,\,)_{\mathbb{V}}$.

###  2.3. Special cycles

We fix a canonical lift
$(\mathcal{G},\iota_{\mathcal{G}},\lambda_{\mathcal{G}})$ of
$(\mathbb{Y},\iota_{\mathbb{Y}},\lambda_{\mathbb{Y}})$ to $O_{\breve{F}}$ in
the sense of [Gro86] such that the action of $O_{F}$ on
$\operatorname{Lie}\mathcal{G}$ is via the inclusion $\bar{i}$. Such lift is
unique up to isomorphism by [How19, Proposition 2.1].

######  Definition 2.3.

For an $O_{F}$-lattice $L$ of $\mathbb{V}$, define $\mathcal{Z}(L)$ to be the
subfunctor of $\mathcal{N}$ such that $\mathcal{N}(S)$ is the set of
isomorphism classes of tuples
$(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{N}(S)$ such that for any $x\in
L$ the quasi-homomorphism

$\rho^{-1}\circ
x\circ\rho_{\mathcal{G}}:\mathbb{Y}\times_{\mathrm{Spec}\,\bar{\kappa}}\bar{S}\rightarrow
X\times_{S}\bar{S}$

entends to a homomorphism
$\mathcal{G}\times_{{\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}}S\rightarrow X$.
By Grothendieck-Messing theory $\mathcal{Z}(L)$ is a closed formal subscheme
of $\mathcal{N}$. For $x\in\mathbb{V}$, we let
$\mathcal{Z}(x)\coloneqq\mathcal{Z}(L)$ when $L=\langle x\rangle$.

###  2.4. Bruhat-Tits stratification of $\mathcal{N}_{\mathrm{red}}$

We say a lattice $\Lambda\subset C$ (resp. $\Lambda\subset\mathbb{V}$) is a
vertex lattice if
$\pi\Lambda^{\sharp}\subseteq\Lambda\subseteq\Lambda^{\sharp}$ where
$\Lambda^{\sharp}$ is dual lattice of $\Lambda$ with respect to $(\,,\,)$
(resp. $(\,,\,)_{\mathbb{V}}$)222Notice that the vertex lattice $\Lambda$ in
the sense of [RTW14], [HSY20] or[HSY21] corresponds to $\Lambda^{\sharp}$ in
our convention.. Using the isomorphism of hermitian spaces (2.5), we often
identify $\Lambda$ with $b^{-1}(\Lambda)$ and use the same notation to denote
both. We call $t=\mathrm{dim}_{\mathbb{F}_{q}}(\Lambda^{\sharp}/\Lambda)$ the
type of $\Lambda$. Recall from [RTW14, Lemma 3.2] that $t$ has to be an even
integer. To each vertex lattice $\Lambda$ of type $2m$, we can associate to it
a subscheme $\mathcal{N}_{\Lambda}$ which is a subscheme of the minuscule
special cycle $\mathcal{Z}(\Lambda)$, see Definition 3.5 below. Let
$V=\Lambda^{\sharp}/\Lambda$, we can define a (modified) Deligne-Lusztig
variety $Y_{V}$ over $\bar{\kappa}$, see (3.2) below. We prove that $Y_{V}$ is
projective and smooth, see Proposition 3.2. When $m\neq 0$ the scheme
$\mathcal{N}_{\Lambda}$ is isomorphic to $Y_{V}$, see Theorem 3.16.

For vertex lattices of type $0$, we define $\mathrm{Exc}_{\Lambda}$ following
the idea of [How19, Appendix].

######  Definition 2.4.

The exceptional divisor $\mathrm{Exc}$ of $\mathcal{N}$ is the set of all
points $z=(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{N}(\bar{\kappa})$ such
that the action

$\iota:O_{F}\rightarrow\operatorname{End}(\operatorname{Lie}X)$

factor through
$O_{F}\xrightarrow[]{i}\mathcal{O}_{\breve{F}}\rightarrow\bar{\kappa}$ where
$\mathcal{O}_{\breve{F}}\rightarrow\bar{\kappa}$ is the quotient map. For a
vertex lattice $\Lambda$ in $C$ of type $0$, define $\mathrm{Exc}_{\Lambda}$
to be is the set of all points
$z=(X,\iota,\lambda,\rho,\mathcal{F})\in\mathrm{Exc}$ such that
$\rho(M(X))=\Lambda\otimes_{O_{F}}O_{\breve{F}}$. Both $\mathrm{Exc}$ and
$\mathrm{Exc}_{\Lambda}$ are closed subset of $\mathcal{N}$ and we endow them
the structure of reduced schemes over $\bar{\kappa}$.

The following is a refinement of [How19, Proposition A.2].

######  Lemma 2.5.

Each $\mathrm{Exc}_{\Lambda}$ is a Cartier divisor of $\mathcal{N}$ isomorphic
to $\mathbb{P}^{n-1}_{\bar{\kappa}}$. The scheme $\mathrm{Exc}$ is a disjoint
union of $\mathrm{Exc}_{\Lambda}$ over all type $0$ lattices $\Lambda$ in $C$.

###### Proof.

Let $z=(X,\iota,\lambda,\rho,\mathcal{F})\in\mathrm{Exc}(\bar{\kappa})$ and
$M=\rho(M(X))\subset N$. Then the action of $\iota(\pi)$ on
$\operatorname{Lie}X$ is trivial. Hence $\Pi M\subset\mathbf{V}M$ as
$\operatorname{Lie}X=M/\mathbf{V}M$. Since
$\dim_{\bar{\kappa}}(M/\mathbf{V}M)=\dim_{\bar{\kappa}}(M/\Pi M)=n$, we know
$\mathbf{V}M=\Pi M$ which is equivalent to $\tau(M)=M$. By [RTW14, Proposition
4.1], $M=\Lambda\otimes_{O_{F}}O_{\breve{F}}$ for some vertex lattice
$\Lambda$. Since $M$ is self-dual, $\Lambda$ is of type $0$. Hence
$z\in\mathrm{Exc}_{\Lambda}(\bar{\kappa})$. Moreover for any
$\bar{\kappa}$-algebra $R$, every rank $n-1$ locally direct summand of
$\operatorname{Lie}X_{R}$ satisfies the Krämer’s signature condition as in
Definition 2.2 and determines a point of $\mathrm{Exc}_{\Lambda}(R)$ uniquely.
So we get an isomorphism
$\mathbb{P}^{n-1}_{\bar{\kappa}}\rightarrow\mathrm{Exc}_{\Lambda}$. Since
$\mathcal{N}$ is regular and $\mathrm{Exc}_{\Lambda}$ has codimension $1$,
$\mathrm{Exc}_{\Lambda}$ is a Cartier divisor in $\mathcal{N}$. By looking at
$\rho(M(X))$, it is clear that
$\mathrm{Exc}_{\Lambda}\cap\mathrm{Exc}_{\Lambda^{\prime}}(k)=\emptyset$ if
$\Lambda\neq\Lambda^{\prime}$. Hence $\mathrm{Exc}$ is a disjoint union of
over all type $0$ lattices $\Lambda$. ∎

######  Remark 2.6.

The proof of Lemma 2.5 shows that the definition of $\mathrm{Exc}_{\Lambda}$
above agrees with that of [HSY21, §2].

By Proposition 3.17 below, $\mathcal{N}_{\Lambda}=\mathrm{Exc}_{\Lambda}$ for
type 0 lattices $\Lambda$. The reduced locus $\mathcal{N}_{\mathrm{red}}$ has
a decomposition (see Theorem 3.19)

$\mathcal{N}_{\mathrm{red}}=\bigcup_{\Lambda}\ \mathcal{N}_{\Lambda},$

where the union is over all vertex lattices. The reduced subscheme
$\mathcal{Z}(L)_{\mathrm{red}}$ is a union of Bruhat-Tits strata (see
Proposition 3.20)

(2.7) $\mathcal{Z}(L)_{\mathrm{red}}=\bigcup_{\Lambda\supset
L}\mathcal{N}_{\Lambda}.$

###  2.5. Horizontal and vertical part

A formal scheme $X$ over $\mathrm{Spf}\,O_{\breve{F}}$ is called horizontal
(resp. vertical) if it is flat over ${\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}$
(resp. $\pi$ is locally nilpotent on $\mathcal{O}_{X}$). For a formal scheme
$X$ over $\mathrm{Spf}\,O_{\breve{F}}$, its horizontal part $X_{\mathscr{H}}$
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 $X_{\mathscr{V}}\subset X$ 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

(2.8) $X=X_{\mathscr{H}}\cup X_{\mathscr{V}}$

as a union of horizontal and vertical formal subschemes. Notice that the
horizontal part $X_{\mathscr{H}}$ is canonically defined while the vertical
part $X_{\mathscr{V}}$ depends on the choice of $m$.

######  Lemma 2.7.

For a lattice $L^{\flat}\subset\mathbb{V}$ of rank greater or equal to $n-1$
with non-degenerate hermitian form, $\mathcal{Z}(L^{\flat})$ is noetherian.

###### Proof.

First we know that $\mathcal{Z}(L)$ is locally noetherian since it is a closed
formal subscheme of $\mathcal{N}$ which is locally noetherian. Since the rank
of $L$ is greater or equal to $n-1$, the number of vertex lattices $\Lambda$
containing $L$ is finite. By (2.7), we know that
$\mathcal{Z}(L)_{\mathrm{red}}$ is a closed subset in finitely many
irreducible components of $\mathcal{N}_{\mathrm{red}}$. Since each irreducible
component of $\mathcal{N}_{\mathrm{red}}$ is quasi-compact, we know that
$\mathcal{Z}(L)$ is quasi-compact, hence noetherian. ∎

######  Lemma 2.8.

For a rank $n-1$ lattice $L^{\flat}\subset\mathbb{V}$ with non-degenerate
hermitian form, $\mathcal{Z}(L)_{\mathscr{V}}$ is supported on the reduced
locus $\mathcal{N}_{\mathrm{red}}$ of $\mathcal{N}$, i.e.,
$\mathcal{O}_{\mathcal{Z}(L)_{\mathscr{V}}}$ is annihilated by a power of the
ideal sheaf of $\mathcal{N}_{\mathrm{red}}$.

###### Proof.

We remark here that $\mathcal{N}_{\mathrm{red}}$ is exactly the supersingular
locus of $\mathcal{N}$. Hence the proof of the lemma is the same as that of
[LZ22a, Lemma 5.1.1]. ∎

###  2.6. 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 $K_{0}^{X}(X)$.
Let $K^{\prime}_{0}(Y)$ be the Grothendieck group of coherent sheaves of
$\mathcal{O}_{Y}$-modules on $Y$. We have a group homomorphism
$K_{0}^{Y}(X)\rightarrow K^{\prime}_{0}(Y)$ which is an isomorphism when $X$
is regular.

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. When
$X$ is regular, we have a cup product $\cdot$ on $K^{Y}_{0}(X)_{\mathbb{Q}}$
defined by tensor product of complexes. Under the identification
$K_{0}^{Y}(X)\xrightarrow{\sim}K^{\prime}_{0}(Y)$, the cup product is nothing
but derived tensor product:

$[A]\cdot[B]=[A\otimes^{\mathbb{L}}_{\mathcal{O}_{X}}B].$

When $X$ is a scheme, the cup product satisfies ([SABK94, Section I.3, Theorem
1.3])

(2.9)
$\mathrm{F}^{i}K_{0}^{Y}(X)_{\mathbb{Q}}\cdot\mathrm{F}^{j}K_{0}^{Y}(X)_{\mathbb{Q}}\subset\mathrm{F}^{i+j}K_{0}^{Y}(X)_{\mathbb{Q}}.$

It is expected that (2.9) is also true when $X$ is a formal scheme, see
[Zha21, (B.3)], however we do not need this fact in this paper. Throughout the
paper, we assume $X=\mathcal{N}$ unless stated otherwise.

Recall that for $x\in\mathbb{V}$, $\mathcal{Z}(x)$ is a Cartier divisor
([How19, Proposition 4.3]).

######  Definition 2.9.

Let $L\subset\mathbb{V}$ be a rank $r$ lattice with a basis
$\\{x_{1},\ldots,x_{r}\\}$. Define ${}^{\mathbb{L}}\mathcal{Z}(L)$ to be

(2.10)
$[\mathcal{O}_{\mathcal{Z}(x_{1})}\otimes_{\mathcal{O}_{N}}^{\mathbb{L}}\cdots\otimes_{\mathcal{O}_{N}}^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x_{r})}]\in
K_{0}^{\mathcal{Z}(L)}(\mathcal{N})$

where $\otimes^{\mathbb{L}}$ is the derived tensor product of complexes of
coherent locally free sheaves on $\mathcal{N}$. By [How19, Corollary C],
${}^{\mathbb{L}}\mathcal{Z}(L)$ is independent of the choice of the basis
$\\{x_{1},\ldots,x_{r}\\}$.

######  Definition 2.10.

When $L$ has rank $n$, we define the intersection number

(2.11) $\mathrm{Int}(L)=\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L)),$

where $\chi$ is the Euler characteristic.

######  Lemma 2.11.

When $L$ is a rank $n$ lattice in $\mathbb{V}$, $\mathcal{Z}(L)$ is a proper
scheme over ${\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}$. In particular,
$\mathrm{Int}(L)$ is finite.

###### Proof.

By Lemma 2.8 $\mathcal{Z}(L)_{\mathscr{V}}$ is a scheme. We show that
$\mathcal{Z}(L)_{\mathscr{H}}$ is empty. If not, there exists
$z\in\mathcal{Z}(L)(\mathcal{O}_{K})$ for some finite extension $K$ of
$\breve{F}$. Let $X$ be the corresponding formal hermitian $O_{F}$-module of
signature $(1,n-1)$ over $\mathcal{O}_{K}$. Since $L$ has rank $n$ and
$\mathcal{G}$ has signature $(0,1)$, this would imply that $X$ has signature
$(0,n)$, which is a contradiction. Hence $\mathcal{Z}(L)$ is a scheme. Since
$\mathcal{Z}(L)_{\mathrm{red}}$ is contained in finitely many irreducible
components of $\mathcal{N}_{\mathrm{red}}$ and each irreducible component is
proper over $\mathrm{Spec}\,\bar{\kappa}$, it follows that $\mathcal{Z}(L)$ is
proper over ${\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}$. The finiteness of
$\operatorname{\mathrm{Int}}(L)$ then follows from the same argument before
[Zha21, (B.4)]. ∎

###  2.7. A geometric cancellation law

Recall that for two lattices $L,L^{\prime}\subset\mathbb{V}$ of rank $n$, we
define

$n(L^{\prime},L)=\\#\\{L^{\prime\prime}\subset L_{F}\mid L\subset
L^{\prime\prime},L^{\prime\prime}\cong L^{\prime}\\}.$

Also recall that $\delta_{\mathrm{odd}}(n)=1$ or $0$ depending on whether $n$
is odd or not.

######  Proposition 2.12.

Let $L=I_{\ell}\obot L_{2}\subset\mathbb{V}$ where $L_{2}$ is of rank $r$,
$I_{\ell}$ is unimodular of rank $\ell$ and $n=\ell+r$. Let $I_{r}$ be a
unimodular lattice that contains $L_{2}$. Then

(2.12) $\displaystyle\operatorname{\mathrm{Int}}(I_{\ell}\obot
L_{2})-\operatorname{\mathrm{Int}}(L_{2})=n(I_{r},L_{2})\cdot(\delta_{\mathrm{odd}}(n)-\delta_{\mathrm{odd}}(r)).$

Moreover,

(2.13)
$\displaystyle\operatorname{\mathrm{Int}}(I_{n})=\delta_{\mathrm{odd}}(n).$

###### Proof.

If $L_{2}$ is unimodular and $r=2$, then
$\operatorname{\mathrm{Int}}(L_{2})=0$ by [Shi20, Theorem 1.3] and [HSY20,
Theorem 1.3]. Combining this with (2.12), we obtain (2.13). In order to prove
(2.12), we prove the following equation,

(2.14) $\operatorname{\mathrm{Int}}(I_{1}\obot
L_{2})-\operatorname{\mathrm{Int}}(L_{2})=(-1)^{r}n(I_{r},L_{2}).$

which is the special case of (2.12) when $\ell=1$. The general case then
follows from an easy induction on $n$ using (2.14) and the fact

(2.15) $n(I_{n},I_{\ell}\obot L_{2})=n(I_{r},L_{2}).$

By Proposition 3.2 of [HSY21], we have the following decomposition of Cartier
divisors on $\mathcal{N}_{n}$

$\mathcal{Z}(I_{1})=\tilde{\mathcal{Z}}(I_{1})+\sum_{\Lambda_{0}\supset
I_{1}}\mathrm{Exc}_{\Lambda_{0}},$

where the summation is over vertex lattices of type $0$ in $\mathbb{V}_{n}$
and $\tilde{\mathcal{Z}}(I_{1})\cong\mathcal{N}_{n-1}$ by [HSY21, Corollary
2.7]. By the same corollary, we know that

$\chi(\mathcal{N}_{n},[\mathcal{O}_{\tilde{\mathcal{Z}}(I_{1})}]\cdot{}^{\mathbb{L}}\mathcal{Z}(L_{2}))=\chi(\mathcal{N}_{r},{}^{\mathbb{L}}\mathcal{Z}(L_{2}))=\operatorname{\mathrm{Int}}(L_{2}).$

Hence we have

$\operatorname{\mathrm{Int}}(L)-\operatorname{\mathrm{Int}}(L_{2})=\sum_{\Lambda_{0}\supset
I_{1}}\chi(\mathcal{N}_{n},[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}]\cdot{}^{\mathbb{L}}\mathcal{Z}(L_{2})).$

If $L_{2}\not\subset\Lambda_{0}$, then
$\mathrm{Exc}_{\Lambda_{0}}\cap\mathcal{Z}(L_{2})$ is empty by Proposition
3.20 below. If $L_{2}\subset\Lambda_{0}$, then by [HSY21, Corollary 3.6], we
have

$\chi(\mathcal{N}_{n},[\mathcal{O}_{\mathrm{Exc}_{\Lambda_{0}}}]\cdot{}^{\mathbb{L}}\mathcal{Z}(L_{2}))=(-1)^{r}.$

Hence

$\operatorname{\mathrm{Int}}(L)-\operatorname{\mathrm{Int}}(L_{2})=\sum_{\Lambda_{0}\supset
I_{1}\obot L_{2}}(-1)^{r}.$

Combining this with (2.15) finishes the proof of (2.14) and the proposition. ∎

##  3\. Bruhat–Tits Stratification of Krämer models

We prove a Bruhat–Tits stratification for the Krämer model (Theorem 3.19),
analogous to the case of the Pappas model treated in [RTW14]. More precisely,
we define closed subschemes $\mathcal{N}_{\Lambda}$ (Definition 3.5) and show
that the reduced locus of $\mathcal{N}$ is stratified by
$\mathcal{N}_{\Lambda}$ (Theorem 3.19). From this stratification we obtain a
stratification of the reduced locus of $\mathcal{Z}(L)$ (Proposition 3.20). We
also show that $\mathcal{N}_{\Lambda}$ is isomorphic to the (modified)
Deligne-Lusztig variety $Y_{V,\bar{\kappa}}$ defined in §3.1 (Theorem 3.16),
and is in particular a smooth projective variety over $\bar{\kappa}$. We
remark here that for the purpose of our main result (Theorem 9.7), only a
weeker version of Proposition 3.20 is needed (namely we do not need the
reducedness of $\mathcal{N}_{\Lambda}$). However we believe the rest of this
section contributes to the theory of Rapoport-Zink space and is of independent
interest.

###  3.1. Deligne-Lusztig varieties

Through out this subsection we assume $m\geq 1$. Let $V$ be a $2m$-dimensional
symplectic space over $\kappa=\mathbb{F}_{q}$ equipped with the symplectic
form $\langle\,,\,\rangle$. Let
$V_{\bar{\kappa}}=V\otimes_{\kappa}\bar{\kappa}$ and denote the bilinear
extension of $\langle\,,\,\rangle$ to $V_{\bar{\kappa}}$ still by
$\langle\,,\,\rangle$. Let $\mathrm{Gr}(i,V)$ be the Grassmannian variety
parametrizing rank $i$ locally direct summands of $V_{R}$ for any
$\kappa$-algebra $R$. Let $L_{V}$ be the subvariety of $\mathrm{Gr}(m,V)$
whose $R$-points are rank $m$ locally direct summands of $V_{R}$ that are
Lagrangian. Let $S_{V}$ be the subvariety of $L_{V}$ as in [RTW14, Equation
(5.3)] whose $\bar{\kappa}$-points are specified by

(3.1) $S_{V}(\bar{\kappa})=\\{U\in
L_{V}(\bar{\kappa})\mid\dim(U\cap\Phi(U))\geq m-1\\},$

where $\Phi$ is the Frobenius endormophism. By [RTW14, Proposition 5.3] and
its remark, $S_{V}$ has isolated singularities which are exactly the
$\kappa$-points of $L_{V}$. By Proposition 5.5 of loc.cit., $S_{V}$ is
irreducible of dimension $m$. To resolve the singularities of $S_{V}$, define
$Y_{V}$ to be the subvariety of $L_{V}\times\mathrm{Gr}(m-1,V)$ whose
$\bar{\kappa}$-points are specified by

(3.2)
$Y_{V}(\bar{\kappa})=\\{(U,U^{\prime})\in(L_{V}\times\mathrm{Gr}(m-1,V))(\bar{\kappa})\mid
U^{\prime}\subseteq U\cap\Phi(U)\\}.$

Then the variety $Y_{V}$ is a projective subscheme of
$\mathrm{Gr}(m,V)\times\mathrm{Gr}(m-1,V)$. The forgetful map
$(U,U^{\prime})\rightarrow U$ defines a morphism $\pi_{m}:Y_{V}\rightarrow
S_{V}$.

######  Lemma 3.1.

The morphism $\pi_{m}$ is a projective morphism. It is an isomorphism outside
the singular locus of $S_{V}$. For a singular point $z$ of $S_{V}$,
$\pi^{-1}_{m}(z)\cong\mathbb{P}^{m-1}_{\bar{\kappa}}$.

###### Proof.

First we know $\pi_{m}$ is projective as it is a morphism between projective
schemes. Consider a $\bar{\kappa}$-point $z=U$ of $S_{V}$ outside its singular
locus. Then $U\cap\Phi(U)$ has dimension $m-1$, this entails
$U^{\prime}=U\cap\Phi(U)$. In fact for any affine open subset
$\mathrm{Spec}\,R$ of $S_{V}$ outside the singular locus, define
$U^{\prime}_{R}:=U_{R}\cap\mathrm{Span}_{R}\\{\Phi(U_{R})\\}\in\mathrm{Gr}(m-1,V_{R})$
where $U_{R}\in L_{V}(R)$ is the universal object over $\mathrm{Spec}\,R$. The
map $U_{R}\mapsto(U_{R},U^{\prime}_{R})$ defines the inverse morphim of
$\pi_{m}$ outside the singular locus of $S_{V}$.

If $z=U$ is a singular $\bar{\kappa}$-point, then $U=\Phi(U)$ and $U^{\prime}$
can be any element in
$\mathrm{Gr}(m-1,U)\cong\mathbb{P}^{m-1}_{\bar{\kappa}}$. This finishes the
proof of the lemma. ∎

######  Proposition 3.2.

The projective variety $Y_{V}$ is smooth of dimension $m$.

###### Proof.

By Lemma 3.1 and the fact that $S_{V}$ has dimension $m$, it suffices to show
that the tangent space $T_{z}(Y_{V})$ of $Y_{V}$ at $z$ has dimension less or
equal to $m$ at any point $z=(U,U^{\prime})$ of $Y_{V}(\bar{\kappa})$ such
that $U=\Phi(U)$ is rational (a singlular point of $S_{V}$). We choose a
$\kappa$-basis $\\{e_{1},\ldots,e_{2m}\\}$ of $V$ such that

$U=\mathrm{Span}_{\bar{\kappa}}\\{e_{1},\ldots,e_{m}\\},\ \langle
e_{i},e_{m+j}\rangle=\delta_{ij},\quad 1\leq i,j\leq m,$

and all other pairings between basis vectors are zero. We assume that
$U^{\prime}$ is the hyperplane of $U$ defined by

$\sum_{i=1}^{m}b_{i}e_{i}^{*}=0$

where $\\{e_{i}^{*}\\}_{i=1}^{m}$ is the dual basis of $\\{e_{i}\\}_{i=1}^{m}$
and $b_{i}$ are not all zero. Without loss of generality we can assume
$b_{m}=1$.

Let $\tilde{V}:=V\otimes_{\kappa}(\bar{\kappa}[\epsilon]/(\epsilon^{2}))$ and
$\tilde{e}_{i}:=e_{i}\otimes 1\in\tilde{V}$. The tangent space $T_{z}(Y_{V})$
can be identified with
$\operatorname{Hom}(\mathcal{O}_{Y_{V},z},\bar{\kappa}[\epsilon]/(\epsilon^{2}))$
which parameterizes lifts of $(U,U^{\prime})$. A lift
$(\tilde{U},\tilde{U}^{\prime})$ of $(U,U^{\prime})$ to $T_{z}(Y_{V})$ can be
determined as follows. First $\tilde{U}$ can be spanned by vectors
$x_{1},\ldots,x_{m}$ defined by

$(x_{1},\ldots,x_{m})=(\tilde{e}_{1},\ldots,\tilde{e}_{2m})\begin{pmatrix}I_{m}\\\
A\epsilon\end{pmatrix}$

where $A\in M_{m\times m}(\bar{\kappa})$ is symmetric as $U$ is Lagrangian.
The filtration $\tilde{U}^{\prime}$ is a hyperplane of $\tilde{U}$ defined by

$\sum_{i=1}^{m}\tilde{b}_{i}x_{i}^{*}=0$

where $\tilde{b}_{i}=b_{i}+\epsilon c_{i}$ for $1\leq i\leq m-1$ and
$\tilde{b}_{m}=1$ where $c_{i}\in\bar{\kappa}$. Then $\tilde{U}^{\prime}$ is
spanned by vectors

$x_{j}-\tilde{b}_{j}x_{m}=\tilde{e}_{j}-\tilde{b}_{j}\tilde{e}_{m}+\epsilon\sum_{i=1}^{m}(a_{ij}-\tilde{b}_{j}a_{im})\tilde{e}_{i+m},\
1\leq j\leq m-1.$

Since $\Phi(\epsilon)=\epsilon^{q}=0$, $\Phi(\tilde{U})$ is spanned by

$(\tilde{e}_{1},\ldots,\tilde{e}_{2m})\begin{pmatrix}I_{m}\\\
0\end{pmatrix}=(\tilde{e}_{1},\ldots,\tilde{e}_{m}).$

Hence the condition $\tilde{U}^{\prime}\subset\Phi(\tilde{U})$ is equivalent
to

$a_{ij}=\tilde{b}_{j}a_{im},\ 1\leq i\leq m,1\leq j\leq m-1.$

In particular $a_{mj}=\tilde{b}_{j}a_{mm}$. Together with the fact
$A={}^{t}A$, we can deduce $a_{jm}=a_{mj}=\tilde{b}_{j}a_{mm}$ for $1\leq
j\leq m$. Hence

$a_{ij}=\tilde{b}_{j}a_{im}=\tilde{b}_{j}\tilde{b}_{i}a_{mm}$

for all $1\leq i,j\leq m$. In conclusion, a point in
$\operatorname{Hom}(\mathcal{O}_{Y_{V},z},\bar{\kappa}[\epsilon]/(\epsilon^{2}))$
is determined by

$c_{1},\ldots,c_{m-1},a_{mm}.$

This shows that $T_{z}(Y_{V})$ of $Y_{V}$ at $z$ has dimension less or equal
to $m$. ∎

######  Remark 3.3.

One can show that $Y_{V}$ is in fact the blow-up of $S_{V}$ along its singular
locus.

For later use, we record the following lemma.

######  Lemma 3.4.

Let $U_{0}\in L_{V}(\bar{\kappa})$. Define a subvariety $L_{V}^{U_{0}}$ of
$L_{V,\bar{\kappa}}$ by specifying its $\bar{\kappa}$-points

$L_{V}^{U_{0}}(\bar{\kappa})=\\{U\in
L_{V}(\bar{\kappa})\mid\dim_{\bar{\kappa}}(U\cap U_{0})\geq m-1\\}.$

Then $\dim_{\bar{\kappa}}T_{z}(L_{V}^{U_{0}})\leq m$ for any $z\in
L_{V}^{U_{0}}(\bar{\kappa})-\\{U_{0}\\}$.

###### Proof.

Fix a point $z=U\in L_{V}^{U_{0}}(\bar{\kappa})-\\{U_{0}\\}$. Take a basis
$\\{e_{1},\ldots,e_{m-1}\\}$ of the isotropic subspace $U\cap U_{0}$. Enlarge
it to a basis $\\{e_{1},\ldots,e_{m}\\}$ of $U$, and further to a basis
$\\{e_{1},\ldots,e_{2m}\\}$ of $V_{\bar{\kappa}}$. We assume that $\langle
e_{i},e_{m+j}\rangle=\delta_{ij}$ for $1\leq i,j\leq m$ and all other pairings
between these basis vectors are zero. This implies that
$U_{0}=\mathrm{Span}_{\bar{\kappa}}\\{e_{1},\ldots,e_{m-1},ae_{m}+be_{2m}\\}$
for some $a,b\in\bar{\kappa}$. By further adjust basis we can assume $a=0$,
$b=1$.

Let $\tilde{V}:=V\otimes_{\kappa}(\bar{\kappa}[\epsilon]/(\epsilon^{2}))$ and
$\tilde{U}_{0}:=U_{0}\otimes_{\bar{\kappa}}(\bar{\kappa}[\epsilon]/(\epsilon^{2}))$.
Let $\tilde{e}_{i}:=e_{i}\otimes 1\in\tilde{V}$. The tangent space
$T_{z}(L_{V}^{U_{0}})$ can be identified with
$\operatorname{Hom}(\mathcal{O}_{L_{V}^{U_{0}},z},\bar{\kappa}[\epsilon]/(\epsilon^{2}))$
which parameterizes lifts of $U$ such that
$\mathrm{rank}_{\bar{\kappa}[\epsilon]/(\epsilon^{2})}(U\cap U_{0})\geq m-1$.
A lift $\tilde{U}$ of $U$ to $T_{z}(L_{V}^{U_{0}})$ can be parameterized as
follows. First $\tilde{U}$ can be spanned by vectors $x_{1},\ldots,x_{m}$
defined by

$(x_{1},\ldots,x_{m})=(\tilde{e}_{1},\ldots,\tilde{e}_{2m})\begin{pmatrix}I_{m}\\\
A\epsilon\end{pmatrix}$

where $A\in M_{m}(\bar{\kappa})$ and $A={}^{t}A$ as $\tilde{U}$ is Lagrangian.
A hyperplane in $\tilde{U}_{0}$ lifting $U\cap U_{0}$ is defined by one single
equation

$\tilde{e}_{2m}^{*}-\sum_{j=1}^{m-1}\epsilon b_{j}\tilde{e}_{j}^{*}=0,$

where $b_{j}\in\bar{\kappa}$. Hence $\tilde{U}\cap\tilde{U}_{0}$ is spanned by
vectors

$\tilde{e}_{j}+\epsilon b_{j}\tilde{e}_{2m},\ 1\leq j\leq m-1.$

If in addition the above vectors are in $\tilde{U}$, there must exists
$\lambda_{j}^{i}\in\bar{\kappa}[\epsilon]/(\epsilon^{2})$ ($1\leq i\leq m$,
$1\leq j\leq m-1$) such that

$\tilde{e}_{j}+\epsilon
b_{j}\tilde{e}_{2m}=\sum_{i=1}^{m}\lambda_{j}^{i}x_{i}=\sum_{i=1}^{m}\lambda_{j}^{i}(\tilde{e}_{i}+\sum_{l=1}^{m}\epsilon
a_{li}\tilde{e}_{m+l}).$

Compare coefficients we see that $\lambda_{j}^{i}=\delta_{ij}$ and

$a_{mj}=b_{j},\ a_{li}=0,$

for $1\leq j\leq m-1$, $1\leq l,i\leq m-1$. Hence a point in
$\operatorname{Hom}(\mathcal{O}_{L_{V}^{U_{0}},z},\bar{\kappa}[\epsilon]/(\epsilon^{2}))$
is determined by

$b_{1},\ldots,b_{m-1},a_{mm},$

Hence the tangent space $T_{z}(L_{V}^{U_{0}})$ has dimension $\leq m$. ∎

###  3.2. Minuscule cycle $\mathcal{N}_{\Lambda}$ and its tangent space

In this subsection, we identify $\Lambda$ with $b^{-1}(\Lambda)$ using the
isomorphism of hermitian spaces (2.5).

######  Definition 3.5.

For a vertex lattice $\Lambda\subset\mathbb{V}$ of type $t(\Lambda)=2m$,
define the subfunctor $\mathcal{N}_{\Lambda}$ to be the subfunctor of
$\mathcal{N}$ such that for a $O_{\breve{F}}$-scheme $S$,
$\mathcal{N}_{\Lambda}(S)$ is the set of isomorphism classes of tuples
$(X,\iota,\lambda,\rho,\mathcal{F})$ satisfying the following conditions.

  1. (1)

$(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{Z}(\Lambda)(S)$.

  2. (2)

If $m\geq 1$, we require in addition that
$x_{*}(\operatorname{Lie}(\mathcal{G}\times_{{\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}}S))\subset\mathcal{F}$
for any $x\in\Lambda$.

We first describe the $\bar{\kappa}$-points of $\mathcal{N}$ and
$\mathcal{N}_{\Lambda}$.

######  Proposition 3.6.

There is a bijection between $\mathcal{N}_{\mathrm{red}}(\bar{\kappa})$ and
the set of pairs of $\mathcal{O}_{\breve{F}}$-lattices $(M,M^{\prime})$ in $N$
satisfying

$M^{\sharp}=M,\quad\Pi\tau(M)\subset M\subset\Pi^{-1}\tau(M),\quad
M^{\prime}\subset\tau^{-1}(M)\cap M,\quad\hbox{and
}\quad\mathrm{length}(M/M^{\prime})=1.$

###### Proof.

Let $(X,\iota,\lambda,\rho,\mathcal{F})$ be a $\bar{\kappa}$-point of
$\mathcal{N}$ and $M(X)$ be the Dieudonné module of $X$. Define
$M=\rho(M(X))\subset N$ and $M^{\prime}=\rho(\Pr^{-1}(\mathcal{F}))\subset N$
where $\Pr:M(X)\rightarrow\operatorname{Lie}X=M(X)/\mathbf{V}M(X)$ is the
natural quotient map. The condition $M^{\sharp}=M$ is equivalent to the fact
that $\lambda$ is a principal polarization. The condition $\Pi\tau(M)\subset
M\subset\Pi^{-1}\tau(M)$ is equivalent to $\pi_{0}M\subset\mathbf{V}M\subset
M$. The condition $M^{\prime}\subset\tau^{-1}(M)\cap M$ and
$\mathrm{length}_{\bar{\kappa}}(M/M^{\prime})=1$ is equivalent to the
condition

$\mathbf{V}M\subset M^{\prime}\subset M,\ \Pi M^{\prime}\subset\mathbf{V}M,\
\dim_{\bar{\kappa}}(M/M^{\prime})=1.$

Hence we have translated all conditions in the definition of $\mathcal{N}$ in
term of Dieudonné modules. The proposition now follows from Dieudonné theory.
∎

For a vertex lattice $\Lambda$ in $C$, define

(3.3) $\breve{\Lambda}:=\Lambda\otimes_{O_{F}}O_{\breve{F}},\
\breve{\Lambda}^{\sharp}:=\Lambda^{\sharp}\otimes_{O_{F}}O_{\breve{F}}.$

######  Corollary 3.7.

Let $\Lambda$ be a vertex lattice in $\mathbb{V}$. There is a bijection
between $\mathcal{N}_{\Lambda}(\bar{\kappa})$ and the set of pairs of
$\mathcal{O}_{\breve{F}}$-lattices $(M,M^{\prime})$ in $N$ satisfying the
conditions in Proposition 3.6 and the following condition.

  1. (1)

If $t(\Lambda)=0$, then $M=\breve{\Lambda}$.

  2. (2)

If $t(\Lambda)\geq 2$, then $\breve{\Lambda}\subset M^{\prime}\subset M$.

###### Proof.

The condition that $\breve{\Lambda}\subset M$ is a direct translation of
Condition (1) in Definition 3.5. When $t(\Lambda)=0$, both $M$ and
$\breve{\Lambda}$ are self-dual, and thus $M=\breve{\Lambda}$. (2) is a direct
consequence of Condition (2) in Definition 3.5. ∎

To study the tangent space of $\mathcal{N}_{\Lambda}$, we recall the
Grothendieck-Messing deformation theory of $\mathcal{N}$ from [How19, §3]. We
remark here that although [How19] deals with the case $F_{0}=\mathbb{Q}_{p}$,
the argument in fact applies to general $F_{0}$ using the relative display
theory of [ACZ16]. Let $R\in\mathrm{Nilp}\,O_{\breve{F}}$. For a strict
$O_{F_{0}}$-module $X$ over $\mathrm{Spec}\,R$, we denote by $D^{\prime}(X)$
the Lie algebra of the universal vector extension of $X$ (a vector extension
of $X$ is an extension of $X$ by $\mathbb{G}_{a}^{n}$). Denote by $D(X)$ the
subspace of $D^{\prime}(X)$ on which $O_{F_{0}}$ acts by the structural
morphism $O_{F_{0}}\rightarrow\mathcal{O}_{S}$. A point $z\in\mathcal{N}(R)$
corresponds to a strict $O_{F_{0}}$-module $(X,\iota,\lambda)$ over $R$
together with filtration $\mathcal{F}\subset\operatorname{Lie}X$ satisfying
Definition 2.2. We have the following exact sequence of locally free
$R$-modules

(3.4) $0\rightarrow\mathrm{Fil}(X)\rightarrow
D(X)\rightarrow\operatorname{Lie}X\rightarrow 0,$

where $\mathrm{Fil}(X)$ and $\operatorname{Lie}X$ are of rank $n$ and $D(X)$
is of rank $2n$. The principal polarization $\lambda$ induces a symplectic
form $\langle\,,\,\rangle$ on $D(X)$ such that

$\langle\iota(a)x,y\rangle=\langle x,\iota(\bar{a})y\rangle$

for all $a\in O_{F}$ and $x,y\in D(X)$. With respect to $\langle\,,\,\rangle$
the Hodge filtration $\mathrm{Fil}(X)$ is maximal isotropic. Hence
$\langle\,,\,\rangle$ induces a perfect pairing (still denoted by
$\langle\,,\,\rangle$):

(3.5) $\langle\,,\,\rangle:\mathrm{Fil}(X)\times\operatorname{Lie}X\rightarrow
R.$

The submodule $\mathcal{F}\subset\operatorname{Lie}X$ and its perpendicular
complement $\mathcal{F}^{\bot}$ (which is locally a direct summand of
$\mathrm{Fil}(X)$ of rank one) with respect to (3.5) determine each other. The
condition on $\mathcal{F}$ in Definition 2.2 is

(3.6) $O_{F}\text{ acts on }\mathcal{F}\text{ by
}O_{F}\xrightarrow[]{\bar{i}}O_{\breve{F}}\rightarrow\mathcal{O}_{S}\text{ and
on }\operatorname{Lie}X/\mathcal{F}\text{ by
}O_{F}\xrightarrow[]{i}O_{\breve{F}}\rightarrow\mathcal{O}_{S}.$

This is equivalent to the condition that $O_{F}$ acts on $\mathcal{F}^{\bot}$
by $O_{F}\xrightarrow[]{\bar{i}}O_{\breve{F}}\rightarrow\mathcal{O}_{S}$ and
on $\mathrm{Fil}(X)/\mathcal{F}^{\bot}$ by
$O_{F}\xrightarrow[]{i}O_{\breve{F}}\rightarrow\mathcal{O}_{S}$. Since
$O_{F_{0}}$ acts on $D(X)$ by $i_{0}$ and $O_{F}=O_{F_{0}}[\pi]$, (3.6) is
further equivalent to

(3.7) $(\Pi+\pi)\cdot\mathcal{F}^{\bot}=0,\
(\Pi-\pi)\cdot\mathrm{Fil}(X)\subset\mathcal{F}^{\bot},$

where we use $\Pi$ to denote the action $\iota(\pi)$ on $D(X)$.

######  Definition 3.8.

Let $\mathscr{C}$ be the following category. Objects in $\mathscr{C}$ are
triples $(\mathcal{O},\mathcal{O}\rightarrow\bar{\kappa},d)$ where
$\mathcal{O}$ is an Artinian $O_{\breve{F}}$-algebra,
$\mathcal{O}\rightarrow\bar{\kappa}$ is an $O_{\breve{F}}$-algebra
homomorphism, and $d$ is a nilpotent divided power structure on
$\mathrm{Ker}(\mathcal{O}\rightarrow\bar{\kappa})$. Morphisms in $\mathscr{C}$
are $O_{\breve{F}}$-algebra homomorphisms compatible with structure maps to
$\bar{\kappa}$ and divided power structures.

Let
$z=(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{Z}(\Lambda)(\bar{\kappa})$
and $M=\rho(M(X))\subset N$. Then $\Lambda\subset M$ by Corollary 3.7. We can
identify (3.4) with

$0\rightarrow\mathbf{V}M/\pi_{0}M\rightarrow M/\pi_{0}M\rightarrow
M/\mathbf{V}M\rightarrow 0.$

Let $\mathcal{F}^{\bot}\subset\mathbf{V}M/\pi_{0}M$ be the perpendicular
complement of $\mathcal{F}$ as described above. Denote by
$\mathcal{Z}(\Lambda)_{z}$ (resp. $\mathcal{N}_{\Lambda,z}$) the completion of
$\mathcal{Z}(\Lambda)$ (resp. $\mathcal{N}_{\Lambda}$) at $z$. For any
$\mathcal{O}\in\mathscr{C}$ and
$\tilde{z}=(\tilde{X},\cdots)\in\mathcal{Z}(\Lambda)_{z}(\mathcal{O})$, we can
identify $D(\tilde{X})$ with
$M_{\mathcal{O}}:=M\otimes_{O_{\breve{F}_{0}}}\mathcal{O}$ and by
Grothendieck-Messing theory $\tilde{z}$ corresponds to a filtration of free
$\mathcal{O}$-module direct summands

$\tilde{\mathcal{F}}^{\bot}\subset\widetilde{\mathrm{Fil}}\subset
M_{\mathcal{O}},$

which lifts the filtration $\mathcal{F}^{\bot}\subset\mathrm{Fil}\subset
M_{\bar{\kappa}}=M/\pi_{0}M$. Let $f_{\mathcal{O}}$ be the map

(3.8)
$f_{\mathcal{O}}:\tilde{z}\mapsto(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}}).$

######  Lemma 3.9.

Let the notations be as above. Denote by $\Lambda_{M,\mathcal{O}}$ the image
of the composition of maps $\breve{\Lambda}\rightarrow M\rightarrow
M_{\mathcal{O}}$, and let $\Lambda_{M,\mathcal{O}}^{\bot}$ be its
perpendicular complement in $M_{\mathcal{O}}$ under the alternating form
$\langle\,,\,\rangle$.

  1. (1)

The map $f_{\mathcal{O}}$ defines a bijection from
$\mathcal{Z}(\Lambda)_{z}(\mathcal{O})$ to the set consisting of pairs
$(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}})$ lifting
$(\mathcal{F}^{\bot},\mathrm{Fil})$ satisfying the following conditions:

    1. (a)

$\tilde{\mathcal{F}}^{\bot}$ and $\widetilde{\mathrm{Fil}}$ are free
$\mathcal{O}$-module direct summands of $M_{\mathcal{O}}$ of rank $1$ and $n$
respectively and $\tilde{\mathcal{F}}^{\bot}\subset\widetilde{\mathrm{Fil}}$;

    2. (b)

$\widetilde{\mathrm{Fil}}$ is isotropic with respect to $\langle\,,\,\rangle$;

    3. (c)

$(\Pi+\pi)\cdot\tilde{\mathcal{F}}^{\bot}=0$ and
$(\Pi-\pi)\cdot\widetilde{\mathrm{Fil}}\subset\tilde{\mathcal{F}}^{\bot}$;

    4. (d)

$\widetilde{\mathrm{Fil}}$ contains
$\widetilde{\mathrm{Fil}}^{-}:=(\Pi+\pi)\cdot\Lambda_{M,\mathcal{O}}$.

  2. (2)

The restriction of $f_{\mathcal{O}}$ to $\mathcal{N}_{\Lambda,z}(\mathcal{O})$
defines a bijection from $\mathcal{N}_{\Lambda,z}(\mathcal{O})$ to the set
consisting of pairs $(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}})$
satisfying the above conditions together with the extra condition:

    1. (e)

$\tilde{\mathcal{F}}^{\bot}\subset\Lambda_{M,\mathcal{O}}^{\bot}$.

###### Proof.

Proof of (1): By the previous discussion,
$(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}})$ satisfies conditions
(a), (b) and (c) for any $\tilde{z}\subset\mathcal{N}_{z}(\mathcal{O})$
($\mathcal{N}_{z}$ is the completion of $\mathcal{N}$ at $z$). Conversely by
Grothendieck-Messing theory any pair
$(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}})$ lifting
$(\mathcal{F}^{\bot},\mathrm{Fil})$ satisfying (a), (b) and (c) gives rise to
a unique point $\tilde{z}\in\mathcal{N}_{z}(\mathcal{O})$. Since the action of
$O_{F}$ on $\operatorname{Lie}\mathcal{G}$ is via the inclusion $\bar{i}$, the
Hodge filtration of $\mathcal{G}_{\mathcal{O}}$ is
$\mathrm{Span}_{\mathcal{O}}\\{(\Pi+\pi)\cdot e\otimes 1\\}$ where $e$ is a
generator of $M(\mathbb{Y})$ as in §2.2. The image of the
$\mathrm{Span}_{\mathcal{O}}\\{(\Pi+\pi)\cdot e\otimes 1\\}$ under $\Lambda$
in $M_{\mathcal{O}}$ is exactly $\widetilde{\mathrm{Fil}}^{-}$. By
Grothendick-Messing theory again,
$\tilde{z}\in\mathcal{Z}(\Lambda)_{z}(\mathcal{O})$ if and only if condition
(d) holds.

(2) is a corollary of (1). For any
$\tilde{z}=(\tilde{X},\ldots,\tilde{\mathcal{F}})\in\mathcal{Z}(\Lambda)_{z}(\mathcal{O})$,
let $\tilde{\mathcal{F}}^{\prime}$ be the preimage of $\tilde{\mathcal{F}}$
under the quotient map $M_{\mathcal{O}}\rightarrow
M_{\mathcal{O}}/\widetilde{\mathrm{Fil}}$. Condition (2) in Definition 3.5 is
equivalent to $\Lambda_{M,\mathcal{O}}\subset\tilde{\mathcal{F}}^{\prime}$.
The perpendicular complement of $\tilde{\mathcal{F}}^{\prime}$ with respect to
$\langle\,,\,\rangle$ is $\tilde{\mathcal{F}}^{\bot}$. Hence condition (2) in
Definition 3.5 is equivalent to condition (e). Hence
$\tilde{z}\in\mathcal{N}_{\Lambda,z}(\mathcal{O})$ if and only if (e) is
satisfied. This finishes the proof of the lemma. ∎

######  Lemma 3.10.

Let $\Lambda$ be a vertex lattice of type $2m$ in $C$ and $M\subset N$ be an
$O_{\breve{F}}$ lattice such that $\Lambda\subset M$ and $M=M^{\sharp}$. Then
there is an $O_{\breve{F}}$-basis $\\{e_{1},\ldots,e_{n}\\}$ of $M$ such that

$(e_{\alpha},e_{\alpha+m})=1,\ (e_{\mu},e_{\mu})\in O_{\breve{F}}^{\times}$

for $1\leq\alpha\leq m$, $2m+1\leq\mu\leq n$, the inner product $(\,,\,)$
between any other basis vectors is zero, and

$\breve{\Lambda}=\mathrm{Span}_{O_{\breve{F}}}\\{\Pi e_{1},\ldots,\Pi
e_{m},e_{m+1},\ldots,e_{n}\\}.$

###### Proof.

By assumption we have $\Pi
M\subset\Pi\breve{\Lambda}^{\sharp}\subset\breve{\Lambda}\subset M$ and
$\dim_{\bar{\kappa}}(M/\breve{\Lambda})=m$. With respect to the
$\bar{\kappa}$-valued quadratic form $(\,,\,)\pmod{\pi}$ on $M/\Pi M$,
$\breve{\Lambda}/\Pi M$ has a decomposition

$\breve{\Lambda}/\Pi M=R\obot W,$

where $R$ is totally isotropic and $W$ is non-degenerate. Then by the
nondegeneracy of $(\,,\,)\pmod{\pi}$ on $M/\Pi M$ we know that there is a
totally isotropic subspace $R^{\prime}$ such that

$M/\Pi M=(R^{\prime}\oplus R)\obot W,$

and $(\,,\,)\pmod{\pi}$ induces a perfect pairing between $R$ and
$R^{\prime}$. Hence we can find a basis $\\{\bar{e}_{1},\bar{e}_{n}\\}$ of
$M/\Pi M$ such that $R^{\prime}=\langle\bar{e}_{1},\ldots,\bar{e}_{m}\rangle$,
$R=\langle\bar{e}_{m+1},\ldots,\bar{e}_{2m}\rangle$,
$W=\langle\bar{e}_{2m+1},\ldots,\bar{e}_{n}\rangle$, and

$(\bar{e}_{\alpha},\bar{e}_{\alpha+m})=1\pmod{\pi},\
(\bar{e}_{\mu},\bar{e}_{\mu})\pmod{\pi}\in\bar{\kappa}^{\times}$

for $1\leq\alpha\leq m$ and $2m+1\leq\mu\leq n$ and the pairing between all
other basis vectors are zero. We can lift the above basis to a basis
$\\{e_{1},\ldots,e_{n}\\}$ of $M$ which will satisfy the assumptions of the
lemma. ∎

######  Proposition 3.11.

The scheme $\mathcal{Z}(\Lambda)$ has no $O_{\breve{F}}/(\pi^{2})$-point. In
other words, $\mathcal{Z}(\Lambda)$ is a subscheme of
$\mathcal{N}\times_{{\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}}\mathrm{Spec}\,\bar{\kappa}$.

###### Proof.

Let $\mathcal{O}=O_{\breve{F}}/(\pi^{2})$ with the reduction map
$\mathcal{O}\rightarrow\bar{\kappa}$ and the natural divided power structure
on $\pi\mathcal{O}$. Then $\mathcal{O}\in\mathscr{C}$. Let
$z=(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{Z}(\Lambda)(\bar{\kappa})$
and $M=\rho(M(X))\subset N$ as in Proposition 3.6. Then by Corollary 3.7
$\breve{\Lambda}\subset M$, and we can assume there is an
$O_{\breve{F}}$-basis $\\{e_{1},\ldots,e_{n}\\}$ of $M$ as in Lemma 3.10.
Denote the image of $e_{i}$ in $M_{\mathcal{O}}$ still by $e_{i}$. Then
$\\{e_{1},\ldots,e_{n},\Pi e_{1},\ldots,\Pi e_{n}\\}$ is an
$\mathcal{O}$-basis of $M_{\mathcal{O}}$. With respect to the alternating form
$\langle\,,\,\rangle$, we have by (2.2)

(3.9) $\langle e_{\alpha},\Pi e_{m+\alpha}\rangle=-1/\delta,\ \langle
e_{m+\alpha},\Pi e_{\alpha}\rangle=-1/\delta,\ \langle e_{\mu},\Pi
e_{\mu}\rangle\in O_{\breve{F}_{0}}^{\times},$

for $1\leq\alpha\leq m$, $2m+1\leq\mu\leq n$, and all other pairings between
basis vectors are zero.

Assume that $z$ can be lifted to a point
$\tilde{z}\in\mathcal{Z}(\Lambda)_{z}(\mathcal{O})$, which corresponds to a
pair $(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}})$ as in Lemma 3.9.
First notice that

(3.10)
$\widetilde{\mathrm{Fil}}^{-}=(\Pi+\pi)\cdot\Lambda_{M,\mathcal{O}}=\mathrm{Span}_{O_{\breve{F}}\otimes_{O_{\breve{F}_{0}}}\mathcal{O}}\\{\pi\Pi
e_{1},\ldots,\pi\Pi e_{m},(\Pi+\pi)e_{m+1},\ldots,(\Pi+\pi)e_{n}\\}.$

With respect to the alternating form $\langle\,,\,\rangle$, its perpendicular
complement $(\widetilde{\mathrm{Fil}}^{-})^{\bot}$ in
$\Lambda_{M,\mathcal{O}}$ is generated by

(3.11) $\\{(\Pi+\pi)e_{1},\ldots(\Pi+\pi)e_{m},\Pi e_{m+1},\pi
e_{m+1},\ldots,\Pi e_{2m},\pi
e_{2m},(\Pi+\pi)e_{2m+1},\ldots,(\Pi+\pi)e_{n}\\}.$

By Lemma 3.9 (c), $\tilde{\mathcal{F}}^{\bot}$ is annihilated by $\Pi+\pi$,
hence it is spanned by a vector

$v=\sum_{i=1}^{n}a_{i}(\Pi-\pi)e_{i},$

where $a_{i}\in\mathcal{O}^{\times}$ for some $i$ as
$\tilde{\mathcal{F}}^{\bot}$ is a direct summand of $M_{\mathcal{O}}$. By
Lemma 3.9, we must have
$\widetilde{\mathrm{Fil}}^{-}\subset\widetilde{\mathrm{Fil}}$,
$\tilde{\mathcal{F}}^{\bot}\subset\widetilde{\mathrm{Fil}}$ and
$\widetilde{\mathrm{Fil}}$ is isotropic. Hence
$\widetilde{\mathrm{Fil}}\subset(\widetilde{\mathrm{Fil}}^{-})^{\bot}$.
Moreover
$\langle\widetilde{\mathrm{Fil}}^{-},\tilde{\mathcal{F}}^{\bot}\rangle=0$,
which implies $a_{i}\in\pi\mathcal{O}$ for $1\leq i\leq m$ and $2m+1\leq i\leq
n$. Hence without loss of generality we can assume that $a_{m+1}=1$.

Since $\widetilde{\mathrm{Fil}}$ is a direct summand of $M_{\mathcal{O}}$ we
have $M_{\mathcal{O}}=\widetilde{\mathrm{Fil}}\oplus S$ where $S$ is an
$\mathcal{O}$-module. We can write $\Pi e_{1}=w+w^{\prime}$ where
$w\in\widetilde{\mathrm{Fil}}$ and $w^{\prime}\in S$. Since $\pi\Pi
e_{1}\in\widetilde{\mathrm{Fil}}^{-}\subset\widetilde{\mathrm{Fil}}$, we must
have $\pi w^{\prime}=0$. This implies that $w^{\prime}\in\pi M_{\mathcal{O}}$
and $w$ is of the form

$w=(\Pi+b\pi)e_{1}+x$

where $b\in\mathcal{O}$ and
$x\in\pi\cdot\mathrm{Span}_{\mathcal{O}}\\{e_{2},\Pi e_{2},\ldots,e_{n},\Pi
e_{n}\\}$. Since
$w\in\widetilde{\mathrm{Fil}}\subset(\widetilde{\mathrm{Fil}}^{-})^{\bot}$, by
(3.11), we must have $b=1$ and $x$ is of the form

$x=\sum_{i=2}^{m}d_{i}(\Pi+\pi)e_{i}+\sum_{i=m+1}^{2m}(c_{i}+d_{i}\Pi)e_{i}+\sum_{i=2m+1}^{n}d_{i}(\Pi+\pi)e_{i},$

where $c_{i}\in\pi\mathcal{O}$ for $m+1\leq i\leq 2m$ and
$d_{i}\in\pi\mathcal{O}$ for $2\leq i\leq n$. Since
$(\Pi+\pi)e_{i}\in\widetilde{\mathrm{Fil}}$ for $2m+1\leq i\leq n$, by
changing $w$ and $x$ at the same time if necessary we can assume that
$d_{i}=0$ for $2m+1\leq i\leq n$. By (3.9), we have

$\langle(\Pi+\pi)e_{1},(\Pi-\pi)e_{m+1}\rangle=2\pi\langle e_{1},\Pi
e_{m+1}\rangle\neq 0.$

Moreover

$\displaystyle\langle x,v\rangle=$ $\displaystyle\sum_{i=2}^{m}\langle
d_{i}(\Pi+\pi)e_{i},a_{m+i}(\Pi-\pi)e_{m+i}\rangle+\sum_{i=m+1}^{2m}\langle(c_{i}+d_{i}\Pi)e_{i},a_{i-m}(\Pi-\pi)e_{i-m}\rangle=0$

Here we have used the fact that $a_{i}\in\pi\mathcal{O}$ for $1\leq i\leq m$,
$c_{i}\in\pi\mathcal{O}$ for $m+1\leq i\leq 2m$ and $d_{i}\in\pi\mathcal{O}$
for $2\leq i\leq 2m$. Then $\langle w,v\rangle\neq 0$ which contradicts the
fact that $\widetilde{\mathrm{Fil}}$ is isotropic. Hence there is no lift of
$z$ into $\mathcal{Z}(\Lambda)(\mathcal{O})$. This proves the lemma. ∎

As $\mathcal{N}_{\Lambda}$ is a formal subscheme of $\mathcal{Z}(\Lambda)$,
the following corollary is immediate.

######  Corollary 3.12.

$\mathcal{N}_{\Lambda}$ is a formal subscheme of
$\mathcal{N}\times_{{\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}}\mathrm{Spec}\,\bar{\kappa}$.

######  Proposition 3.13.

Let $\Lambda$ be a vertex lattice of type $2m$ ($m\geq 1$) in $\mathbb{V}$ and
$z\in\mathcal{N}_{\Lambda}(\bar{\kappa})$. Then the tangent space
$T_{z}(\mathcal{N}_{\Lambda})$ has dimension less or equal to $m$.

###### Proof.

Let
$z=(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{N}_{\Lambda}(\bar{\kappa})$
and $M=\rho(M(X))\subset N$ as in Proposition 3.6. Let
$\mathcal{O}=\bar{\kappa}[\epsilon]/(\epsilon^{2})$, then $\mathcal{O}$ is an
$O_{\breve{F}}$-algebra through the map
$O_{\breve{F}}\rightarrow\bar{\kappa}\rightarrow\mathcal{O}$ and the ideal
$(\epsilon)\subset\mathcal{O}$ is equipped with a natural divided power
structure. Then $\mathcal{O}\in\mathscr{C}$. Any point $\tilde{z}\in
T_{z}(\mathcal{N}_{\Lambda})=\mathcal{N}_{\Lambda,z}(\mathcal{O})$ corresponds
to a unique pair $(\tilde{\mathcal{F}}^{\bot},\widetilde{\mathrm{Fil}})$
lifting $(\mathcal{F}^{\bot},\mathrm{Fil})$ as in Lemma 3.9. We prove the
lemma in two cases.

Case (a): $\mathrm{Fil}\neq\Pi\cdot M_{\bar{\kappa}}$. Since $M_{\mathcal{O}}$
is a free $O_{F}\otimes_{O_{F_{0}}}\mathcal{O}$-modules of rank $n$, we have
the following exact sequence

$0\rightarrow\Pi\cdot M_{\mathcal{O}}\rightarrow
M_{\mathcal{O}}\xrightarrow{\Pi}\Pi\cdot M_{\mathcal{O}}\rightarrow 0,$

where $\Pi\cdot M_{\mathcal{O}}$ is a free $\mathcal{O}$-module of rank $n$
and the first arrow is the natural injection. This implies the following
sequence is exact.

(3.12) $0\rightarrow(\Pi\cdot
M_{\mathcal{O}})\cap\widetilde{\mathrm{Fil}}\rightarrow\widetilde{\mathrm{Fil}}\xrightarrow{\Pi}\Pi\cdot\widetilde{\mathrm{Fil}}\rightarrow
0.$

Since $\widetilde{\mathrm{Fil}}\neq\Pi\cdot M_{\mathcal{O}}$, by (3.12) we
know that $\Pi\cdot\widetilde{\mathrm{Fil}}\neq\\{0\\}$. Since by Lemma 3.9,
$\Pi\cdot\widetilde{\mathrm{Fil}}\subset\tilde{\mathcal{F}}^{\bot}$ and
$\tilde{\mathcal{F}}^{\bot}$ has rank $1$, we know that
$\Pi\cdot\widetilde{\mathrm{Fil}}=\tilde{\mathcal{F}}^{\bot}$ by Nakayama’s
lemma. Hence $\tilde{\mathcal{F}}^{\bot}$ is determined by
$\widetilde{\mathrm{Fil}}$. Moreover $\widetilde{\mathrm{Fil}}$ is determined
by its image in the $\mathcal{O}$-module
$(\widetilde{\mathrm{Fil}}^{-})^{\bot}/\widetilde{\mathrm{Fil}}^{-}$ where
$\widetilde{\mathrm{Fil}}^{-}=\Pi\cdot\Lambda_{M,\mathcal{O}}$ as in Lemma
3.9. Equation (3.10) is still true and implies that
$\widetilde{\mathrm{Fil}}^{-}$ is an isotropic free $\mathcal{O}$-module
direct summand of $M_{\mathcal{O}}$ of rank $n-m$ (notice that $\pi=0$ in
$\mathcal{O}$). Moreover

(3.13)
$(\widetilde{\mathrm{Fil}}^{-})^{\bot}=\mathrm{Span}_{\mathcal{O}}\\{\Pi
e_{1},\ldots,\Pi e_{m},\Pi e_{m+1},e_{m+1},\ldots,\Pi e_{2m},e_{2m},\Pi
e_{2m+1},\ldots,\Pi e_{n}\\},$

and
$(\widetilde{\mathrm{Fil}}^{-})^{\bot}/\widetilde{\mathrm{Fil}}^{-}=V_{\mathcal{O}}$
where $V=\Pi\cdot\breve{\Lambda}^{\sharp}/\Pi\cdot\breve{\Lambda}$ is a
symplectic vector space over $\bar{\kappa}$ of dimension $2m$. The proposition
in this case follows from Lemma 3.4 if we define $U_{0}=\Pi\cdot
M/\Pi\cdot\breve{\Lambda}\subset V$.

Case (b): $\mathrm{Fil}=\Pi\cdot M_{\bar{\kappa}}$. By Lemma 3.10, we know
$\Pi\cdot M_{\mathcal{O}}\subset\Lambda_{M,\mathcal{O}}$ and
$\Lambda_{M,\mathcal{O}}$ is a free $\mathcal{O}$-module direct summand of
$M_{\mathcal{O}}$ of corank $m$. Hence $(\Lambda_{M,\mathcal{O}})^{\bot}$ is a
free $\mathcal{O}$-module of rank $m$ and is in $(\Pi\cdot
M_{\mathcal{O}})^{\bot}=\Pi\cdot M_{\mathcal{O}}$. As in [Krä03], we assume
that we have a $\mathcal{O}\otimes_{O_{F_{0}}}O_{F}$-basis
$\\{e_{1},\ldots,e_{n}\\}$ of $M_{\mathcal{O}}$ such that $\langle
e_{i},\Pi\cdot e_{j}\rangle=\delta_{ij}$ for $1\leq i,j\leq n$ and all other
pairings between these basis vectors are zero. The lift
$\widetilde{\mathrm{Fil}}$ is spanned by $x_{1},\ldots,x_{n}$ where

$(x_{1},\ldots,x_{n})=(e_{1},\ldots,e_{n},\Pi e_{1},\ldots,\Pi
e_{n})\begin{pmatrix}A\epsilon\\\ I_{n}\end{pmatrix}$

where $A\in M_{n}(\bar{\kappa})$ and $A={}^{t}A$ since
$\widetilde{\mathrm{Fil}}$ is isotropic. Assume
$\mathcal{F}^{\bot}\subset\mathrm{Fil}=\Pi\cdot M_{\bar{\kappa}}$ is spanned
by

$\sum_{i=1}^{n}b_{n+i}\Pi\cdot e_{i}.$

Then $b_{n+i}\neq 0$ for some $i$ and we can assume without loss of generality
$b_{n+1}=1$. The lift $\tilde{\mathcal{F}}^{\bot}$ is spanned by

$\sum_{i=1}^{n}\tilde{b}_{i}e_{i}+\sum_{i=n+1}^{2n}\tilde{b}_{i}\Pi\cdot
e_{i},$

where $\tilde{b}_{n+1}=1$ and $\tilde{b}_{n+i}=b_{n+i}+\epsilon c_{i}$ for
$2\leq i\leq n$ and some $c_{i}\in\bar{\kappa}$. Let

$\lambda={}^{t}(\tilde{b}_{n+1},\ldots,\tilde{b}_{2n}).$

Equations (4.7), (4.8) and (4.10) of [Krä03] tell us that

$A=\gamma_{1}\lambda\cdot{}^{t}\lambda$

for some $\gamma_{1}\in\bar{\kappa}$. Equation (4.5) of loc.cit. tells us

${}^{t}(\tilde{b}_{1},\ldots,\tilde{b}_{n})=A\lambda,$

which is equal to $\gamma_{1}\lambda\cdot{}^{t}\lambda\cdot\lambda=0$ as
${}^{t}\lambda\cdot\lambda=0$ by (4.9) of loc.cit.. In particular
$\tilde{\mathcal{F}}^{\bot}\subset\Pi M_{\mathcal{O}}$ and a point in
$T_{z}(\mathcal{N}_{\Lambda})$ is determined by the $n-1$ parameters $c_{i}$
for $2\leq i\leq n$ together with the additional parameter $\gamma_{1}$. Now
the condition
$\tilde{\mathcal{F}}^{\bot}\subset(\Lambda_{M,\mathcal{O}})^{\bot}$ (condition
(e) of Lemma 3.9) imposes further $n-m$ independent linear equations on the
parameters $c_{i}$ for $2\leq i\leq n$. This shows that the tangent space
$T_{z}(\mathcal{N}_{\Lambda})$ has dimension less or equal to $m$. This
finishes the proof of the proposition. ∎

###  3.3. Isomorphism between $\mathcal{N}_{\Lambda}$ and $Y_{V,\bar{\kappa}}$

By [RTW14, Lemma 6.1], the lattices $\breve{\Lambda}$ and
$\breve{\Lambda}^{\sharp}$ (see (3.3)) are closed under $\Pi$, $\mathbf{V}$
and $\mathbf{F}$, hence determine formal hermitian $O_{F}$-modules (see
Definition 2.1) $X_{-}$ and $X_{+}$ (denoted by $X_{\Lambda^{-}}$ and
$X_{\Lambda^{+}}$ resp. in [RTW14, §6]) of dimension $n$ over $\bar{\kappa}$
together with quasi-isogenies $\rho_{-}:X_{-}\rightarrow\mathbb{X}$,
$\rho_{+}:X_{+}\rightarrow\mathbb{X}$. The inclusion
$\breve{\Lambda}\subset\breve{\Lambda}^{\sharp}$ also defines an isogeny
$\rho_{\Lambda}:X_{-}\rightarrow X_{+}$ of height $2m$. Both $\rho_{-}$ and
$\rho_{+}$ have height $m$. Since $X_{-}\cong\mathbb{Y}^{n}$ as an
$O_{F}$-module, for any $\bar{\kappa}$-scheme $S$, condition (1) in Definition
3.5 is equivalent to the condition

(3.14) $(1)^{\prime}:\text{ The quasi-isogeny
}\rho_{X,-}:=\rho^{-1}\circ(\rho_{-})_{S}:(X_{-})_{S}\rightarrow X\text{ is an
isogeny}.$

This is further equivalent by loc.cit. to the condition

(3.15) $(1)^{\prime\prime}:\text{ The quasi-isogeny
}\rho_{X,+}:=(\rho_{+})^{-1}_{S}\circ\rho:X\rightarrow(X_{+})_{S}\text{ is an
isogeny}.$

######  Lemma 3.14.

The functor $\mathcal{N}_{\Lambda}$ is representable by a projective scheme
over $\bar{\kappa}$. The functor monomorphism
$\mathcal{N}_{\Lambda}\rightarrow\mathcal{N}$ is a closed immersion.

###### Proof.

By Proposition 3.11, $\mathcal{Z}(\Lambda)$ is defined over $\bar{\kappa}$. It
is clearly a closed formal subscheme of $\mathcal{N}$. Since for any
$\bar{\kappa}$-scheme $S$, condition (1) in Definition 3.5 is equivalent to
(3.14), the functor $\mathcal{Z}(\Lambda)$ can be represented by a projective
scheme over $\bar{\kappa}$ by exact the same argument as that of [VW11, Lemma
3.2]. Condition (2) defines $\mathcal{N}_{\Lambda}$ as a closed subscheme of
$\mathcal{Z}(\Lambda)$, hence is itself projective over $\bar{\kappa}$ and a
closed formal subscheme of $\mathcal{N}$. This finishes the proof of the
lemma. ∎

In the following discussion we assume that $\Lambda$ has type greater or equal
to $2$. Let $V=\Lambda^{\sharp}/\Lambda$ and define a symplectic form
$\langle\,,\,\rangle_{V}$ on $V$ as follows. For $\bar{x},\bar{y}\in V$ with
lifts $x,y\in\Lambda^{\sharp}$, define $\langle\bar{x},\bar{y}\rangle_{V}$ by
the image of $\pi_{0}\delta\langle x,y\rangle$ in $\mathbb{F}_{q}$ (see §2.2).
Extend this form bilinearly to $V_{\bar{\kappa}}$. Note that $\tau$ induces
identity on $V$ and the Frobenius $\Phi$ on $V_{\bar{\kappa}}$. Let $R$ be a
$\bar{\kappa}$-algebra and
$(X,\iota,\lambda,\rho,\mathcal{F})\in\mathcal{N}_{\Lambda}(R)$. As in the
proof of[VW11, Corollary 3.9], $\mathrm{Image}(D(\rho_{X,+}))$ and
$\mathrm{Image}(D((\rho_{\Lambda})_{R}))$ are locally free direct summands of
$D((X_{+})_{R})$ of corank $m$ and $2m$ respectively. Define

$U(X):=\mathrm{Image}(D(\rho_{X,+}))/\mathrm{Image}(D((\rho_{\Lambda})_{R})).$

Then $U(X)$ is a locally free direct summand of
$D((X_{+})_{R})/\mathrm{Image}(D((\rho_{\Lambda})_{R}))\cong\breve{\Lambda}^{\sharp}/\breve{\Lambda}\otimes_{\bar{\kappa}}R=V_{R}$
of corank $m$. Define

$\mathcal{F}(X):=(D(\rho_{X,+}))(q_{X}^{-1}(\mathcal{F}))/\mathrm{Image}(D((\rho_{\Lambda})_{R})),$

where $q_{X}:D(X)\rightarrow\operatorname{Lie}X$ is the natural quotient
homomorphism of $R$-modules. Then $\mathcal{F}(X)$ is a locally free direct
summand of $U(X)$ of rank $m-1$. We define a map
$\phi:\mathcal{N}_{\Lambda}\rightarrow\mathrm{Gr}(m,V_{\bar{\kappa}})\times\mathrm{Gr}(m-1,V_{\bar{\kappa}})$
by

$\phi:(X,\iota,\lambda,\rho,\mathcal{F})\mapsto(U(X),\mathrm{Span}_{R}\\{\Phi(\mathcal{F}(X))\\})\in(\mathrm{Gr}(m,V_{\bar{\kappa}})\times\mathrm{Gr}(m-1,V_{\bar{\kappa}}))(R).$

######  Lemma 3.15.

$\phi$ defines a bijection between $\mathcal{N}_{\Lambda}(\bar{\kappa})$ and
$Y_{V}(\bar{\kappa})$.

###### Proof.

A point $z\in\mathcal{N}_{\Lambda}(\bar{\kappa})$ corresponds to a pair
$(M,M^{\prime})$ as in Proposition 3.6. By definition we have

$\phi(z)=(U,U^{\prime}):=(M/\breve{\Lambda},\Phi(M^{\prime}/\breve{\Lambda})).$

By the definition of $\langle\,,\,\rangle_{V}$, the condition $M=M^{\sharp}$
is equivalent to the fact that $U$ is Lagrangian. The condition
$M^{\prime}\subset\tau^{-1}(M)\cap M$ is equivalent to $U^{\prime}\subset
U\cap\Phi(U)$. This shows that $\phi(z)\in Y_{V}(\bar{\kappa})$.

Conversely assume $(U,U^{\prime})\in Y_{V}(\bar{\kappa})$ and let
$M=\Pr^{-1}(U)$ and $M^{\prime}=\Pr^{-1}(\Phi^{-1}(U^{\prime}))$ where
$\Pr:\breve{\Lambda}^{\sharp}\rightarrow\breve{\Lambda}^{\sharp}/\breve{\Lambda}$
is the natural quotient map. Then we have

$\Pi\tau(M)\subset\Pi\tau(\breve{\Lambda}^{\sharp})=\Pi\breve{\Lambda}^{\sharp}\subset\breve{\Lambda}\subset
M,$

and

(3.16)
$M\subset\breve{\Lambda}^{\sharp}\subset\Pi^{-1}\breve{\Lambda}=\Pi^{-1}\tau\breve{\Lambda}\subset\Pi^{-1}\tau
M.$

This shows that $(M,M^{\prime})$ satisfies the conditions in Proposition 3.6
and Corollary 3.7. This defines the inverse of $\phi$ on the level of
$\bar{\kappa}$-points. Hence $\phi$ defines a bijection between
$\mathcal{N}_{\Lambda}(\bar{\kappa})$ and $Y_{V}(\bar{\kappa})$. ∎

######  Theorem 3.16.

Let $\Lambda$ be a vertex lattice of type $2m$ ($m\geq 1$) in $\mathbb{V}$.
The morphism $\phi$ defines an isomorphism $\mathcal{N}_{\Lambda}\rightarrow
Y_{V,\bar{\kappa}}$. In particular $\mathcal{N}_{\Lambda}$ is smooth of
dimension $m$ over $\bar{\kappa}$.

###### Proof.

Let $\mathcal{N}^{\mathrm{red}}_{\Lambda}$ be the underlying reduced
$\bar{\kappa}$-scheme of $\mathcal{N}_{\Lambda}$. Lemma 3.15 shows that $\phi$
induces a morphism
$\phi^{\mathrm{red}}:\mathcal{N}^{\mathrm{red}}_{\Lambda}\rightarrow
Y_{V,\bar{\kappa}}$. In fact the argument of Lemma 3.15 shows that
$\phi^{\mathrm{red}}$ is bijection on any $\kappa^{\prime}$-valued points for
any perfect extension $\kappa^{\prime}$ of $\bar{\kappa}$. Using the theory of
relative display and windows developed in [ACZ16], we can show that
$\phi^{\mathrm{red}}$ induces a bijection on $\kappa^{\prime}$-valued points
for any field extension $\kappa^{\prime}$ of $\bar{\kappa}$. Hence
$\phi^{\mathrm{red}}$ is an isomorphism on fibers, in particular quasi-finite.
Since it is a morphism between projective varieties, it is proper. Hence
$\phi^{\mathrm{red}}$ is finite by (the Grothendieck version of) Zariski’s
main theorem. As $\mathcal{N}^{\mathrm{red}}_{\Lambda}$ is reduced, [Bou, §3
$n^{0}$2 Proposition 7] implies that $\phi^{\mathrm{red}}$ is flat. As
$\phi^{\mathrm{red}}$ is finite, flat and an isomorphism on fibers, we can use
Lemma [Krä03, Lemma 4.4] to conclude that $\phi^{\mathrm{red}}$ is an
isomorphism. In particular $\mathcal{N}^{\mathrm{red}}_{\Lambda}$ is a smooth
$\bar{\kappa}$-variety of dimension $m$. Now Proposition 3.13 shows that
$\mathcal{N}^{\mathrm{red}}_{\Lambda}=\mathcal{N}_{\Lambda}$, this finishes
the proof of the theorem. ∎

######  Proposition 3.17.

Let $\Lambda$ be a vertex lattice of type $0$ in $\mathbb{V}$. Then
$\mathcal{N}_{\Lambda}$ is the exceptional divisor $\mathrm{Exc}_{\Lambda}$
and is isomorphic to $\mathbb{P}^{n-1}_{\bar{\kappa}}$.

###### Proof.

Let $R$ be any $\bar{\kappa}$-algebra and $z$ be any point in
$\mathcal{N}_{\Lambda}(R)$ and $(X,\iota,\lambda,\rho,\mathcal{F})$ be the
pullback of the universal object of $\mathcal{N}$ to $z$. As $\Lambda$ is a
self-dual lattice, the quasi-isogeny $\rho_{-}$ has height $0$. Thus the
isogeny

$\rho_{X,-}=\rho^{-1}\circ(\rho_{-})_{R}:(X_{-})_{R}\rightarrow X$

has height $0$ and is an isomorphism, hence we can identify $(X,\ldots,\rho)$
with $((X_{-})_{R},\ldots,(\rho_{-})_{R})$. As
$\pi|_{\Lambda}=\mathbf{V}_{\Lambda}$ for any vertex lattice $\Lambda$, and
$\operatorname{Lie}X_{-}=M(X_{-})/\mathbf{V}M(X_{-})$, the action of
$\iota(\pi)$ on $\operatorname{Lie}X_{-}$ is trivial. The point $z$ is
uniquely determined by the filtration $\mathcal{F}\subset\operatorname{Lie}X$.
Hence $\mathcal{F}$ can be any rank $n-1$ locally free $R$-module on
$\operatorname{Lie}X$. This shows that $\mathcal{N}_{\Lambda}$ is isomorphic
to $\mathbb{P}^{n-1}_{\bar{\kappa}}$ and is in particular reduced. Moreover if
$R=\bar{\kappa}$, then $\rho(M(X))=\breve{\Lambda}$. This shows that
$\mathcal{N}_{\Lambda}$ is a subscheme of $\mathrm{Exc}_{\Lambda}$ according
Definition 2.4. By the proof of Lemma 2.5, we know that
$\mathcal{N}_{\Lambda}$ and $\mathrm{Exc}_{\Lambda}$ have the same
$\bar{\kappa}$-points. As they are both reduced subscheme of $\mathcal{N}$,
they must be the same. This proves the proposition. ∎

###  3.4. Bruhat-Tits stratification

######  Lemma 3.18.

For any pair $(M,\mathcal{F})$ satisfying the condition in Proposition 3.6,
there is a unique vertex lattice $\Lambda(M)$ such that $\Lambda(M)\subset M$
and $\Lambda(M)$ is maximal among all such vertex lattices.

###### Proof.

This is essentially [RTW14, Proposition 4.1] as such $M$ satisfies the
conditions in Proposition 2.4 of loc.cit.. ∎

######  Theorem 3.19.

There is a stratification of $\mathcal{N}_{\mathrm{red}}$ by closed strata
$\mathcal{N}_{\Lambda}$ given by

(3.17) $\mathcal{N}_{\mathrm{red}}=\bigcup_{\Lambda}\ \mathcal{N}_{\Lambda}.$

where the union is over all vertex lattices in $\mathbb{V}$. We call this the
Bruhat-Tits stratification of $\mathcal{N}_{\mathrm{red}}$. In the following,
assume that $\Lambda$ and $\Lambda^{\prime}$ are vertex lattices of type
greater or equal to $2$, and $\Lambda_{0}$ and $\Lambda^{\prime}_{0}$ are
vertex lattices of type $0$.

  1. (1)

If $\Lambda\subset\Lambda^{\prime}$, then $\mathcal{N}_{\Lambda^{\prime}}$ is
a subscheme of $\mathcal{N}_{\Lambda}$.

  2. (2)

The intersection of $\mathcal{N}_{\Lambda^{\prime}}\cap\mathcal{N}_{\Lambda}$
is nonempty if and only if $\Lambda^{\prime\prime}=\Lambda+\Lambda^{\prime}$
is a vertex lattice, in which case we have
$\mathcal{N}_{\Lambda^{\prime}}\cap\mathcal{N}_{\Lambda}=\mathcal{N}_{\Lambda^{\prime\prime}}$.

  3. (3)

The intersection of
$\mathcal{N}_{\Lambda^{\prime}_{0}}\cap\mathcal{N}_{\Lambda_{0}}$ is always
empty if $\Lambda_{0}\neq\Lambda^{\prime}_{0}$.

  4. (4)

The intersection $\mathcal{N}_{\Lambda}\cap\mathcal{N}_{\Lambda_{0}}$ is
nonempty if and only if $\Lambda\subset\Lambda_{0}$ in which case
$\mathcal{N}_{\Lambda}\cap\mathcal{N}_{\Lambda_{0}}$ is isomorphic to
$\mathbb{P}_{\bar{\kappa}}^{m-1}$ where $2m$ is the type of $\Lambda$.

###### Proof.

To prove (3.17), it suffices to check this on $\bar{\kappa}$-points. A point
$z\in\mathcal{N}_{\mathrm{red}}(\bar{\kappa})$ corresponds to a pair
$(M,M^{\prime})$ as in Proposition 3.6. Take $\Lambda=\Lambda(M)$ as in Lemma
3.18. If $\Lambda$ has type $0$, then both $\breve{\Lambda}$ and $M$ are self-
dual and $\breve{\Lambda}\subset M$, so they have to be equal. Hence
$z\in\mathcal{N}_{\Lambda}$ by Corollary 3.7. If $\Lambda$ is not of type $0$,
then $M$ is not $\tau$-invariant, hence $M^{\prime}=M\cap\tau^{-1}(M)$ is
uniquely determined. Since $\Lambda$ is $\tau$-invariant,
$\breve{\Lambda}\subset M^{\prime}$. Hence
$z\in\mathcal{N}_{\Lambda}(\bar{\kappa})$ by Corollary 3.7. This proves
(3.17).

(1) follows immediately from Definition 3.5.

(2). If $\Lambda^{\prime\prime}$ is a vertex lattice, then
$\mathcal{N}_{\Lambda}\cap\mathcal{N}_{\Lambda^{\prime}}=\mathcal{N}_{\Lambda^{\prime\prime}}$
by Definition 3.5. Conversely if
$\mathcal{N}_{\Lambda^{\prime}}\cap\mathcal{N}_{\Lambda}(\bar{\kappa})$ is
nonempty, let
$(M,M^{\prime})\in\mathcal{N}_{\Lambda^{\prime}}\cap\mathcal{N}_{\Lambda}(\bar{\kappa})$.
Then $\Lambda(M)\supset\Lambda+\Lambda^{\prime}$ by the maximality of
$\Lambda(M)$. Then
$\Lambda+\Lambda^{\prime}\subset\Lambda(M)\subset\Lambda(M)^{\sharp}\subset\Lambda^{\sharp}\cap(\Lambda^{\prime})^{\sharp}=(\Lambda+\Lambda^{\prime})^{\sharp}$.
Hence $\Lambda+\Lambda^{\prime}$ is a vertex lattice.

(3) follows directly from Corollary 3.7.

(4). By Corollary 3.7, a point $(M,M^{\prime})\in\mathcal{N}(\bar{\kappa})$ is
in $\mathcal{N}_{\Lambda}\cap\mathcal{N}_{\Lambda_{0}}$ if and only if
$M=\Lambda_{0}\otimes_{O_{F}}O_{\breve{F}}$ and $\Lambda\subset
M^{\prime}\subset M$. This show that $\Lambda\subset\Lambda_{0}$ and
$\mathcal{N}_{\Lambda}\cap\mathcal{N}_{\Lambda_{0}}(\bar{\kappa})=\mathbb{P}(\Lambda_{0}/\Lambda)(\bar{\kappa})$.
Similarly one can show that

$\mathcal{N}_{\Lambda}\cap\mathcal{N}_{\Lambda_{0}}(R)=\mathbb{P}(\Lambda_{0}/\Lambda)(R)$

for any $\bar{\kappa}$-algebra $R$ using relative display theory. This
finishes the proof of (4). ∎

######  Proposition 3.20.

For a rank $r$ lattice $L\subset\mathbb{V}$, the reduced subscheme
$\mathcal{Z}(L)_{\mathrm{red}}$ of $\mathcal{Z}(L)$ is a union of Bruhat-Tits
strata:

(3.18)
$\mathcal{Z}(L)_{\mathrm{red}}=\bigcup_{L\subset\Lambda}\mathcal{N}_{\Lambda},$

where the union is taken over all vertex lattices $\Lambda$ such that
$L\subset\Lambda$. Moreover, the intersection of $\mathcal{Z}(L)$ with
$\mathcal{N}_{\Lambda}$ is nonempty if and only if
$L\subseteq\Lambda^{\sharp}$.

###### Proof.

The proof of (3.18) is the same as that of [Shi18, Proposition 3.8].

If $L\subseteq\Lambda^{\sharp}$ and $L$ is integral, define
$\Lambda^{\prime}:=L+\Lambda$. Then $\Lambda^{\prime}$ is a vertex lattice and
$\Lambda\subseteq\Lambda^{\prime}$. By (3.18),
$\mathcal{N}_{\Lambda^{\prime}}$ is in the intersection of $\mathcal{Z}(L)$
and $\mathcal{N}_{\Lambda}$.

Conversely if the intersection of $\mathcal{Z}(L)$ and $\mathcal{N}_{\Lambda}$
is not empty, then by (3.18) and Theorem 3.19, there exists a vertex lattice
$\Lambda^{\prime}$ such that $\Lambda\subseteq\Lambda^{\prime}$ and
$L\subseteq\Lambda^{\prime}$. Since
$\Lambda^{\prime}\subseteq(\Lambda^{\prime})^{\sharp}\subseteq\Lambda^{\sharp}$,
we know that $L\subseteq\Lambda^{\sharp}$. This finishes the proof of the
lemma. ∎

##  4\. Fourier transform: the geometric side

###  4.1. Horizontal and vertical part of
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})$

######  Definition 4.1.

Let $L^{\flat}$ be a rank $n-1$ integral lattice in $\mathbb{V}$. We say that
$L^{\flat}$ is horizontal if one of the following conditions is satisfied

  1. (1)

$L^{\flat}$ is unimodular.

  2. (2)

$L^{\flat}$ is of the form $L^{\flat}=M\obot L^{\prime}$ 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 nonsplit.

We denote the set of horizontal lattices by $\mathrm{Hor}$.

######  Lemma 4.2.

Let $L^{\flat}$ be a rank $n-1$ lattice in $\mathbb{V}$. Then $L^{\flat}$ is
horizontal if and only if there is a unique vertex lattice $\Lambda$ which
contains $L^{\flat}$. If this is the case, $\Lambda$ is of type $0$.

###### Proof.

We first prove the “only if” direction. Let $\Lambda$ be any vertex lattice
containing $L^{\flat}$. If $L^{\flat}$ is unimodular, then $\Lambda$ has to be
of the form $L^{\flat}\obot L^{\prime}$ where $L^{\prime}$ is the unique
unimodular lattice in $(L^{\flat}_{F})^{\bot}$. If $L^{\flat}$ is of the form
$M\obot L^{\prime}$ such that $M$ is of rank $n-2$ and $(M_{F})^{\bot}$ is
nonsplit, then the proof of [Shi18, Theorem 3.10] implies that there is a
unique vertex lattice $\Lambda^{\prime}$ 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 a only one point). Then $\Lambda$ must be of the form
$M\obot\Lambda^{\prime}$. In both cases, $\Lambda$ is unique and is of type
$0$.

We now prove the “if” direction. If $t(L^{\flat})\geq 2$, then there exist a
type $2$ vertex lattice $\Lambda_{2}$ containing $L^{\flat}$ and any type $0$
vertex lattice containing $\Lambda_{2}$ (there are $q+1$ of them) also
contains $L^{\flat}$. Hence $t(L^{\flat})\leq 1$ and $L^{\flat}$ is of the
form $M\obot L^{\prime}$ such that $M$ is of rank $n-2$. If $(M_{F})^{\bot}$
is split and ${\mathrm{val}}(L^{\prime})>0$, then [HSY20, Corollary 3.11]
implies that there are more than one type $0$ vertex lattices
$\Lambda^{\prime}$ in $(M_{F})^{\bot}$ containing $L^{\prime}$. For any such
$\Lambda^{\prime}$, $M\obot\Lambda^{\prime}$ is a vertex lattice of type $0$
containing $L^{\flat}$. This shows that in order for such $\Lambda$ to be
unique, $L^{\flat}$ must satisfies the conditions in Definition 4.1. The lemma
is proved. ∎

For a rank $n-1$ lattice $L^{\flat}$ in $\mathbb{V}$, define

(4.1) $\mathrm{Hor}(L^{\flat})\coloneqq\\{M^{\flat}\in\mathrm{Hor}\mid
L^{\flat}\subseteq M^{\flat}\\}.$

When $\mathrm{dim}(\mathbb{V})=2$ and $\chi(\mathbb{V})=-1$, for
$y\in\mathbb{V}$, define

$\tilde{\mathcal{Z}}(y)^{o}\coloneqq\left\\{\begin{array}[]{cc}\mathcal{Z}_{{\mathrm{val}}(y)}^{+}\sqcup\mathcal{Z}_{{\mathrm{val}}(y)}^{-}&\text{
if }{\mathrm{val}}(y)>0,\\\ \mathcal{Z}_{0}&\text{ if
}{\mathrm{val}}(y)=0.\end{array}\right.$

Here $\mathcal{Z}_{0}\cong{\mathrm{Spf}\,\mathcal{O}_{\breve{F}}}$ and
$\mathcal{Z}_{s}^{+}\cong\mathcal{Z}_{s}^{-}\cong\mathrm{Spf}\,W_{s}$ are
quasi-canonical lifting cycles defined in [Shi20, §3] where $W_{s}$ is a
totally ramified abelian extension of $\mathcal{O}_{\breve{F}}$ of degree
$q^{s}$. When $\mathrm{dim}(\mathbb{V})=2$ and $\chi(\mathbb{V})=1$, for
$y\in\mathbb{V}$, define $\tilde{\mathcal{Z}}(y)^{o}$ to be
$\mathcal{Z}^{h}(y)$, where
$\mathcal{Z}^{h}(y)\cong\mathrm{Spf}\,O_{\breve{F}}$ is as in [HSY20, Theorem
4.1]. In all cases, $\tilde{\mathcal{Z}}(y)^{\circ}$ is a closed subscheme of
$\mathcal{N}_{2}$.

Now for a rank $n-1$ lattice $L^{\flat}$ in $\mathbb{V}$ and
$M^{\flat}\in\mathrm{Hor}(L^{\flat})$, we can decompose $M^{\flat}$ as
$M\obot\mathrm{Span}\\{y\\}$ where $M$ is unimodular and ${\mathrm{val}}(y)$
has to be zero if $(M_{F})^{\bot}$ is split. By [HSY21, Proposition 2.6], the
unimodular lattice $M$ induces a closed embedding
$\mathcal{N}_{2}\hookrightarrow\mathcal{N}_{n}$. We define
$\tilde{\mathcal{Z}}(M^{\flat})^{\circ}$ to be the image of the composed
embedding
$\tilde{\mathcal{Z}}(y)^{\circ}\hookrightarrow\mathcal{N}_{2}\hookrightarrow\mathcal{N}_{n}$
where $\tilde{\mathcal{Z}}(y)^{\circ}$ is the closed formal subscheme of
$\mathcal{N}_{2}$ defined above. The following is [HSY21, Theorem 4.1].

######  Theorem 4.3.

Let $L^{\flat}$ be a rank $n-1$ integral lattice in $\mathbb{V}$, then

(4.2)
$\mathcal{Z}(L^{\flat})_{\mathscr{H}}=\bigcup_{M^{\flat}\in\mathrm{Hor}(L^{\flat})}\tilde{\mathcal{Z}}(M^{\flat})^{\circ}.$

In particular, $\mathcal{Z}(L^{\flat})_{\mathscr{H}}$ is of pure dimension
$1$. We have the following identity in
$\mathrm{Gr}^{n-1}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})$:

(4.3)
$[\mathcal{O}_{\mathcal{Z}(L^{\flat})_{\mathscr{H}}}]=\sum_{M^{\flat}\in\mathrm{Hor}(L^{\flat})}[\mathcal{O}_{\tilde{\mathcal{Z}}(M^{\flat})^{\circ}}].$

######  Lemma 4.4.

For any formal subscheme $Z$ of $\mathcal{N}$ and $0\leq i\leq n$,
$F^{i}K_{0}^{Z}(\mathcal{N})$ is an ideal in $K_{0}(\mathcal{N})$.

###### Proof.

By definition $F^{i}K_{0}^{Z}(\mathcal{N})$ is generated by elements of the
form $[\mathcal{F}^{\bullet}]$ where $\mathcal{F}^{\bullet}$ is a finite
complex of locally free coherent $\mathcal{O}_{\mathcal{N}}$-modules acyclic
outside a sub formal scheme $Y$ of $Z$ such that the codimension of $Y$ in
$\mathcal{N}$ is greater or equal to $i$. By Kunneth formula for chain
complexes, the product complex
$\mathcal{F}^{\bullet}\otimes_{\mathcal{O}_{\mathcal{N}}}\mathcal{K}^{\bullet}$
is acyclic outside $Y$ as well for any finite complexes of locally free
coherent $\mathcal{O}_{\mathcal{N}}$-modules $\mathcal{K}^{\bullet}$. This
proves the lemma. ∎

By Lemma 4.4, for any formal subscheme $Z$ of $\mathcal{N}$, we can define a
quotient ring (not necessary with identity)

(4.4)
$\mathrm{Gr}^{\prime}K_{0}^{Z}(\mathcal{N}):=K_{0}^{Z}(\mathcal{N})/F^{n}K_{0}^{Z}(\mathcal{N}).$

In particular
$\mathrm{Gr}^{n-1}K_{0}^{Z}(\mathcal{N})=F^{n-1}K_{0}^{Z}(\mathcal{N})/F^{n}K_{0}^{Z}(\mathcal{N})$
is a subgroup of $\mathrm{Gr}^{\prime}K_{0}^{Z}(\mathcal{N})$.

Since $\mathcal{Z}(L^{\flat})_{\mathscr{H}}$ is one-dimensional, the
intersection
$\mathcal{Z}(L^{\flat})_{\mathscr{H}}\cap\mathcal{Z}(L^{\flat})_{\mathscr{V}}$
must be $0$-dimensional if nonempty. It follows that there is a decomposition

(4.5)
$\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})=\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})_{\mathscr{H}}}(\mathcal{N})\oplus\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})_{\mathscr{V}}}(\mathcal{N}).$

Under this decomposition, we have

(4.6)
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})={}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}+{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N}),$

where we denote by the same notation the image of
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})$ under the natural quotient map
$K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})\rightarrow\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})$.
It follows that the element
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})$
is canonically defined although $\mathcal{Z}(L^{\flat})_{\mathscr{V}}$ depends
on the choice of a large integer $m\gg 0$.

Since $\mathcal{Z}(L^{\flat})_{\mathscr{H}}$ has expected dimension,
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}$ is in fact in
$\mathrm{Gr}^{n-1}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})$ and is
represented by the structure sheaf of $\mathcal{Z}(L^{\flat})_{\mathscr{H}}$.
In order to match the analytic side of our conjecture, we need to slightly
modify ${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}$.

######  Definition 4.5.

Let $L^{\flat}$ be a horizontal lattice in $\mathbb{V}$. Define
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})^{\circ}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})$
by

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})^{\circ}=\begin{cases}[\mathcal{O}_{\tilde{\mathcal{Z}}(L^{\flat})^{\circ}}]+\frac{1-(-1)^{n-1}}{2}[\mathcal{O}_{\mathbb{P}_{\Lambda}}]&\text{
if }L^{\flat}\text{ is unimodular},\\\
[\mathcal{O}_{\tilde{\mathcal{Z}}(L^{\flat})^{\circ}}]+[\mathcal{O}_{\mathbb{P}_{\Lambda}}]&\text{
otherwise},\end{cases}$

where $\Lambda$ is the unique type $0$ vertex lattice containing $L^{\flat}$
as in Lemma 4.2 and $\mathbb{P}_{\Lambda}$ is a projective line in
$\mathrm{Exc}_{\Lambda}$.

######  Remark 4.6.

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})^{\circ}$ is the difference cycle
$\mathcal{D}(L^{\flat})$ defined in [HSY21, Definition 2.15].

######  Definition 4.7.

Let $L^{\flat}$ be a rank $n-1$ lattice in $\mathbb{V}$. Define
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}^{*}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N})$
by

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}^{*}:=\sum_{M^{\flat}\in\mathrm{Hor}(L^{\flat})}{}^{\mathbb{L}}\mathcal{Z}(M^{\flat})^{\circ},$

where $\mathcal{Z}(M^{\flat})_{\mathscr{H}}^{\circ}$ is as in Definition 4.5.
Define the modified vertical part of the derived special cycle
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})$ by

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}^{*}:={}^{\mathbb{L}}\mathcal{Z}(L^{\flat})-{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}^{*}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})}(\mathcal{N}).$

For any $x\in\mathbb{V}\setminus L^{\flat}_{F}$, define

(4.7)
$\displaystyle\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{H}}(x):=\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}^{*}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]),\hbox{
and
}\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}(x):=\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}^{*}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]).$

######  Lemma 4.8.

If $L^{\flat}$ is a horizontal lattice of rank $n-1$ in $\mathbb{V}$, then

(4.8)
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}^{*}={}^{\mathbb{L}}\mathcal{Z}(L^{\flat}).$

In particular for any $x\in\mathbb{V}\setminus L_{F}^{\flat}$ we have

$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{H}}(x)=\operatorname{\mathrm{Int}}_{L^{\flat}}(x).$

###### Proof.

Let $\Lambda$ be the unique type $0$ vertex lattice containing $L^{\flat}$ as
indicated by Lemma 4.2. Then $\Lambda\cap L^{\flat}_{F}$ is the unique
unimodular lattice in $\mathrm{Hor}(L^{\flat})$. By Theorem 4.3, we have

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}^{*}-{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})=(m-1+\frac{1-(-1)^{n-1}}{2})[\mathcal{O}_{\mathbb{P}_{\Lambda}}]-{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}},$

where $m:=|\mathrm{Hor}(L^{\flat})|$. By Proposition 3.20 and Lemma 4.2, we
know that
${}^{\mathbb{L}}Z(L^{\flat})\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{N}_{\Lambda}}(\mathcal{N})$.
[HSY21, Corollary 3.5] implies

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}=m^{\prime}[\mathcal{O}_{\mathbb{P}_{\Lambda}}]$

for some integer $m^{\prime}$. Hence in order to prove (4.8), it suffices to
show

(4.9) $m^{\prime}=m-\frac{1+(-1)^{n-1}}{2}.$

Now assume $L^{\flat}=M\obot L^{\prime}$ where $M$ is unimodular and of rank
$n-2$ and ${\mathrm{val}}(L^{\prime})=a$. Then $m=a+1$. By [HSY21, Lemma 4.3],
we know that

$\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}\cdot[\mathcal{O}_{\mathcal{N}_{\Lambda}}])=2a+1=2m-1.$

By [HSY21, Corollary 3.7], we know

$\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}\cdot[\mathcal{O}_{\mathcal{N}_{\Lambda}}])=m^{\prime}\cdot\chi(\mathcal{N},[\mathcal{O}_{\mathbb{P}_{\Lambda}}]\cdot[\mathcal{O}_{\mathcal{N}_{\Lambda}}])=-2m^{\prime}.$

On the other hand, by [HSY21, Corollary 3.6],

$\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{H}}\cdot[\mathcal{O}_{\mathcal{N}_{\Lambda}}])+\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}\cdot[\mathcal{O}_{\mathcal{N}_{\Lambda}}])=\chi(\mathcal{N},{}^{\mathbb{L}}\mathcal{Z}(L^{\flat})\cdot[\mathcal{O}_{\mathcal{N}_{\Lambda}}])=(-1)^{n-1}.$

Combine the above equations, we get (4.9). ∎

######  Lemma 4.9.

We have

${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}^{*}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})_{\mathscr{V}}}(\mathcal{N}).$

###### Proof.

It follows immediately from Theorem 4.3, Lemma 4.8, and the definition of
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}^{*}$. ∎

###  4.2. Hermitian lattices and Fourier transform

We fix an additive character $\psi:F_{0}\rightarrow\mathbb{C}^{\times}$ whose
conductor is $O_{F_{0}}$. Recall that the Fourier transform with respect to
$\psi$ is defined by

(4.10)
$\widehat{\varphi}(x)=\int_{\mathbb{V}}\varphi(y)\cdot\psi(\operatorname{Tr}_{F/F_{0}}(x,y))d\mu(y),$

where $d\mu$ is the unique self-dual Haar measure on $\mathbb{V}$ with respect
to this transform. For a lattice $L$ in $\mathbb{V}$ we use $L^{\vee}$ to
denote its dual under the quadratic form
$\operatorname{Tr}_{F/F_{0}}((\,,\,))$. The following lemma is well-known and
easy to check.

######  Lemma 4.10.

Let $L\subset\mathbb{V}$ be a lattice of rank $n$ and
$1_{L}\in\mathscr{S}(\mathbb{V})$ be its characteristic function. Then

$\widehat{1_{L}}=\operatorname{vol}(L,d\mu)\cdot 1_{L^{\vee}}.$

######  Lemma 4.11.

Let $L$ be a rank $n$ lattice in $\mathbb{V}$. A function
$\varphi\in\mathscr{S}(\mathbb{V})$ is $L$-invariant (invariant under the
translation of $L$) if and only if its Fourier transform $\widehat{\varphi}$
is supported on $L^{\vee}$.

###### Proof.

We first prove the “only if” direction. For any $\mu\in\mathbb{V}$,let

$L(\bar{\mu}):=\mu+L.$

Any $L$-invariant function $\phi\in\mathscr{S}(\mathbb{V})$ is a linear
combination of the characteristic functions $1_{L(\bar{\mu})}$. So it suffices
to assume $\phi=1_{L(\mu)}$. In this case,

$\widehat{\phi}(x)=\psi(\operatorname{Tr}_{F/F_{0}}(x,\mu))\cdot\widehat{1_{L}}(x)$

is supported on $L^{\vee}$ by Lemma 4.10. This proves the “only if” direction.

For the “if” direction, it suffices to show that if $\varphi$ is supported on
$L^{\vee}$, then $\widehat{\varphi}$ is $L$-invariant. For any $z\in L$ we
have

$\widehat{\varphi}(x+z)=\int_{\mathbb{V}}\varphi(y)\cdot\psi(\operatorname{Tr}_{F/F_{0}}(x,y))\cdot\psi(\operatorname{Tr}_{F/F_{0}}(z,y))d\mu(y),$

Since $\psi(\operatorname{Tr}_{F/F_{0}}(z,y))=1$ for any $z\in L$ and $y\in
L^{\vee}$ and $\varphi$ is supported on $L^{\vee}$, the above is equal to
$\widehat{\varphi}(x)$. This finishes the proof of the lemma. ∎

For an integer $m$, recall that

$\mathbb{V}^{\geq m}=\\{x\in\mathbb{V}\mid{\mathrm{val}}(x)\geq m\\}.$

######  Definition 4.12.

Define $\mathscr{S}(\mathbb{V})^{\geq m}$ to be the subspace of
$\mathscr{S}(\mathbb{V})$ consisting of functions $\varphi$ such that
$\widehat{\varphi}$ is supported on $\mathbb{V}^{\geq m}$.

######  Lemma 4.13.

Let $\Lambda$ be a vertex lattice. Any $\Lambda$-invariant function in
$\mathscr{S}(\mathbb{V})$ is in $\mathscr{S}(\mathbb{V})^{\geq-1}$.

###### Proof.

By Lemma 4.11, it suffices to show that
$\Lambda^{\vee}\subset\mathbb{V}^{\geq-1}$. Since $\Lambda$ is a vertex
lattice, we have

$\Lambda^{\sharp}=H^{t}\obot I_{n-2t},$

for some $t$. Simple calculation gives then

$\Lambda^{\vee}=\frac{1}{\pi}\Lambda^{\sharp}=\frac{1}{\pi}H^{t}\obot\frac{1}{\pi}I_{n-2t}\subset\mathbb{V}^{\geq-1}.$

∎

###  4.3. Fourier transform of
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}$

######  Theorem 4.14.

Let $\Lambda$ be a vertex lattice and $\mathcal{K}\in
K^{\mathcal{N}_{\Lambda}}_{0}(\mathcal{N})$. For any
$x\in\mathbb{V}\setminus\\{0\\}$, the class
$\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]\in
K^{\mathcal{N}_{\Lambda}}_{0}(\mathcal{N})$ is $\Lambda$-invariant. To be more
precise for any $y\in\Lambda$ such that $x+y\neq 0$, we have

(4.11)
$\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]=\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x+y)}].$

Moreover, the function

$\operatorname{\mathrm{Int}}_{\mathcal{K}}(x):=\chi(\mathcal{N},\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x)}])$

extends to a $\Lambda$-invariant function in
$\mathscr{S}(\mathbb{V})^{\geq-1}$.

###### Proof.

Any element $\mathcal{K}\in K^{\mathcal{N}_{\Lambda}}_{0}(\mathcal{N})\cong
K^{\prime}_{0}(\mathcal{N}_{\Lambda})$ is a sum of elements of the form
$[\mathcal{F}]$ where $[\mathcal{F}]$ is a coherent sheave of
$\mathcal{O}_{\mathcal{N}_{\Lambda}}$-module. Hence it suffices to prove the
theorem for $\mathcal{K}=[\mathcal{F}]$. By [How19, Corollary C], we know

$[\mathcal{O}_{\mathcal{Z}(y)}\otimes_{\mathcal{O}_{\mathcal{N}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x)}]=[\mathcal{O}_{\mathcal{Z}(y)}\otimes_{\mathcal{O}_{\mathcal{N}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x+y)}].$

For any $y\in\Lambda$ with $x+y\neq 0$, $\mathcal{N}_{\Lambda}$ is a subscheme
of $\mathcal{Z}(y)$ by Proposition 3.20. Hence we have

$\displaystyle\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]=$
$\displaystyle[\mathcal{F}\otimes_{\mathcal{O}_{\mathcal{N}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x)}]$
$\displaystyle=$
$\displaystyle[\mathcal{F}\otimes_{\mathcal{O}_{\mathcal{N}_{\Lambda}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{N}_{\Lambda}}\otimes_{\mathcal{O}_{\mathcal{Z}(y)}}\mathcal{O}_{\mathcal{Z}(y)}\otimes_{\mathcal{O}_{\mathcal{N}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x)}]$
$\displaystyle=$
$\displaystyle[\mathcal{F}\otimes_{\mathcal{O}_{\mathcal{N}_{\Lambda}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{N}_{\Lambda}}\otimes_{\mathcal{O}_{\mathcal{Z}(y)}}\mathcal{O}_{\mathcal{Z}(y)}\otimes_{\mathcal{O}_{\mathcal{N}}}^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(x+y)}]$
$\displaystyle=$
$\displaystyle\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x+y)}]$

We have proved the $\Lambda$-invariance of
$\mathcal{K}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]$. It follows that
$\operatorname{\mathrm{Int}}_{\mathcal{K}}(x)$ is also $\Lambda$-invariant.
Hence we can define $\operatorname{\mathrm{Int}}_{\mathcal{K}}(0)$ to be
$\operatorname{\mathrm{Int}}_{\mathcal{K}}(x)$ for any $0\neq x\in\Lambda$ and
obtain a (unique) $\Lambda$-invariant function (still denoted by
$\operatorname{\mathrm{Int}}_{\mathcal{K}}(x)$) for all $x\in\mathbb{V}$. In
particular $\operatorname{\mathrm{Int}}_{\mathcal{K}}(x)$ is locally constant.
If $x\notin\Lambda^{\sharp}$, by Proposition 3.20, the intersection of
$\mathcal{Z}(x)$ with $\mathcal{N}_{\Lambda}$ is empty, which implies
$\operatorname{\mathrm{Int}}_{\mathcal{K}}(x)=0$. This shows that the function
$\operatorname{\mathrm{Int}}_{\mathcal{K}}(x)$ is compactly supported. Hence
it is in $\mathscr{S}(\mathbb{V})$ and is in fact in
$\mathscr{S}(\mathbb{V})^{\geq-1}$ by Lemma 4.13. This finishes the proof of
the theorem. ∎

######  Theorem 4.15.

For every non-degenerate lattice $L^{\flat}$ of $\mathbb{V}$ of rank $n-1$,
the function $\mathrm{Int}_{L^{\flat},\mathscr{V}}$ on $\mathbb{V}\setminus
L^{\flat}_{F}$ can be extended to an element in
$\mathscr{S}(\mathbb{V})^{\geq-1}$ which we denote by the same notation.

###### Proof.

Lemmas 4.9 and 2.8 imply that
${}^{\mathbb{L}}\mathcal{Z}(L^{\flat})_{\mathscr{V}}^{*}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{N}_{\mathrm{red}}}(\mathcal{N})\cap\mathrm{Gr}^{\prime}K_{0}^{\mathcal{Z}(L^{\flat})_{\mathscr{V}}}(\mathcal{N})$.
Lemma 2.7 implies that there exist finitely many classes
$\mathcal{K}_{i}\in\mathrm{Gr}^{\prime}K_{0}^{\mathcal{N}_{\mathrm{red}}}(\mathcal{N})_{\mathbb{Q}}$
together with $C_{i}\in\mathbb{Q}$ such that

$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}(x)\\\
=\sum_{i}C_{i}\cdot\chi(\mathcal{N},\mathcal{K}_{i}\cdot[\mathcal{O}_{\mathcal{Z}(x)}]).$

By Theorem 3.19 we may assume that $\mathcal{K}_{i}$ is supported on some
$\mathcal{N}_{\Lambda}$. Now we can apply Theorem 4.14 to conclude the proof.
∎

###  4.4. Partial Fourier transform

Let $L^{\flat}$ be a rank $n-1$ non-degenerate lattice in $\mathbb{V}$. Let
$\mathbb{W}=(L^{\flat}_{F})^{\bot}$. For any function $\varphi$ defined on
$\mathbb{V}\setminus L^{\flat}_{F}$, we define its partial Fourier transform
$\varphi^{\bot}$ as a function on $\mathbb{W}\setminus\\{0\\}$ by

(4.12)
$\varphi^{\bot}(x):=\int_{L^{\flat}_{F}}\varphi(x+y)dy,\quad\forall\,x\in\mathbb{W}.$

######  Theorem 4.16.

The partial Fourier transform
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\bot}\in\mathscr{S}(\mathbb{W})^{\geq-1}$
and is $\mathbb{W}^{\geq 0}$-invariant. In particular it is constant on
$\mathbb{W}^{\geq 0}$.

###### Proof.

It is easy to see that partial Fourier transform maps
$\mathscr{S}(\mathbb{V})$ to $\mathscr{S}(\mathbb{W})$. It remains to show
that the partial Fourier transform of
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\bot}\in\mathscr{S}(\mathbb{W})$
is supported on $\mathbb{W}^{\geq-1}$. For $x\in\mathbb{W}$, we have

$\widehat{\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\bot}}(x)=\widehat{\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}}(x),$

where $\widehat{\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}}(x)$ is
the Fourier transform of
$\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}\in\mathscr{S}(\mathbb{V})$.
Since $\widehat{\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}}$ is
supported on $\mathbb{V}^{\geq-1}$ by Theorem 4.15, we know that
$\widehat{\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\bot}}(x)$ is
supported on $\mathbb{W}^{\geq-1}$.

Since $\mathbb{W}$ is one-dimensional, $\mathbb{W}^{\geq m}$ is a full rank
lattice in $\mathbb{W}$ for any $m\in\mathbb{Z}$. By Lemma 4.11 and what we
just proved, $\operatorname{\mathrm{Int}}_{L^{\flat},\mathscr{V}}^{\bot}$ is
invariant under the translation of
$(\mathbb{W}^{\geq-1})^{\vee}=\mathbb{W}^{\geq 0}$. ∎

##  5\. Review of local densities and primitive local densities

In this section, we recall various explicit formulas of local density
polynomials following Section 5 of [HSY21].

###  5.1. Basic properties of local density and primitive local density
polynomials

######  Definition 5.1.

Let $M$ and $L$ be two hermitian $O_{F}$-lattices of rank $m$ and $n$
respectively. We use $\mathrm{Herm}_{L,M}$ to denote the scheme of hermitian
$O_{F}$-module homomorphisms from $L$ to $M$, which is a scheme of finite type
over $O_{F_{0}}$. Moreover, we define

$\mathrm{Den}(M,L):=\lim_{d\rightarrow\infty}\frac{\left|\mathrm{Herm}_{L,M}(O_{F_{0}}/(\pi_{0}^{d}))\right|}{q^{N\cdot
d_{L,M}}},$

where $d_{L,M}$ is the dimension of
$\mathrm{Herm}_{L,M}\otimes_{O_{F_{0}}}F_{0}$. More specifically, for a
hermitian $O_{F}$-lattice $L$ with hermitian from $(\,,\,)$, we define
$L_{(d)}\coloneqq L\otimes_{O_{F_{0}}}O_{F_{0}}/(\pi_{0}^{d})$ and
$(x,y)_{(d)}=\pi(\tilde{x},\tilde{y})\in O_{F_{0}}/(\pi_{0}^{d})$ where
$x,y\in L_{(d)}$ and $\tilde{x},\tilde{y}\in L$ are any lifts of $x,y$. Then

$\displaystyle\mathrm{Herm}_{L,M}(O_{F_{0}}/(\pi_{0}^{d}))=\\{\phi\in\mathrm{Hom}$
${}_{O_{F}}(L_{(d)},M_{(d)})\mid(\phi(x),\phi(y))_{(d)}\equiv(x,y)_{(d)}\text{
for all }x,y\in L_{(d)}\\}.$

It is well-known that there is a local density polynomial
$\mathrm{Den}(M,L,X)\in\mathbb{Q}[X]$ such that

(5.1) $\mathrm{Den}(M,L,q^{-2k})=\mathrm{Den}(M\obot H^{k},L).$

Moreover, we denote $\mathrm{Den}(M,L)=\mathrm{Den}(M,L,1)$ and

(5.2) $\mathrm{Den}^{\prime}(M,L)\coloneqq-2\cdot\frac{\partial}{\partial
X}\mathrm{Den}(M,L,X)|_{X=1}.$

Similarly, the primitive local density polynomial $\mathrm{Pden}(M,L,X)$ is
defined to be the polynomial in $\mathbb{Q}[X]$ such that

(5.3)
$\mathrm{Pden}(M,L,q^{-2k})=\lim_{d\to\infty}q^{-dn(2(m+2k)-n)}|\mathrm{Pherm}_{L,M\obot
H^{k}}(O_{F_{0}}/(\pi_{0}^{d}))|,$

where

$\displaystyle\mathrm{Pherm}_{L,M\obot H^{k}}(O_{F_{0}}/(\pi_{0}^{d}))$
$\displaystyle\coloneqq\\{\phi\in\mathrm{Herm}_{L,M\obot
H^{k}}(O_{F_{0}}/(\pi_{0}^{d}))\mid\text{ $\phi$ is primitive}\\}.$

Here we recall that $\phi\in\mathrm{Herm}_{L,M\obot
H^{k}}(O_{F_{0}}/(\pi_{0}^{d}))$ is primitive if
$\dim_{\mathbb{F}_{q}}((\phi(L)+\pi(M\obot H^{k}))/\pi(M\obot H^{k})=n$. In
particular, we have $\mathrm{Pden}(M,M)=\mathrm{Den}(M,M)$ since any
$\phi\in\mathrm{Herm}_{M,M}(O_{F_{0}}/(\pi_{0}^{d})$ is primitive.

Recall that without explicit mentioning, we assume $\epsilon=\chi(L)$. As an
analogue of (1.3) and (1.4), we define

(5.4)
$\displaystyle\mathrm{Pden}^{\prime}(L)\coloneqq-2\cdot\frac{\frac{\mathrm{d}}{\mathrm{d}X}\big{|}_{X=1}\mathrm{Pden}(I_{n},L,X)}{\mathrm{Den}(I_{n},I_{n})}\text{
and
}\mathrm{Pden}_{t}(L)\coloneqq\frac{\mathrm{Pden}(\Lambda_{t}^{\sharp},L)}{\mathrm{Den}(\Lambda_{t}^{\sharp},\Lambda_{t}^{\sharp})}.$

To save notation, we simply denote $\mathrm{Pden}_{0}(L)$ (resp.
$\mathrm{Pden}_{0}^{\prime}(L)$) by $\mathrm{Pden}(L)$ (resp.
$\mathrm{Pden}^{\prime}(L)$). We define

(5.5)
$\partial\mathrm{Pden}(L)\coloneqq\mathrm{Pden}^{\prime}(L)+\displaystyle\sum_{j=1}^{\frac{t_{\max}}{2}}c_{2j}\cdot\mathrm{Pden}_{2j}(L),$

where $c_{2j}$ is as in (1.5).

######  Lemma 5.2.

Let $L$ be a lattice. If there exists $x\in L$ such ${\mathrm{val}}(x)\leq-1$,
then
$\partial\mathrm{Den}(L)=\mathrm{Den}^{\prime}(L)=\partial\mathrm{Pden}(L)=\mathrm{Pden}^{\prime}(L)=0$.

###### Proof.

Assume $M\cong I_{n}$ or $M\cong\Lambda_{2t}$ for some $t$. Then
$\mathrm{Den}(M\obot H^{k},L)=0$ and $\mathrm{Pden}(M\obot H^{k},L)=0$ since
there is no vector in $M$ with valuation less than or equal to $-1$. ∎

Now we record several results that describe the relation between local density
and primitive local density polynomials.

######  Lemma 5.3.

[HSY21, Lemma 5.1] Let $M$ and $L$ be lattices of rank $m$ and $n$. Then we
have

$\mathrm{Den}(M,L,X)=\sum_{L\subset L^{\prime}\subset
L_{F}}(q^{n-m}X)^{\ell(L^{\prime}/L)}\mathrm{Pden}(M,L^{\prime},X),$