# Non-reductive special cycles and twisted arithmetic fundamental lemma

Zhiyu Zhang  Department of Mathematics, Stanford University, 450 Jane Stanford
Way Building 380, Stanford, CA 94305<EMAIL_ADDRESS>
###### Abstract.

We consider arithmetic analogs of the relative Langlands program and
applications of new non-reductive geometry. Firstly, we introduce mirabolic
special cycles, which produce special cycles to many Hodge type Rapoport-Zink
spaces via pullbacks e.g. Kudla–Rapoport cycles. Secondly, we formulate
arithmetic intersection problems for these cycles and formulate a method of
arithmetic induction. As a main example, we formulate arithmetic twisted
Gan–Gross–Prasad conjectures on unitary Shimura varieties and prove a key
twisted arithmetic fundamental lemma using mirabolic special cycles,
arithmetic inductions and Weil type relative trace formulas.

###### Contents

  1. 1 Introduction
  2. 2 Mirabolic special cycles on Rapoport–Zink spaces
  3. 3 Twisted orbits, orbital integrals and fundamental lemmas
  4. 4 Twisted arithmetic fundamental lemmas
  5. 5 Arithmetic induction systems and reductions
  6. 6 Twisted unitary Shimura varieties and special cycles
  7. 7 Local–global decomposition formulas and transfers
  8. 8 The proof of twisted AFL
  9. 9 Arithmetic twisted Gan-Gross-Prasad conjectures for Asai L-functions

##  1\. Introduction

###  1.1. Arithmetic relative Langlands program and non-reductive $p$-adic
geometry

Non-reductive groups are used a lot in the theory of modular forms and
automorphic forms for reductive groups, see e.g. Fourier coefficients,
parabolic inductions, Jacobi groups and Jacobi forms. In the relative
Langlands program (see e.g. recent work of Ben-Zvi–Sakellaridis–Venkatesh
[BZSV]), we have a lot of interesting examples of period integrals and
$L$-functions for a reductive group $G$ acting on a nice space $X$. In pursuit
of an arithmetic relative Langlands program relating algebraic cycles and
$L$-functions (see e.g. the survey [Zhang2024survey]), we would like to have
geometric objects similar to Shimura varieties attached to these $G$-spaces. A
test case will be $X=G/H$ for a subgroup $H\leq G$. However, non-reductive $H$
may not even give complex algebraic varieties. A typical example is the
embedding (related to Fourier coefficients for modular forms and horocycle
flows for dynamic systems)

$\mathbb{G}_{a}=\begin{pmatrix}1&*\\\ 0&1\end{pmatrix}\to\mathrm{SL}_{2}$

which on adelic quotients gives real circles $S^{1}$ on complex modular curves
$X_{0}(N)$.

One motivation of the paper is new non-reductive geometry in the
nonarchimedean world (or how to find $p$-adic analogs of circles). Via
mirabolic special cycles and pullbacks, in this paper we construct local
geometric cycles for _a nice representation_ $X=V$ of $G$.

We consider the following local arithmetic relative Langlands program. Let $p$
be a prime and $F_{0}/\mathbb{Q}_{p}$ be a local field. Let
$(H,\mu_{H},b_{H})\to(G,\mu_{G},b_{G})$ be an embedding of local Shimura data
over $F_{0}$. For compatible level $K_{H}\leq H(F_{0}),K_{G}\leq G(F_{0})$, we
expect an embedding of relevant Rapoport–Zink spaces (which exists in the
parahoric Hodge type cases [HP17][kim2018][HK2019Hodgeparahoric])

$\mathcal{N}_{H}:=\mathcal{N}_{H,\mu_{H},b_{H},K_{H}}\to\mathcal{N}_{G}:=\mathcal{N}_{G,\mu_{G},b_{G},K_{G}}$

as formal schemes over $p$-adic integers, compatible with actions of
$J_{b_{H}}(F_{0})\to J_{b_{G}}(F_{0})$. If $b_{G}$ is basic, then $J_{b_{G}}$
is an inner form of $G$. For $g\in J_{b_{G}}(F_{0})/J_{b_{H}}(F_{0})$, we have
the special cycle

$\mathcal{Z}_{G/H}(g):=g\mathcal{N}_{H}\to\mathcal{N}_{G}.$

Let $X$ be a $G$-variety with a companion $J_{b_{G}}$-variety $X_{b_{G}}$ over
$F_{0}$. In good cases, for regular $x\in X_{b_{G}}(F_{0})$ we expect the
existence of _special Rapoport–Zink cycles_ (with level $K_{G}$)

$\mathcal{Z}_{X}(x)\to\mathcal{N}_{G}$

such that $g\mathcal{Z}_{X}(x)=\mathcal{Z}_{X}(gx),\,\forall g\in
J_{b_{G}}(F_{0})$. When $X=G/H$ is affine spherical, we recover the cycle
$\mathcal{Z}_{H,G}(g)=g\mathcal{N}_{H}$ as above. In the group case $G=H\times
H$ and $X=H$, we recover the graph of any automorphism $g\in J_{b_{H}}$ on
$\mathcal{N}_{H}$. In this paper, we will also encounter the Galois twisted
group case and study twisted graphs. We expect such constructions for a good
class of $G$-spaces (see e.g. [BZSV]) and admits a derived enhancement to
general $x\in X(F_{0})$.

In this paper, we focus on the case $X=V$ is a nice representation of $G$. Our
first main result is a discovery of new algebraic cycles on general linear
Rapoport–Zink spaces — the mirabolic special cycles. Then, we may achieve the
above construction via pullbacks from universal mirabolic special cycles
attached to $V$. For instance, we recover Kudla–Rapoport type cycles on basic
unitary Rapoport–Zink spaces [KR2011local] for $X_{b_{G}}=\mathbb{V}$ where
$J_{b_{G}}=\mathrm{U}(\mathbb{V})$. A more general theory of Rapoport–Zink
cycles will be studied in a future work [ZZZ].

Let $F/F_{0}$ be the unramified quadratic extension. Let $V$ be a
$n$-dimensional $F$-vector space. Choosing a framing object
$(\mathbb{X},\iota_{\mathbb{X}})$, consider the basic general linear
Rapoport–Zink space for $\mathrm{Res}_{F/F_{0}}\mathrm{GL}(V)$:

$\mathcal{N}^{\mathrm{GL}}_{n}\to\mathrm{Spf}O_{\breve{F_{0}}}$

as the moduli space of $(X,\iota,\rho)$ over test schemes $S$, where $X$ is a
strict formal $O_{F_{0}}$-module over $S$ of relative $O_{F_{0}}$-height $2n$
and dimension $n$, $\iota$ is a $O_{F}$-action on $X$ of Kottwitz signature
$(n-1,1)$, $\rho$ is a $O_{F}$-linear height $0$ quasi-isogeny $\rho:X\times
S/p\to\mathbb{X}\times S/p$. Let $\mathbb{V}$ be the $n$-dimensional
$F$-vector space of special quasi-homomorphisms attached to
$(\mathbb{X},\iota_{\mathbb{X}})$ as in the work of Kudla-Rapoport
[KR2011local]. For non-zero $u\in\mathbb{V}$, the work of Kudla–Rapoport
[KR2011local] introduces Kudla–Rapoport divisors $\mathcal{Z}(u)$ on the
unitary Rapoport zink space $\mathcal{N}_{n}$, which can be thought as an
arithmetic geometrization of spherical Weil representations.

######  Theorem 1.1.1 (Theorem 2.3.2).

For non-zero $u\in\mathbb{V}$, we have mirabolic special cycles

$\mathcal{Z}^{\mathrm{GL}}(u)\to\mathcal{N}^{\mathrm{GL}}_{n}$

which are relative Cartier divisors. For a natural embedding of the unitary
Rapoport–Zink space $\mathcal{N}_{n}\to\mathcal{N}_{n}^{\mathrm{GL}}$, we
recover the Kudla–Rapoport divisors via the pullback diagram:

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See Theorem 2.3.2 for more geometric properties, which is similar to the work
of Kudla–Rapoport [KR2011local, Theorem 1.1]. Let
$\mathrm{GL}(u)\leq\mathrm{GL}(\mathbb{V})$ be the mirabolic subgroup defined
as the stabilizer of $u$, then the generic fiber of
$\mathcal{Z}^{\mathrm{GL}}(u)$ is an analog of local Shimura varieties for the
(non-reductive) mirabolic group $\mathrm{GL}(u)$. Given any Rapoport–Zink
space $\mathcal{N}_{G}$ and a $n$-dimensional representation $V$ of $G$
inducing an embedding $\mathcal{N}_{G}\to\mathcal{N}_{n}^{\mathrm{GL}}$, we
obtain _special linear cycles_ $Z_{V}(u)\to\mathcal{N}_{G}$ ($u\not=0$) as
(derived) pull backs of $Z^{\mathrm{GL}}(u)$, see Remark 2.3.3. These cycles
are arithmetic analogs of fibers for the variety of triples in the theory of
Coulomb branches [BFN], which may be explored further. From moduli definitions
of mirabolic special cycles, we have

$\mathcal{Z}_{V_{1}\oplus
V_{2}}(u_{1},u_{2})=\mathcal{Z}_{V_{1}}(u_{1})\cap_{\mathcal{N}_{G}}\mathcal{Z}_{V_{2}}(u_{2}),\,\,\forall
u_{i}\not=0.$

In this way, Kudla–Rapoport conjectures [KR2011local][LiZhangKR2019] can be
viewed as studying arithmetic degrees of (a derived version of) special linear
cycles attached to the representation $V^{\oplus\dim V}$ of $\mathrm{U}(V)$.
Therefore, we expect these special linear cycles will be helpful to formulate
and study more arithmetic intersection problems, which will be studied in a
future work. For instance, see [WZhangBesselAFL2022] and the role of Kudla-
Rapoport cycles in arithmetic fundamental lemmas [AFL-Wei2019][AFL-
JEMS][ZZhang2021] and Kudla–Rapoport conjectures [LiZhangKR2019]. In this
paper, they also play a key role in the proof of a twisted arithmetic
fundamental lemma whose statement does not involve such cycles!

###  1.2. Arithmetic intersections and arithmetic induction systems

Now, we consider arithmetic intersection problems from the above program.
Assume that $\mathcal{N}_{G}$ is regular. Let $X_{1},...,X_{m}$ be varieties
with $G$-actions such that we have constructed above special cycles
$\mathcal{Z}_{X_{i}}(x_{i})\to\mathcal{N}_{G}$ on an open $J_{b_{G}}$-stable
subset $x_{i}\in U_{i}\subseteq X_{i,b_{G}}$. Assume that there exists a dense
open $J_{b_{G}}$-stable subset
$U_{X_{1},\ldots,X_{m}}\subseteq\prod_{i=1}^{m}U_{X_{i}}$, such that for any
$(x_{i})\in U_{X_{1},\ldots,X_{m}}$, the intersection
$\cap_{i=1}^{m}\mathcal{Z}_{X_{i}}(x_{i})$ is a proper scheme, and
$\sum_{i=1}^{m}\mathrm{codim}\mathcal{Z}_{X_{i}}(x_{i})=\dim\mathcal{N}_{G}$.
Then we consider the $J_{b_{G}}$-invariant arithmetic invariant functional

$\mathrm{Int}_{X_{1},\ldots,X_{m}}^{G}:U_{X_{1},\ldots,X_{m}}\to\mathbb{Q},$
$\mathrm{Int}_{X_{1},\ldots,X_{m}}^{G}(x_{1},\ldots,x_{m}):=\mathcal{Z}_{X_{1}}(x_{1})\cap^{\mathbb{L}}\ldots\cap^{\mathbb{L}}\mathcal{Z}_{X_{m}}(x_{m}):=\chi(\mathcal{N}_{G},\mathcal{O}_{\mathcal{Z}_{X_{1}}(x_{1})}\otimes^{\mathbb{L}}\ldots\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}_{X_{m}}(x_{m})}).$

Moreover, we expect _arithmetic fundamental lemmas_ (AFL) and _arithmetic
transfers_ (AT) relating $\mathrm{Int}_{X_{1},\ldots,X_{m}}^{G}$ with analytic
integrals of certain test functions on these orbits. In good cases these
special cycles may be globalized and used to formulate Gross–Zagier type
formulas [GrossZagier] for Shimura varieties, with applications to
Beilinson–Bloch–Kato conjectures on $L$-functions and motives. The unitary
arithmetic Gan–Gross–Prasad conjectures [GGP-AGGP][AFL-Invent] and arithmetic
Siegel–Weil formulas in the Kudla program [Kudla-Annals97][KRY-
book][LiZhangKR2019][Li-Liu][LiZhangGSpin] are well-studied examples. See also
the work of [Madapusi-derived] on derived special cycles and some extensions
of the Kudla program.

####  1.2.1. Arithmetic induction systems

To understand the above arithmetic intersection questions, we introduce the
notion of arithmetic induction systems for good tuples $(G,X=X_{1},Y=X_{2})$
(Definition 5.1.2), which allows an inductive understanding of arithmetic /
analytic functionals. The idea is already crucially used in the recent proof
of Kudla–Rapoport conjectures [LiZhangKR2019] and arithmetic fundamental
lemmas [AFL-Wei2019][AFL-JEMS][ZZhang2021].

In this paper, guided by the above program, we mainly study arithmetic analogs
of the twisted Gan–Gross–Prasad conjectures [twistedGGP][Wang-TGGP] on twisted
Asai L-functions. As a key local conjecture, we study and prove twisted
arithmetic fundamental lemmas (Conjecture 4.0.8) via local arithmetic
inductions and global methods.

######  Theorem 1.2.1 (Proposition 5.3.2, Proposition 5.4.1).

Both sides of the twisted arithmetic fundamental lemma Conjecture 4.0.8 are
arithmetic induction systems.

The proof uses above mirabolic special cycles. In the sequel [ZZZ], we will
study more examples of arithmetic intersection questions and prove they are
arithmetic induction systems.

###  1.3. Twisted arithmetic fundamental lemma and arithmetic Gan-Gross-Prasad
conjectures

Let $F/F_{0}$ be an unramified quadratic extension. We consider

  * •

(arithmetic unitary side) Let $V=(V,h_{0})$ be a split $n$-dimensional
hermitian space over $F$. The group $G:=\mathrm{GL}(V)$ acts naturally on the
set of (non-degenerate) hermitian structures $h\in X=\mathrm{Herm}(V)$ on $V$
with two orbits – split orbits and non-split orbits (determined by isomorphism
classes of $(V,h)$). Let $H_{V,h}:=\mathrm{U}(V,h)$, which acts naturally on
$X_{V,h}:=\mathrm{GL}(V)/\mathrm{U}(V,h)$ (resp. $Y_{V,h}=V$). For a non-split
hermitian space $\mathbb{V}=(V,h_{1})$ and
$H_{\mathbb{V}}=\mathrm{U}(\mathbb{V})$, we consider unramified unitary
Rapoport–Zink spaces $\mathcal{N}_{n}$ for $(V,h_{0})$ with a natural
embedding $\mathcal{N}_{n}\to\mathcal{N}_{n}^{\mathrm{GL}}$. On
$\mathcal{N}_{n}$, we have _twisted fixed cycles_
$\mathcal{N}_{n}^{\mathrm{Herm}}(g)=g\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{N}_{n},\forall
g\in\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})$ (resp. _Kudla–Rapoport
divisors_ $\mathcal{Z}(u),\forall 0\not=u\in\mathbb{V}$ ).

  * •

(analytic general linear side) Let $V_{0}$ be a $n$-dimensional vector space
over $F_{0}$ and $V_{0}^{*}$ be its linear dual. Let
$H^{\prime}=\mathrm{GL}(V_{0})$. Let $X^{\prime}=\mathrm{GL}(V_{0})$ with the
conjugacy action of $H^{\prime}$, and $Y^{\prime}=V_{0}^{\prime}:=V_{0}\times
V_{0}^{*}$ with the natural action of $H^{\prime}$.

We have a matching of regular semi-simple orbits

$[H^{\prime}\backslash(X^{\prime}\times
Y^{\prime})]_{\mathrm{rs}}\cong[H_{V}\backslash(X_{V}\times
Y_{V})]_{\mathrm{rs}}\coprod[H_{\mathbb{V}}\backslash(X_{\mathbb{V}}\times
Y_{\mathbb{V}})]_{\mathrm{rs}},\,\,(\gamma,u^{\prime})\leftrightarrow(g,u).$

with matching characteristic polynomials
$\mathrm{charpol}(\gamma)=\mathrm{charpol}(g)$ and inner products
$u_{2}\gamma^{i}u_{1}=(g^{i}u,u),i\in\mathbb{Z}$. Choose a self-dual lattice
$L$ in $V$. Choose an orthogonal basis $\\{e_{i}\\}_{i=1}^{n}$ of $L$, and let
$L_{0}=\sum_{i=1}^{n}O_{F_{0}}e_{i}$. We may assume that
$V_{0}=L_{0}\otimes\mathbb{Q}$. We consider the lattice
$L^{\prime}=L_{0}\times L_{0}^{*}\subseteq Y^{\prime}=V_{0}^{\prime}$. Define
standard test functions (10)

$\Phi_{L^{\prime}}^{\prime}=1_{\mathrm{GL}(L_{0})}\times 1_{L^{\prime}}\in
C_{c}^{\infty}(X^{\prime}\times
Y^{\prime}),\,\Phi_{L}=1_{\mathrm{Herm}(L)}\times 1_{L}\in
C_{c}^{\infty}(X_{V}\times Y_{V}).$

######  Theorem 1.3.1 (Twisted Fundamental Lemma, Theorem 3.2.3).

For any regular semi-simple orbits
$(\gamma,u^{\prime})\in(\mathrm{GL}_{n}(V_{0})\times
V_{0}^{\prime})(F_{0})_{\mathrm{rs}}$, we have

$\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{L^{\prime}})=\begin{cases}\mathrm{Orb}((g,u),\Phi_{L}),&\text{if
$(\gamma,u^{\prime})$ matches $(g,u)\in(\mathrm{GL}(V)/\mathrm{U}(V)\times
V)(F_{0})_{\mathrm{rs}}$.}\\\ 0,&\text{if $(\gamma,u^{\prime})$ matches
$(g,u)\in(\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})\times\mathbb{V})(F_{0})_{\mathrm{rs}}$.}\\\
\end{cases}$

For
$(g,u)\in(\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})\times\mathbb{V})(F_{0})_{\mathrm{rs}}$,
consider the arithmetic intersection number (12)

$\mathrm{Int}^{\mathrm{Herm},\mathbb{V}}(g,u):=(g\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{N}_{n})\cap^{\mathbb{L}}_{\mathcal{N}_{n}}\mathcal{Z}(u)\in\mathbb{Q}.$

######  Theorem 1.3.2 (Twisted Arithmetic Fundamental Lemma, Conjecture 4.0.8,
Theorem 4.0.8).

For any
$(g,u)\in(\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})\times\mathbb{V})(F_{0})_{\mathrm{rs}}$
matching $(\gamma,u^{\prime})\in(\mathrm{GL}_{n}\times
V_{0}^{\prime})(F_{0})_{\mathrm{rs}}$, we have

$\partial\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{L^{\prime}})=-\mathrm{Int}^{\mathrm{Herm},\mathbb{V}}(g,u)\log
q.$

Compared to the semi-Lie AFL [AFL-Wei2019] which uses the diagonal
$\mathrm{U}(V)\to\mathrm{U}(V)\times\mathrm{U}(V)$, our twisted AFL uses the
twisted diagonal $\mathrm{U}(V)\to\mathrm{GL}(V)$ which is quite different.

Such twisted AFL is a crucial ingredient for a relative trace formula approach
towards arithmetic analogs of twisted Gan–Gross–Prasad conjectures
[twistedGGP] using twisted Fourier-Jacobi cycles following [Liu-FJcycles], for
which we give a formulation at the end of this paper.

######  Conjecture 1.3.3 (Arithmetic Twisted Gan-Gross-Prasad conjecture,
Conjecture 9.0.3).

Let $F/F_{0}$ (resp. $E/F_{0}$) be a CM (resp. totally real) quadratic
extension of number fields. Let $\mu$ be a conjugate symplectic automorphic
character of $\mathbb{A}_{F}^{\times}/F^{\times}$ of weight one. Let $\pi$ be
a cuspidal automorphic representation of
$\mathrm{U}(V_{E_{0}})(\mathbb{A}_{E_{0}})$. Let $\Pi$ be the automorphic
representation of $\mathrm{GL}_{n}(\mathbb{A}_{E})$ as the base change of
$\pi$. Assume that $\Pi_{\infty}$ is relevant [Liu-FJcycles, Definition 1.2].
Then the following statements are equivalent.

  1. (1)

The twisted arithmetic functional $\mathrm{FJ}_{\epsilon}\not=0$.

  2. (2)

We have $L^{\prime}(1/2,\Pi,\mathrm{As}_{E/F}\otimes\mu)\not=0$ and
$\mathrm{Hom}_{\mathbb{C}[\mathrm{U}(V)(\mathbb{A}_{f})]}(\pi_{f}\otimes\Omega(\mu,\epsilon),\mathbb{C})\not=0.$

Allowing such twisting in arithmetic Gan–Gross–Prasad conjectures will be
useful to handle arithmetics of motives (e.g. Asai motives), see for instance
works of [liu2016hirzebruch] [loeffler2023euler][loeffler2020p] related to
modular forms and elliptic curves.

The proof of our twisted AFL is based on induction and globalization, using
above mirabolic special cycles, arithmetic induction systems and the method of
[AFL-Wei2019]. Firstly, we show that both sides of twisted AFL are arithmetic
induction systems under new relative Cayley maps on hermitian structures.
Secondly, we consider globalizations over a totally real field $F_{0}$, which
are expected to be automorphic distributions for $\mathrm{SL}_{2,F_{0}}$. We
show the adelic difference is automorphic, using the modularity of geometric
theta series on unitary Shimura varieties with hyperspecial levels
[BHKRY][AFL-JEMS]. We prove local-global decomposition formulas, which
indicate the global sides are correct generalizations of the local sides. By
such local-global compatibility and arithmetic induction, we deduce many
Fourier coefficients of the adelic difference of two global sides vanish. We
deduce the difference is a constant by modularity. Hence the local difference
vanishes, and we obtain the twisted AFL identity.

Such globalization can be understood in general via what we call _Weil type
relative trace formulas_. In a general setup, let $(H_{1},G,H_{2})$ be a tuple
of algebraic groups over a number field $F_{0}$ where $H_{i}\subseteq G$
($i=1,2$) are subgroups and $G$ is reductive. Consider the kernel function

$K_{f}(x,y)=\sum_{\gamma\in G(F)}f(x^{-1}\gamma y),\quad
f\in\mathcal{S}(G(\mathbb{A})),x,y\in G(\mathbb{A}).$

Compared to the usual relative trace formula, we need an algebraic
representation $V$ of $H_{1}$. Then the _Weil type relative trace formula_ for
$(H_{1},G,H_{2},V)$ is the distribution (which may be not convergent in
general)

(1)
$I(f,\phi)=\int_{[H_{1}]\times[H_{2}]}K_{f}(h_{1},h_{2})\theta_{\phi}(h_{1})dh_{1}dh_{2},\quad
f\in\mathcal{S}(G(\mathbb{A})),\phi\in\mathcal{S}(V(\mathbb{A}))$

where theta series $\theta_{\phi}(g)=\sum_{u\in V(F_{0})}\phi(g^{-1}u)$ gives
an automorphic realization of $\phi\in\mathcal{S}(V(\mathbb{A}))$. If
$f\otimes\phi$ has regular supports at a place of $F_{0}$, then we have a
decomposition into orbital integrals

$I(f,\phi)=\sum_{(\gamma,u)\in[H\backslash(G/H\times
V)]_{\mathrm{rs}}(F_{0})}\mathrm{Orb}((\gamma,u),f\otimes\phi).$

In this paper, we consider similar _arithmetic Weil type relative formulas_

$I^{1}(f,\phi):=(Z_{V}(\phi),R(f)_{*}\mathrm{Sh}_{H})_{\mathrm{Sh}_{G}}$

where $Z_{V}(\phi),\phi\in\mathcal{S}(V(\mathbb{A}_{f}))$ are global versions
of special linear cycles and $R(f),f\in\mathcal{S}(G(\mathbb{A}_{f}))$ are
Hecke translations on the Shimura variety $\mathrm{Sh}_{G}$, and
$(-,-)_{\mathrm{Sh}_{G}}$ is an Arakelov type arithmetic intersection pairing
defined using regular integral models.

###  1.4. More applications and further questions

It is interesting to study the geometry of these new special linear cycles
(and relations to [BFN][BZSV]), which could be used in arithmetic relative
Langlands program and may have globalization in good cases. We expect our
method to give a uniform way of formulating and proving arithmetic fundamental
lemmas for other good tuples $(G,X,Y)$ combining the idea of twisted fixed
cycles and linear special cycles. The same method can give an arithmetic
transfer result for vertex lattices as in [ZZhang2021]. We could formulate
similar questions for $H=\mathrm{U}(V)\to G=\mathrm{U}(V_{E_{0}})$ where
$E_{0}/F_{0}$ is any quadratic étale algebra. For simplicity of notations, we
choose not to include arithmetic transfers and general twisted cases.

Finally, we give an overview of the paper. Firstly we introduce the local
setup and formulate a twisted AFL. In Section 2, we introduce and study
mirabolic special cycles on Rapoport–Zink spaces. In Section 3, we formulate
the matching of orbits and orbital integrals and recall the twisted
fundamental lemma. In Section 4, we introduce twisted fixed cycles and
formulate our twisted AFL. In Section 5, we introduce arithmetic induction
systems and new relative Cayley maps, and show both sides of the twisted AFL
are such examples using mirabolic special cycles. In Section 6, we introduce
global Kudla–Rapoport divisors and twisted complex multiplication cycles on
unitary Shimura varieties. In Section 7, we form global analytic and geometric
generating functions and prove local-global decomposition formulas for non-
singular Fourier coefficients via basic uniformizations. In Section 8, we
establish the twisted AFL. In Section 9, we formulate twisted arithmetic Gan-
Gross-Prasad conjecture and indicate its relation to our twisted AFL.

### Acknowledgements

I thank M. Rapoport, C. Li, A. Mihatsch and W. Zhang for helpful discussions.
I am also grateful for the hospitality of the Simons Laufer Mathematical
Sciences Institute in Spring 2023, where the project began.

###  1.5. Notations and Conventions

  * •

Let $X$ be an algebraic variety with an action of an algebraic group $H$ over
a field $F_{0}$. We say $x\in X(F_{0})$ is called regular semi-simple if the
stabilizer of $x$ inside $G$ is trivial, and the orbit of $x$ is closed in
$X$. We often denote by $(-)_{\mathrm{rs}}$ (resp. $[-]_{\mathrm{rs}}$) the
space of regular semi-simple elements (resp. orbits).

  * •

Let $p$ be an odd prime.

  * •

Let $F_{0}/\mathbb{Q}_{p}$ be a finite extension. Choose a uniformizer
$\varpi$ of $F_{0}$. Denote by $\breve{F_{0}}$ the $p$-adic completion of a
maximal unramified extension of $F_{0}$ and $O_{\breve{F_{0}}}$ be its ring of
integers.

  * •

Denote by $\mathbb{F}_{q}$ (resp. $\mathbb{F}$) the residue field of $F_{0}$
(resp. $\breve{F_{0}}$). Let
$\sigma_{\breve{F_{0}}/F_{0}}\in\mathrm{Gal}(\breve{F_{0}}/F_{0})$ be the
canonical lifting of $q$-Frobenius $x\rightarrow x^{q}$ on $\mathbb{F}$.

  * •

$F/F_{0}$ is an unramified quadratic extension of $p$-adic local fields. We
fix an embedding $F\to\breve{F_{0}}$. Denote by
$\overline{(-)}\in\mathrm{Gal}(F/F_{0})$ the non-trivial Galois involution.

  * •

Denote by $\mathrm{val}=\mathrm{val}_{F_{0}}:F^{\times}_{0}\to\mathbb{Z}$ the
standard valuation of $F_{0}$, in particular $\mathrm{val}_{F_{0}}(\varpi)=1$.
Let $\eta=\eta_{F/F_{0}}$ be the quadratic character
$\eta(x)=(-1)^{\mathrm{val}_{F_{0}}(x)}:F^{\times}_{0}\to\\{\pm 1\\}$.

  * •

Let $t\in\mathbb{R}$ and $n\in\mathbb{Z}$. For
$h\in\mathrm{SL}_{2}(\mathbb{R})$ under the Iwasawa decomposition

$h=\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}\begin{pmatrix}a^{1/2}&0\\\
0&a^{-1/2}\end{pmatrix}\kappa_{\theta},\quad a\in\mathbb{R}_{+},\quad
b\in\mathbb{R},$

and

$\kappa_{\theta}=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}\in\mathrm{SO}_{2}(\mathbb{R}),$

The standard weight $n$ Whittaker function on
$h\in\mathrm{SL}_{2}(\mathbb{R})$ is defined as

(2) $W_{t}^{(n)}(h):=|a|^{n/2}e^{2\pi it(b+ai)}e^{in\theta}.$

  * •

In global set up, denote by $F_{0}$ a totally real number field. Let $F_{0,+}$
be the cone of totally positive elements in $F_{0}$. Let $F/F_{0}$ be a CM
quadratic extension. Denote by $\Phi$ a CM type of $F$. The CM type $\Phi$ is
called unramified at $p$, if
$\Phi\otimes\mathbb{Q}_{p}:F\otimes\mathbb{Q}_{p}\to\overline{\mathbb{Q}}_{p}$
is induced from a CM type of the maximal unramified (over $\mathbb{Q}_{p}$)
subalgebra of $F\otimes\mathbb{Q}_{p}$.

  * •

In global set up, we denote by $\mathbb{A}$, $\mathbb{A}_{0}$ for the adele
rings of $\mathbb{Q}$ and $F_{0}$ respectively. Denote by $(-)_{f}$ the terms
of finite adeles. Denote by $(-)_{p}$ (resp. $(-)_{v}$) the local term at a
place $p$ of $\mathbb{Q}$ (resp. a place $v$ of $F_{0}$). Let $\Delta$ be a
finite collection of places of $F_{0}$. Denote by $(-)_{\Delta}$ the product
of $(-)_{v}$ over all places $v\in\Delta$.

  * •

In global set up, for Weil representations, we use the additive character
$\psi_{F_{0}}:F_{0}\backslash\mathbb{A}_{F_{0}}\rightarrow\mathbb{C}$ given by
$\psi_{F_{0}}:=\psi_{\mathbb{Q}}\circ\mathrm{Tr}_{F_{0}/\mathbb{Q}}$, where
$\psi_{\mathbb{Q}}:\mathbb{Q}\backslash\mathbb{A}\rightarrow\mathbb{C}$ is the
standard additive character.

  * •

For a compact open subgroup $K_{0}\subseteq\mathrm{SL}_{2}(\mathbb{A}_{0,f})$,
denote by $\mathcal{A}_{\rm
hol}(\mathrm{SL}_{2}(\mathbb{A}_{F_{0}}),K_{0},n)_{\overline{\mathbb{Q}}}$ the
space of holomorphic automorphic forms on
$\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})$ that are left invariant under
$\mathrm{SL}_{2}(F_{0})$, right invariant under $K_{0}$, of parallel weight
$n$ and with Fourier coefficients in the algebraic closure
$\overline{\mathbb{Q}}$ of $\mathbb{Q}$.

##  2\. Mirabolic special cycles on Rapoport–Zink spaces

Let $F/F_{0}$ be an unramified quadratic extension of $p$-adic local fields.
For $n\geq 1$, there are two isomorphism classes of $n$-dimensional $F/F_{0}$
hermitian spaces, and we denote by $V$ the split one and $\mathbb{V}$ the non-
split one.

###  2.1. The Rapoport–Zink spaces

Let $r,s\geq 0$ be two integers such that $r+s=n$. For any
$\mathrm{Spf}O_{\breve{F}_{0}}$-scheme $S$, _a basic tuple $(X,\iota)$ of
signature $(r,s)$_ over $S$ is the following data:

  * •

$X$ is a formal $O_{F_{0}}$-module over $S$ of relative $O_{F_{0}}$-height
$2n$ and dimension $n$ which is strict, i.e. the induced action of $O_{F_{0}}$
on $\mathrm{Lie}X$ is via the structure morphism
$O_{F_{0}}\to\mathcal{O}_{S}$.

  * •

$\iota:O_{F}\to\mathrm{End}(X)$ is an $O_{F}$-action on $X$ that extends the
$O_{F_{0}}$-action such that the _Kottwitz condition_ of signature $(r,s)$
holds for all $a\in O_{F}$:

(3)
$\mathrm{charpol}(\iota(a)\mid\mathrm{Lie}X)=(T-a)^{r}(T-\overline{a})^{s}\in\mathcal{O}_{S}[T].$

_A principally polarized basic tuple of signature $(r,s)$_ is a tuple
$(X,\iota,\lambda)$ where $(X,\iota)$ is a basic tuple of sign $(r,s)$ and
$\lambda:X\to X^{\vee}$ is a principal polarization that is $O_{F}/O_{F_{0}}$
semi-linear, i.e. the Rosati involution $\mathrm{Ros}_{\lambda}$ on
$\iota:O_{F}\to\mathrm{End}(X)$ agrees with $a\mapsto\overline{a}$. An
isomorphism $(X_{1},\iota_{1})\cong(X_{2},\iota_{2})$ (resp.
$(X_{1},\iota_{1},\lambda_{1})\cong(X_{2},\iota_{2},\lambda_{2})$) between two
such basic tuples (resp. principally polarized basic tuples) is an
$O_{F}$-linear isomorphism $\varphi:X_{1}\cong X_{2}$ (resp.
$\varphi:X_{1}\cong X_{2}$ such that $\varphi^{*}(\lambda_{2})=\lambda_{1}$).

Up to $O_{F}$-linear quasi–isogenies, there exists a unique such tuple
$(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$ of sign $(n-1,1)$ over
$\mathbb{F}$. Fix one schoice of
$(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$ as the framed
principally polarized basic tuple.

######  Definition 2.1.1.

The general linear Rapoport–Zink space is the functor

$\mathcal{N}_{n}^{\mathrm{GL}}\to\mathrm{Spf}O_{\breve{F}_{0}}$

sending $S$ to the set of isomorphism classes of tuples $(X,\iota,\rho)$,
where

  * •

$(X,\iota)$ is a basic tuple of signature $(n-1,1)$ over $S$.

  * •

$\rho:X\times_{S}{\overline{S}}\to\mathbb{X}\times_{\mathbb{F}}\overline{S}$
is an $O_{F}$-linear quasi-isogeny of height $0$ over
$\overline{S}:=S\times_{O_{\breve{F}_{0}}}\mathbb{F}$.

######  Definition 2.1.2.

The unitary Rapoport–Zink space is the functor

$\mathcal{N}_{n}\to\mathrm{Spf}O_{\breve{F}_{0}}$

sending $S$ to the set of isomorphism classes of tuples
$(X,\iota,\lambda,\rho)$, where

  * •

$(X,\iota,\lambda)$ is a principally polarized basic tuple of signature
$(n-1,1)$ over $S$.

  * •

$\rho:X\times_{S}{\overline{S}}\to\mathbb{X}\times_{\mathbb{F}}\overline{S}$
is an $O_{F}$-linear quasi-isogeny of height $0$ over the reduction
$\overline{S}:=S\times_{O_{\breve{F}_{0}}}\mathbb{F}$ such that
$\rho^{*}(\lambda_{\mathbb{X},\overline{S}})=\lambda_{\overline{S}}$.

######  Proposition 2.1.3.

The functor $\mathcal{N}_{n}^{\mathrm{GL}}$ (resp. $\mathcal{N}_{n}$) is
representable by a formal scheme locally formally of finite type, which is
formally smooth over $\mathrm{Spf}O_{\breve{F}_{0}}$ of relative dimension
$2(n-1)$ (resp. $n-1$).

###### Proof.

The representability follows from the method of Rapoport–Zink [RZ96, Thm.
2.16]. The formal smoothness and dimension formulas follow from
Grothendieck–Messing theory. ∎

###  2.2. Symmetries and involutions on Rapoport–Zink spaces

Let $\mathbb{E}$ be a formal $O_{F_{0}}$-module over $\mathbb{F}$ of relative
height $2$ and dimension $1$. Complete $\mathbb{E}$ into a principally
polarized framing basic triple
$(\mathbb{E},\iota_{\mathbb{E}},\lambda_{\mathbb{E}})$ of _signature $(1,0)$_
over $\mathbb{F}$. Denote by
$(\mathcal{E},\iota_{\mathcal{E}},\lambda_{\mathcal{E}})$ the basic
principally polarized triple over $\mathrm{Spf}O_{\breve{F}_{0}}$ obtained
from the canonical lifting of $\mathbb{E}$.

Let _the space of special quasi-homomorphisms_ be the $F$-vector space

(4)
$\mathrm{Hom}_{O_{F}}^{\circ}(\mathbb{E},\mathbb{X}):=\mathrm{Hom}_{O_{F}}(\mathbb{E},\mathbb{X})\otimes\mathbb{Q}$

equipped with the hermitian form:

(5) $(x,y):=\lambda_{\mathbb{E}}^{-1}\circ
y^{\vee}\circ\lambda_{\mathbb{X}}\circ
x\in\mathrm{Hom}_{O_{F}}^{\circ}(\mathbb{E},\mathbb{E})\cong
F,\quad\quad\forall
x,\,y\in\mathrm{Hom}_{O_{F}}^{\circ}(\mathbb{E},\mathbb{X}).$

Then $\mathrm{Hom}_{O_{F}}^{\circ}(\mathbb{E},\mathbb{X})$ is a non-split
hermitian space of dimension $n$ and we may write
$\mathbb{V}=\mathrm{Hom}_{O_{F}}^{\circ}(\mathbb{E},\mathbb{X})$. The linear
dual of $\mathbb{V}$ is
$\mathbb{V}^{*}=\mathrm{Hom}_{O_{F}}^{\circ}(\mathbb{X},\mathbb{E})$. The
hermitian form on $\mathbb{V}$ gives an isomorphism
$\lambda_{\mathbb{V}}:\mathbb{V}\to\mathbb{V}^{*}$, hence an involution
$\sigma(-)=\lambda_{\mathbb{V}}^{-1}\circ(-)^{*}\circ\lambda_{\mathbb{V}}:\mathrm{GL}(\mathbb{V})\to\mathrm{GL}(\mathbb{V})$
with fixed subgroup $\mathrm{U}(\mathbb{V})$. There is an identification
$\mathrm{GL}(\mathbb{V})(F_{0})\cong\mathrm{Aut}^{\circ}(\mathbb{X},\iota_{\mathbb{X}})$
where $\sigma$ is identified with

$\sigma(g)=\lambda_{\mathbb{X}}^{-1}\circ(g^{\vee})^{-1}\circ\lambda_{\mathbb{X}},\quad\forall
g\in\mathrm{Aut}^{\circ}(\mathbb{X},\iota_{\mathbb{X}}).$

The general linear group
$\mathrm{GL}(\mathbb{V})(F_{0})\cong\mathrm{Aut}^{\circ}(\mathbb{X},\iota_{\mathbb{X}})$
acts on $\mathcal{N}_{n}^{\mathrm{GL}}$ by acting on the framing:

$g\cdot(X,\iota,\rho)=(X,\iota,g\circ\rho).$

Similarly, the unitary group
$\mathrm{U}(\mathbb{V})(F_{0})\cong\mathrm{Aut}^{\circ}(\mathbb{X},\iota_{\mathbb{X}},\lambda_{\mathbb{X}})$
acts on $\mathcal{N}_{n}$ by acting on the framing. By Drinfeld’s rigidity of
quasi-isogenies, we have a $\mathrm{U}(\mathbb{V})(F_{0})$-equivariant
embedding

$i_{\mathrm{BC}}:\mathcal{N}_{n}\to\mathcal{N}_{n}^{\mathrm{GL}}.$

######  Definition 2.2.1.

Consider the Galois involution

$\sigma:\mathcal{N}^{\mathrm{GL}}_{n}\to\mathcal{N}^{\mathrm{GL}}_{n}$

sending
$(X,\iota,\rho)\to(X^{\vee},\overline{\iota^{\vee}},\lambda_{\mathbb{X}}^{-1}\circ(\rho^{\vee})^{-1}).$

######  Proposition 2.2.2.

The action of $\mathrm{GL}(\mathbb{V})(F_{0})$ on
$\mathcal{N}^{\mathrm{GL}}_{n}$ is $\sigma$-equivariant. The fixed point locus
of $\sigma$ on $\mathcal{N}^{\mathrm{GL}}_{n}$ recovers $\mathcal{N}_{n}$:

$(\mathcal{N}^{\mathrm{GL}}_{n})^{\sigma=\mathrm{id}}=\mathcal{N}_{n}.$

###### Proof.

We have
$\sigma(g.(X,\iota,\rho))=(X^{\vee},\overline{\iota^{\vee}},\lambda_{\mathbb{X}}^{-1}\circ((g\circ\rho)^{\vee})^{-1})=(X^{\vee},\overline{\iota^{\vee}},\lambda_{\mathbb{X}}^{-1}\circ(g^{\vee})^{-1}\circ(\rho^{\vee})^{-1})=\sigma(g)(\sigma(X,\iota,\rho)).$Hence
the $\mathrm{GL}(\mathbb{V})(F_{0})$-action on $\mathcal{N}^{\mathrm{GL}}_{n}$
is $\sigma$-equivariant. If
$(X,\iota)\in(\mathcal{N}^{\mathrm{GL}}_{n})^{\sigma=\mathrm{id}}$, then there
is an $O_{F}$-linear isomorphism $\varphi:X\cong X^{\vee}$ compatible with the
framing $\rho$ and $\lambda_{\mathbb{X}}^{-1}\circ(\rho^{\vee})^{-1}$. This is
equivalent to that $\varphi$ is a lifting of $\lambda_{\mathbb{X}}$ to $X$.
Hence $\varphi$ is a principal polarization on $X$ and we see that
$(X,\iota,\varphi)\in\mathcal{N}_{n}$. Hence
$(\mathcal{N}^{\mathrm{GL}}_{n})^{\sigma=\mathrm{id}}=\mathcal{N}_{n}$. ∎

######  Remark 2.2.3.

The above proposition can be regarded as a geometrization of base change and
Galois descent for representations of $\mathrm{U}(V)$ and $\mathrm{GL}(V)$.

###  2.3. Geometry of mirabolic special cycles and Kudla–Rapoport cycles

Consider a non-zero vector $u\in\mathbb{V}-0$. The Kudla–Rapoport cycle
[KR2011local]

$\mathcal{Z}(u)\to\mathcal{N}_{n}$

is the closed formal subscheme of $\mathcal{N}$ sending a
$\mathrm{Spf}O_{\breve{F_{0}}}$-scheme $S$ to the subset
$(X,\iota_{X},\lambda_{X},\rho_{X})\in\mathcal{N}(S)$ such that

$(\rho_{X})^{-1}\circ u:\mathbb{E}\times_{\mathbb{F}}\overline{S}\to
X\times_{S}\overline{S}$

lifts to a homomorphism from $\mathcal{E}$ to $X$ over $S$. By [KR2011local,
Prop 3.5], $\mathcal{Z}(u)$ is a relative Cartier divisor in
$\mathcal{N}_{n}$.

We now introduce two kinds of mirabolic special cycles on
$\mathcal{N}^{\mathrm{GL}}_{n}$, which recover the notion of Kudla–Rapoport
cycles after base change to
$i_{\mathrm{BC}}:\mathcal{N}_{n}\to\mathcal{N}^{\mathrm{GL}}_{n}$.

######  Definition 2.3.1.

For $u\in\mathbb{V}-0$, the mirabolic special cycle

$\mathcal{Z}^{\mathrm{GL}}(u)\to\mathcal{N}^{\mathrm{GL}}_{n}$

is the subfunctor of $\mathcal{N}^{\mathrm{GL}}$ sending a
$\mathrm{Spf}O_{\breve{F_{0}}}$-scheme $S$ to
$(X,\iota_{X},\rho_{X})\in\mathcal{N}^{\mathrm{GL}}_{n}(S)$ such that

$(\rho_{X})^{-1}\circ u:\mathbb{E}\times_{\mathbb{F}}\overline{S}\to
X\times_{S}\overline{S}$

lifts to a homomorphism from $\mathcal{E}$ to $X$ over $S$. For
$u^{*}\in\mathbb{V}^{*}-0$, the mirabolic special cycle

$\mathcal{Z}^{\mathrm{GL}^{*}}(u^{*})\to\mathcal{N}^{\mathrm{GL}}_{n}$

is the subfunctor of $\mathcal{N}^{\mathrm{GL}}$ sending a
$\mathrm{Spf}O_{\breve{F_{0}}}$-scheme $S$ to
$(X,\iota_{X},\rho_{X})\in\mathcal{N}^{\mathrm{GL}}_{n}(S)$ such that
$u^{*}\circ\rho_{X}:X\times_{S}\overline{S}\to\mathbb{E}\times_{\mathbb{F}}\overline{S}$
lifts to a homomorphism from $X$ to $\mathcal{E}$ over $S$.

######  Theorem 2.3.2.

  1. (1)

The subfunctors $\mathcal{Z}^{\mathrm{GL}}(u)$ and
$\mathcal{Z}^{\mathrm{GL}^{*}}(u^{*})$ are represented by formal schemes, and
are relative Cartier divisors in $\mathcal{N}_{n}^{\mathrm{GL}}$.

  2. (2)

(The pullback formula) The diagram

$\textstyle{\mathcal{Z}(u)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Z}^{\mathrm{GL}}(u)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}^{\mathrm{GL}}_{n}}$

is Cartesian, in other words
$\mathcal{Z}^{\mathrm{GL}}(u)|_{\mathcal{N}_{n}}=\mathcal{Z}(u)$.

  3. (3)

(Galois duality)
$\sigma(\mathcal{Z}^{\mathrm{GL}}(u^{*}))=\mathcal{Z}^{\mathrm{GL}}(u),\,\mathrm{if}\,\lambda_{\mathbb{X}}^{-1}\circ(u^{*})^{\vee}\circ\lambda_{\mathbb{E}}=u.$

  4. (4)

(Galois descent)
$[\mathcal{Z}^{\mathrm{GL}}(u)]^{\sigma=\mathrm{id}}=\mathcal{Z}(u)$.

  5. (5)

(Translation laws) For $a\in O_{F}^{\times}$,
$\mathcal{Z}^{\mathrm{GL}}(au)=\mathcal{Z}^{\mathrm{GL}}(u)$. For
$g\in\mathrm{GL}(\mathbb{V})(F_{0})$, we have

$g.\mathcal{Z}^{\mathrm{GL}}(u)=\mathcal{Z}^{\mathrm{GL}}(gu),\,\,g.\mathcal{Z}^{\mathrm{GL}^{*}}(u^{*})=\mathcal{Z}^{\mathrm{GL}^{*}}(gu^{*}).$

  6. (6)

(Cancellation laws) If $(u,u^{*})=1$, then
$\mathcal{Z}(u)\cap\mathcal{Z}(u^{*})\cong\mathcal{N}_{n-1}^{\mathrm{GL}}$.

  7. (7)

(Functoriality) Let $n,m\geq 1$. Consider a natural embedding
$\mathcal{N}_{n}^{\mathrm{GL}}\times\mathcal{N}_{m}^{\mathrm{GL}}\to\mathcal{N}_{n+m}^{\mathrm{GL}}$
given by taking direct sums. For
$(u_{1},u_{2})\in\mathbb{V}_{n}\oplus\mathbb{V}_{m}=\mathbb{V}_{n+m}$ such
$u_{i}\not=0$, we have

$\mathcal{Z}^{\mathrm{GL}}_{n+m}(u_{1},u_{2})|_{\mathcal{N}_{n}^{\mathrm{GL}}\times\mathcal{N}_{m}^{\mathrm{GL}}}=\mathcal{Z}^{\mathrm{GL}}_{n}(u_{1})\times\mathcal{Z}^{\mathrm{GL}}_{m}(u_{2}).$

###### Proof.

We prove the first part following the proof of [KR2011local, Prop 3.5]. The
first part follows from the proof that Rapoport–Zink spaces are represented by
formal schemes in [RZ96, Prop 2.9].

We now prove that $\mathcal{Z}^{\mathrm{GL}}(u)$ is locally defined by the
vanishing of one equation. Let $R$ be a $\mathrm{Spf}O_{\breve{F}}$-algebra
and $I$ is an ideal of $R$ such that $I^{2}=0$. We equip $I$ with trivial
divided powers and let $R_{0}=R/I$. Given a morphism
$\phi:\mathrm{Spec}R\to\mathcal{N}^{\mathrm{GL}}_{n}$ whose restriction to
$\mathrm{Spec}R_{0}$ factors through $\mathcal{Z}^{\mathrm{GL}}(u)$. Then we
need to show the obstruction to factoring the given morphism $\phi$ through
$\mathcal{Z}^{\mathrm{GL}}(u)$ is given by the vanishing of one element in
$I$. Let $X$ be the $A$-point of $\mathcal{N}$ corresponding to $\phi$. The
condition implies that there is an $O_{F}$-linear homomorphism
$\alpha:D_{\mathcal{E}}\to D_{X}$ of relative Dieudonne crystals over
$\mathrm{Spec}A$, which maps the Hodge filtration $\mathcal{F}_{\mathcal{E}}$
to $\mathcal{F}_{X}$ after tensoring with $A_{0}$. By Grothendieck–Messing
theory, the obstruction is exactly the condition
$\alpha(\mathcal{F}_{\mathcal{E}})\subseteq\mathcal{F}_{X}$. As $\alpha$ is
$O_{F}$-linear and $\mathcal{F}_{\mathcal{E}}$ is generated by one element
$\bar{1}_{0}$ as $O_{F}\otimes\mathbb{A}$ module, the obstruction is
equivalent to that $\alpha(\bar{1}_{0})\in(F_{X})_{0}$, i.e.
$\alpha(\bar{1}_{0})=0\in(\mathrm{Lie}X)_{0}$. By the Kottwitz signature
condition, $(\mathrm{Lie}X)_{0}$ is a locally free $A$-module of rank $1$.
After choosing a local generator of $\mathrm{Lie}X_{0}$, we may identify
$\alpha(\bar{1}_{0})$ with an element in $A$ with zero image in $A_{0}$, i.e.
an element in $I$. This shows that $\mathcal{Z}^{\mathrm{GL}}(u)$ is a
divisor.

To show that $\mathcal{Z}^{\mathrm{GL}}(u)$ is a relative Cartier divisor, we
only need to show any local generator of the divisor is non-zero and not
divisible by $p$. As
$\mathcal{Z}^{\mathrm{GL}}(u)|_{\mathcal{N}_{n}}=\mathcal{Z}(u)$, this follows
from the corresponding proposition for $\mathcal{Z}(u)$ proved in
[KR2011local, Prop 3.5].

The following four properties follow by definition. For the cancellation law,
if $(u,u^{*})=1$ we can obtain a splitting of
$\mathbb{X}=\mathbb{X}^{\flat}\oplus\mathbb{E}$ where
$\mathbb{X}^{\flat}=\mathrm{Ker}u^{*}:\mathbb{X}\to\mathbb{E}$ and the direct
summand $\mathbb{E}$ is the image of $u$. Since $u,u^{*}$ lift to
$\mathcal{Z}(u)\cap\mathcal{Z}(u^{*})$, similarly we obtain a splitting of the
universal tuple $X=X^{\flat}\oplus\mathcal{E}$ over
$\mathcal{Z}(u)\cap\mathcal{Z}(u^{*})$. Therefore
$\mathcal{Z}(u)\cap\mathcal{Z}(u^{*})\cong\mathcal{N}_{n-1}^{\mathrm{GL}}$.
Finally, the functoriality follows by moduli definitions. ∎

For any subset $S\subseteq\mathbb{V}$, we define mirabolic special cycle
$\mathcal{Z}(S)^{\mathrm{GL}}\subseteq\mathcal{N}^{\mathrm{GL}}_{n}$ as the
lifting locus of $S$.

######  Remark 2.3.3 (Special linear cycles on Hodge type Rapoport-Zink
spaces).

Let $F/F_{0}$ be a quadratic extension. Let $V$ be a $n$-dimension $F$-vector
space. We extend the notion of mirabolic special cycles to any (relative)
Rapoport-Zink spaces
$\mathcal{N}_{V}:=\mathcal{N}_{\mathrm{Res}_{F/F_{0}}\mathrm{GL}(V),b_{V},\mu_{V},K_{p}}$
of EL type over $F_{0}$. For simplicity, assume $K_{p}\leq\mathrm{GL}(V)$ is
hyperspecial.

Let $\mathbb{X}$ be a framing object for $\mathcal{N}_{V}$, and $\mathbb{E}$
be a formal Lubin-Tate $O_{F}$-module of dimension $1$ (and $O_{F}$-height 1)
over $\mathbb{F}$ with its canonical lifting $\mathcal{E}$ over
$O_{\breve{F}}$. Define the space of special quasi-homomorphism to be the
$F$-vector space
$\mathbb{V}=\mathrm{Hom}_{O_{F}}(\mathbb{E},\mathbb{X})\otimes\mathbb{Q}$.
Then for non-zero $u\in\mathbb{V}$ we define

$Z_{V}(u)\to\mathcal{N}_{V}$

as the closed formal subscheme of lifting locus of $u$ to
$\mathcal{E}\to\mathcal{X}$. Because of the $O_{F}/O_{F_{0}}$-signature
condition given by $\mu$, it is still representable by a formal subscheme. Via
pullback, we get _special linear cycles_

$Z_{V}(u)\to\mathcal{N}_{G}:=\mathcal{N}_{G,b,\mu,K}$

on general unramified Hodge type Rapoport-Zink spaces $(G,b,\mu)$
[HP17][kim2018] associated to a representation $V$ of $G$ which induces a
embedding of local Shimura datum $(G,b,\mu)\to(\mathrm{GL}(V),b_{V},\mu_{V})$.

##  3\. Twisted orbits, orbital integrals and fundamental lemmas

In this section, we introduce orbits and orbital integrals in our twisted
setup. We recall the twisted fundamental lemma, which is a motivation to the
twisted arithmetic fundamental lemma, and will be used in our global set up.

###  3.1. Orbits and orbital integrals

On the unitary side, let $(V,h_{V})$ be a $n$-dimensional hermitian space over
$F$. Consider reductive groups over $F_{0}$:

$H_{V}=\mathrm{U}(V),\,G_{V}=\mathrm{GL}(V).$

Let $X_{V}=\mathrm{Herm}(V,h_{V})$ be the set of hermitian structures $A$ on
$V$ such that $(V,A)\cong(V,h_{V})$. We identify

(6) $G_{V}/H_{V}\cong X_{V},\quad g\mapsto A_{g}:=gh_{V}g^{-1}.$

Let $Y_{V}=V$. Consider the action of $g\in H_{V}$ on

$X_{V}\times Y_{V}=\mathrm{Herm}(V,h_{V})\times V$

given by $h.(g,u)=(h^{-1}gh,h^{-1}u)$. The pair $(g,u)$ is regular semi-simple
(with respect to $H_{V}$) if and only if $u,A_{g}u,\ldots,A_{g}^{n-1}u$ are
linearly independent. Choose a Haar measure on $H_{V}(F_{0})$. For
$\Phi\in\mathcal{S}(X_{V}\times Y_{V})$, define its orbital integral on
$(g,u)\in X_{V}\times Y_{V}$ by

(7) $\mathrm{Orb}((g,u),\Phi):=\int_{H_{V}(F_{0})}\Phi(h.(g,u))dh$

which converges absolutely if $(g,u)$ is regular semi-simple.

On the general linear side, let $V_{0}$ be a $n$-dimensional vector space over
$F_{0}$. Let

$H^{\prime}=\mathrm{GL}(V_{0}),\,G^{\prime}=\mathrm{GL}(V_{0})\times\mathrm{GL}(V_{0}),\,H^{\prime}_{20}=\mathrm{GL}(V_{0}).$

We consider diagonal embeddings $H^{\prime}\rightarrow G^{\prime}\leftarrow
H^{\prime}_{20}$. The space $X^{\prime}=G^{\prime}/H^{\prime}_{20}$ can be
naturally identified with $\mathrm{GL}(V_{0})$, where the natural left action
of $H^{\prime}$ on $X^{\prime}$ is identified with conjugacy action on
$\mathrm{GL}(V_{0})$. Let $Y^{\prime}=V_{0}^{\prime}:=V_{0}\times V_{0}^{*}$
with the natural action of $H^{\prime}$.

Similarly, we consider the action of $H^{\prime}$ on

$X^{\prime}\times Y^{\prime}=\mathrm{GL}(V_{0})\times V_{0}\times V_{0}^{*}$

given by $h.(\gamma,u^{\prime}=(u_{1},u_{2}))=(h^{-1}\gamma
h,h^{-1}u_{1},u_{2}h)$. We use the _transfer factor_

(8)
$\omega(\gamma,u^{\prime}):=(-1)^{\mathrm{val}_{F_{0}}(\det(\gamma^{i}u_{1})_{i=0}^{n-1})}\in\\{\pm
1\\}.$

Choose a Haar measure on $H^{\prime}(F_{0})$. For
$\Phi^{\prime}\in\mathcal{S}(X^{\prime}\times Y^{\prime})$, define the
$s$-variable orbital integral ($s\in\mathbb{C}$)

(9)
$\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime},s):=\omega(\gamma,u^{\prime})\int_{H^{\prime}(F_{0})}\Phi^{\prime}(h.(\gamma,u^{\prime}))|h|^{s}\eta(h)dh$

which converges absolutely if $(\gamma,u^{\prime})$ is regular semi-simple.
Here we write $|h|=|\det h|,\eta(h)=\eta(\det h)$ for short. Set

$\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime})=\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime},0),\quad\partial\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime})=\frac{d}{ds}|_{s=0}\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime},s).$

Consider the embedding $X_{V}\times V\to\mathrm{End}(V)\times V\times V^{*}$
given by $(g,u)\to(g,\,u,\,u^{*}:=(x\mapsto h_{V}(x,u)))$. We may choose a
basis of $V$ and assume that $V=V_{0}\otimes_{F_{0}}F$.

######  Definition 3.1.1.

A pair $(g,u)\in X_{V}\times Y_{V}$ matches a pair $(\gamma,u^{\prime})\in
X^{\prime}\times Y^{\prime}$ if they are conjugated by $\mathrm{GL}(V)$ inside
$\mathrm{End}(V)\times V\times V^{*}$.

The matching relation is equivalent to the matching of the following
invariants:

$\det(T\mathrm{id}_{V}+g)=\det(T\mathrm{id}_{V}+\gamma)\in
F^{\prime}[T],\quad(g^{i}u,u)=u_{2}(\gamma^{i}u_{1}),\quad 0\leq i\leq n-1.$

######  Proposition 3.1.2.

The matching relation gives a natural bijection of regular semi-simple orbits:

$[H^{\prime}\backslash(X^{\prime}\times
Y^{\prime})]_{\mathrm{rs}}\cong[H_{V}\backslash(X_{V}\times
Y_{V})]_{\mathrm{rs}}\coprod[H_{\mathbb{V}}\backslash(X_{\mathbb{V}}\times
Y_{\mathbb{V}})]_{\mathrm{rs}},\,\,(\gamma,u^{\prime})\leftrightarrow(A,u).$

###### Proof.

This follows from the same proof in the Jacquet-Rallis case. See
[leslie2019endoscopic, Section 4.1]. ∎

###  3.2. Transfers and Twisted Fundamental lemmas

Choose a self-dual lattice $L$ in $V$. Choose an orthogonal basis
$\\{e_{i}\\}$ of $L$, and let $L_{0}=\sum_{i}O_{F_{0}}e_{i}$. We may assume
that $V_{0}=L_{0}\otimes\mathbb{Q}$. We consider the lattice
$L^{\prime}=L_{0}\times L_{0}^{*}\subseteq Y^{\prime}=V_{0}^{\prime}$. We
normalize the Haar measure on $\mathrm{U}(V)(F_{0})$ (resp.
$\mathrm{GL}(V_{0})(F_{0})$) such that the stabilizer $\mathrm{U}(L)$ (resp.
$\mathrm{GL}(L_{0})$) is of volume $1$.

######  Definition 3.2.1.

We say $\Phi^{\prime}\in\mathcal{S}(X^{\prime}\times Y^{\prime})$ and a pair
$(\Phi_{V}\in\mathcal{S}(X_{V}\times
Y_{V})),\Phi_{\mathbb{V}}\in\mathcal{S}(X_{\mathbb{V}}\times Y_{\mathbb{V}})$
are _transfers_ to each other, if for any regular semisimple pair
$(\gamma,u^{\prime})\in X^{\prime}\times Y^{\prime}$, we have

$\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime})=\begin{cases}\mathrm{Orb}((g,u),\Phi_{V}),\quad\text{if
$(\gamma,u^{\prime})$ matches $(g,u)\in(X_{V}\times Y_{V})_{\mathrm{rs}}$.}\\\
\mathrm{Orb}((g,u),\Phi_{\mathbb{V}}),\quad\text{if $(\gamma,u^{\prime})$
matches $(g,u)\in(X_{\mathbb{V}}\times Y_{\mathbb{V}})_{\mathrm{rs}}$.}\\\
\end{cases}$

######  Remark 3.2.2.

Let $E_{0}/F_{0}$ be a étale quadratic algebra. Let $E=F\otimes_{F_{0}}E_{0}$.
Let $F^{\prime}/F_{0}$ be the third quadratic extension inside $E$. We can
consider the following generalization of the above orbits ($E_{0}=F$) and the
Jacquet–Rallis case ($E_{0}=F_{0}\times F_{0}$). Let

$H_{V}=\mathrm{U}(V),\quad G_{V}^{E_{0}}=\mathrm{U}(V_{E_{0}}).$

Instead of $X_{V}$, consider the hermitian symmetric space

$X_{V}^{E_{0}}=\mathrm{U}(V_{E_{0}})^{-}:=\\{g\in\mathrm{U}(V_{E_{0}})|g\sigma_{E/F}(g)=\mathrm{id}.\\}$

Let

$H^{\prime}=\mathrm{GL}(V_{0}),\quad
G^{{}^{\prime}E_{0}}=\mathrm{GL}(V_{0},F^{\prime}),\quad
H^{\prime}_{20}=\mathrm{GL}(V_{0}).$

Instead of $X^{\prime}$, consider the $F^{\prime}/F_{0}$-symmetric space over
$F_{0}$:

$X^{{}^{\prime}E_{0}}=S_{n,F^{\prime}/F_{0}}=\\{\gamma\in\mathrm{GL}_{n}(F^{\prime})|\gamma\sigma_{F^{\prime}/F_{0}}(\gamma)=\mathrm{id}\\}.$

By Hilbert 90, we have
$\mathrm{U}(V_{E_{0}})^{-}\cong\mathrm{U}(V_{E_{0}})/\mathrm{U}(V),\,\,S_{n,F^{\prime}/F_{0}}=\mathrm{GL}_{n}(F^{\prime})/\mathrm{GL}_{n}(F_{0}).$
Note that the Lie algebra of $X_{V,E_{0}}$ and $X^{\prime}_{E_{0}}$ can be
identified after multiplying suitable elements in $F^{\prime}$. Then we have a
similar matching of regular semi-simple orbits

$[H^{\prime}\backslash(X^{{}^{\prime}E_{0}}\times
Y^{\prime})]_{\mathrm{rs}}\cong[H_{V}\backslash(X_{V}^{E_{0}}\times
Y_{V})]_{\mathrm{rs}}\coprod[H_{\mathbb{V}}\backslash(X_{\mathbb{V}}^{E_{0}}\times
Y_{\mathbb{V}})]_{\mathrm{rs}},\,\,(\gamma,u^{\prime})\leftrightarrow(A,u).$

similar to [leslie2019endoscopic, Section 4.1].

Let $\mathrm{GL}(L_{0})$ be the stabilizer of $L_{0}$ in $\mathrm{GL}(V_{0})$.
Let $\mathrm{Herm}(L)$ be the subset of $A\in X_{V}=\mathrm{Herm}(V,h_{V})$
such that $L$ is integral under $A$. Define standard test functions

(10) $\Phi_{L^{\prime}}^{\prime}=1_{\mathrm{GL}(L_{0})}\times
1_{L^{\prime}}\in C_{c}^{\infty}(X^{\prime}\times
Y^{\prime}),\,\Phi_{L}=1_{\mathrm{Herm}(L)}\times 1_{L}\in
C_{c}^{\infty}(X_{V}\times Y_{V})$

We have the following twisted fundamental lemma which appears in the work of
Danielle Wang [Wang-TGGP].

######  Theorem 3.2.3 (Twisted Fundamental Lemmas).

$\Phi^{\prime}_{L^{\prime}}$ is a transfer of $(\Phi_{L},0)$.

###### Proof.

Using the Cayley map, we reduce to similar transfer results on lie algebra
case, which is equivalent to the Jacquet–Rallis fundamental lemma proved in
[beuzart2021new]. ∎

##  4\. Twisted arithmetic fundamental lemmas

In this section, we introduce twisted derived fixed cycles and formulate a
twisted arithmetic fundamental lemma. Recall we have a
$\mathrm{U}(\mathbb{V})$-equivariant embedding

$\mathcal{N}_{n}\to\mathcal{N}_{n}^{\mathrm{GL}}.$

######  Definition 4.0.1.

For $g\in\mathrm{GL}(\mathbb{V})(F_{0})$, the twisted fixed cycle of $g$ is
the derived special cycle (in the Grothendieck group of $\mathcal{N}_{n}$)
which only depends on $g\in
X_{\mathbb{V}}=\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})$:

(11)
$\mathcal{N}_{n}^{\mathrm{Herm}}(g)=g\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{N}_{n}\in
K_{0}^{\prime}(\mathcal{N}_{n}),$

which is a derived $1$-cycle by [AFL-Wei2019, Lemma B.2]. We identify
$X_{\mathbb{V}}=\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})=\mathrm{Herm}(\mathbb{V},h_{\mathbb{V}})$
as the set of hermitian structures $A$ on $\mathbb{V}$ such that
$(\mathbb{V},A)\cong(\mathbb{V},h_{\mathbb{V}})$.

######  Proposition 4.0.2.

For a regular semi-simple pair
$(g,u)\in\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})\times\mathbb{V}$, the
intersection
$g\mathcal{N}_{n}\cap_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{Z}(u)$ is a
proper scheme.

###### Proof.

Let
$(X,\iota,\lambda)\in(g\mathcal{N}_{n}\cap_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{N}_{n})(S)$.
Then there exists an $O_{F}$-linear morphism $\varphi:X\to X$ lifting $g$. If
$(X,\iota,\lambda)\in
g\mathcal{N}_{n}\cap_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{Z}(u)$, using the
lifting $\varphi$ we see that $(X,\iota,\lambda)\in\mathcal{Z}(L_{g,u})$ where
$L_{g,u}$ is the $O_{F}$-span of $u,gu,\ldots,g^{n-1}u$. As $(g,u)$ is regular
semi-simple, $L_{g,u}$ has full rank $n$ hence $\mathcal{Z}(L_{g,u})$ is a
proper scheme by [LiZhangKR2019, Lemma 2.10.1]. As
$g\mathcal{N}_{n}\cap_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{Z}(u)$ is a a
closed formal subscheme of $\mathcal{Z}(L_{g,u})$, it is also a proper scheme.
∎

For any regular semi-simple pair
$(g,u)\in(X_{\mathbb{V}}\times\mathbb{V})(F_{0})_{\mathrm{rs}}$, consider the
derived intersection number

(12)
$\mathrm{Int}^{\mathrm{Herm},\mathbb{V}}(g,u)=\mathcal{N}_{n}^{\mathrm{Herm}}(g)\cap^{\mathbb{L}}_{\mathcal{N}_{n}}\mathcal{Z}(u)=g\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{Z}(u)=\chi(\mathcal{N}_{n}^{\mathrm{GL}},\mathcal{O}_{\mathcal{N}_{n}}\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(u)}).$

By Proposition 4.0.2, $\mathrm{Int}(g,u)\in\mathbb{Z}$ is well-defined and
finite.

######  Proposition 4.0.3.

We have

$\mathrm{Int}^{\mathrm{Herm},\mathbb{V}}(g,u)=\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}_{n}^{\mathrm{GL}}}g\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{Z}^{\mathrm{GL}}(u).$

###### Proof.

As $\mathcal{Z}^{\mathrm{GL}}(u),\mathcal{Z}(u)$ are Cartier divisors, the
intersection
$\mathcal{Z}(u)=\mathcal{Z}^{\mathrm{GL}}(u)\cap_{\mathcal{N}^{\mathrm{GL}}_{n}}\mathcal{N}_{n}$
has correct codimension hence is equal to
$\mathcal{Z}^{\mathrm{GL}}(u)\cap_{\mathcal{N}^{\mathrm{GL}}_{n}}^{\mathbb{L}}\mathcal{N}_{n}$
by [AFL-Wei2019, Lemma B.2]. The result follows. ∎

Now we formulate our twisted arithmetic fundamental lemma.

######  Conjecture 4.0.4 (Twisted arithmetic fundamental lemma).

For any
$(g,u)\in(\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})\times\mathbb{V})(F_{0})_{\mathrm{rs}}$
matching $(\gamma,u^{\prime})\in(\mathrm{GL}_{n}\times
V_{0}^{\prime})(F_{0})_{\mathrm{rs}}$, we have

$\partial\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{L^{\prime}})=-\mathrm{Int}^{\mathrm{Herm},\mathbb{V}}(g,u)\log
q.$

######  Remark 4.0.5.

The original arithmetic fundamental lemma [AFL-Invent] study similar
arithmetic intersection problems via the embedding of
$\mathcal{N}_{n}\to\mathcal{N}_{n}\times\mathcal{N}_{n}$ and graphs of
automorphisms on $\mathcal{N}_{n}\times\mathcal{N}_{n}$. In above conjecture,
we use the embedding $\mathcal{N}_{n}\to\mathcal{N}_{n}^{\mathrm{GL}}$, hence
we call it a twisted arithmetic fundamental lemma.

######  Proposition 4.0.6.

The twisted AFL conjecture holds when $n=1$.

###### Proof.

In this case, the (twisted) fixed cycles will be the whole $\mathcal{N}_{1}$.
The twisted AFL agrees with the original AFL [AFL-Invent] and follows from
Gross’s theory of quasi-canonical lifting. ∎

######  Proposition 4.0.7.

The twisted AFL conjecture holds in the maximal order case, i.e., when
$O_{F}[g]$ ($g\in\mathrm{Herm}(\mathbb{V},h_{\mathbb{V}})$)is a product of
discrete valuation rings.

###### Proof.

We follow the proof in the AFL case [Mihatsch-comparison, Section 9][AFL-
Wei2019, Prop 2.6]. Write $F[g]=\prod_{i=1}^{m}F_{i}$ as a product of fields.
Write $E_{0}=F_{0}$. Then above identities is reduced to the twisted AFL in
the case $n=1$ for $F_{i}/F_{i,0}$ (and $E_{i,0}=F_{i,0}$). ∎

In the rest of the paper, we will prove the following main theorem.

######  Theorem 4.0.8 (Twisted AFL).

The twisted AFL conjecture holds for any regular semi-simple $(g,u)$ and
$n\geq 1$.

##  5\. Arithmetic induction systems and reductions

In this section, we develop and apply the idea of arithmetic induction systems
(Definition 5.1.2). We show the geometric side and analytic side in the
twisted AFL conjecture 4.0.8 are both arithmetic induction systems, which will
be used in our proof of twisted AFL. The mirabolic special cycle is a key
ingredient for arithmetic induction on the geometric side.

###  5.1. Arithmetic induction systems

Let $F_{0}$ be a $p$-adic field. Consider tuples $(G,X,Y)$ of following data:

  * •

$G$ is a reductive group over $F_{0}$, $X$ is an affine variety over $F_{0}$
with $G$-action and $Y=(\mathbb{V},q)$ be a quadratic space over $F_{0}$ with
an isometric linear algebraic action of $G$.

For $y_{0}\in Y$, denote its stabilizer in $G$ by $G_{y_{0}}$. We have
$G(F_{0})\backslash(X(F_{0})\times G(F_{0}).y_{0})=G_{y_{0}}(F_{0})\backslash
X(F_{0})$.

######  Definition 5.1.1.

A tuple $(G,X,Y)$ is good if for any $y_{0}\in Y$ with $q(y_{0})\in
O_{F_{0}}^{\times}$, there exists a tuple $(G_{y_{0}},X_{y_{0}},Y_{y_{0}})$
and a rational isomorphism

$\mathfrak{c}_{y_{0}}:G_{y_{0}}\backslash X\to
G_{y_{0}}\backslash(X_{y_{0}}\times Y_{y_{0}}).$

We call $\mathfrak{c}_{y_{0}}$ a relative Cayley map for $(G,X,Y,y_{0})$.

######  Definition 5.1.2.

For a good tuple $(G,X,Y)$, we say a locally constant function $F=F_{(G,X,Y)}$
on an open dense subset $U\subseteq X\times Y$ forms an arithmetic induction
system for $(G,X,Y)$ if

  1. (1)

$F$ is $G$-invariant.

  2. (2)

We have $F(x,y)=0$ unless $q(y)\in O_{F_{0}}$.

  3. (3)

For $y_{0}\in Y$, denote by $F_{y_{0}}=F(-,y_{0}):X\to\mathbb{C}$. If
$(x,y_{0})\in U$ with $q(y_{0})\in O_{F_{0}}^{\times}$, then
$\mathfrak{c}_{y_{0}}$ is defined on $U\times y_{0}$, and there exists a good
tuple $(G_{y_{0}},X_{y_{0}},Y_{y_{0}})$ and a locally constant function
$F_{(G_{y_{0}},X_{y_{0}},Y_{y_{0}})}$ on an open dense subset of
$X_{y_{0}}\times Y_{y_{0}}$ such that

$F_{y_{0}}=F_{(G_{y_{0}},X_{y_{0}},Y_{y_{0}})}\circ\mathfrak{c}_{y_{0}}.$

  4. (4)

The function $F_{(G_{y_{0}},X_{y_{0}},Y_{y_{0}})}$ satisfies $(1)(2)(3)$ for
the good tuple $(G_{y_{0}},X_{y_{0}},Y_{y_{0}})$.

Arithmetic induction systems for $(G,X,Y)$ form a $\mathbb{C}$-vector space.

###  5.2. Relative Cayley maps and matchings

We first consider the unitary side. Let $\mathbb{V}=\mathbb{V}^{\flat}\oplus
Fu^{0}$ be a splitting of hermitian spaces where $(u^{0},u^{0})=1$. The
element $u^{0}$ gives a splitting of the framing object
$\mathbb{X}=\mathbb{X}^{\flat}\oplus\mathbb{E}$, which induces embeddings

$i_{u_{0}}:\mathcal{N}_{n-1}\cong\mathcal{Z}(u^{0})\hookrightarrow\mathcal{N}_{n},\quad
X^{\flat}\mapsto X^{\flat}\oplus\mathcal{E},$
$i_{u_{0},u_{0}^{*}}:\mathcal{N}_{n-1}^{\mathrm{GL}}\cong\mathcal{Z}(u^{0},(u^{0})^{*})\hookrightarrow\mathcal{N}_{n-1}^{\mathrm{GL}},\quad
X^{\flat}\mapsto X^{\flat}\oplus\mathcal{E}.$

Any hermitian structure $A\in\mathrm{Herm}(\mathbb{V})$ could be written as

$A=\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix},\quad
a\in\mathrm{Herm}(\mathbb{V}^{\flat}),b\in\mathbb{V}^{\flat},c\in(\mathbb{V}^{\flat})^{*},d\in
F.$

######  Definition 5.2.1.

Consider the $\mathrm{U}(V^{\flat})$-equivariant rational relative Cayley map

(13)
$c_{\mathbb{V},\mathrm{Herm}}:\mathrm{Herm}(\mathbb{V})\to\mathrm{Herm}(\mathbb{V}^{\flat})\times\mathbb{V}^{\flat},\quad
A=\begin{pmatrix}a&b\\\ c&d\\\
\end{pmatrix}\mapsto(A^{\flat}=\frac{a}{1-d},u^{\flat}=\frac{b}{1-d}).$

We then consider the analytic side. Let $V_{0}=V_{0}^{\flat}\oplus Fe$ be a
splitting of vector spaces over $F_{0}$. We have the induced splitting
$V_{0}^{*}=(V_{0}^{\flat})^{*}\oplus Fe^{*}$ where
$(e^{*},e)=1,e^{*}|_{V_{0}^{\flat}}=0$. Any $\gamma\in\mathrm{GL}(V_{0})$
could be written as

$\gamma=\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix},\quad
a\in\mathrm{End}(V_{0}^{\flat}),b\in\mathbb{V}^{\flat}_{0},c\in(\mathbb{V}^{\flat}_{0})^{*},d\in
F_{0}.$

######  Definition 5.2.2.

Consider the $\mathrm{GL}(V_{0}^{\flat})$-equivariant rational relative Cayley
map

(14)
$c_{V_{0},\mathrm{GL}}:\mathrm{GL}(V_{0})\to\mathrm{GL}(V_{0}^{\flat})\times
V_{0}\times V_{0}^{*},\quad\gamma=\begin{pmatrix}a&b\\\ c&d\\\
\end{pmatrix}\mapsto(\gamma^{\flat}=\frac{a}{1-d},u^{\flat}_{1}=\frac{b}{1-d},u^{\flat}_{2}=\frac{c}{1-d}).$

######  Proposition 5.2.3.

  1. (1)

If $(g,u^{0})$ (resp. $(\gamma,e,e^{*})$ ) is regualr semi-simple, then
$c_{\mathbb{V},\mathrm{Herm}}(g)$ (resp. $c_{V_{0},\mathrm{GL}}(\gamma)$ )is
regular semi-simple.

  2. (2)

If $(g,u^{0})$ matches $(\gamma,e,e^{*})$, then
$c_{\mathbb{V},\mathrm{Herm}}(g)$ matches $c_{V_{0},\mathrm{GL}}(\gamma)$.

###### Proof.

As regular semi-simpleness is a condition on stabilizer of the group action,
the first part follows. If $(g,u^{0})$ matches $(\gamma,e,e^{*})$, then by
definition $(g,u^{0},(u^{0})^{*})$ and $(\gamma,e,e^{*})$ is conjugate in
$\mathrm{End}(V)\times V\times V^{*}$. The relative Cayley map
$c_{V}:\mathrm{End}(V)\to\mathrm{End}(V^{\flat})\times V\times V^{*}$ given by
$A\to(a,b/(1-d),c/(1-d))$ restricts to $c_{V_{0},\mathrm{GL}}$ and
$c_{\mathbb{V},\mathrm{Herm}}$. As $c_{V}$ is $\mathrm{GL}(V)$-equivariant,
the result follows. ∎

###  5.3. Arithmetic induction on geometric side

######  Proposition 5.3.1.

Assume that $1-d\in O_{F}^{\times}$. Then for $g\in\mathrm{GL}(V)(F_{0})$ such
that $(g,u^{0})$ is regular semi-simple. We have an equality of closed formal
subscheme in $\mathcal{N}_{n-1}^{\mathrm{GL}}$:

(15)
$g\mathcal{N}_{n}\cap_{\mathcal{N}_{n}^{\mathrm{GL}}}\mathcal{N}_{n-1}^{\mathrm{GL}}=g^{\flat}\mathcal{N}_{n-1}\cap_{\mathcal{N}_{n-1}^{\mathrm{GL}}}\mathcal{Z}^{\mathrm{GL}}(u^{\flat}).$

###### Proof.

The group $\mathrm{GL}(\mathbb{V})$ acts on the set of principal polarizations
on $\mathbb{X}$, whose stabilizer at $\lambda_{\mathbb{X}}$ can be identified
with $\mathrm{U}(\mathbb{V})$. There
$g\in\mathrm{GL}(\mathbb{V})/\mathrm{U}(\mathbb{V})$ can be regarded as a
principal polarization $\lambda_{g}=g.\lambda_{\mathbb{X}}$ on $\mathbb{X}$.
Then $g\mathcal{N}_{n}\subseteq\mathcal{N}_{n}^{\mathrm{GL}}$ is the lifting
locus of the principal polarization $\lambda_{g}$. We study $(X,\iota,\rho)\in
g\mathcal{N}_{n}\cap_{\mathcal{N}_{n}^{\mathrm{GL}}}\mathcal{N}_{n-1}^{\mathrm{GL}}$.

Firstly the new polarization $\lambda_{g}$ on
$\mathbb{X}=\mathbb{X}^{\flat}\times\mathbb{E}$ gives a polarization on
$\mathbb{X}^{\flat}$ which corresponds to $g^{\flat}$. Regard $\lambda_{g}$ as
an endomorphism from $X^{\flat}\times\mathbb{E}$ to
$(X^{\flat})^{\vee}\times\mathbb{E}^{\vee}$, which gives a lifting from
$\mathbb{E}$ to $(X^{\flat})^{\vee}$ which gives a lifting of
$u^{\flat}:\mathbb{E}\to\mathbb{X}$ via the lifting of the polarization $X\to
X^{\vee}$ on $g\mathcal{N}_{n}$. This shows
$g\mathcal{N}_{n}\cap_{\mathcal{N}_{n}^{\mathrm{GL}}}\mathcal{N}_{n-1}^{\mathrm{GL}}\subseteq
g^{\flat}\mathcal{N}_{n-1}\cap_{\mathcal{N}_{n-1}^{\mathrm{GL}}}\mathcal{Z}^{\mathrm{GL}}(u^{\flat}).$
The other direction follows similarly. ∎

######  Proposition 5.3.2.

Assume that $1-d\in O_{F}^{\times}$. Hence for $g\in\mathrm{GL}(V)(F_{0})$
such that $(g,u^{0})$ is regular semi-simple, we have

$\mathrm{Int}^{\mathrm{Herm},\mathbb{V}}(g,u^{0})=\mathrm{Int}^{\mathrm{Herm},\mathbb{V}^{\flat}}(c_{\mathbb{V},\mathrm{Herm}}(g))\in\mathbb{Q}.$

###### Proof.

By Proposition 4.0.3,

$\mathrm{Int}^{\mathrm{Herm},\mathbb{V}^{\flat}}(g^{\flat},u^{\flat})=\mathcal{N}_{n-1}\cap^{\mathbb{L}}_{\mathcal{N}_{n-1}^{\mathrm{GL}}}g^{\flat}\mathcal{N}_{n-1}\cap^{\mathbb{L}}_{\mathcal{N}^{\mathrm{GL}}_{n-1}}\mathcal{Z}^{\mathrm{GL}}(u^{\flat}).$

The above intersection
$g\mathcal{N}_{n}\cap_{\mathcal{N}_{n}^{\mathrm{GL}}}\mathcal{N}_{n-1}^{\mathrm{GL}}=g^{\flat}\mathcal{N}_{n-1}\cap_{\mathcal{N}_{n-1}^{\mathrm{GL}}}\mathcal{Z}^{\mathrm{GL}}(u^{\flat})$
has correct codimension hence by [AFL-Wei2019, Lemma B.2] we have

$g\mathcal{N}_{n}\cap_{\mathcal{N}_{n}^{\mathrm{GL}}}^{\mathbb{L}}\mathcal{N}_{n-1}^{\mathrm{GL}}=g^{\flat}\mathcal{N}_{n-1}\cap_{\mathcal{N}_{n-1}^{\mathrm{GL}}}^{\mathbb{L}}\mathcal{Z}^{\mathrm{GL}}(u^{\flat}).$

Intersecting both sides with $\mathcal{N}_{n}$ inside
$\mathcal{N}_{n}^{\mathrm{GL}}$ and using identification
$\mathcal{Z}(u_{0})\cong\mathcal{N}_{n-1}$ and above Proposition 5.3.1, we
obtain the result. ∎

###  5.4. Arithmetic induction on analytic side

Assume that the self-dual lattice $L=L^{\flat}\oplus O_{F}e$ has an orthogonal
decomposition where $(e,e)=1$, which induces the splitting
$V_{0}=V_{0}^{\flat}\oplus F_{0}e$.

######  Proposition 5.4.1.

Assume that $1-d\in O_{F}^{\times}$. Then for
$\gamma\in\mathrm{GL}(V_{0})(F_{0})$ such that $(\gamma,e,e^{*})$ is regular
semi-simple, write
$c_{V_{0},\mathrm{GL}}(\gamma)=(\gamma^{\flat},u_{1}^{\flat},u_{2}^{\flat})$.
We have

(16)
$\mathrm{Orb}(\gamma,e,e^{*},\Phi^{\prime}_{L^{\prime}},s)=\mathrm{Orb}(\gamma^{\flat},u_{1}^{\flat},u_{2}^{\flat},\Phi^{\prime}_{(L^{\prime})^{\flat}},s)$

In particular, we have
$\partial\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{L^{\prime}})=\partial\mathrm{Orb}(c_{V_{0},\mathrm{GL}}(\gamma),\Phi^{\prime}_{(L^{\prime})^{\flat}})\in\mathbb{Q}\log
q$.

###### Proof.

This follows from the proof of [ZZhang2021, Corollary 3.22.] and the integral
transitivity lemma [ZZhang2021, Lemma 4.12]. Note in our case, no twisting
element is needed. ∎

######  Remark 5.4.2.

We expect arithmetic induction systems to arise from orbital integrals in
general. Consider the orbital integral of
$f^{\prime}\otimes\phi^{\prime}\in\mathcal{S}(X^{\prime}\times Y^{\prime})$.
Choose a compact open subgroup $K_{H^{\prime}}\leq H^{\prime}$ such that
$f^{\prime}$ is $K_{H^{\prime}}$-invariant. Fix $x_{0}\in Y^{\prime}$ with
$(x_{0},x_{0})=1$. We know $H^{\prime}$ acts transitively on $\\{x\in
Y^{\prime}|(x,x)=1\\}$ whose stabilizer at $x_{0}$ is $H^{\prime\flat}$. If we
assume the transitivity for the action of $K_{H^{\prime}}$ on $\Omega:=\\{x\in
Y^{\prime}|(x,x)=1,\phi^{\prime}(x)\not=0\\}$, then we have
$\mathrm{Orb}_{H^{\prime}}(g,x_{0},(f^{\prime}\otimes\phi^{\prime}))=\mathrm{vol}(K_{H^{\prime}})\mathrm{Orb}_{H^{\prime\flat}}(g,f)$.
If $f\in\mathcal{S}(X^{\prime})$ goes to
$f^{\flat}\times\phi^{\flat}\in\mathcal{S}(X^{\prime\flat}\times
Y^{\prime\flat})$ under the relative Cayley map, then generically we have
$\mathrm{Orb}_{H^{\flat}}(g,f)=\mathrm{Orb}_{H^{\flat}}(c_{V_{0}}(g),f^{\flat}\otimes\phi^{\flat})$.

##  6\. Twisted unitary Shimura varieties and special cycles

Starting with this section, we give the proof of the twisted AFL conjecture,
i.e. Theorem 4.0.8 via global methods and Weil type relative trace formulas.
Moreover, we introduce and study twisted CM cycles on unitary Shimura
varieties. It turns out that for our local purpose on twisted AFL, we could
work with punctured integral models away from infinitely many places and do
not need to handle the constant term of global distributions.

We switch to global set up. Let $F_{0}$ be a totally real number field, and
$F/F_{0}$ be a CM quadratic extension. For the properness of our Shimura
varieties, we always assume that $[F_{0}:\mathbb{Q}]>1$. Let $E_{0}/F_{0}$ be
a totally real quadratic extension. Then $E=E_{0}\otimes_{F_{0}}F$ is a CM
quadratic extension of $E_{0}$. Let $F^{\prime}/F_{0}$ be the third subfield
of $E$ fixed by $\sigma_{E_{0}/F_{0}}\otimes\sigma_{F/F_{0}}$, where
$\sigma_{E_{0}/F_{0}}$ (resp. $\sigma_{F/F_{0}}$) the non-trivial Galois
involution on $E_{0}/F_{0}$ (resp. $F/F_{0}$). We will study the embedding of
algebraic groups

$H=\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V)\to
G=\mathrm{Res}_{E_{0}/\mathbb{Q}}\mathrm{U}(V_{E_{0}})$

over $\mathbb{Q}$ for a suitable $F/F_{0}$-hermitian space $V$. In order to
use moduli interpretations of Shimura varieties, we will work with the RSZ
variant of unitary Shimura varieties [RSZ-AGGP].

###  6.1. Shimura varieties for $V$ and $V_{E_{0}}$

Choose a CM type $\Phi\subseteq\mathrm{Hom}(F,\overline{\mathbb{Q}})$ of $F$
with a distinguished element $\varphi_{0}\in\Phi$. Let $V$ be a
$F/F_{0}$-hermitian space of dimension $n\geq 1$ with signature
$\\{(n-1,1)_{\varphi_{0}},(n,0)_{\varphi\in\Phi-\\{\varphi_{0}\\}}\\}$.
Consider the following reductive groups over $\mathbb{Q}$:

$Z^{\mathbb{Q}}:=\\{x\in\mathrm{Res}_{F/\mathbb{Q}}\mathbb{G}_{m}|\,\mathrm{Nm}_{F/F_{0}}x\in\mathbb{G}_{m}\\},$
$H^{\mathbb{Q}}:=\\{g\in\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{GU}(V)|\,c(g)\in\mathbb{G}_{m}\\},$
$H=\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V),\quad\widetilde{H}=Z^{\mathbb{Q}}\times
H\cong Z^{\mathbb{Q}}\times_{\mathbb{G}_{m}}H^{\mathbb{Q}}.$

Here
$c:\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{GU}(V)\to\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathbb{G}_{m}$
is the similitude character. For neat compact open subgroups
$K_{Z^{\mathbb{Q}}}\subseteq Z^{\mathbb{Q}}(\mathbb{A}_{f}),K\subseteq
H(\mathbb{A}_{f})$, consider the the _RSZ unitary Shimura variety_ with level
$\widetilde{K}=K_{Z^{\mathbb{Q}}}\times K$ associated to the Shimura datum
$(\widetilde{H},\\{h_{\widetilde{H}}\\})$:

(17)
$\mathrm{Sh}_{\widetilde{K}}(\widetilde{H},\\{h_{\widetilde{H}}\\})(\mathbb{C})\cong\mathrm{Sh}_{K_{Z^{\mathbb{Q}}}}(Z^{\mathbb{Q}},h_{\Phi})(\mathbb{C})\times\mathrm{Sh}_{K}(H,\\{h_{H}\\})(\mathbb{C}).$

The canonical model of
$\mathrm{Sh}_{K_{\widetilde{H}}}(\widetilde{H},\\{h_{\widetilde{H}}\\})$

$M:=M_{\widetilde{H},\widetilde{K}}\to\mathrm{Spec}F^{\mathrm{RSZ}}$

is a $n-1$ dimensional smooth _projective_ ($[F_{0}:\mathbb{Q}]>1$) variety
over the reflex field $F^{\mathrm{RSZ}}$.

Now we consider the CM type
$\Phi_{E_{0}}\subseteq\mathrm{Hom}(E,\overline{\mathbb{Q}})$ of $E$ induced by
$\Phi$, i.e. $\varphi\in\Phi_{E_{0}}$ iff $\varphi|_{F}\in\Phi$. We denote the
element of $\Phi_{E_{0}}$ over $\varphi_{0}\in\Phi$ by
$\\{\varphi_{0,E_{0}},\varphi_{0,E_{0}}^{c}\\}$, where
$\varphi_{0,E_{0}}^{c}=\varphi_{0,E_{0}}\circ\sigma_{E/F}$. The base change
$V_{E_{0}}$ is a $E/E_{0}$-hermtian space of dimension $n\geq 1$ with
signature
$\\{(n-1,1)_{\varphi_{0,E_{0}}},(n-1,1)_{\varphi_{0,E_{0}}^{c}},(n,0)_{\varphi\in\Phi_{E_{0}}-\\{\varphi_{0,E_{0}},\varphi_{0,E_{0}}^{c}\\}}\\}$.
Similar to $V$, we consider the following reductive groups for $V_{E_{0}}$:

$Z^{\mathbb{Q},E_{0}}:=\\{x\in\mathrm{Res}_{E/\mathbb{Q}}\mathbb{G}_{m}|\,\mathrm{Nm}_{E/E_{0}}x\in\mathbb{G}_{m}\\},$
$G^{\mathbb{Q}}:=\\{g\in\mathrm{Res}_{E_{0}/\mathbb{Q}}\mathrm{GU}(V_{E_{0}})|\,c(g)\in\mathbb{G}_{m}\\},$
$G=\mathrm{Res}_{E_{0}/\mathbb{Q}}\mathrm{U}(V_{E_{0}}),\quad\widetilde{G}=Z^{\mathbb{Q},E_{0}}\times
G\cong Z^{\mathbb{Q},E_{0}}\times_{\mathbb{G}_{m}}G^{\mathbb{Q}}.$

For neat compact open subgroups $K_{Z^{\mathbb{Q},E_{0}}}\subseteq
Z^{\mathbb{Q}}(\mathbb{A}_{f}),K^{E_{0}}\subseteq G(\mathbb{A}_{f})$, we have
the canonical model of RSZ Shimura varieties for the Shimura datum
$(\widetilde{G},\\{h_{\widetilde{G}}\\})$:

$M^{E_{0}}=M_{\widetilde{G},\widetilde{K^{E_{0}}}}\to\mathrm{Spec}E^{\mathrm{RSZ}}$

which is a $2n-1$ dimensional smooth _projective_ variety over the reflex
field $E^{\mathrm{RSZ}}$. Base change gives a natural embedding of Shimura
datum
$(\widetilde{H},\\{h_{\widetilde{H}}\\})\to(\widetilde{G},\\{h_{\widetilde{G}}\\})$.
Hence for compatible levels $\widetilde{K}$ and $\widetilde{K^{E_{0}}}$ (i.e.
$\widetilde{K^{E_{0}}}\cap\widetilde{H}=\widetilde{K}$), we have a twisted
diagonal embedding of Shimura varieties

(18)
$M\times_{\mathrm{Spec}F^{\mathrm{RSZ}}}\mathrm{Spec}{E^{\mathrm{RSZ}}}\to
M^{E_{0}}.$

###  6.2. Punctured PEL type integral models

For RSZ unitary Shimura varieties, we have moduli interpretations. Choose a
hermitian lattice $L\subseteq V$ and a finite collection $\Delta$ of finite
places for $F_{0}$ such that

  * •

all $2$-adic places are in $\Delta$ and $F/F_{0}$ is unramified outside
$\Delta$.

  * •

$E_{0}/F_{0}$ is unramified outside $\Delta$.

  * •

$L$ is self-dual away from $\Delta$.

  * •

$\Phi$ is unramified away from $\Delta$.

######  Definition 6.2.1.

Let $K(L)^{\Delta}$ (resp. $K^{E_{0}}(L)^{\Delta}$) be the stabilizer of $L$
(resp. $L_{O_{E_{0}}}$) in $\mathrm{U}(V)(\mathbb{A}_{f}^{\Delta})$ (resp.
$\mathrm{U}(V_{E_{0}})(\mathbb{A}_{f}^{\Delta})$) and
$K_{Z^{\mathbb{Q}}}^{\Delta}$ (resp. $K_{Z^{\mathbb{Q}}}^{E_{0},\Delta}$) be
the unique maximal compact open subgroup of
$Z^{\mathbb{Q}}(\mathbb{A}_{f}^{\Delta})$ (resp.
$Z^{\mathbb{Q},E_{0}}(\mathbb{A}_{f}^{\Delta})$). And we call

(19) ${\widetilde{K}=K_{Z^{\mathbb{Q}},\Delta}\times
K_{Z^{\mathbb{Q}}}^{\Delta}\times K_{\Delta}\times
K(L)^{\Delta}\subseteq\widetilde{H}(\mathbb{A}_{f}).}$

a level structure for $L$ and $\Delta$, where $K_{Z^{\mathbb{Q}},\Delta}$
(resp. $K_{\Delta}$) is a sufficiently small compact open subgroup of
$Z^{\mathbb{Q}}(F_{0,\Delta})$ (resp. $\mathrm{U}(V)(F_{0,\Delta})$).

We call

(20) ${\widetilde{K^{E_{0}}}=K_{Z^{\mathbb{Q}},\Delta}^{E_{0}}\times
K_{Z^{\mathbb{Q}}}^{E_{0},\Delta}\times K_{\Delta}^{E_{0}}\times
K^{E_{0}}(L)^{\Delta}\subseteq\widetilde{G}(\mathbb{A}_{f}).}$

a level structure for $L_{O_{E_{0}}}$ and $\Delta$, where
$K_{Z^{\mathbb{Q}},\Delta}^{E_{0}}$ (resp. $K_{\Delta}^{E_{0}}$) is a
sufficiently small enough compact open subgroup of
$Z^{\mathbb{Q}}(E_{0,\Delta})$ (resp. $\mathrm{U}(V)(E_{0,\Delta})$).

Following [RSZ-AGGP, Section 6.1], we introduce PEL type integral models over
$O_{E^{\mathrm{RSZ}}}[\Delta^{-1}]$ for $M$ and $M^{E_{0}}$. For $?=F_{0}$
(resp. $?=E_{0}$), we set $O_{?,0}=O_{F_{0}}$ (resp. $O_{?,0}=O_{E_{0}}$),
$O_{?}=O_{F}$ (resp. $O_{?}=O_{E}$). Now consider the functor

$\mathcal{M}_{0}^{?}\to\mathrm{Spec}O_{E^{\mathrm{RSZ}}}[\Delta^{-1}]$

sending a locally noetherian scheme $S$ to the groupoid
$\mathcal{M}_{0}^{?}(S)$ of tuples
$(A_{0},\iota_{0},\lambda_{0},\overline{\eta}_{0})$ where

  * •

$A_{0}$ is an abelian scheme over $S$ of dimension $[?:\mathbb{Q}]$.

  * •

$\iota_{0}:O_{?}[\Delta^{-1}]\to\mathrm{End}(A_{0})[\Delta^{-1}]$ is an
$O_{?}[\Delta^{-1}]$-action satisfying the _Kottwitz condition_ of signature
$\\{(1,0)_{\varphi\in\Phi^{?}}\\}$:

(21)
$\mathrm{charpol}(\iota_{0}(a)\mid\mathrm{Lie}A_{0})=\prod_{\varphi\in\Phi^{?}}(T-\varphi(a))\in\mathcal{O}_{S}[T].$

  * •

$\lambda_{0}:A_{0}\to A_{0}^{\vee}$ is an away-from-$\Delta$ principal
polarization such that for all $a\in O_{?}[\Delta^{-1}]$ we have
$\lambda^{-1}_{0}\circ\iota_{0}(a)^{\vee}\circ\lambda_{0}=\iota_{0}(\overline{a})$.

  * •

$\overline{\eta}_{0}$ is a $K_{Z^{\mathbb{Q}},\Delta}^{?}$-level structure.

An isomorphism between two tuples is a quasi-isogeny preserving the
polarization and $K_{Z^{\mathbb{Q}},\Delta}$-level structure. The functor
$\mathcal{M}_{0}^{?}$ is representable, finite and étale. We assume that
$\mathcal{M}_{0}^{?}$ is non-empty, which holds if $\Delta$ is large enough.
The generic fiber $M_{0}^{?}$ of $\mathcal{M}_{0}^{?}$ is a disjoint union of
copies of
$\mathrm{Sh}_{K_{Z^{\mathbb{Q}}}^{?}}(Z^{\mathbb{Q},?},h_{\Phi^{?}})$, see
[RSZ-AGGP, Lemma 3.4]. We work with one copy and still denote its étale
integral model by $\mathcal{M}_{0}^{?}$.

######  Definition 6.2.2.

The integral RSZ Shimura variety with level $K^{?}$ for $L$ and $\Delta$ is
the functor

$\mathcal{M}^{?}\to\mathrm{Spec}O_{E^{\mathrm{RSZ}}}[\Delta^{-1}]$

sending a locally noetherian scheme $S$ to the groupoid of tuples
$(A_{0},\iota_{0},\lambda_{0},\overline{\eta}_{0},A,\iota,\lambda,\overline{\eta}_{\Delta})$
where

  * •

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

  * •

$A$ is an abelian scheme over $S$ of dimension $n[?:\mathbb{Q}]$.

  * •

$\iota:O_{?}[\Delta^{-1}]\to\mathrm{End}(A)[\Delta^{-1}]$ is an
$O_{?}[\Delta^{-1}]$-action satisfying the _Kottwitz condition_ of the
signature of $V_{?}$.

  * •

$\lambda:A\to A^{\vee}$ is a prime-to-$\Delta$ principal polarization such
that for all $a\in O_{?}[\Delta^{-1}]$ we have
$\lambda^{-1}\circ\iota(a)^{\vee}\circ\lambda=\iota(\overline{a}).$

  * •

$\overline{\eta}_{\Delta}$ is a $K_{\Delta}^{?}$-orbit of isometries of
hermitian modules

$\eta_{\Delta}:V_{\Delta}(A_{0},A)=\prod_{v\in\Delta}V_{v}(A_{0},A)\cong
V_{?,\Delta}$

as smooth $F_{0,\Delta}$-sheaves on $S$. Here
$V_{v}(A_{0},A)=\mathrm{Hom}_{F\otimes_{F_{0}}F_{0,v}}(V_{v}A_{0},V_{v}A)$ is
the Hom space between rational Tate modules of $A_{0}$ and $A$, with the
hermitian form

$(x,y):=\lambda_{0}^{-1}\circ y^{\vee}\circ\lambda\circ x.$

  * •

For any finite place $v\not\in\Delta$ of $F_{0}$, we put the sign condition
and Eisenstein condition at $v$ as in [RSZ-AGGP, Section 4.1]. By [RSZ-AGGP,
Section 4.1] and [RSZ-AGGP, Rem. 6.5 (i)], the Eisenstein condition is
automatic when the places of $?\otimes_{F_{0}}F$ above $v$ are unramified over
$p$, and the sign condition is automatic when $v$ is split in $F$.

An isomorphism between two tuples is a pair of quasi-isogenies
$(\phi_{0},\phi):(A_{0},A)\to(A_{0}^{\prime},A^{\prime})$ preserving
polarizations and the $K_{Z^{\mathbb{Q}},\Delta}\times K_{\Delta}$-orbit of
level structures.

######  Theorem 6.2.3.

The functor $\mathcal{M}^{?}$ is representable by a separated scheme flat and
of finite type over $\mathrm{Spec}O_{E}^{\mathrm{RSZ}}[\Delta^{-1}]$.
Moreover, $\mathcal{M}^{?}$ is smooth and projective over
$\mathrm{Spec}O_{E^{\mathrm{RSZ}}}[\Delta^{-1}]$.

###### Proof.

This follows from [RSZ-Shimura, Theorem 6.2]. Because of our assumption on
$\Delta$, $E^{\mathrm{RSZ}}/F^{\mathrm{RSZ}}$ is also unramified away form
$\Delta$, hence $\mathcal{M}^{?}$ is still smooth over
$\mathrm{Spec}O_{E^{\mathrm{RSZ}}}[\Delta^{-1}]$. ∎

From now on, for simplicity we write $(A_{0},A,\overline{\eta})$ (resp.
$A_{0}^{E_{0}},A^{E_{0}},\overline{\eta}^{E_{0}}$) for a $S$-point of
$\mathcal{M}$ (resp. $\mathcal{M}^{E_{0}}$). Serre tensor construction gives a
natural embedding

(22)
$\mathcal{M}\to\mathcal{M}^{E_{0}},\,\,(A_{0},A,\overline{\eta})\mapsto(A_{0}\otimes_{O_{F}}O_{E},A\otimes_{O_{F}}O_{E},\overline{\eta}\otimes_{F}E).$

which recovers the embedding of Shimura varieties (18) over
$E^{\mathrm{RSZ}}$.

###  6.3. Arithmetic theta series of Kudla–Rapoport divisors

####  6.3.1. On the unitary Shimura variety $M$

For any $(A_{0},A,\overline{\eta})\in M(S)$, endow the $F$-vector space
$\mathrm{Hom}^{\circ}_{F}(A_{0},A)=\mathrm{Hom}_{O_{F}}(A_{0},A)\otimes\mathbb{Q}$
with the hermitian form

(23) $(u_{1},u_{2}):=\lambda_{0}^{-1}\circ u_{2}^{\vee}\circ\lambda\circ
u_{1}\in\mathrm{End}^{\circ}_{F}(A_{0})\simeq F.$

######  Definition 6.3.1.

Consider non-zero $\xi\in F_{0}$ and $\mu\in V(\mathbb{A}_{0,f})/K$. The
Kudla–Rapoport cycle $Z(\xi,\mu)$ is the functor sending any locally
noetherian $E$-scheme $S$ to the groupoid of tuples
$(A_{0},A,\overline{\eta},u)$ where

  * •

$(A_{0},A,\overline{\eta})\in M(S)$.

  * •

$u\in\mathrm{Hom}^{\circ}_{F}(A_{0},A)$ such that $(u,u)=\xi$.

  * •

$\overline{\eta}(u)$ is in the $K$-orbit $\mu$.

An isomorphism between two tuples $(A_{0},A,\overline{\eta},u)$ and
$(A_{0}^{\prime},A^{\prime},\overline{\eta}^{\prime},u^{\prime})$ is an
isomorphism of tuples
$(\phi_{0},\phi):(A_{0},A,\overline{\eta})\simeq(A_{0}^{\prime},A^{\prime},\overline{\eta}^{\prime})$
such that $\phi\circ u=u^{\prime}\circ\phi_{0}$.

By positivity of the Rosati involution, $Z(\xi,\mu)$ is empty unless $\xi\in
F_{0,+}$ is totally positive. The natural forgetful morphism $i:Z(\xi,\mu)\to
M$ is finite and unramified, and is étale locally a Cartier divisor. We view
$Z(\xi,\mu)$ as elements in the Chow group (of $\mathbb{Q}$-coefficients)
$\mathrm{Ch}^{1}(M)$.

For a Schwartz function $\phi\in\mathcal{S}(V(\mathbb{A}_{0,f}))^{K}$ and
$\xi\in F_{0,+}$, the $\phi$-averaged Kudla–Rapoport divisor is the finite
summation

(24) $Z(\xi,\phi)\colon=\sum_{\mu\in
V_{\xi}(\mathbb{A}_{0,f})/K}\phi(\mu)\,Z(\xi,\mu)\in\mathrm{Ch}^{1}(M).$

Here $V_{\xi}=\\{x\in V|(x,x)=\xi\\}$ is the hyperboloid in $V$ of length
$\xi$. We put

$Z(0,\phi):=-\phi(0)c_{1}(\omega)\in\mathrm{Ch}^{1}(M)$

where $\omega$ is the automorphic line bundle (Hodge bundle) on $M$, which is
a descent of the tautological line bundle on the hermitian symmetric domain
$X$ of $M$ to $E$.

For $\xi\in F_{0}$, consider the weight $n$ Whittaker function (2) on
$\mathrm{SL}_{2}(F_{0,\infty})$:

$W^{(n)}_{\xi}(h_{\infty})=\prod_{v|\infty}W^{(n)}_{\xi}(h_{v}).$

Consider the Weil representation $\omega=\omega_{V,\psi}$ of
$\mathrm{SL}_{2}(\mathbb{A}_{0,f})$ on $\mathcal{S}(V(\mathbb{A}_{0,f}))$
which commutes with the natural action of $\mathrm{U}(V)(\mathbb{A}_{0,f})$.
For a Schwartz function $\phi\in\mathcal{S}(V(\mathbb{A}_{0,f}))^{K}$, we form
_the geometric generating series of Kudla–Rapoport divisors_ on $M$ by

(25) $Z(h,\phi):=W^{(n)}_{0}(h_{\infty})Z(0,\omega(h_{f})\phi)+\sum_{\xi\in
F_{0,+}}W^{(n)}_{\xi}(h_{\infty})Z(\xi,\omega(h_{f})\phi)\in\mathrm{Ch}^{1}(M),$

for any
$h=(h_{\infty},h_{f})\in\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})=\mathrm{SL}_{2}(F_{0,\infty})\times\mathrm{SL}_{2}(\mathbb{A}_{0,f})$.
For any $h_{f}\in\mathrm{SL}_{2}(\mathbb{A}_{0,f})$, we have

(26) $Z(h,\omega(h_{f})\phi)=Z(hh_{f},\phi).$

The modularity of geometric theta series of divisors is known by the thesis
work of Yifeng Liu [Liu-Thesis].

######  Proposition 6.3.2.

[AFL-Wei2019, Thm. 8.1] For any $\phi\in\mathcal{S}(V(\mathbb{A}_{0,f}))^{K}$,
$Z(h,\phi)$ is a weight $n$ holomorphic automorphic form valued in
$\mathrm{Ch}^{1}(M_{\widetilde{H},\widetilde{K}})$. More precisely, we have

$Z(h,\phi)\in\mathcal{A}_{\rm
hol}(\mathrm{SL}_{2}(\mathbb{A}_{F_{0}}),K_{0},n)_{\overline{\mathbb{Q}}}\otimes_{\overline{\mathbb{Q}}}\mathrm{Ch}^{1}(M)_{\overline{\mathbb{Q}}}$

where $K_{0}\subseteq\mathrm{SL}_{2}(\mathbb{A}_{0,f})$ is any compact open
subgroup that fixes $\phi$ under the Weil representation.

####  6.3.2. On integral model $\mathcal{M}$

To do arithmetic intersection, we consider the arithmetic theta series on
$\mathcal{M}$. Now $K$ is a level structure for $L$ and $\Delta$ (Definition
(6.2.1)).

######  Definition 6.3.3.

Let $\xi\in F_{0,+}$ and $\mu_{\Delta}\in V(F_{0,\Delta})/K_{\Delta}$. _The
Kudla–Rapoport cycle_ $\mathcal{Z}(\xi,\mu_{\Delta})\to\mathcal{M}$ is the
functor sending any locally noetherian scheme $S$ over
$O_{E^{\mathrm{RSZ}}}[1/\Delta]$ to the groupoid of tuples
$(A_{0},A,\overline{\eta},u)$ where

  * •

$(A_{0},A,\overline{\eta})\in\mathcal{M}(S)$;

  * •

$u\in\mathrm{Hom}_{O_{F}}(A_{0},A)[1/\Delta]$ with $(u,u)=\xi$.

  * •

$\overline{\eta}(u)$ is in the $K_{\Delta}$-orbit $\mu_{\Delta}$.

######  Definition 6.3.4.

A Schwartz function $\phi\in\mathcal{S}(V(\mathbb{A}_{f}))^{K}$ is standard
for $L$, if $\phi=\phi_{\Delta}\otimes\phi^{\Delta}$ where where
$\phi^{\Delta}=1_{\widehat{L}^{\Delta}}$ agrees with the indicator of $L$ in
$\mathcal{S}(V(\mathbb{A}_{f}^{\Delta}))^{K}$ away from $\Delta$.

Consider $\phi\in\mathcal{S}(V(\mathbb{A}_{0,f}))^{K}$ that is standard for
$L$. For $\xi\in F_{0,+}$, the $\phi$-averaged Kudla–Rapoport divisor is the
finite summation

(27) $\mathcal{Z}(\xi,\phi)\colon=\sum_{\mu_{\Delta}\in
V_{\xi}(F_{0,\Delta})/K_{\Delta}}\phi_{\Delta}(\mu_{\Delta})\,\mathcal{Z}(\xi,\mu_{\Delta}),$

viewed as elements in the Chow group $\mathrm{Ch}^{1}(\mathcal{M})$.

Let $\mu$ be an archimedean place of $E^{\mathrm{RSZ}}$. we have two kinds of
Green functions [AFL-Wei2019, Section 3] for the Kudla–Rapoport divisor
$Z(\xi,\phi)_{\mathbb{C}}$ on $M\otimes_{E^{\mathrm{RSZ}},\nu}\mathbb{C}$:

  * •

The _Kudla Green function_ $\mathcal{G}^{\bf K}(\xi,h_{\infty},\phi)$ with a
variable $h_{\infty}\in\mathrm{SL}_{2}(F_{0,\infty})$, which occurs in the
analytic side naturally via computations of archimedean orbital integrals.

  * •

The _automorphic Green function_ [JBGreen] $\mathcal{G}^{\bf
B}(\xi,h_{\infty},\phi)=\mathcal{G}^{\bf B}(\xi,\phi)$ for $\xi\in F_{0,+}$
which is admissible by [JBGreen, Cor. 5.16].

We consider _arithmetic Kudla–Rapoport divisors_ ($?=\bf K,\bf B$)

(28)
$\widehat{\mathcal{Z}}^{?}(\xi,h_{\infty},\phi)=(\mathcal{Z}(\xi,\phi),(\mathcal{G}^{?}_{\nu}(\xi,h_{\nu},\phi))_{\nu})\in\widehat{\mathrm{Ch}}^{1}(\mathcal{M}).$

###  6.4. Derived twisted CM cycles

We use the twisted diagonal embedding to define derived twisted CM cycles on
$\mathcal{M}$.

######  Definition 6.4.1.

For $\mu_{\Delta}^{E_{0}}\in
K_{\Delta}^{E_{0}}\backslash\mathrm{U}(V)(E_{0,\Delta})/K_{\Delta}^{E_{0}}$,
the _Hecke correspondence away from $\Delta$_

$\mathrm{Hk}_{\mu_{\Delta}^{E_{0}}}\to\mathcal{M}^{E_{0}}\times\mathcal{M}^{E_{0}}$

is the functor sending any locally noetherian
$O_{E^{\mathrm{RSZ}}}[\Delta^{-1}]$-scheme $S$ to the groupoid of tuples
$(A_{0}^{E_{0}},A_{1}^{E_{0}},\overline{\eta}_{A_{1}^{E_{0}}},A_{2}^{E_{0}},\overline{\eta}_{A_{2}^{E_{0}}},\varphi)$
where

  1. (1)

$(A_{0}^{E_{0}},A_{1}^{E_{0}},\overline{\eta}_{A_{1}^{E_{0}}}),\,(A_{0},A_{2}^{E_{0}},\overline{\eta}_{A_{2}^{E_{0}}})\in\mathcal{M}^{E_{0}}(S)$.

  2. (2)

$\varphi\in\mathrm{Hom}_{O_{E}}(A_{1}^{E_{0}},A_{2}^{E_{0}})[1/\Delta]$ is a
prime-to-$\Delta$ quasi-isogeny such that
$\varphi^{*}\lambda_{2}=\lambda_{1}$.

  3. (3)

We have
$\eta_{A_{2}^{E_{0}}}\varphi\eta_{A_{1}^{E_{0}}}^{-1}\in\mu_{\Delta}^{E_{0}}\subseteq\mathrm{U}(V)(E_{0,\Delta})$
for some $\eta_{A_{1}^{E_{0}}}\in\overline{\eta}_{A_{1}^{E_{0}}}$ and
$\eta_{A_{2}^{E_{0}}}\in\overline{\eta}_{A_{2}^{E_{0}}}.$

Recall above $\eta_{A_{i}^{E_{0}}}$ ($i=1,2$) is an isometry of hermitian
modules $\eta_{A_{i}^{E_{0}}}:V_{\Delta}(A_{0},A)\cong V_{E_{0},\Delta}$. So
naturally we have
$\eta_{A_{2}^{E_{0}}}\varphi\eta_{A_{1}^{E-0}}^{-1}\in\mathrm{U}(V)(E_{0,\Delta})$.
Recall we have an embedding
$H=\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V)\leq
G=\mathrm{Res}_{E_{0}/\mathbb{Q}}\mathrm{U}(V_{E_{0}})$. We now pullback these
Hecke correspondences from $G\times G$ to $H\times H$ via Serre tensor
constructions and define the _twisted CM cycles_.

Recall over $E^{\mathrm{RSZ}}$, we may attach a complex point
$(A_{0}^{E_{0}},A^{E_{0}},\overline{\eta})\in M^{E_{0}}(\mathbb{C})$ a
$F/F_{0}$ hermitian space via Betti cohomology

$V(A_{0}^{E_{0}},A^{E_{0}}):=\mathrm{Hom}_{F}(H_{1}(A_{0},\mathbb{Q}),H_{1}(A,\mathbb{Q}))$

The existence of $\overline{\eta}$ implies that we have an isomorphism
$V(A_{0}^{E_{0}},A^{E_{0}})\cong V_{E_{0}}$ as hermitian spaces.

######  Definition 6.4.2.

For an element
$\overline{\alpha}\in\mathrm{U}(V)(F_{0})\backslash\mathrm{U}(V)(E_{0})/\mathrm{U}(V)(F_{0})$,
the twisted CM cycle for $\alpha$ and $\mu_{\Delta}^{E_{0}}$ over
$E^{\mathrm{RSZ}}$

(29) $CM^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})\to M\times M$

is the functor sending any locally noetherian $E^{\mathrm{RSZ}}$-scheme $S$ to
the groupoid of tuples
$(A_{0},A_{1},\overline{\eta}_{A_{1}},A_{2},\overline{\eta}_{A_{2}},\varphi)$
where

  1. (1)

$(A_{0},A_{1},\overline{\eta}_{A_{1}}),\,(A_{0},A_{2},\overline{\eta}_{A_{2}})\in
M(S)$.

  2. (2)

$\varphi\in\mathrm{Hom}_{O_{E}}(A_{1}\otimes_{O_{F}}O_{E},A_{2}\otimes_{O_{F}}O_{E})[1/\Delta]$
is a prime-to-$\Delta$ quasi-isogeny such that
$\varphi^{*}\lambda_{2}=\lambda_{1}$.

  3. (3)

The induced map $\varphi_{*}:H_{1}(A_{1}^{E_{0}},\mathbb{Q})\cong
H_{1}(A_{2}^{E_{0}},\mathbb{Q})$ on each complex point transforms to
$\alpha\in\mathrm{U}(V_{E_{0}})$ for some $\alpha\in\overline{\alpha}$, under
an identification $V(A_{0}^{E_{0}},A^{E_{0}}_{i})\cong V_{E_{0}}$ of hermitian
spaces that induces $\eta_{A_{i}}:V_{\Delta}(A_{0}^{E_{0}},A^{E_{0}})\cong
V_{E_{0},\Delta}$ for some
$\eta_{A_{i}^{E_{0}}}\in\overline{\eta}_{A_{i}^{E_{0}}}$.

######  Remark 6.4.3.

If $E_{0}=F_{0}\times F_{0}$, then by [AFL-Wei2019, (2.11.))] the categorical
quotient $\mathrm{U}(V)\backslash\mathrm{U}(V_{E_{0}})/\mathrm{U}(V)$ over
$F_{0}$, could be identified via characteristic polynomials with an affine
subscheme $\mathcal{A}_{n}\subseteq\mathrm{Res}_{F/F_{0}}F[T]_{\deg=n}$
consist of conjugate self-reciprocal monic polynomials.

By considering the relative position of $\varphi$ over generic fibers, we have
a decomposition of Hecke cycles into _twisted CM cycles_

(30)
$(\mathcal{M}\times\mathcal{M})\cap_{\mathcal{M}^{E_{0}}\times\mathcal{M}^{E_{0}}}\mathrm{Hk}_{\mu^{\Delta}}=\coprod_{\overline{\alpha}}\mathcal{CM}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})$

where $\overline{\alpha}$ runs over
$\mathrm{U}(V)(F_{0})\backslash\mathrm{U}(V)(E_{0})/\mathrm{U}(V)(F_{0})$,
such that
$\mathcal{CM}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})_{E^{\mathrm{RSZ}}}=CM^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})$.

######  Definition 6.4.4.

We form the derived intersection in the K-group of $\mathcal{M}^{E_{0}}$ and
decomposition as above:

$(\mathcal{M}\times\mathcal{M})\cap^{\mathbb{L}}_{\mathcal{M}^{E_{0}}\times\mathcal{M}^{E_{0}}}\mathrm{Hk}_{\mu^{\Delta}}=\coprod_{\alpha}{}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta}),$

and
${}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})\in
F_{1}^{\prime}K^{\prime}_{0,\mathcal{M}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})}(\mathcal{M}\times\mathcal{M})$
is called the derived twisted CM cycle for
$(\overline{\alpha},\mu^{E_{0}}_{\Delta})$.

######  Definition 6.4.5.

An element
$\overline{\alpha}\in\mathrm{U}(V)(F_{0})\backslash\mathrm{U}(V)(E_{0})/\mathrm{U}(V)(F_{0})$
is called irreducible over $F$, if its image in
$\mathrm{GL}(V)\backslash\mathrm{GL}(V_{E_{0}})/\mathrm{GL}(V)$ contains some
element $\alpha^{E_{0}}$ whose characteristic polynomial over $E$ is
irreducible, i.e. $E[\alpha^{E_{0}}]$ is a field.

By the theory of complex multiplication (for $A^{E_{0}}$), if
$\overline{\alpha}$ is irreducible, then
${}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\alpha,\mu^{E_{0}}_{\Delta})$ is
$1$-dimensional away from finitely many primes with $0$-dimensional generic
fiber $CM^{E_{0}}(\alpha,\mu^{E_{0}}_{\Delta})$ over $E^{\mathrm{RSZ}}$. See
also [AFL-Wei2019, Prop. 7.9].

######  Remark 6.4.6.

It is possible to give moduli interpretations of integral twisted CM cycles
$\mathcal{CM}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})$ without using
Hecke correspondences, by requiring the induced map
$\eta_{A_{2}^{E_{0}}}\varphi_{*}\eta_{A_{1}^{E_{0}}}^{-1}:V_{\Delta}\cong
V_{\Delta}(A_{0}^{E_{0}},A^{E_{0}}_{1})\to
V_{\Delta}(A_{0}^{E_{0}},A^{E_{0}}_{2})\cong V_{\Delta}$ agrees with the image
of $\alpha$ in
$\mathrm{U}(V)(F_{0,\Delta})\backslash\mathrm{U}(V)(E_{0,\Delta})/H(F_{0,\Delta})$.
This agrees with the above definition because we put the sign condition on
integral model $\mathcal{M}$ using sign invariants as in [RSZ-Shimura, Section
4.1]. For our purpose, knowing the information on generic fibers already gives
the desired local-global decomposition of global intersection numbers into
local orbital integrals and intersection numbers.

Let $\mathcal{C}_{1}(\mathcal{M})$ be the $\mathbb{Q}$-vector space spanned by
$1$-cycles on $\mathcal{M}$, and $\mathcal{C}_{1}(\mathcal{M})_{\sim}$ be the
quotient of $\mathcal{C}_{1}(\mathcal{M})$ by the subspace generated by
$1$-cycles on $\mathcal{M}$ that are contained in a closed fiber and
rationally trivial within that fiber.

######  Definition 6.4.7 (derived twisted CM cycles).

For a Schwartz function $f=f_{\Delta}\otimes
1_{(K^{E_{0}})^{\Delta}}\in\mathcal{S}(G(\mathbb{A}_{f}))^{K^{E_{0}}\times
K^{E_{0}}}$, the derived twisted CM cycle for $f$ is the finite summation

(31)
${}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\alpha,f)=\sum_{\mu_{\Delta}^{E_{0}}\in
K^{E_{0}}_{\Delta}\backslash\mathrm{U}(V)(E_{0,\Delta})/K^{E_{0}}_{\Delta}}f_{\Delta}(\mu_{\Delta}^{E_{0}})^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\alpha,\mu^{E_{0}}_{\Delta})\in\mathcal{C}_{1}(\mathcal{M})_{\sim}.$

Here we regard
${}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\alpha,\mu^{E_{0}}_{\Delta})\in\mathcal{C}_{1}(\mathcal{M})_{\sim}$
via the the projection map
$\mathrm{pr}_{1}:\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ to the first
factor.

##  7\. Local–global decomposition formulas and transfers

In this section, we globalize two sides of the twisted AFL. Let $\xi\in
F_{0}$. For $\phi=\phi_{\Delta}\otimes
1_{\widehat{L}^{\Delta}}\in\mathcal{S}(V(\mathbb{A}_{f}))^{K}$ and
$f=f_{\Delta}\otimes
1_{K^{E}_{0}}\in\mathcal{S}(\mathrm{U}(V_{E_{0}})(\mathbb{A}_{f}))^{K_{E_{0}}\times
K_{E_{0}}}$, we consider arithmetic intersection number
$\mathrm{Int}(\xi,f\otimes\phi)$ and study its local-global decomposition
formulas following the argument of [AFL-Invent, Theorem 3.9] [RSZ-AGGP,
Theorem 8.15].

###  7.1. Geometric generating functions

We globalize the geometric side in Section 4. Consider the quotient
$\mathbb{Q}$-vector space

$\mathbb{R}_{\Delta}:=\mathbb{R}\,/\sum_{v|\ell,v\in\Delta}{\mathbb{Q}\log\ell}.$

Consider the $\Delta$-punctured arithmetic intersection pairing between
$\mathbb{Q}$-vector spaces:

(32)
$(\cdot,\cdot):\widehat{\mathrm{Ch}}^{1}(\mathcal{M})\times\mathcal{C}_{1}(\mathcal{M})\to\mathbb{R}_{\Delta}.$

Recall the volume factor
$\tau(Z^{\mathbb{Q}})=\\#Z^{\mathbb{Q}}(\mathbb{Q})\backslash
Z^{\mathbb{Q}}(\mathbb{A}_{f})$.

######  Definition 7.1.1 ($\xi$-part of global arithmetic intersection
numbers).

For $\xi\in F_{0}$ and
$\Phi=f\otimes\phi\in\mathcal{S}((\mathrm{U}(V_{E_{0}})\times
V)(\mathbb{A}_{0,f}))^{(K_{E_{0}}\times K_{E_{0}})\times K}$ as above, we
define _global arithmetic intersection numbers_

(33)
$\mathrm{Int}^{?}(\xi,\Phi):=\frac{1}{\tau(Z^{\mathbb{Q}})[E^{\mathrm{RSZ}}:F]}(\widehat{\mathcal{Z}}^{?}(\xi,\phi),{}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\alpha,f))\in\mathbb{R}_{\Delta}.$

We set $\mathrm{Int}(\xi,\Phi)=\mathrm{Int}^{\bf B}(\xi,\Phi)$.

If $\xi\not=0$, we may lift $\mathrm{Int}(\xi,\Phi)$ to $\mathbb{R}$ using the
physical divisor $\widehat{\mathcal{Z}}^{\bf B}(\xi,\phi)$. The global
intersection decomposes into semi-global intersection numbers at each local
place $v$ of $F_{0}$:

(34) $\mathrm{Int}(\xi,\Phi)=\sum_{v}\mathrm{Int}_{v}(\xi,\Phi).$

We now connect the local intersection pairing $\mathrm{Int}_{v}(\xi,\Phi)$ via
basic uniformization to our local set up for the twisted AFL.

###  7.2. Basic uniformization of special cycles at inert places

Assume that $v_{0}\not\in\Delta$ is inert in $F$. Let $V^{(v_{0})}$ be the
nearby $F/F_{0}$-hermitian space of $V$ at $v_{0}$, which is positively
definite at all $v|\infty$ and locally isomorphic to $V$ at all finite places
$v\not=v_{0}$ of $F_{0}$.

Let $\nu$ be a finite place of $E^{\mathrm{RSZ}}$ with $\nu|_{F}=w$,
$\nu|_{F_{0}}=v_{0}$. By our assumption that $\Phi$ is unramifeid over $p$, we
have $\breve{E}^{\mathrm{RSZ}}_{\nu}=\breve{F_{w}}=\breve{F_{0,v}}$.

Denote by $\mathcal{M}^{\wedge}$ be the formal completion of
$\mathcal{M}_{O_{\breve{E}^{\mathrm{RSZ}}_{\nu}}}$ along the basic locus of
the geometric special fiber $\mathcal{M}\otimes\mathbb{F}_{\nu}$. The formal
scheme $\mathcal{M}^{\wedge}$ is formally smooth and _formally of finite type_
over $O_{\breve{E}^{\mathrm{RSZ}}_{\nu}}$. Consider the (relative) unitary
Rapoport–Zink space

$\mathcal{N}_{n}\to\mathrm{Spf}O_{\breve{F}}$

(associated to $L_{v_{0}}$) as in Section 4.

Consider reductive groups $H^{(v_{0})}:=\mathrm{U}(V^{(v_{0})})$ over $F_{0}$
and
$\widetilde{H}^{(v_{0})}:=Z^{\mathbb{Q}}\times\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V^{(v_{0})})$
over $\mathbb{Q}$. We have the following basic uniformization theorem.

######  Proposition 7.2.1.

There is a natural isomorphism of formal schemes over
$O_{\breve{E}^{\mathrm{RSZ}}_{\nu}}=O_{\breve{F_{0}}}$:

(35)
$\Theta:\,\,\mathcal{M}^{\wedge}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{H}^{(v_{0})}(\mathbb{Q})\backslash(\mathcal{N}^{\prime}\times\widetilde{H}(\mathbb{A}_{f}^{p})/\widetilde{K}^{p}),$

where

$\mathcal{N}^{\prime}\simeq(Z^{\mathbb{Q}}\left(\mathbb{Q}_{p}\right)/K_{Z^{\mathbb{Q}},p})\times\mathcal{N}_{n}\times\prod_{v|p,v\not=v_{0}}\mathrm{U}(V)(F_{0,v})/K_{v}.$

There is a natural projection map

(36) $\mathcal{M}^{\wedge}\to
Z^{\mathbb{Q}}(\mathbb{Q})\backslash(Z^{\mathbb{Q}}(\mathbb{A}_{0,f})/K_{Z^{\mathbb{Q}}})$

with fibers isomorphic to

$\Theta_{0}:\mathcal{M}^{\wedge}_{0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{(v_{0})}(F_{0})\backslash[\mathcal{N}_{n}\times
H(\mathbb{A}_{0,f}^{v_{0}})/K^{v_{0}}].$

###### Proof.

See [RSZ-AGGP, Thm 8.15]. ∎

Hence we denote by $\mathcal{M}^{\wedge}_{0}$ any fiber of
$\mathcal{M}^{\wedge}$ under the projection map (36).

Note that $v_{0}$ is split or inert in $E_{0}$. Consider the Rapoport–Zink
space $\mathcal{N}_{n}^{E_{0,v_{0}}}:=\mathcal{N}^{\mathrm{GL}}_{n}$ (resp.
$\mathcal{N}_{n}\times\mathcal{N}_{n}$ ) when $v_{0}$ is inert in $E_{0}$
(resp. $v_{0}$ splits in $E_{0}$). Similarly, we have basic uniformization of
the formal completion $\mathcal{M}^{E_{0},\wedge}$ of $\mathcal{M}^{E_{0}}$
along its basic locus.

######  Proposition 7.2.2.

There is a natural isomorphism of formal schemes over
$O_{\breve{E}^{\mathrm{RSZ}}_{\nu}}=O_{\breve{F_{0}}}$:

(37)
$\Theta^{E_{0}}:\,\,\mathcal{M}^{E_{0},\wedge}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{G}^{(v_{0})}(\mathbb{Q})\backslash(\mathcal{N}^{E_{0,v_{0}}\prime}\times\widetilde{G}(\mathbb{A}_{f}^{p})/\widetilde{K^{E_{0}}}^{p}),$

where

$\mathcal{N}^{E_{0,v_{0}}\prime}\simeq(Z^{\mathbb{Q},E_{0}}(\mathbb{Q}_{p})/K_{Z^{\mathbb{Q},E_{0}},p})\times\mathcal{N}^{E_{0,v_{0}}}_{n}\times\prod_{v|p,v\not=v_{0}}\mathrm{U}(V)(E_{0,v})/K_{v}^{E_{0}}.$

There is a natural projection map

(38) $\mathcal{M}^{E_{0},\wedge}\to
Z^{\mathbb{Q},E_{0}}(\mathbb{Q})\backslash(Z^{\mathbb{Q},E_{0}}(\mathbb{A}_{0,f})/K_{Z^{\mathbb{Q}},E_{0}})$

with fibers isomorphic to

$\Theta_{0}^{E_{0}}:\mathcal{M}^{E_{0},\wedge}_{0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{(v_{0})}(F_{0})\backslash[\mathcal{N}_{n}^{E_{0,v_{0}}}\times
H(\mathbb{A}_{0,f}^{v_{0}})/K^{v_{0}}].$

###### Proof.

This is a special case of basic uniformization theorem for general Hodge type
Shimura varieties, see [PappasRapoport-GlobalShimura, Theorem 4.10.6]. The
same construction of uniformization maps in [RSZ-AGGP, Thm 8.15] applies here
and gives the desired isomorphism. ∎

Set $V_{\xi}:=\\{u\in V|(u,u)=\xi\\}$ be the subset of $V$ consists of
elements of norm $\xi$.

######  Proposition 7.2.3.

[AFL-Wei2019, Prop 7.4.] Assume that $\xi\not=0$. The restriction of special
divisor $\mathcal{Z}(\xi,\phi)$ on each fiber $\mathcal{M}^{\wedge}_{0}$ of
the projection is identified with

(39)
$\sum_{(g,u)\in\mathrm{U}(V)(F_{0})\backslash(\mathrm{U}(V)(\mathbb{A}^{v}_{f})/K^{v}\times
V_{\xi})}\phi^{v}(g^{-1}u)[\mathcal{Z}(u)\times 1_{gK^{v}}].$

Recall the Hecke correspondence (Definition 6.4.1)

$\mathrm{Hk}_{\mu_{\Delta}^{E_{0}}}\to\mathcal{M}^{E_{0}}\times\mathcal{M}^{E_{0}}.$

Let $\mathrm{Hk}_{\mu_{\Delta}^{E_{0}}}^{\wedge}$ be the locally noetherian
formal scheme over $O_{\breve{F_{0}}}$ obtained as pullback of
$\mathrm{Hk}_{\mu_{\Delta}^{E_{0}}}$ on
$\mathcal{M}^{E_{0},\wedge}\times\mathcal{M}^{E_{0},\wedge}$. Consider
reductive groups
$G^{(v_{0})}=\mathrm{Res}_{E_{0}/F_{0}}\mathrm{U}(V^{(v_{0})}_{E_{0}})$ over
$F_{0}$ and
$\widetilde{G^{(v_{0})}}=Z^{\mathbb{Q},E_{0}}\times\mathrm{Res}_{E_{0}/\mathbb{Q}}\mathrm{U}(V^{(v_{0})}_{E_{0}})$
over $\mathbb{Q}$.

Denote by $\mathrm{Hk}_{\mu_{\Delta}^{E_{0}},0}^{\wedge}$ any fiber of the
natural projection map sending $(A_{0},A^{E_{0}},\overline{\eta}^{E_{0}})$ to
$A_{0}$

$\mathrm{Hk}_{\mu_{\Delta}^{E_{0}}}^{\wedge}\to
Z^{E_{0},\mathbb{Q}}(\mathbb{Q})\backslash
Z^{E_{0},\mathbb{Q}}(\mathbb{A}_{f})/K_{Z^{E_{0},\mathbb{Q}}}.$

Consider the discrete set

$\mathrm{Hk}_{\mu^{E_{0}}_{\Delta},0}^{(v_{0})}=\\{(g_{1},g_{2})\in
G(\mathbb{A}_{f}^{v_{0}})/K^{E_{0},v_{0}}\times
G(\mathbb{A}_{f}^{v_{0}})/K^{E_{0},v_{0}}|g_{1}^{-1}g_{2}\in
K^{E_{0}}\mu_{\Delta}^{E_{0}}K^{E_{0}}\\}.$

We recall the compatibility of Hecke correspondences and basic uniformization.

######  Theorem 7.2.4.

We have a natural isomorphism of formal schemes

$\mathrm{Hk}_{\mu^{E_{0}}_{\Delta},0}^{\wedge}\cong
G^{(v_{0})}(F_{0})\backslash[\mathcal{N}^{E_{0},v_{0}}_{n}\times\mathrm{Hk}_{\mu_{\Delta}^{E_{0}},0}^{(v_{0})}]$

compatible with the projection under the isomorphism $\Theta$ (c.f. (35)).

###### Proof.

Although we work over $\mathcal{M}^{E_{0}}$, the same proof of [AFL-Wei2019,
Prop. 7.16] (on $\mathcal{M}$) applies. ∎

Assume $v_{0}$ is inert in $E_{0}$. For $\delta\in
G^{(v_{0})}(F_{0,v})/H^{(v_{0})}(F_{0,v})$, consider the twisted fixed cycles
of $\delta$ on $\mathcal{N}_{n}$ (see Definition 4.0.1)

${}^{\mathbb{L}}\mathcal{N}^{\mathrm{Herm},v_{0}}(\delta)=\delta\mathcal{N}_{n}\cap^{\mathbb{L}}_{\mathcal{N}_{n}^{E_{0},v_{0}}}\mathcal{N}_{n}\in
K_{0}^{\prime}(\mathcal{N}_{n}).$

For $(\delta,h)\in B^{(v_{0})}(F_{0})\times
H(\mathbb{A}_{0,f}^{v_{0}})/K^{v_{0}}$, consider the twisted CM cycle

(40)
${}^{\mathbb{L}}\mathcal{CN}^{E_{0}}(\delta,h)_{K^{E_{0},v_{0}}}:=[{}^{\mathbb{L}}\mathcal{N}^{\mathrm{Herm}}(\delta)\times
1_{hK^{E_{0},v_{0}}}]\rightarrow[\mathcal{N}_{n}\times
H(\mathbb{A}_{f}^{v_{0}})/K^{v_{0}}].$

The summation

$\sum_{(\delta^{\prime},h^{\prime})}{}^{\mathbb{L}}\mathcal{CN}^{E_{0}}(\delta^{\prime},h^{\prime})_{K^{v_{0}}}$

over the $H^{(v_{0})}(F_{0})$-orbit of $(\delta,h)$ in
$G^{(v_{0})}(F_{0})/H^{(v_{0})}(F_{0})\times
H(\mathbb{A}_{0,f}^{v_{0}})/K^{v_{0}}$ descends to a cycle
$[{}^{\mathbb{L}}\mathcal{CM}(\delta,h)]_{K^{v_{0}}}$ on the quotient
$\mathcal{M}^{\wedge}$ by above Theorem 7.2.2. As in the proof of [AFL-Invent,
Thm. 3.9], we have

######  Proposition 7.2.5.

For a Schwartz function $f=1_{K^{E_{0}}}\otimes
f_{\Delta}\in\mathcal{S}(\mathrm{U}(V_{E_{0}})(\mathbb{A}_{0,f}),K^{E_{0}})$,
the restriction of the basic uniformization
${}^{\mathbb{L}}\mathcal{CM}^{E_{0}}(\alpha,f)^{\wedge}$ of twisted CM cycles
to $\mathcal{M}^{E_{0},\wedge}_{0}$ is the sum

(41) $\sum_{(\delta,h)\in H^{(v_{0})}(F_{0})\backslash
B^{(v_{0})}(F_{0})\times
H(\mathbb{A}_{0,f}^{v_{0}})/K^{v_{0}}}f^{v_{0}}(h.\delta)[{}^{\mathbb{L}}\mathcal{CM}(\delta,h)]_{K_{G}^{v_{0}}}.$

###  7.3. Analytic generating functions

We now globalize the analytic side in Section 3. Let $F^{\prime}/F_{0}$ be the
third quadratic extension inside $E$ fixed by
$\sigma_{E_{0}/F_{0}}\otimes\sigma_{F/F_{0}}$. Consider

$H^{\prime}_{1}=\mathrm{GL}_{n,F_{0}}\rightarrow
G^{\prime}=\mathrm{GL}_{n,F^{\prime}}\leftarrow
H^{\prime}_{20}=\mathrm{GL}_{n,F_{0}}.$

Consider the symmetric space

(42)
$S_{n,F^{\prime}/F_{0}}=\\{\gamma\in\mathrm{GL}_{n,F^{\prime}}|\gamma\sigma_{F^{\prime}/F_{0}}(\gamma)=id\\}$

By Hilbert 90, we have $S_{n,F^{\prime}/F_{0}}\cong
G^{\prime}/H^{\prime}_{20}$. Consider the natural quadratic space
$V^{\prime}=F_{0}^{n}\times(F_{0}^{n})^{*}$.

Let $\eta=\eta_{F/F_{0}}:\mathbb{A}_{F_{0}}\to\mathbb{C}^{\times}$ be the
quadratic character associated to $F/F_{0}$ by global class field theory.
Similar to the local set up, consider $\eta$-twisted orbital integrals for the
diagonal action of $h\in H^{\prime}_{1}(F_{0})=\mathrm{GL}_{n}(F_{0})$ on the
product $(\gamma,u_{1},u_{2})\in S_{n,F^{\prime}/F_{0}}\times V^{\prime}$ by

(43) $h.(\gamma,u_{1},u_{2})=(h^{-1}\gamma h,h^{-1}u_{1},u_{2}h).$

######  Proposition 7.3.1.

. There is a natural identification of categorical quotients

$[H_{1}^{\prime}\backslash
G^{\prime}/H_{20}^{\prime}]=[\mathrm{U}(V)\backslash\mathrm{U}(V_{E_{0}})/\mathrm{U}(V)]$

compatible with determinant maps to
$[\mathbb{G}_{m,F}^{Nm_{F/F_{0}}=1}\backslash\mathbb{G}_{m,E}^{Nm_{E/E_{0}}=1}].$
In particular, we get a matching of regular semi-simple orbits

$[H_{1}^{\prime}\backslash
G^{\prime}/H_{20}^{\prime}]_{\mathrm{rs}}=\coprod_{V}[\mathrm{U}(V)\backslash\mathrm{U}(V_{E_{0}})/\mathrm{U}(V)]_{\mathrm{rs}}$

where $V$ ranges over $n$-dimensional $F/F_{0}$-hermitian spaces.

###### Proof.

This is a globalization of Proposition 3.1.2. The Lie algebra version is
straightforward. As this is a geometric statement, we may base change to
$E_{0}$ and the result follows from the Jacquet Rallis case (computed via
Igusa’s criterion) [zhang2014fourier, Lemma 3.1]. ∎

Let $\overline{\alpha}$ be an element in
$\mathrm{U}(V)(F_{0})\backslash\mathrm{U}(V)(E_{0})/\mathrm{U}(V)(F_{0})$.
Assume that $\overline{\alpha}$ is irreducible. Consider the subscheme
$S_{n,F^{\prime}/F_{0}}(\alpha)\subseteq S_{n,F^{\prime}/F_{0}}$ of elements
corresponds to $\alpha$ under above identification of categorical quotients.
Then $S_{n,F^{\prime}/F_{0}}(\alpha)(F_{0})$ consists of exactly one
$\mathrm{GL}_{n}(F_{0})$-orbit.

Consider a decomposable function

$\Phi^{\prime}=\otimes_{v}\Phi^{\prime}_{v}\in\mathcal{S}((S_{n,F^{\prime}/F_{0}}\times
V^{\prime})(\mathbb{A}_{F_{0}}))$

such that for every infinite place $v|\infty$ of $F_{0}$, $\Phi^{\prime}_{v}$
is the chosen partial Gaussian test function (relative to $\alpha$) introduced
in [AFL-Wei2019, Section 12.4].

For $(\gamma,u^{\prime})\in(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime})(F_{0})$ that is regular semisimple (equivalently
$u^{\prime}\not=0$ as $\alpha$ is irreducible), define the global
$\eta$-orbital integral ($v$ runs over all places of $F_{0}$):

(44)
$\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime},s)=\prod_{v}\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{v},s)$

where $\eta$ is the quadratic character associated to $F/F_{0}$ by class field
theory.

Denote by $[(S_{n,F^{\prime}/F_{0}}(\alpha)\times V^{\prime})(F_{0})]$ (resp.
$[(S_{n,F^{\prime}/F_{0}}(\alpha)\times V^{\prime})(F_{0})]_{\mathrm{rs}}$)
the set of (resp. regular semisimple) $\mathrm{GL}_{n}(F_{0})$-orbits in
$(S_{n,F^{\prime}/F_{0}}(\alpha)\times V^{\prime})(F_{0})$. For
$h\in\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})$ and $s\in\mathbb{C}$, consider the
regularized integral as in [AFL-Wei2019, Section 11.4]:

(45)
$\mathbb{J}(h,\Phi^{\prime},s)=\int_{g\in[\mathrm{GL}_{n}(F_{0})\backslash\mathrm{GL}_{n}(\mathbb{A}_{F_{0}})}\left(\sum_{(\gamma,u^{\prime})\in(S_{n,F^{\prime}/F}(\alpha)\times
V^{\prime})(F_{0})}\omega(h)\Phi^{\prime}(g^{-1}.(\gamma,u^{\prime}))\right)|g|^{s}\eta(g)dg.$

By [AFL-Wei2019, Thm. 12.14], we see that $\mathbb{J}(h,\Phi^{\prime},s)$ is a
smooth function of $(h,s)$, entire in $s\in\mathbb{C}$ and is left invariant
under $h\in\mathrm{SL}_{2}(F_{0})$ by Poisson summation formula. We have a
decomposition:

(46)
$\mathbb{J}(h,\Phi^{\prime},s)=\mathbb{J}(h,\Phi^{\prime},s)_{0}+\sum_{(\gamma,u^{\prime})\in[(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime})(F_{0})]_{\mathrm{rs}}}\mathrm{Orb}((\gamma,u^{\prime}),\omega(h)\Phi^{\prime},s),$

where $\mathbb{J}(h,\Phi^{\prime},s)_{0}$ is the term over the two regular
nilpotent orbits as in [AFL-Wei2019, Section 12.6].

For $\xi\in F_{0}^{\times}$, let $V^{\prime}_{\xi}$ be the $F_{0}$-subscheme
of $V^{\prime}$ defined by $\\{(u_{1},u_{2})\in
V^{\prime}|u_{2}(u_{1})=\xi\\}$. The $\xi$-th Fourier coefficient of
$\mathbb{J}(\cdot,\Phi^{\prime},s)$ is equal to

(47)
$\displaystyle\sum_{(\gamma,u^{\prime})\in[(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime}_{\xi})(F_{0})]_{\mathrm{rs}}}\mathrm{Orb}((\gamma,u^{\prime}),\omega(h)\Phi^{\prime},s).$

Then we introduce

(48)
$\partial\mathbb{J}(h,\Phi^{\prime}):=\frac{d}{ds}\Big{|}_{s=0}\mathbb{J}(h,\Phi^{\prime},s).$
(49) $\operatorname{\partial
Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{v}):=\frac{d}{ds}\Big{|}_{s=0}\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{v},s).$

By Leibniz’s rule, we have the standard decomposition:

(50)
$\partial\mathbb{J}(h,\Phi^{\prime})=\partial\mathbb{J}(h,\Phi^{\prime})_{0}+\sum_{v}\partial\mathbb{J}_{v}(h,\Phi^{\prime}),$

where $v$ runs over all places of $F_{0}$ and

(51)
$\partial\mathbb{J}_{v}(h,\Phi^{\prime}):=\sum_{(\gamma,u^{\prime})\in[(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime})(F_{0})]_{\mathrm{rs}}}\operatorname{\partial
Orb}((\gamma,u^{\prime}),\omega(h)\Phi^{\prime}_{v})\cdot\mathrm{Orb}((\gamma,u^{\prime}),\omega(h)\Phi^{\prime
v}).$

And the nilpotent term $\partial\mathbb{J}(h,\Phi^{\prime})_{0}$ is part of
the $0$-th Fourier coefficient of $\partial\mathbb{J}(h,\Phi^{\prime})$. We do
not need its precise formula in this paper. Consider the Fourier expansion

(52) $\partial\mathbb{J}_{v}(h,\Phi^{\prime})=\sum_{\xi\in
F_{0}}\partial\mathbb{J}_{v}(\xi,h,\Phi^{\prime}),$
$\partial\mathbb{J}_{v}(\xi,h,\Phi^{\prime})=\sum_{(\gamma,u^{\prime})\in[(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime}_{\xi})(F_{0})]_{\mathrm{rs}}}\operatorname{\partial
Orb}((\gamma,u^{\prime}),\omega(h)\Phi^{\prime}_{v})\cdot\mathrm{Orb}((\gamma,u^{\prime}),\omega(h)\Phi^{\prime
v}).$

######  Proposition 7.3.2.

The function $\partial\mathbb{J}(h,\Phi^{\prime})$ is a smooth function on
$h\in\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})$, left invariant under
$\mathrm{SL}_{2}(F_{0})$, and of parallel weight $n$. If $\Phi^{\prime}$ is
$K_{0}$-invariant under the Weil representation, then
$\partial\mathbb{J}(h,\Phi^{\prime})$ is right $K_{0}$-invariant.

###### Proof.

This follows from similar properties of $\mathbb{J}(h,\Phi^{\prime},s)$ from
above discussion. ∎

######  Proposition 7.3.3 (local-global decomposition: analytic side).

Assume that $\xi\not=0$. Then

  1. (1)

If $v\in\mathrm{Inert}(F/F_{0})$, then

$\partial\mathbb{J}_{v}(\xi,1,\Phi^{\prime})=\sum_{(\gamma,u^{\prime})\in[(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime}_{\xi})(F_{0})]_{\mathrm{rs}}}\operatorname{\partial
Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{v})\cdot\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime
v}).$

  2. (2)

If $v$ is an archimedean place of $F_{0}$, then

$\partial\mathbb{J}_{v}(\xi,1,\Phi^{\prime})=\sum_{(\gamma,u^{\prime})\in[(S_{n,F^{\prime}/F_{0}}(\alpha)\times
V^{\prime}_{\xi})(F_{0})]_{\mathrm{rs}}}\operatorname{\partial
Orb}((\gamma,u^{\prime}),\Phi^{\prime}_{v})\cdot\mathrm{Orb}((\gamma,u^{\prime}),\Phi^{\prime
v}).$

###### Proof.

This follows from above discussion by taking $h=id$. ∎

We will see in next subsection that if $\Phi^{\prime}$ is a good transfer of
$\Phi$ then $\partial\mathbb{J}_{v}(\xi,\Phi^{\prime})=0$ for
$v\in\mathrm{Split}(F/F_{0})$.

###  7.4. Local-global comparison and transfers

######  Definition 7.4.1.

Let $v$ be a finite place $v$ of $F_{0}$. For Schwartz function
$f_{v}\in\mathcal{S}(\mathrm{U}(V_{E_{0},v_{0}})$ and
$\phi_{v}\in\mathcal{S}(V_{v_{0}})$, the orbital integral of
$\Phi_{v}=f_{v}\otimes\phi_{v}$ at a regular semi-simple orbit of
$(g,u)\in[\mathrm{U}(V)\backslash(\mathrm{U}(V_{E_{0}})(\alpha)/\mathrm{U}(V)\times
V_{\xi})](F_{0,v_{0}})$ is the following integral:

(53)
$\mathrm{Orb}((g,u),\Phi_{v})=\int_{h\in\mathrm{U}(V)(F_{0,v})}(\int_{h_{2}\in\mathrm{U}(V)(F_{0,v})}f_{v}(h^{-1}gh_{2})dh_{2})\phi_{v}(h^{-1}u)dh,$

where the Haar measure on $\mathrm{U}(V)(F_{0,v})$ is normalized such that a
hyperspecial maximal compact open subgroup has volume $1$.

Let $\Phi=f\otimes\phi\in\mathcal{S}(\mathrm{U}(V_{E_{0}})\times
V)(\mathbb{A}_{0,f}))$ be a stanford decomposable function with level
$K_{E_{0}}$ and $K$. In particular,
$\Phi^{\Delta}=1_{\mathrm{U}(L_{O_{E_{0}}})^{\Delta}}\times
1_{\widetilde{L}^{\Delta}}$.

######  Proposition 7.4.2 (local-global decomposition: geometric side).

Assume that $\xi\not=0$. Let $v|p$ be a place of $F_{0}$. Then

  1. (1)

If $v\in\mathrm{Split}(F/F_{0})$, then $\mathrm{Int}_{v}(\xi,\Phi)=0$.

  2. (2)

If $v\in\mathrm{Inert}(F/F_{0})$, then

$\mathrm{Int}_{v}(\xi,\Phi)=\log
q_{v}^{2}\sum_{(g,u)\in[\mathrm{U}(V)\backslash(\mathrm{U}(V_{E_{0}})(\alpha)/\mathrm{U}(V)\times
V_{\xi})](F_{0})]}\mathrm{Int}^{\mathrm{Herm},v_{0}}(g,u)\mathrm{Orb}((g,u),\Phi^{v_{0}}).$

###### Proof.

We follow the proof of [AFL-Wei2019, Theorem 9.4]. The first part is the same:
if $\mathrm{Int}_{v}(\xi,\Phi)$ is non-zero, then for any intersection point
$(A_{0},A,\overline{\eta})\in\mathcal{CM}^{E_{0}}(\overline{\alpha},\mu^{E_{0}}_{\Delta})\cap\mathcal{Z}(\xi,\mu_{\Delta})(\mathbb{F}_{p}^{alg})$,
we could construct isogeny from $A_{0}^{n}$ to $A$ to show that is in the
supersingular locus of the mod $v$ fiber of $\mathcal{M}$, and $v$ is inert or
ramified in $F$.

The second part follows by combining the above uniformization (37). See also
[AFL-Invent, Theorem 3.9] [RSZ-AGGP, Theorem 8.15]. ∎

To compare the above local-global decompositions, we now define the notion of
transfers (relative to $\alpha$).

######  Definition 7.4.3.

We say functions $\Phi=f\otimes\phi\in\mathcal{S}(\mathrm{U}(V_{E_{0}})\times
V)(\mathbb{A}_{0,f}))$ and
$\Phi^{\prime}=f^{\prime}\otimes\phi^{\prime}\in\mathcal{S}((S_{n,F^{\prime}/F_{0}}\times
V^{\prime})(\mathbb{A}_{F_{0}}))$ are _partial transfers_ (relative to
$\alpha$), if

  * •

for any archimedean place $v$ of $F_{0}$, $\Phi^{\prime}_{v}$ is a Gaussian
test function (relative to $\alpha$).

  * •

for any finite place $v$ of $F_{0}$, $\Phi_{v}^{\prime}$ and $\Phi_{v}$ are
required to have same orbital integrals (53) for matching regular semi-simple
orbits (from Proposition 7.3.1)

$(\gamma,u_{1},u_{2})\in[\mathrm{GL}_{n}(F_{0,v})\backslash
S_{n,F^{\prime}_{v}/F_{v}}(\alpha)\times V^{\prime})(F_{0,v})]_{\mathrm{rs}}$

and

$(g,u)\in[\mathrm{U}(V)\backslash\mathrm{U}(V_{E_{0}})(\alpha)/\mathrm{U}(V)\times
V^{\prime})(F_{0,v})]_{\mathrm{rs}}.$

Let $\Phi=f\otimes\phi$ and $\Phi^{\prime}=f^{\prime}\otimes\phi^{\prime}$ be
a pair of partial transfers (relative to $\alpha$).

######  Proposition 7.4.4.

If $v\in\mathrm{Split}(F/F_{0})$, then $\partial\mathbb{J}_{v}(\xi,\Phi)=0$.

###### Proof.

This follows from the same proof of [AFL-Wei2019, Lemma 14.2]. ∎

######  Proposition 7.4.5.

Assume that $\xi\not=0$ and $v|\infty$ is an archimedean place of $F_{0}$.
Then

$2\partial\mathbb{J}_{v}(\xi,\Phi^{\prime})+\mathrm{Int}^{\mathbf{K}}_{v}(\xi,\Phi)=0.$

###### Proof.

As $E_{0}/F_{0}$ is split at archimedean places, this is the same as [AFL-
Wei2019, Theorem 10.1] by our choice of partial Gaussian test function
$\Phi^{\prime}_{\infty}$. ∎

######  Corollary 7.4.6.

Assume that the twisted AFL conjecture 4.0.8 for all $(g,u)\in
O_{V^{(v)}}(\alpha,\xi)(F_{0})$ at $v$ holds true, and $\Phi^{v},\Phi^{\prime
v}$ are transfers on $O_{V^{(v)}}(\alpha,\xi)(F_{0})$, then we have

$\mathrm{Int}_{v}(\xi,\Phi)+2\partial\mathbb{J}_{v}(\xi,\Phi^{\prime})=0.$

###### Proof.

This follows from comparison of right hand sides of Proposition 7.3.3 and
Proposition 7.4.2. The archimedean comparison is done by Proposition 7.4.5. ∎

######  Proposition 7.4.7.

Assume that $\mathrm{Orb}((g,u),1_{K_{w}}\otimes 1_{L_{w}})\not=0$ for all
finite places $w\nmid v\Delta$ (which always holds after enlarging $\Delta$).
Fix a compact subset $\Omega_{v}$ of
$[\mathrm{U}(V^{v}_{E_{0}})/\mathrm{U}(V^{v})\times V^{v}](F_{0,v})$ (in
$v$-adic topology). Then after possibly shrinking $K_{\Delta}$, there exists
choice for $\phi^{v}=\phi_{\Delta}\otimes
1_{\widehat{L}^{v\Delta}}\in\mathcal{S}(V(\mathbb{A}_{f}^{v}))$ and
$f=f_{\Delta}\otimes
1_{K^{E_{0},v\Delta}}\in\mathcal{S}(K^{E_{0},v}\backslash\mathrm{U}(V)(\mathbb{A}_{f}^{v})/K^{E_{0},v})$
such that for
$(g_{1},u_{1})\in[\mathrm{U}(V)\backslash\mathrm{U}(V_{E_{0}})(\alpha)/\mathrm{U}(V)\times
V^{\prime}_{\xi})(F_{0})]_{\mathrm{rs}}$ the following are equivalent

  1. (1)

We have that $\mathrm{Orb}((g_{1},u_{1}),\Phi^{v})\not=0$ where
$\Phi^{v}=f^{v}\otimes\phi^{v}$, and that $(g_{1}.u_{1})$ is in the
$\mathrm{U}(V^{v})(F_{0,v})$-orbit of $\Omega_{v}$.

  2. (2)

The pair $(g_{1},u_{1})$ is in the $\mathrm{U}(V^{v})(F_{0,v})$-orbit of
$(g,u)$.

###### Proof.

This follows from the same proof of [AFL-Wei2019, Lemma 13.7]. ∎

From Proposition 7.4.7, we obtain the converse of the Corollary 7.4.6. Choose
$\Omega_{v}$ such that non-zero terms in right-hand side of Proposition 7.3.3
and Proposition 7.4.5 always intersects with $\Omega_{v}$.

######  Corollary 7.4.8.

The twisted AFL conjecture 4.0.8 for $(g,u)$ holds if and only if
($\xi_{0}=(u,u)$)

$2\partial_{v}\mathbb{J}(\xi_{0},\Phi^{\prime})+\mathrm{Int}_{v}(\xi_{0},\Phi)=0.$

##  8\. The proof of twisted AFL

In this section, we finish the proof of twisted AFL (Theorem 4.0.8) using
Corollary 7.4.8.

###  8.1. Globalization

Let $F_{v_{0}}/F_{0,v_{0}}$ be an unramified quadratic extension of $p$-adic
local fields with residue fields $\mathbb{F}_{q^{2}}/\mathbb{F}_{q}$. Now we
give a proof of twisted AFL Conjecture 4.0.8 for $F_{v_{0}}/F_{0,v_{0}}$ and a
regular semi-simple orbit

$(g_{0},u_{0})\in[\mathrm{U}(\mathbb{V}_{v_{0}})(F_{v_{0}})\backslash(\mathrm{GL}(\mathbb{V}_{v_{0}})(F_{v_{0}})/\mathrm{U}(\mathbb{V}_{v_{0}})(F_{v_{0}})\times\mathbb{V}_{v_{0}})]_{\mathrm{rs}}.$

We can find a CM quadratic extension $F/F_{0}$ of a totally real field $F_{0}$
with a distinguished embedding $\varphi_{0}:F_{0}\hookrightarrow\mathbb{R}$, a
totally real quadratic extension $E_{0}/F_{0}$, a $n$-dimensional
$F/F_{0}$-hermitian space $V$, a CM type $\Phi$ of $F$ such that

  * •

$\Phi$ is unramified at $p$.

  * •

There exists a place $v_{0}|p$ of $F_{0}$ inert in $F$ and split in $E_{0}$
such that the completion of $F/F_{0}$ at $v_{0}$ is exactly
$F_{v_{0}}/F_{0,v_{0}}$. All places $v\not=v_{0}$ of $F_{0}$ above $p$ is
split in $F$.

  * •

$V$ is of signature $(n-1,1)$ at $\varphi_{0}$, and $(n,0)$ at all other real
places of $F_{0}$.

  * •

The localization $V\otimes_{F}F_{v_{0}}$ is isomorphic to the hermitian space
$V_{v_{0}}$ we start with. The space $V_{p}=\prod_{v|p}V_{v}$ contains a
lattice $L_{p}=\prod_{v|p}L_{v}$ such that $L_{v}$ is self-dual.

Let $V^{(v_{0})}$ be the nearby $F/F_{0}$ hermitian space of $V$ at $v_{0}$ so
the localization of $V^{(v_{0})}$ at $v_{0}$ is isomorphic to
$V^{(v_{0})}_{v_{0}}=\mathbb{V}_{v_{0}}$. As in [AFL-Wei2019, Section 10.1],
by local constancy of intersection numbers [AFL-constant] (whose argument is
quite general and applies to our Rapoport–Zink space
$\mathcal{N}_{n}^{\mathrm{GL}}$), we can find a pair
$(g,u)\in(\mathrm{U}(V^{(v_{0})})\backslash\mathrm{U}(V^{(v_{0})}_{E_{0}})/\mathrm{U}(V^{(v_{0})})\times
V^{v_{0}})(F_{0})_{\mathrm{rs}}$ such that

  * •

$(g,u)$ is $v_{0}$-closely enough to $(g_{0},u_{0})$ such that two sides of
Conjecture 4.0.8 are the same for $(g,u)$ and $(g_{0},u_{0})$. The
characteristic polynomial $\alpha$ of $g$ is irreducible over $F$.

  * •

The norm $\xi_{0}=(u,u)_{V^{v_{0}}}\in F_{0}$ is non-zero.

  * •

For all places $v\not=v_{0}$ of $F_{0}$ above $p$, the orbital integral for
the $\mathrm{U}(V_{v})$-orbit of $(g,u)$

$\mathrm{Orb}((g,u),1_{\mathrm{U}(L_{v})}\times 1_{L_{v}})\not=0.$

Choose a finite collection $\Delta$ of finite places of $F_{0}$ such that
$p\nmid\Delta$, i.e. all places $v$ of $F_{0}$ above $p$ are not in $\Delta$.

We can enlarge $\Delta$ (keeping $p\nmid\Delta$) to assume that $\Delta$ and
$L$ are in the global setup (c.f. 6.2) and $\Phi$ is unramified at all
$\ell\not\in\Delta$. In particular, $E_{0}/F_{0}$ is unramified from $\Delta$
(but there are infinitely many places of $F_{0}$ inert in $E_{0}$). Consider a
level structure $\widetilde{K}$ for $L$ and $\Delta$.

###  8.2. The final proof

We choose $\Phi=f\otimes\phi$ and
$\Phi^{\prime}=f^{\prime}\otimes\phi^{\prime}$ be a pair of partial transfers
(relative to $\alpha$) satisfying the requirement of Proposition 7.4.7.

As in the proof of AFL in [AFL-Wei2019], the twisted AFL identity for $(g,u)$
will follow from

(54)
$2\partial\mathbb{J}(\xi_{0},\Phi^{\prime})+\mathrm{Int}(\xi_{0},\Phi)+\mathrm{Int}^{\mathbf{K-B}}_{\infty}(\xi_{0},\Phi)=0\,\text{in
$\mathbb{R}_{\Delta}$.}$

Consider the difference

$\textbf{Diff}(h)=2\partial\mathbb{J}(h,\Phi^{\prime})+\mathrm{Int}(h,\Phi)+\mathrm{Int}^{\mathbf{K-B}}_{\infty}(h,\Phi),$

with Fourier coefficients ($\xi\not=0$)

$\textbf{Diff}(\xi)=2\partial\mathbb{J}(\xi,\Phi^{\prime})+\mathrm{Int}(\xi,\Phi)+\mathrm{Int}^{\mathbf{K-B}}_{\infty}(\xi,\Phi).$

As the modularity of arithmetic theta series is currently unknown for general
$F_{0}\not=\mathbb{Q}$, we finish the proof via a modification following [AFL-
JEMS, Section 10.6]. We may enlarge $\Delta$ and assume there exists a
characteristic polynomial $\alpha_{m}$ such that $\alpha_{m}$ is of maximal
order away from $\Delta$. By the maximal order case of twisted AFL
(Proposition 4.0.7) and 7.4.6, we have

(55)
$2\partial\mathbb{J}_{\alpha_{m}}(\xi_{0},\Phi^{\prime})+\mathrm{Int}_{\alpha_{m}}(\xi_{0},\Phi)+\mathrm{Int}^{\mathbf{K-B}}_{\alpha_{m},\infty}(\xi_{0},\Phi)=0,$

Here the notions $\partial\mathbb{J}_{\alpha_{m}},\mathrm{Int}_{\alpha_{m}}$
means we replace $\alpha$ by $\alpha_{m}$ in the definition of
$\partial\mathbb{J},\mathrm{Int}_{\alpha_{m}}$.

Then the twisted AFL identity for $(g,u)$ will follow from the equation

(56)
$2\partial\mathbb{J}(\xi_{0},\Phi^{\prime})^{\circ}+\mathrm{Int}(\xi_{0},\Phi)^{\circ}+\mathrm{Int}^{\mathbf{K-B}}_{\infty}(\xi_{0},\Phi)^{\circ}=0\,\text{in
$\mathbb{R}_{\Delta}$.}$

where each term is the difference between corresponding terms in equation (54)
for $\alpha$ and $\alpha_{m}$. By the geometric modularity result [Liu-Thesis]
we know $\mathrm{Int}(h,\Phi)^{\circ}$ is a homolophirc automorphic forms on
$h\in\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})$.

Togethere with Proposition 7.3.2 and [AFL-JEMS, Proposition 10.5], we see that
the difference

$\textbf{Diff}^{\circ}(h)=2\partial\mathbb{J}(h,\Phi^{\prime})^{\circ}+\mathrm{Int}(h,\Phi)^{\circ}+\mathrm{Int}^{\mathbf{K-B}}_{\infty}(h,\Phi)^{\circ}$

is a homolophirc automorphic forms on
$h\in\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})$.

By arithmetic inductions on twisted AFL and local-global decomposition
formulas (Proposition 7.3.3 and Proposition 7.4.5), similar to [AFL-Wei2019,
Lemma 13.6] we have

(57) $\textbf{Diff}(\xi)=0,\,\text{if $(\xi,\Delta)=1$.}$

Similarly,

(58)
$2\partial\mathbb{J}_{\alpha_{m}}(\xi,\Phi^{\prime})+\mathrm{Int}_{\alpha_{m}}(\xi,\Phi)+\mathrm{Int}^{\mathbf{K-B}}_{\alpha_{m},\infty}(\xi,\Phi)=0,\text{if
$(\xi,\Delta)=1$.}$

Hence

(59) $\textbf{Diff}^{\circ}(\xi)=\textbf{Diff}(\xi)=0,\,\text{if
$(\xi,\Delta)=1$.}$

As $\textbf{Diff}^{\circ}$ is a homolophirc automorphic forms on
$h\in\mathrm{SL}_{2}(\mathbb{A}_{F_{0}})$, by [Lemma 13.6][AFL-Wei2019] we
deduce

(60) $\textbf{Diff}^{\circ}(h)=0\,\text{in $\mathbb{R}_{\Delta}$.}$

In particular, $\textbf{Diff}^{\circ}(\xi_{0})=0$ hence
$\textbf{Diff}(\xi_{0})=0$ from (55). This finishes the proof of twisted AFL
identity by Corollary 7.4.8.

##  9\. Arithmetic twisted Gan-Gross-Prasad conjectures for Asai L-functions

In this section, we formulate arithmetic analogs of the twisted Gan-Gross-
Prasad conjecture [twistedGGP][Wang-TGGP] and discuss a relative trace formula
approach following [Liu-FJcycles, Section 5], where the twisted arithmetic
fundamental lemma plays a key role. The related cycles for global height
pairings are produced by Hecke actions on twisted unitary Shimura varieties
and certain doubling/magic divisors (extending Kudla’s geometric theta series)
on self-product of unitary Shimura varieties. The construction is certain
“motivic dedoubling” of Kudla-Rapoport divisors.

We return to the set up of Section 6.1. So $F/F_{0}$ (resp. $E/F_{0}$) is a CM
(resp. totally real) quadratic extension of number fields. And $V$ is a
$F/F_{0}$-hermitian space of dimension $n\geq 1$ with signature
$\\{(n-1,1)_{\varphi_{0}},(n,0)_{\varphi\in\Phi-\\{\varphi_{0}\\}}\\}$. The
embedding of algebraic groups

$H=\mathrm{Res}_{F_{0}/\mathbb{Q}}\mathrm{U}(V)\to
G=\mathrm{Res}_{E_{0}/\mathbb{Q}}\mathrm{U}(V_{E_{0}})$

induces the embedding of (RSZ) unitary Shimura varieties over
$E^{\mathrm{RSZ}}$

$X_{K}:=M_{\widetilde{H},\widetilde{K}}\to
Y_{K}:=M_{\widetilde{G},\widetilde{K_{E_{0}}}}$

for compatible levels $K\leq H(\mathbb{A}_{f}),K_{E_{0}}\leq
G(\mathbb{A}_{f})$. In particular, $\dim Y_{K}=2\dim X_{K}=2(n-1)$.

Let $\mu$ be a conjugate symplectic automorphic character of
$\mathbb{A}_{F}^{\times}/F^{\times}$ of weight one with associated number
field $M_{\mu}$ [Liu-FJcycles, Definition 4.1, 4.3]. Let $\mu^{c}$ be the
complex conjugate of $\mu$. Fix a CM data
$D_{\mu}=(A_{\mu},i_{\mu},\lambda_{\mu},r_{\mu})$ [Liu-FJcycles, Definition
4.5]. Here $A_{\mu}$ is an abelian variety over $F$ with a CM structure
$i_{\mu}:M_{\mu}\to\mathrm{End}_{F}(A)_{\mathbb{Q}}$, a polarization
$\lambda_{\mu}:A_{\mu}\to A^{\vee}_{\mu}$ and an isomorphism
$r_{\mu}:M_{\mu}\otimes_{\mathbb{Q}}F\cong H_{1}^{dR}(A/F)$ of
$M_{\mu}\otimes_{\mathbb{Q}}F$-modules. In particular, $\dim
A_{\mu}=[M_{\mu}:\mathbb{Q}]/2$.

Let $A_{K}$ be the Albanese variety of $X_{K}$, and $A_{\infty}=\lim_{K}A_{K}$
(resp. $Y_{\infty}=\lim_{K}Y_{K}$)be the inverse limit of $A_{K}$ over all
levels $K$.Consider the $M_{\mu}[\mathrm{U}(V)(\mathbb{A}_{f})]$-module
$\Omega(\mu):=\mathrm{Hom}_{F}(A_{\infty},A_{\mu})_{\mathbb{Q}}.$

######  Theorem 9.0.1.

([Liu-FJcycles, Theorem 1.1]) There is an isomorphism

$\Omega(\mu)\otimes_{M_{\mu}}\mathbb{C}=\oplus_{\epsilon}\oplus_{\chi}\omega(\epsilon,\mu,\chi)$

Here $\omega(\epsilon,\mu,\chi)$ is the finite part of the Weil representation
of $\mathrm{U}(V)$ determined by the index $(\epsilon,\mu,\chi)$ [Liu-
FJcycles, Appendix D] appearing in the (twisted) Fourier-Jacobi GGP conjecture
[twistedGGP].

Consider the twisted diagonal embedding

$\Delta^{E_{0}\times F_{0}}:X_{K}\to Y_{K}\times X_{K}.$

Recall the specific Hecke correspondence $T_{\mu}^{\mathrm{can}}$ on $A_{\mu}$
[Liu-FJcycles, Definition 4.9]. Let $\alpha_{K}:X_{K}\to A_{K}$ be the
Albanese morphism sending the zero-dimensional cycle $D^{n-1}$ to zero, where
$D_{K}$ is the canonical extension of the Hodge divisor on $X_{K}$.

######  Definition 9.0.2.

For $f\in S(K_{E_{0}}\backslash G(\mathbb{A}_{f})/K_{E_{0}})$ and
$\phi\in\mathrm{Hom}_{F}(A_{K},A_{\mu})$, let $T^{f}_{K}$ be the Hecke
correspondence associated to $f$ on $Y_{K}$. The twisted Fourier-Jacobi cycle
for $(f,\phi)$ is the cohomological trivial cycle on $Y_{K}\times A_{\mu}$

(61)
$\mathrm{FJ}(f,\phi)_{K}:=|\pi_{0}^{\mathrm{geo}}(X_{K})|(T^{f}_{K}\otimes
T_{\mu}^{\mathrm{can}})^{*}(\mathrm{id}_{Y_{K}}\times(\phi\circ\alpha_{K}))_{*}(\Delta^{E_{0}\times
F_{0}}X_{K})\in\mathrm{CH}^{n-1+[M_{\mu}:\mathbb{Q}]/2}(Y_{K}\times
A_{\mu})^{0}_{\mathbb{C}}.$

as an element in the cohomological trivial Chow group
$\mathrm{CH}^{n-1+[M_{\mu}:\mathbb{Q}]/2}(Y_{K}\times
A_{\mu})^{0}_{\mathbb{C}}$ (with $\mathbb{C}$-coefficients). Here we add the
number $|\pi_{0}^{\mathrm{geo}}(X_{K})|$ of geometric connected components of
$X_{K}$ to make sure that $\mathrm{FJ}(f,\phi)_{K}$ is compatible with change
of levels.

Let $\Omega(\mu,\epsilon):=\oplus_{\chi}\omega(\epsilon,\mu,\chi)$ and
$\mathrm{Hom}_{E}(A_{K},A_{\mu},\epsilon):=\mathrm{Hom}_{E}(A_{K},A_{\mu})\cap\Omega(\mu,\epsilon)$.

Let $\pi$ be a cuspidal automorphic representation of
$\mathrm{U}(V_{E_{0}})(\mathbb{A}_{E_{0}})$. Let $\Pi$ be the automorphic
representation of $\mathrm{GL}_{n}(\mathbb{A}_{E})$ as the base change of
$\pi$. Assume that $\Pi$ is relevant [Liu-FJcycles]. Following [twistedGGP],
we have the twisted Asai L-function

$L(s,\Pi,\mathrm{As}_{E/F}\otimes\mu)$

as a memorphic functions on $s\in\mathbb{C}$ with central point $s=1/2$.

By the same argument of [Liu-FJcycles, Theorem 1.3], assuming the conjecture
on the injectivity of the $\ell$-adic Abel–Jacobi map, the assignment
$(f,\phi)\to\mathrm{FJ}(f,\phi)_{K}$ induces a complex linear map

$\mathrm{FJ}_{\epsilon}:\pi_{f}\otimes_{\mathbb{C}}\Omega(\mu,\epsilon)\to\mathrm{Hom}_{\mathbb{C}[\mathrm{U}(V_{E_{0}})(\mathbb{A}_{f})]}(\pi_{f},\mathrm{CH}^{n-1+[M_{\mu}:\mathbb{Q}]/2}(Y_{\infty}\times
A_{\mu})^{0}_{\mathbb{C}})$

We now formulate the (unrefined) arithmetic twisted Gan-Gross-Prasad
conjecture (similar to [Liu-FJcycles, Conjecture 1.4]).

######  Conjecture 9.0.3 (Arithmetic Twisted Gan-Gross-Prasad Conjecture).

Let $F/F_{0}$ (resp. $E/F_{0}$) be a CM (resp. totally real) quadratic
extension of number fields. Let $\mu$ be a conjugate symplectic automorphic
character of $\mathbb{A}_{F}^{\times}/F^{\times}$ of weight one. Let $\pi$ be
a cuspidal automorphic representation of
$\mathrm{U}(V_{E_{0}})(\mathbb{A}_{E_{0}})$. Let $\Pi$ be the automorphic
representation of $\mathrm{GL}_{n}(\mathbb{A}_{E})$ as the base change of