# Intersection formulas on moduli spaces of unitary shtukas

Yongyi Chen and Benjamin Howard  Department of Mathematics, Boston College,
140 Commonwealth Ave, Chestnut Hill, MA 02467, USA<EMAIL_ADDRESS>Department
of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467,
USA<EMAIL_ADDRESS>
###### Abstract.

Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-
Weil formula, relating special cycles on moduli spaces of unitary shtukas to
higher derivatives of Eisenstein series. We prove an extension of this formula
in a low-dimensional case, and deduce from it a Gross-Zagier style formula
expressing intersection multiplicities of cycles in terms of higher
derivatives of base-change $L$-functions.

B.H. was supported in part by NSF grant DMS-2101636.

##  1\. Introduction

Feng-Yun-Zhang have defined special cycle classes on moduli spaces of unitary
shtukas, and proved an arithmetic Siegel-Weil formula (in the sense of Kudla
and Kudla-Rapoport) relating them to higher derivatives of Fourier
coefficients of Siegel Eisenstein series.

One would like to exploit this connection between arithmetic geometry and
automorphic forms to prove a Gross-Zagier style formula relating intersection
multiplicities of these cycles to higher derivatives of Langlands
$L$-functions, but this is difficult because the moduli spaces on which the
special cycles live are almost never proper.

The goal of this paper is to formulate and prove such a Gross-Zagier style
formula in a low-dimensional case, in a way that circumvents this lack of
properness.

###  1.1. The Feng-Yun-Zhang classes

Let $k$ be a finite field of odd cardinality $q$, and fix a finite étale
double cover

$X^{\prime}\to X$

of smooth, projective, geometrically connected curves over $k$. Denote by
$\sigma\in\operatorname{Aut}(X^{\prime}/X)$ the nontrivial Galois
automorphism.

Fix integers $n\geq 1$ and $r\geq 0$. Our étale double cover determines a
moduli stack $\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ of
$\mathrm{U}(n)$-shtukas with $r$ legs. It is a Deligne-Mumford stack over $k$,
locally of finite type, equipped with a smooth morphism

$\operatorname{Sht}^{r}_{\mathrm{U}(n)}\to(X^{\prime})^{r}$

of relative dimension $r(n-1)$. In particular,
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ is smooth over $k$ of dimension $rn$.
It is nonempty if and only if $r$ is even ([FYZa, Lemma 6.7]), which we assume
from now on.

To a pair $(\mathcal{E},a)$ consisting of a vector bundle $\mathcal{E}$ on
$X^{\prime}$ of rank $m\leq n$ and a hermitian morphism
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$ (in the sense of §1.4),
Feng-Yun-Zhang [FYZa, §7] associate a _naive special cycle_
$\mathcal{Z}^{r}_{\mathcal{E}}(a)$. This is a Deligne-Mumford stack equipped
with a finite morphism

$\mathcal{Z}^{r}_{\mathcal{E}}(a)\to\operatorname{Sht}^{r}_{\mathrm{U}(n)}.$

The naive special cycle has expected dimension $r(n-m)$, but this expectation
is rarely fulfilled. To correct for this, one finds in [FYZb, Definition 4.8]
the construction of a cycle class

(1.1.1)
$[\mathcal{Z}^{r}_{\mathcal{E}}(a)]\in\mathrm{Ch}_{r(n-m)}(\mathcal{Z}^{r}_{\mathcal{E}}(a))$

on $\mathcal{Z}^{r}_{\mathcal{E}}(a)$ of the desired dimension. When no
confusion can arise, we denote in the same way the image of this class under
the pushforward

(1.1.2)
$\mathrm{Ch}_{r(n-m)}(\mathcal{Z}^{r}_{\mathcal{E}}(a))\to\mathrm{Ch}^{rm}(\operatorname{Sht}^{r}_{\mathrm{U}(n)}).$

These special cycles are the function field analogues of the Kudla-Rapoport
cycles [KR14] on Shimura varieties for unitary groups of signature $(n-1,1)$.

It sometimes happens that the naive special cycle
$\mathcal{Z}^{r}_{\mathcal{E}}(a)$ is a proper $k$-stack. When this is the
case the above pushforward can be refined to a homomorphism

(1.1.3)
$\mathrm{Ch}_{r(n-m)}(\mathcal{Z}^{r}_{\mathcal{E}}(a))\to\mathrm{Ch}_{c}^{rm}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})$

valued in the Chow group with proper support, and so (1.1.1) determines a
class there as well. Using the intersection pairing

(1.1.4)
$\operatorname{Ch}^{r(n-m)}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})\times\operatorname{Ch}_{c}^{rm}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})\to\operatorname{Ch}_{c}^{rn}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})$

and the degree map

(1.1.5)
$\deg\colon\mathrm{Ch}_{c}^{rn}(\operatorname{Sht}_{\mathrm{U}(n)}^{r})\to\mathbb{Q},$

we obtain a $\mathbb{Q}$-valued intersection between
$[\mathcal{Z}^{r}_{\mathcal{E}}(a)]$ and any codimension $r(n-m)$ cycle class
on $\operatorname{Sht}^{r}_{\mathrm{U}(n)}$.

###  1.2. The modularity conjecture

Let us restrict to the case $n=2m$, so that the special cycles (1.1.1)
determine classes in the middle codimension Chow group of
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$.

Denote by $F=k(X)$ the field of rational functions on $X$, and similarly for
$F^{\prime}=k(X^{\prime})$. Let

$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$

be an unramified Hecke character whose restriction to
$\mathbb{A}_{F}^{\times}$ is trivial.

Let $\mathcal{A}(H_{m})$ be the space of automorphic forms on the rank $2m$
quasi-split unitary group $H_{m}$ over $F$. Any form $f\in\mathcal{A}(H_{m})$
fixed by the standard maximal compact open subgroup $K_{m}\subset
H_{m}(\mathbb{A}_{F})$ is determined by its _geometric Fourier coefficients_
$f_{(\mathcal{E},a)}$. These are defined in §2.2, but at the moment all the
reader needs to know is that this is a collection of complex numbers indexed
by pairs $(\mathcal{E},a)$ consisting of a vector bundle $\mathcal{E}$ of rank
$m$ on $X^{\prime}$ and a hermitian morphism
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$.

The following is the modularity conjecture of [FYZb, Conjecture 4.15],
restricted to the case of cycles in middle codimension. Abbreviate

(1.2.1)
$d(\mathcal{E})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{\deg(\mathcal{E}^{\vee})-\deg(\mathcal{E})}{2}=\mathrm{rank}(\mathcal{E})\deg(\omega_{X})-\deg(\mathcal{E})$

for any vector bundle $\mathcal{E}$ on $X^{\prime}$.

######  Conjecture 1.2.1 (Feng-Yun-Zhang).

There is an automorphic form

$Z^{r,\chi}\colon H_{m}(F)\backslash
H_{m}(\mathbb{A}_{F})/K_{m}\to\mathrm{Ch}^{rm}(\mathrm{Sht}^{r}_{U(n)})_{\mathbb{C}}$

whose geometric Fourier coefficients are the rescaled special cycles

$Z^{r,\chi}_{(\mathcal{E},a)}=\frac{\chi(\det(\mathcal{E}))}{q^{md(\mathcal{E})}}\,[\mathcal{Z}^{r}_{\mathcal{E}}(a)]\in\mathrm{Ch}^{rm}(\mathrm{Sht}^{r}_{U(n)})_{\mathbb{C}}.$

See §2.3 for the meaning of $\chi(\det(\mathcal{E}))$.

Our first result provides some evidence toward this conjecture in a low
dimensional case, in the same style as [FYZb, Part 3].

######  Theorem 1.2.2.

Suppose $\mathcal{E}_{2}$ is a line bundle on $X^{\prime}$ (so $m=1$ and
$n=2$) and $a_{2}\colon\mathcal{E}_{2}\cong\sigma^{*}\mathcal{E}_{2}^{\vee}$
is a hermitian isomorphism. The naive special cycle

$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})\to\operatorname{Sht}^{r}_{\mathrm{U}(2)}$

is proper over $k$, and there exists a $K_{1}$-fixed automorphic form
$\mathscr{D}\in\mathcal{A}(H_{1})$, depending on $(\mathcal{E}_{2},a_{2})$,
whose geometric Fourier coefficients are given by

$\mathscr{D}_{(\mathcal{E}_{1},a_{1})}=\frac{\chi(\mathcal{E}_{1})}{q^{d(\mathcal{E}_{1})}}\,\deg\big{(}[\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1})]\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}$

for every line bundle $\mathcal{E}_{1}$ on $X^{\prime}$ and every hermitian
map $a_{1}:\mathcal{E}_{1}\to\sigma^{*}\mathcal{E}_{1}^{\vee}$.

This result appears in the text as part (3) of Theorem 3.3.1. The automorphic
form $\mathscr{D}$ is essentially the kernel function appearing in the _new
way_ integrals of [PSR88, Qin07]; see Remark 2.4.4. Our second main result
exploits this observation to prove a higher derivative Gross-Zagier style
intersection formula.

###  1.3. Intersections with arithmetic theta lifts

We continue to assume that $n=2m$ is even, and that
$\chi:\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$ is an unramified
Hecke character whose restriction to $\mathbb{A}_{F}^{\times}$ is trivial.

Assuming Conjecture 1.2.1, for a $K_{m}$-fixed form $f$ in an irreducible
cuspidal automorphic representation $\pi\subset\mathcal{A}(H_{m})$ we define
the _arithmetic theta lift_

(1.3.1) $\vartheta^{r,\chi}(f)=\int_{H_{m}(F)\backslash
H_{m}(\mathbb{A}_{F})}f(h)Z^{r,\chi}(h)\,dh\in\operatorname{Ch}^{rm}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})_{\mathbb{C}}.$

Following ideas of Kudla [Kud04, §8] in the context of Shimura varieties,
developed further by Liu [Liu11] and Li-Liu [LL21], one could hope for an
_arithmetic Rallis inner product formula_

(1.3.2)
$\deg\big{(}\vartheta^{r,\chi}(f)\cdot\vartheta^{r,\chi}(f)\big{)}\stackrel{{\scriptstyle?}}{{=}}C\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)$

(for some explicit constant $C$) relating the self-intersection multiplicity
of $\vartheta^{r,\chi}(f)$ to the $r^{\mathrm{th}}$ central derivative of the
twisted base-change $L$-function at the center of its functional equation.

Unfortunately, in the present context the left hand side of (1.3.2) is not
defined, as the moduli space $\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ is not
proper as soon as $n\geq 2$. While one can form the self-intersection of
$\vartheta^{r,\chi}(f)$ as an element in the usual Chow group (i.e. not the
Chow group with proper support) of $0$-cycles on
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$, there is no analogue of the degree
map (1.1.5) on this Chow group.

To circumvent this, fix a pair $(\mathcal{E}_{2},a_{2})$ consisting of a rank
$m$ vector bundle $\mathcal{E}_{2}$ on $X^{\prime}$ and a hermitian morphism
$a_{2}:\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$, and _assume_ that
the associated naive special cycle
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})\to\operatorname{Sht}^{r}_{\mathrm{U}(n)}$
is proper over $k$. Using (1.1.4) and (1.1.5) to intersect

$[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\in\operatorname{Ch}^{rm}_{c}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})$

against the arithmetic theta lift (1.3.1), we propose the following Gross-
Zagier style intersection formula.

######  Conjecture 1.3.1.

For any $K_{m}$-invariant $f\in\pi\subset\mathcal{A}(H_{m})$ as above, we have

$\displaystyle\deg\big{(}\vartheta^{r,\chi}(f)\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}$
$\displaystyle=f_{(\mathcal{E}_{2},-a_{2})}\frac{q^{md(\mathcal{E}_{2})}}{(\log
q)^{r}}\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\left(q^{ns\deg(\omega_{X})}L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)\right),$

where $f_{(\mathcal{E}_{2},-a_{2})}$ is the geometric Fourier coefficient in
the sense of §2.2.

The properness assumption imposed on
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})$ is very strong. Although no proof
is provided, it is claimed in [FYZb, Example 4.20] that it holds whenever
$m=1$ and $a_{2}:\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$ is
injective. There is circumstantial evidence that properness may also hold for
some middle codimension special cycles on
$\operatorname{Sht}^{r}_{\mathrm{U}(4)}$. For $m>2$ there is no expectation
that any middle codimension special cycles on
$\operatorname{Sht}^{r}_{\mathrm{U}(2m)}$ are proper. The point we wish to
convey is that the above conjecture predicts that _every_ instance of a proper
special cycle in middle dimension has its own higher derivative intersection
formula, and so even these low-dimensional cases are of interest.

######  Remark 1.3.2.

Conjecture 1.3.1 is perhaps less like the Gross-Zagier theorem and more like a
hybrid of the arithmetic Rallis inner product formula and the Gross-Kohnen-
Zagier theorem. An analogue of it on quaternionic Shimura curves over
$\mathbb{Q}$ can be deduced easily from [KRY06, Corollary 1.0.7].

Our second main result is the proof of the above conjecture in the simplest
case, stated as part (4) of Theorem 3.3.1 in the body of the text. This result
is conditional on Conjecture 1.2.1, which is needed to even define the
arithmetic theta lift (1.3.1).

######  Theorem 1.3.3.

Conjecture 1.3.1 holds in the case where $\mathcal{E}_{2}$ is a line bundle on
$X^{\prime}$ (so $m=1$ and $n=2$) and
$a_{2}\colon\mathcal{E}_{2}\cong\sigma^{*}\mathcal{E}_{2}^{\vee}$ is a
hermitian isomorphism.

At the core of the above results is Theorem 3.2.6, which extends the Feng-Yun-
Zhang arithmetic Siegel-Weil formula on
$\operatorname{Sht}^{r}_{\mathrm{U}(2)}$. Any further generalization of this
theorem (more precisely, any new cases of Conjecture 3.1.2) would lead
immediately to new cases of Conjecture 1.3.1.

For example, one could attempt to prove Theorems 1.2.2 and 1.3.3 under the
weaker hypothesis that
$a_{2}\colon\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$ is injective.
Even more interesting would be to find examples of proper special cycles of
middle codimension on $\operatorname{Sht}^{r}_{\mathrm{U}(4)}$, and extend our
results to that setting.

###  1.4. Notation

The double cover $X^{\prime}\to X$ over $k$ fixed above remains fixed through
the paper, $\sigma\in\operatorname{Aut}(X^{\prime}/X)$ is its nontrivial
automorphism, and the nontrivial automorphism of $F^{\prime}=k(X^{\prime})$
over $F=k(X)$ is denoted the same way. Denote by

(1.4.1) $\eta=\eta_{F^{\prime}/F}\colon\mathbb{A}_{F}^{\times}\to\\{\pm 1\\}$

the associated quadratic character.

Let $\omega_{X}$ be the sheaf of Kähler differentials on $X$. By the
assumption that $X^{\prime}\to X$ is étale, its pullback is the sheaf of
Kähler differentials $\omega_{X^{\prime}}$ on $X^{\prime}$. Denote by
$\omega_{F}$ the stalk of $\omega_{X}$ at the generic point of $X$. In other
words, $\omega_{F}$ is the $1$-dimensional $F$ vector space of rational Kähler
differentials on $X$.

As in [FYZa], for any vector bundle $\mathcal{E}$ on $X^{\prime}$ we denote by

$\mathcal{E}^{\vee}=\underline{\operatorname{Hom}}_{\mathcal{O}_{X^{\prime}}}(\mathcal{E},\omega_{X^{\prime}})$

the _Serre dual_ of $\mathcal{E}$. A _hermitian morphism_
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$ is a morphism (in the
category of coherent sheaves on $X^{\prime}$) satisfying the hermitian
condition $\sigma^{*}a^{\vee}=a$. As usual, $a$ is _injective_ if the coherent
sheaf $\ker(a)$ is trivial.

If $R$ is an $F$-algebra we abbreviate $R^{\prime}=R\otimes_{F}F^{\prime}$. If
$L$ is a one-dimensional $F$-vector space (e.g. $\omega_{F}$), denote by

$\operatorname{Herm}_{n}(R,L)\subset M_{n}(L\otimes_{F}R^{\prime})$

the $R$-submodule of hermitian matrices. In particular,

$\operatorname{Herm}_{n}(R)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\operatorname{Herm}_{n}(R,F)$

is the usual $R$-module of $n\times n$ hermitian matrices with entries in
$R^{\prime}$.

For any group scheme $G$ over $F$ we abbreviate $[G]=G(F)\backslash
G(\mathbb{A}_{F})$. In particular, this applies to the scheme
$\operatorname{Herm}_{n}$ over $F$ with functor of points
$R\mapsto\operatorname{Herm}_{n}(R)$.

###  1.5. Acknowledgements

Yongyi Chen thanks Keerthi Madapusi and Tony Feng for helpful conversations.

Ben Howard thanks Jan Bruinier, who suggested (some years ago) that the
integral representations of [PSR88] might be used to obtain interesting Gross-
Zagier style intersection formulas. This suggestion was the inspiration for
the present work. He also thanks Tony Feng, Aaron Pollack, Chris Skinner, and
Wei Zhang for helpful conversations during the SLMath/MSRI semester _Algebraic
cycles, $L$-values, and Euler systems_, where part of this work was carried
out.

##  2\. Eisenstein series on unitary groups

This section begins by establishing some basic notation for quasi-split
unitary groups, and the Siegel Eisenstein series on them. Once that is done we
prove two results.

The first, Theorem 2.4.2 is a modest generalization of the famous doubling
formula of Piatetski-Shapiro and Rallis [PSR87], providing the link between
Siegel Eisenstein series and base-change $L$-functions. This will be needed in
the proof of Theorem 3.3.1.

The second, Proposition 2.5.1, is an explicit formula relating singular
Fourier coefficients of an Eisenstein series on a unitary group of rank $4$ to
non-singular Fourier coefficients of an Eisenstein series on a unitary group
of rank $2$. This is one of the main ingredients in the proof of Theorem
3.2.6.

###  2.1. The quasi-split unitary group

For an integer $n\geq 1$, abbreviate

$w_{n}=\begin{pmatrix}&I_{n}\\\
-I_{n}&\end{pmatrix}\in\operatorname{GL}_{2n}(F).$

We endow the space of column vectors $W_{n}=(F^{\prime})^{2n}$ with the
standard skew-hermitian form $h_{n}(x,y)={}^{t}x\cdot w_{n}\cdot\sigma(y)$.
The associated rank $2n$ quasi-split unitary group over $F$ is denoted

$H_{n}=\mathrm{U}(W_{n})\subset\mathrm{Res}_{F^{\prime}/F}\operatorname{GL}_{2n}.$

The _standard Siegel parabolic_ $P_{n}\subset H_{n}$ is the subgroup of
matrices whose lower left $n\times n$ block vanishes. Its unipotent radical is
denoted

$N_{n}=\left\\{n(b)=\begin{pmatrix}I_{n}&b\\\
&I_{n}\end{pmatrix}:b\in\operatorname{Herm}_{n}\right\\}$

while its Levi factor is denoted

$M_{n}=\left\\{m(\alpha)=\begin{pmatrix}\alpha&\\\
&\sigma({}^{t}\alpha^{-1})\end{pmatrix}:\alpha\in\mathrm{Res}_{F^{\prime}/F}\operatorname{GL}_{n}\right\\}.$

The _standard compact open subgroup_ of $H_{n}(\mathbb{A}_{F})$ is
$K_{n}=\prod_{v}K_{n,v}$, where for any place $v$ of $F$ we set

$K_{n,v}=H_{n}(F_{v})\cap\operatorname{GL}_{2n}(\mathcal{O}_{F^{\prime}_{v}}).$

######  Remark 2.1.1.

The Haar measure on $H_{n}(\mathbb{A}_{F})$ is always normalized so that the
standard compact open $K_{n}$ has volume $1$, and $[H_{n}]$ is given the
induced quotient measure. The Haar measure on
$\operatorname{Herm}_{n}(\mathbb{A}_{F})\cong N(\mathbb{A}_{F})$ is given the
self-dual Haar measure; explicitly, this is the one for which the compact open
subgroup

$\prod_{v}\operatorname{Herm}_{n}(\mathcal{O}_{F_{v}})\subset\operatorname{Herm}_{n}(\mathbb{A}_{F})$

of hermitian matrices with integral entries has volume
$q^{-n^{2}\deg(\omega_{X})/2}$. The induced quotient measure on
$[\operatorname{Herm}_{n}]\cong[N]$ has volume $1$.

###  2.2. Fourier coefficients

Fix a nontrivial additive character $\psi_{0}\colon k\to\mathbb{C}^{\times}$.
As in [FYZa, §2.2], we the $k$-linear _residue map_
$\mathrm{Res}\colon\omega_{F}\to k$ to define a canonical pairing

$\langle\cdot,\cdot\rangle\colon\operatorname{Herm}_{n}(\mathbb{A}_{F},\omega_{F})\times\operatorname{Herm}_{n}(\mathbb{A}_{F})\to
k$

by $\langle T,b\rangle=\mathrm{Res}(-\mathrm{Tr}(Tb))$. Its composition with
$\psi_{0}$ is denoted

$\langle\cdot,\cdot\rangle_{\psi_{0}}\colon\operatorname{Herm}_{n}(\mathbb{A}_{F},\omega_{F})\times\operatorname{Herm}_{n}(\mathbb{A}_{F})\to\mathbb{C}^{\times}.$

Any automorphic form $f\in\mathcal{A}(H_{n})$ has a Fourier expansion

(2.2.1) $f(g)=\sum_{T\in\operatorname{Herm}_{n}(F,\omega_{F})}f_{T}(g),$

in which the $T$-coefficient is

$f_{T}(g)=\int_{[\operatorname{Herm}_{n}]}f(n(b)g)\,\langle
T,b\rangle_{\psi_{0}}\,db,$

with Haar measure as in Remark 2.1.1.

Suppose $f\in\mathcal{A}(H_{n})$ is right invariant under $K_{n}$. By the
Iwasawa decomposition, $f$ is determined by its restriction to
$P_{n}(\mathbb{A}_{F})\subset H_{n}(\mathbb{A}_{F})$, and hence is determined
by the collection of functions
$\\{f_{T}|_{M_{n}(\mathbb{A}_{F})}\\}_{T\in\operatorname{Herm}_{n}(F,\omega_{F})}.$
As explained in [FYZa, §2.6], this gives rise to a collection of _geometric
Fourier coefficients of $f$_: complex numbers $f_{(\mathcal{E},a)}$ indexed by
pairs $(\mathcal{E},a)$ consisting of a rank $n$ vector bundle $\mathcal{E}$
on $X^{\prime}$ and a hermitian morphism
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}.$

The precise definition of $f_{(\mathcal{E},a)}$ is as follows. The generic
fiber of $\mathcal{E}$ is a free $F^{\prime}$-module of rank $n$, and upon
choosing an isomorphism $\mathcal{E}_{F^{\prime}}\cong(F^{\prime})^{n}$ the
hermitian morphism $a$ is given by a hermitian matrix
$T\in\operatorname{Herm}_{n}(F,\omega_{F})$. For every place $v$ of
$F^{\prime}$ there is an $\alpha_{v}\in\operatorname{GL}_{n}(F^{\prime}_{v})$
such that the completed stalk of $\mathcal{E}$ at $v$ is identified with the
lattice
$\alpha_{v}\cdot\mathcal{O}_{F^{\prime}_{v}}^{n}\subset(F_{v}^{\prime})^{n}$.
Setting
$\alpha=\prod_{v}\alpha_{v}\in\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})$,
we define

$f_{(\mathcal{E},a)}=f_{T}(m(\alpha)).$

######  Remark 2.2.1.

We identify the set of rank $n$ vector bundles on $X^{\prime}$ with the double
quotient

$\operatorname{GL}_{n}(F^{\prime})\backslash\operatorname{GL}_{n}(\mathbb{A}_{F^{\prime}})/\prod_{v}\operatorname{GL}_{n}(\mathcal{O}_{F^{\prime}_{n}})$

by sending $\mathcal{E}\mapsto\alpha$ with $\alpha$ as above. In particular,

$q^{\deg(\det(\mathcal{E}))}=|\det(\alpha)|_{F^{\prime}}.$

###  2.3. Siegel Eisenstein series

Fix an unramified Hecke character
$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$, and denote
by $\chi_{0}$ its restriction to $\mathbb{A}_{F}^{\times}$. We always assume

(2.3.1) $\chi_{0}=1\quad\mbox{or}\quad\chi_{0}=\eta,$

where $\eta$ is the quadratic character (1.4.1). We often use the isomorphism

$\mathrm{Pic}(X^{\prime})\cong(F^{\prime})^{\times}\backslash\mathbb{A}_{F^{\prime}}^{\times}/\prod_{v}\mathcal{O}_{F^{\prime}_{v}}^{\times}$

of Remark 2.2.1 to view $\chi$ as a character
$\chi\colon\mathrm{Pic}(X^{\prime})\to\mathbb{C}^{\times}.$

######  Remark 2.3.1.

We will follow the discussion of [FYZa] for Eisenstein series on
$H_{n}(\mathbb{A}_{F})$, but warn the reader that throughout _loc. cit._ it is
assumed that the Hecke character $\chi$ satisfies $\chi_{0}=\eta^{n}$. Thus
some of the formulas of _loc. cit._ require small modifications to account for
our weaker assumption (2.3.1). This weaker assumption is the only one imposed
in [Tan99], and everything we need to know about Eisenstein series can be
deduced from the results found there.

For a complex variable $s$, denote by

$I_{n}(s,\chi)=\mathrm{Ind}_{P_{n}(\mathbb{A}_{F})}^{H_{n}(\mathbb{A}_{F})}(\chi|\cdot|_{F^{\prime}}^{s+\frac{n}{2}})$

the unnormalized smooth induction. Here we are viewing both $\chi$ and
$|\cdot|_{F^{\prime}}$ as characters of $P_{n}(\mathbb{A}_{F})$ by composing
them with the homomorphism

(2.3.2) $P_{n}(\mathbb{A}_{F})\to\mathbb{A}_{F^{\prime}}^{\times}$

sending an element of
$P_{n}(\mathbb{A}_{F})\subset\operatorname{GL}_{2n}(\mathbb{A}_{F^{\prime}})$
to the determinant of its upper left $n\times n$ block.

Given any $\Phi_{s}\in I_{n}(s,\chi)$ with $\mathrm{Re}(s)>n/2$, the summation

$E(g,\Phi_{s})=\sum_{\gamma\in P_{n}(F)\backslash H_{n}(F)}\Phi_{s}(\gamma g)$

is absolutely convergent and defines a Siegel Eisenstein series on
$H_{n}(\mathbb{A}_{F})$. Suppose $T\in\operatorname{Herm}_{n}(F,\omega_{F})$
is nonsingular, in the sense that $\det(T)\neq 0$ after fixing some
trivialization $\omega_{F}\cong F$. As in [FYZa, §2.2 ], modified to account
for the choice of Haar measure in Remark 2.1.1, the corresponding Fourier
coefficient is

(2.3.3)
$E_{T}(g,\Phi_{s})=\int_{\operatorname{Herm}_{n}(\mathbb{A}_{F})}\Phi_{s}\big{(}w_{n}^{-1}n(b)g\big{)}\,\langle
T,b\rangle_{\psi_{0}}\,db.$

There is a unique standard section $s\mapsto\Phi_{s}^{\circ}\in I_{n}(s,\chi)$
satisfying $\Phi_{s}^{\circ}(k)=1$ for all $k\in K_{n}$. We call this the
_normalized spherical section_ , and denote the associated _spherical
Eisenstein series_ by

$E(g,s,\chi)=E(g,\Phi_{s}^{\circ}).$

The standard $L$-function $L(s,\eta^{i}\chi_{0})$ satisfies the functional
equation

(2.3.4)
$q^{\frac{s}{2}\deg(\omega_{X})}L(s,\eta^{i}\chi_{0})=q^{\frac{1-s}{2}\deg(\omega_{X})}L(1-s,\eta^{i}\chi_{0}).$

The product of $L$-functions

$\mathscr{L}_{n}(s,\chi_{0})=\prod_{i=1}^{n}L(2s+i,\eta^{i-n}\chi_{0})$

agrees with the $\mathscr{L}_{n}(s)$ defined in [FYZa, §2.6] when
$\chi_{0}=\eta^{n}$.

Because of the assumption (2.3.1), there is an intertwining operator

$M_{n}(s)\colon I_{n}(s,\chi)\to I_{n}(-s,\chi)$

defined for $\mathrm{Re}(s)>n/2$ by

(2.3.5)
$(M_{n}(s)\Phi_{s})(g)=\int_{\operatorname{Herm}_{n}(\mathbb{A}_{F})}\Phi_{s}\left(w_{n}^{-1}n(b)g\right)\,db.$

By combining the local calculation of [Tan99, Proposition 2.1], as simplified
on page 170 of _loc. cit._ , with the global function equation (2.3.4), one
finds that the unramified spherical section satisfies

(2.3.6)
$M_{n}(s)\Phi_{s}^{\circ}=q^{-2ns\deg(\omega_{X})}\frac{\mathscr{L}_{n}(-s,\chi_{0})}{\mathscr{L}_{n}(s,\chi_{0})}\cdot\Phi^{\circ}_{-s}.$

It follows that $E(g,s,\chi)$ has meromorphic continuation, and that the
renormalized Eisenstein series

(2.3.7)
$\widetilde{E}(g,s,\chi)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}q^{ns\deg(\omega_{X})}\cdot\mathscr{L}_{n}(s,\chi_{0})\cdot
E(g,s,\chi)$

of [FYZb, §9.5] satisfies the functional equation

(2.3.8) $\widetilde{E}(g,s,\chi)=\widetilde{E}(g,-s,\chi).$

For nonsingular $T$, the integral (2.3.3) determined by the normalized
spherical section $\Phi_{s}^{\circ}$ can be expressed as products of local
representation densities. Hence the same is true of the geometric Fourier
coefficient $E_{(\mathcal{E},a)}(s,\chi)$ indexed by a rank $n$ vector bundle
$\mathcal{E}$ on $X^{\prime}$ and an _injective_ hermitian morphism
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$. We only need the simplest
case of these formulas, in which $a$ is an isomorphism.

######  Proposition 2.3.2.

If $\mathcal{E}$ is a rank $n$ vector bundle on $X^{\prime}$, and
$a\colon\mathcal{E}\cong\sigma^{*}\mathcal{E}^{\vee}$ is a hermitian
isomorphism, then

$q^{ns\deg(\omega_{X})}\cdot\mathscr{L}_{n}(s,\chi_{0})\cdot
E_{(\mathcal{E},a)}(s,\chi)=\chi(\det(\mathcal{E})).$

###### Proof.

This is a special case of [FYZa, Theorem 2.8], slightly extended (see Remark
2.3.1) using the calculation of unramified local Whittaker functions found in
[Tan99, Proposition 3.2]. ∎

###  2.4. The duplication formula

In this subsection we assume that $n=2m$ is even. We continue to work with an
unramified Hecke character
$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$, but now
assume $\chi_{0}=1$ instead of the weaker (2.3.1).

Consider the Eisenstein series

$E(g,s,\chi)=\sum_{\gamma\in P_{n}(F)\backslash
H_{n}(F)}\Phi^{\circ}_{s}(\gamma g)$

on $H_{n}(\mathbb{A}_{F})$ associated to the normalized spherical section
$\Phi_{s}^{\circ}\in I_{n}(s,\chi)$. We are interested in its pullback via the
_standard embedding_ $i_{0}\colon H_{m}\times H_{m}\to H_{n}$ defined by

(2.4.1) $i_{0}\left(\begin{pmatrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{pmatrix},\begin{pmatrix}a_{2}&b_{2}\\\
c_{2}&d_{2}\end{pmatrix}\right)=\begin{pmatrix}a_{1}&&b_{1}&\\\
&a_{2}&&b_{2}\\\ c_{1}&&d_{1}&\\\ &c_{2}&&d_{2}\end{pmatrix}.$

######  Definition 2.4.1.

The _doubling kernel_ is the two-variable automorphic form

(2.4.2)
$D(g_{1},g_{2},s,\chi)=E(i_{0}(g_{1},g_{2}),s,\chi)\in\mathcal{A}(H_{m}\times
H_{m}).$

For any $T_{2}\in\operatorname{Herm}_{m}(F,\omega_{F})$ define the _new way
kernel_

(2.4.3)
$D_{\square,T_{2}}(g_{1},g_{2},s,\chi)=\int_{[\operatorname{Herm}_{m}]}D(g_{1},n(b)g_{2},s,\chi)\,\langle
T_{2},b\rangle_{\psi_{0}}\,db$

as the $T_{2}$-coefficient of the doubling kernel with respect to the variable
$g_{2}$.

As in (2.2.1), any automorphic form $f\in\mathcal{A}(H_{m}\times H_{m})$ in
two variables has a double Fourier expansion

$f(g_{1},g_{2})=\sum_{T_{1},T_{2}\in\operatorname{Herm}_{m}(F,\omega_{F})}f_{T_{1},T_{2}}(g_{1},g_{2}).$

The double Fourier coefficients of the doubling kernel (2.4.2) are related to
the Fourier coefficients of $E(g,s,\chi)$ by

$D_{T_{1},T_{2}}(g_{1},g_{2},s,\chi)=\sum_{T=\left(\begin{smallmatrix}T_{1}&*\\\
*&T_{2}\end{smallmatrix}\right)\in\operatorname{Herm}_{n}(F,\omega_{F})}E_{T}(i_{0}(g_{1},g_{2}),s,\chi),$

and so the new way kernel (2.4.3) has Fourier expansion

$\displaystyle D_{\square,T_{2}}(g_{1},g_{2},s,\chi)$
$\displaystyle=\sum_{T_{1}\in\operatorname{Herm}_{m}(F)}D_{T_{1},T_{2}}(g_{1},g_{2},s,\chi)$
$\displaystyle=\sum_{T=\left(\begin{smallmatrix}*&*\\\
*&T_{2}\end{smallmatrix}\right)\in\operatorname{Herm}_{n}(F,\omega_{F})}E_{T}(i_{0}(g_{1},g_{2}),s,\chi).$

As in §2.2, we can take geometric Fourier coefficients of the doubling kernel
(2.4.2) in the second variable. The result is, for every pair
$(\mathcal{E}_{2},a_{2})$ consisting of a rank $m$ vector bundle
$\mathcal{E}_{2}$ on $X^{\prime}$ and a hermitian map
$a_{2}\colon\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$, an
automorphic form

(2.4.4)
$D_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi)\in\mathcal{A}(H_{m})$

in the variable $g_{1}$ whose geometric Fourier coefficients are related to
those of the Eisenstein series $E(g,s,\chi)$ by

(2.4.5)
$D_{(\mathcal{E}_{1},a_{1}),(\mathcal{E}_{2},a_{2})}(s,\chi)=\sum_{a=\left(\begin{smallmatrix}a_{1}&*\\\
*&a_{2}\end{smallmatrix}\right)}E_{(\mathcal{E},a)}(s,\chi).$

On the right hand side $\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$, and
the sum is over all hermitian maps
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$ for which the composition

$\mathcal{E}_{i}\xrightarrow{\mathrm{inc.}}\mathcal{E}_{1}\oplus\mathcal{E}_{2}\xrightarrow{a}\sigma^{*}\mathcal{E}_{1}^{\vee}\oplus\sigma^{*}\mathcal{E}_{2}^{\vee}\xrightarrow{\mathrm{proj.}}\sigma^{*}\mathcal{E}_{i}^{\vee}$

agrees with $a_{i}$ for both $i\in\\{1,2\\}$.

######  Theorem 2.4.2 (The duplication formula).

If $f\in\pi$ is a $K_{m}$-fixed vector in an irreducible cuspidal automorphic
representation $\pi\subset\mathcal{A}(H_{m})$, then for all $g_{2}\in
H_{m}(\mathbb{A}_{F})$ we have

$\displaystyle\int_{[H_{m}]}D(g_{1},g_{2},s,\chi)f(g_{1})\,dg_{1}=\frac{L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)}{\mathscr{L}_{n}(s,\chi_{0})}\cdot\chi(\det(g_{2}))\cdot
f(g_{2}^{\dagger}).$

Here $g\mapsto g^{\dagger}$ is the automorphism of $H_{m}(\mathbb{A}_{F})$
defined on block matrices by

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}^{\dagger}=\begin{pmatrix}a&-b\\\
-c&d\end{pmatrix},$

where each block $a,b,c,d$ has size $m\times m$. The $L$-function on the right
hand side is the twisted base-change $L$-function as defined in [Clo91]. See
also [Min11].

######  Remark 2.4.3.

In the equality of Theorem 2.4.2, if one takes the Petersson inner product of
both sides against any cusp form $\breve{f}\in\mathcal{A}(H_{m})$ the
resulting formula is the usual _doubling formula_ of Piatetski-Shapiro and
Rallis [PSR87], extended to the unitary case by Li [Li92]; see also [Liu11].
By varying the auxiliary form $\breve{f}$, the stronger duplication formula
above would follow immediately from the doubling formula if one knew a priori
that the integral on the left hand side of Theorem 2.4.2 defined a _cuspidal_
automorphic form in the variable $g_{2}$. Thus the only real new information
in Theorem 2.4.2 is the cuspidality of the left hand side.

Before giving the proof, we restate Theorem 2.4.2 in the precise form in which
we will use it. If the automorphic form $f(g)$ in Theorem 2.4.2 has
$T^{\mathrm{th}}$ Fourier coefficient $f_{T}$, then the automorphic form
$f(g^{\dagger})$ has $T^{\mathrm{th}}$ Fourier coefficient
$f_{-T}(g^{\dagger})$. Thus taking the $T_{2}$-coefficient in both sides of
Theorem 2.4.2 yields the formula

(2.4.6)
$\displaystyle\int_{[H_{m}]}D_{\square,T_{2}}(g_{1},g_{2},s,\chi)f(g_{1})\,dg_{1}$
$\displaystyle=\frac{L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)}{\mathscr{L}_{n}(s,\chi_{0})}\cdot\chi(\det(g_{2}))\cdot
f_{-T_{2}}(g_{2}^{\dagger}).$

Equivalently, expressed in the language of geometric Fourier coefficients,

(2.4.7)
$\displaystyle\int_{[H_{m}]}D_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi)f(g_{1})\,dg_{1}$
$\displaystyle=\frac{L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)}{\mathscr{L}_{n}(s,\chi_{0})}\cdot\chi(\det(\mathcal{E}_{2}))\cdot
f_{(\mathcal{E}_{2},-a_{2})}.$

######  Remark 2.4.4.

Extending ideas of Böcherer [Bo83, Bo85] from symplectic groups to unitary
groups, Ikeda [Ike94] showed that the new way kernel (2.4.3) is a linear
combination of Eisenstein series and theta series for the hermitian space
determined by $T_{2}$. As explained in [GS20], if one substitutes this linear
combination into the integral on the left hand side of (2.4.6) and sets
$g_{2}=1$, the resulting formula is essentially the _new way integral_ of
[PSR88, Qin07]. This explains our choice of terminology for (2.4.3).

Now we turn to the proof of Theorem 2.4.2. In addition to the standard
embedding (2.4.1), we will make use of the twisted embedding

(2.4.8) $i\colon H_{m}\times H_{m}\to H_{n}$

defined by $i(g_{1},g_{2})=i_{0}(g_{1},g_{2}^{\dagger})$. Set

$\delta=\begin{pmatrix}&&I_{m}&\\\ &I_{m}&&\\\ -I_{m}&I_{m}&&\\\
&&I_{m}&I_{m}\end{pmatrix}\in H_{n}(F)\cap
K_{n}\subset\operatorname{GL}_{2n}(F^{\prime}).$

######  Lemma 2.4.5.

Consider the doubled space $W_{n}^{\Delta}=W_{m}\oplus W_{m}$ endowed with the
skew-hermitian form $h_{n}^{\Delta}=h_{m}\oplus-h_{m}$. There is an
isomorphism

$H_{n}^{\Delta}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}U(W_{n}^{\Delta})\cong
H_{n}$

identifying the canonical inclusion $i^{\Delta}\colon H_{m}\times H_{m}\to
H_{n}^{\Delta}$ with the twisted embedding (2.4.8), and identifying the
stabilizer $P_{n}^{\Delta}\subset H_{n}^{\Delta}$ of the diagonal Lagrangian
$\\{(x,x):x\in W_{m}\\}\subset W_{n}^{\Delta}$ with the conjugate
$\delta^{-1}P_{n}\delta\subset H_{n}$ of the standard Siegel parabolic.

###### Proof.

Using the standard basis
$e_{1},\ldots,e_{m},f_{1},\ldots,f_{m}\in(F^{\prime})^{2m}=W_{m}$, we define a
basis
$e^{\Delta}_{1},\ldots,e^{\Delta}_{n},f^{\Delta}_{1},\ldots,f^{\Delta}_{n}\in
W_{n}^{\Delta}$ by

$\displaystyle e^{\Delta}_{i}=\begin{cases}(e_{i},0)&\mbox{if }1\leq i\leq
m\\\ (0,e_{i-m})&\mbox{if }m<i\leq n\end{cases}\quad\mbox{and}\quad
f^{\Delta}_{i}=\begin{cases}(f_{i},0)&\mbox{if }1\leq i\leq m\\\
(0,-f_{i-m})&\mbox{if }m<i\leq n.\end{cases}$

The $F^{\prime}$-linear isomorphism
$W_{n}^{\Delta}\cong(F^{\prime})^{2n}=W_{n}$ determined by this basis is an
isometry, and the induced isomorphism of unitary groups has the desired
properties. ∎

######  Proposition 2.4.6.

For any cuspidal automorphic form $f\in\mathcal{A}(H_{m})$ and any standard
section $\Phi_{s}\in I_{n}(s,\chi)$ we have the equality

$\int_{[H_{m}]}E(i(g_{1},g_{2}),\Phi_{s})f(g_{1})\,dg_{1}=\int_{H_{m}(\mathbb{A}_{F})}\Phi_{s}(\delta
i(g_{1},g_{2}))f(g_{1})\,dg_{1}$

for all $g_{2}\in H_{m}(\mathbb{A}_{F})$.

###### Proof.

The proof follows a similar line of reasoning as in [PSR87], where it is shown
that the two sides of the desired equality agree after taking the Petersson
inner product against any cusp form in $\mathcal{A}(H_{m})$. See Remark 2.4.3.

By mild abuse of notation, we denote by

(2.4.9) $H_{m}\times H_{m}^{\prime}\subset H_{n}$

the image of the twisted embedding (2.4.8). In other words, abbreviate
$H_{m}=i(H_{m},1)$ and $H_{m}^{\prime}=i(1,H_{m})$. Using Lemma 2.4.5, one
sees that (2.4.9) is the stabilizer of an orthogonal splitting

$W_{n}=V\oplus V^{\prime}$

with $V$ isometric to $(W_{m},h_{m})$ and $V^{\prime}$ isometric to
$(W_{m},-h_{m})$.

Inserting the definition of the Eisenstein series and unfolding shows that

(2.4.10)
$\displaystyle\int_{[H_{m}]}E(i(g_{1},g_{2}),\Phi_{s})\,f(g_{1})\,dg_{1}$
$\displaystyle=\sum_{\gamma\in P_{n}(F)\backslash
H_{n}(F)/H_{m}(F)}\int_{I_{\gamma}(F)\backslash
H_{m}(\mathbb{A}_{F})}\Phi_{s}(\gamma i(g_{1},g_{2}))\,f(g_{1})\,dg_{1}$

where $I_{\gamma}=H_{m}\cap\gamma^{-1}P_{n}\gamma$. It follows from Lemma
2.4.5 that $I_{\delta}=1$, and we are reduced to proving the vanishing of the
integrals indexed by all $\gamma\neq\delta$.

In practice, this amounts to upgrading the analysis of right $(H_{m}\times
H_{m}^{\prime})$-orbits in $P_{n}\backslash H_{n}$ found in [PSR87, §2] to an
analysis of the orbits for the action of the subgroup $H_{m}\subset
H_{m}\times H_{m}^{\prime}$. Say that a subgroup $I\subset H_{m}$ is
_negligible_ if there exists a proper parabolic subgroup $Q\subset H_{m}$ with
unipotent radical $N$ such that $N\subset I\subset Q.$

######  Lemma 2.4.7.

Consider the action of the subgroup $H_{m}\subset H_{n}$ on the set of all
Lagrangian subspaces $L\subset W_{n}$.

  1. (1)

The set of Lagrangians $L\subset W_{n}$ satisfying $L\cap V=0$ is a single
$H_{m}$-orbit, and the stabilizer in $H_{m}$ of any such $L$ is trivial.

  2. (2)

If $L\subset W_{n}$ is a Lagrangian for which $L\cap V\neq 0$, its stabilizer
in $H_{m}$ is negligible.

###### Proof.

The proof of [PSR87, Lemma 2.1] establishes a bijection between the set of
Lagrangians $L\subset W_{n}$ and the set of triples $(Y,Y^{\prime},\alpha)$ in
which

  * •

$Y\subset V$ and $Y^{\prime}\subset V^{\prime}$ are totally isotropic
subspaces,

  * •

$\alpha\colon Y^{\perp}/Y\cong(Y^{\prime})^{\perp}/Y^{\prime}$ is an isometry.

The bijection sends a Lagrangian $L$ to the triple defined by $Y=L\cap V$ and
$Y^{\prime}=L\cap V^{\prime}$ together with the isometry characterized by

$\alpha(y)=y^{\prime}\iff y+y^{\prime}\in L$

for all $y\in Y^{\perp}$ and $y^{\prime}\in(Y^{\prime})^{\perp}$. Under this
bijection, the action of the subgroup $H_{m}\times H_{m}^{\prime}\subset
H_{n}$ on the set of Lagrangians translates to an action on triples: an
element $(h,h^{\prime})$ acts by

$(Y,Y^{\prime},\alpha)\mapsto(hY,h^{\prime}Y^{\prime},h^{\prime}\alpha
h^{-1}),$

in which $h^{\prime}\alpha h^{-1}$ is the composition

$(hY)^{\perp}/hY\xrightarrow{h^{-1}}Y^{\perp}/Y\xrightarrow{\alpha}(Y^{\prime})^{\perp}/Y^{\prime}\xrightarrow{h^{\prime}}(h^{\prime}Y^{\prime})^{\perp}/h^{\prime}Y^{\prime},$

and we are identifying $H_{m}\cong\mathrm{U}(V)$ and
$H_{m}^{\prime}\cong\mathrm{U}(V^{\prime})$.

Consider the set of all Lagrangians $L\subset W_{n}$ for which $L\cap V=0$.
These correspond to triples of the form $(0,0,\alpha)$, where $\alpha\colon
V\cong V^{\prime}$ is an isometry. The action of $H_{m}\cong\mathrm{U}(V)$ on
this set is simply transitive, proving (1).

To prove (2), suppose $L$ corresponds to a triple $(Y,Y^{\prime},\alpha)$ with
$Y\neq 0$. The stabilizer of $L$ in $H_{m}$ is

$\displaystyle\\{h\in H_{m}:(hY,Y^{\prime},\alpha
h^{-1})=(Y,Y^{\prime},\alpha)\\}$ $\displaystyle=\\{h\in H_{m}:hY=Y\mbox{ and
}h=\mathrm{id}\mbox{ on }Y^{\perp}/Y\\},$

which is contained in the proper parabolic subgroup stabilizing $Y$, and
contains its unipotent radical. Hence this stabilizer is negligible. ∎

To complete the proof of Proposition 2.4.6, we make use of the bijection

$P_{n}(F)\backslash H_{n}(F)\cong\\{\mbox{Lagrangian subspaces }L\subset
W_{n}\\}.$

If $\gamma\in P_{n}(F)\backslash H_{n}(F)$ corresponds to $L\subset W_{n}$,
then $I_{\gamma}\subset H_{m}$ is equal to the stabilizer of $L$ in $H_{m}$.
By the lemma such an $I_{\gamma}$ is either trivial or negligible, and the
$\gamma$ for which $I_{\gamma}=1$ form a single orbit under $H_{m}(F)$.

Having noted above that $I_{\delta}=1$, we have now proved that
$I_{\gamma}\subset H_{m}$ is negligible for any double coset $\gamma\in
P_{n}(F)\backslash H_{n}(F)/H_{m}(F)$ with $\gamma\neq\delta$. In other words,
there exists a proper parabolic subgroup $Q_{\gamma}$ with unipotent radical
$N_{\gamma}$ satisfying $N_{\gamma}\subset I_{\gamma}\subset Q_{\gamma}.$ The
integral

$\int_{[N_{\gamma}]}\Phi_{s}(\gamma
i(g_{1},g_{2}))\,f(g_{1})\,dg_{1}=\Phi_{s}(\gamma
i(1,g_{2}))\int_{[N_{\gamma}]}f(g_{1})\,dg_{1}$

vanishes by the cuspidality of $f$, and therefore so does the integral indexed
by $\gamma$ on the right hand side of (2.4.10). ∎

###### Proof of Theorem 2.4.2.

Proposition 2.4.6 implies the first equality in

(2.4.11)
$\displaystyle\chi(\det(g_{2}))^{-1}\int_{[H_{m}]}D(g_{1},g_{2}^{\dagger},s,\chi)f(g_{1})\,dg_{1}$
$\displaystyle=\chi(\det(g_{2}))^{-1}\int_{H_{m}(\mathbb{A}_{F})}\Phi^{\circ}_{s}(\delta
i(g_{1},g_{2}))f(g_{1})\,dg_{1}$
$\displaystyle=\chi(\det(g_{2}))^{-1}\int_{H_{m}(\mathbb{A}_{F})}\Phi^{\circ}_{s}(\delta
i(g_{2}g_{1},g_{2}))f(g_{2}g_{1})\,dg_{1}$
$\displaystyle=\int_{H_{m}(\mathbb{A}_{F})}\Phi^{\circ}_{s}(\delta
i(g_{1},1))f(g_{2}g_{1})\,dg_{1}.$

For the final equality we have used the observation that the composition

$H_{m}(\mathbb{A}_{F})\xrightarrow{g\mapsto\delta
i(g,g)\delta^{-1}}P_{n}(\mathbb{A}_{F})\xrightarrow{\eqref{parabolic
character}}\mathbb{A}_{F^{\prime}}^{\times}$

agrees with the determinant on $H_{m}(\mathbb{A}_{F})$, a consequence of Lemma
2.4.5.

Viewing (2.4.11) as an automorphic form in the variable $g_{2}\in
H_{m}(\mathbb{A}_{F})$, it is a $K_{m}$-fixed vector in the space of $\pi$.
Indeed, the second integral in (2.4.11) is right $K_{m}$-invariant because
$\Phi_{s}^{\circ}$ is right $K_{n}$-invariant, and $i(K_{m},K_{m})\subset
K_{n}$. The final integral in (2.4.11) is a linear combination of right
translates of $f$, and hence lies in $\pi$. Up to scaling, $f\in\pi$ is the
unique $K_{m}$-fixed vector, and we deduce that

$\int_{H_{m}(\mathbb{A}_{F})}\Phi^{\circ}_{s}(\delta
i(g_{1},1))\,f(g_{2}g_{1})\,dg_{1}=c(s)\cdot f(g_{2})$

for some $c(s)$ independent of $g_{2}$.

To compute the scalar $c(s)$, let $\breve{f}\in\breve{\pi}$ be the
$K_{m}$-fixed vector in the contragredient representation of $\pi$, normalized
by $\langle f,\breve{f}\rangle=1$. Pairing both sides of the above equality
against $\breve{f}$ results in

$\int_{H_{m}(\mathbb{A}_{F})}\Phi^{\circ}_{s}(\delta
i(h,1))\,\langle\pi(h)f,\breve{f}\rangle\,dh=c(s).$

The left hand side is the well-known doubling zeta integral studied in [Li92,
PSR87], see also [HKS96, Liu11], and we deduce

$c(s)=\frac{L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)}{\mathscr{L}_{n}(s,\chi_{0})}.$

This completes the proof of the duplication formula. ∎

###  2.5. Degenerate coefficients in low rank

We return to the assumption that the unramified Hecke character
$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$ satisfies
(2.3.1).

We are interested in degenerate coefficients of the Eisenstein series
$E_{T}(g,s,\chi)$ on $H_{2}(\mathbb{A}_{F})$ determined by the normalized
spherical section $\Phi_{s}^{\circ}\in I_{2}(s,\chi)$. More precisely, for a
fixed

$T=\begin{pmatrix}0&0\\\
0&t\end{pmatrix}\in\operatorname{Herm}_{2}(F,\omega_{F})$

with $t\in\operatorname{Herm}_{1}(F,\omega_{F})$ nonzero, we will relate the
Fourier coefficient

$\displaystyle E_{T}(g,s,\chi)$
$\displaystyle=\int_{[\operatorname{Herm}_{2}]}E(n(b)g,s,\chi)\,\langle
T,b\rangle_{\psi_{0}}\,db$

to the $t^{\mathrm{th}}$ coefficient of the Eisenstein series

$\underline{E}(h,s,\chi)=\sum_{\gamma\in P_{1}(F)\backslash
H_{1}(F)}\underline{\Phi}_{s}^{\circ}(\gamma h)$

on $H_{1}(\mathbb{A}_{F})$ determined by the normalized spherical section
$\underline{\Phi}_{s}^{\circ}\in I_{1}(s,\chi)$.

######  Proposition 2.5.1.

Fix $h\in H_{1}(\mathbb{A}_{F})$ and

$m(\alpha)=\left(\begin{smallmatrix}\alpha\\\
&\sigma(\alpha^{-1})\end{smallmatrix}\right)\in H_{1}(\mathbb{A}_{F})$

with $\alpha\in\mathbb{A}_{F^{\prime}}^{\times}$. Recalling the standard
embedding $i_{0}\colon H_{1}\times H_{1}\to H_{2}$ from (2.4.1), we have the
equality

$\displaystyle q^{2s\deg(\omega_{X})}\mathscr{L}_{2}(s,\chi_{0})\cdot
E_{T}\big{(}i_{0}(m(\alpha),h),s,\chi\big{)}$
$\displaystyle=\chi(\alpha)|\alpha|_{F^{\prime}}^{1+s}q^{2s\deg(\omega_{X})}\mathscr{L}_{2}(s,\chi_{0})\cdot\underline{E}_{t}\left(h,\frac{1}{2}+s,\chi\right)$
$\displaystyle\quad+\chi(\alpha)|\alpha|_{F^{\prime}}^{1-s}q^{-2s\deg(\omega_{X})}\mathscr{L}_{2}(-s,\chi_{0})\cdot\underline{E}_{t}\left(h,\frac{1}{2}-s,\chi\right).$

Note that the right hand side is invariant under $s\mapsto-s$, in accordance
with the functional equation (2.3.8) satisfied by the left hand side.

###### Proof.

By [GS19, Lemma 5.2.5], for all $g\in H_{2}(\mathbb{A}_{F})$ we have

$\displaystyle E_{T}(g,s,\chi)$
$\displaystyle=\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}\Phi_{s}^{\circ}\left(i_{0}\left(1,w_{1}^{-1}n(b)\right)\cdot
g\right)\,\langle t,b\rangle_{\psi_{0}}\,db$ (2.5.1)
$\displaystyle\quad+\int_{\operatorname{Herm}_{2}(\mathbb{A}_{F})}\Phi_{s}^{\circ}\left(w_{2}^{-1}n(b)g\right)\,\langle
T,b\rangle_{\psi_{0}}\,db.$

The first integral in (2.5.1) is easy to deal with. After noting that

(2.5.2) $\underline{\Phi}_{s+1/2}^{\circ}(h)=\Phi_{s}^{\circ}(i_{0}(1,h))$

for all $h\in H_{1}(\mathbb{A}_{F})$, we find

$\displaystyle\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}\Phi^{\circ}_{s}\left(i_{0}\left(m(\alpha),w_{1}^{-1}n(b)h\right)\right)\,\langle
t,b\rangle_{\psi_{0}}\,db$
$\displaystyle=\chi(\alpha)|\alpha|_{F^{\prime}}^{s+1}\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}\Phi^{\circ}_{s}\left(i_{0}\left(1,w_{1}^{-1}n(b)h\right)\right)\,\langle
t,b\rangle_{\psi_{0}}\,db$
$\displaystyle=\chi(\alpha)|\alpha|_{F^{\prime}}^{s+1}\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}\underline{\Phi}^{\circ}_{s+1/2}(w_{1}^{-1}n(b)h)\,\langle
t,b\rangle_{\psi_{0}}\,db$ (2.5.3)
$\displaystyle=\chi(\alpha)|\alpha|_{F^{\prime}}^{s+1}\cdot\underline{E}_{t}(h,s+1/2).$

The second integral in (2.5.1) is not quite so elementary, but is made
explicit by the following lemma.

######  Lemma 2.5.2.

For all $h\in H_{1}(\mathbb{A}_{F})$ we have

$\displaystyle\int_{\operatorname{Herm}_{2}(\mathbb{A}_{F})}\Phi^{\circ}_{s}\left(w_{2}^{-1}n(b)\cdot
i_{0}(m(\alpha),h)\right)\,\langle T,b\rangle_{\psi_{0}}\,db$
$\displaystyle=\chi(\alpha)|\alpha|_{F^{\prime}}^{-s+1}q^{-4s\deg(\omega_{X})}\frac{\mathscr{L}_{2}(-s,\chi_{0})}{\mathscr{L}_{2}(s,\chi_{0})}\cdot\underline{E}_{t}(h,-s+1/2,\chi),$

###### Proof.

Define a subgroup of $\operatorname{Herm}_{2}(\mathbb{A}_{F})$ by

$U=\left\\{\begin{pmatrix}x&z\\\
\sigma(z)&0\end{pmatrix}:x\in\mathbb{A}_{F}\mbox{ and
}z\in\mathbb{A}_{F^{\prime}}\right\\}.$

There is an obvious decomposition
$\operatorname{Herm}_{2}(\mathbb{A}_{F})\cong
U\times\operatorname{Herm}_{1}(\mathbb{A}_{F})$, where we view
$\operatorname{Herm}_{1}(\mathbb{A}_{F})\subset\operatorname{Herm}_{2}(\mathbb{A}_{F})$
as the subgroup of matrices with all entries outside the lower right corner
equal to $0$. The choices of Haar measures on
$\operatorname{Herm}_{1}(\mathbb{A}_{F})$ and
$\operatorname{Herm}_{2}(\mathbb{A}_{F})$ from Remark 2.1.1 then determine a
Haar measure on $U$ compatible with this product decomposition.

Consider the function of $h\in H_{1}(\mathbb{A}_{F})$ defined by

$f^{\alpha}_{s}(h)=\int_{U}\Phi^{\circ}_{s+1/2}\left(w_{2}^{-1}n(u)\cdot
i_{0}(m(\alpha),w_{1}^{-1}h)\right)\,du.$

As in [GS19, Lemma 5.2.6(i)], this function satisfies $f^{\alpha}_{s}\in
I_{1}(s,\chi)$, allowing us to form the associated Eisenstein series
$\underline{E}(h,f^{\alpha}_{s})$ on $H_{1}(\mathbb{A}_{F})$. It is clear from
the definition that $f^{\alpha}_{s}$ is right invariant under $K_{1}$, and so
we must have

$f^{\alpha}_{s}(h)=c(s)\cdot\underline{\Phi}_{s}^{\circ}(h)$

for some function $c(s)$ independent of $h\in H_{1}(\mathbb{A}_{F})$.

We begin by computing

$\displaystyle\int_{\operatorname{Herm}_{2}(\mathbb{A}_{F})}\Phi^{\circ}_{s}\left(w_{2}^{-1}n(b)\cdot
i_{0}(m(\alpha),h)\right)\,\langle T,b\rangle_{\psi_{0}}\,db$
$\displaystyle=\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}\int_{U}\Phi^{\circ}_{s}\left(w_{2}^{-1}n(u)n\begin{pmatrix}0&0\\\
0&b^{\prime}\end{pmatrix}\cdot i_{0}(m(\alpha),h)\right)\,\langle
T,b^{\prime}\rangle_{\psi_{0}}\,du\,db^{\prime}$
$\displaystyle=\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}\int_{U}\Phi^{\circ}_{s}\left(w_{2}^{-1}n(u)\cdot
i_{0}\left(m(\alpha),n(b^{\prime})h\right)\right)\,\langle
T,b^{\prime}\rangle_{\psi_{0}}\,du\,db^{\prime}$
$\displaystyle=\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}f^{\alpha}_{s-1/2}(w_{1}^{-1}n(b^{\prime})h)\,\langle
T,b^{\prime}\rangle_{\psi_{0}}\,db^{\prime}.$

By (2.3.3), the final integral is the $t^{\mathrm{th}}$ coefficient of
$\underline{E}(h,f^{\alpha}_{s-1/2})$, and hence

(2.5.4)
$\displaystyle\int_{\operatorname{Herm}_{2}(\mathbb{A}_{F})}\Phi^{\circ}_{s}\left(w_{2}^{-1}n(b)\cdot
i_{0}(m(\alpha),h)\right)\,\langle T,b\rangle_{\psi_{0}}\,db$
$\displaystyle=c(s-1/2)\cdot\underline{E}_{t}(h,s-1/2,\chi).$

It remains to compute $c(s)$. Recalling the intertwining operator (2.3.5), the
same calculation as above (but with $T=0$) shows that

$\displaystyle(M_{2}(s)\Phi^{\circ}_{s})(i_{0}(m(\alpha),h))$
$\displaystyle=\int_{\operatorname{Herm}_{2}(\mathbb{A}_{F})}\Phi^{\circ}_{s}\left(w_{2}^{-1}n(b)\cdot
i_{0}(m(\alpha),h)\right)\,db$
$\displaystyle=\int_{\operatorname{Herm}_{1}(\mathbb{A}_{F})}f^{\alpha}_{s-1/2}(w_{1}^{-1}n(b^{\prime})h)\,db^{\prime}$
$\displaystyle=(M_{1}(s-1/2)f^{\alpha}_{s-1/2})(h).$

On the other hand

$\displaystyle(M_{2}(s)\Phi^{\circ}_{s})(i_{0}(m(\alpha),h))$
$\displaystyle=\chi(\alpha)|\alpha|_{F^{\prime}}^{-s+1}(M_{2}(s)\Phi^{\circ}_{s})(i_{0}(1,h))$
$\displaystyle\stackrel{{\scriptstyle\eqref{spherical
intertwining}}}{{=}}\chi(\alpha)|\alpha|_{F^{\prime}}^{-s+1}q^{-4s\deg(\omega_{X})}\frac{\mathscr{L}_{2}(-s,\chi_{0})}{\mathscr{L}_{2}(s,\chi_{0})}\Phi^{\circ}_{-s}(i_{0}(1,h))$
$\displaystyle\stackrel{{\scriptstyle\eqref{spherical genus
drop}}}{{=}}\chi(\alpha)|\alpha|_{F^{\prime}}^{-s+1}q^{-4s\deg(\omega_{X})}\frac{\mathscr{L}_{2}(-s,\chi_{0})}{\mathscr{L}_{2}(s,\chi_{0})}\underline{\Phi}_{-s+1/2}^{\circ}(h).$

Combining these shows that

$M_{1}(s-1/2)f^{\alpha}_{s-1/2}=\chi(\alpha)|\alpha|_{F^{\prime}}^{-s+1}q^{-4s\deg(\omega_{X})}\frac{\mathscr{L}_{2}(-s,\chi)}{\mathscr{L}_{2}(s,\chi)}\underline{\Phi}_{-s+1/2}^{\circ},$

and comparing with

$M_{1}(s-1/2)\underline{\Phi}_{s-1/2}^{\circ}\stackrel{{\scriptstyle\eqref{spherical
intertwining}}}{{=}}q^{(-2s+1)\deg(\omega_{X})}\frac{\mathscr{L}_{1}(-s+1/2,\chi_{0})}{\mathscr{L}_{1}(s-1/2,\chi_{0})}\cdot\underline{\Phi}^{\circ}_{-s+1/2}$

one deduces

$c(s-1/2)=\chi(\alpha)|\alpha|_{F^{\prime}}^{-s+1}q^{-(2s+1)\deg(\omega_{X})}\frac{\mathscr{L}_{2}(-s,\chi_{0})}{\mathscr{L}_{2}(s,\chi_{0})}\frac{\mathscr{L}_{1}(s-1/2,\chi_{0})}{\mathscr{L}_{1}(-s+1/2,\chi_{0})}.$

To complete the proof of the lemma, substitute this formula into (2.5.4) and
use the functional equation (2.3.8) for $\underline{E}(h,s,\chi)$. ∎

To complete the proof of Proposition 2.5.1, take $g=i_{0}(m(a),h)$ in the
equality (2.5.1), replace the first integral with the expression from (2.5.3),
and replace the second integral with the expression from Lemma 2.5.2. ∎

We now translate Proposition 2.5.1 into the language of geometric Fourier
coefficients from §2.2. Fix two line bundles
$\mathcal{E}_{1},\mathcal{E}_{2}\in\mathrm{Pic}(X^{\prime})$ and an injective
hermitian morphism
$a_{2}\colon\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$. This data
determines a pair $(\mathcal{E},a)$, in which

$\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}\xrightarrow{a=0\oplus
a_{2}}\sigma^{*}\mathcal{E}_{1}^{\vee}\oplus\sigma^{*}\mathcal{E}_{2}^{\vee}=\sigma^{*}\mathcal{E}^{\vee}.$

Proposition 2.5.1 is equivalent to the relation

(2.5.5) $\displaystyle q^{2s\deg(\omega_{X})}\mathscr{L}_{2}(s,\chi_{0})\cdot
E_{(\mathcal{E},a)}(s,\chi)$
$\displaystyle=\chi(\mathcal{E}_{1})q^{(1+s)\deg(\mathcal{E}_{1})}q^{2s\deg(\omega_{X})}\mathscr{L}_{2}(s,\chi_{0})\cdot\underline{E}_{(\mathcal{E}_{2},a_{2})}\left(\frac{1}{2}+s,\chi\right)$
$\displaystyle\quad+\chi(\mathcal{E}_{1})q^{(1-s)\deg(\mathcal{E}_{1})}q^{-2s\deg(\omega_{X})}\mathscr{L}_{2}(-s,\chi_{0})\cdot\underline{E}_{(\mathcal{E}_{2},a_{2})}\left(\frac{1}{2}-s,\chi\right)$

for all pairs $(\mathcal{E},a)$ of the above form.

##  3\. Cycles on moduli spaces of shtukas

Fix an integer $n\geq 1$. Now we turn to the study of special cycle classes on
the moduli space $\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ of unitary shtukas,
as in the introduction.

First we recall the arithmetic Siegel-Weil formula of Feng-Yun-Zhang, relating
the degrees of special $0$-cycles on $\operatorname{Sht}^{r}_{\mathrm{U}(n)}$
to the Siegel Eisenstein series from §2. In Conjecture 3.1.2 we propose a
modest extension of this formula. This conjectural extension seems to be the
minimum extra information that one needs to extract a Gross-Zagier style
intersection formula from the arithmetic Siegel-Weil formula.

In §3.2 we prove Conjecture 3.1.2 in a simple case, for special $0$ cycles on
$\operatorname{Sht}^{r}_{\mathrm{U}(2)}$ of a particular type. Finally, in
§3.3, we put everything together to prove Theorems 1.2.2 and 1.3.3 of the
introduction.

###  3.1. The arithmetic Siegel-Weil formula

We consider the special cycle classes (1.1.1) of maximal codimension in
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$. As in [FYZa, §7], let

$\mathcal{Z}^{r}_{\mathcal{E}}(a)\to\operatorname{Sht}_{\mathrm{U}(n)}^{r}$

be the naive special cycle determined by a rank $n$ vector bundle
$\mathcal{E}$ on $X^{\prime}$, and a hermitian morphism
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$.

The following _arithmetic Siegel-Weil formula_ is the main result of [FYZa].
It is the function field analogue of a formula conjectured by Kudla-Rapoport
[KR14], and proved by Li-Zhang [LZ22], relating the degrees of special cycles
on (integral models of) unitary Shimura varieties to derivatives of Siegel
Eisenstein series on quasi-split unitary groups.

######  Theorem 3.1.1 (Feng-Yun-Zhang).

Let $(\mathcal{E},a)$ be as above. If the hermitian morphism $a$ is injective,
then the naive special cycle $\mathcal{Z}^{r}_{\mathcal{E}}(a)$ is proper over
$k$, and the special $0$-cycle class
$[\mathcal{Z}^{r}_{\mathcal{E}}(a)]\in\mathrm{Ch}_{c}^{rn}(\operatorname{Sht}_{\mathrm{U}(n)}^{r})$
satisfies

(3.1.1) $\deg\,[\mathcal{Z}^{r}_{\mathcal{E}}(a)]=\frac{1}{(\log
q)^{r}}\cdot\frac{q^{\frac{n}{2}d(\mathcal{E})}}{\chi(\det(\mathcal{E}))}\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\widetilde{E}_{(\mathcal{E},a)}(s,\chi)$

for any unramified Hecke character

$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$

whose restriction to $\mathbb{A}_{F}^{\times}$ is $\eta^{n}$. On the right
hand side,

  * •

$d(\mathcal{E})$ is defined by (1.2.1),

  * •

$\widetilde{E}(g,s,\chi)$ is the renormalized Eisenstein series on
$H_{n}(\mathbb{A}_{F})$ of (2.3.7),

  * •

$\widetilde{E}_{(\mathcal{E},a)}(s,\chi)$ is its geometric Fourier coefficient
in the sense of §2.2.

###### Proof.

The properness assertion is [FYZa, Proposition 7.13]. The equality (3.1.1)
follows by combining the equalities

$\deg([\mathcal{Z}^{r}_{\mathcal{E}}(a)])=\frac{1}{(\log
q)^{r}}\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}q^{sd(\mathcal{E})}\mathrm{Den}(q^{-2s},\mathrm{coker}(a))$

and

$\mathrm{Den}(q^{-2s},\mathrm{coker}(a))=\frac{q^{(\frac{n}{2}-s)d(\mathcal{E})}}{\chi(\det(\mathcal{E}))}\widetilde{E}_{(\mathcal{E},a)}(s,\chi)$

of [FYZb, §9.3 and §9.5]. ∎

As explained in the introduction, properness of the naive special cycle
$\mathcal{Z}^{r}_{\mathcal{E}}(a)$ is needed to make sense of the degree
(1.1.5) appearing in the theorem. Because lack of properness is the only
obstruction to formulating the equality (3.1.1) in general (i.e. without the
assumption that $a$ is injective), we make the following conjecture.

######  Conjecture 3.1.2.

The equality (3.1.1) holds whenever the naive special cycle
$\mathcal{Z}^{r}_{\mathcal{E}}(a)$ is proper over $k$.

Proving this in any case with $a$ not injective seems to to require methods
substantially different from those that go into the proof of Theorem 3.1.1.

###  3.2. A low rank case of Conjecture 3.1.2

We will prove Conjecture 3.1.2 when $\mathcal{E}$ is a vector bundle on
$X^{\prime}$ of rank $n=2$, and $a\colon\mathcal{E}\to\mathcal{E}^{\vee}$ is a
(not necessarily injective) hermitian morphism of a particular type. Some of
the preparatory results needed for this result work in greater generality, so
for the moment we work with any positive integers $m\leq n$.

######  Lemma 3.2.1.

Let $\mathscr{X}$ be a connected quasi-smooth derived Artin stack over $k$
with classical truncation $i\colon X\hookrightarrow\mathscr{X}$. If $X$ is a
local complete intersection over $k$ of dimension
$\dim(X)=\operatorname{vd}(\mathscr{X})$, where
$\operatorname{vd}(\mathscr{X})$ denotes the virtual dimension of
$\mathscr{X}$, then $\mathscr{X}$ is classical and $i\colon
X\hookrightarrow\mathscr{X}$ is an isomorphism.

###### Proof.

The derived pullback of $i$ along any derived affine smooth chart
$\operatorname{Spec}(R)\to\mathscr{X}$ is the canonical inclusion

$\operatorname{Spec}(\pi_{0}(R))\hookrightarrow\operatorname{Spec}(R),$

where $\pi_{0}(R)$ is a local complete intersection and
$\dim(\pi_{0}(R))=\operatorname{vd}(R)$. As the hypotheses and conclusion of
the lemma are both smooth-local in $\mathscr{X}$, we may assume
$\mathscr{X}=\operatorname{Spec}(R)$ and $X=\operatorname{Spec}(\pi_{0}(R))$.

In this setting, the canonical map $R\twoheadrightarrow\pi_{0}(R)$
corresponding to $i$ induces an isomorphism on $\pi_{0}$, so the fiber of this
map is 1-connective. By [Lur, Corollary 25.3.6.4], the relative cotangent
complex $\mathbb{L}_{\pi_{0}(R)/R}=\mathbb{L}_{i}$ is 2-connective, i.e. of
Tor-amplitude in degrees $\leq-2$. On the other hand, both $R$ and
$\pi_{0}(R)$ are quasi-smooth, so $\mathbb{L}_{R/k}$ and
$\mathbb{L}_{\pi_{0}(R)/k}$ are perfect of Tor-amplitude in degrees $\geq-1$,
which implies that $\mathbb{L}_{i}$ is perfect of Tor-amplitude in degrees
$\geq-2$. Combining these two observations, it follows that
$\mathbb{L}_{i}\simeq M[2]$ for some locally free $\pi_{0}(R)$-module $M$.
Finally,

$\operatorname{rank}(M)=\chi(\mathbb{L}_{i})=\chi(\mathbb{L}_{\pi_{0}(R)/k})-\chi(\mathbb{L}_{R/k})=\dim(\pi_{0}(R))-\operatorname{vd}(R)=0,$

so $M$ is the zero module, and $\mathbb{L}_{i}\simeq 0$. As $i$ also clearly
induces an isomorphism on classical truncations, it follows by [Lur, Corollary
25.3.6.6] that $i$ is an isomorphism. ∎

######  Lemma 3.2.2.

Suppose $\mathcal{E}$ is rank $m$ vector bundle on $X^{\prime}$, and
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$ is a hermitian morphism.
If the naive special cycle $\mathcal{Z}^{r}_{\mathcal{E}}(a)$ on
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ is a local complete intersection of
dimension $r(n-m)$, then the cycle class

$[\mathcal{Z}_{\mathcal{E}}^{r}(a)]\in\operatorname{Ch}_{r(n-m)}(\mathcal{Z}_{\mathcal{E}}^{r}(a))$

is the fundamental class of $\mathcal{Z}_{\mathcal{E}}^{r}(a)$ in the usual
sense.

###### Proof.

The content of [FYZb, Theorem 6.6, Remark 6.7] is that the naive special cycle
$\mathcal{Z}^{r}_{\mathcal{E}}(a)$ can be realized as the classical truncation
of a quasi-smooth derived Artin stack $\mathscr{Z}^{r}_{\mathcal{E}}(a)$ of
virtual dimension $r(n-m)$, in such a way that the canonical isomorphism

$\operatorname{Ch}_{r(n-m)}(\mathcal{Z}^{r}_{\mathcal{E}}(a))\cong\operatorname{Ch}_{r(n-m)}(\mathscr{Z}^{r}_{\mathcal{E}}(a))$

identifies the cycle class $[\mathcal{Z}_{\mathcal{E}}^{r}(a)]$ of [FYZb,
Definition 4.8] with the virtual fundamental class of
$\mathscr{Z}^{r}_{\mathcal{E}}(a)$. The claim follows immediately from this,
as $\mathcal{Z}^{r}_{\mathcal{E}}(a)\cong\mathscr{Z}^{r}_{\mathcal{E}}(a)$ by
Lemma 3.2.1. ∎

######  Lemma 3.2.3.

The canonical morphism
$\operatorname{Sht}_{\mathrm{U}(1)}^{r}\to(X^{\prime})^{r}$ recording the legs
is finite étale.

###### Proof.

The cartesian diagram [FYZb, (2.2)] expresses this morphism as a pullback of
the (finite étale) Lang isogeny $\operatorname{Prym}\to\operatorname{Prym}$,
where $\operatorname{Prym}$ is the kernel of the norm map
$\operatorname{Pic}_{X^{\prime}}\to\operatorname{Pic}_{X}$. ∎

######  Lemma 3.2.4.

Suppose $\mathcal{E}$ is rank $m$ vector bundle on $X^{\prime}$, and
$a\colon\mathcal{E}\cong\sigma^{*}\mathcal{E}^{\vee}$ is a hermitian
isomorphism. Let
$\mathcal{Z}^{r}_{\mathcal{E}}(a)\to\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ be
the associated naive special cycle. There is an isomorphism of $k$-stacks

$\operatorname{Sht}_{\mathrm{U}(n-m)}^{r}\cong\mathcal{Z}^{r}_{\mathcal{E}}(a).$

###### Proof.

For a $k$-scheme $S$, an object of
$\operatorname{Sht}_{\mathrm{U}(n-m)}^{r}(S)$ is a chain of modifications

(3.2.1)
$\mathcal{G}_{0}\dashrightarrow\mathcal{G}_{1}\dashrightarrow\cdots\dashrightarrow\mathcal{G}_{r}\cong{}^{\tau}\mathcal{G}_{0}$

of vector bundles rank $n-m$ on $X^{\prime}_{S}$, endowed with compatible
hermitian isomorphisms
$h_{i}\colon\mathcal{G}_{i}\cong\sigma^{*}\mathcal{G}_{i}^{\vee}$, and
satisfying some extra conditions [FYZa, Definition 6.6]. Abbreviating
$\mathcal{F}_{i}=\mathcal{E}_{S}\oplus\mathcal{G}_{i}$ for the orthogonal
direct sum, the chain of modifications

(3.2.2)
$\mathcal{F}_{0}\dashrightarrow\mathcal{F}_{1}\dashrightarrow\cdots\dashrightarrow\mathcal{F}_{r}\cong{}^{\tau}\mathcal{F}_{0}$

defines an object of $\operatorname{Sht}_{\mathrm{U}(n)}^{r}(S)$. The
canonical inclusions $\mathcal{E}_{S}\to\mathcal{F}_{i}$ define a lift of this
point to $\mathcal{Z}^{r}_{\mathcal{E}}(a)$, and this defines a morphism

$\operatorname{Sht}_{\mathrm{U}(n-m)}^{r}\to\mathcal{Z}^{r}_{\mathcal{E}}(a).$

Conversely, if one starts with a chain of modifications (3.2.2) defining an
object of $\operatorname{Sht}_{\mathrm{U}(n)}^{r}(S)$, and a compatible family
of isometric embeddings $\mathcal{E}_{S}\to\mathcal{F}_{i}$ defining a lift to
$\mathcal{Z}_{\mathcal{E}}(a)(S)$, then
$\mathcal{E}_{S}\subset\mathcal{F}_{i}$ splits off an orthogonal complement
$\mathcal{G}_{i}=\mathcal{E}_{S}^{\perp}$. These define a chain of
modifications (3.2.1), and hence an object of
$\operatorname{Sht}_{\mathrm{U}(n-m)}^{r}(S)$. This provides an inverse to the
above morphism. ∎

Recall from [FYZb, §4.3] that the moduli space
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$ carries a collection of _tautological
line bundles_
$\ell_{1},\ldots,\ell_{r}\in\operatorname{Pic}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})$,
as well as a Chern class map

$c_{1}\colon\operatorname{Pic}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})\to\operatorname{Ch}^{1}(\operatorname{Sht}^{r}_{\mathrm{U}(n)}).$

Denote by $p_{1},\ldots,p_{r}\colon\operatorname{Sht}^{r}_{\mathrm{U}(n)}\to
X^{\prime}$ the morphisms recording the legs.

######  Lemma 3.2.5.

Suppose $\mathcal{E}_{1}$ is a line bundle on $X^{\prime}$, $\mathcal{E}_{2}$
is a vector bundle of rank $n-1$, and
$a_{2}\colon\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$ is an
injective hermitian morphism. If we endow the rank $n$ vector bundle
$\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$ with the hermitian morphism

$a=\begin{pmatrix}0&0\\\
0&a_{2}\end{pmatrix}\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee},$

then there is an isomorphism of
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$-stacks
$\mathcal{Z}^{r}_{\mathcal{E}}(a)\cong\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})$
such that the induced isomorphism of Chow groups identifies

(3.2.3)
$[\mathcal{Z}^{r}_{\mathcal{E}}(a)]\in\operatorname{Ch}_{0}(\mathcal{Z}^{r}_{\mathcal{E}}(a))$

with

$\left(\prod_{i=1}^{r}c_{1}(p_{i}^{*}\sigma^{*}\mathcal{E}_{1}^{-1}\otimes\ell_{i})\right)\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\in\operatorname{Ch}_{0}(\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})).$

The intersection pairing

$\operatorname{Ch}_{r(n-1)}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})\times\operatorname{Ch}_{r}(\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2}))\to\operatorname{Ch}_{0}(\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2}))$

on the left is that of [FYZa, §7.7].

###### Proof.

As in the proof of Lemma 3.2.4, an object of
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})(S)$ is a chain of modifications

(3.2.4)
$\mathcal{F}_{0}\dashrightarrow\mathcal{F}_{1}\dashrightarrow\cdots\dashrightarrow\mathcal{F}_{r}\cong{}^{\tau}\mathcal{F}_{0}$

of rank $n$ vector bundles on $X^{\prime}_{S}$, endowed with compatible
hermitian isomorphisms
$h_{i}\colon\mathcal{F}_{i}\cong\sigma^{*}\mathcal{F}_{i}^{\vee}$, and a
compatible family of morphisms $\mathcal{E}_{2,S}\to\mathcal{F}_{i}$
respecting the hermitian structures. If we extend the domain of these
morphisms to $\mathcal{E}_{S}$ by

(3.2.5)
$\mathcal{E}_{S}=\mathcal{E}_{1,S}\oplus\mathcal{E}_{2,S}\xrightarrow{\mathrm{proj}}\mathcal{E}_{2,S}\to\mathcal{F}_{i},$

we obtain an object of $\mathcal{Z}^{r}_{\mathcal{E}}(a)(S)$. This defines
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})\to\mathcal{Z}^{r}_{\mathcal{E}}(a)$.

For the inverse, if we start with an object of
$\mathcal{Z}^{r}_{\mathcal{E}}(a)(S)$ defined by a chain of modifications
(3.2.4) and a compatible family of maps $\mathcal{E}_{S}\to\mathcal{F}_{i}$
respecting hermitian structures, the image of
$\mathcal{E}_{1,S}\to\mathcal{F}_{i}$ must be orthogonal both to itself and to
the image of the (injective, as $a_{2}$ is injective) morphism
$\mathcal{E}_{2,S}\to\mathcal{F}_{i}$. The subsheaf
$\mathcal{E}_{2,S}^{\perp}\subset\mathcal{F}_{i}$ is a line bundle on which
the hermitian form on $\mathcal{F}_{i}$ is nondegenerate, which therefore has
no nonzero isotropic local sections. Thus the morphism
$\mathcal{E}_{1,S}\to\mathcal{F}_{i}$ is zero, and
$\mathcal{E}_{S}\to\mathcal{F}_{i}$ has the form (3.2.5) for some morphism
$\mathcal{E}_{2,S}\to\mathcal{F}_{i}$. This morphism defines an object of
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})(S)$.

The identification of cycle classes is now an exercise in unpacking [FYZb,
Definition 4.8]. Using the notation found there, there is a commutative
diagram

$\textstyle{{\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}_{\mathcal{E}}^{r}(a)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}^{r}_{\mathcal{E}_{2}}}}$$\textstyle{{\mathcal{Z}_{\mathcal{E}_{2}}^{r}[0]^{\circ}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}_{\mathcal{E}/\mathcal{E}_{1}}^{r,\circ}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}_{\mathcal{E}}^{r}[\mathcal{E}_{1}]^{\circ}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}_{\mathcal{E}}^{r}}}$

in which every arrow is an open and closed immersion, and the top horizontal
equality is the isomorphism we have constructed above. In [FYZb, Definition
4.4] one finds the definition of a cycle class

$[\mathcal{Z}_{\mathcal{E}/\mathcal{E}_{1}}^{r,\circ}]\in\operatorname{Ch}_{r}(\mathcal{Z}_{\mathcal{E}/\mathcal{E}_{1}}^{r,\circ})$

whose restriction to the open and closed substack
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})$ is
$[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]$. On the other hand, the class
(3.2.3) is defined as the restriction of

$[\mathcal{Z}_{\mathcal{E}}^{r}[\mathcal{E}_{1}]^{\circ}]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\prod_{i=1}^{r}c_{1}(p_{i}^{*}\sigma^{*}\mathcal{E}_{1}^{-1}\otimes\ell_{i})\right)\cdot[\mathcal{Z}_{\mathcal{E}/\mathcal{E}_{1}}^{r,\circ}]\in\operatorname{Ch}_{0}(\mathcal{Z}_{\mathcal{E}/\mathcal{E}_{1}}^{r,\circ})$

to the open and closed substack $\mathcal{Z}_{\mathcal{E}}^{r}(a)$. The
desired equality of cycle classes is immediate from this. ∎

We now specialize to the case $n=2$, and combine Proposition 2.5.1 with
calculations of Feng-Yun-Zhang on $\operatorname{Sht}^{r}_{\mathrm{U}(1)}$ to
prove cases of Conjecture 3.1.2 on $\operatorname{Sht}^{r}_{\mathrm{U}(2)}$.

######  Theorem 3.2.6.

Suppose $\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$ is a direct sum of
line bundles on $X^{\prime}$, endowed with a hermitian morphism of the form

$a=\begin{pmatrix}*&*\\\
*&a_{2}\end{pmatrix}\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$

with $a_{2}\colon\mathcal{E}_{2}\cong\sigma^{*}\mathcal{E}_{2}^{\vee}$ an
isomorphism. The naive special cycle

$\mathcal{Z}^{r}_{\mathcal{E}}(a)\to\operatorname{Sht}^{r}_{\mathrm{U}(2)}$

is proper over $k$, and the degree of its associated $0$-cycle class

(3.2.6)
$[\mathcal{Z}^{r}_{\mathcal{E}}(a)]\in\operatorname{Ch}_{0}(\mathcal{Z}^{r}_{\mathcal{E}}(a))$

is given by the formula (3.1.1) predicted by Conjecture 3.1.2.

###### Proof.

If $a$ is injective, this is Theorem 3.1.1. If $a$ is not injective, the
hypothesis that $a_{2}$ is an isomorphism implies that
$\mathrm{ker}(a)\subset\mathcal{E}$ is a complementary summand to
$\mathcal{E}_{2}$. As both sides of (3.1.1) only depend on the isomorphism
class of the pair $(\mathcal{E},a)$, we may replace $\mathcal{E}_{1}$ by
$\mathrm{ker}(a)$, and thereby reduce to the case

$a=\begin{pmatrix}0&0\\\ 0&a_{2}\end{pmatrix}.$

For $a$ of this form, Lemmas 3.2.4 and 3.2.5 provide us with isomorphisms

(3.2.7)
$\operatorname{Sht}_{\mathrm{U}(1)}^{r}\cong\mathcal{Z}_{\mathcal{E}_{2}}^{r}(a_{2})\cong\mathcal{Z}_{\mathcal{E}}^{r}(a)$

of $k$-stacks, all proper and smooth of dimension $r$ by Lemma 3.2.3.

The stack $\operatorname{Sht}^{r}_{\mathrm{U}(1)}$ has its own naive special
cycle

$\mathcal{Z}_{\mathcal{E}_{1}}(0)\cong\operatorname{Sht}^{r}_{\mathrm{U}(1)},$

whose associated $0$-cycle class is defined [FYZa, Definition 4.8] by

(3.2.8)
$[\mathcal{Z}^{r}_{\mathcal{E}_{1}}(0)]=\prod_{i=1}^{r}c_{1}(p_{i}^{*}\sigma^{*}\mathcal{E}_{1}^{-1}\otimes\ell_{i})\in\operatorname{Ch}^{r}(\operatorname{Sht}^{r}_{\mathrm{U}(1)}).$

By Lemma 3.2.2, the cycle class

$[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\in\operatorname{Ch}_{r}(\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2}))$

is just the usual fundamental class of the smooth $k$-stack
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})$, and it follows that the first
isomorphism in (3.2.7) identifies (3.2.8) with

$\left(\prod_{i=1}^{r}c_{1}(p_{i}^{*}\sigma^{*}\mathcal{E}_{1}^{-1}\otimes\ell_{i})\right)\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\in\operatorname{Ch}_{0}(\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})).$

In this last equality $\ell_{1},\ldots,\ell_{r}$ are the tautological line
bundles on $\operatorname{Sht}^{r}_{\mathrm{U}(n)}$, which pull-back under

$\operatorname{Sht}^{r}_{\mathrm{U}(1)}\cong\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})\to\operatorname{Sht}^{r}_{\mathrm{U}(n)}$

to the eponymous line bundles on $\operatorname{Sht}^{r}_{\mathrm{U}(1)}$
appearing in (3.2.8).

By Lemma 3.2.5 and the preceding paragraph, the composition (3.2.7) identifies
(3.2.8) with (3.2.6). In particular these $0$-cycles have the same degree. The
degree of (3.2.8) is computed in [FYZb, Theorem 10.2], and we deduce the
explicit formula

(3.2.9) $\deg\,[\mathcal{Z}^{r}_{\mathcal{E}}(a)]=\frac{2}{(\log
q)^{r}}\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\left(q^{d(\mathcal{E}_{1})s}L(2s,\eta)\right).$

Proposition 2.5.1, rewritten as (2.5.5), expresses the
$(\mathcal{E},a)$-coefficient of the spherical Eisenstein series $E(g,s,\chi)$
on $H_{2}(\mathbb{A}_{F})$ in terms of the
$(\mathcal{E}_{2},a_{2})$-coefficient of the spherical Eisenstein series
$\underline{E}(g,s,\chi)$ on $H_{1}(\mathbb{A}_{F})$. This latter Fourier
coefficient is known by the explicit formula of Proposition 2.3.2. Combining
these formulas, and simplifying using the global functional equation (2.3.4),
one finds

$\frac{q^{d(\mathcal{E}_{1})}}{\chi(\mathcal{E}_{1})\chi(\mathcal{E}_{2})}\cdot\widetilde{E}_{(\mathcal{E},a)}(s,\chi)=q^{d(\mathcal{E}_{1})s}L(2s,\eta)+q^{-d(\mathcal{E}_{1})s}L(-2s,\eta).$

Taking the $r^{\mathrm{th}}$ central derivative of both sides, and using the
standing assumption that $r$ is even, we deduce

(3.2.10)
$\frac{q^{d(\mathcal{E})}}{\chi(\det(\mathcal{E}))}\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\widetilde{E}_{(\mathcal{E},a)}(s,\chi)=2\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\left(q^{d(\mathcal{E}_{1})s}L(2s,\eta)\right).$

Note we have used $d(\mathcal{E}_{1})=d(\mathcal{E})$, which follows from
$\mathcal{E}_{2}\cong\sigma^{*}\mathcal{E}_{2}^{\vee}$.

Comparing (3.2.9) with (3.2.10) completes the proof. ∎

###  3.3. Connection with the doubling kernel

We now study intersections of middle codimension cycles on
$\operatorname{Sht}^{r}_{\mathrm{U}(n)}$, and so assume $n=2m$ is even. Let

$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$

be an unramified Hecke character whose restriction to
$\mathbb{A}_{F}^{\times}$ is trivial.

Fix a pair $(\mathcal{E}_{2},a_{2})$ consisting of a rank $m$ vector bundle
$\mathcal{E}_{2}$ on $X^{\prime}$ and a hermitian morphism
$a_{2}\colon\mathcal{E}_{2}\to\sigma^{*}\mathcal{E}_{2}^{\vee}$. Assume that
the associated naive special cycle

(3.3.1)
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})\to\operatorname{Sht}^{r}_{\mathrm{U}(n)}$

is proper over $k$, so that it determines a special cycle class

$[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\in\operatorname{Ch}^{rm}_{c}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})$

in the middle codimension Chow group with proper support.

Holding $(\mathcal{E}_{2},a_{2})$ fixed, consider pairs
$(\mathcal{E}_{1},a_{1})$ with $\mathcal{E}_{1}$ another rank $m$ vector
bundle on $X^{\prime}$ and
$a_{1}\colon\mathcal{E}_{1}\to\sigma^{*}\mathcal{E}_{1}^{\vee}$ a hermitian
morphism. As in [FYZa, Lemma 7.7], there is a decomposition of the fiber
product

(3.3.2)
$\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1})\times_{\operatorname{Sht}^{r}_{\mathrm{U}(n)}}\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})\cong\bigsqcup_{a=\left(\begin{smallmatrix}a_{1}&*\\\
*&a_{2}\end{smallmatrix}\right)}\mathcal{Z}^{r}_{\mathcal{E}}(a)$

into open and closed substacks, in which
$\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$, and the disjoint union is
over the same hermitian morphisms
$a\colon\mathcal{E}\to\sigma^{*}\mathcal{E}^{\vee}$ as in (2.4.5). As
explained in [FYZa, §7.7], there is an intersection pairing

$\operatorname{Ch}_{rm}(\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1}))\times\operatorname{Ch}_{rm}(\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2}))\to\bigoplus_{a=\left(\begin{smallmatrix}a_{1}&*\\\
*&a_{2}\end{smallmatrix}\right)}\operatorname{Ch}_{0}(\mathcal{Z}^{r}_{\mathcal{E}}(a)),$

under which the special cycle classes of (1.1.1) satisfy the expected (but not
obvious) intersection relation

(3.3.3)
$[\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1})]\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]=\sum_{a=\left(\begin{smallmatrix}a_{1}&*\\\
*&a_{2}\end{smallmatrix}\right)}[\mathcal{Z}^{r}_{\mathcal{E}}(a)]$

of [FYZb, Theorem 7.1].

The fiber product on the left hand side of (3.3.2) is finite over
$\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})$, and so is proper over $k$. Hence
each $\mathcal{Z}^{r}_{\mathcal{E}}(a)$ appearing on the right hand side is
also proper. Using the pushforwards (1.1.2) and (1.1.3), we now view (3.3.3)
as an equality of cycle classes in the Chow group
$\operatorname{Ch}_{c}^{rn}(\operatorname{Sht}^{r}_{\mathrm{U}(n)})$ of
$0$-cycles with proper support, with the intersection pairing on the left hand
side now understood as (1.1.4).

Applying the degree map (1.1.5) to both sides of (3.3.3), Conjecture 3.1.2
implies the first equality in the (conjectural) intersection formula

(3.3.4)
$\displaystyle\deg\big{(}[\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1})]\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\frac{1}{(\log
q)^{r}}\cdot\frac{q^{md(\mathcal{E})}}{\chi(\det(\mathcal{E}))}\sum_{a=\left(\begin{smallmatrix}a_{1}&*\\\
*&a_{2}\end{smallmatrix}\right)}\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\widetilde{E}_{(\mathcal{E},a)}(s,\chi)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:geometric doubling
coefficients}}}{{=}}\frac{1}{(\log
q)^{r}}\cdot\frac{q^{md(\mathcal{E}_{1})}q^{md(\mathcal{E}_{2})}}{\chi(\det(\mathcal{E}_{1}))\chi(\det(\mathcal{E}_{2}))}\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\widetilde{D}_{(\mathcal{E}_{1},a_{1}),(\mathcal{E}_{2},a_{2})}(s,\chi)$

for any Hecke character
$\chi\colon\mathbb{A}_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$ whose
restriction to $\mathbb{A}_{F}^{\times}$ is trivial. On the right hand side is
a (double) Fourier coefficient of the doubling kernel

$\widetilde{D}(g_{1},g_{2},s,\chi)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}q^{ns\deg(\omega_{X})}\cdot\mathscr{L}_{n}(s,\chi_{0})\cdot
D(g_{1},g_{2},s,\chi)$

from (2.4.2), renormalized as in (2.3.7).

######  Theorem 3.3.1.

Suppose $\mathcal{E}_{2}$ is a line bundle on $X^{\prime}$ (so $m=1$ and
$n=2$), and $a_{2}\colon\mathcal{E}_{2}\cong\sigma^{*}\mathcal{E}_{2}^{\vee}$
is a hermitian isomorphism.

  1. (1)

The naive special cycle (3.3.1) is proper over $k$.

  2. (2)

The equality (3.3.4) holds for every pair $(\mathcal{E}_{1},a_{1})$.

  3. (3)

There exists a $K_{1}$-fixed automorphic form
$\mathscr{D}\in\mathcal{A}(H_{1})$ whose geometric Fourier coefficients are
given by

$\mathscr{D}_{(\mathcal{E}_{1},a_{1})}=\frac{\chi(\mathcal{E}_{1})}{q^{d(\mathcal{E}_{1})}}\,\deg\big{(}[\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1})]\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}$

for every pair $(\mathcal{E}_{1},a_{1})$ consisting of a line bundle
$\mathcal{E}_{1}$ on $X^{\prime}$ and a hermitian morphism
$a_{1}:\mathcal{E}_{1}\to\sigma^{*}\mathcal{E}_{1}^{\vee}$.

  4. (4)

Assuming Conjecture 1.2.1, the equality

$\displaystyle\deg\big{(}\vartheta^{r,\chi}(f)\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}$
$\displaystyle=f_{(\mathcal{E}_{2},-a_{2})}\frac{q^{d(\mathcal{E}_{2})}}{(\log
q)^{r}}\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\left(q^{2s\deg(\omega_{X})}L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)\right),$

of Conjecture 1.3.1 holds for every $K_{1}$-fixed
$f\in\pi\subset\mathcal{A}(H_{1})$.

###### Proof.

The properness claim (1) follows from Lemmas 3.2.3 and 3.2.4.

For (2), we have explained above that (3.3.4) is a consequence of Conjecture
3.1.2, but one does not need to know that conjecture in full generality: for a
fixed $(\mathcal{E}_{2},a_{2})$, to deduce (3.3.4) one only needs to know
Conjecture 3.1.2 for those pairs $(\mathcal{E},a)$ of the form
$\mathcal{E}=\mathcal{E}_{1}\oplus\mathcal{E}_{2}$ with

$a=\left(\begin{matrix}a_{1}&*\\\ *&a_{2}\end{matrix}\right).$

For $(\mathcal{E}_{2},a_{2})$ satisfying our current hypotheses, this is
Theorem 3.2.6.

Analogues of (3) and (4) hold for any $n=2m$ and any pair
$(\mathcal{E}_{2},a_{2})$ for which (3.3.4) holds, so that is the generality
in which we prove them.

For any pair $(\mathcal{E}_{1},a_{1})$ the function
$\widetilde{D}_{(\mathcal{E}_{1},a_{1}),(\mathcal{E}_{2},a_{2})}(s,\chi)$
appearing in (3.3.4) is the $(\mathcal{E}_{1},a_{1})$ Fourier coefficient of
the renormalized new way kernel

$\widetilde{D}_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi)=q^{ns\deg(\omega_{X})}\cdot\mathscr{L}_{n}(s,\chi_{0})\cdot
D_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi)$

of (2.4.4), an automorphic form in the variable $g_{1}\in
H_{m}(\mathbb{A}_{F})$. This proves that (3) holds with

$\mathscr{D}(g_{1})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{(\log
q)^{r}}\cdot\frac{q^{md(\mathcal{E}_{2})}}{\chi(\det(\mathcal{E}_{2}))}\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\widetilde{D}_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi),$

as the equality (3.3.4) says precisely that

$\mathscr{D}_{(\mathcal{E}_{1},a_{1})}=\frac{\chi(\det(\mathcal{E}_{1}))}{q^{md(\mathcal{E}_{1})}}\deg\big{(}[\mathcal{Z}^{r}_{\mathcal{E}_{1}}(a_{1})]\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}.$

This last equality also shows that the automorphic form $Z^{r,\chi}$ of
Conjecture 1.2.1 satisfies

$\mathscr{D}(g_{1})=\deg\big{(}Z^{r,\chi}(g_{1})\cdot[\mathcal{Z}_{\mathcal{E}_{2}}^{r}(a_{2})]\big{)},$

because both sides have the same geometric Fourier coefficients. Recalling the
definition (1.3.1) of the arithmetic theta lift, for any $K_{m}$-fixed cusp
form $f\in\pi\subset\mathcal{A}(H_{m})$ we now compute

$\displaystyle\deg\big{(}\vartheta^{r,\chi}(f)\cdot[\mathcal{Z}_{\mathcal{E}_{2}}^{r}(a_{2})]\big{)}$
$\displaystyle=\int_{[H_{m}]}f(g_{1})\,\deg\big{(}Z^{r,\chi}(g_{1})\cdot[\mathcal{Z}_{\mathcal{E}_{2}}^{r}(a_{2})]\big{)}\,dg_{1}$
$\displaystyle=\int_{[H_{m}]}f(g_{1})\mathscr{D}(g_{1})\,dg_{1}$
$\displaystyle=\frac{1}{(\log
q)^{r}}\cdot\frac{q^{md(\mathcal{E}_{2})}}{\chi(\det(\mathcal{E}_{2}))}\cdot\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\int_{[H_{m}]}f(g_{1})\widetilde{D}_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi)\,dg_{1}.$

Theorem 2.4.2, restated as (2.4.7), shows that

$\displaystyle\int_{[H_{m}]}f(g_{1})\widetilde{D}_{\square,(\mathcal{E}_{2},a_{2})}(g_{1},s,\chi)\,dg_{1}$
$\displaystyle=\chi(\det(\mathcal{E}_{2}))\cdot
f_{(\mathcal{E}_{2},-a_{2})}\cdot
q^{ns\deg(\omega_{X})}L(s+1/2,\mathrm{BC}(\pi)\otimes\chi),$

and we deduce

$\displaystyle\deg\big{(}\vartheta^{r,\chi}(f)\cdot[\mathcal{Z}^{r}_{\mathcal{E}_{2}}(a_{2})]\big{)}$
$\displaystyle=f_{(\mathcal{E}_{2},-a_{2})}\frac{q^{md(\mathcal{E}_{2})}}{(\log
q)^{r}}\frac{d^{r}}{ds^{r}}\Big{|}_{s=0}\left(q^{ns\deg(\omega_{X})}L(s+1/2,\mathrm{BC}(\pi)\otimes\chi)\right).$

This both proves (4), and shows more generally that Conjecture 1.3.1 is a
consequence of the conjectural equality (3.3.4). ∎

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