1 Introduction
Let
$E $
be a CM field, V a hermitian space over E of signature
$(n,1), (n+1,0),.. . ,(n+1,0)$
, and X a Shimura variety for
$U(V)$
. Let F be the maximal totally real subfield and
$F_{>0}$
the set of totally positive elements of F. For
$t \in F_{>0}$
, we have a special divisor
$Z_t$
on X, following Kudla’s work [Reference KudlaKud97a] for orthogonal Shimura varieties. Let
$[ Z_t]$
be the class of
$Z_t$
in the Chow group
${\mathrm {Ch}}^1(X)_{\mathbb {C}}$
of divisors on X with
${\mathbb {C}}$
-coefficients. By Liu [Reference LiuLiu11a], the generating series

with a suitable constant term, is a
${\mathrm {Ch}}^1(X)_{\mathbb {C}}$
-valued holomorphic modular form. Here,
$q=\prod _{k=1}^{[F:{\mathbb {Q}}]} e^{2\pi i\tau _k}$
with
$\tau =(\tau _k )_{k=1}^{[F:{\mathbb {Q}}]}\in {\mathcal {H}}^{[F:{\mathbb {Q}}]}$
, where
${\mathcal {H}}$
is the usual upper half plane. This is an analog of the theorem of Borcherds [Reference BorcherdsBor99], Yuan, S. Zhang and W. Zhang [Reference Yuan, Zhang and ZhangYZZ09] for orthogonal Shimura varieties, which was originally conjectured by Kudla [Reference KudlaKud97a]. In [Reference KudlaKud02, Reference KudlaKud03, Reference KudlaKud04], Kudla also raised the problem of finding (canonical) arithmetic extensions of special divisors on integral models of Shimura varieties to obtain a modular generating series, which is crucial for Kudla’s program on arithmetic theta lifting.
The main result of this paper provides a solution to Kudla’s modularity problem in the case that X is proper with arbitrary level structures at split places and certain lattice level structures at nonsplit places. The arithmetic extensions are defined using S. Zhang’s theory of admissible arithmetic divisors. Slightly more explicitly, we construct a regular integral model
${\mathcal {X}}$
of X proper flat over
${\mathcal {O}}_E$
. An admissible arithmetic divisor on
${\mathcal {X}}$
is an analog of an admissible Green function (i.e., one with harmonic curvature). Consider the normalized admissible extension
$ Z_t^{\overline {\mathcal {L}}}$
of
$Z_t$
, which is the Zariski closure at every finite place of E where the model is smooth. Let
$[ Z_t^{\overline {\mathcal {L}}}]$
be its class in the arithmetic Chow group. Then the generating series

with a suitable constant term, is a holomorphic modular form. Here,
$ {\mathfrak {e}}_t $
is formed using coefficients of an explicit Eisenstein series and its derivative.
Previous to our work, solutions to Kudla’s modularity problem were obtained using different methods by Kudla, Rapoport and Yang [Reference KudlaKud03] [Reference Kudla, Rapoport and YangKRY06] for quaternionic Shimura curves, by Bruinier, Burgos Gil, and Kühn [Reference Bruinier, Burgos Gil and KühnBBGK07] for Hilbert modular surfaces, over
${\mathbb {Q}}$
with minimal level structures, by Howard and Madapusi Pera [Reference Howard and PeraHMP20] for orthogonal Shimura varieties over
${\mathbb {Q}}$
, and by Bruinier, Howard, Kudla, Rapoport and Yang [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20a] for unitary Shimura varieties over imaginary quadratic fields, with self-dual lattice level structures. Compared to these results, we expect that the greater generality of the level structures in our result could be more useful for some purposes – for example, to approach modularity in higher codimensions following the inductive process in [Reference Yuan, Zhang and ZhangYZZ09] for the generic fibers.
In the other direction, S. Zhang [Reference ZhangZha20] introduced the notion of
$\mathsf {L}$
-liftings of divisor classes (on general polarized arithmetic varieties), and then deduced a solution to Kudla’s modularity problem directly from the modularity results for the generic fibers in the first paragraph, regardless of level structures. The
$\mathsf {L}$
-lifting of a divisor class is also admissible but ‘normalized’ in the level of arithmetic divisor classes using the Faltings heights. Our approach is an explicit alternative of S. Zhang’s. In some applications, an explicit modular generating series as our (1.2) is necessary. For example, W. Zhang’s proof of the arithmetic fundamental lemma [Reference ZhangZha21a] used the explicit result of [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20a].
The main ingredient in the proof of our main result is an arithmetic mixed Siegel-Weil formula, which identifies the arithmetic intersection between the generating series (1.2) with a CM 1-cycle on
${\mathcal {X}}$
(associated to a 1-dimensional hermitian subspace of V) and an explicit modular form constructed from theta series and (derivatives of) Eisenstein series.
Arithmetic mixed Siegel-Weil formulas appeared in the literature in different contexts. The first one appeared in the work of Gross and Zagier [Reference Gross and ZagierGZ86, p 233, (9.3)] for generating series of Hecke operators on the square of a modular curve, and implies their celebrated formula relating heights of heegner points and derivatives of L-functions. This arithmetic mixed Siegel-Weil formula was partially generalized to quaternionic Shimura curves over totally real fields in the work of Yuan, S. Zhang and W. Zhang [Reference Yuan, Zhang and ZhangYZZ13, 1.5.6] on the general Gross-Zagier formula. For certain orthogonal Shimura varieties over
${\mathbb {Q}}$
, an arithmetic mixed Siegel-Weil formula was conjectured by Bruinier and Yang [Reference Bruinier and YangBY09, Conjecture 1.3]. Its analog for unitary Shimura varieties over imaginary quadratic fields with certain self-dual lattice level structures was proved by Bruinier, Howard and Yang [Reference Bruinier, Howard and YangBHY15, Theorem C].
In the rest of this introduction, we first state our main result in more detail. Then we discuss its proof. Finally, we mention two non-holomorphic modular variants of (1.2).
1.1 Main result
To state our main result, we need some preliminaries.
1.1.1 Admissible divisors
Let E be a number field,
${\mathcal {X}}$
a regular scheme (or more generally Deligne-Mumford stack) proper flat over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_E$
and
$\overline {\mathcal {L}}=({\mathcal {L}},\|\cdot \|)$
an ample hermitian line bundle on
${\mathcal {X}}$
. At each infinite place v of E, equip the complex manifold
$ {\mathcal {X}}_{E_v} $
with the Kähler form that is the curvature form
${\mathrm {curv}}(\overline {\mathcal {L}}_{E_v})$
. First, a Green function is admissible (introduced by Gillet and Soulé [Reference Gillet and SouléGS90, 5.1] following Arakelov [Reference ArakelovAra74]) if its curvature form
$\alpha $
is harmonic; equivalently, on each connected component of
${\mathcal {X}}_{E_v}$
,
${\mathrm {curv}}(\overline {\mathcal {L}}_{E_v})^{n-1}\wedge \alpha $
is proportional to
${\mathrm {curv}}(\overline {\mathcal {L}}_{E_v})^{n}$
, where
$n=\dim {\mathcal {X}}_{E_v}$
. It is further normalized if on each connected component of
${\mathcal {X}}_{E_v}$
, its pairing with (i.e., integration against)
${\mathrm {curv}}(\overline {\mathcal {L}}_{E_v})^{n}$
is 0. Second, at each finite place v, a divisor Y on
${\mathcal {X}}_{ {\mathcal {O}}_{E_v}}$
is admissible if it has ‘harmonic curvature’ with respect to
$\overline {\mathcal {L}}_{ {\mathcal {O}}_{E_v}}$
, in the sense that on each connected component of
${\mathcal {X}}_{{\mathcal {O}}_{E_v}}$
, the linear form on the space of vertical divisors defined by intersecting with
$Y\cdot c_1( {\mathcal {L}}_{ {\mathcal {O}}_{E_v}})^{n-1}$
is proportional to the linear form defined by intersecting with
$ c_1( {\mathcal {L}}_{ {\mathcal {O}}_{E_v}})^{n}$
. We further call Y normalized if its vertical part has intersection pairing 0 with
$c_1({\mathcal {L}})^{n}$
. Finally, an arithmetic divisor on
${\mathcal {X}}$
is (normalized) admissible if it is (normalized) admissible at every finite place and its Green function is (normalized) admissible. For a divisor Z on
${\mathcal {X}}_E $
, we have the unique normalized admissible extension
$Z^{\overline {\mathcal {L}}}$
on
${\mathcal {X}}$
(called the Arakelov lifting of Z in [Reference ZhangZha20]).
Let
$\widehat {\mathrm {Ch}}_{{\overline {\mathcal {L}}},{\mathbb {C}}} ^1({\mathcal {X}}) $
be the space of admissible arithmetic divisors with
${\mathbb {C}}$
-coefficients, modulo the
${\mathbb {C}}$
-span of the principal ones. If
${\mathcal {X}}$
is connected, then the natural map

is surjective and has a 1-dimensional kernel. It is the pullback of
$ \widehat {\mathrm {Ch}}_{\mathbb {C}}^1\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E} \right ){\ \simeq \ } {\mathbb {C}}$
, where the isomorphism is by taking degrees. Then the
${\mathfrak {e}}_t\in {\mathbb {C}}\subset \widehat {\mathrm {Ch}}_{{\overline {\mathcal {L}}},{\mathbb {C}}}^1({\mathcal {X}})$
in (1.2) is understood in this way.
1.1.2 Shimura varieties and integral models
Let
$V({\mathbb {A}}_E^\infty )$
be the space of finite adelic points of V. For an open compact subgroup
$K\subset U(V({\mathbb {A}}_E^\infty ))$
, we have a
$U(V)$
-Shimura variety
${\mathrm {Sh}}(V)_K$
(which could be stacky) of level K defined over E. We assume that
${\mathrm {Sh}}(V)_K$
is proper; equivalently,
$F\neq {\mathbb {Q}}$
or
$F={\mathbb {Q}},n=1$
and
$V $
is nonsplit at some finite place.
Let
$\Lambda \subset V({\mathbb {A}}_E^\infty )$
be a hermitian lattice with stabilizer
$K_\Lambda \subset U(V({\mathbb {A}}_E^\infty ))$
. Let
$K\subset K_\Lambda $
such that
$K_v=K_{\Lambda ,v}$
for v nonsplit in E. We construct a regular integral model
${\mathcal {X}}_E$
of
${\mathrm {Sh}}(V)_K$
proper flat over
${\mathcal {O}}_E$
under some conditions on
$E,F,\Lambda $
(Theorem 4.4.4). Our construction is largely suggested by Liu.
We have two constructions according to different conditions on
$E,F$
. First, assume that
$E/{\mathbb {Q}}$
is tamely ramified. We have the normalization in
${\mathrm {Sh}}(V)_K$
of the flat model of
${\mathrm {Sh}}(V)_{K_\Lambda }$
of Kisin [Reference KisinKis10], Kisin and Pappas [Reference Kisin and PappasKP18] over
${\mathcal {O}}_{E,(v)}$
, for every finite place v. We want to show their regularity and glue them to obtain a regular integral model
${\mathcal {X}}_K$
over
${\mathcal {O}}_E$
. For this purpose, we use a certain regular PEL moduli space for a group closely related to
$U(V)$
over the ring of integers of a reflex field
$E'/E$
, constructed by Rapoport, Smithling and W. Zhang [Reference Rapoport, Smithling and ZhangRSZ20]. Expectably, the moduli space and our integral models are closely related, as shown by Xu in Appendix B (the proof for the general level at split places was suggested by Liu). Second, replacing the tameness assumption by the assumption that
$E/{\mathbb {Q}}$
is Galois or
$E $
is the composition of F with some imaginary quadratic field, which implies that
$E'=E$
, we can construct a regular integral model over
${\mathcal {O}}_{E,(v)}$
from the above moduli space, following [Reference Liu, Tian, Xiao, Zhang and ZhuLTX+22]. Moreover, if both the tameness assumption and the replacement hold, the two constructions give the same model.
We remark that by our choice of
$\Lambda $
,
${\mathcal {X}}_{K_\Lambda }$
is smooth over
${\mathcal {O}}_E$
so that the finite part of the normalized admissible extension of a divisor on
${\mathrm {Sh}}(V)_{K_\Lambda }$
is the Zariski closure on
${\mathcal {X}}_{K_\Lambda }$
.
1.1.3 Hodge bundles and CM cycles
Let
${\mathcal {L}}_{K_\Lambda }$
be an arbitrarily line bundle on
${\mathcal {X}}_{K_\Lambda }$
extending the Hodge (line) bundle on
${\mathrm {Sh}}(V)_{K_\Lambda }$
. Let
${\mathcal {L}}_K$
, denoted by
${\mathcal {L}}$
if K is clear from the context, be the pullback of
${\mathcal {L}}_{K_\Lambda }$
to
${\mathcal {X}}_K$
. Let
$\overline {\mathcal {L}}=({\mathcal {L}},\|\cdot \|)$
, where
$\|\cdot \|$
is the descent of the natural hermitian metric on the hermitian symmetric domain uniformizing
${\mathrm {Sh}}(V)_K$
. It is compatible under pullbacks as K shrinks. Changing
$\overline {\mathcal {L}}$
,
$c_1(\overline {\mathcal {L}})\in \widehat {\mathrm {Ch}}_{\mathbb {C}}^1\left ( {\mathcal {X}} \right )$
changes by an element in the pullback of
$ \widehat {\mathrm {Ch}}_{\mathbb {C}}^1\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E}\right ) $
. (It is a special feature due to the smoothness of
${\mathcal {X}}_{K_\Lambda }$
over
${\mathcal {O}}_E$
.) In particular, changing
$\overline {\mathcal {L}}$
will not change the generating series (1.2). However, this fact does not play a role in our proof.
For a 1-dimensional hermitian subspace
$ W\subset V$
, we have an associated 0-dimensional Shimura subvariety of
${\mathrm {Sh}}(V)$
. On
${\mathcal {X}}_{K_\Lambda }$
, let the 1-cycle
${\mathcal {P}}_{K_\Lambda }$
be its Zariski closure, divided by the degree of its genetic fiber (so that
$\deg {\mathcal {P}}_{K_\Lambda ,E}=1$
). Then
${\mathcal {P}}_{K_\Lambda }$
is independent of the choice of this subspace (Proposition 5.1.9). We do not need CM cycles at other levels.
1.1.4 Generating series
We start with the non-constant terms of the generating series (1.2).
For
$x\in {\mathbb {V}}^{\infty }$
with norm in
$ F_{>0}$
, the orthogonal complement of
${\mathbb {A}}_E ^\infty x $
in
${\mathbb {V}}^\infty $
defines a (shifted) unitary Shimura subvariety
$Z(x)_K$
of
${\mathrm {Sh}}(V)_K$
of codimension 1. For
$t\in F_{>0}$
and a Schwartz function
$\phi $
on
$V({\mathbb {A}}_E^\infty )$
invariant by K, the weighted special divisor is

It is compatible under pullbacks as K shrinks.
Now we define
${\mathfrak {e}}_t$
. Let
$E(s,\tau )$
be the Siegel-Eisenstein series on
${\mathcal {H}}^{[F:{\mathbb {Q}}]}$
associated to
$\phi $
. Its t-th Whittaker function
$E_t(s,\tau )$
has a decomposition

into the infinite component and the finite component. Here, we choose the s-variable so that
$E(0,\tau )$
is holomorphic of weight
$n+1$
, equivalently, the infinite component
$W_{\infty ,t} (0, \tau )$
is a multiple of
$q^t$
. (Note that
$s=0$
is the critical point for the Siegel-Weil formula, but not the center for the functional equation.) The t-th Fourier coefficient of
$E(0,\tau )$
is
$\frac {E_t(0,\tau )} {q^t}$
. Define

Then
${\mathfrak {e}}_t$
is independent of
$\tau $
.
We introduce a number that will appear in the constant term of the generating series. Let

where
${\mathrm {Disc}}_E\in {\mathbb {Z}}$
is the discriminant of
$E/{\mathbb {Q}}$
, and
${\mathfrak {b}} =-({1+\log 4})$
when
$ n=1$
and more complicated in general. See (3.30) and the remarking following it. We hope to compute the Faltings height
$c_1(\overline {\mathcal {L}}_{K_\Lambda })\cdot {\mathcal {P}}_{ K_\Lambda }$
based on [Reference YuanYZ18] in a future work. And we expect cancellation among the terms defining
${\mathfrak {a}}$
so that the definition of
${\mathfrak {a}}$
will be elementary and transparent.
Theorem 1.1.1 (Theorem 4.4.21, (4.28)).
If
$\phi _v=1_{\Lambda _v}$
at ramified places, the generating series (1.2) with the constant term being
$\phi (0)\left ( c_1(\overline {\mathcal {L}}^\vee )+{\mathfrak {a}}\right )$
is a holomorphic modular form on
${\mathcal {H}}^{[F:{\mathbb {Q}}]}$
of parallel weight
$n+1$
valued in
$\widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}( {\mathcal {X}}_K)$
. Here, we understand
${\mathfrak {a}},{\mathfrak {e}}_t\in {\mathbb {C}}\subset \widehat {\mathrm {Ch}}_{{\overline {\mathcal {L}}},{\mathbb {C}}}^1({\mathcal {X}}_K)$
as discussed below (1.3).
Since the formation of normalized admissible extension is compatible under flat pullbacks, the generating series (1.2) is compatible under pullbacks as K shrinks.
We note that the sum of the normalized admissible Green function for
$Z_t$
and
${\mathfrak {e}}_t$
recovers the Bruinier-Borcherds Green function used by Bruinier, Howard, Kudla, Rapoport and Yang [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20a] for
$F={\mathbb {Q}}$
, so that Theorem 1.1.1 is an analog of [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20a, Theorem B]. Though the Bruinier-Borcherds construction can not be directly extended to a general F (as explained to us by Bruinier), our construction could be considered as an alternative generalization.
1.2 Sketch of the proof
Now we discuss the proof of Theorem 1.1.1. By the 1-dimensionality of the kernel of (1.3) and the modularity of the generic fiber of the generating series (1.2) (i.e., the generating series (1.1)), the modularity of (1.2) is equivalent to the modularity of the generating series of arithmetic intersection numbers between
$[Z_t^{\overline {\mathcal {L}}}]+{\mathfrak {e}}_t $
’s and a 1-cycle on
${\mathcal {X}}_K$
whose generic fiber has nonzero degree. (A similar strategy was used in [Reference KudlaKud03, Reference Kudla, Rapoport and YangKRY06].)
Assume that
$K=K_\Lambda $
for simplicity and let us take the 1-cycle to be
${\mathcal {P}}_{K_\Lambda }$
. Then this generating series of arithmetic intersection numbers
$ \left ( [Z_t^{\overline {\mathcal {L}}}]+{\mathfrak {e}}_t\right )\cdot {\mathcal {P}}_{K_\Lambda }$
is the arithmetic analog of the integration of a theta series of
$U(1,1)\times U(V) $
along
$U(W)\backslash U(W({\mathbb {A}}_E))$
, where
$W\subset V$
is the 1-dimensional hermitian subspace defining
${\mathcal {P}}_{K_\Lambda }$
. By the Siegel-Weil formula for
$U(1,1)\times U(W) $
, this integration is a theta-Eisenstein series (i.e., a linear combination of products of theta series and Eisenstein series).Footnote
1
Let
${\theta E}(s,\tau )$
be the theta-Eisenstein series associated to
$\phi $
and let
${\theta E}_{t}(s,\tau )$
be its t-th Whittaker coefficient. We study the holomorphic projections of
${\theta E}'(0,\tau )$
in order to match the above generating series of arithmetic intersection numbers, which is supposed to be holomorphic in view of our goal (Theorem 1.1.1). A priori, there are two holomorphic projections. One is the projection of
${\theta E}'(0,\tau )$
to the space of holomorphic cusp forms, which we call ‘cuspidal holomorphic projection’ and denote by
${\theta E}^{\prime }_{{\mathrm {chol}}}(0,\tau )$
. Let
${\theta E}^{\prime }_{{\mathrm {chol}},t}(0,\tau )$
be its t-th Whittaker coefficient. The other is for
${\theta E}_{t}'(0,\tau )$
and purely at infinite places, which we call ‘quasi-holomorphic projection’ following [Reference Yuan, Zhang and ZhangYZZ13] and denote by
${\theta E}_{t,{\mathrm {qhol}}}'(0,\tau )$
. However, neither of them could be the desired match, since
${\theta E}^{\prime }_{{\mathrm {chol}}}(0,\tau )$
has no constant term and
${\theta E}_{t,{\mathrm {qhol}}}'(0,\tau )$
is (in general) not the t-th Whittaker coefficient of a modular form. Thus, we compute their difference and find that
${\theta E}_{t,{\mathrm {qhol}}}'(0,\tau )-{\theta E}^{\prime }_{{\mathrm {chol}},t}(0,\tau )$
is the sum of
$2{\mathfrak {e}}_t q^t$
and the t-th Whittaker coefficient of a holomorphic Eisenstein series. See (3.31). In the case
$n=1$
, the computations of holomorphic projections have their roots in [Reference Gross and ZagierGZ86]. The strategy we follow is outlined in [Reference Yuan, Zhang and ZhangYZZ13, 6.4.3], and the explicit computation was done by Yuan [Reference YuanYua22]. We largely follow [Reference YuanYua22].
Let f be the sum of
$-\frac {1}{2}{\theta E}^{\prime }_{{\mathrm {chol}}}(0,\tau )$
and the negative of the holomorphic Eisenstein series in the last sentence. Then f is a holomorphic modular form on
${\mathcal {H}}^{[F:{\mathbb {Q}}]}$
of parallel weight
$n+1$
. The t-th Whittaker coefficient of f is
$ -\frac {1}{2}\frac {{\theta E}_{t,{\mathrm {qhol}}}'(0,\tau ) }{q^t}+{\mathfrak {e}}_t $
.
In 5.2.1, we define some explicit Schwartz functions
$\phi _v'$
, for every ramified place v, which are ‘error functions’ due to ramification. Let g be sum of the theta-Eisenstein series associated to
$\phi ^v\phi _v'$
’s. The following is our arithmetic mixed Siegel-Weil formula (Theorem 5.2.5) in the case
$K=K_\Lambda $
.
Theorem 1.2.1. Assume that
$\phi _v=1_{\Lambda _v}$
for v nonsplit in E. The arithmetic intersection number
$ \left ( [Z_t^{\overline {\mathcal {L}}}]+{\mathfrak {e}}_t\right )\cdot {\mathcal {P}}_{K_\Lambda }$
is the t-th Fourier coefficient of
$f-g-\frac {1}{n}E(0,\tau )$
.
In Theorem 5.2.5 for a general K, we use the pullback of
${\mathcal {P}}_{K_\Lambda }$
to
${\mathcal {X}}_K$
, instead of natural CM cycles on
${\mathcal {X}}_K$
, to simplify certain local computations. See Remark 5.2.6.
We remind the reader that in Theorem 5.2.5, we actually use the automorphic Green function for
$Z_t$
constructed by Bruinier [Reference BruinierBru02, Reference BruinierBru12] and Oda and Tsuzuki [Reference Oda and TsuzukiOT03] (for
$n=1$
and
$F={\mathbb {Q}}$
, it was well known and used by Gross and Zagier [Reference Gross and ZagierGZ86]). Its difference with the normalized admissible Green function is
$\frac {1}{n}E_t(0,\tau )$
by Lemma 4.2.4 and the remark following it.
Let us remark on the innovation in proving the arithmetic mixed Siegel-Weil formula. We consider the difference of two CM cycles. The generic fiber of the difference has degree 0. Then the generating series of arithmetic intersection numbers is modular by the admissibility. (A similar observation was used in [Reference Mihatsch and ZhangMZ21] to generalize the arithmetic fundamental lemma. See also [Reference ZhangZha21b].) This modularity enables us to ‘switch CM cycles’ and thus avoid computing improper intersections directly. This idea is inspired by [Reference Yuan, Zhang and ZhangYZZ13] and [Reference ZhangZha21a].
1.3 Non-holomorphic variants
We obtain a non-holomorphic modular variant of the generating series (1.2), where the sum of the normalized admissible Green function for
$Z_t$
and
${\mathfrak {e}}_t$
is replaced by Kudla’s Green function [Reference KudlaKud97b]. See Theorem 4.4.24. This is an analog of [Reference Kudla, Rapoport and YangKRY06, Theorem A] [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20a, Theorem 7.4.1]. Theorem 4.4.24 follows from Theorem 1.1.1, and the modularity of the differences between the generating series of two kinds of Green functions (Theorem 4.2.10). The latter (Theorem 4.2.10) is an analog of the main result of Ehlen and Sankaran [Reference Ehlen and SankaranES18] for
$F={\mathbb {Q}}$
.
Note that Kudla’s Green function is not admissible. We also obtain a non-holomorphic modular generating series with admissible Green functions (Theorem 4.4.21, (4.29)). This has not appeared in the literature yet, as far as we know.
2 Some notations and conventions
2.1
For a number field F, let
${\mathbb {A}}_F$
be the ring of adeles of F and
${\mathbb {A}}_F^\infty $
the ring of finite adeles of F. For a finite place v of a number field F, let
$\varpi _{F_v}$
be a uniformizer of
$F_v$
. Let
$q_{F_v}$
be the cardinality of
${\mathcal {O}}_{F_v}/\varpi _{F_v}$
. The discrete valuation is
$v(\varpi _{F_v})=1$
and the absolute value
$| \cdot |_{F_v}$
is
$|\varpi _v|_{F_v}=q_{F_v}^{-1}$
. For an infinite place, v is understood as a pair of complex embeddings. If v is real, the absolute value
$| \cdot |_{F_v}$
is the usual one; if v is complex, the absolute value
$| \cdot |_{F_v}$
is the square of the usual one. Their product is
$|\cdot |_{{\mathbb {A}}_F}$
. The symbol
$|\cdot |$
without a subscript means the usual real or complex absolute value.
Below in this paper,
$E/F$
is always a CM extension. Let
$\infty $
be the set of infinite places of F. Let
$F_{>0}\subset F$
be the subset of totally positive elements. For a place v of F,
$E_v$
is understood as
$E\otimes _F F_v$
. The nontrivial Galois action will be denoted by
$x\mapsto \overline x$
, and the norm map
${\mathrm {Nm}}_{E/F}$
or its local version is abbreviated as
${\mathrm {Nm}}$
. Let
$\eta $
be the associated quadratic Hecke character of
$ F^\times \backslash {\mathbb {A}}_F^\times $
via the class field theory.
For a set of place S of F and a decomposable adelic object X over
${\mathbb {A}}_F$
, we use
$X_S$
(resp.
$X^S$
) to denote the S-component (resp. component away from S) of X if the decomposition of X into the product of
$X_S$
and
$X^S$
is clear from the context. For example,
${\mathbb {A}}_F={\mathbb {A}}_{F,S}{\mathbb {A}}_{F}^S$
and
${\mathbb {A}}_E={\mathbb {A}}_{E,S}{\mathbb {A}}_{E}^S$
by regarding
${\mathbb {A}}_E$
as over
${\mathbb {A}}_F$
. Here is another example which is ubiquitous in the paper: a function
$\phi $
on the space of
${\mathbb {A}}_{F}$
-points of an algebraic group over F that can be decomposed as
$\phi =\phi _S\otimes \phi ^S$
, where
$\phi _S$
(resp.
$\phi ^S$
) is a function on the set of
${\mathbb {A}}_{F,S}$
-points (resp.
${\mathbb {A}}_{F}^{S}$
-points) of the group. Note that such a decomposition of
$\phi $
is not unique. By using these notations, we understand that we have fixed such a decomposition. See 2.3 for an example. If
$S=\{v\}$
, we write
$X_v$
(resp.
$X^v$
) for
$X_S$
(resp.
$X^S$
).
2.2
All hermitian spaces are assumed to be nondegenerate. We always use
$\langle {\cdot ,\cdot } \rangle $
to denote a hermitian pairing and
$q(x)=\langle {x,x} \rangle $
the hermitian norm if the underlying hermitian space (over E,
$E_v$
or
${\mathbb {A}}_E$
) is indicated in the context. For a hermitian space V over E, we use V to denote
$V(E)$
to lighten the notation if there is no confusion. For
$t\in F$
, let
$V^t=\{v\in V:q(v)=t\}.$
The same notation applies to a local or adelic hermitian space. We use
$U(V)$
for both the algebraic group
$U(V)$
and its group of F-points. Define

A hermitian space
${\mathbb {V}}/{\mathbb {A}}_E$
is called coherent (resp. incoherent) if its determinant belongs (resp. does not belong) to
$ F^\times {\mathrm {Nm}} ( {\mathbb {A}}_E^\times )$
; equivalently,
${\mathbb {V}}{\ \simeq \ } V({\mathbb {A}}_E)$
for some (resp. no) hermitian space
$V/E$
. If
${\mathbb {V}}$
is incoherent of dimension
$1$
, for a place v of F nonsplit in E, there is a unique hermitian space
$V/E$
such
$V({\mathbb {A}}_E^v){\ \simeq \ } {\mathbb {V}}^v$
. We call V the v-nearby hermitian space of
${\mathbb {W}}$
.
2.3
Let
${\mathcal {S}}({\mathbb {V}})$
be the space of
${\mathbb {C}}$
-valued Schwartz functions. For
$v\in \infty $
such that
${\mathbb {V}}(E_v)$
is positive definite, the standard Gaussian function on
${\mathbb {V}}_\infty $
is
$e^{-2\pi {q(x)}}\in {\mathcal {S}}({\mathbb {V}}(E_v)) $
. If a hermitian space
${\mathbb {V}}/{\mathbb {A}}_E$
is totally positive-definite, let

be the subspace of functions of the form
$\phi = \phi _\infty \otimes \phi ^\infty $
, where
$\phi _\infty $
is the pure tensor of standard Gaussian functions over all infinite places and
$ \phi ^\infty \in {\mathcal {S}}({\mathbb {V}} ^\infty ) $
taking values in
${\mathbb {C}}$
. For
$\phi \in {\mathcal {S}}({\mathbb {V}} ^\infty ) $
, we always fix such a decomposition.
2.4
Fix the additive character of
$F\backslash {\mathbb {A}}_F$
to be
$\psi :=\psi _{\mathbb {Q}}\circ {\mathrm {Tr}}_{F/ {\mathbb {Q}}} $
, where
$\psi _{\mathbb {Q}}$
is the unique additive character of
${\mathbb {Q}}\backslash {\mathbb {A}}_{\mathbb {Q}}$
such that
$\psi _{{\mathbb {Q}},\infty }(x)=e^{2\pi ix}$
. The additive character of
${\mathbb {A}}_E$
is
$\psi _{E}:=\psi \circ {\mathrm {Tr}}_{E/ F} $
. For
$ t\in {\mathbb {A}}_F $
(we in fact only use
$t\in F$
), let
$\psi _{t}(b)=\psi (tb) .$
For a place v of F and
$t\in F_v$
, Let
$\psi _{v,t}(b)=\psi _v (tb) .$
Then
$\psi _{v,t}=\psi _{v,t_v}$
for
$ t\in {\mathbb {A}}_F $
.
Fix the self-dual Haar measures for
$F_v$
and
$E_v$
. Then

are the induced Haar measures on
$F_v^\times $
and
$E_v^\times $
. The subscripts will be omitted later in the paper. They induce the quotient measure on
$E_v^\times /F_v^\times {\ \simeq \ } U(1)({F_v})$
.
For
$\phi \in {\mathcal {S}}({\mathbb {V}}(E_v))$
, the Fourier transform of
$\phi $
(with respect to
$\psi $
and a Haar measure) is

We fix the self-dual Haar measure on
${\mathbb {V}}(E_v)$
.
2.5
Let
$G=U(1,1) $
be the unitary group over F of the standard skew-hermitian space over E of dimension
$2$
; that is, the skew-hermitian form is given by the matrix

Then
$w\in G(F)$
. We use
$w_v$
to denote the same matrix in
$G(F_v)$
For
$b\in {\mathbb {G}}_{a,F}$
, let

For
$a\in {\mathrm {Res}}_{E/F}{\mathbb {G}}_{m,E} $
, let

Let
$N=\{n(b):b \in {\mathbb {G}}_{a,F}\}\subset G$
,
$M=\{m(a):a \in {\mathrm {Res}}_{E/F}{\mathbb {G}}_{m,E} \}\subset G,$
and
$P=MN$
the subgroup of upper triangular matrices. Then G is generated P and w. The isomorphism
$N{\ \simeq \ } {\mathbb {G}}_{a,F}$
induces an additive character and a Haar measure on
$N({\mathbb {A}}_F)$
which we fix in this paper.
Let
$K^{\max }_v$
be the intersection of
$G(F_v) $
with the standard maximal compact subgroup
${\mathrm {GL}}_{2}(E_v)$
. Then
$K^{\max }_v$
is a maximal compact subgroup of
$G(F_v) $
. For
$v\in \infty $
,
$K^{\max }_v$
is the group of matrices

where
$k_1,k_2\in E_v$
are of norm 1. We have the Iwasawa decomposition

2.6
For a place v of F, the local modulus character of
$G(F_v) $
is given by

if
$g=n(b)m(a)k$
with
$k\in K^{\max }_v$
under the Iwasawa decomposition. The global modulus character
$\delta $
of
$G({\mathbb {A}}_F) $
is the product of the local ones. Since we will use results in [Reference YuanYZ18, Reference Yuan, Zhang and ZhangYZZ13], where the subgroup
${\mathrm {SL}}_2 \subset G $
is used, we remind the reader that our modulus character, when restricted to
${\mathrm {SL}}_2(F_v)\subset G(F_v)$
, is the square of the one in loc. cit..
2.7
For
${\mathfrak {w}}=({\mathfrak {w}}_v)_{v\in \infty }$
, where
${\mathfrak {w}}_v$
is a pair of integers, let
$ {\mathcal {A}} (G,{\mathfrak {w}}) $
be the space of smooth automorphic forms for G of weight
${\mathfrak {w}}$
. Let
${\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$
be the subspace of holomorphic automorphic forms. A characterization is as follows. For
$v\in \infty $
,
$t\in F_{v, \geq 0}$
and a pair of integers
$(w_1,w_2) $
, the standard holomorphic
$\psi _{v,t}$
-Whittaker function on
$G(F_v)$
of weight
$(w_1,w_2)$
is

for
$g=n(b)m(a)[k_1,k_2]$
under the Iwasawa decomposition. For
$t\in {\mathbb {A}}_{F,\infty }$
, let

An automorphic form f on
$G({\mathbb {A}}_F)$
is holomorphic of weight
${\mathfrak {w}}$
if for
$t\in F_{>0}\cup \{0\}$
, its
$\psi _t$
-Whittaker function is a tensor of the finite and infinite component:
$f_t= f_t^\infty \otimes W^{{\mathfrak {w}}}_{\infty , t}$
, and for other
$t\in F$
, its
$\psi _t$
-Whittaker function is 0. In this case, we call the locally constant function
$f_t^\infty $
on
$G({\mathbb {A}}_F^\infty )$
the t-th Fourier coefficient of f. (Then
$f_t^\infty (1)$
is the t-th Fourier coefficient in the sense of classical modular forms.)
For a subfield
$C\subset {\mathbb {C}}$
, let

be the C-subspace of automorphic forms f whose Fourier coefficients take values in C. (In the sense of classical modular forms, it means that the coefficients of the q-expansion of f along all cusps are in C.) Taking Fourier coefficients defines an embedding of C-vector spaces

where
${\mathrm {LC}}\left ( G({\mathbb {A}}_F^\infty ), C\right )$
means locally constant functions on
$G({\mathbb {A}}_F^\infty )$
valued in
$ C$
. For a C-vector space X, we have the induced embedding

Define the t-th Fourier coefficient of an element in
${\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})_{C}\otimes _C X$
to be the t-th component of its image.
2.8
Let
${\mathbb {V}}/{\mathbb {A}}_E$
be a hermitian space. For a character
$\chi _{_{\mathbb {W}}}$
of
$E^\times \backslash {\mathbb {A}}_E^\times $
such that
$\chi _{_{\mathbb {V}},v}|_{ F_v^\times }=\eta _v^{\dim {\mathbb {V}}}$
for every place v of F, the Weil representation
$\omega =\omega _{_{\mathbb {V}}}$
on
${\mathcal {S}}( {{\mathbb {V}}})$
is the restricted tensor product of local Weil representations of
$G(F_v)\times U({{\mathbb {V}}(E_v)})$
on
${\mathcal {S}}({{\mathbb {V}}(E_v)})$
. The local Weil representation (which we still denote by
$ \omega $
instead of
$ \omega _v$
if the meaning is clear from the context) of
$G(F_v)$
is defined as follows: for
$\phi \in {\mathcal {S}}( {{\mathbb {V}}(E_v)}) $
,

Here,
$\gamma _{{\mathbb {V}}(E_v)}$
is the Weil index associated to
$\psi _v$
and
${{\mathbb {V}}(E_v)}$
.
2.9
For
$v\in \infty $
, define
${\mathfrak {k}}^{\chi _{_{\mathbb {V}},v}}$
to be the unique integer such that

for
$z\in E_v$
of norm 1. Define
${\mathfrak {w}}_{\chi _{_{\mathbb {V}}}}=({\mathfrak {w}}_{\chi _{_{\mathbb {V}}},v})_{v\in \infty }$
, where

3 Theta-Eisenstein series
First, we recall basic knowledge about Eisenstein series and theta series. Then we set up basic properties of theta-Eisenstein series (i.e., linear combinations of products of theta series and Eisenstein series). Finally, we study two kinds of holomorphic projections of theta-Eisenstein series. The origin of theta-Eisenstein series is in the work of Gross and Zagier[Reference Gross and ZagierGZ86]. We largely follow the works of Yuan, S. Zhang and W. Zhang [Reference YuanYua22, Reference YuanYZ18, Reference Yuan, Zhang and ZhangYZZ13].
3.1 Eisenstein series and theta series
Let
${\mathbb {W}}$
be a hermitian space over
$ {\mathbb {A}}_E$
(with respect to the extension
$E/F$
). Let
$\chi _{_{\mathbb {W}}}$
be a character of
$E^\times \backslash {\mathbb {A}}_E^\times $
such that
$\chi _{_{\mathbb {V}}}|_{{\mathbb {A}}_F^\times }=\eta $
. We have the Weil representation
$\omega _{_{\mathbb {W}}}$
, which we simply denote by
$\omega $
.
3.1.1 Local Whittaker integrals
Let v be a place of F. For
$t\in F_v$
,
$\phi \in {\mathcal {S}}( {\mathbb {W}}_v )$
and
$g\in G(F_v)$
, define the Whittaker integral

We immediately have the following equations:


Since
$wn(b)m(a)=m(\overline a^{-1})wn(b{\mathrm {Nm}}(a)^{-1})$
,
$ m(\overline a^{-1})n(b)=n(b')m(\overline a^{-1})$
for some
$b'$
, and
$\chi _{_{\mathbb {W}},v}({\mathrm {Nm}}(a))=1$
, a direct computation gives

Lemma 3.1.1 [Reference IchinoIch04, Proposition 6.2][Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.7 (2)].
Let
$t\neq 0$
.
(1) The set
$ {\mathbb {W}}_v^t$
is either empty or consists of one orbit of
$U( {\mathbb {W}}_v)$
.
(2) If
$ {\mathbb {W}}_v^t$
is empty, then
$W_{v,t}(0,g,\phi )=0$
. Otherwise, for
$x\in {\mathbb {W}}_v^t$
, we have

for a nonzero constant
$\kappa $
.
(3) If
$\dim {\mathbb {W}}=1$
, with the measure fixed in 2.4,
$\kappa =\frac {\gamma _{{\mathbb {W}}_v}}{L(1,\eta _v)}$
.
Assume
$\dim {\mathbb {W}}=1$
.
Lemma 3.1.2. Assume that v is a finite place and
$\phi (0)=0$
. Then for t small enough,
$W_{v,t} (s, g, \phi )=W_{v,0} (s, g, \phi )$
and is a holomorphic function.
Proof. The proof is by the reasoning as the proof of [Reference QiuQiu21, Lemma 4.2.4 (2)]
We define the following normalization (following [Reference Yuan, Zhang and ZhangYZZ13, 6.1.1]):

For
$t=0$
,
$W_{v,t} (s, g, \phi ) $
has a possible pole at
$s=0$
. And we take a different normalization (following [Reference Yuan, Zhang and ZhangYZZ13, 6.1.1], and taking care of the difference between the modulus characters mentioned in 2.6),

Here, if v is a finite place,
${\mathrm {Diff}}_v$
is the different of
$F_v/{\mathbb {Q}}_v$
and
${\mathrm {Disc}}_v$
is the discriminant of
$E_v/F_v$
, and if
$v\in \infty $
,
${\mathrm {Diff}}_{v}={\mathrm {Disc}}_v=1$
.
Lemma 3.1.3.
(1) There is an analytic continuation of
$ W^\circ _0(s,g,\phi )$
to
${\mathbb {C}}$
such that
$ W^\circ _0(0,g,\phi )=\omega (g)\phi (0)$
.
(2) If
$E_v/{\mathbb {Q}}_{v}$
is unramified where
${\mathbb {W}}_v=E_v$
with
$q={\mathrm {Nm}}$
, and
$\phi =1_{{\mathcal {O}}_{E_v}}$
, then
$ W^\circ _0(s,g,\phi )=\delta _v(g)^{-s}\omega (g)\phi (0).$
(3) If
$v\in \infty $
and
$\phi $
is the standard Gaussian function, then
$ W^\circ _0(s,g,\phi )=\delta _v(g)^{-s}\omega (g)\phi (0).$
Proof. (1) follows from [Reference Yuan, Zhang and ZhangYZZ13, Proposition 6.1]. (2) follows from [Reference TanTan99, Proposition 2.1]. (3) follows from [Reference YuanYZ18, Lemma 7.6 (1)] (or its proof).
3.1.2 Siegel-Eisenstein series
Now we come back to a general
${\mathbb {W}}$
. For
$\phi \in {\mathcal {S}}({\mathbb {W}})$
, we have a Siegel-Eisenstein series of G:

which is absolutely convergent if
${\mathrm {Re}} s>1-\dim {\mathbb {W}}/2$
and has a meromorphic continuation to the entire complex plane [Reference TanTan99]. Moreover, it is holomorphic at
$s=0$
[Reference TanTan99, Proposition 4.1].
Let
$E_t(s,g,\phi )$
be the
$\psi _t$
-Whittaker function of
$E(s,g,\phi )$
. Let
$W_{t} (s, g, \phi )$
be the global counterpart of the Whittaker integral (3.1) so that if
$\phi $
is a pure tensor, then
$W_{t} (s, g, \phi )$
is the product of
$W_{v,t} (s, g_v, \phi _v)$
over all places of F. By the non-vanishing of
$L(1,\eta )$
and Lemma 3.1.3,
$W_{0} (s, g, \phi )$
has a meromorphic continuation to the entire complex plane, which is holomorphic at
$s=0$
. Then we have


By Lemma 3.1.1 (2) and (3.8), for
$h\in U({\mathbb {W}})$
, we have

For
$t\neq 0$
and a pure tensor
$\phi $
, define

Extend the definition to all Schwartz functions by linearity. Then

3.1.3 Coherent case: Siegel-Weil formula
Assume that
${\mathbb {W}}=W({\mathbb {A}}_E)$
is coherent. For
$\phi \in {\mathcal {S}}(W ({\mathbb {A}}_E))$
and
$(g,h)\in G({\mathbb {A}}_F ) \times U({\mathbb {W}} ) $
, we define a theta series, which is absolutely convergent:

Then
$ \theta (g,h,\phi )$
is smooth, slowly increasing and
$ G(F)\times U(W) $
-invariant.
Assume that W is anisotropic. Then
$E(s,g,\phi )$
is holomorphic at
$s=0$
, and the following equation is a special case of the regularized Siegel-Weil formula [Reference IchinoIch04, Theorem 4.2]:

where
$\kappa =2$
if
$ \dim W=1$
and
$\kappa =1$
if
$ \dim W>1$
.
3.1.4 Incoherent case: derivative
Assume that
${\mathbb {W}}$
is incoherent. By Lemma 3.1.1 (1), for
$t\neq 0$
, the summand in (3.12) corresponding to v is nonzero only if t is represented by
${\mathbb {W}}^v$
. By the incoherence,
${\mathbb {W}}_v$
does not represent t. In particular, we have the following lemma.
Lemma 3.1.4. For v split in E, the summand in (3.12) corresponding v is
$0$
.
Assume that
$\dim {\mathbb {W}}=1$
. Assume that
$\phi $
is a pure tensor.
First, assume that
$t\neq 0$
. By the product formula for Hasse invariant and the Hasse principle, if t is represented by
${\mathbb {W}}^v$
, then v is nonsplit in E and t is represented by the v-nearby hermitian space W of
${\mathbb {W}}$
. See 2.2. By Lemma 3.1.1 (2) and Lemma 3.1.4, we have

for
$h\in U({\mathbb {W}}_v)$
, where v is split in E.
Now we consider the constant term. Let

where
${\mathrm {Disc}}_E,{\mathrm {Disc}}_F\in {\mathbb {Z}}$
are the discriminants of E and F over
${\mathbb {Q}}$
. By [Reference YuanYZ18, p 586] (note the difference by
$2$
between the s-variables in the L-factors in loc. cit. and (3.6)),

3.2 Theta-Eisenstein series
From now on, always assume
$\dim {\mathbb {W}}=1$
.
3.2.1 Definition
Let
$V^\sharp /E$
be a hermitian space of dimension
$n>0$
. Let
$\chi _{V^\sharp }$
be a character of
$E^\times \backslash {\mathbb {A}}_E^\times $
such that
$\chi _{V^\sharp }|_{{\mathbb {A}}_F^\times }=\eta ^{n}$
. Let
${\mathbb {V}}= {\mathbb {W}}\oplus V^\sharp ({\mathbb {A}}_E)$
be the orthogonal direct sum and
$\chi _{\mathbb {V}}=\chi _{_{\mathbb {W}}}\chi _{V^\sharp }$
. We have the corresponding Weil representations. Below, we shall use
$\omega $
to denote a Weil representation if the hermitian space is indicated in the context – for example, by the function that it acts on.
For
$ \phi \in {\mathcal {S}}\left ( {\mathbb {V}}\right )$
, we define a theta-Eisenstein series
$ {\theta E}(s,g,\phi )$
on G associated the to the orthogonal decomposition
${\mathbb {V}}= {\mathbb {W}}\oplus V^\sharp ({\mathbb {A}}_E)$
:

If
$\phi =\phi _1\otimes \phi _2$
with
$\phi _1\in {\mathcal {S}}\left ( {\mathbb {W}}\right )$
and
$\phi _2\in {\mathcal {S}}\left ( V^\sharp ({\mathbb {A}}_E)\right )$
, then

For
$t\in F$
, let
${\theta E} _t (s,g,\phi )$
be the
$\psi _t$
-Whittaker function of
${\theta E} (s,g,\phi )$
.
3.2.2 Coherent case
Assume that
${\mathbb {W}}=W({\mathbb {A}}_E)$
is coherent. The regularized Siegel-Weil formula (3.13) immediately implies the following ‘mixed Siegel-Weil formula’:

(We will prove an arithmetic analog of (3.19) for
${\mathbb {W}}$
being incoherent in 5.2.) Then for
$t\in F$
,

For
$\phi $
invariant by
$U(W(E_v)), v\in \infty $
, the integration in (3.20) is a finite sum.
Lemma 3.2.1. Let
$t\in F_{>0}$
and
$\phi $
a pure tensor. Let u be a finite place of F,
$O\subset {\mathbb {V}}_u$
an open compact neighborhood of
$0$
and
$\phi ^O= \phi ^u\otimes (\phi _u 1_{{\mathbb {V}}_u-O})$
. Given
$g\in G({\mathbb {A}}_F^{u })P(F_u)$
, for O small enough, we have
$ {\theta E} _t (0,g,\phi ) = {\theta E} _t (0,g,\phi ^O) .$
Proof. Write
$g_u=m(a)n(b)$
, where
$a\in E_u^\times $
. See 2.5. Then
$\{a h_u^{-1}x:x\in V^t,h\in U({\mathbb {V}})\}\subset {\mathbb {V}}_u^{a^2t} $
. The latter is closed in
${\mathbb {V}}_u$
and does not contain 0. Thus,
$O\cap \{ah_u^{-1}x:x\in V^t,h\in U({\mathbb {V}})\}=\emptyset $
if O is small enough. Then the lemma follows from (3.20), the remark below it, and the definition of the Weil representation in 2.8.
3.2.3 Incoherent case
Assume that
${\mathbb {W}}$
is incoherent.
For a place v of F nonsplit in E, let W be the v-nearby hermitian space of
${\mathbb {W}}$
. For
$\phi =\phi _{1}\otimes \phi _{2}$
with
$\phi _{1}\in {\mathcal {S}}\left ( {\mathbb {W}}_v \right )$
,
$\phi _2\in {\mathcal {S}}\left ( V^\sharp (E_v)\right )$
and
$x=(x_1,x_2)\in V:= W(E_v)\oplus V^\sharp (E_v)$
with
$ x_1\neq 0,$
let

This is a local analog of (3.18). Extend this definition to
$ {\mathcal {S}}\left ( {\mathbb {W}}_v \right )\otimes {\mathcal {S}}\left ( V^\sharp (E_v)\right )\subset {\mathcal {S}}\left ( {\mathbb {V}}(E_v) \right )$
by linearity. The inclusion is an equality unless
$v\in \infty $
. However, this subspace is enough for our purpose. (Besides, there is another definition of
${W\theta }_{v,x}$
for the whole
$ {\mathcal {S}}\left ( {\mathbb {V}}(E_v) \right )$
. We will not need it.) For v nonsplit in E and
$t\neq 0$
, define

Note that the analog of (3.12) does not hold.
We study
${W\theta }_{v,x} '(0,g,\phi )$
following [Reference Yuan, Zhang and ZhangYZZ13]. Indeed, the computation is only on the Eisenstein (i.e., Whittaker) part. We remind the reader of the difference between the modulus characters mentioned in 2.6. By (3.3), (3.4) and Lemma 3.1.1, we have the following lemma, which says that under the action of
$P(F_v)$
,
${W\theta }_{v,x} '(s_0, g , \phi )$
behaves in the same was as the Weil representation.
Lemma 3.2.2 [Reference Yuan, Zhang and ZhangYZZ13, Lemma 6.6].
The following relations hold:


Corollary 3.2.3. For
$a\in E^\times $
,
${\theta E}^{\prime }_{t} (0,m(a)g,\phi ) (v)={\theta E}^{\prime }_{a^2t} (0,g,\phi ) (v)$
.
3.3 Holomorphic projections
We define quasi-holomorphic projection and cuspidal holomorphic projection, and compare them for theta-Eisenstein series (Lemma 3.3.3). After imposing Gaussian condition at infinite places in 3.3.4, we make the comparison more explicit in (3.31). Finally, after imposing the incoherence condition, we explicitly compute the quasi-holomorphic projection (Proposition 3.3.13).
3.3.1 Definitions
For
$v\in \infty $
, let
${\mathfrak {w}}_v$
be a pair of integers whose sum
$|{\mathfrak {w}}_v|$
is
$\geq 2$
. For
$t\in F_{v,>0}$
, let
$W^{{\mathfrak {w}}_v}_{v, t}$
be the standard holomorphic Whittaker function of weight
${\mathfrak {w}}_v$
as in 2.7. Then

For
$t\in F_{v,>0}$
, a
$\psi _{v,t}$
-Whittaker function W on
$G(F_v)$
, and
$g\in G(F_v)$
, define

If
$W_{s}$
has a meromorphic continuation to
$s=0$
, define the quasi-holomorphic projection

of W of weight
${\mathfrak {w}}_v$
. Here,
$\widetilde {\lim \limits _{s\to 0}}$
denotes the constant term at
$s=0$
.
Let
${\mathfrak {w}}=({\mathfrak {w}}_v)_{v\in \infty }$
, where
${\mathfrak {w}}_v$
is a pair of integers. For a continuous function
$f:N(F)\backslash G({\mathbb {A}}_F)\to {\mathbb {C}}$
with
$\psi _t$
-Whittaker function
$f_t$
for
$t\in F_{>0}$
, let
$f_{t,{\mathrm {qhol}}}$
be the quasi-holomorphic projection of
$f_t$
of weight
${\mathfrak {w}}$
at all infinite places (if it is well defined).
For an automorphic form f on
$G({\mathbb {A}}_F)$
, the cuspidal holomorphic projection
$f_{{\mathrm {chol}}}$
of weight
${\mathfrak {w}}$
of f is the
$L^2$
-orthogonal projection of f to the subspace
${\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$
of cusp forms – that is, for every cusp form
$\phi \in {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$
, the Petersson inner product
$\langle {f,\phi } \rangle $
equals
$\langle {f_{{\mathrm {chol}}},\phi } \rangle $
.
Lemma 3.3.1 [Reference LiuLiu11b, Proposition 6.2][Reference Yuan, Zhang and ZhangYZZ13, Proposition 6.12].
Assume that there exists
$\epsilon>0$
such that for
$v\in \infty $
and
$a \in E_v^\times $
with
$|a|_{E_v}\to \infty $
, we have

where
$m(a)$
is as in 2.5. Then for
$t\in F_{>0}$
,
$f_{t,{\mathrm {qhol}}}$
is well defined and
$f_{t,{\mathrm {qhol}}}=f_{{\mathrm {chol}},t}$
.
3.3.2 Holomorphic projections of
${\theta E} ' (0,g,\phi )$
Let
${\mathfrak {w}}={\mathfrak {w}}_{\chi _{_{\mathbb {V}}}}$
be as in 2.9. Then
$|{\mathfrak {w}}_v|=n+1$
. Holomorphic projections below are of weight
${\mathfrak {w}}$
. Retrieve the notations in 3.2.1. Let
${\theta E}^{\prime }_{{\mathrm {chol}}}(0,g,\phi )$
be the cuspidal holomorphic projection of the derivative
${\theta E}'(0,g,\phi )$
. For
$t\in F_{>0}$
, let
${\theta E}^{\prime }_{{\mathrm {chol}},t}(0,g,\phi )$
be its
$\psi _t$
-Whittaker function. Let
${\theta E}_t(s,g,\phi )$
be the
$\psi _t$
-Whittaker function of
${\theta E}(s,g,\phi )$
. Let
${\theta E}_{t,{\mathrm {qhol}}}'(0,g,\phi )$
be the quasi-holomorphic projection of
$ {\theta E}_t'(0,g,\phi )$
if it is well defined. The difference between
${\theta E}^{\prime }_{{\mathrm {chol}},t}(0,g,\phi )$
and
${\theta E}_{t,{\mathrm {qhol}}}'(0,g,\phi )$
is given as follows.
For
$\phi =\phi _1\otimes \phi _2\in \overline {\mathcal {S}}\left ( {\mathbb {V}}\right )$
with
$\phi _1\in {\mathcal {S}}\left ( {\mathbb {W}}\right )$
and
$\phi _2\in \overline {\mathcal {S}}\left ( V^\sharp ({\mathbb {A}}_E)\right )$
, define

which is the product of
$E_0(s,g,\phi _1)$
and the constant term of
$\theta (g,\phi _2)$
. See (3.9). The definition extends to general
$\phi \in {\mathcal {S}}\left ( {\mathbb {V}}\right )$
by linearity. By (3.3) and (3.4), we can define an Eisenstein series

For
$t\in F_{>0}$
, let
$J_t(s,g,\phi )$
be its
$\psi _t$
-Whittaker function. Let
$J_{t,{\mathrm {qhol}}}'(0,g,\phi )$
be the quasi-holomorphic projection of the derivative
$J_t'(0,g,\phi )$
if it is well defined.
Remark 3.3.2. In the notations of [Reference TanTan99],
$\delta (g)^s \omega (g)\phi (0)$
is in the degenerate principal series
$I(n/2+s,\chi _{_{\mathbb {V}}})$
, while
$W_0(s,g,\phi _1) \omega (g)\phi _2(0)$
is in
$I(n/2-s,\chi _{_{\mathbb {V}}})$
by (3.3) and (3.4).
Lemma 3.3.3. If one of
$ {\theta E}_{t,{\mathrm {qhol}}}'(0,g,\phi )$
and
$J_{t,{\mathrm {qhol}}}'(0,g,\phi )$
is well defined, then so is the other one. In this case,
${\theta E}^{\prime }_{{\mathrm {chol}},t}(0,g,\phi )={\theta E}_{t,{\mathrm {qhol}}}'(0,g,\phi )-J_{t,{\mathrm {qhol}}}'(0,g,\phi )$
.
Proof. By the same proof of [Reference Yuan, Zhang and ZhangYZZ13, Lemma 6.13],
${\theta E}' (0,m(a)g,\phi )-{\theta E}_{00}'(0,m(a)g,\phi )$
is exponentially decay, and
$J' (0,m(a)g,\phi )-{\theta E}_{00}'(0,m(a)g,\phi )$
is exponentially decay up to the derivative at
$s=0$
of the intertwining part of the constant term. The intertwining part of the constant term lies in
$I(-n/2\pm s,\chi _{_{\mathbb {V}}})$
so that its derivative at
$s=0$
has growth rate
$O_g\left (|a | _{E_v}^{-n/2+1/2+\epsilon } \right )$
(in the notations in Lemma 3.3.1). In particular, both differences satisfy the growth condition in Lemma 3.3.1. Thus,
${\theta E}' (0,g,\phi )- J' (0,g,\phi ) $
satisfies the growth condition in Lemma 3.3.1. Since the cuspidal holomorphic projection of the Eisenstein series
$J(s,g,\phi )$
is
$0$
, the lemma follows.
3.3.3 A new Eisenstein series
We introduce a new Eisenstein series in order to compute
$J'(0,g,\phi )$
. For
$\phi _v=\phi _{v,1}\otimes \phi _{2,v}\in {\mathcal {S}}\left ( {\mathbb {V}}(E_v)\right )$
with
$\phi _{v,1}\in {\mathcal {S}}\left ( {\mathbb {W}}_v\right )$
and
$\phi _{2,v}\in {\mathcal {S}}\left ( V^\sharp (E_v)\right )$
, define a function on
$G(F_v)\times V^\sharp (E_v)$
:

For the moment, we only need

Extend this definition to the whole
$ {\mathcal {S}}\left ( {\mathbb {V}}(E_v)\right )$
linearly.
By (3.3), (3.4) and Lemma 3.1.3 (1), a direct computation shows the following lemma.
Lemma 3.3.4. The function
$c(g,\phi _v)$
on
$G(F_v)$
is in the same principal series as
$ \omega (g)\phi _v(0)$
; that is,
(1)
$c(m(a) g , \phi _v )= \chi _{_{\mathbb {V}},v} (a)|\det a|_{E_v }^{\dim {\mathbb {V}}/2} c(g,\phi _v)$
for
$ a\in E_v^\times $
;
(2)
$c(n(b) g , \phi _v )= c(g,\phi _v)$
for
$ b\in F_v$
.
Thus, we can define the following Eisenstein series in the case that
$\phi $
is a pure tensor:

Lemma 3.3.5. For all but finitely many finite places,
$c(g,\phi _v)=0$
for all g.
Proof. If
$v\in \infty $
, by Lemma 3.1.3 (3) which says
$ W^\circ _0(s,g,\phi )=\delta _v(g)^{-s}\omega (g)\phi (0),$
we clearly have
$c(g,\phi _v) = 0$
. The same is true if
$E_v/{\mathbb {Q}}_{v}$
is unramified and
$\phi =1_{{\mathcal {O}}_{E_v}}$
by Lemma 3.1.3 (2). These cases cover all but finitely many finite places.
Let

where the sum is over these finite places of F. Let
$C_t(s,g,\phi )$
be the
$\psi _t$
-Whittaker function of
$C(0,g,\phi )$
. The definitions can be obviously extended to the whole
$ {\mathcal {S}}\left ( {\mathbb {V}}\right )$
by linearity.
By (3.16), a direct computation shows that

3.3.4 Gaussian functions and holomorphy
Below in this section, assume that
${\mathbb {V}} $
is totally positive definite and
$\phi = \phi _\infty \otimes \phi ^\infty \in \overline {\mathcal {S}}\left ( {\mathbb {V}}\right )$
. (So
$ \phi _\infty $
is Gaussian. See 2.3.) Let
$v\in \infty .$
Then

for
$[k_1,k_2]\in K_v^{\max }$
as in 2.5 if
${\mathfrak {w}}_v=(w_1,w_2)$
. (Indeed, first check (3.26) for
$g=w_v$
and
$g\in K_v^{\max }$
being diagonal, then for
$g\in K_v^{\max }$
being anti-diagonal, and finally for general
$g\in K_v^{\max }$
.) Then by the Iwasawa decomposition, it is easy to check that for
$g\in G(F_v), x\in {\mathbb {V}}(E_v)$
,

By (3.2) combined with (3.25), (3.3) and (3.4),
$W_{v,t} (0, \cdot , \phi _v)$
is a multiple of
$ W^{{\mathfrak {w}}_v}_{v, t} $
. Then by [Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.11 (2) (4)], we have


Lemma 3.3.6. Both
$E(0,g,\phi )$
and
$C(0,g,\phi )$
are holomorphic of weight
${\mathfrak {w}}$
.
Proof. For
$E(0,g,\phi )$
, use (3.8), (3.9), (3.26), (3.27) and (3.28). For
$C(0,g,\phi )$
, by Lemma 3.1.3 (3),
$c(g,\phi _v)=0$
for
$v\in \infty $
. Thus,
$C(s,g,\phi )(v)=0$
for
$v\in \infty $
. The rest of the proof is the same as the proof for
$E(0,g,\phi )$
.
For
$t\in F^\times $
, let
$ E_t'(0,g,\phi )(v)$
be as in (3.11) so that we have the decomposition (3.12). Let

By (3.27) and (3.28), if
$t\in F_{>0}$
, then
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$
is a multiple of
$W^{{\mathfrak {w}}}_{\infty , t}(g_\infty )$
; otherwise,
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )=0$
. We call
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$
the holomorphic part of the Whittaker function
$E_t'(0,g,\phi )$
.
3.3.5 Properties of Eisenstein series
We list some properties of the above Eisenstein series for later use. The reader may skip these properties for the moment.
By [Reference YuanYZ18, Lemma 7.6 (2)] (or its proof) and taking care of the difference between the modulus characters mentioned in 2.6, we have the following lemma.
Lemma 3.3.7. Let
$E_v/F_v$
be split,
${\mathbb {W}}_v=E_v$
and
$q={\mathrm {Nm}}$
. Assume
$\phi _v=\phi _{v,1}\otimes \phi _{v,2}$
, where
$\phi _{v,1}=1_{{\mathcal {O}}_{E_v}}$
and
$\phi _{v,2}\in {\mathcal {S}}\left ( V^\sharp (E_v)\right )$
. Then
$c(1,\phi _v) =2 \log |{\mathrm {Diff}}_{v}|_v \phi _v(0),$
where
${\mathrm {Diff}}_{v}$
is the different of
$F_v/{\mathbb {Q}}_{v}$
.
We omit the routine proof of the following analog of (3.2).
Lemma 3.3.8. For a place v of F and
$k\in K^{\max }_v$
, we have
$E(s,g,\omega (k)\phi )=E(s,gk,\phi )$
. The same relation holds for
$C(s,g,\phi )$
,
${\theta E}(s,g,\phi ) $
,
${\theta E}^{\prime }_{{\mathrm {chol}}}(0,g,\phi ),$
and their t-th Whittaker/Fourier coefficients, and
$E^{\prime }_{t}(0,g,\phi ) (v)$
(thus,
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$
) for
$t\in F^\times $
.
Lemma 3.3.9.
(1) We have
$E_0(0,g,\phi )= \omega ( g)\phi (0)$
.
(2) If, moreover,
${\mathbb {V}}$
is incoherent, then for a finite place v,
$C_0(0,g,\phi )(v)= c(g_v,\phi _v)\omega ( g^v)\phi ^v (0)$
.
(3) In (2), assume that
$\phi _{{{v}}}$
is supported outside
$V^\sharp (E_{v})$
for v in a set S of two places of F and
$g\in P ({\mathbb {A}}_{F,S })G({\mathbb {A}}_F^{S })$
. Then
$E_0(0,g,\phi )=0$
and
$C_0(0,g,\phi )=0$
.
Proof. (1) If
$F={\mathbb {Q}},n=1$
and
${\mathbb {V}}^\infty $
is not split at some finite place, it is proved in [Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.9 (3)]. If
$F\neq {\mathbb {Q}}$
so that we have at least 2 infinite places, its proof is similar to the one in [Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.9 (3)] by using (3.9) and (3.28) (for
$t=0$
).
(2) The proof is similar, with the fact that
${\mathbb {V}}$
is not split at (at least) 2 places outside v by the incoherence. See also [Reference YuanYua22, p 65-66].
(3) follows from (1)(2) directly.
3.3.6 Compute
$J_{t,{\mathrm {qhol}}}'(0,g,\phi )$
Let
$t\in F_{>0}$
. For
$v\in \infty $
, let
$E_{t,{\mathrm {qhol}}}'(0,g,\phi )(v)$
be the quasi-holomorphic projection of
$ E_t'(0,g,\phi )(v)$
. By (3.24), Lemma 3.3.6 and the discussion below it, to compute
$J_{t,{\mathrm {qhol}}}'(0,g,\phi )$
, we only need to compute
$E_{t,{\mathrm {qhol}}}'(0,g,\phi )(v)$
.
Consider the quasi-holomorphic projection
$W_{v,t,{\mathrm {qhol}}}' (0, g, \phi _v)$
of
$W_{v,t}' (0, g, \phi _v)$
. By definition, it is a multiple of
$W^{{\mathfrak {w}}_v}_{v, t}(g)$
. Then by (3.27),

is a well defined constant
$b_{v,t}$
. We define

Remark 3.3.10. The constant
${{\mathfrak {b}}}$
can be explicitly computed using [Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.11] and [Reference YuanYua22, Lemma 3.3] in principle. For example, if
$ n=1$
, then
${\mathfrak {b}} =-({1+\log 4})$
. (This is twice of the corresponding number in [Reference YuanYua22, Lemma 3.3 (2)] due to the difference between the modulus characters mentioned in 2.6.) It is more complicated in general. The full computation could be tedious, and the result in a previous version of our paper actually contains a mistake. (Fortunately, we will not need the explict number of
${\mathfrak {b}}$
.) Ziqi Guo (student of Yuan, author of [Reference YuanYua22]) pointed this out to us and informed us that he will give full details on this in his upcoming work.
Lemma 3.3.11. We have
$b_{v,t}=b_{v,1}+\log | t|_v$
and
$b_{v,1}$
is independent of v.
Proof. The lemma follows direct computations with the following ingredients. For the equation, use (3.4) and (3.27). Note that the dependence on v is on the weight
${\mathfrak {w}}_v$
. We use (3.2) and (3.26) for
$g\in K_v^{\max }$
.
Then
$E_{t,{\mathrm {qhol}}}'(0,g,\phi )(v)=({{\mathfrak {b}}} + \log |t|_v) E_t(0,g,\phi ). $
Since both
$E(0,g,\phi )$
and
$C(0,g,\phi )$
in (3.24) are holomorphic of weight
${\mathfrak {w}}$
, we have

Combined with Lemma 3.3.3, we have

3.3.7 Quasi-holomorphic projection of
${\theta E} ' _t (0,g,\phi )$
Assume that
${\mathbb {W}}/{\mathbb {E}}$
is incoherent. For
$v\in \infty $
and W the v-nearby hermitian space of
${\mathbb {W}}$
, let
$V=W\oplus V^\sharp $
. For
$x\in V(E_v)-V^\sharp (E_v)$
, define

where
$x_1$
is the projection of x to
$W(E_v)$
(so that
$x_1\neq 0$
), and

(For
$s \in {\mathbb {C}}$
with
$ {\mathrm {Re}} s>-n$
,
$P_s(t)$
converges absolutely.) Define

Lemma 3.3.12. For
$s\in {\mathbb {C}}$
with
${\mathrm {Re}} s>0$
, (3.33) converges absolutely and
${\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$
is holomorphic on s. Moreover,
$ {\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$
admits a meromorphic continuation to
$\{s\in {\mathbb {C}}, {\mathrm {Re}} s>-1\}$
with at most a simple pole at
$s=0$
.
Then the constant term
$\widetilde {\lim \limits _{s\to 0}}{\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$
of
$ {\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$
at
$s=0$
is well defined (and used in the following proposition). We will prove Lemma 3.3.12 after Lemma 6.1.10, by comparing
${\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$
to a Green function with s-variable, which has a meromorphic continuation.
Proposition 3.3.13. Let
$t\in F_{>0}$
and let
$\phi \in \overline {\mathcal {S}}({\mathbb {V}})$
be a pure tensor.
(1) We have

Here,
$L_{\mathrm {f}}(s,\eta )$
is the finite part of
$L(s,\eta )$
.
(2) Assume that
$\phi _{{{v}}}$
is supported outside
$V^\sharp (E_{v})$
for v in a set S of two places of F and
$g\in P ({\mathbb {A}}_{F,S })G({\mathbb {A}}_F^{S })$
. Then we have

Proof. The proof of (1) is almost identical with [Reference YuanYZ18, Theorem 7.2] and is omitted. (Note that [Reference YuanYZ18, Assumption 7.1] in [Reference YuanYZ18, Theorem 7.2] is only used to identify the quasi-holomorphic projection with the cuspidal holomorphic projection and does not play a role in computing the quasi-holomorphic projection). (2) follows from (1) immediately.
Lemma 3.3.14. Let
$t\in F_{>0}$
and let
$\phi $
be a pure tensor such that
$\phi ^\infty $
is
${\overline {\mathbb {Q}}}$
-valued. Let u be a finite place of F of residue characteristic p. Let
$O\subset {\mathbb {V}}_u$
be an open compact neighborhood of
$0$
and
$\phi ^O= \phi ^u\otimes (\phi _u 1_{{\mathbb {V}}_u-O})$
. Given
$g\in G({\mathbb {A}}_F^{u })P(F_u)$
, for O small enough, we have

Proof. The proof is similar to the one of Lemma 3.2.1, except that we further need (3.22), (3.26), (3.33) and (3.34).
By (3.34), using (3.22) and (3.33), for
$h\in U({\mathbb {W}}_v)$
where v is split in E,

4 Special divisors
This section is about special divisors on unitary Shimura varieties. It consists of 4.1-4.4. First, we define their generating series which are modular. Second, we introduce their Green functions and show the modularity of the differences between the generating series of different kinds of Green functions. Third, we raise two modularity problems for their admissible extensions on integral models. Finally, we propose a precise conjecture and state our modularity theorems.
4.1 Generating series
Let
${\mathbb {V}} $
be a totally positive-definite incoherent hermitian space over
${\mathbb {A}}_E$
(with respect to the extension
$E/F$
) of dimension
$n+1$
where
$n>0$
. Fix an infinite place
$v_0 \in \infty $
of F. Let
$V_0$
be the unique hermitian space over E such that
$V_0({\mathbb {A}}_E^v){\ \simeq \ } {\mathbb {V}}^v$
and
$V_0(E_v)$
is of signature
$(n,1)$
. For an open compact subgroup K of
$U({\mathbb {V}}^\infty )$
, let
${\mathrm {Sh}}({\mathbb {V}})_K$
be the n-dimensional smooth unitary Shimura variety associated to
$U(V_0)$
of level K over E (see [Reference Liu, Tian, Xiao, Zhang and ZhuLTX+22, 3.2]), which we allow to be a Deligne-Mumford stack. (It is expected that
${\mathrm {Sh}}({\mathbb {V}})_K$
does not depend on the choice of
$v_0$
. See [Reference Li and LiuLL21, Remark 1.2].) See (4.4) for its usual complex uniformization.
-
• From now on, we always assume that
${\mathrm {Sh}}({\mathbb {V}})_K$ is proper.
Equivalently,
$F\neq {\mathbb {Q}}$
, or
$F={\mathbb {Q}},n=1$
and
${\mathbb {V}}^\infty $
is not split at some finite place.
4.1.1 Simple special divisors
Let
${\mathbb {V}}^{\infty }_{>0}\subset {\mathbb {V}}^{\infty }$
be the subset of x’s such that
${q(x)}\in F_{>0}$
. For
$x\in {\mathbb {V}}^{\infty }_{>0}$
, let
$ x^\perp $
be the orthogonal complement of
${\mathbb {A}}_E ^\infty x $
in
${\mathbb {V}}^\infty $
. Regard
$U\left ( x^{\perp } \right )$
as a subgroup of
$U({\mathbb {V}}^\infty )$
. Then we have a finite morphism

explicated in [Reference KudlaKud97a, (2.4)] and [Reference Li and LiuLL21, Definition 4.1]. The proper pushforward defines a divisor
$Z(x)_K $
on
${\mathrm {Sh}}({\mathbb {V}})_K$
that is called a simple special divisor. The following observation is trivial.
Lemma 4.1.1. We have
$Z(x)_K=Z(kxa)_K$
for every
$a\in E^\times , k\in K$
.
Let
$L_K$
be the Hodge line bundle on
${\mathrm {Sh}}({\mathbb {V}})_K$
. See 4.2.1 for the description in terms of the complex uniformization of
${\mathrm {Sh}}({\mathbb {V}})_K$
. Let
$c_1(L_K^\vee )$
be the first Chern class of the dual of
$ L_K$
, and
$[Z(x)_K]\in {\mathrm {Ch}}^{1}({\mathrm {Sh}}({\mathbb {V}})_K) $
the class of
$Z(x)_K$
. For
$ \phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$
the subspace of K-invariant functions, define a formal generating series of divisor classes:

where
$ x_\infty \in {\mathbb {V}}_\infty $
such that
$q(x_\infty )={q(x)}\in F_{>0}$
.
Let
${\mathfrak {w}}={\mathfrak {w}}_{\chi _{_{\mathbb {V}}}}$
which is defined in 2.9.
Theorem 4.1.2 [Reference LiuLiu11a, Theorem 3.5].
For every
$ \phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K} $
, we have

Note that
$ \dim {\mathrm {Ch}}^{1}({\mathrm {Sh}}({\mathbb {V}})_K)_{{\mathbb {C}}} <\infty .$
4.1.2 Weighted special divisors
For
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$
and
$t\in F_{>0}$
, define the weighted special divisor

which is a finite sum. Lemma 4.1.1 implies the following lemma.
Lemma 4.1.3. For every
$a\in E^\times $
,
$Z_t(\phi )_K=Z_{a^2t}(\omega (a)\phi )_K.$
Let
$\phi = \phi _\infty \otimes \phi ^\infty $
be as in 2.3. We define another weighted special divisor

By (3.26), for
$g\in G({\mathbb {A}}_F)$
, we have

Then
$[Z_t\left ( \omega (\cdot )\phi ^\infty \right )_K] $
is the t-th Fourier coefficient of
$z\left (\omega (\cdot )\phi \right )_K$
. See 2.7.
Lemma 4.1.4. Let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) $
be a pure tensor. Let u be a finite place of F,
$O\subset {\mathbb {V}}_u$
an open compact neighborhood of
$0$
and
$\phi ^O=\phi ^u\otimes (\phi _u 1_{{\mathbb {V}}_u-O})$
. Given
$g\in G({\mathbb {A}}_F^{u })P(F_u)$
, for O small enough, if K fixes
$\phi ^O$
, then
$ Z_t(\omega (g)\phi ) _K= Z_t(\omega (g)\phi ^O) _K.$
Proof. The lemma is an analog to Lemma 3.2.1, and the proof is also similar. We record the proof for the reader’s convenience. Write
$g_u=m(a)n(b)$
, where
$a\in E_u^\times $
. See 2.5. Then
$\{a x:x\in {\mathbb {V}}_u^t\}\subset {\mathbb {V}}_u^{a^2t} $
. The latter is closed in
${\mathbb {V}}_u$
and does not contain 0. Thus,
$O\cap \{ax:x\in {\mathbb {V}}_u^t\}=\emptyset $
if O is small enough. Then the lemma follows from the definition of
$ Z_t(\omega (g)\phi ) _K$
and the Weil representation formula in 2.8.
4.1.3 Change level
For
$K\subset K'$
, let
$\pi _{_K,_{K'}}:{\mathrm {Sh}}({\mathbb {V}})_{K}\to {\mathrm {Sh}}({\mathbb {V}})_{K'}$
be the natural projection.
Lemma 4.1.5 [Reference KudlaKud97a, PROPOSITION 5.10][Reference LiuLiu11a, Corollary 3.4].
We have
$\pi _{_K,_{K'}}^{*}Z_t(\phi )_{K'}=Z_t(\phi )_K$
and
$\pi _{_K,_{K'}}^{*}L_{K '}=L_K$
. In particular,
$\pi _{_K,_{K'}}^{*}z(\phi )_{K'}=z(\phi )_K$
.
Remark 4.1.6. We have
$\pi _{_K,_{K'},*}Z(x)_K =d(x) Z(x)_{K'}$
where
$d(x)$
is the degree of
$Z(x)_K$
over
$Z(x)_{K'}$
. It is easy to check that
$d(x)$
is not constant near 0. In particular, it does not extend to a smooth function on
${\mathbb {V}}^\infty $
. Thus, for a general
$\phi $
, there seems no
$\phi '\in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K'}$
such that
$\pi _{_K,_{K'},*}Z_t(\phi )_{K}$
is of the form
$Z_t(\phi ')_{K'}$
for every t.
Below, let
$L=L_K$
,
$Z(x)=Z(x)_K$
,
$Z_t(\phi )=Z_t(\phi )_K$
and
$z(\phi )=z(\phi )_K$
for simplicity if K is clear from the context.
4.2 Green functions
We use complex uniformization to defined automorphic Green functions for special divisors. They are admissible. We compare them with the normalized admissible Green functions and show the modularity of the differences between their generating series. Then we recall Kudla’s Green functions and prove the modularity of the differences between the generating series of normalized admissible Green functions and Kudla’s Green functions.
4.2.1 Complex uniformization
For nonnegative integers
$p,q$
, let
${\mathbb {C}}^{p,q}$
be the
$p+q$
dimensional hermitian space associated to the hermitian matrix
${\mathrm {diag}}(-1_p,1_q)$
. Let
$U\left ({\mathbb {C}}^{1,n}\right )$
be the unitary group of
${\mathbb {C}}^{1,n}$
(so of signature
$(n,1)$
in the usual convention). Let
${\mathbb {B}}_n$
be the complex open unit ball of dimension n. Embed
${\mathbb {B}}_n$
in
${\mathbb {P}}({\mathbb {C}}^{1,n})$
as the set of negative lines as follows:
$[z_1,\ldots ,z_n]\in {\mathbb {B}}_{n}$
, where
$\sum _{i=1}^{n}|z_i|^2<1$
, is the line containing the vector
$(1,z_1\ldots ,z_n) $
. Then
$U\left ({\mathbb {C}}^{1,n}\right )$
acts on
${\mathbb {B}}_n$
naturally and transitively. Let
$\Omega $
be the tautological bundle of negative lines on
${\mathbb {B}}_n$
, and
$\overline \Omega $
the hermitian line bundle with the metric induced from the negative of the hermitian form. The Chern form of
$\overline \Omega $
is a
$U\left ({\mathbb {C}}^{1,n}\right )$
-invariant Kähler form

For an arithmetic subgroup
$\Gamma \subset U({\mathbb {C}}^{1,n})$
, on (the orbifold)
$\Gamma \backslash {\mathbb {B}}_n $
, we have the descent
$\overline \Omega _\Gamma $
of
$\overline \Omega $
whose (orbifold) Chern curvature form is the descent of
$c_1(\overline \Omega )$
. Define the degree

If
$\Gamma \backslash {\mathbb {B}}_n$
is compact, this degree is the usual degree via intersection theory.
For
$x\in {\mathbb {C}}^{1,n}$
, let
${\mathbb {B}}_x\subset {\mathbb {B}}_n$
be the subset of negative lines perpendicular to x. Later, we will use the following function on
${\mathbb {B}}_n $
measuring ‘the distance to
${\mathbb {B}}_x$
’:

where
$\widetilde z $
is a nonzero vector contained in the line z. In particular,
$R_x>0$
outside
${\mathbb {B}}_x$
. If
$q(x)<0$
, then
$x^\perp $
is of signature
$(n,0)$
so that
${\mathbb {B}}_x=\emptyset $
. Below, until 4.2.3, assume
${q(x)}>0$
. Then
$x^\perp $
is of signature
$(n-1,1)$
. So
${\mathbb {B}}_x$
is a complex unit ball of dimension
$n-1$
. Assume that
$\Gamma \cap U (x^\perp )$
is an arithmetic subgroup of
$U (x^\perp )$
. Let
$C(x,\Gamma )$
be the pushforward of the fundamental cycle by

Define

Now we can uniformize Shimura varieties and special divisors. Let v be an infinite place of F. Let V be the unique hermitian space over E such that
$V({\mathbb {A}}_E^v){\ \simeq \ } {\mathbb {V}}^v$
and
$V(E_v){\ \simeq \ } {\mathbb {C}}^{1,n}$
. By [Reference Li and LiuLL21, Lemma 5.5], we have the complex uniformization:

Then the Hodge bundle
$L_{E_v}$
is the descent of
$\Omega \times 1_{U({\mathbb {V}}^\infty )/K}$
. Let
$\overline {L}_{E_v}$
be the descent of
$\overline \Omega \times 1_{U({\mathbb {V}}^\infty )/K}$
. Let
$h_1,.. .,h_m$
be a set of representatives of
$U(V)\backslash U({\mathbb {V}}^\infty )/K$
. Let
$\Gamma _{h_j}=U(V)\cap h_jKh_j^{-1}$
. Then (4.4) decomposes as the disjoint union of
$\Gamma _{h_j} \backslash {\mathbb {B}}_n$
’s. Then for
$t\in F_{>0}$
, we have (see [Reference KudlaKud97a, PROPOSITION 5.4])

4.2.2 Admissible Green functions
Admissible Green functions are Green functions with harmonic curvatures. See Appendix A.3. Admissible Green functions for special divisors are constructed by Bruinier [Reference BruinierBru02, Reference BruinierBru12], Oda and Tsuzuki [Reference Oda and TsuzukiOT03]. For
$F={\mathbb {Q}}$
and
$n=1$
, it appeared in the work of Gross and Zagier [Reference Gross and ZagierGZ86] for
$n=1$
. We learned the following explicit computation from S. Zhang.
First, we start by working on
${\mathbb {B}}_n$
. Let
$x_0=(0,\ldots ,0,1)\in {\mathbb {C}}^{1,n}$
. Then
${\mathbb {B}}_{x_0}\subset {\mathbb {B}}_n$
consists of points
$ (z_1,\ldots ,z_{n-1},0) $
’s. We want a
$U\left ( x_0 ^\perp \right )$
-invariant smooth function G on
${\mathbb {B}}_n-{\mathbb {B}}_{x_0}$
such that
$G(z_1,\ldots ,z_n)+\log |z_n|^2 $
extends smoothly to
${\mathbb {B}}_n$
,
$\lim _{|z|\to 1}G(z) =0,$
and G is a solution of the following Laplacian equation:

The quotient of
${\mathbb {B}}_n-{\mathbb {B}}_{x_0}$
by
$U\left ( x_0 ^\perp \right )$
is isomorphic to
$(1,\infty )$
via

Thus, we look for
$G=Q(t(z))$
, where Q is a smooth function on
$(1,\infty )$
such that
$ Q(t)+ \log ( t-1) $
extends to a smooth function on
${\mathbb {R}}$
. By (4.3) and the
$U\left ( x_0 ^\perp \right )$
-invariance, (4.6) is reduced to the following hypergeometric differential equation

For
${\mathrm {Re}} s>-1$
, there is a unique solution
$Q_s$
such that
$Q_s(t)+\log (t-1)$
extends to a smooth function on
${\mathbb {R}}$
and
$\lim _ {t\to \infty }Q_s(t)=0$
:

where F is the hypergeometric function. (See also [Reference Oda and TsuzukiOT03, 2.5.3]. When
$n=1$
, our
$Q_s$
is the Legendre function of the second kind in [Reference Gross and ZagierGZ86, 238] up to shifting s by 1).
For a general x with
$q(x)>0$
, we have the following Green function for
${\mathbb {B}}_x$
:

We will need the following explicit formula later: if
$ x=(x_1,x_2)\in {\mathbb {C}}^{1,n}= {\mathbb {C}}^{1,0}\oplus {\mathbb {C}}^{0,n} $
, then

Second, we define Green functions for
$ C( x ,\Gamma )$
on arithmetic quotient of
${\mathbb {B}}_n.$
Let
$\Gamma $
be an arithmetic subgroup of
$U\left ({\mathbb {C}}^{1,n}\right )$
. Let

Lemma 4.2.1 [Reference Oda and TsuzukiOT03, Proposition 3.1.1, Remark 3.1.1, Remark 3.2.1].
For
$s \in {\mathbb {C}}$
with
${\mathrm {Re}} s>0$
, the sum
$g_s$
converges absolutely and defines a smooth function on
$ \Gamma \backslash {\mathbb {B}}_{n}-C( x ,\Gamma )$
. Moreover,
$g_s$
is holomorphic on s.
It is easy to see that
$g_s$
with
${\mathrm {Re}} s>0$
is a Green function for
$ C( x ,\Gamma )$
.
Theorem 4.2.2 [Reference Oda and TsuzukiOT03, Theorem 7.8.1].
(1) There is a meromorphic continuation of
$g_s$
to
$s\in {\mathbb {C}}$
with a simple pole at
$s=0$
and residue
$-\frac {\deg _{\overline \Omega _\Gamma }(C( x ,\Gamma ))}{\deg (\overline \Omega _ \Gamma )}$
.
(2) The function
$\widetilde {\lim \limits _{s\to 0}} g_s$
is an admissible Green function for
$ C( x ,\Gamma )$
.
Recall that
$\widetilde {\lim \limits _{s\to 0}}$
denotes taking the constant term at
$s=0$
.
Remark 4.2.3. (1) We read the residue
$\kappa $
in [Reference Oda and TsuzukiOT03, Theorem 7.8.1 (3)] as follows. Beside the obvious differences between the choices of Green function here and in [Reference Oda and TsuzukiOT03] (more precisely, s-variables and signs), the Kähler form and ‘volume form’ here and in [Reference Oda and TsuzukiOT03] are different. First, the Kähler form on the bottom of [Reference Oda and TsuzukiOT03, 514] is
$\pi c_1(\overline \Omega )$
in our notation. Second, [Reference Oda and TsuzukiOT03, Theorem 7.8.1 (3)] uses volumes to express the residue, while we use degrees of line bundles to express the residue. The volume form for
$ {\mathbb {B}}_n$
on the top of [Reference Oda and TsuzukiOT03, 515] is
$\frac {\pi ^n}{n!}c_1(\overline \Omega )^n$
in our notation. Third, there is a
$\pi $
missing in (the numerator of) the residue
$\kappa $
in [Reference Oda and TsuzukiOT03, Theorem 7.8.1 (3)]. It should be easy to spot from [Reference Oda and TsuzukiOT03, Proposition 3.1.2, Lemma 7.2.2].
(2) When
$n=1$
and
$\Gamma ={\mathrm {SL}}_2({\mathbb {Z}})$
via
${\mathrm {SU}}({\mathbb {C}}^{1,1}){\ \simeq \ } {\mathrm {SL}}_2({\mathbb {R}})$
, we have
$\deg (\overline \Omega _ \Gamma )=\frac {1}{12}$
. See [Reference KühnK01, 4.10]. Thus, for
$\Gamma _0(N)$
a standard congruence subgroup of
$\Gamma $
, Theorem 4.2.2 (1) coincides with the residue
$\frac {-12}{[\Gamma :\Gamma _0(N)]}$
in [Reference Gross and ZagierGZ86, p 239, (2.13)]. Theorem 4.2.2 (1) is not used in any other place of the paper.
By [Reference Oda and TsuzukiOT03, Proposition 3.1.2] and taking care of the differences in the remark, we have

Thus,

Finally, we define Green functions for
$Z_t(\phi )$
,
$t\in F_{>0}$
. Let
$v\in \infty $
and V as in 4.2.1. For
$s \in {\mathbb {C}}$
with
${\mathrm {Re}} s> 0$
, consider the following formal sum for
$(z,h)\in {\mathbb {B}}_n \times U({\mathbb {V}}^\infty )$
:

By (4.5) and Lemma 4.2.1, if
$(z,h)$
is not over
$Z_t(\phi ^\infty )_{E_v}$
via (4.4), the formal sum is defined and absolutely convergent. Thus,
$ {\mathcal {G}}_{Z_t(\phi ^\infty )_{E_v},s}$
descends to
${\mathrm {Sh}}({\mathbb {V}})_{K,E_v}-Z_t(\phi ^\infty )_{E_v} $
via (4.4), which we still denote by
${\mathcal {G}}_{Z_t(\phi ^\infty )_{E_v},s}$
. By (4.5) and Theorem 4.2.2,
${\mathcal {G}}_{Z_t(\phi ^\infty )_{E_v},s} $
is a Green function for
$Z_t(\phi ^\infty )_{E_v}$
. For
$g\in G({\mathbb {A}}_F)$
, let

which is a Green function for
$Z_t(\omega (g)\phi )_{E_v}$
by (4.2). Define the automorphic Green functions for
$Z_t(\phi ^\infty )_{E_v}$
and
$Z_t(\omega (g)\phi )_{E_v}$
to be

respectively, which are admissible by Theorem 4.2.2.
Let
$ {\mathcal {G}}^{\overline L_{E_v}}_{Z_t(\phi )_{E_v}}$
be the normalized admissible Green function for
$Z_t(\phi )_{E_v}$
with respect to
$\overline L_{E_v}$
as in Definition A.3.3. In particular, its integration against
$c_1(\overline L_{E_v})^{n}$
is 0.
Lemma 4.2.4. (1) We have

Both sides are independent of K.
(2) We have

Proof. The equation in (1) follows from (4.9) and the independence of the right-hand side follows from the projection formula. (2) follows from Theorem 4.1.2.
Remark 4.2.5. The automorphic form in Lemma 4.2.4 (2) can be made explicit:

This is a geometric version of the Siegel-Weil formula (3.13). It is stated in [Reference KudlaKud97a, COROLLARY 10.5] for the orthogonal case; the proof carries over to the unitary case.
4.2.3 Kudla’s Green function
We recall Kudla’s Green functions for special divisors [Reference KudlaKud97b], following [Reference LiuLiu11a, 4C] in the unitary case. We consider simple special divisors
$Z(x)$
’s, instead of weight special divisors
$Z_t(\phi )$
’s. We extend the definition of special divisors as follows. For
$x\in {\mathbb {V}}^{\infty } $
such that
${q(x)}\in F ^\times -F_{>0}$
, let
$Z(x)= 0$
.
First, we work on
${\mathbb {B}}_n$
. For
$v\in \infty $
, V as in the end of 4.2.1 with respect to v,
$g\in G(F_v)$
and
$x\in V$
, define

where the exponential integral
${\mathrm {Ei}}(t)=\int _{-\infty }^t\frac {e^s}{s}ds$
on
$t\in (-\infty ,0)$
has a log-singularity at 0. If
$q(x)\neq 0$
so that
${\mathbb {B}}_x$
is either empty or a complex unit ball of dimension
$n-1$
,
$G^{\mathrm {Kud}}(x,g)$
is a Green function for
${\mathbb {B}}_x$
. If
$q(x)<0$
, equivalently
${\mathbb {B}}_x$
is empty, then
$G^{\mathrm {Kud}}$
is smooth.
Now we work on
${\mathrm {Sh}}({\mathbb {V}})_K$
. Let
$x\in {\mathbb {V}}^\infty $
with
$q(x)\in F^\times $
. For
$v\in \infty $
, if
$u(q(x))>0$
for every
$u\in \infty -\{v\}$
, by the Hasse-Minkowski theorem and Witt’s theorem, there exists
$h\in U({\mathbb {V}}^\infty )$
and
$x^{(v)}\in V-\{0\}$
, where V is as in the last paragraph, such that
$ x=h^{-1}x^{(v)}$
. Define

where
$h_1,.. .,h_m$
is a set of representatives of
$U(V)\backslash U({\mathbb {V}}^\infty )/K$
. By the decomposition analogous to (4.5) (see the second equation on [Reference KudlaKud97a, p 56]) and [Reference LiuLiu11a, Proposition 4.9]),
${\mathcal {G}}^{\mathrm {Kud}}_{Z( x)_{E_v}}(g)$
is absolutely convergent and descends to
${\mathrm {Sh}}({\mathbb {V}})_{K,E_v}$
via (4.4). And it is a Green function for
$Z(x)_{E_v}$
.
Besides Kudla’s Green functions, we will need their projections to the constant function
$1$
to modify the normalized Green function. See 4.4.3.
Definition 4.2.6. For
$x\in {\mathbb {V}}^\infty $
with
$q(x)\in F^\times $
,
$g\in G({\mathbb {A}}_{F,\infty })$
and
$v\in \infty $
, let

Note that if
$u(q(x))<0$
for some
$u\in \infty -\{v\}$
, then
${\mathcal {G}}^{\mathrm {Kud}}_{Z(x)_{E_v}}(g)=0$
. Thus, if
$q(x) $
is negative at more than one infinite places, then
${\mathcal {G}}^{\mathrm {Kud}}_{Z(x)_{E_v}}(g)=0$
and
$ {\mathfrak {k}}(x,g_v ) =0$
for every
$v\in \infty $
.
The following can be read from [Reference Garcia and SankaranGS19, (1.12), Theorem 1.2, (1.18), (1.19), Proposition 5.9].
Theorem 4.2.7. Let
$ E_t'(0,g,\phi )(v)$
be as in (3.11). For
$t\in F_{>0}$
and
$v\in \infty $
, we have

For
$t\in F^\times $
with
$v(t)<0$
for exactly one infinite place v, we have

And for
$t=0$
, we have

We remind the reader that
$\infty $
is the set of infinite places of F. And our formulas differ from [Reference Garcia and SankaranGS19] by a factor
$1/2$
since in loc. cit., the authors use the set of infinite places of E.
4.2.4 Modularity of difference of Green functions
We need a more general notion of modular forms.
Definition 4.2.8. Let V be a topological
${\mathbb {C}}$
-vector space and
$V^{*}$
the continuous dual. Let
$ {\mathcal {A}}(G,{\mathfrak {w}}, V) $
be the space of smooth V-valued function f on
$G({\mathbb {A}})$
such that for every
$l\in V^{*}$
, we have
$l\circ f\in {\mathcal {A}}(G,{\mathfrak {w}}).$
Remark 4.2.9. Note that
$l\circ f$
is automatically smooth.
Clearly, if V is the topological direct sum of
$V_1, V_2$
and
$f_i\in {\mathcal {A}}(G,{\mathfrak {w}}, V_i) $
, then
$f_1+f_2\in {\mathcal {A}}(G,{\mathfrak {w}}, V) $
.
Now we define the formal generating series of Green functions. Recall that
$\infty $
is the set of infinite places of F, of cardinality
$[F:{\mathbb {Q}}]$
. Let

which is the product of
$[F:{\mathbb {Q}}]$
many copies of
${\mathbb {C}}$
. Then

is the (disconnected) complex manifold that is the disjoint union of all base changes of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
to
$E_v$
’s (each base change itself may not be connected either!).
Let


which are formal generating series of smooth functions on
${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$
with logarithmic singularities along the same formal generating series of special divisors. Then
$ {\mathcal {G}}^{{\overline {\mathcal {L}}}}(g,\phi )- {\mathcal {G}}^{\mathrm {Kud}}(g,\phi )$
is a formal generating series valued in
${C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }),$
the space of smooth
${\mathbb {C}}$
-valued functions on
${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$
.
Let
$E_1:={\mathcal {O}}_{{\mathrm {Sh}}({\mathbb {V}})_{K}}\left ( {\mathrm {Sh}}({\mathbb {V}})_{K}\right )$
, which is a finite field extension of E (since
${\mathrm {Sh}}({\mathbb {V}})_{K}$
is connected). Then we have a morphism
${\mathrm {Sh}}({\mathbb {V}})_{K }\to \operatorname {\mathrm {Spec}} E_1$
. By Stein factorization,
${\mathrm {Sh}}({\mathbb {V}})_{K }$
, as a variety over
$E_1$
, is geometrically connected. So the connected components of
${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty } ={\mathrm {Sh}}({\mathbb {V}})_{K}\otimes _{\mathbb {Q}}{\mathbb {R}} $
are exactly indexed by the underlying set of
$\operatorname {\mathrm {Spec}} E_1\otimes _{\mathbb {Q}}{\mathbb {R}}$
, equivalently, the set of conjugate pairs of infinite places of
$E_1$
(which, as a finite field extension of the CM field E, has only complex embeddings but no real embeddings). Let
$ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
be the space of locally constant functions on
${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$
. Then we have the canonical isomorphism from
$ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
to the product copies of
${\mathbb {C}}$
indexed by the set of conjugate pairs of infinite places of
$E_1$
. Now we can embed
$ {\mathcal {O}}_{E_1}^\times $
in
${\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
via this isomorphism and the Dirichlet regulator map, so that the
${\mathbb {C}}$
-span of the image of
$ {\mathcal {O}}_{E_1}^\times $
, denoted by
${\mathbb {C}} {\mathcal {O}}_{E_1}^\times $
is of codimension 1 in
$ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
by Dirichlet’s unit theorem. Let
${\mathbb {C}} {\mathcal {O}}_{E_1}^\times $
be this span. Let

Equip
$\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $
with the quotient of the
$L^\infty $
-topology. Define an embedding

by mapping
$a\in {\mathbb {C}}$
to the constant function a on
${\mathrm {Sh}}({\mathbb {V}})_{K,E_v}$
for some
$v\in \infty $
(rather than
${\mathrm {Sh}}({\mathbb {V}})_{K,E_{1,w}}$
for some infinite place w of
$E_1$
). Below, by a complex number in
$ \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $
, we understand it as the image by (4.13).
Theorem 4.2.10. Let
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$
be as in (3.29). For
$g\in G({\mathbb {A}})$
, the generating series of
$\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $
-valued functions on
$G({\mathbb {A}})$

pointwise converges to an element in
$ {\mathcal {A}}\left ( G,{\mathfrak {w}}, \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })\right ) .$
Remark 4.2.11. (1) It might be interesting to study whether Theorem 4.2.10 still holds if we put Fréchet topology on
$\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
, and if we require stronger convergence.
(2) Theorem 4.2.10 is a strengthened analog of the main result of Ehlen and Sankaran [Reference Ehlen and SankaranES18], which is for
$F={\mathbb {Q}}$
.
Proof. We follow [Reference Kudla, Rapoport and YangKRY06] and [Reference Mihatsch and ZhangMZ21]. Let
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $
be the
$L^2$
-orthogonal complement of
${\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
in
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
, endowed with
$L^\infty $
-topology. Then
$\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $
is the topological direct sum of
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $
and
$ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) /{\mathbb {C}} {\mathcal {O}}_{E_1}^\times $
.
Let the generating series
${\mathcal {D}}^\circ $
be the projection of
${\mathcal {D}}$
to
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $
. By [Reference Mihatsch and ZhangMZ21, Lemma 3.8]Footnote
2
and the same argument in the proof [Reference Kudla, Rapoport and YangKRY06, Theorem 4.4.4],Footnote
3
for every
$g\in G({\mathbb {A}})$
,
${\mathcal {D}}^\circ (g)$
converges in
$L^\infty \left ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right ) $
. Particularly,
${\mathcal {D}}^\circ (g)$
converges in
$L^2\left ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )$
since
${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$
is compact. So we can consider
${\mathcal {D}}^\circ $
(identified as its limit) as a function on
$G({\mathbb {A}}) \times {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$
. Then by the argument in the proof of [Reference Mihatsch and ZhangMZ21, Theorem 3.9],
${\mathcal {D}}^\circ \in C^\infty \left ( G({\mathbb {A}}) \times {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )$
. Also note that for every
$g\in G({\mathbb {A}})$
,
${\mathcal {D}}^\circ (g)\in C^\infty \left ( {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )^\circ $
. Thus,
${\mathcal {D}}^\circ $
is a smooth
$C^\infty \left ( {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )^\circ $
-valued function on
$G({\mathbb {A}})$
. See [Reference TrévesTré67, Theorem 40.1 and Corollary]. Then the argument in the proof of either [Reference Kudla, Rapoport and YangKRY06, Theorem 4.4.4] or [Reference Mihatsch and ZhangMZ21, Theorem 3.9] shows that
${\mathcal {D}}^\circ \in {\mathcal {A}}\left ( G,{\mathfrak {w}}, \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ \right ) .$
Consider the projection from
$\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $
to
$ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) /{\mathbb {C}} {\mathcal {O}}_{E_1}^\times {\ \simeq \ } {\mathbb {C}}$
– that is,

By Theorem 4.2.7 and that
$E_0(0,g,\phi )= \omega ( g)\phi (0)$
(see Lemma 3.3.9 (1)), the projection of
${\mathcal {D}}$
(defined on each of its terms) pointwise converges to an element in
$ {\mathcal {A}}\left ( G,{\mathfrak {w}} \right ).$
Since
${\mathcal {D}}$
is the sum of this projection and
${\mathcal {D}}^\circ $
, the theorem follows.
4.3 Modularity problems
We will raise two modularity problems for admissible extensions of special divisors. Before that, we recall some notions and Kudla’s modularity problem.
4.3.1 Preliminaries
A (regular) integral model of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
over an integral domain R with fraction field E is a (regular) Deligne-Mumford stack proper flat over
$\operatorname {\mathrm {Spec}} R$
with a fixed isomorphism of its generic fiber to
$ {\mathrm {Sh}}({\mathbb {V}})_{K }.$
An isomorphism between integral models is an isomorphism over
$\operatorname {\mathrm {Spec}} R$
that respects the fixed isomorphisms to
$ {\mathrm {Sh}}({\mathbb {V}})_{K }.$
Let
${\mathcal {X}}_K$
be a regular integral model of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_E$
. Let
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}({\mathcal {X}}_K) $
be the Chow group of arithmetic divisors with
${\mathbb {C}}$
-coefficients. See Definition A.3.4. In particular, we have an isomorphism

by taking degrees (see Remark A.3.5), and an arithmetic intersection pairing

See Appendix A.4. Here, we recall that
$Z_1({\mathcal {X}}_K)$
is the group of 1-cycles on
${\mathcal {X}}_K$
.
Let
$ {\mathcal {L}}= {\mathcal {L}}_K$
be an extension of
$L=L_K $
to
${\mathcal {X}}_K$
, which we allow to be a line bundle. Let
$ \overline {\mathcal {L}} $
be
${\mathcal {L}}$
equipped with a hermitian metric. Let
$c_1(\overline {\mathcal {L}}^\vee )\in \widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K) $
be the first arithmetic Chern class of the dual of
$\overline {\mathcal {L}}$
. See Example A.3.7.
4.3.2 Kudla’s problem
We consider the following modularity problem of Kudla [Reference KudlaKud02, Reference KudlaKud03, Reference KudlaKud04]: find an arithmetic divisor
$\widehat {\mathcal {Z}}(x)$
on
${\mathcal {X}}$
extending
$Z(x)$
, explicitly and canonically, such that

where
$ x_\infty \in {\mathbb {V}}_\infty $
such that
$q(x_\infty )={q(x)}\in F_{>0}$
lies in
$ {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}}) \otimes \widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K) $
. The existence of such
$\widehat {\mathcal {Z}}(x)$
is obvious, by choosing a section of the natural surjection
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}({\mathcal {X}}_K)\to {\mathrm {Ch}}^1({\mathrm {Sh}}({\mathbb {V}})_K)_{\mathbb {C}}$
. However, it is only defined at the level of divisor classes, and not explicit.
4.3.3 Admissible extensions
We consider the above modularity problem for admissible extensions. In particular, we assume Assumption A.1.1 for
${\mathcal {X}}_K$
. (Also recall that
${\mathcal {X}}_K$
is connected, as
${\mathrm {Sh}}({\mathbb {V}})_K$
is.) Assume that
$ {\mathcal {L}}$
is ample. Let
$ \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) \subset \widehat {\mathrm {Ch}}^1_{\mathbb {C}}({\mathcal {X}}_K)$
be the subgroup of arithmetic divisors that are admissible with respect to
$\overline {\mathcal {L}}$
. See Definition A.3.4. By Lemma A.3.6, the natural map

is surjective, and the kernel is the image of the pullback

In particular,
$ \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $
is finite dimensional.
Definition 4.3.1. Define an embedding
$ {\mathbb {C}}\hookrightarrow \widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $
as the composition of the inverse
$\deg ^{-1}: {\mathbb {C}}{\ \simeq \ } \widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E} \right )$
of taking degree and (4.15). Below, by a complex number in
$\widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)$
, we understand it as the image by this embedding.
Remark 4.3.2. The intersection
$ \widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)\cap \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $
in
$ \widehat {\mathrm {Ch}}_{\mathbb {C}}({\mathcal {X}}_K) $
is
${\mathbb {C}}$
, where
${\mathbb {C}}$
is in
$\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
via (4.13).
Let
${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)$
the group of admissible arithmetic divisors. See Definition A.3.1. For a divisor Z on
${\mathrm {Sh}}({\mathbb {V}})_{K}{\ \simeq \ } {\mathcal {X}}_{K,E}$
, let

be the normalized admissible extension of Z with respect to
$\overline {\mathcal {L}}$
. See Definition A.3.3. Let
$[ Z(x)^{\overline {\mathcal {L}}}] $
be its class in
$\widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)$
. Then a preiamge of
$[ Z(x)] $
via (4.14) is of the form
$[ Z(x)^{\overline {\mathcal {L}}}] +e(x)$
for some
$e(x)\in {\mathbb {C}}$
. Note that
$c_1(\overline {\mathcal {L}}^\vee )\in \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $
. See Example A.3.7.
Problem 4.3.3. Find
$a\in {\mathbb {C}}$
and
$e=\{e(x) \in {\mathbb {C}}\}_{x\in K\backslash {\mathbb {V}}^{ \infty }_{>0}}$
explicitly such that for every
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$
, the generating series

where
$ x_\infty \in {\mathbb {V}}_\infty $
such that
$q(x_\infty )={q(x)}\in F_{>0}$
, lies in
$ {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}}) \otimes \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $
. Moreover,
$e(x)$
should to be naturally decomposed into a sum of ‘local components’ such that the v-component should be 0 at all but finitely many places and
$\infty $
-component should be independent of the choice of the regular integral model.
In other words, we want a modular generating series by modifying each
$[Z(x)^{\overline {\mathcal {L}}}]$
by an explicit constant once for all
$\phi $
. A weaker statement is to allow the modification to depend on
$g,\phi $
.
Problem 4.3.4. Find
$a\in {\mathbb {C}}$
and a smooth function
$e_t(g,\phi )$
on
$G({\mathbb {A}}_F)$
for
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$
explicitly such that the generating series

lies in
$ {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}}) \otimes \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $
. Moreover,
$e_t(g,\phi )$
should to be naturally decomposed into a sum of ‘local components’ such that the v-component should be 0 at all but finitely many places and
$\infty $
-component should be independent of the choice of the regular integral model.
Remark 4.3.5. In (4.17) and (4.18), one may replace
$\omega (g)\phi (0)\left ( c_1(\overline {\mathcal {L}}^\vee )+a\right )$
by
$\omega (g)\phi (0)c_1(\overline {\mathcal {L}}^\vee )$
by adding a suitable multiple of the degree of
$z\left (\omega (g)\phi \right )$
, which is modular by Theorem 4.1.2 and can be made explicit by (4.11). However, we keep the freedom to have ‘a’ to get the decomposition of
$e_t(g,\phi )$
as we will see in Section 4.4.
By Theorem 4.1.2 and Lemma A.3.6 (1), we immediately have the following lemma.
Lemma 4.3.6. For
${\mathcal {P}}\in Z_1({\mathcal {X}}_K )_{\mathbb {C}}$
such that
$\deg {\mathcal {P}}_E=0$
,

Remark 4.3.7. (1) This lemma is called almost modularity in [Reference Mihatsch and ZhangMZ21, Theorem 4.3].
(2) In the case that
$K=K_{\Lambda }$
with a different
$\Lambda $
, for Kudla-Rapoport arithmetic divisors and
${\mathcal {P}}\in Z_1({\mathcal {X}}_K )_{\mathbb {C}}$
with
${\mathcal {P}}_E=0$
, the analogous statement is proved in [Reference ZhangZha21b, Theorem 14.6].
When
$\deg {\mathcal {P}}_E\neq 0$
, the truth of (4.19) is in fact equivalent to the modularity of
$ z\left (\omega (g)\phi \right )_{e,a}^{\overline {\mathcal {L}}}$
by Lemma 4.3.8 below. We will not use exactly this ‘numerical criterion’, but use Lemma 4.3.8 in a more sophisticated way to prove our modularity results in 5.2.4.
Lemma 4.3.8. Let X be a
${\mathbb {C}}$
-vector space,
$x\in X$
nonzero, and l a linear form on X such that
$l(x)=1$
. Let f be a formal generating series of functions on
$G({\mathbb {A}})$
valued in X and
$\overline f$
the corresponding formal generating series of functions on
$G({\mathbb {A}})$
valued in
$X/{\mathbb {C}} x$
. Assume that
$\overline f\in {\mathcal {A}}(G,{\mathfrak {w}}) \otimes X/{\mathbb {C}} x$
. Then
$ f\in {\mathcal {A}}(G,{\mathfrak {w}}) \otimes X $
if and only if
$l\circ f\in {\mathcal {A}}(G,{\mathfrak {w}}). $
Proof. Define a section
${\mathfrak {s}}$
of the projection
$X\to X/{\mathbb {C}} x$
by
${\mathfrak {s}}:z\mapsto s(z)- l\left ( s(z)\right ) \cdot x,$
where s is any section (and
${\mathfrak {s}}$
is independent of the choice of s). Then
$ f= {\mathfrak {s}}(\overline f)+ \left ( l\circ f \right ) \cdot x $
, and the lemma follows.
4.4 Conjecture and theorems
First, we define specific integral models. Then we define explicit admissible extensions. Then we will propose a precise conjecture. Finally, we state our modularity theorems.
4.4.1 Integral models
Let us at first set up some notations and assumptions that are needed to construct our integral models. For a finite place v of F and an
${\mathcal {O}}_{E_v}$
-lattice
${\Lambda _v}$
of
${\mathbb {V}}(E_v)$
, the dual lattice is defined as
$\Lambda _v^\vee =\{x\in {\mathbb {V}}(E_v):\langle {x,{\Lambda _v}} \rangle \subset {\mathcal {O}}_{E_v}\}$
. Then
${\Lambda _v}$
is called
-
• self-dual if
${\Lambda _v}=\Lambda _v^\vee $ ;
-
•
$\varpi _{E_v}$ -modular if
$ \Lambda _v^\vee =\varpi _{E_v}^{-1}{\Lambda _v}$ ;
-
• almost
$\varpi _{E_v}$ -modular if
$ \Lambda _v^\vee \subset \varpi _E^{-1}{\Lambda _v}$ and the inclusion is of colength 1.
Assume the following assumption in the rest of the paper.
Assumption 4.4.1. (1) At least one of the following three conditions hold:
-
(1.a) every finite place of E is at most tamely ramified over
$ {\mathbb {Q}}$ ;
-
(1.b)
$E/{\mathbb {Q}}$ is Galois;
-
(1.c)
$E $ is the composition of F with some imaginary quadratic field.
(2) Every finite place v of F ramified over
$ {\mathbb {Q}}$
or of residue characteristic 2 is unramified in E.
(3) At every finite place v of F inert in E, there is a self-dual lattice
$\Lambda _v$
in
${\mathbb {V}}(E_v)$
.
(4) At every finite place v of F ramified in E, there is a
$\pi _v$
-modular (resp. almost
$\pi _v$
-modular) lattice
$\Lambda _v$
in
${\mathbb {V}}(E_v)$
if n is odd (resp. n is even).
We will classify
${\mathbb {V}}$
containing such
$\Lambda $
in Remark 5.1.2 below.
At every place v split in E, let
$\Lambda _v$
be a self-dual lattice in
${\mathbb {V}}(E_v)$
. Let

Let
$K_\Lambda \subset U({\mathbb {V}}^\infty )$
be the stabilizer of
$\Lambda $
.
Definition 4.4.2. Let
${{\widetilde K_\Lambda }}$
be the directed poset of compact open subgroups
$K\subset K_\Lambda $
, under the inclusion relation, such that for a finite place v of F,
-
(1) if
$v $ is non-split in E, then
$K_v=K_{\Lambda ,v}$ ;
-
(2) if
$v $ is split in E, then
$K_v$ is a principal congruence subgroup of
$K_{\Lambda ,v}$ .
Under Assumption 4.4.1 (1.a), for
$K\in {{\widetilde K_\Lambda }}$
and a finite place v of E, we have an integral model
$\mathscr {S}_v$
of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E,(v)}$
, as constructed in (B.2.11), B.2.25 and B.2.28. (Note that we can always choose a CM type satisfying the matching condition (B.2.20).) We remind the reader of the differences between the notations on the fields here and in Appendix B.
Lemma 4.4.3.
(1) The integral model
$\mathscr {S}_v$
is regular.
(2) If
$K= {K_{\Lambda }}$
, there is an ample
${\mathbb {Q}}$
-line bundle
$\mathscr {P}_v$
on
$\mathscr {S}_v$
extending
$L_{K_\Lambda }$
.
Proof. (1) We apply B.2.26. We use M to denote the reflex field in B.2.26, which is defined in (B.1.15) (and denoted by E there), to avoid confusion. We use N to denote the field extension of the reflex field in B.2.26 (denoted by L there). By the finite étaleness of the moduli space of relative dimenison 0 as in B.1.23, we may choose
$N/M$
to be unramified at
$\nu $
[Reference ProjectSta18, Tag 04GL]. Then by [Reference Rapoport, Smithling and ZhangRSZ20] and B.2.26,
$\mathscr {S}_{v,{\mathcal {O}}_{N,(\nu )}}$
is regular for a place
$\nu $
of N over v. By the descent of regularity under faithfully flat morphism [Reference ProjectSta18, Tag 033D], the lemma follows.
(2) Let
$N/E$
be a finite Galois extension such that v is unramified in N and every connected component of
$\mathscr {S}_{v,{\mathcal {O}}_{N,(v)}}$
is geometrically connected. By its construction (B.2.11), every connected component is a quotient of a connected component of an integral model of Hodge type over
${\mathcal {O}}_{N,(v)}$
by a finite group action. The integral model of Hodge type is a closed subscheme of the integral Siegel moduli space [Reference XuXu21, Reference XuXu25], on which we have a well-known ample Hodge line bundle. The restriction is an ample line bundle on each geometrically connected component of the integral model of Hodge type. Taking norm along the quotient map by the finite group action, we get an an ample line bundle on every component of
$\mathscr {S}_{v,{\mathcal {O}}_{N,(v)}}$
. See [Reference Bosch, Lütkebohmert and RaynaudBLR90, Section 6, Theorem 7 and Example B] and [Reference VistoliVis04] for the stack case. Then taking norm map along the quotient map by the
${\mathrm {Gal}}(N/E)$
, we have an ample line bundle
$\mathscr {P}'$
on
$\mathscr {S}_{v}$
. Dividing
$\mathscr {P}'$
by the product of the order of the finite group and
$[N:E]$
, we get the desired ample line bundle on
$\mathscr {S}_{v}$
.
Under Assumption 4.4.1 (1.b) or (1.c), the reflex field (B.1.15), in our notations, is E. For
$K\in {{\widetilde K_\Lambda }}$
and a finite place v of E, both sides of the morphism (B.1.30) (and the generalizations as in B.2.17 or B.2.28) are equipped with a natural action of the finite group
$Z^{{\mathbb {Q}}}\left ( {\mathbb {A}}_{\mathbb {Q}}^{\{p\}\cup \infty } \right )/Z^{{\mathbb {Q}}}({\mathbb {Z}}_{(p)})K_{Z^{{\mathbb {Q}}}}^p $
compatible with the morphism. See [Reference Liu, Tian, Xiao, Zhang and ZhuLTX+22, Definition 4.2.2]. Moreover, taking quotients by this finite group, we get a new morphism whose target is isomorphic to
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E,(v)}$
. Then the source of the new morphism is an integral model of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E,(v)}$
, whose regularity is assured by [Reference Rapoport, Smithling and ZhangRSZ20] and faithfully flat descent of regularity. We also denote this integral model by
$\mathscr {S}_v$
. Moreover, if Assumption 4.4.1 (1.a) also holds, then the construction here and in Lemma 4.4.3 coincides, by B.2.26.
We want to glue these models to obtain a regular integral model of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E}$
.
Theorem 4.4.4. Assume Assumption 4.4.1.
(1) For
$K\in {{\widetilde K_\Lambda }}$
, there is a regular integral model
${\mathcal {X}}_K$
of
${\mathrm {Sh}}({\mathbb {V}})_{K}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E}$
such that
${\mathcal {X}}_{K, {\mathcal {O}}_{E,(v)}}{\ \simeq \ } \mathscr {S}_v$
as integral models. See 4.3.2. Moreover, Assumption A.1.1 holds for
${\mathcal {X}}_K$
.
(2) For
$K\subset K'$
in
${{\widetilde K_\Lambda }}$
, there is a unique finite flat morphism
$\pi _{K,K' }:{\mathcal {X}}_{K }\to {\mathcal {X}}_{K' }$
extending the natural morphism
${\mathrm {Sh}}({\mathbb {V}})_{K }\to {\mathrm {Sh}}({\mathbb {V}})_{K' }$
.
(3) Regard
${\mathcal {X}}_{K }$
as an
$ {\mathcal {X}}_{K_{\Lambda } }$
-scheme and
${\mathrm {Sh}}({\mathbb {V}})_{K }$
as an
$ {\mathrm {Sh}}({\mathbb {V}})_{K_{\Lambda } }$
-scheme via
$\pi _{K,K' }$
. There is a unique action of
$K_\Lambda /K$
(note that K is normal in
$K_{\Lambda }$
) on the
$ {\mathcal {X}}_{K_{\Lambda } }$
-scheme
${\mathcal {X}}_{K }$
extending the standard action of
$K_\Lambda /K$
on the
$ {\mathrm {Sh}}({\mathbb {V}})_{K_{\Lambda } }$
-scheme
${\mathrm {Sh}}({\mathbb {V}})_{K }$
by ‘right translation’.
Proof. The construction of
${\mathcal {X}}_K$
is as follows. Continue to use the notation M in the proof of Lemma 4.4.3. First, consider the analog of the morphism (B.1.30) over
${\mathcal {O}}_M[R^{-1}]$
, where R is a finite set of finite places of
$ {\mathbb {Q}}$
such that
$K_v=K_{\Lambda ,v}$
for v not over R (in particular, we do not need the generalizations as in B.2.17 or B.2.28). See, for example, [Reference Rapoport, Smithling and ZhangRSZ20, 5.1] with extra level structures over R (similar to (B.1.29)). Denote this morphism by
${\mathcal {M}}\to {\mathcal {M}}_0$
. We use Galois descent to construct a model of
${\mathrm {Sh}}({\mathbb {V}})_K$
outside finitely many finite places. Let
$N/M$
be a finite extension, Galois over E, such that the base change of
${\mathcal {M}}_0$
to
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{N}[R^{-1}]$
is a finite disjoint union of
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{N}[R^{-1}]$
. Let
$S\supset R$
be a finite set of finite places of
${\mathbb {Q}}$
such that
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{N}[S^{-1}]\to \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E}[S^{-1}] $
is unramified. Then the fiber of
${\mathcal {M}}_{{\mathcal {O}}_{N}[S^{-1}]}\to {\mathcal {M}}_{0,{\mathcal {O}}_{N}[S^{-1}]}$
over a chosen
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{N}[S^{-1}]$
is a regular Deligne-Mumford stack
${\mathcal {M}}^S$
proper over
$ \operatorname {\mathrm {Spec}} {\mathcal {O}}_{N}[S^{-1}]$
with generic fiber
${\mathrm {Sh}}({\mathbb {V}})_{K,N}$
. By Zariski’s main theorem (for stacks which easily follows from the scheme version), after possibly enlarging S, we may assume that the action of the finite group
${\mathrm {Gal}}(N/E)$
on
${\mathrm {Sh}}({\mathbb {V}})_{K,N}$
extends to an action on
${\mathcal {M}}^S$
. By [Reference Bosch, Lütkebohmert and RaynaudBLR90, Section 6, Example] (and [Reference VistoliVis04] for the stack case), after possibly enlarging S,
${\mathcal {M}}^S$
descends to
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E}[S^{-1}]$
. Let
${\mathcal {Y}} ^S$
be the resulted Deligne-Mumford stack. By construction and B.2.26, we have
${\mathcal {Y}} ^S_{{\mathcal {O}}_{E,(v)}}{\ \simeq \ } \mathscr {S}_v$
for
$v\not \in S$
. Now, let
${\mathcal {X}}_K$
be the glueing of
${\mathcal {Y}}^S$
and
$\mathscr {S}_v$
with
$v\in S$
. Then
${\mathcal {X}}_{K, {\mathcal {O}}_{E,(v)}}{\ \simeq \ } \mathscr {S}_v$
for every finite place v.
We check Assumption A.1.1. For
$K'\subset K$
small enough with
$K^{\prime S}=K^S$
, the resulted Deligne-Mumford stack
${\mathcal {Y}}^{\prime S}$
is representable by [Reference Rapoport, Smithling and ZhangRSZ20, 5.2]. Note that
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{N}[S^{-1}]\to \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E}[S^{-1}] $
is unramified (so finite étale). Then Assumption A.1.1 (1) holds (with the current S) by construction. Similarly, Assumption A.1.1 (2) holds.
The uniqueness in (2) and (3) follows from the separatedness of the models. The existence follows from the construction and corresponding properties of the PEL type integral models in [Reference Rapoport, Smithling and ZhangRSZ20, Theorem 5.4]. We omit the details.
Remark 4.4.5. If we drop condition (2) in Definition 4.4.2 on K, Theorem 4.4.4 (1)(2) still holds. Indeed, to construct
${\mathcal {X}}_K$
, choose
$K_1\in \widetilde K_\Lambda $
such that
$K_1\subset K$
. Let
${\mathcal {X}}_K$
be the quotient of
${\mathcal {X}}_{K_1}$
by
$K/K_1$
where is the action is Theorem 4.4.4 (3).
By [Reference Rapoport, Smithling and ZhangRSZ20, Theorem 5.2] and our construction, we deduce the following lemma.
Lemma 4.4.6. If
$K_v=K_{\Lambda ,v}$
, then
${\mathcal {X}}_{K,{\mathcal {O}}_{E}, (v)}$
is smooth over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{{E}, (v)}$
.
Remark 4.4.7. We may relax Assumption 4.4.1 (3) by allowing
$\Lambda _v$
to be almost self-dual. See B.2.28. Then Lemma 4.4.3 still holds. However, Lemma 4.4.6 does not hold any more.
Corollary 4.4.8. A
${\mathbb {Q}}$
-line bundle (in particular a line bundle)
${\mathcal {L}}_{K_{\Lambda }}$
on
${\mathcal {X}}_{K_{\Lambda }}$
extending
$L_{K_{\Lambda }}$
is ample. Moreover, it is unique as a
${\mathbb {Q}}$
-line bundle.
Proof. By Lemma 4.4.6, the corresponding divisors of two different extensions differ by a
${\mathbb {Q}}$
-linear combination of special fibers of
${\mathcal {X}}_{K_{\Lambda }}$
, which is 0 in
${\mathrm {Ch}}^1\left ( {\mathcal {X}}_{K_{\Lambda }}\right )_{\mathbb {Q}}$
. The uniqueness follows. Now the ampleness follows from Lemma 4.4.3 (2).
Below, by a line bundle, we mean a
${\mathbb {Q}}$
-line bundle.
4.4.2 Admissible extensions
We want to define admissible extensions that are compatible as the level changes. So we consider a system of integral models.
Definition 4.4.9. (1) Let
$\widetilde {\mathcal {X}}$
be the system
$\{{\mathcal {X}}_K\}_{K\in {{\widetilde K_\Lambda }}}$
of integral models with transition morphisms
$\pi _{_K,_{K'}}$
as in Theorem 4.4.4 (1)(2).
(2) For
$h\in K_\Lambda $
, define the ‘right translation by h’ automorphism on
${\mathcal {X}}_K$
to be the action of h as in Theorem 4.4.4 (3).
(3) Fix an arbitrarily (
${\mathbb {Q}}$
-)line bundle
${\mathcal {L}}_{K_{\Lambda }}$
on
${\mathcal {X}}_{K_\Lambda }$
extending
$L_{K_{\Lambda }}.$
Let
$ {\mathcal {L}}_K=\pi _{_K,{_K}_{\Lambda }}^{*} {\mathcal {L}}_{K_{\Lambda }}$
.
We use
${\mathcal {L}}$
to denote the compatible-under-pullback system
$\{{\mathcal {L}}_K\}_{K\in {{\widetilde K_\Lambda }}}$
of ample line bundles on
$\widetilde {\mathcal {X}}=\{{\mathcal {X}}_K\}_{K\in {{\widetilde K_\Lambda }}}$
. The ampleness follows from the ampleness of
${\mathcal {L}}_{K_{\Lambda }}$
and finiteness of
$\pi _{_K,{_K}_{\Lambda }}$
. While using
${\mathcal {L}}$
for
${\mathcal {X}}_K$
, we mean
${\mathcal {L}}_K$
so that it has the same meaning as before (we previously used
${\mathcal {L}}$
as the abbreviation of
${\mathcal {L}}_K$
). Let
$\overline {\mathcal {L}}_K$
be the corresponding hermitian line bundle with the hermitian metric as in 4.2.1, and define
$\overline {\mathcal {L}}$
accordingly.
By Proposition A.3.8 – that is, the compatibility of the formation of admissible (Chow) cycles under flat pullback – we can define the direct limit group of admissible arithmetic divisors along the directed poset
${\widetilde K_\Lambda }$


where the transition maps are
$\pi _{_K,_{K'}}^{*}$
’s. By the projection formula (which gives that the composition of pullback by pushforward is the multiplication by the degree of the finite flat morphism), the natural maps

are injective and we understand the formers as subspaces of the latters, respectively. The embeddings
$ {\mathbb {C}}\hookrightarrow \widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)$
(Definition 4.3.1) are then the same embedding

Remark 4.4.10. If we drop condition (2) in Definition 4.4.2 as in Remark 4.4.10, (4.20)–(4.22) do not change.
Definition 4.4.11. Let
$\overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}\subset \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) $
consist of functions invariant by some
$K\in {{{\widetilde K_\Lambda }}}$
.
For
$t\in F_{>0}$
,
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}$
and
$K\in {{\widetilde K_\Lambda }}$
stabilizing
$\phi $
, consider the normalized admissible extension
$ Z_t( \omega (g)\phi )_K^{\overline {\mathcal {L}}} $
of
$ Z_t( \omega (g)\phi )_K$
(as in (4.16)). By Lemma 4.1.5 and Proposition A.3.8,

is independent of K. And by definition,

is independent of K.
For
$h\in K_\Lambda $
, the ‘right translation by h’ in
${\mathrm {Aut}}\left ( {\mathcal {X}}_{K}/{\mathcal {X}}_{K_{\Lambda } }\right )$
fixes
${\mathcal {L}}_K$
by definition. Thus, it induces an automorphism on
$\widehat Z^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}})$
and
$ \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}}) $
. Since it sends
$Z_t(g ,\phi ) $
to
$Z_t(g ,\omega (h)\phi ) $
, by Proposition A.3.8, we have the following lemma.
Lemma 4.4.12. The ‘right translation by h’ automorphism is the identity map on the image of
$ {\mathbb {C}}\hookrightarrow \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}})$
, fixes
$c_1(\overline {\mathcal {L}}^\vee ) $
, and sends
$Z_t(g ,\phi ) ^{\overline {\mathcal {L}}}$
to
$Z_t(g ,\omega (h)\phi ) ^{\overline {\mathcal {L}}}$
.
4.4.3 Conjecture
We define generating series of admissible arithmetic divisors in two ways, essentially by choosing different Green functions. (In fact, there will be a third one in 5.2.2.) The first (resp. second) definition is made toward Problem 4.3.4 (resp. Problem 4.3.3).
Recall the holomorphic part
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$
of the derivative of
$E_t(s,g,\phi )$
at
$s=0$
. See (3.29).
Definition 4.4.13. For
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{{{\widetilde K_\Lambda }}}$
,
$t\in F_{>0}$
and
$g\in G({\mathbb {A}}_F )$
, let

For
$a\in {\mathbb {C}}$
and
$g\in G({\mathbb {A}}_F)$
, let

Remark 4.4.14. (1) We rewrite the definition of
$z_t(g, \phi )_{{\mathfrak {e}}}^{{\overline {\mathcal {L}}}}$
as follows, which is closer to the notation ‘
$\left ( [ Z_t^{\overline {\mathcal {L}}}]+{\mathfrak {e}}_t\right ) q^t $
’ in (1.2). For
$t\in F_{>0}$
, we have the t-th Fourier coefficient
$ \frac {E_t(0,g,\phi )} {W^{{\mathfrak {w}}}_{\infty , t}(1)}$
,
$g\in G({\mathbb {A}}_F^\infty ),$
of
$E(0,g,\phi )$
. See 2.7. We similarly and formally consider
$ \frac {E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi ) } {W^{{\mathfrak {w}}}_{\infty , t}(1)},\ g\in G({\mathbb {A}}_F^\infty )$
as the ‘t-th Fourier coefficient’ of
$E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$
. Let

Then
$ {\mathfrak {e}}_t(1 ,\phi ^\infty )$
is the constant
${\mathfrak {e}}_t $
in (1.4). And we can rewrite

(2) We also expect that the (proposed) infinite component of
$e_t(g,\phi )$
in Problem 4.3.4 is
$ E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi ) +E_t(0,g,\phi ) \log {\mathrm {Nm}}_{F/{\mathbb {Q}}}t$
for any unitary Shimura variety. In particular, in the situation of this section, the v-component is 0 for every finite place v. It is related to Lemma 4.4.6. (The smoothness is ‘exotic’ at places where
$E/F$
is ramified.)
By (3.10), (3.14) and Lemma 4.4.12, we have the following lemma.
Lemma 4.4.15. For
$h\in K_{\Lambda }$
, the ‘’right translation by h’ automorphism on
${\mathcal {X}}_{K}$
sends
$z(g, \phi )_{{\mathfrak {e}},a}^{{\overline {\mathcal {L}}}}$
to
$z(g, \omega (h)\phi )_{{\mathfrak {e}},a}^{{\overline {\mathcal {L}}}}$
.
Now we specify the correct constant a to be used in
$z(g, \phi )_{{\mathfrak {e}},a}^{{\overline {\mathcal {L}}}}$
. Let

where
${\mathfrak {b}}$
is as in (3.30),
${\mathfrak {c}}$
is as in (3.15),
${\mathrm {Disc}}_F$
is the discriminant of
$F/ {\mathbb {Q}}$
, and
${\mathcal {P}}_{K_\Lambda }\in Z^1({\mathcal {X}}_{K_\Lambda })$
is a CM cycle to be precisely defined in Definition 5.1.10.
Remark 4.4.16. (1) The definition of
${\mathfrak {a}}$
is rather complicated. The reader may skip it for the moment.
(2) Looking at [Reference YuanYZ18, Theorem 1.7] and [Reference YuanYZ18, p 590], one can deduce cancellation between the terms in the definition of
${\mathfrak {a}}$
if one can, expectably, relate the Shimura varieties here and in loc. cit..
Conjecture 4.4.17. Assume Assumption 4.4.1. Let
$ \phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}$
. For
$g\in G({\mathbb {A}})$
, we have

For the second definition, we modify the normalized admissible extensions of special divisors
$Z(x)$
’s instead of weighted special divisors
$Z_t( \omega (g)\phi )^{\overline {\mathcal {L}}}$
. Unfortunately, the modification does not only depend on x, and the result generating series is not holomorphic.
Definition 4.4.18. For
$x\in {\mathbb {V}}^\infty $
with
$q(x)\in F^\times $
,
$g\in G({\mathbb {A}}_{F,\infty })$
and
$K\in {{\widetilde K_\Lambda }}$
, let
$ Z(x)^{\mathcal {L}} $
be the divisor on
${\mathcal {X}}_K$
that is the normalized admissible extension of
$Z(x)$
with respect to
${\mathcal {L}}$
, and
$ {\mathcal {G}}_{Z(x)_{E_v}}^{\overline L_{E_v}}$
the normalized admissible Green function for
$Z(x)_{E_v}$
with respect to
$\overline L_{E_v}$
. Let

Here,
$ {\mathfrak {k}}(x,g_v)$
is defined in Definition 4.2.6. For
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$
,
$a\in {\mathbb {C}}$
and
$g\in G({\mathbb {A}}_F)$
, let

Though
$Z(x,g)_{K,{\mathfrak {k}}}^{\overline {\mathcal {L}}}$
, as an element in
$\widehat Z^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}}) $
, depends on K,
$z(g, \phi )_{{\mathfrak {k}},a}^{{\overline {\mathcal {L}}}}$
does not. Indeed, by Lemma 3.3.9 (1),
$E_0(0,g,\phi )= \omega ( g)\phi (0)$
. Then by Theorem 4.2.7, we have

Then since
$z(g, \phi )_{{\mathfrak {e}},a }^{{\overline {\mathcal {L}}}}$
does not depend on K, neither does
$z(g, \phi )_{{\mathfrak {k}},a }^{{\overline {\mathcal {L}}}}$
.
The above reasoning also shows that (4.26) is equivalent to

4.4.4 Modularity theorems
We need some notations to state our theorems. Let
${\mathfrak {Ram}}$
be the set of finite places of F nonsplit in E, that are ramified in E or over
${\mathbb {Q}}$
. Let

Let

be the span of pure tensors
$\phi $
such that for every finite place v of F nonsplit in E,
$\phi _v=\omega (g)1_{\Lambda _v}$
for some
$g\in {\mathbb {G}}_v$
. In particular, we have no condition for
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}_{\mathfrak {Ram}}$
over split places. The following proposition and remark further show that we have no condition for
$\phi $
outside
${\mathfrak {Ram}}$
.
Proposition 4.4.19. For a finite place v of F inert in E (so that
$\Lambda _v$
is self-dual by Assumption 4.4.1) such that
$\chi _{_{\mathbb {V}},v}$
is unramified, the span of
$ \{\omega (g)1_{\Lambda _v}, g\in G(F_v)\}$
is
${\mathcal {S}}({\mathbb {V}}(E_v))^{K_{\Lambda ,v}}$
.
Proof. If v has residue characteristic
$\neq 2$
, this is a special case of [Reference HoweHow79, Theorem 10.2]. In general, let
$K^{\max }_v\subset G(F_v)$
be as in (2.5). Embedding
${\mathcal {S}}({\mathbb {V}}(E_v))^{K_{\Lambda ,v}\times K^{\max }_v}$
in an induced representation as [Reference RallisRal82, (3.1)]. It is routine to show that
${\mathcal {S}}({\mathbb {V}}(E_v))^{K_{\Lambda ,v}\times K^{\max }_v}$
is generated by
$1_{\Lambda _v}$
as a module over the Hecke algebra of bi-
$K^{\max }_v$
-invariant Schwartz functions on
$G(F_v)$
. Then the proposition follows from Kudla’s supercuspidal support theorem for big theta lift. See [Reference Gan and IchinoGI14, Proposition 5.2]
Remark 4.4.20. We may choose
$\chi _{_{\mathbb {V}}}$
such
$\chi _{_{\mathbb {V}},v}$
is unramified if v is inert in E.
Let
$ {\mathcal {A}}_{{\mathrm {hol}}}\left ( {\mathbb {G}},{\mathfrak {w}}\right ) $
and
$ {\mathcal {A}} \left ( {\mathbb {G}},{\mathfrak {w}}\right ) $
be the restrictions of
$ {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$
and
$ {\mathcal {A}} (G,{\mathfrak {w}})$
to
${\mathbb {G}}$
, respectively.
Theorem 4.4.21. Assume Assumption 4.4.1. Let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}_{\mathfrak {Ram}} $
. For
$g\in {\mathbb {G}}$
, we have

and

In fact, by (4.27), (4.28) and (4.29) are equivalent. Theorem 4.4.21 will be proved in 5.2.4.
Remark 4.4.22. (1) The restriction of (4.28) to
$g\in G({\mathbb {A}}_\infty )\subset {\mathbb {G}}$
gives Theorem 1.1.1, with a less strict condition on
$\phi _v$
,
$v\in {\mathfrak {Ram}}$
. Remark 4.4.10 allows
$K_v$
to be arbitrary for v split in E.
(2) By the density of
${\mathbb {G}}\subset G(F)\backslash G({\mathbb {A}}_F)$
, the restriction gives

In particular,
$ z(g, \phi )_{{\mathfrak {e}},{\mathfrak {a}}}^{{\overline {\mathcal {L}}}}$
with
$ g\in {\mathbb {G}}$
extends uniquely to an element in
$ {\mathcal {A}}_{{\mathrm {hol}}}\left ( G,{\mathfrak {w}}\right )\otimes \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}})$
. Conjecture 4.4.17 predicts that this extension is
$ z(g, \phi )_{{\mathfrak {e}},{\mathfrak {a}}}^{{\overline {\mathcal {L}}}}$
with
$ g\in G({\mathbb {A}})$
. However, it seems not trivial to check this prediction. The same discussion applies to
$z(g, \phi )_{{\mathfrak {k}},{\mathfrak {a}} }^{{\overline {\mathcal {L}}}}$
.
Definition 4.4.23. For
$x\in {\mathbb {V}}^\infty $
with
$q(x)\in F^\times $
and
$g\in G({\mathbb {A}}_{F,\infty })$
, let

Here, Kudla’s Green function
$ {\mathcal {G}}_{Z(x)_{E_v}}^{\mathrm {Kud}}$
is defined in (4.12).
For
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$
,
$a\in {\mathbb {C}}$
and
$g\in G({\mathbb {A}}_F)$
, let

It is directly to check that
$z(g, \phi )_{a}^{{\mathcal {L}},\mathrm {Kud}}$
is compatible under pullbacks by
$\pi _{_K,_{K'}}$
’s.
Define the topology on
$\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)$
as follows. Let
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $
be the
$L^2$
-orthogonal complement of
${\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
in
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$
, endowed with
$L^\infty $
-topology. Then
$\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)$
is the direct sum of
$C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $
and the finite dimensional subspace of cycles with harmonic curvatures at
$\infty $
. Endow
$\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)$
with the direct sum topology. Let
$ {\mathcal {A}} \left ( {\mathbb {G}},{\mathfrak {w}},\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)\right )$
be the restrictions of
$ {\mathcal {A}} \left ( G,{\mathfrak {w}},\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)\right )$
to
${\mathbb {G}}$
.
Theorem 4.2.10 and (4.28) of Theorem 4.4.21 imply the following theorem.
Theorem 4.4.24. Let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{K}_{\mathfrak {Ram}} $
, where
$K\in {{\widetilde K_\Lambda }}$
. The generating series
$z(g, \phi )_{ {\mathfrak {a}}}^{{\mathcal {L}},\mathrm {Kud}}$
of
$\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)$
-valued functions on
${\mathbb {G}}$
pointwise converges to an element in
$ {\mathcal {A}} \left ( {\mathbb {G}},{\mathfrak {w}},\widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K)\right ).$
5 Arithmetic mixed Siegel-Weil formula
In this section, we prove our modularity theorem (Theorem 4.4.21) above using an arithmetic analog of the mixed Siegel-Weil formula (3.19). First, we define CM cycles. Then, we state the formula and use this formula to prove Theorem 4.4.21. We end this section by discussing some possible future generalizations and applications of our results. We continue to use notations and assumptions in 4.4.
5.1 CM cycles
In this subsection, we define an orthogonal decomposition

where
${\mathbb {W}}$
is an incoherent hermitian space over
${\mathbb {A}}_E$
of dimension 1, and
$V^\sharp $
is a hermitian space over E of dimension n. Then we define a CM cycle

associated to the 0-dimensional Shimura variety for
${\mathbb {W}}$
, normalized to be of generic degree 1.
5.1.1 Lattices at unramified places
For a finite place v of F unramified in E, by [Reference JacobowitzJac62, Section 7], there exists
$e_v^{(0)},\ldots ,e_v^{(n)}\in \Lambda _v$
of unit norms, such that
$\Lambda _v$
is their orthogonal direct sum.
5.1.2 Lattices at ramified places
Let
${\mathfrak {Ram}}_{E/F}$
be the set of finite places of F ramified in E. For
$v\in {\mathfrak {Ram}}_{E/F}$
, let
${\mathbb {M}}_v$
be the
$ {\mathcal {O}}_{E_v}$
-lattice of rank 2 with an isotropic basis
$\{X,Y\}$
such that
$\langle {X,Y} \rangle =\varpi _{E_v}$
. Then
${\mathbb {M}}_v$
is a
$\varpi _{E_v}$
-modular lattice in
${\mathbb {M}}_v\otimes {E_v}$
, and its determinant with respect to this basis is
$-{\mathrm {Nm}}(\varpi _{E_v})$
. The hermitian space
${\mathbb {M}}_v\otimes {E_v}$
has determinant
$-1\in F_v^\times /{\mathrm {Nm}}( {E_v^\times })$
. In the other direction, starting with a 2-dimensional hermitian space H over
$E_v$
of determinant
$-1\in F_v^\times /{\mathrm {Nm}}( {E_v^\times })$
, let
$e_v^{(0)},e_v^{(1)}\in H$
be orthogonal such that
$q\left ( e_v^{(0)}\right ), q\left ( e_v^{(1)}\right )\in {\mathcal {O}}_{F_v}^\times .$
Then one can choose
${\mathbb {M}}_v\subset {\mathcal {O}}_{E_v}e_v^{(0)}\oplus {\mathcal {O}}_{E_v}e_v^{(1)}$
to be the preimage of one of the two isotropic lines in
$( {\mathcal {O}}_{E_v}e_v^{(0)}\oplus {\mathcal {O}}_{E_v}e_v^{(1)})/\varpi _{E_v}.$
In particular, one easily sees that

and

These two relations will only be used in the proof of Lemma 5.2.3.
Recall that for
$v\in {\mathfrak {Ram}}_{E/F}$
,
$v\nmid 2$
by Assumption 4.4.1 (1), and
$\Lambda _v$
is (almost)
$\varpi _{E_v}$
-modular as in Assumption 4.4.1 (4). Recall that the rank of
$\Lambda _v$
is
$n+1$
.
Lemma 5.1.1 [Reference JacobowitzJac62, Section 8].
(1) If n is odd, then
$\Lambda _v{\ \simeq \ } {\mathbb {M}}_v^{\oplus (n+1)/2}$
.
(2) If n is even, then
$\Lambda _v$
is the orthogonal direct sum of
$n/2$
-copies of
${\mathbb {M}}_v$
and a rank-1 hermitian
${\mathcal {O}}_{E_v}$
-module with determinant in
${\mathcal {O}}_{F_v}^\times $
.
5.1.3 CM cycles
For a finite place
$v\not \in {\mathfrak {Ram}}_{E/F}$
, let
$e_v^{(0)}$
be as in 5.1.1. For
$v \in {\mathfrak {Ram}}_{E/F}$
, let
$\Lambda _{v,1}\subset \Lambda _v$
be a copy of
${\mathbb {M}}_v$
. See Lemma 5.1.1. Let
$e_v^{(0)}\in E_v \Lambda _{v,1}$
such that

Let
${\mathbb {W}}$
be the restricted tensor product of
$E_v e_{v}^{(0)}$
, for every
$v\not \in \infty $
, and a
$1$
-dimensional subspace of
${\mathbb {V}}(E_v)$
, for every
$v\in \infty $
. Note that since

${\mathbb {W}}$
is incoherent. The orthogonal complement of
${\mathbb {W}}$
in
${\mathbb {V}}$
is coherent of dimension n. We denote the corresponding hermitian space over E by
$V^\sharp $
. This gives (5.1).
Let
$K_{{\mathbb {W}}}= K\cap U({\mathbb {W}}^\infty )$
. The morphism
${\mathrm {Sh}}({\mathbb {W}})_{K_{\mathbb {W}}}\to {\mathrm {Sh}}({\mathbb {V}})_{K}$
, analogous to (4.1), defines a zero cycle on
${\mathrm {Sh}}({\mathbb {V}})_{K}$
. (Indeed, this is a ‘simple special zero cycle’ compared with 4.1.1.)
Definition 5.1.3. Let
$d_{{\mathbb {W}},K} $
be the degree of this zero cycle. Let
$ {\mathcal {P}}_{{\mathbb {W}},K}\in Z_1({\mathcal {X}}_K)_{\mathbb {Q}}$
be
$1/ d_{{\mathbb {W}},K}$
times the Zariski closure of this cycle.
Recall that K is normal in
$K_{\Lambda }$
(Definition 4.4.2), and for
$h\in K_\Lambda $
, there is the ‘right translation by h’ automorphism on
${\mathcal {X}}_{K}$
(Definition 4.4.9 (3)).
Lemma 5.1.4. The ‘right translation by h’ automorphism on
${\mathcal {X}}_{K}$
sends
$ {\mathcal {P}}_{{\mathbb {W}},K}$
to
$ {\mathcal {P}}_{h{\mathbb {W}},K}$
.
Proof. When restricted to the generic fiber, the proof goes in the same way as [Reference KudlaKud97a, LEMMA 2.2 (iv)]. Taking Zariski closure, we get the lemma.
In particular, for
$k\in K$
,
$d_{{\mathbb {W}},K} =d_{k {\mathbb {W}},K},$
and
$ {\mathcal {P}}_{{\mathbb {W}},K} ={\mathcal {P}}_{k {\mathbb {W}},K}. $
Then by the flatness of
$\pi _{_K,_{K'}}$
(and the commutativity of taking Zariski closure and flat pullback) and [Reference LiuLiu11a, Proposition 3.2],

where each summand is independent of the choice of the representatives k. We shall later abbreviate
$(U({\mathbb {W}})\cap K')\backslash K'/K$
as
$U({\mathbb {W}})\backslash K'/K$
.
5.1.4 Another description
We will give another description of
${\mathcal {P}}_{{\mathbb {W}},K}$
that shows the independence of
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }$
on
${\mathbb {W}}$
. (It will also be used in 6.1.2 to compute intersection numbers.) Before that, we introduce new lattices, open compact subgroups and Shimura varieties.
For
$v\in {\mathfrak {Ram}}_{E/F}$
, let
$e_v^{(1)}\in \Lambda _{v,1}$
be orthogonal to
$e_v^{(0)}$
such that
$ q\left ( e_v^{(1)}\right )\in {\mathcal {O}}_{F_v}^\times ,$
and let
$\Lambda _{v,1}^\perp \subset \Lambda _v$
be the orthogonal complement of
$\Lambda _{v,1}$
. Then we have

where each inclusion is of colength 1, and
$\Lambda _v=\Lambda _{v,1}\oplus \Lambda _{v,1}^\perp .$
Let
$K_v^\dagger \subset U({\mathbb {V}}_{v})$
be the stabilizer of
${\mathcal {O}}_{E_v}e_v^{(0)}\oplus {\mathcal {O}}_{E_v}e_v^{(1)}\oplus \Lambda _{v,1}^\perp $
, and
$K^\dagger =K_v^\dagger \prod _{u\neq v}K_u$
.
Lemma 5.1.5. We have
$[K_v^\dagger :K_v^\dagger \cap K_v]=2$
.
Proof. The index is the cardinality of the isotropic lines in
$\left ({\mathcal {O}}_{E_v}e_v^{(0)}\oplus {\mathcal {O}}_{E_v}e_v^{(1)}\right ) /\varpi _{E_v}$
, which is 2.
Let
$K_{{\mathbb {W}},v}^{(0)}\subset U({\mathbb {W}}_v)$
be the stabilizer of
${\mathcal {O}}_{E_v}e_v^{(0)}$
, that is,
$K_{{\mathbb {W}},v}^{(0)}=U({\mathbb {W}}_v)$
, and
$K_{\mathbb {W}}^{(0)}=K_{{\mathbb {W}},v}^{(0)} \prod _{u\neq v }K_{{\mathbb {W}},u}$
. Then we have a diagram of morphisms of Shimura varieties

Applying pushfoward, pullback and pushfoward along the diagram (5.5) to the fundamental cycle of
${\mathrm {Sh}}({\mathbb {W}})_{K_{\mathbb {W}}^{(0)}}$
, we obtain a zero cycle on
${\mathrm {Sh}}({\mathbb {V}})_{K}$
. Divide it by its degree to obtain a zero cycle of degree 1.
Lemma 5.1.6. The Zariski closure of this degree 1 zero cycle is
${\mathcal {P}}_{{\mathbb {W}},K}$
.
Proof. By Lemma 5.1.5,
$[K^\dagger :K^\dagger \cap K]=2$
. Since
$K_{{\mathbb {W}},v}^{(0)}\not \subset K$
,
$K_{{\mathbb {W}},v}^{(0)} (K^\dagger \cap K) =K^\dagger $
. Then the fiber product of the first two morphisms in (5.5) is
${\mathrm {Sh}}({\mathbb {W}})_{ (K^\dagger \cap K)\cap U({\mathbb {W}}^\infty ) }$
(an analog of (5.4)). The natural morphism
${\mathrm {Sh}}({\mathbb {W}})_{ (K^\dagger \cap K) \cap U({\mathbb {W}}^\infty ) } \to {\mathrm {Sh}}({\mathbb {V}})_{ K }$
factors through
${\mathrm {Sh}}({\mathbb {W}})_{ K_{\mathbb {W}}}$
. The lemma follows.
Remark 5.1.7. One may define
${\mathcal {P}}_{{\mathbb {W}},K}$
via a diagram of integral models similar to (5.5), as in [Reference Rapoport, Smithling and ZhangRSZ20, (4.30)].
Lemma 5.1.8. Let
$h\in U(E_v \Lambda _{v,1})\times \{1_{E_v\Lambda _{v,1}^\perp }\}\subset U({\mathbb {V}}(E_v))$
such that
$\Lambda _{v,1}\subset h\left ( {\mathcal {O}}_{E_v} e_v^{(0)}\oplus {\mathcal {O}}_{E_v} e_v^{(1)}\right )$
and is the preimage of one of the two isotropic lines in the reduction modulo
$\varpi _{E_v}$
. Then
$h\in K_v^\dagger $
.
Proof. Let
$K^c=K^\dagger \backslash K$
. The two preimages of the two isotropic lines in the reduction of
$h\left ( {\mathcal {O}}_{E_v} e_v^{(0)}\oplus {\mathcal {O}}_{E_v} e_v^{(1)}\right )$
are
$h \Lambda _{v,1}$
and
$hK^c \Lambda _{v,1}$
. Then either
$h \Lambda _{v,1}=\Lambda _{v,1}$
or
$hK^c \Lambda _{v,1}= \Lambda _{v,1}$
. Each implies
$h\in K_v^\dagger $
.
Proposition 5.1.9. The CM cycle
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }$
does not depend on the choice of
${\mathbb {W}}$
.
Proof. To define
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }$
, we specify
$e_v^{(0)}\in {\mathbb {W}}$
as in 5.1.3. For
$v\not \in {\mathfrak {Ram}}_{E/F}$
, the choices of
$e_v^{(0)}$
differ by
$K_{\Lambda ,v}$
actions, which do not change
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }$
by Lemma 5.1.4. For
$v \in {\mathfrak {Ram}}_{E/F}$
, the choices of
$\Lambda _{v,1}$
differ by
$K_{\Lambda ,v}$
actions. See Lemma 5.1.1. By Lemma 5.1.8, we only need to show that
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }= {\mathcal {P}}_{h{\mathbb {W}},K_\Lambda }$
for
$h\in K_v^\dagger $
, where
$v\in {\mathfrak {Ram}}_{E/F}$
. We use Lemma 5.1.6. The pushforward of the fundamental cycle of
${\mathrm {Sh}}({\mathbb {W}})_{K_{\mathbb {W}}^{(0)}}$
by the first map in (5.5) is the same as the one obtained by replacing
${\mathbb {W}}$
by
$h{\mathbb {W}}$
, by the analog of Lemma 5.1.4 (on the generic fiber).
Definition 5.1.10. We denote
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }$
by
${\mathcal {P}}_{K_\Lambda }$
.
This definition is only used in (4.25). Later, we will still use
${\mathcal {P}}_{{\mathbb {W}},K_\Lambda }$
for the uniformity of the notation as the level changes. (For
$K\neq K_\Lambda $
,
${\mathcal {P}}_{{\mathbb {W}},K}$
depends on the choice of
${\mathbb {W}}$
.)
5.2 Formula
The arithmetic mixed Siegel-Weil formula compares the generating series of arithmetic intersection numbers between arithmetic special divisors and CM 1-cycles on the integral models with an explicit automorphic form. We use this formula to prove our main theorem Theorem 4.4.21.
5.2.1 Error functions
Both sides of the arithmetic mixed Siegel-Weil formula will have decompositions into local components (we will see in the proof in Section 6). We define some functions measuring the difference between these local components, and they will appear in the explicit automorphic form.
For a place v of F nonsplit in E, let W be the v-nearby hermitian space of
${\mathbb {W}}$
. See 2.2. Define the orthogonal direct sum

Then we have isomorpisms

and similar isomorphisms for W and
${\mathbb {W}}$
. Consider

Let
$\Lambda _v^\sharp \subset {\mathbb {W}}_v$
be the orthogonal complement of
${\mathcal {O}}_{E_v}e_v^{(0)}$
in
$\Lambda _v$
. For
$x=(x_1,x_2)\in V(E_v)-V^\sharp (E_v) ,$
let

where the smooth function
${W\theta }_{v,x} '(0, 1, \phi _v)$
on
$V(E_v)-V^\sharp (E_v)$
is as in (3.21).
Note that
$\Lambda _v={\mathcal {O}}_{E_v}e_v^{(0)}\oplus \Lambda _v^\sharp $
. So the computation for
$\phi _v'(x)$
is only on the component
${\mathcal {O}}_{E_v}e_v^{(0)}$
, and we can apply computations in [Reference YuanYZ18, Reference Yuan, Zhang and ZhangYZZ13].
Lemma 5.2.1 [Reference Yuan, Zhang and ZhangYZZ13, Proposition 6.8].
Assume that
$v\not \in {\mathfrak {Ram}}$
. Then
$\phi _v'=0$
.
Let
$c(g,\phi _v)$
be as below (3.23). Recall that
${\mathrm {Diff}}_{v}$
is the different of
$F_v$
over
${\mathbb {Q}}_{v}$
.
Lemma 5.2.2 [Reference YuanYZ18, Lemma 9.2].
Assume that
$v\in {\mathfrak {Ram}}-{\mathfrak {Ram}}_{E/F}$
. Then
$\phi _v'$
extends to a Schwartz function on
$ V(E_v)$
such that
$c(1,\phi _v)-2\phi _v'(0)=2\log |{\mathrm {Diff}}_{v}|_{v}$
.
For
$v \in {\mathfrak {Ram}}_{E/F}$
, as in 5.1.4, we have

and
$\Lambda _v=\Lambda _{v,1}\oplus \Lambda _{v,1}^\perp $
. For
$x=(x_1,x_2)\in V(E_v)-V^\sharp (E_v) ,$
let

Lemma 5.2.3. Assume that
$v\in {\mathfrak {Ram}}_{E/F}$
. Then
$\phi _v'$
extends to a Schwartz function on
$ V(E_v)$
such that
$c(1,\phi _v)-2\phi _v'(0)=2 \log |{\mathrm {Diff}}_v|_{F_v} $
.
Proof. The computation for
$\phi _v'(x)$
is indeed only on the component
$\Lambda _{v,1}$
of
$\Lambda _v$
. For simplicity, we assume that
$n=2$
so that we do not have the component
$ \Lambda _{v,1}^\perp $
.
Consider a larger lattice
$A={\mathcal {O}}_{E_v}e_v^{(0)}\oplus {\mathcal {O}}_{E_v}e_v^{(1)}$
. For
$x=(x_1,x_2)\in V(E_v)-V^\sharp (E_v) ,$
let

By [Reference YuanYZ18, Lemma 9.2],
$ \varphi $
extends to a Schwartz function on
$ V(E_v)$
such that
$c(1,1_{A})-2 \varphi (0)=2\log |{\mathrm {Diff}}_v|_{F_v}$
. It is enough to extend
$\varphi (x) 1_{{ \varpi _{E_v}{\mathcal {O}}_{E_v} e_v^{(1)} } } (x_2)-\phi _v'(x)$
to a Schwartz function on
$ V(E_v)$
, and show that the twice of its value at
$0$
is
$c(1,1_{A})-c(1,\phi _v) $
. First,

By Lemma 3.1.2 and (5.3),
${W\theta }_{v,x} '(0, 1, \phi _v)1_{{ {\mathcal {O}}_{E_v}^\times e_v^{(1)} } } (x_2)$
extends to a Schwartz function on
$ V(E_v)$
. It is supported on
$\{x_2\in {\mathcal {O}}_{E_v}^\times e_v^{(1)}\} $
so that its value at 0 is 0. By (3.5), (3.21) and (5.2),

Here, we used that
$L(s,\eta _v)=1$
due to the ramification of v in E. By Lemma 3.1.2, (5.8) extends to a Schwartz function on
$ V(E_v)$
.
Second, by a direct computation using (3.6), Lemma 3.1.2 and (5.2), we have

By [Reference Yuan, Zhang and ZhangYZZ13, p 23], which says
${\mathrm {Vol}}(U(W(E_v))=2|{\mathrm {Diff}}_v{\mathrm {Disc}}_v|_{F_v}^{1/2} $
, the lemma follows.
5.2.2 Generating series with automorphic Green functions
For
$t\in F_{>0}$
, let
$Z_t(\phi )^{{\mathcal {L}},{\mathrm {aut}}}\in \widehat Z^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}( {\mathcal {X}}_K)$
be the admissible extension of
$Z_t(\phi ) $
that is normalized at all finite places with respect to
${\mathcal {L}}$
and equals the automorphic Green function (4.10) at all infinite places. Comparing with (4.23), by Lemma 4.2.4 (1),

only depends on
$\phi $
, but not on K. For
$g\in G({\mathbb {A}}_F )$
and
$a\in {\mathbb {C}}$
, define
$z_t(g, \phi )_{{\mathfrak {e}}}^{{\mathcal {L}},{\mathrm {aut}}}$
(resp.
$z(g, \phi )_{{\mathfrak {e}},a}^{{\mathcal {L}},{\mathrm {aut}}}$
) by the formula defining
$z_t(g, \phi )_{{\mathfrak {e}}}^{{\overline {\mathcal {L}}}}$
(resp.
$z(g, \phi )_{{\mathfrak {e}},a}^{{\overline {\mathcal {L}}}}$
) in Definition 4.4.13, replacing
$Z_t(\phi )^{\overline {\mathcal {L}}}$
by
$Z_t(\phi )^{{\mathcal {L}},{\mathrm {aut}}}$
. By Lemma 4.2.4 (2), (4.26) is equivalent to

Formally define the ‘t-th Fourier coefficient’ (and compare with Remark 4.4.14 (1))

5.2.3 Arithmetic mixed Siegel-Weil formula
Let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}$
. See Definition 4.4.11. Assume that
$\phi $
is a pure tensor for simplicity. Define the automorphic form to appear in the formula as follows. For
$K\in {{\widetilde K_\Lambda }}$
stabilizing
$\phi $
, and
$K'\in {{\widetilde K_\Lambda }}$
containing K, let

Here, the index set and coefficients are the ones in (5.4). We choose a representative k such that
$k_v=1$
if
$K_v=K^{\prime }_v$
. In particular,
$\omega (k)(\phi ^{v}\otimes \phi _v')=\left ( \omega (k)\phi ^{v}\right )\otimes \phi _v'$
for v nonsplit in E, which is what we wrote
$\omega (k)\phi ^{v}\otimes \phi _v'$
for. Inside the bracket, we have 4 automorphic forms in
${\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$
. Here, we use the orthogonal decomposition
${\mathbb {V}}={\mathbb {W}}\oplus V^\sharp ({\mathbb {A}}_E)$
to define
${\theta E}^{\prime }_{{\mathrm {chol}}}(\dots )$
and
$C(\dots )$
, and for a given
$v\in {\mathfrak {Ram}}$
, we use the orthogonal decomposition
$ V=W\oplus V^\sharp $
as in (5.6) to define
$ {\theta E} \left (\dots \right )$
. See 3.2.1, 3.3.1 and 3.3.3 for their definitions. In particular,
$f_{{\mathbb {W}},K'}^K\in {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$
. Finally, the constant
${\mathfrak {b}}$
is as in (3.30),
${\mathfrak {c}}$
is as in (3.15) and the term
$2{{\mathfrak {b}}}[F:{\mathbb {Q}}] -{{\mathfrak {c}}}$
appears in both (3.31) and (4.25).
Remark 5.2.4. The bracketed automorphic form indexed by
$k\in U({\mathbb {W}})\backslash K' /K$
on the right-hand side of (5.9) is independent of the choice of k. Indeed, the 4th automorphic form
$ {\theta E} \left (0,g,\omega (k)\phi ^{v}\otimes \phi _v'\right ) $
is independent of the choice of k, by the
$K^v$
-invariance of
$\phi ^v$
and the mixed Siegel-Weil formula (3.19). By (3.31), for
$t\in F_{>0}$
, the
$\psi _t$
-Whittaker function of the rest 3 terms becomes

Then the independence of the choice of k follows from (3.10), (3.14) and (3.36). Then the independence of the choice of k follows from the automorphy.
Theorem 5.2.5. Let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}$
be a pure tensor such that
$\phi _v=1_{\Lambda _v}$
for every finite place v of F nonsplit in E. Let
$K\in {{\widetilde K_\Lambda }}$
stabilize
$\phi $
. For
$t\in F_{>0}$
and
$g\in P\left ({\mathbb {A}}_{F,{{\mathfrak {Ram}}} }\right ) G\left ({\mathbb {A}}_F^{{{\mathfrak {Ram}}} \cup \infty }\right ) $
, we have

where the arithmetic intersection on the left-hand side is taken on
${\mathcal {X}}_K$
, and
$ f_{{\mathbb {W}},{K_\Lambda },t}^{K,\infty }$
is the t-th Fourier coefficient of
$f_{{\mathbb {W}},{K_\Lambda }}^K$
. See 2.7.
Theorem 5.2.5 will be proved in (the end of) Section 6.
Remark 5.2.6. (1) By the projection formula, given
$\phi $
and
$K'$
, the truth of (5.10) does not depend on the choice of K (stabilizing the given
$\phi $
).
(2) We use
$\pi _{_K,_{K'}}^{*}{\mathcal {P}}_{{\mathbb {W}},K'} $
with
$K'=K_{\Lambda }$
, instead of the more natural CM cycle
${\mathcal {P}}_{{\mathbb {W}},K}$
(i.e.,
$K'=K$
), in order to apply Corollary A.2.6 to avoid computing normalized admissible extensions. (One may expect to reduce the whole Theorem 5.2.5 to the level
$K_\Lambda $
by the projection formula. However, this is not possible due to Remark 4.1.6.)
(3) We can also consider Theorem 5.2.5 for
$K'=K$
. Taking advantage of Theorem 4.4.21, the only difficulty in proving Theorem 5.2.5 for
$K'=K$
is computing normalized admissible extensions at split place. By considering admissible 1-cycles, the difficulty could be solved as in [Reference Yuan, Zhang and ZhangYZZ13, 8.5.1]. Note that we do not need Assumption 5.4 in loc. cit. Rather we will get an extra Eisenstein series.
5.2.4 Proof of Theorem 4.4.21
Assuming Theorem 5.2.5, we prove Theorem 4.4.21 as follows. Recall that as we have discussed immediately after Theorem 4.4.21, that (4.28) and (4.29) of Theorem 4.4.21 are equivalent. We will prove (4.28), assuming that
$\phi $
is a pure tensor such that
$\phi _{v}=1_{\Lambda _v} $
for every finite place v nonsplit in E (so that Theorem 5.2.5 applies to
$\phi $
). So (4.29) holds. It follows from Definition 4.4.18 that (4.29) holds for
$ \omega (g)\phi $
with
$g\in {\mathbb {G}}^\infty $
. Thus, Theorem 4.4.21 holds by the definition of
$ \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}_{\mathfrak {Ram}}$
above it.
So now we assume that
$\phi $
is a pure tensor such that
$\phi _{v}=1_{\Lambda _v} $
for every finite place v nonsplit in E and want to prove (4.28); that is,

Let
$K\in \widetilde K_\Lambda $
such that
$\phi $
is K-invariant. Let
$ f_{{\mathbb {W}},K_\Lambda ,0}^{K} $
be the
$0$
-th Whittaker coefficient of
$f_{{\mathbb {W}},K_\Lambda }^{K}$
. Let
$ A(\cdot , \phi )_K$
be the
${\mathbb {C}}$
-valued function on
$G({\mathbb {A}})$
, understood as valued
$\widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}})$
via
$ {\mathbb {C}}\hookrightarrow \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}})$
as in (4.22), defined by

that is,

By Lemma 4.3.8 (with G and
$G({\mathbb {A}})$
replaced by
${\mathbb {G}}$
) and Theorem 5.2.5,

Then by Lemma 4.2.4 (2), we have

Thus, (5.11) is reduced to the following lemma, whose proof requires some preparations.
Lemma 5.2.7. We have
$ A(g, \phi )_K = \omega (g)\phi (0) {\mathfrak {a}}.$
To determine
$ A(g, \phi )_K$
, one a priori needs to compute
$ f_{{\mathbb {W}},K_\Lambda ,0}^{K}(g) $
, which could be complicated. Indeed, by (3.20) for
$t=0$
and Lemma 3.3.9 (1) (2), we have

The terms in the second line cause the complicatedness.
We take a different approach. We study the invariance properties of
$ A(g, \phi )_K$
. We need a notation. Let
$W_0({\mathbb {G}})$
be the space of
$\psi _0$
-Whittaker functions on
${\mathbb {G}}$
(i.e., smooth
${\mathbb {C}}$
-valued functions f such that
$f(ng)=f(g) $
for
$ n\in N({\mathbb {A}})).$
Then

By (5.14) and Lemma 3.3.4 (2), we have
$ f_{{\mathbb {W}},K_\Lambda ,0}^{K} \in W_0 ({\mathbb {G}})$
. Then by (5.12), we have
$A(g, \phi )_K\in W_0({\mathbb {G}}).$
For
$K'\in \widetilde K_\Lambda $
such that
$\phi $
is
$K'$
-invariant, by (5.13),
$A(g,\phi )_K-A(g,\phi )_{K'}\in {\mathcal {A}}_{{\mathrm {hol}}}({\mathbb {G}},{\mathfrak {w}})$
. Thus, by (5.15),
$A(g,\phi )_K$
does not depend on K. Let us denote
$A(g,\phi )_K$
by
$ A(g, \phi )$
.
Lemma 5.2.8.
(1) For
$g'\in {\mathbb {G}}^\infty $
,
$ A(gg', \phi )=A(g ,\omega (g')\phi )$
.
(2) For
$h\in K_{\Lambda }$
,
$ A(g,\omega (h)\phi )=A(g,\phi )$
.
Proof. (1) Since
$ A(g,\phi ), \omega (g )\phi (0)\in W_0({\mathbb {G}})$
,

Claim:

Indeed, by Theorem 4.2.7, the difference between the generating series in (5.13) and (5.16) equals

(This is the analog of (4.27).) Then the claim follows from (5.13).
Now we prove (1). Apply (5.16) in two ways: first, replace
$\phi $
by
$\omega (g')\phi $
and call the resulted series
$S_1$
; second, replace g by
$gg'$
and call the resulted series
$S_2$
. Then
$S_1,S_2\in {\mathcal {A}} ({\mathbb {G}},{\mathfrak {w}}) \otimes \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}}) $
. So

Thus, it must be 0 by (5.15). This gives (1).
(2) By Lemma 4.4.12, after the ‘right translation by h’, (5.13) becomes

(This is similar to Lemma 4.4.15.) Since (5.13) holds with
$\phi $
replaced by
$\omega (h) \phi $
, taking the difference, we have
$ A(g,\omega (h)\phi )-A(g,\phi )\in {\mathcal {A}}_{{\mathrm {hol}}}({\mathbb {G}},{\mathfrak {w}})$
. It is 0 by (5.15).
To prove Lemma 5.2.7, we need a final ingredient, whose proof is easy and left to the reader.
Lemma 5.2.9. For v split in E, identify
${\mathbb {W}}_v=E_v $
and
$K_{\Lambda ,v}\cap U({\mathbb {W}}_v)={\mathcal {O}}_{E_v}^\times $
. Then
${\mathcal {S}}({\mathbb {W}}_v)^{K_{\Lambda ,v}\cap U({\mathbb {W}}_v)}$
is spanned by
$f_a$
’s, where
$f_a(x):= 1_{{\mathcal {O}}_{E_v}}( xa), a\in E_v^\times $
.
Proof of Lemma 5.2.7.
We have some reduction steps about
$\phi $
. By Lemma 5.2.8 (2), we may assume that
$\phi $
is
$K_\Lambda $
-invariant. In particular, for v split in E,
$\phi _v$
is
$K_{\Lambda ,v}\cap U({\mathbb {W}}_v)$
-invariant. Identify
${\mathbb {W}}_v=E_v$
and
$q={\mathrm {Nm}}$
. By Lemma 5.2.8 (1), Lemma 5.2.9 and the first Weil representation formula in 2.8, we may further assume that
$\phi _v=\phi _{v,1}\otimes \phi _{v,2}$
where
$\phi _{v,1}=1_{{\mathcal {O}}_{E_v}}$
and
$\phi _{v,2}\in {\mathcal {S}}\left ( V^\sharp (E_v)\right )$
(recall that
$V^\sharp (E_v)$
is the orthogonal complement of
${\mathbb {W}}_v$
).
Now we look at
$f_{{\mathbb {W}},K_\Lambda ,0}^{K}(g)$
given by (5.14). By the definition (3.23) and Lemma 3.1.3 (2) (3),
$c(g_v,\phi _v)=0$
unless
$v\in {\mathfrak {Ram}}$
or v split in E (by the same argument as in Lemma 3.3.4). By Lemma 5.2.8 (1), we may assume
$g^\infty =1$
. Then by Lemma 3.3.7, Lemma 5.2.2, Lemma 5.2.3,

Since
$c_1(\overline {\mathcal {L}}_{K }^\vee )=\pi _{_K,{_K}_{\Lambda }}^{*}{\mathcal {L}}_{K_\Lambda }^\vee $
, by the projection formula,

Now by the definition of
$ A(g, \phi )_K$
in (5.12) and the definition of
${\mathfrak {a}}$
in the first line of (4.25), the lemma follows.
5.3 Generalizations and applications
5.3.1 Higher codimensions
Based on their modularity result for generating series of special divisors, Yuan, S. Zhang and W. Zhang [Reference Yuan, Zhang and ZhangYZZ09] proved the modularity for higher-codimensional special cycles inductively, assuming the convergence. One would like to mimic their proof in the arithmetic situation. Then one needs a modularity theorem for divisors with general level structures and test functions, even if the given test function is very good. Thus, the generality of our results is necessary toward modularity in the arithmetic situation in higher codimensions.
In the codimension n case (i.e., for arithmetic 1-cycles), S. Zhang’s theory of admissible cycles is unconditional modulo vertical 1-cycles that are numerically trivial [Reference ZhangZha20]. Then the method in the current paper is still applicable to approach the modularity in the arithmetic situation.
5.3.2 Almost-self dual lattice
There is another lattice level structure at a finite place considered in [Reference Rapoport, Smithling and ZhangRSZ20], defined by an almost-self dual lattice. The integral model is not smooth, but is explicitly described in [Reference Liu, Tian, Xiao, Zhang and ZhuLTX+22]. We hope to include this level structure in a future work. In fact, if we also use admissible 1-cycles, our approach combined with a recent result of Z. Zhang [Reference ZhangZha21b, Theorem 14.6] is already applicable to prove the analog of Theorem 4.4.21, after replacing normalized admissible extensions of special divisors by the ‘admissible projections’ of the Kudla-Rapoport divisors at these new places (provided that they can also be suitably defined on our models). However, the difference between two extensions is not clear so far.
5.3.3 Faltings heights of Shimura varieties and Arithmetic Siegel-Weil formula
Following Kudla [Reference KudlaKud03, Reference KudlaKud04], we propose the arithmetic analog of the geometric Siegel Weil formula (4.11).
Problem 5.3.1. Match
$c_1(\overline {\mathcal {L}})^n\cdot z(g, \phi )_{ {\mathfrak {a}}}^{{\mathcal {L}},\mathrm {Kud}}$
with a linear combination of
$E(0,g, \phi )$
and
$ E'(0,g, \phi )$
(possibly up to some error terms).
The modularity of
$z(g, \phi )_{ {\mathfrak {a}}}^{{\mathcal {L}},\mathrm {Kud}}$
helps to attack this problem as follows. The constant term of
$c_1(\overline {\mathcal {L}})^n\cdot z(g, \phi )_{ {\mathfrak {a}}}^{{\mathcal {L}},\mathrm {Kud}}$
is indeed the Faltings height of
${\mathcal {X}}_{K}$
itself, while the non-constant terms are given by Faltings heights of Shimura subvarieties with the numbers in Definition 4.2.6. (Here, by the Faltings height of
${\mathcal {X}}$
, we mean
$ \deg c_1(\overline {\mathcal {L}})^{n+1}$
.) There is clearly an inductive scheme to compute the Faltings heights/attack arithmetic Siegel-Weil formula, by applying the modularity of the generating series. Moreover, we only need to compute enough terms. This might enable us to avoid dealing with Shimura subvarieties of general level structures from the inductive steps. We can also use
$ z(g, \phi )_{{\mathfrak {k}},{\mathfrak {a}} }^{{\overline {\mathcal {L}}}}$
to attack Problem 5.3.1, since by a direct computation,

For quaternionic Shimura curves, Faltings heights are computed in [Reference Kudla, Rapoport and YangKRY06] [Reference YuanYua22]. For unitary Shimura varieties, in the case
$F={\mathbb {Q}}$
, the Faltings height of
${\mathcal {X}}_{K_\Lambda }$
(for a different lattice
$\Lambda $
) was computed in [Reference Bruinier and HowardBH21] using Borcherds’ theory. See [Reference Bruinier and HowardBH21] for other related results.
5.3.4 Arithmetic theta lifting and Gross-Zagier type formula
Consider the Petersson inner product between the modular generating series of special divisors on the generic fiber and a cusp form f of G [Reference KudlaKud03, Reference KudlaKud04]. When
$n=1$
, it is cohomological trivial and its Beilinson-Bloch height was studied in [Reference LiuLiu11a, Reference LiuLiu11b]. However, when
$n>1$
, the Picard group of
${\mathrm {Sh}}({\mathbb {V}})_K$
is CM by [Reference Murty and RamakrishnanMR92], so that the inner product is 0 in most cases after cohomological trivialization. Thus, it is necessary to consider arithmetic intersection pairing on an integral model (without cohomological trivialization).
The arithmetic intersection pairing with our CM 1-cycle as in Theorem 5.2.5 is simply the Petersson inner product between (5.9) and f. Such a pairing appeared in the work of Gross and Zagier [Reference Gross and ZagierGZ86] and leads to their celebrated formula. We hope to get a Gross-Zagier type formula. In the case
$F={\mathbb {Q}}$
and
$\phi ^\infty =1_{\Lambda }$
(for a different
$\Lambda $
), a Gross-Zagier type formula was obtained in [Reference Bruinier, Howard and YangBHY15] and [Reference Bruinier, Howard, Kudla, Rapoport and YangBHK+20b]. See also [Reference Bruinier and YangBY09]. For general F and test functions as in our case, a general theory of Shimura-type integrals is to be developed.
6 Intersections
In this section, we prove the arithmetic mixed Siegel-Weil formula (Theorem 5.2.5). First, we prove local analogs of the formula, under some regularity assumption which forces improper intersections to disappear. Then, to prove Theorem 5.2.5, we use a global argument involving admissibility of our arithmetic divisors and modularity on the generic fiber (more precisely, Lemma 4.3.6).
6.1 Proper intersections
In order to state the local analogs of the Arithmetic mixed Siegel-Weil formula (Proposition 6.1.4), we need some preliminaries.
We use the CM cycles in 5.1, as well as the notations there – in particular, the orthogonal decomposition
${\mathbb {V}}={\mathbb {W}}\oplus V^\sharp ({\mathbb {A}}_E). $
Definition 6.1.1. For a finite place v of F, a Schwartz function on
${\mathbb {V}}(E_v)$
is
${\mathbb {W}}_v$
-regular if it is supported outside the orthogonal complement
${\mathbb {W}}_v^\perp =V^\sharp (E_{v})$
.
Let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) $
be a pure tensor such that
$\phi _v=1_{\Lambda _v}$
for every finite place v nonsplit in E. Let
$K\in \widetilde K_\Lambda $
such that
$\phi $
is K-invariant.
We assume the following assumption throughout this subsection.
Assumption 6.1.2. There is a nonempty set R of places of F such that
$\phi _v$
is
${\mathbb {W}}_v$
-regular for
$v\in R$
.
Since
$\phi _v=1_{\Lambda _v}$
for every finite place v nonsplit in E, R necessarily contains only places split in E.
As we have seen in the proof of Lemma 4.4.3, for a finite place v of E, there exists a finite unramified extension
$N/E_v$
such that
${\mathcal {X}}_{K ,{\mathcal {O}}_{N}} $
is (part of) a PEL moduli space, and the supersingular locus is thus defined and obviously independent of the choice of N. Let
$t\in F_{>0}$
.
Lemma 6.1.3. For
$g\in P \left ({\mathbb {A}}_{F,R }\right ) G\left ( {\mathbb {A}}_F^{R }\right )$
, the supports of the Zariski closure of
$ Z_t(\omega (g)\phi )^{\mathrm {zar}} $
and
$ {\mathcal {P}}_{{\mathbb {W}},K }$
on
${\mathcal {X}}_{K } $
only intersect on the supersingular loci at finite places of E nonsplit over F.
Proof. The regularity of
$\phi $
at R is preserved by the action of
$P \left ({\mathbb {A}}_{F,R }\right )$
on
$\phi $
. By the regularity of
$\omega (g)\phi $
, the lemma follows from the same proof as [Reference Kudla and RapoportKR14, Lemma 2.21]. Or one may reduce the lemma to (the version over a general CM field of) [Reference Kudla and RapoportKR14, Lemma 2.21] as follows. Choose n vectors
$x_1,\ldots ,x_n$
spanning
$V^\sharp $
. Then
$ {\mathcal {P}}_{{\mathbb {W}},K ,E }\subset Z(x_1)\cap \ldots \cap Z(x_n)$
. Since
$ Z_t(\omega (g)\phi ) $
is a finite sum of
$Z(x)$
’s with
$x $
outside
$V^\sharp ({\mathbb {A}}_E) $
, [Reference Kudla and RapoportKR14, Lemma 2.21] applies.
For
$k\in U({\mathbb {V}}^{\infty \cup R})$
, that is,
$k_v=1$
for
$v\in R$
. Then
${\mathbb {W}}_v=(k{\mathbb {W}})_v$
and
$\phi _v$
is
$(k{\mathbb {W}})_v$
-regular. By Lemma 6.1.3, the intersection number
$\left ( Z_t(\omega (g) \phi ) ^{\mathrm {zar}} \cdot {\mathcal {P}}_{k^{-1}{\mathbb {W}},K} \right )_{{\mathcal {X}}_{K,{\mathcal {O}}_{E_v}}}$
of the restrictions of the cycles to
${\mathcal {X}}_{K,{\mathcal {O}}_{E_v}}$
is well defined as in (A.1).
Proposition 6.1.4. Recall the set
${\mathfrak {Ram}}$
of finite places of F nonsplit in E, and ramified in E or over
${\mathbb {Q}}$
. Let
$g\in P\left ({\mathbb {A}}_{F,{\mathfrak {Ram}}\cup R}\right ) G\left ({\mathbb {A}}_F^{{\mathfrak {Ram}}\cup R}\right ) $
. Let v be a place of F nonsplit in E and
$k\in U({\mathbb {V}}^{\infty \cup R\cup \{v\}})$
. For
$v\not \in {\mathfrak {Ram}}\cup \infty $
, resp.
$v\in {\mathfrak {Ram}} $
, resp.
$v\in \infty $
, we respectively have



We will prove Proposition 6.1.4 for
$k=1$
, in Proposition 6.1.6 (
$v\not \in {\mathfrak {Ram}}\cup \infty $
), Proposition 6.1.7 and Proposition 6.1.9 (
$v\in {\mathfrak {Ram}} $
), Proposition 6.1.11 (
$v\in \infty $
). The proof for the general k is the same, except one needs to keep track of k. For simplicity, let

We prepare more notations for later computations. For a finite place v of F nonsplit in E, let
$E_v^{\mathrm {ur}}$
be the complete maximal unramified extension of
$E_v$
. Let
${\mathbb {E}}$
be the unique formal
${\mathcal {O}}_{F_v}$
-module of relative height
$2$
and dimension 1 over
$\operatorname {\mathrm {Spec}} E_v^{\mathrm {ur}}/\varpi _{E_v}$
. The endomorphism ring of
${\mathbb {E}}$
is the maximal order of the unique division quaternion algebra B over
$F_v $
. Fixing an embedding
$\iota : E_v \hookrightarrow B$
such that
$\iota ( {\mathcal {O}}_{E_v})$
is in the maximal order of B. Then
${\mathbb {E}}$
becomes a formal
${\mathcal {O}}_{E_v}$
-module of relative height
$1$
and dimension 1, which we still donote by
${\mathbb {E}}$
. Let
$\overline \iota $
be
$\iota $
precomposed with the nontrivial
${\mathrm {Gal}}(E_v/{F_v})$
-conjugation. It produces another
${\mathcal {O}}_{E_v}$
-module
$\overline {\mathbb {E}}$
. Fix an
${\mathcal {O}}_{F_v}$
-linear principal polarization
$\lambda _{{\mathbb {E}}}$
on
${\mathbb {E}}$
. Let
${\mathcal {E}}$
and
$\overline {\mathcal {E}}$
be the canonical liftings of
${\mathbb {E}}$
and
$\overline {\mathbb {E}}$
respectively, as
${\mathcal {O}}_{E_v}$
-modules. They are isomorphic as formal
${\mathcal {O}}_{F_v}$
-modules, and equipped with a unique
${\mathcal {O}}_{F_v}$
-linear principal polarization
$\lambda _{\mathcal {E}}$
lifting
$\lambda _{{\mathbb {E}}}$
.
6.1.1 Finite places of F inert in E
For such a v,
$\Lambda _v$
is self-dual.
Before we can compute the intersection number, we need to uniformize the integral model, CM cycle and special divisors using Rapoport-Zink spaces.
For a nonnegative integer m, let
${\mathcal {N}}_m $
be the unramified relative unitary Rapoport-Zink space of signature
$(m,1)$
[Reference Kudla and RapoportKR11] [Reference Li and ZhangLZ21, 2.1] over
$\operatorname {\mathrm {Spf}} {\mathcal {O}}_{E_v^{\mathrm {ur}}}$
. It is the deformation space of the polarized hermitian
${\mathcal {O}}_{E_v}$
-module
${\mathbb {X}}_m:=\overline {\mathbb {E}}\times {\mathbb {E}}^m$
with the product polarization
$\lambda _m$
. It is formally smooth of relative dimension m. The space
${\mathrm {Hom}}_{{\mathcal {O}}_{E_v}}({\mathbb {E}},{\mathbb {X}}_m)_{{\mathbb {Q}}}$
carries a natural hermitian pairing

For
$m=n$
, we let
${\mathcal {N}}={\mathcal {N}}_n$
. By [Reference Li and ZhangLZ21, 2.2], we have
${\mathrm {Hom}}_{{\mathcal {O}}_{E_v}}({\mathbb {E}},{\mathbb {X}}_n)_{{\mathbb {Q}}}{\ \simeq \ } V(E_v).$
And
$U(V(E_v))$
is isomorphic to the group of
${\mathcal {O}}_{E_v}$
-linear self-quasi-isogenies of
${\mathbb {X}}_n$
preserving
$\lambda _n$
[Reference Rapoport, Smithling and ZhangRSZ18, (4.3)]. In particular,
$U(V(E_v))$
acts on
${\mathcal {N}}$
. For every
$x\in V(E_v)-\{0\} $
, we have the Kudla-Rapoport divisor
${\mathcal {Z}}(x)$
on
$ {\mathcal {N}} $
[Reference Kudla and RapoportKR11] [Reference Li and ZhangLZ21, 2.3] that is the locus where x lifts to a quasi-isogeny. It is a (possibly empty) relative Cartier divisor. See [Reference Kudla and RapoportKR11, Proposition 3.5], which is only stated for
$F_v={\mathbb {Q}}_p$
but holds in the general case.
Let
$\widehat {{\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}^{ss}} $
be the formal completion of
${\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}$
along the supersingular locus. Then we have the following formal uniformization [Reference Li and ZhangLZ21, 13.1]:

For
$x\in V(E_v)-\{0\} $
and
$h\in U\left ( {\mathbb {V}}^{\infty ,v}\right )$
, we have a relative cartier divisor
$[{\mathcal {Z}}(x),h]$
of
$\widehat {{\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}^{ss}} $
.
The Rapoport-Zink space
${\mathcal {N}}_0 $
is naturally a closed formal subscheme of
${\mathcal {N}}$
by adding canonical liftings; that is, the morphism
${\mathcal {N}}_0\to {\mathcal {N}}$
is given by
$X\mapsto X\times {\mathcal {E}}^n$
. The subspace

becomes the subspace
$W(E_v)$
of
$V(E_v)$
, and the subgroup
$U(W(E_v))$
stabilizes
${\mathcal {N}}_0$
. We have

where
$d_{{\mathbb {W}},K} $
is the degree of the fundamental cycle of
${\mathrm {Sh}}({\mathbb {W}})_{K_{\mathbb {W}}}$
. See Definition 5.1.3), and the right-hand side is defined using the formal uniformization (6.5) of
$\widehat {{\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}^{ss}}$
with
${\mathcal {N}}_{0}$
understood as a 1-cycle on
${\mathcal {N}}$
.
Recall that
$V^t$
is the subset of
$V(E)$
of elements of norm t.
Proposition 6.1.5. Under the formal uniformization (6.5) of
$\widehat {{\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}^{ss}}$
, for
$g\in G({\mathbb {A}}_F^{ v})$
, we have

Proof. This follows from [Reference Li and LiuLL21, (8.3)] and the flatness of
${\mathcal {Z}}(x)$
.
Let
$ \left ({\mathcal {Z}}(x)\cdot {\mathcal {N}}_0\right )_{{\mathcal {N}}}$
be the Euler-Poincaré characteristic of the derived tensor product
${\mathcal {O}}_{{\mathcal {Z}}(x) }\otimes ^{\mathbb {L}} {\mathcal {O}}_{{\mathcal {N}}_0}$
. Since
$h\in U(W(E_v))$
stabilizes
${\mathcal {N}}_0$
,
$\left ({\mathcal {Z}}(x)\cdot {\mathcal {N}}_0\right )_{{\mathcal {N}}}= \left ({\mathcal {Z}}(hx)\cdot {\mathcal {N}}_0\right )_{{\mathcal {N}}}$
. By Lemma 6.1.3, (6.6), Proposition 6.1.5 and a direct computation, for
$g\in P ({\mathbb {A}}_{F,R\cup \{v\}})G({\mathbb {A}}_F^{ R\cup \{v\}})$
,

Proposition 6.1.6. If v is unramified over
${\mathbb {Q}}$
, for
$g\in P ({\mathbb {A}}_{F,R})G({\mathbb {A}}_F^{ R})$
and
$k=1$
, (6.1) holds.
Proof. First, we compute
$ \left ( {\mathcal {Z}}(x)\cdot {\mathcal {N}}_0\right )_{{\mathcal {N}}}$
(i.e., we need to compute the length of the locus on
${\mathcal {N}}_0$
to which x lifts). Recall that
$\Lambda _v^\sharp \subset {\mathbb {V}}(E_v)$
is the orthogonal complement of
${\mathcal {O}}_{E_v}e_v^{(0)}$
in
$\Lambda _v$
. Under the isomorphism
${\mathrm {Hom}}_{{\mathcal {O}}_{E_v}}({\mathbb {E}},{\mathbb {X}}_n)_{{\mathbb {Q}}}{\ \simeq \ } V(E_v)$
, the image of
${\mathrm {Hom}}_{{\mathcal {O}}_{E_v}}({\mathbb {E}}, {\mathbb {E}}^n)$
is
$\Lambda _v^\sharp $
. Then for
$ x=(x_1,x_2)\in W(E_v)\oplus V^\sharp (E_v) $
with
$x_1\neq 0$
, if it lifts, then
$x_2\in \Lambda _v^\sharp $
. Moreover, by Gross’ result on canonical lifting [Reference GrossGro86, Proposition3.3], we have

Second, if
$g_v= 1$
, express the left-hand side of (6.1) by (6.8) and (6.9). Compare it with the expression of the right-hand side of (6.1) given by (3.22). By Lemma 5.2.1, (6.1) follows.
Finally, we reduce the general case to the case
$g_v=1$
in two reduction steps. (I) We claim the following: replacing
$g_v$
by
$g_vn(b_v)$
for
$b_v\in F_v$
or by
$g_vk_v $
for
$k_v\in K_v^{\max }$
(see 2.5), both sides of (6.1) are multiplied by the same constant. Indeed, for the left-hand side of (6.1), we directly use the definition of the Weil representation. For the right-hand side, besides the definition, we further need (3.2) and Lemma 3.2.2. (II), By Corollary 3.2.3 and Lemma 4.1.3, we can replace g by
$m(a)g$
for some
$a\in E^\times $
. By the Iwasawa decomposition, the fact that
$E_v^\times =E^\times {\mathcal {O}}_{E_v}^\times $
, and the claim in the first reduction step, we may assume that
$g_v= 1$
. The proposition is proved.
Proposition 6.1.7. If v is ramified over
${\mathbb {Q}}$
, for
$g\in P ({\mathbb {A}}_{F,R\cup \{v\}})G({\mathbb {A}}_F^{ R\cup \{v\}})$
, (6.2) holds.
Proof. The proof is the same with the proof of Proposition 6.1.6 with the following exceptions:
-
• in the claim in the reduction step (I), remove ‘or by
$g_vk_v $ for
$k_v\in K_v^{\max }$ ’;
-
• in the reduction step (II), ‘Corollary 3.2.3’ should be replaced by ‘Corollary 3.2.3 and that
$ {\theta E} \left (0,m(a)g,\phi ^{v}\otimes \phi _v'\right )= {\theta E} \left (0,g,\phi ^{v}\otimes \phi _v'\right ) $ for
$a\in E^\times $ ’.
6.1.2 Finite places of F ramified in E
For such a v,
$\Lambda _v$
is a
$\pi _v$
-modular or almost
$\pi _v$
-modular lattice. This case is more complicated.
We still need formal uniformization using Rapoport-Zink spaces. For a nonnegative integer m, let
${\mathcal {N}}_m$
be the exotic smooth relative unitary Rapoport-Zink space of signature
$(m,1)$
over
$\operatorname {\mathrm {Spf}} {\mathcal {O}}_{E_v^{\mathrm {ur}}}$
. See [Reference Rapoport, Smithling and ZhangRSZ18, Section 6,7], [Reference Rapoport, Smithling and ZhangRSZ17, 3.5] and [Reference Li and LiuLL22, 2.1]. It will also be specified below. It is formally smooth over
$\operatorname {\mathrm {Spf}} {\mathcal {O}}_{E_v^{\mathrm {ur}}}$
of relative dimension m. (Note that the case
$m=0$
is not covered in either [Reference Rapoport, Smithling and ZhangRSZ17] or [Reference Li and LiuLL22], but is specifically indicated in [Reference Rapoport, Smithling and ZhangRSZ18, Example 7.2].) Let
${\mathcal {N}}={\mathcal {N}}_n$
. We will use
${\mathcal {N}} $
for the formal uniformization of
${\mathcal {X}}_K$
. The analog of the formal uniformization (6.6) of
${\mathcal {P}}$
using
${\mathcal {N}}_0$
is more subtle: we will use
${\mathcal {N}}_1$
to define morphisms
${\mathcal {N}}_0\to {\mathcal {N}}_1\to {\mathcal {N}}$
, which will lead us to the formal uniformization (6.11) of
${\mathcal {P}}$
.
Remark 6.1.8. The reason for this subtlety might be explained as follows. Recall that the construction of
${\mathcal {P}}$
requires an additional rank 2 sub-lattice
$\Lambda _{v,1}$
as in 5.1.3 at each finite place, which is a direct summand of
$\Lambda _v$
. And
$E_v\Lambda _{v,1}$
contains a distinguished vector
$e_v^{(0)}$
of unit norm. One might consider
${\mathcal {O}}_{E_v}e_v^{(0)}\subset E_v\Lambda _{v,1}$
and
$ \Lambda _{v,1}\subset \Lambda _v$
as being parallel to the morphisms
${\mathcal {N}}_0\to {\mathcal {N}}_1$
and
${\mathcal {N}}_1\to {\mathcal {N}}$
. Note that
${\mathcal {O}}_{E_v}e_v^{(0)}$
is not contained in
$\Lambda _{v,1}$
( or
$\Lambda _v$
). This makes it nontrivial to define a morphism
${\mathcal {N}}_0\to {\mathcal {N}}_1$
or
${\mathcal {N}}_0\to {\mathcal {N}}$
. See also [Reference Rapoport, Smithling and ZhangRSZ18, Remark 12.3]. The morphism
${\mathcal {N}}_0\to {\mathcal {N}}_1$
we use is given by [Reference Rapoport, Smithling and ZhangRSZ18, Section 12].
We have 6 steps before the main result Proposition 6.1.9 of this 6.1.2.
First, we specify
${\mathcal {N}}_1$
,
${\mathcal {N}}$
and
${\mathcal {N}}_1\to {\mathcal {N}}$
. Assume that
$\varpi _{E_v}^2=\varpi _{F_v}$
. The framing object
${\mathbb {X}}_1$
for the deformation space
${\mathcal {N}}_1$
is the Serre tensor
${\mathcal {O}}_{E_v}\otimes _{{\mathcal {O}}_{F_v}}\overline {\mathbb {E}}$
, which is the conjugate of the framing object [Reference Rapoport, Smithling and ZhangRSZ17, (3.5)], with the polarization conjugate to the one in [Reference Rapoport, Smithling and ZhangRSZ17, (3.6)]. In the case that n is odd (the case of Lemma 5.1.1 (1)), the framing object for
${\mathcal {N}}$
is
${\mathbb {X}}_n:={\mathbb {X}}_1\times ({\mathbb {E}}^2)^{(n-1)/2}$
with the product polarization
$\lambda _n$
, where the polarization on
${\mathbb {E}}^2$
is given by

In the same way, we have a polarization
$\widetilde {\lambda } $
on
${\mathcal {E}}^2$
using
$\lambda _{\mathcal {E}}$
. This gives us a morphism
${\mathcal {N}}_1\to {\mathcal {N}}$
by
$X\mapsto X\times ({\mathcal {E}}^2)^{(n-1)/2}$
with the polarization
$\widetilde {\lambda } $
on each of
${\mathcal {E}}^2$
. In the case that n is even (the case of Lemma 5.1.1 (2)), the framing object for
${\mathcal {N}}$
is
${\mathbb {X}}_n:={\mathbb {X}}_1\times ({\mathbb {E}}^2)^{(n-2)/2} \times {\mathbb {E}}$
where the polarization
$\lambda _{{\mathbb {E}}}'$
on the last component is a multiple of
$\lambda _{{\mathbb {E}}}$
so that the induced hermitian pairing on
${\mathrm {Hom}}({\mathbb {E}},{\mathbb {E}})_{{\mathbb {Q}}}$
(defined as in (6.4)) has determinant
$q(e_v)$
as in Lemma 5.1.1 (2). This gives us a morphism
${\mathcal {N}}_1\to {\mathcal {N}}$
by
$X\mapsto X\times ({\mathcal {E}}^2)^{(n-1)/2}\times {\mathcal {E}}$
with the unique lifting of
$\lambda _{{\mathbb {E}}}'$
on the last component
${\mathcal {E}}$
.
Second, the uniformizations of
$\widehat {{\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}^{ss}}$
and
$Z_t(\phi )^{\mathrm {zar}}$
are as follows. By [Reference Rapoport, Smithling and ZhangRSZ17, (3.10)],
$V(E_v){\ \simeq \ } {\mathrm {Hom}}_{{\mathcal {O}}_{E_v}}({\mathbb {E}},{\mathbb {X}}_n)_{{\mathbb {Q}}} $
and
$U(V(E_v))$
is isomorphic to the group of
${\mathcal {O}}_{E_v}$
-linear self-quasi-isogenies of
${\mathbb {X}}_n$
preserving
$\lambda _n$
. In particular,
$U(V)$
acts on
${\mathcal {N}}$
. The analog of the formal uniformization (6.5) of
$\widehat {{\mathcal {X}}_{K, {\mathcal {O}}_{E_v^{\mathrm {ur}}}}^{ss}}$
holds by [Reference Li and LiuLL22, (4.9)]. For every
$x\in V(E_v)-\{0\} $
, we have the Kudla-Rapoport divisor
${\mathcal {Z}}(x)$
on
$ {\mathcal {N}} $
, which is a (possibly empty) relative Cartier divisor [Reference Li and LiuLL22, Lemma 2.41]. The analog of Proposition 6.1.5 holds by [Reference Li and LiuLL22, Proposition 4.29] combined with the argument in the proof of Proposition 6.1.5. Though [Reference Li and LiuLL22] only uses even dimensional hermitian spaces, the specific results that we cite hold in the general case by the same proof.
Third, we recall the morphisms
${\mathcal {N}}_0\to {\mathcal {N}}_1$
defined in [Reference Rapoport, Smithling and ZhangRSZ18, Section 12]. This is rather complicated for general
${\mathcal {N}}_{2m}\to {\mathcal {N}}_{2m+1}$
. Fortunately, in our case, we have the following convenient description. By [Reference Rapoport, Smithling and ZhangRSZ18, Example 12.2],
$ {\mathcal {N}}_1$
is isomorphic to the disjoint union of two copies of the Lubin-Tate deformation space for the formal
${\mathcal {O}}_{F_v}$
-module
${\mathbb {E}}$
. We write
${\mathcal {N}}_1={\mathcal {N}}_1^+\coprod {\mathcal {N}}_1^{-}$
. Recall that B is the unique division quaternion algebra over
$F_v $
, and its maximal order
${\mathcal {O}}_B$
is the endomorphism ring of
${\mathbb {E}}$
. For
$c\in B^\times $
, we have two closed embeddings (moduli of the canonical lifting)
${\mathcal {N}}_0\to {\mathcal {N}}_1^{\pm }$
associated to
$c\iota c^{-1}:E_v\hookrightarrow B$
. Let
${\mathcal {N}}_0^{c,\pm }$
be the union of the images.
Fourth, we need to specify c so that we can use
${\mathcal {N}}_0^{c,\pm }$
to uniformize
${\mathcal {P}}$
. See (6.11) below. Let
$e_v^{(1)}$
be as in 5.1.4. Then

where the middle is the adjunction isomorphism, and the last isomorphism is compatible with
${\mathrm {Hom}}_{{\mathcal {O}}_{E_v}}({\mathbb {E}},{\mathbb {X}}_n)_{{\mathbb {Q}}}{\ \simeq \ } V(E_v)$
. And the hermitian form on
$W(E_v)\oplus E_v e_v^{(1)}$
corresponds to
$-2\varpi _{F_v}{\mathrm {Nm}}_B $
(see the proof of [Reference Rapoport, Smithling and ZhangRSZ17, Lemma 3.5]), where
${\mathrm {Nm}}_B$
is the reduced norm on B. Let c correspond to
$\varpi _{E_v} e_v^{(1)}$
. Since
$q\left ( e_v^{(1)}\right )\in {\mathcal {O}}_{F_v}^\times $
(see 5.1.2),
$c\in {\mathcal {O}}_B^\times $
(note that
$v\nmid 2$
here).
Fifth, we uniformize
${\mathcal {P}}$
. By Lemma 5.1.6, we have another description of
${\mathcal {P}}$
via the diagram (5.5) of morphisms of Shimura varieties. See also Remark 5.1.7. Comparing it with the moduli interpretation of
${\mathcal {N}}_0\to {\mathcal {N}}_1$
in [Reference Rapoport, Smithling and ZhangRSZ18, Proposition 12.1], we have the following analog of (6.6):

where
$d_{K_{\mathbb {W}}^{(0)}} $
is the degree of the fundamental cycle of
${\mathrm {Sh}}({\mathbb {W}})_{K_{\mathbb {W}}^{(0)}}$
. See (5.5). Here, the extra factor 2 comes from Lemma 5.1.5.
Finally, we compute
$ \left ( {\mathcal {Z}}(x)\cdot {\mathcal {N}}_{0}^{c,\pm }\right )_{{\mathcal {N}}}$
. Since
$c\in {\mathcal {O}}_B^\times $
, by [Reference Rapoport, Smithling and ZhangRSZ17, Lemma 6.5, Proposition 7.1],

Here,
${\mathcal {Y}}\left (\varpi _{E_v}e_v^{(1)}\right )$
is the Kudla-Rapoport divisor on
${\mathcal {N}}_1$
, where
$\varpi _{E_v}e_v^{(1)}$
lifts. Let
${\mathbb {X}}^\perp $
be the direct complement of
${\mathbb {X}}_1$
in
${\mathbb {X}}_n$
, that is,
${\mathbb {X}}^\perp =({\mathbb {E}}^2)^{(n-1)/2}$
if n is odd, and
${\mathbb {X}}^\perp =({\mathbb {E}}^2)^{(n-2)/2} \times {\mathbb {E}}$
is n is even. Let
$ \Lambda _{v,1}^\perp $
be as in 5.1.4. By Lemma 5.1.1 and (6.10),

corresponds to
$\Lambda _v^\perp $
. Then (similar to the deduction of (6.9)) by (6.12) and Gross’ result on canonical lifting [Reference GrossGro86, Proposition 3.3], we have

for
$ x=(x_1,x_2)\in W(E_v)\oplus V^\sharp (E_v) $
with
$x_1\neq 0$
. Here, the extra factor 2 comes from that
${\mathcal {N}}_0^{c,\pm }$
has 2 components.
Proposition 6.1.9. Assume that
$g\in P ({\mathbb {A}}_{F,R\cup \{v\}})G({\mathbb {A}}_F^{ R\cup \{v\}})$
and
$k=1$
. Then (6.2) holds.
Proof. As an analog of (6.8), we have

The rest of the proof is the same as the proof of Proposition 6.1.7, after replacing Lemma 5.2.2 by Lemma 5.2.3.
6.1.3 Infinite places of F
Let
$v\in \infty $
. Under the complex uniformization (4.4) of
$ {\mathrm {Sh}}({\mathbb {V}})_{K,E_v}$
,

where
$o:=[0,.. .,0]\in {\mathbb {B}}_n$
, and
$d_{{\mathbb {W}},K} $
is the degree of the fundamental cycle of
${\mathrm {Sh}}({\mathbb {W}})_{K_{\mathbb {W}}}$
. For
$g\in P ({\mathbb {A}}_{F,R})G({\mathbb {A}}_F^{ R})$
, by the definition of
${\mathcal {G}}_{Z_t( \cdot )_{E_v},s}$
(above (4.10)), a direct computation gives

Now we compare the inner sums in (3.33) and (6.14). Recall the involved functions
$P_s$
and
$Q_s$
. See (3.32) and (4.7), respectively. From (3.32), we have

where F is the hypergeometric function. In particular,

where the constant for
$O(\cdot )$
is uniform near
$s=0$
. We also have

From (4.7), we have

where the constant for
$O(\cdot )$
is uniform near
$s=0$
. We also have

Lemma 6.1.10. For
$s_0>-1$
, on
$\{s\in {\mathbb {C}},{\mathrm {Re}} s>s_0\}$
, the sum

converges uniformly and absolutely. And its value at
$s=0$
is 0.
Proof. We compare the sum in the lemma with the sum in (6.14), which is absolutely convergent by Lemma 4.2.1. By (6.15) and (6.17), the sum in the lemma for s is dominated by a multiple of the sum in (6.14) with s replaced by
$s+1$
. The convergence in the lemma follows. By (6.16) and (6.18), the value at
$s=0$
is 0.
By Theorem 4.2.2, the integration on the right-hand side of (6.14) admits a meromorphic continuation to s around 0 with a simple pole at
$s=0$
.
Proof of Lemma 3.3.12.
For this moment, we consider a general
$ \phi \in \overline {\mathcal {S}}({\mathbb {V}}) $
(without any regularity assumption). The above discussion still holds replacing
$Z_t(\phi )$
by

Then Lemma 3.3.12 follows from (6.14) and the first part of Lemma 6.1.10.
Recall the definition of
${\mathcal {G}}_{Z_t( \cdot ) _{E_v}}^{\mathrm {aut}}$
in (4.10). Then

where both multipliers
$(\cdot )^{-1}$
go to 1 as
$s\to 0$
(this gives the first and third ‘=’), and the second ‘=’ follows from Lemma 6.1.10. Thus, we have proved the following proposition.
Proposition 6.1.11. For
$g\in P ({\mathbb {A}}_{F,R})G({\mathbb {A}}_F^{ R})$
, (6.3) holds for v.
6.2 Improper intersections
In this subsection, we prove the arithmetic mixed Siegel-Weil formula (Theorem 5.2.5). The proof starts in 6.2.1. Before that, let us discuss the strategy.
Lemma 6.2.1. Let
$Y=X\oplus X'$
be the orthogonal direct sum of two non-degenerate quadratic spaces over a non-archimedean local field of characteristic
$\neq 2$
. Let
$\widehat {\mathcal {S}}(Y- X')_{\overline {\mathbb {Q}}}$
be the space of the Fourier transforms of functions in the space
${\mathcal {S}}(Y- X')_{\overline {\mathbb {Q}}}$
of
$\overline {\mathbb {Q}}$
-valued Schwartz functions on Y supported on
$Y-X'$
(the Fourier transforms are clearly also
$\overline {\mathbb {Q}}$
-valued). Then

Proof. Since Fourier transform respects orthogonal direct sum, one may assume that
$X'=\{0\}$
. Then the lemma is well known and also easy to check directly.
We will use the following notation. For a finite set
$S $
of finite places of F, let

The main difficulty in proving Theorem 5.2.5 is from improper-intersections. However, if we choose a
$\overline {\mathbb {Q}}$
-valued pure tensor
$\phi $
that is
${\mathbb {W}}_v$
-regular at some places v in S, we can prove a
$\operatorname {\mathrm {mod}} \overline {\mathbb {Q}}\log S$
-version of Theorem 5.2.5 for
$\phi $
, see Lemma 6.2.3. Note that both the CM cycle and the regularity assumption are associated to
${\mathbb {W}}$
. The same result holds if we replace
${\mathbb {W}}$
by some
${\mathbb {W}}'$
(and use the corresponding CM cycle and regularity assumption). Accordingly, we make Table 1.

The table displays the
$\operatorname {\mathrm {mod}} \overline {\mathbb {Q}}\log S$
-version of Theorem 5.2.5 under different conditions. The second row indicates that we consider the CM cycle associated to
${\mathbb {W}}$
or
${\mathbb {W}}'$
. The second column indicates that we impose the regularity assumption on
$\phi $
associated to
${\mathbb {W}}$
or
${\mathbb {W}}'$
, or no regularity assumption. Then a cell indexed by them indicates the
$\operatorname {\mathrm {mod}} \overline {\mathbb {Q}}\log S$
-version of Theorem 5.2.5 for the corresponding CM cycle under the corresponding regularity assumption on
$\phi $
.
Lemma 6.2.3 gives the cell L of Table 1. Replacing
${\mathbb {W}}$
by
${\mathbb {W}}'$
, we get L
$'$
.
We want to arrive at the cell C (proved in Corollary 6.2.9), the
$\operatorname {\mathrm {mod}} \overline {\mathbb {Q}}\log S$
-version of Theorem 5.2.5 for the general
$\phi $
. We use the cell B the bridge from L to C. The relation between this cell B and L
$'$
is the ‘switch CM cycles’ indicated in the last paragraph above 1.3. The relation between B and L could be considered as ‘switch regularity assumptions’.
Instead of considering this cell B directly, we consider the generating series of arithmetic intersection numbers with the difference of the two CM cycles, which is modular by Lemma 4.3.6. Then using Lemma 6.2.1, we prove the
$\operatorname {\mathrm {mod}} \overline {\mathbb {Q}}\log S$
-version of Theorem 5.2.5 for the general
$\phi $
, after replacing the CM cycle by the difference. See Proposition 6.2.5. This gives P of Table 2. Then the combination of L
$'$
and P/
${\mathbb {W}}'$
proves the cell B. Here, P/
${\mathbb {W}}'$
is the special case of P under the extra regularity assumption associated to
${\mathbb {W}}'$
.

We need to remove ‘
$(\operatorname {\mathrm {mod}} \overline {\mathbb {Q}}\log S)$
’. We will use the following theorem. It is a corollary of Baker’s celebrated theorem on transcendence of logarithms of algebraic numbers (see [Reference WaldschmidtWal03, Theorem 1.1]), and the fact that logarithms of prime numbers are
${\mathbb {Q}}$
-linearly independent.
Theorem 6.2.2. Let
$p_1,\ldots ,p_m$
be distinct prime numbers, then
$\log p_1,\ldots ,\log p_m$
are
$ \overline {\mathbb {Q}}$
-linearly independent.
6.2.1 Set-up
We need the following convenient notation. For a set T of finite places of F, let

For example,
$ {\mathbb {G}}_{ \emptyset }=G\left ({\mathbb {A}}_F^{{{\mathfrak {Ram}}} \cup \infty }\right ) P\left ({\mathbb {A}}_{F,{{\mathfrak {Ram}}}}\right ) $
is the group appearing in Theorem 5.2.5.
Below, let
$\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right ) ^{{{\widetilde K_\Lambda }}}$
be a pure tensor such that
$\phi ^\infty $
is
$\overline {\mathbb {Q}}$
-valued and
$\phi _v=1_{\Lambda _v}$
for every finite place v of F nonsplit in E. Let
$K\in {{\widetilde K_\Lambda }}$
stabilize
$\phi $
. Let S be a set of finite places of F, and
$K'=K_S K_\Lambda ^S$
. Let

Let
$f_{{\mathbb {W}}}=f_{{\mathbb {W}},K'}^K$
be defined as in (5.9). We remind the reader that we will use other incoherent hermitian spaces
${\mathbb {W}}'$
over
${\mathbb {A}}_E$
of dimension 1. These notations apply to
${\mathbb {W}}'$
in the same way.
Let
$t\in F_{>0}$
. Our goal is to prove that for
$g\in {\mathbb {G}}_{ \emptyset } $
and a suitable set S of finite places of F,

We also introduce an equation equivalent to (6.20), and both will play roles in the proof of Theorem 5.2.5. The
$\psi _t$
-Whittaker function of the right-hand side of (5.9) (which is the definition of
$f_{{\mathbb {W}}}$
), coincides with the right-hand side of (3.31) up to the last term. Comparing the definition of
$z_t(g, \phi )_{\mathfrak {e}}^{{\mathcal {L}},{\mathrm {aut}}}$
with the left-hand side of (3.31), (6.20) is equivalent to the following equation:

6.2.2 Regular test functions
We use Assumption 6.1.2 on regularity here.
Lemma 6.2.3. Assume that S contains a nonempty subset R such that
$\phi _v$
is
${\mathbb {W}}_v$
-regular for
$v\in R$
(i.e., Assumption 6.1.2 holds). Then for
$g\in {\mathbb {G}}_{R}$
, (6.20) holds. Equivalently, (6.21) holds.
Remark 6.2.4. The statement in Lemma 6.2.3 becomes more transparent if
$S=R$
. However, we need the flexibility to vary such R in S later.
Proof. Recall (5.4),

Here, we choose k such that
$k_v=1$
for
$v\in S$
or nonsplit in E. (This is possible since
$K'=K_S K_\Lambda ^S$
and
$K_v=K_{\Lambda ,v}$
for v nonsplit in E.) Then Assumption 6.1.2 still holds with
${\mathbb {W}}$
replaced by
$k^{-1}{\mathbb {W}}$
. So by Lemma 6.1.3,
$Z_t(\omega (g) \phi ^\infty ) $
and
${\mathcal {P}}_{k^{-1}{\mathbb {W}},K}$
do not meet on the generic fiber. Then by Lemma 4.4.6, we can apply Corollary A.2.6 at finite places
$v\not \in S$
(nothing happens if
$K_v=K_{\Lambda ,v}$
). Then we have

Comparing this equation with (3.35), (6.21) is implied by Proposition 6.1.4.
6.2.3 CM cycles of degree 0
Let
${\mathbb {W}}'$
be another incoherent hermitian subspace of
${\mathbb {W}}$
and
${\mathcal {P}}_{{\mathbb {W}}'}$
the CM cycle defined accordingly as in 5.1. Since the automorphic Green function is admissible and
$ {\mathcal {P}}_{{\mathbb {W}} ,E}-{\mathcal {P}}_{{\mathbb {W}}',E} $
has degree 0, by Lemma 4.3.6,

Moreover, (6.22) is independent of the choice of a. We abbreviate
$z(g, \phi )_{{\mathfrak {e}},a}^{{\overline {\mathcal {L}}}}$
to
$z(g, \phi )_{{\mathfrak {e}}}^{{\overline {\mathcal {L}}}}$
. The 0-th Fourier coefficient of (6.22) is 0. Indeed, by Lemma 4.4.12, the action of
$K_\Lambda $
on
$ \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}(\widetilde {\mathcal {X}})$
fixes
$c_1(\overline {\mathcal {L}}_K^\vee )$
. The vanishing of the 0-th Fourier coefficient follows from Lemma 5.1.4.
Proposition 6.2.5. Assume that the cardinality of S is at least 2. Given
$\phi $
as in 6.2.1, if
$K_S$
is small enough (depending on
$\phi _S$
), then for all
$g\in G({\mathbb {A}}_{F}^\infty )$
,

Proof. For
$G^{\mathrm {der}}=SU(1,1)$
,
$G(F_v)=G^{\mathrm {der}}(F_v) K^{\max }_v$
. By Lemma 3.3.8, it is enough to prove (6.23) for
$g\in G^{\mathrm {der}}({\mathbb {A}}_F) $
.
We need a lemma whose statement requires some more notations. Since
$G^{\mathrm {der}} {\ \simeq \ } {\mathrm {SL}}_{2,F}$
, by the q-expansion principle for
${\mathrm {SL}}_{2,F}$
[Reference ChaiCha90], we have

Here,
${\mathcal {A}}_{{\mathrm {hol}}}(G^{\mathrm {der}},{\mathfrak {w}})_{\overline {\mathbb {Q}}}$
is as in 2.7 with G replaced by
$G^{\mathrm {der}}$
, and
${\mathfrak {w}}$
is understood as the restriction of
${\mathfrak {w}}$
to
$G^{\mathrm {der}}(F_\infty )\cap K^{\max }_v$
for
$v\in \infty $
. Thus, we have Fourier coefficients as in 2.7. For
$f\in {\mathcal {A}}_{{\mathrm {hol}}}(G^{\mathrm {der}},{\mathfrak {w}})$
, let
$[f ]$
be its image in
$ {\mathcal {A}}_{{\mathrm {hol}}}(G^{\mathrm {der}},{\mathfrak {w}})_{\overline {\mathbb {Q}}}\otimes _{\overline {\mathbb {Q}}} {\mathbb {C}}/ \overline {\mathbb {Q}}\log {{S}} $
. Then the
$ {\mathbb {C}}/ \overline {\mathbb {Q}}\log {{S}}$
-valued locally constant function
$f_t^\infty \operatorname {\mathrm {mod}}\overline {\mathbb {Q}}\log {{S}}$
on
$G^{\mathrm {der}}({\mathbb {A}}_F^\infty )$
coincides with the t-th Fourier coefficient of
$[f ]$
.
Lemma 6.2.6. Assume that
$\phi _R$
is
${\mathbb {W}}_R$
-regular, where
$R\subset S$
consists of a single element, and
$\phi _{R'}$
is
${\mathbb {W}}^{\prime }_{R'}$
-regular, where
$R' \subset S\backslash R$
consists of a single element. Then we have the following equality in
$ {\mathcal {A}}_{{\mathrm {hol}}}(G^{\mathrm {der}},{\mathfrak {w}})_{\overline {\mathbb {Q}}}\otimes _{\overline {\mathbb {Q}}} {\mathbb {C}}/ \overline {\mathbb {Q}}\log {{S}} $
after restriction from G to
$G^{\mathrm {der}}$
:

Proof. Consider the difference

of the two sides of (6.24), before passing to
${\mathcal {A}}_{{\mathrm {hol}}}(G^{\mathrm {der}},{\mathfrak {w}})_{\overline {\mathbb {Q}}}\otimes _{\overline {\mathbb {Q}}} {\mathbb {C}}/ \overline {\mathbb {Q}}\log {{S}} $
. By the cuspidality of (6.22) and Lemma 3.3.9 (3), the
$0$
-th Fourier coefficient
$ f_0^\infty (g) =0$
for
$g\in {\mathbb {G}}_{\{v_1,v_2\}} .$
Write

as a finite sum, where
$f_i\in {\mathcal {A}}_{{\mathrm {hol}}}(G^{\mathrm {der}},{\mathfrak {w}})_{\overline {\mathbb {Q}}}$
and
$a_i\in {\mathbb {C}}/ \overline {\mathbb {Q}}\log {{S}}$
are
$\overline {{\mathbb {Q}}}$
-linearly independent. Then for
$t\in F_{>0}\cup \{0\}$
, the t-th Fourier coefficient of (6.25) is
$\sum _{i} f_{i,t}^\infty a_i $
. For
$g\in {\mathbb {G}}_{R\cup R'} \cap G^{\mathrm {der}}({\mathbb {A}}_F)$
,
$\sum _{i} f_{i,t}^\infty (g) a_i=0 $
for
$t\in F_{>0}$
by Lemma 6.2.3 (applied to
${\mathbb {W}},{\mathbb {W}}'$
respectively), and also for
$t=0$
by the above discussion for the constant term. Thus,
$f_{i,t}^\infty (g)=0$
by the
$\overline {{\mathbb {Q}}}$
-linear independence of
$a_i$
’s. So
$f_i(g)=0$
. By the density of
${\mathbb {G}}_{R\cup R'} \cap G^{\mathrm {der}}({\mathbb {A}}_F) $
in
$G^{\mathrm {der}}(F)\backslash G^{\mathrm {der}}({\mathbb {A}}_F)$
,
$f_i(g)=0$
for
$G^{\mathrm {der}}({\mathbb {A}}_F)$
. So (6.24) holds.
Remark 6.2.7. The density argument can not be applied directly to
$[f|_{G^{\mathrm {der}}({\mathbb {A}}_F)}].$
Now we continue the proof of the proposition. Recall that
$w_{v} \in G^{\mathrm {der}}(F_{v})\subset G(F_{v})$
as in 2.5 acts on
${\mathcal {S}}({\mathbb {V}}_{v}) $
by Fourier transform (multiplied by the Weil index) via the Weil representation
$\omega $
. See 2.8. By Lemma 6.2.1, for a finite place v of F, there exists a
${\mathbb {W}}_v$
-regular Schwartz function
$\Phi _{{{v}}} $
on
${\mathbb {V}}(E_v)$
such that
$\phi _{{{v}}}= \Phi _{{{v}}}+\omega (w_{v}) \Phi _{{{v}}}.$
Choose
$K_R , K_{ R'}$
small enough to stabilize
$ \Phi _{{R}}, \Phi _{{R'}}$
. By Lemma 6.2.6, (6.23) with
$\phi _{{R}},\phi _{{R'}}$
replaced by
$\Phi _{{R}},\Phi _{{R'}}$
holds for
$g\in G^{\mathrm {der}}({\mathbb {A}}_F) $
, and thus, it holds for
$gw_{R},gw_{R'},gw_{R}w_{R'}\in G^{\mathrm {der}}({\mathbb {A}}_F) $
replacing g as well. Then by Lemma 3.3.8, (6.23) with one or both of
$\phi _{{R}},\phi _{{R'}}$
replaced by
$ \omega (w_{R}) \Phi _{{R}},\omega (w_{R'}) \Phi _{{R'}}$
respectively holds for
$g\in G^{\mathrm {der}}({\mathbb {A}}_F) $
. Thus, including (6.23), we have four equations in total. Taking their sum, we have (6.23) for the original
$\phi $
and
$g\in G^{\mathrm {der}}({\mathbb {A}}_F) $
.
6.2.4 Remove regularity and log S
Lemma 6.2.8. Assume that the cardinality of S is at least 2. Assume that
$R\subset S$
consists of a single element and
$\phi _R$
is
${\mathbb {W}}^{\prime }_R$
-regular. If
$K_S$
is small enough (depending on
$\phi _S$
), then for
$g\in {\mathbb {G}}_{R} $
, (6.20) holds (literally, for
${\mathbb {W}}$
rather than
${\mathbb {W}}'$
). Equivalently, (6.21) holds.
Proof. By Lemma 6.2.3 (with
${\mathbb {W}}$
replaced by
${{\mathbb {W}}'}$
), for
$g\in {\mathbb {G}}_R $
, we have

Taking the difference between (6.23) and (6.26), (6.20) follows for
$g\in {\mathbb {G}}_{ R} $
.
Corollary 6.2.9. If K is small enough, then for
$g\in {\mathbb {G}}_{\emptyset } $
, (6.20) holds.
Proof. We prove (6.21), which is equivalent to (6.20). Let
$R\subset S$
consist of a single element. By Lemma 3.3.8 and the Iwasawa decomposition, it is enough to prove (6.21) for
$g\in {\mathbb {G}}_{R}$
. Then by Lemma 3.2.1, Lemma 3.3.14 and Lemma 4.1.4, we may assume that
$\phi _{R}(0)=0$
. Such a
$\phi _{R}$
can be written as a sum of
${\mathbb {W}}^{\prime }_{R}$
-regular functions for finitely many
${\mathbb {W}}'$
’s (in fact, only depending on
${\mathbb {W}}^{\prime }_{R}$
). Since (6.21) is linear on
$\phi _{R}$
, the corollary follows from Lemma 6.2.8 with
${\mathbb {W}}^{\prime R} ={\mathbb {W}}^R$
and
${\mathbb {W}}^{\prime }_{R}$
varying.
Proof of Theorem 5.2.5.
We may assume that
$\phi ^\infty $
is
$\overline {\mathbb {Q}}$
-valued. It is enough to prove (5.10) modulo
$\overline {\mathbb {Q}}\log {{S}}$
. Indeed, choosing another set
$S'$
of 4 places split in E modulo
$\overline {\mathbb {Q}}\log {{S'}}$
and requiring S and
$S'$
to have no same residue characteristics, then (5.10) follows from Theorem 6.2.2 (i.e., the
$\overline {{\mathbb {Q}}}$
-linear independence of
$\log p$
’s).
Now we prove (5.10) modulo
$\overline {\mathbb {Q}}\log {{S}}$
by decomposing it into equations established in Corollary 6.2.9 for
$l^{-1}{\mathbb {W}}$
’s where
$l\in U({\mathbb {W}})\backslash K_\Lambda / K'$
(instead of a single
${\mathbb {W}}$
, and the double coset is clarified below (5.4)). Note that by Remark 5.2.6, we may shrink K freely. For the left-hand side of (5.10), that is,
$2 z_t(g, \phi ^\infty )_{\mathfrak {e}}^{{\mathcal {L}},{\mathrm {aut}}}\cdot \pi _{_K,{_K}_{\Lambda }}^{*}{\mathcal {P}}_{{\mathbb {W}},{K_\Lambda }}$
, by (5.4), we have

Now we consider the right-hand side of (5.10); that is,
$ f_{{\mathbb {W}},{K_\Lambda },t}^{K,\infty }(g) $
. We choose l such that
$l_v=1$
for
$v\not \in S$
(this is possible since
$K'=K_S K_\Lambda ^S$
). In particular,
$(l{\mathbb {W}})_v={\mathbb {W}}_v$
for
$v\in {\mathfrak {Ram}}$
so that
$\phi _v'$
defined in (5.7) in terms of
${\mathbb {W}}_v$
is the same as that defined in (5.7) in terms of
$(l{\mathbb {W}})_v$
. By the coset decomposition

and

we deduce from the definition (5.9) of
$ f_{{\mathbb {W}},{K_\Lambda }}^{K} $
that

By Corollary 6.2.9, (6.27) and (6.28) imply (5.10) modulo
$\overline {\mathbb {Q}}\log {{S}}$
.
A Admissible divisors
We recall S. Zhang’s theory of admissible cycles on a polarized arithmetic variety [Reference ZhangZha20]. They are cycles with ‘harmonic curvatures’. We only consider admissible divisors [Reference ZhangZha20, 2.5, Admissible cycles]. In particular, for a divisor on the generic fiber, we have its admissible extensions. With an extra local condition, we have the normalized admissible extension. The functoriality of (normalized) admissibile cycles under flat morphisms is important for us.
It is worth mentioning that while the normalized admissible extension is defined purely locally and at the level of divisors, a global lifting of a divisors class on the generic fiber is defined (and it is called
$\mathsf {L}$
-lifting) in [Reference ZhangZha20, Corollary 2.5.7]. It is an admissible extension [Reference ZhangZha20, Corollary 2.5.7 (1)] with vanishing Faltings height [Reference ZhangZha20, Corollary 2.5.7 (2)]. They will not be further discussed in this appendix, and are not needed in this paper.
A.1 Deligne-Mumford stacks over a Dedekind domain
Let
${\mathcal {O}}$
be a Dedekind domain. Let
${\mathcal {M}}$
be a connected regular Deligne-Mumford stack proper flat over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
of relative dimension n. Let M be its generic fiber. Definitions of cycles, rational equivalence, proper pushforward and flat pullback for (Chow) cycles are applicable to Deligne-Mumford stacks over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
. See [Reference GilletGil09]. Let
$Z^{*}(M)$
(resp.
$Z^{*}({\mathcal {M}})$
) be the graded
${\mathbb {Q}}$
-vector space of cycles on M (resp.
$ {\mathcal {M}}$
) with
${\mathbb {Q}}$
-coefficients. Let
${\mathrm {Ch}}^{*}(M)$
and
${\mathrm {Ch}}^{*}({\mathcal {M}})$
be the
${\mathbb {Q}}$
-vector spaces of Chow cycles.
We shall only work under the following convenient assumption, which simplifies the local intersection theory. For every closed point
$s\in \operatorname {\mathrm {Spec}} {\mathcal {O}}$
, let
${\mathcal {O}}_s$
be the completed local ring.
Assumption A.1.1. (1) There is a finite subset
$S\subset \operatorname {\mathrm {Spec}} {\mathcal {O}}$
, a regular scheme
$\widetilde {\mathcal {M}}$
proper flat over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}-S$
and a finite étale morphism
$\pi :\widetilde {\mathcal {M}}\to {\mathcal {M}}|_{\operatorname {\mathrm {Spec}} {\mathcal {O}}-S}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}-S$
.
(2) For every
$s\in S$
, there is a regular scheme
$\widetilde {\mathcal {M}}$
proper flat over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_s$
and a finite étale morphism
$\pi :\widetilde {\mathcal {M}}\to {\mathcal {M}}_{\operatorname {\mathrm {Spec}} {\mathcal {O}}_s}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_s$
.
In either case (1) or (2), we call
$\widetilde {\mathcal {M}}$
a covering of
$ {\mathcal {M}}|_{\operatorname {\mathrm {Spec}} {\mathcal {O}}-S}$
or
$ {\mathcal {M}}_{\operatorname {\mathrm {Spec}} {\mathcal {O}}_s}$
. For another covering
$\widetilde {\mathcal {M}}'$
, the fiber product is regular and proper flat over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}-S$
or
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_s$
, making a third covering.
A line bundle
${\mathcal {L}}$
on
${\mathcal {M}}$
is ample if in both cases (1) and (2), its pullback to some covering is ample. And it is relatively positive if
$ \deg {\mathcal {L}}|_ C> 0$
for every closed curve (1-dimensional closed substack) C in every special fiber of
${\mathcal {M}}$
. It is routine to check that the definition does not depend on the choice of covering by making a third covering. The following notions, which are used in the whole paper, are also defined via coverings: intersection number, Chern class and Zariski closure.
A.2 Local cycles
Assume that
${\mathcal {O}}$
is a completed local ring (so a DVR). Let s be the unique closed point of
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
. We will use two intersection pairings. First, for
$X\in Z ^i({\mathcal {M}} )$
and
$Y\in Z^{n+1-i}({\mathcal {M}} )$
with disjoint supports on M, define

where
$\pi $
is a covering morphism and the latter intersection number is a usual one, defined either using Serre’s
${\mathrm {Tor}}$
-formula (equivalently rephrased as the Euler-Poincaré characteristic of the derived tensor product
${\mathcal {O}}_{\pi ^{*}(X)}\bigotimes ^{\mathbb {L}} {\mathcal {O}}_{\pi ^{*}(Y)}$
([Reference Gillet and SouléGS90, 4.3.8 (iv)]), or as a cohomological pairing.
Second, let
${\mathcal {M}}_s$
be the special fiber of
${\mathcal {M}}$
and
$Z_s^1({\mathcal {M}} )\subset Z^1({\mathcal {M}})$
the subspace of divisors supported on
${\mathcal {M}}_s$
. We use the intersection pairing between
$ Z_s^1({\mathcal {M}} )$
and an n-tuple of
${\mathbb {Q}}$
-Cartier divisors as in [Reference FultonFul84, Example 6.5.1] (defined using a covering as in (A.1)):

It only depends on the rational equivalence classes of the Cartier divisors. (In particular, by fixing
$n-1$
rational equivalence classes of Cartier divisors, we get a pairing between
$Z_s^1({\mathcal {M}} )$
and
$ Z^{1}({\mathcal {M}} )$
. This view point might be helpful.) In this subsection, we will use this second intersection pairing until Corollary A.2.6.
Let
${\mathcal {L}}$
be a line bundle on
${\mathcal {M}}$
. Let
$B^1_{\mathcal {L}}({\mathcal {M}})\subset Z_s^{1}( {{\mathcal {M}} }$
be the kernel of the linear form
$Z_s^{1}( {{\mathcal {M}} }) \to {\mathbb {Q}}$
defined by intersection with
$ c_1({\mathcal {L}})^{n}$
. Assume that the generic fiber of
${\mathcal {L}} $
is ample and
${\mathcal {L}}$
is relatively positive. The local index theorem [Reference ZhangZha20, Lemma 2.5.1] (see also [Reference Yuan and ZhangYZ17]) implies the following lemma.
Lemma A.2.1. The pairing
$(X,Y)\mapsto X\cdot c_1({\mathcal {L}})^{n-1} \cdot Y$
on
$B_{\mathcal {L}}^1({\mathcal {M}})$
is negative definite.
Let
$Z_{\mathcal {L}}^1({\mathcal {M}} )$
be the orthogonal complement of
$B_{\mathcal {L}}^1({\mathcal {M}})$
under the pairing
$X\cdot c_1({\mathcal {L}})^{n-1} \cdot Y$
; that is,

Then by definition, we have a decomposition

and an exact sequence

For a prime cycle X on M, let
$X^{\mathrm {zar}}$
be its Zariski closure on
${\mathcal {M}}$
. Extend the definition by linearity.
Definition A.2.2. (1) We call
$Z_{\mathcal {L}}^1({\mathcal {M}} )$
the space of admissible divisors with respect to
${\mathcal {L}}$
.
(2) For
$X\in Z^1(M)$
, an admissible extension with respect to
${\mathcal {L}}$
is an element in its preimage by
$ Z_{\mathcal {L}}^1({\mathcal {M}} )\to Z^1(M)$
. Define the normalized admissible extension
$X^{\mathcal {L}}$
of X with respect to
${\mathcal {L}}$
to be the projection of
$X^{\mathrm {zar}}$
to
$Z_{\mathcal {L}}^1({\mathcal {M}} )$
in (A.2).
Remark A.2.3. In terms of [Reference ZhangZha20, Corollary 2.5.7 (1)],
$Z_{\mathcal {L}}^1({\mathcal {M}} )\subset Z ^1({\mathcal {M}} )$
is the subspace of cycles X with ‘harmonic curvatures’ (compare with Remark A.3.2 (3)); that is, the element in
$ {\mathrm {Hom}}_{\mathbb {Q}}\left ( Z_s^{1}({\mathcal {M}} ),{\mathbb {Q}}\right )$
defined by intersection with
$ X \cdot c_1({\mathcal {L}})^{n-1} $
is a multiple of the one defined by intersection with
$ c_1({\mathcal {L}})^{n} $
.
Then by definition, we have the following lemma.
Lemma A.2.4. Assume that
${\mathcal {M}}$
is smooth over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
. Then
$B_{\mathcal {L}}^1({\mathcal {M}})=0$
. In particular, for
$X\in Z^1(M)$
,
$X^{\mathrm {zar}}$
is the normalized admissible extension.
By the projection formula (and the commutativity of taking Zariski closure and closed pushforward/flat pullback), we easily deduce the following lemma.
Lemma A.2.5. Let
${\mathcal {M}}'$
be a regular Deligne-Mumford stack and
$ f: {\mathcal {M}}'\to {\mathcal {M}}$
a finite flat morphism. Let
${\mathcal {L}}'$
be the pullback of
${\mathcal {L}}$
to
${\mathcal {M}}'$
. Consider
${\mathcal {M}}'$
as a Deligne-Mumford stack over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
via f and
${\mathcal {M}}$
. Then
$f^{*}\left ( B_{\mathcal {L}}^1({\mathcal {M}})\right )\subset B_{{\mathcal {L}}'}^1({\mathcal {M}}') $
,
$f_*\left ( B_{{\mathcal {L}}'}^1({\mathcal {M}}')\right )= B_{{\mathcal {L}}}^1({\mathcal {M}}) $
,
$f^{*}\left ( Z_{\mathcal {L}}^1({\mathcal {M}})\right )\subset Z_{{\mathcal {L}}'}^1({\mathcal {M}}') $
and
$f_*\left ( Z_{{\mathcal {L}}'}^1({\mathcal {M}}')\right )= Z_{{\mathcal {L}}}^1({\mathcal {M}}) .$
In particular, the decomposition (A.2) and the formation of normalized admissible extension are preserved under pullback and pushforward by f.
Corollary A.2.6. In Lemma A.2.5 with f finite, let
$M'$
be the generic fiber of
${\mathcal {M}}'$
and assume that
${\mathcal {M}}\to \operatorname {\mathrm {Spec}} {\mathcal {O}}$
is smooth. Let
$X\in Z^1(M')$
and
$Y\in Z^{n}({\mathcal {M}})$
such that X and
$f^{*}(Y)$
have disjoint supports. Then we have
$X^{\mathcal {L}}\cdot f^{*}(Y)=X^{\mathrm {zar}}\cdot f^{*}(Y)$
.
A.3 Admissible arithmetic Chow group of divisors
Now let
${\mathcal {O}}$
be the ring of integers of a number field. In particular,
${\mathcal {M}}_{\mathbb {C}}:={\mathcal {M}} \otimes _{\mathbb {Z}} {\mathbb {C}}=M\otimes _{\mathbb {Q}}{\mathbb {C}}$
is a complex orbifold. Let
$\overline {\mathcal {L}}=({\mathcal {L}},\|\cdot \|)$
be a hermitian line bundle on
${\mathcal {M}}$
such that the generic fiber of
${\mathcal {L}} $
is ample,
${\mathcal {L}}$
is relatively positive, and the hermitian metric
$\|\cdot \|$
is invariant under the involution induced by complex conjugation. See [Reference Gillet and SouléGS90, 3.1.2]. In particular, endowing
${\mathcal {M}}_{\mathbb {C}} $
with the Chern curvature form
${\mathrm {curv}}\left ( \overline {\mathcal {L}}_{\mathbb {C}}\right )$
, it is a smooth Kähler orbifold.
Definition A.3.1. (1) The group
${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}})$
of admissible (with respect to
$\overline {\mathcal {L}}$
) arithmetic divisors on
${\mathcal {M}}$
with
${\mathbb {C}}$
-coefficients is the
${\mathbb {C}}$
-vector space of pairs
$(X,g)$
where
-
•
$X\in Z^1({\mathcal {M}}) _{\mathbb {C}} $ such that for every closed point
$s\in \operatorname {\mathrm {Spec}} {\mathcal {O}}$ , the restriction of X to
$ {\mathcal {M}}_{{\mathcal {O}}_s} $ is contained in
$Z^1_{{\mathcal {L}}_{{\mathcal {O}}_s} }\left ( {\mathcal {M}}_{{\mathcal {O}}_s}\right )_{\mathbb {C}}$ ,
-
• g is a Green function for
$ X_ {\mathbb {C}}$ on
${\mathcal {M}}_{\mathbb {C}}$ , admissible with respect to
$\overline {\mathcal {L}}_{\mathbb {C}}$ , and invariant under the involution induced by complex conjugation. Here, admissibility means that the curvature form
$ \delta _{X}+\frac {i }{2\pi }\partial \bar \partial g$ is harmonic.
(2) For
$X\in Z^1(M)_{\mathbb {C}}$
, an admissible extension of X with respect to
$\overline {\mathcal {L}}$
is an element in the preimage of X by the natural surjection
${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}}) \to Z^1(M)_{\mathbb {C}}$
.
Remark A.3.2. (1) A Green current on Deligne-Mumford stacks is defined in [Reference GilletGil09, Section 1]. In our situation, a Green function is simply an orbifold function whose pullback to the finite étale cover by a smooth variety is a usual Green function.
(2) Admissible Green functions for
$X_{\mathbb {C}}$
always exists and are the same modulo locally constant functions, and (A.3) is the non-archimedean analog of this fact.
(3) By [Reference ZhangZha20, 2.2], a closed
$(1,1)$
-form
$\alpha $
is harmonic if and and only if on each connected component of
${\mathcal {M}}_{\mathbb {C}}$
,
$ \alpha \wedge {\mathrm {curv}}\left ( \overline {\mathcal {L}}_{\mathbb {C}}\right ) ^{n-1} $
is a constant multiple of
$ {\mathrm {curv}}\left ( \overline {\mathcal {L}}_{\mathbb {C}}\right ) ^{n}.$
Definition A.3.3. (1) An admissible Green function is normalized with respect to
$\overline {\mathcal {L}}_{\mathbb {C}}$
if it has vanishing harmonic projection; that is, on each connected component of
${\mathcal {M}}_{\mathbb {C}}$
, its integration against
${\mathrm {curv}}(\overline {\mathcal {L}}_{\mathbb {C}})^{n}$
is 0.
(2) An element in
${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}})$
is normalized with respect to
$\overline {\mathcal {L}}$
if it is normalized at every place. For
$X\in Z^1(M)_{{\mathbb {C}}}$
, let
$X^{\overline {\mathcal {L}}}\in {\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}}) $
be its normalized admissible extension with respect to
$\overline {\mathcal {L}}$
.
Then the normalized admissible extension of a divisor on M exists and is unique.
For every nonzero rational function f on M,
$({\mathrm {div}}(f),-\log |f|^2)$
is contained in
${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}})$
.
Definition A.3.4. (1) Let
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}({\mathcal {M}}) $
be the quotient of the space of arithmetic divisors with
${\mathbb {C}}$
-coefficients by the
${\mathbb {C}}$
-span of
$({\mathrm {div}}(f),-\log |f|^2)$
’s for all nonzero rational functions.
(2) Let
$\widehat {\mathrm {Ch}}_{{\overline {\mathcal {L}}},{\mathbb {C}}}^1({\mathcal {M}}) $
be the quotient of
${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}})$
by the
${\mathbb {C}}$
-span of
$({\mathrm {div}}(f),-\log |f|^2)$
’s.
Remark A.3.5. Let
$ \widehat {\mathrm {Ch}}^{1}({\mathcal {M}})$
be the Chow group of arithmetic divisors with
${\mathbb {Z}}$
-coefficients defined by Gillet and Soulé [Reference Gillet and SouléGS90] for schemes, which is extended to the stacky case in [Reference GilletGil09]. Then
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}({\mathcal {M}}) $
is the quotient of
$\widehat {\mathrm {Ch}}^1({\mathcal {M}}) _{{\mathbb {C}}}$
by the pullback of the kernel of the degree map
$ \widehat {\mathrm {Ch}}^1\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}} \right )_{{\mathbb {C}}}\to {\mathbb {C}}$
. In particular, we have an isomorphism
$ \widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}} \right ){\ \simeq \ } {\mathbb {C}}$
by taking degrees.
Lemma A.3.6. (1) The natural map
$\widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}}) \to {\mathrm {Ch}}^1(M) _{\mathbb {C}}$
is surjective. Its kernel is generated by connected components of special fibers of
${\mathcal {M}}$
at all finite places and locally constant functions on
${\mathcal {M}}_{\mathbb {C}}$
.
(2) Assume that
${\mathcal {M}}$
is connected. Then the kernel of
$\widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {M}})\to {\mathrm {Ch}}^1(M)_{\mathbb {C}}$
is 1-dimensional, and is the pullback of
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}}\right )$
.
Proof. By (A.3) and Remark A.3.2 (2), (1) holds. If
${\mathcal {M}}$
is connected, then
$E_1:={\mathcal {O}}_M(M)$
is a finite field extension of the fraction field of
${\mathcal {O}}$
. Then
${\mathcal {M}}$
over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}_{E_1}$
has geometrically connected fibers by Stein factorization. By (1), the kernel is the pullback of
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}}_{E_1}\right ) $
, which is 1-dimensional by the finiteness of the class number of
$E_1$
and Dirichlet’s unit theorem. See [Reference Gillet and SouléGS90, 3.4.3]. And it equals the pullback of
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}}\right )$
.
Example A.3.7. The arithmetic first Chern class
$c_1(\overline {\mathcal {L}})$
of
$\overline {\mathcal {L}}$
is the class of
$ ({\mathrm {div}}(s),-\log \|s\| ^2)$
for a nonzero rational section s. By Remark A.2.3 (or one may follow our definition), one immediately sees that
${\mathrm {div}}(s)$
has ‘harmonic curvature’ at every finite place. The curvature form of
$-\log \|s\| ^2$
is by definition the Kähler form. So
$c_1(\overline {\mathcal {L}})\in \widehat {\mathrm {Ch}}^1_{{\mathbb {C}},{\overline {\mathcal {L}}}}({\mathcal {M}})$
.
Now we consider the functoriality. By Lemma A.2.5, we have the following proposition.
Proposition A.3.8. Let
${\mathcal {M}}'$
be a regular Deligne-Mumford stack and
$ f: {\mathcal {M}}'\to {\mathcal {M}}$
a finite flat morphism over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
, such that the restriction of f to the generic fibers is finite étale. Let
${\mathcal {L}}'$
be the pullback of
${\mathcal {L}}$
to
${\mathcal {M}}'$
. Consider
${\mathcal {M}}'$
as a Deligne-Mumford stack over
$\operatorname {\mathrm {Spec}} {\mathcal {O}}$
via f and
${\mathcal {M}}$
. Then the formation of
${\widehat {Z}}^1_{{\mathbb {C}},{\overline {\mathcal {L}}}}({\mathcal {M}})$
,
$\widehat {\mathrm {Ch}}^1_{{\mathbb {C}},{\overline {\mathcal {L}}}}({\mathcal {M}}) $
and normalized admissible extension with respect to
$\overline {\mathcal {L}}$
is preserved under pullback and pushforward by f.
A.4 Arithmetic intersection pairing
Let
$Z_1({\mathcal {M}})_{\mathbb {C}}$
be the group of 1-cycles on
${\mathcal {M}}$
. We define an arithmetic intersection pairing following [Reference Bost, Gillet and SouléBGS94, 2.3.1]:

We reduce the pairing to the arithmetic intersection pairing between
$\widehat {\mathrm {Ch}}^1 ({\mathcal {M}}) _{{\mathbb {C}}}$
and
$\widehat {\mathrm {Ch}}^n ({\mathcal {M}}) _{{\mathbb {C}}}$
, which is defined in [Reference GilletGil09] for general Deligne-Mumford stacks (without Assumption A.1.1). Let
$\omega ({\widehat x})$
be the curvature form of
$\widehat x$
, which is a smooth
$(1,1)$
form on the orbifold
$M_{{\mathbb {C}}}$
independent of the choice of a representative of
$\widehat x$
. Choose
$\widehat y=(Y,g_Y)\in \widehat {\mathrm {Ch}}^n_{{\mathbb {C}}}({\mathcal {M}})$
. Then
$ \widehat x\cdot Y$
is the arithmetic intersection number
$\widehat x\cdot \widehat y $
minus
$\int _{{\mathcal {M}}_{\mathbb {C}}}\omega ({\widehat x}) g_Y$
.
Now assume Assumption A.1.1 and that
$(X,g_X)$
is a representative of
$\widehat x$
such that
$X\cap Y$
is empty on the generic fiber of
${\mathcal {M}}$
. Then
$ \widehat x\cdot Y $
is the sum of the intersection numbers of the restrictions of X and Y to
${\mathcal {M}}_{{\mathcal {O}}_s}$
over all closed points
$s\in \operatorname {\mathrm {Spec}} {\mathcal {O}}$
defined in (A.1), and
$\int _{Y_{\mathbb {C}}}g_X$
.
The pullback of the kernel of the degree map
$ \widehat {\mathrm {Ch}}^1\left ( \operatorname {\mathrm {Spec}} {\mathcal {O}} \right )_{{\mathbb {C}}}\to {\mathbb {C}}$
to
$\widehat {\mathrm {Ch}}^1({\mathcal {M}})_{{\mathbb {C}}}$
is annihilated by the arithmetic intersection pairing with
$Z_1({\mathcal {M}})_{\mathbb {C}}$
. By Remark A.3.5, the above arithmetic intersection pairing factors through an arithmetic intersection pairing

Similar to Corollary A.2.6, Proposition A.3.8 implies the following result.
Corollary A.4.1. In Proposition A.3.8, let
$M'$
be the generic fiber of
${\mathcal {M}}'$
and assume that
${\mathcal {M}}\to \operatorname {\mathrm {Spec}} {\mathcal {O}}_E$
is smooth. Then for
$X\in Z^1(M')_{{\mathbb {C}}}$
and
$Y\in Z^{n}_{{\mathbb {C}}}({\mathcal {M}})$
, we have

where
$g_{X}^{\overline {\mathcal {L}}}$
is the normalized admissible Green function for X.
B A comparison of the ‘closure’ model with Rapoport–Smithling–Zhang model (appendix by Yujie Xu)
B.1 Preliminaries
B.1.1
Let F be a CM field and
$F_0$
its maximal totally real subfield of index
$2$
. Let
$a\mapsto \overline {a}$
be the nontrivial automorphism of
$F/F_0$
. We fix a presentation
$F=F_0(\sqrt {\Delta })$
for some totally negative element
$\Delta \in F_0$
. Let
$\Phi $
denote the CM type for F determined by
$\sqrt {\Delta }$
; that is,

Let W be a non-degenerate
$F/F_0$
-hermitian space of dimension
$n\geq 2$
. Let

As in [Reference Rapoport, Smithling and ZhangRSZ20,
$\S$
2.1], we use the symbol c to denote the similitude factor of a point on a unitary similitude group. We consider the following algebraic groups over
$\mathbb {Q}$
:



Note that
$Z^{\mathbb {Q}}$
is naturally a central subgroup of
$G^{\mathbb {Q}}$
, and we have the following product decompositions

B.1.8
From now on, we assume, moreover, that the hermitian space W has the following signatures at the archimedean places of
$F_0$
: for a distinguished element
$\varphi _0\in \Phi $
, the signature of
$W_{\varphi _0}$
is
$(1,n-1)$
; and for all other
$\varphi \in \Phi $
, the signature of
$W_{\varphi }$
is
$(0,n)$
. In order to define a Shimura datum
$(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
, by the canonical inclusions
$G_{\mathbb {R}}^{\mathbb {Q}}\subset \prod \limits _{\varphi \in \Phi }\operatorname {GU}(W_{\varphi })$
, it suffices to define the components
$h_{G^{\mathbb {Q}},\varphi }$
of
$h_{G^{\mathbb {Q}}}$
. Consider the matrices

We also choose bases
$W_{\varphi }\ \simeq \ \mathbb {C}^n$
such that the hermitian form on
$W_{\varphi }$
is given by
$J_{\varphi }$
. Consider the
$\mathbb {R}$
-algebra homomorphisms

which induce our desired component maps
$h_{G^{\mathbb {Q}},\varphi }:\mathbb {C}^{\times }\to \operatorname {GU}(W_{\varphi })(\mathbb {R})$
. This gives us our desired Shimura datum
$(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
.
B.1.11
For the group
$Z^{\mathbb {Q}}$
defined in B.1.4, the CM type
$\Phi $
induces an identification

which allows us to define
$h_{Z^{\mathbb {Q}}}:\mathbb {C}^{\times }\to Z^{\mathbb {Q}}(\mathbb {R})$
as the diagonal embedding (via the identification B.1.12) precomposed with complex conjugation. This gives us a Shimura datum
$(Z^{\mathbb {Q}},\{h_{Z^{\mathbb {Q}}}\})$
with reflex field

which is the reflex field for the CM type
$\Phi $
. Recall that this can be computed as the fixed field in
$\mathbb {C}$
of the group
$\{\sigma \in {\mathrm {Aut}}(\mathbb {C})=|\sigma \circ \Phi =\Phi \}$
.
For the group
$\widetilde {G}$
defined in B.1.6, we consider the map

which gives us a Shimura datum
$(\widetilde {G},\{h_{\widetilde {G}}\})$
. Let

be the reflex field for
$(\widetilde {G},\{h_{\widetilde {G}}\})$
, by definition it is computed via

Note that E is the compositum of
$E_{\Phi }$
(as in B.1.13) and F. By [Reference DeligneDel71], we have canonical models
${\mathrm {Sh}}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
over E, for compact open subgroups
$K_{\widetilde {G}}\subset \widetilde {G}(\mathbb {A}_f)$
. By [Reference KisinKis10] (resp. [Reference Kisin and PappasKP18] depending on the level structure), we have integral models
$\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
over
$\mathcal {O}_{E,(v)}$
.
B.1.17
We introduce a Shimura datum
$(G,\{h_G\})$
. Let
$h_G$
be the map

defined by composing
$h_{\widetilde {G}}$
(from B.1.14) with the projection onto the second factor in the map B.1.7. The reflex field for
$(G,\{h_G\})$
is F, embedded into
$\mathbb {C}$
via
$\varphi _0$
.
Moreover, the decomposition in B.1.7 induces a decomposition of Shimura data

Let
$K_{\widetilde {G}}$
be decomposed via B.1.7 into

The natural projections in B.1.19 then induce morphisms of Shimura varieties


Note that the Shimura variety
${\mathrm {Sh}}_{K_G}(G,\{h_G\})$
, which originally appeared in [Reference Gan, Gross and PrasadGGP12], is not of PEL type. However, it is of abelian type, and we have an integral model
$\mathscr {S}_{K_G}(G,\{h_G\})$
defined over
$\mathcal {O}_{F,(\nu )}$
by [Reference KisinKis10] (resp. [Reference Kisin and PappasKP18] depending on the level structureFootnote
4
).
B.1.23
Let
$\nu |p$
be a place of E. Let
$\mathcal {M}_0$
be the moduli functor which associates to each locally Noetherian
$\mathcal {O}_{E,\nu }$
-scheme S the groupoid of tuples
$\mathcal {M}_0(S):=(A_0,\lambda _0,\iota _0 )$
where
-
(i)
$A_0$ is an abelian scheme over S;
-
(ii)
$\iota _0: \mathcal {O}_F\to {\mathrm {End}}_S(A_0)$ is an
$\mathcal {O}_F$ -endomorphism structure on
$A_0$ satisfying the Kottwitz condition of signature
$((0,1)_{\varphi \in \Phi })$ ; that is,
$$\begin{align*}{\mathrm{Char}}(\iota_0(a)|{\mathrm{Lie}} A_0)=\prod\limits_{\varphi\in\Phi}(T-\overline{\varphi}(a))\quad\text{ for all }a\in \mathcal{O}_F;\end{align*}$$
-
(iii)
$\lambda _0$ is a principal polarization of
$A_0$ such that the associated Rosati involution induces the nontrivial Galois automorphism of
$F/F_0$ on
$\mathcal {O}_F$ via
$\iota _0$ .
Then
$\mathcal {M}_0$
is representable by a Deligne-Mumford stack
$\mathcal {M}_0$
finite étale over
$\operatorname {\mathrm {Spec}}\mathcal {O}_{E,\nu }$
. Moreover, we assume
$K^p$
is small enough, so that
$\mathcal {M}_0$
is nonempty. We shall assume throughout the rest of this appendix that
$\mathcal {M}_0$
is nonempty.
Lemma B.1.24 [Reference Rapoport, Smithling and ZhangRSZ20, Lemma 3.4].
The stack
$\mathcal {M}_0$
admits the following decompositon into open and closed substacksFootnote
5
:

such that the generic fiber of
$\mathcal {M}_0^{\xi }$
is canonically isomorphic to
$ {\mathrm {Sh}}_{K_{Z^{\mathbb {Q}}}}(Z^{\mathbb {Q}},\{h_{Z^{\mathbb {Q}}}\})_E$
.
B.1.26
Let
$F_{0,v}$
be the v-adic completion of
$F_0$
, and we set
$F_v:=F\otimes _{F_0}F_{0,v}$
. Suppose for now that the place
$v_0$
of
$F_0$
is unramified over p, and that
$v_0$
either splits in F or is inert in F. Suppose, moreover, that the hermitian space
$W_{v_0}$
is split. If there exists a prime v of
$F_0$
above p that is non-split in F, we assume additionally that
$p\neq 2$
. We choose a vertex lattice
$\Lambda _v$
in the
$F_v/F_{0,v}$
-hermitian space
$W_v$
. For now, we assume that
$\Lambda _{v_0}$
is self-dual. We recall that an
$\mathcal {O}_{F,v}$
-lattice
$\Lambda $
in an
$F_v/F_{0,v}$
-hermitian space is called a vertex lattice of type r if
$\Lambda \subset ^r\Lambda ^{*}\subset \pi _v^{-1}\Lambda $
.Footnote
6
An
$\mathcal {O}_{F,v}$
-lattice
$\Lambda $
in an
$F_v/F_{0,v}$
-hermitian space is called a vertex lattice if it is a vertex lattice of type r for some r. Here,
$\pi _v$
is a uniformizer in
$F_v:=F\otimes _FF_{0,v}$
, where
$F_{0,v}$
is the v-adic completion of
$F_0$
for a place v of
$F_0$
. In particular, a self-dual lattice is simply a vertex lattice of type
$0$
. Assume that
$K_{G,v}=\mathrm {Stab}(\Lambda _v)\subset G(F_{0,v})$
.
B.1.27
Let
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G})$
be the moduli functor which associates to each locally Noetherian
$\mathcal {O}_{E,(\nu )}$
-scheme S the groupoid of triples
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G})(S):=(A_0,\iota _0,\lambda _0, A,\iota ,\lambda ,\overline {\eta }^p)$
, where
-
•
$(A_0,\iota _0,\lambda _0)\in \mathcal {M}_0^{\xi }(S)$ as is defined in B.1.25;
-
•
$(A,\iota )$ is an abelian scheme over S, equipped with an
$\mathcal {O}_F\otimes \mathbb {Z}_{(p)}$ -endomorphism structure
$\iota $ satisfying the Kottwitz condition of signature
$((1,n-1)_{\varphi _0},(0,n)_{\varphi \in \Phi \setminus \{\varphi _0\}})$ ; that is,
$$\begin{align*}{\mathrm{Char}}(\iota(a)|{\mathrm{Lie}} A)=(T-\varphi_0(a))(T-\varphi_0(\overline{a}))^{n-1}\prod\limits_{\varphi\in\Phi\setminus\{\varphi_0\}}(T-\varphi(\overline{a}))^n\quad\text{ for all }a\in F;\end{align*}$$
-
•
$\lambda $ is a polarization of A such that the associated Rosati involution induces the nontrivial Galois automorphism of
$F/F_0$ on
$\mathcal {O}_F\otimes \mathbb {Z}_{(p)}$ via
$\iota $ , and such that the following additional assumption in [Reference Rapoport, Smithling and ZhangRSZ20, (4.2)] is also satisfied: the action of
$\mathcal {O}_{F_0}\otimes \mathbb {Z}_p{\ \simeq \ }\prod \limits _{v|p}\mathcal {O}_{F_0,v}$ on the p-divisible group
$A[p^{\infty }]$ induces a decomposition
$ A[p^{\infty }]=\prod \limits _{v|p}A[v^{\infty }],$ where v ranges over the places of
$F_0$ above p; the polarization
$\lambda $ then induces a polarization
(B.1.28)for each v; we require$$ \begin{align} \lambda_v: A[v^{\infty}]\to A^{\vee}[v^{\infty}]{\ \simeq \ } A[v^{\vee}]^{\vee} \end{align} $$
$\ker \lambda _v$ to be contained in
$A[\iota (\pi _v)]$ of rank
$\#(\Lambda _v^{*}/\Lambda _v)$ for each place v of
$F_0$ above p;
-
•
$\overline {\eta }^p$ is a
$K_G^p$ -orbit of the
$\mathbb {A}^p_{F,f}$ -linear isometry
(B.1.29)where the hermitian form on the left-hand side is$$ \begin{align} {\mathrm{Hom}}_F(\widehat{V}^p(A_0),\widehat{V}^p(A))\ \simeq \ -W\otimes_F \mathbb{A}^p_{F,f}, \end{align} $$
$(x,y)\mapsto \lambda _0^{-1}\circ y^{\vee }\circ \lambda \circ x.$
-
• For each
$v\neq v_0$ over p, we impose the sign condition and Eisenstein condition at v [Reference Rapoport, Smithling and ZhangRSZ20, (4.4), (4.10)].
By [Reference Rapoport, Smithling and ZhangRSZ20, Theorem 4.1], the forgetful map
$(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline {\eta }^p)\mapsto (A_0,\iota _0,\lambda _0)$
is representable and induces a morphism of
$\mathcal {O}_{E,(\nu )}$
-schemes

On the level of generic fibres, (B.1.30) recovers the map (B.1.21).
B.2 Comparison of integral models
B.2.1
Let
$\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
be the integral model defined over
$\mathcal {O}_{E,(\nu )}$
for
${\mathrm {Sh}}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
as constructed in [Reference Kisin and PappasKP18,
$\S$
4.6]. Recall that the abelian type integral model
$\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
is built out of the integral model
$\mathscr {S}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
for a corresponding Hodge type Shimura variety associated to the abelian type
${\mathrm {Sh}}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
, which, as in [Reference Kisin and PappasKP18, 4.6.21] (see also [Reference KisinKis10, 3.4.13] and [Reference DeligneDel79] for more details), is
${\mathrm {Sh}}_{K_{G^{\mathbb {Q}}}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
. Recall that
$\mathscr {S}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
is the
$\mathcal {O}_{F,(\nu )}$
-scheme constructed by taking the flat closure
$\mathscr {S}^-_{}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
of the generic fibre
${\mathrm {Sh}}_{}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
inside some suitable Siegel integral model
$\mathscr {S}_{K'}({\mathrm {GSp}},S^{\pm })_{\mathcal {O}_{F,(\nu )}}$
for some prime
$\nu $
of F above a fixed prime p. For convenience of expositions, we shall fix a symplectic embedding
$i:(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})\hookrightarrow ({\mathrm {GSp}}(V,\psi ),S^{\pm })$
.Footnote
7
By [Reference XuXu21, Reference XuXu25],Footnote
8
this flat closure
$\mathscr {S}^-_{}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\}){\ \simeq \ } \mathscr {S}_{}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
is the desired integral model.Footnote
9
We shall use the model for
${\mathrm {Sh}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
as building blocks for integral models for
${\mathrm {Sh}}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
and
${\mathrm {Sh}}_{K_G}(G,\{h_G\})$
.
Fix a connected component
$\{h_{G^{\mathbb {Q}}}\}^+\subset \{h_{G^{\mathbb {Q}}}\}$
, and let
${\mathrm {Sh}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})^+\subset {\mathrm {Sh}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})$
be the geometrically connected component which is the image of
$\{h_{G^{\mathbb {Q}}}\}^+\times 1$
. Let
$F^p\subset \overline {F}$
be the maximal extension of F that is unramified at primes dividing p. By [Reference DeligneDel79, Theorem 2.6.3], the action of
${\mathrm {Gal}}(\overline {F}/F)$
on
${\mathrm {Sh}}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})^+$
factors through
${\mathrm {Gal}}(\overline {F}^p/F)$
. We abuse the notation and still denote
${\mathrm {Sh}}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})^+$
as the
$F^p$
-scheme obtained via descent. Let
$\mathscr {S}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}})^+$
be the closure of
${\mathrm {Sh}}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})^+$
in
$\mathscr {S}_{K_{G^{\mathbb {Q}},p}}^-(G^{\mathbb {Q}},\{h_{G^{\mathbb {Q}}}\})\otimes _{\mathcal {O}_{F,(\nu )}}\mathcal {O}_{F^p,(\nu )}$
. Here, the notation
$\mathcal {O}_{F^p,(\nu )}$
denotes the ring of integers of
$F^p$
localized at
$(p)$
.Footnote
10
Let
$\mathscr {A}(\widetilde {G}_{\mathbb {Z}_{(p)}})$
(resp.
$\mathscr {A}(G^{\mathbb {Q}}_{\mathbb {Z}_{(p)}})^{\circ }$
) be the group defined in [Reference Kisin and PappasKP18, 4.6.8] for
$\widetilde {G}$
(resp.
$G^{\mathbb {Q}}$
), which was originally defined in [Reference DeligneDel79]. We recall that

and
$\mathscr {A}(G^{\mathbb {Q}}_{\mathbb {Z}_{(p)}})^{\circ }:=(G^{\mathbb {Q}})^{\circ }(\mathbb {Z}_{(p)})_+^-/Z(\mathbb {Z}_{(p)})^-*_{(G^{\mathbb {Q}})^{\circ }(\mathbb {Z}_{(p)})_+/Z(\mathbb {Z}_{(p)})}(G^{\mathbb {Q}})^{{\mathrm {ad}}\circ }(\mathbb {Z}_{(p)})^+$
, where
$(G^{\mathbb {Q}})^{\circ }(\mathbb {Z}_{(p)})_+^-$
is the closure of
$(G^{\mathbb {Q}})^{\circ }(\mathbb {Z}_{(p)})_+$
in
$(G^{\mathbb {Q}})(\mathbb {A}_f^p)$
. By [Reference Kisin and PappasKP18, Lemma 4.6.10], we have an inclusion

Here,
$\mathscr {A}(\widetilde {G}):=\widetilde {G}(\mathbb {A}_f)/\widetilde {Z}(\mathbb {Q})^-*_{\widetilde {G}(\mathbb {Q})_+/\widetilde {Z}(\mathbb {Q})}\widetilde {G}^{{\mathrm {ad}}}(\mathbb {Q})^+$
, where
$\widetilde {Z}(\mathbb {Q})^-$
denotes the closure of
$Z_{\widetilde {G}}(\mathbb {Q})$
in
$\widetilde {G}(\mathbb {A}_f)$
, and

where
$G^{\mathbb {Q}}(\mathbb {Q})_+^-$
denotes the closure of
$G^{\mathbb {Q}}(\mathbb {Q})_+$
in
$G^{\mathbb {Q}}(\mathbb {A}_f)$
. Let
$\widetilde {J}\subset \widetilde {G}(\mathbb {Q}_p)$
denote a set which maps bijectively to a set of coset representatives for the image of
$\mathscr {A}(\widetilde {G}_{\mathbb {Z}_{(p)}})$
in
$\mathscr {A}(G^{\mathbb {Q}})^{\circ }\backslash \mathscr {A}(\widetilde {G})/K_{\widetilde {G},p}$
under (B.2.3). Recall from [Reference Kisin and PappasKP18, 4.6.15], we have

Note that by analogous arguments as loc.cit., the right-hand side of (B.2.4) has a natural structure of a
$\mathcal {O}_{E,(\nu )}:=\mathcal {O}_E\otimes _{\mathcal {O}_F}\mathcal {O}_{F,(\nu )}$
-scheme with
$\widetilde {G}(\mathbb {A}_f^p)$
-action and is a model for
${\mathrm {Sh}}_{K_{\widetilde {G},p}}(\widetilde {G},\{h_{\widetilde {G}}\})$
. Moreover, for sufficiently small
$K_{\widetilde {G}}^p$
, the quotient
$\mathscr {S}_{K_{\widetilde {G},p}}(\widetilde {G},\{h_{\widetilde {G}}\})/K_{\widetilde {G}}^p:=\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
is a finite type
$\mathcal {O}_{E,(\nu )}$
-scheme extending
${\mathrm {Sh}}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
.
Lemma B.2.5.
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ }\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
as
$\operatorname {\mathrm {Spec}}\mathcal {O}_{E,(\nu )}$
-schemes.
Proof. The moduli description for
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
in B.1.26 induces a natural map


by simply forgetting the component
$(A_0,\iota _0,\lambda _0,\overline {\eta }^p)$
in the tuple. Note that the map (B.2.6) is proper, and in particular closed. We fix an arbitrary
$(A^{\star },\iota ^{\star },\lambda ^{\star })\in \mathscr {S}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}})^+$
, and suppose

under the map (B.2.6). Take any
$(h,\gamma ^{-1})\in \mathscr {A}(\widetilde {G}_{\mathbb {Z}_{(p)}})$
. As in [Reference Kisin and PappasKP18, 4.5.3], let
$\widetilde {\mathcal {P}}_{\gamma }\subset \widetilde {G}$
be the torsor given by the fibre over
$\gamma \in \widetilde {G}^{{\mathrm {ad}}}(\mathbb {Z}_{(p)})$
. First we check that

gives another point in the fibre over
$(A^{\star },\iota ^{\star },\lambda ^{\star })$
under the map (B.2.6). This is clear as we only need to check that
$(\eta ^{\star })^{\widetilde {\mathcal {P}}_{\gamma }}$
are
$\mathbb {A}_{F,f}$
-linear isometries

but this is simply given by the composite
$\widetilde {\gamma }^{-1}\circ \eta ^{\star }\circ \iota _{\widetilde {\gamma }}^{-1}$
, where
$\iota _{\widetilde {\gamma }}$
is as defined in [Reference Kisin and PappasKP18, 4.5.3].
It then remains to check that
$\ker (\mathscr {A}(G^{\mathbb {Q}}_{\mathbb {Z}_{(p)}})^{\circ }\to \mathscr {A}(\widetilde {G}_{\mathbb {Z}_{(p)}}))$
acts freely on
$\mathscr {S}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}})^+$
, and this follows from [Reference Kisin and PappasKP18, 4.6.17] and the fact that
$\ker (\mathscr {A}(G^{\mathbb {Q}}_{\mathbb {Z}_{(p)}})^{\circ }\to \mathscr {A}(\widetilde {G}_{\mathbb {Z}_{(p)}}))$
is a subgroup of
$\Delta (G^{\mathbb {Q}},(G^{\mathbb {Q}})^{{\mathrm {ad}}}):=\ker (\mathscr {A}(G^{\mathbb {Q}}_{\mathbb {Z}_{(p)}})\to \mathscr {A}(G^{\mathbb {Q}{\mathrm {ad}}}_{\mathbb {Z}_{(p)}}))$
. Thus,

In particular,
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ }\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
.
B.2.10
For an arbitrary extension
$L/E$
, taking the fibre in (B.1.30) over a fixed
$\mathcal {O}_{L,(\nu )}$
-point
$(A_0^{\star },\iota _0^{\star },\lambda _0^{\star })$
of
$\mathcal {M}_0^{\mathcal {O}_F,\xi }$
gives a flat integral model
$\mathcal {M}^{\star }_{K_G}(G,\{h_G\})$
over
$\mathcal {O}_L$
. Here, we use the upper script
$\star $
to emphasize that the model
$\mathcal {M}^{\star }_{K_G}(G,\{h_G\})$
thus obtained a priori depends on the choice of a base point
$(A_0^{\star },\iota _0^{\star },\lambda _0^{\star })$
. However, recall from B.1.17 that the reflex field for
$(G,\{h_G\})$
is F, by [Reference Kisin and PappasKP18] we also have a normal integral model
$\mathscr {S}_{K_G}(G,\{h_G\})$
over
$\operatorname {\mathrm {Spec}} \mathcal {O}_{F,(v)}$
, which is given by

Here,
$\mathscr {A}(G_{\mathbb {Z}_{(p)}})$
is the analogous group for G as defined in (B.2.2), and
$J\subset G(\mathbb {Q}_p)$
denotes the set analogous to
$\widetilde {J}$
defined above (B.2.4), using the analogous map for G as in (B.2.3).
Lemma B.2.12.
$\mathcal {M}_{K_G}(G,\{h_G\}){\ \simeq \ }\mathscr {S}_{K_G}(G,\{h_G\})_{\mathcal {O}_{L,(\nu )}}$
as
$\operatorname {\mathrm {Spec}}\mathcal {O}_{L,(\nu )}$
-schemes.
Proof. Consider the map


given by forgetting the component
$(A_0^{\star },\iota _0^{\star },\lambda _0^{\star })$
in the tuple. Let
$\mathscr {S}_{K_{G^{\mathbb {Q}},p}}^{\star }(G^{\mathbb {Q}}) _{\mathcal {O}_{L,(\nu )}}$
denote the image of the map (B.2.13). We take an arbitrary
$(A,\iota ,\lambda )\in \mathscr {S}_{K_{G^{\mathbb {Q}},p}}^{\star }(G^{\mathbb {Q}})^+_{\mathcal {O}_{L,(\nu )}}$
, and thus by construction of
$\mathcal {M}^{\star }_{K_{G}}(G,\{h_{G}\})$
, we clearly have

Take any
$(h,\gamma ^{-1})\in \mathscr {A}(G_{\mathbb {Z}_{(p)}})$
. In particular,
$\gamma \in G^{{\mathrm {ad}}}{\ \simeq \ } (G^{\mathbb {Q}})^{{\mathrm {ad}}}$
. Again as in [Reference Kisin and PappasKP18, 4.5.3], let
$\mathcal {P}_{\gamma }\subset G^{\mathbb {Q}}$
be the torsor given by the fibre over
$\gamma \in (G^{\mathbb {Q}})^{{\mathrm {ad}}}(\mathbb {Z}_{(p)})$
. By the same reasoning as in the proof of Lemma B.2.5, we also have

gives another point in the fibre over
$(A_0^{\star },\iota _0^{\star },\lambda _0^{\star })$
under the map (B.1.30). The rest of the argument proceeds similarly as in the proof of Lemma B.2.5 – that is, the kernel
$\ker (\mathscr {A}(G^{\mathbb {Q}}_{\mathbb {Z}_{(p)}})^{\circ }\to \mathscr {A}(G_{\mathbb {Z}_{(p)}}))$
acts freely on
$\mathscr {S}_{K_{G^{\mathbb {Q}},p}}(G^{\mathbb {Q}})^+$
. In particular, we have

and thus,
$\mathcal {M}^{\star }_{K_G}(G){\ \simeq \ }\mathscr {S}_{K_G}(G)_{\mathcal {O}_{L,(\nu )}}$
. (Since the choice of base point
$\star $
does not affect the proof, we may drop the upper script
$\star $
from our notations.)
B.2.17
We consider the Drinfeld level structure integral models analogous to those in [Reference Rapoport, Smithling and ZhangRSZ20,
$\S$
4.3]. Consider the embedding
$\widetilde {\nu }: \overline {\mathbb {Q}}\hookrightarrow \overline {\mathbb {Q}}_p$
, which identifies

The above identification (B.2.18) then gives an identification

where
$w_{\varphi }$
denotes the p-adic place in F induced by
$\widetilde {\nu }\circ \varphi $
.
We fix a place
$v_0$
of F over p that is split in F (and possibly ramified over p) into
$w_0$
and another place
$\overline {w}_0$
in F. We require, moreover, that the CM type
$\Phi $
considered in (B.1.2) and the chosen place
$\nu $
of E above p satisfy the following matching condition:

This condition (B.2.20) only depends on the place
$\nu $
of E induced by
$\widetilde {\nu }$
.
Now we introduce a Drinfeld level structure at
$v_0$
. Recall the level structure subgroup
$K_G$
from (B.1.20). We define a variant compact open subgroup
$K_G^m\subset G(\mathbb {A}_{F_0,f})$
in exactly the same way as
$K_G$
, except that, in the
$v_0$
-factor, we require
$K_{G,v_0}^m\subset G(F_{0,v_0})$
to be the principal congruence subgroup modulo
$\mathfrak {p}_{v_0}^m$
inside
$K_{G,v_0}$
. Clearly,
$K_G=K_G^{m=0}$
. As in (B.1.20), we define
$K_{\widetilde {G}}^m=K_{Z^{\mathbb {Q}}}\times K_G^m$
.
Let
$\Lambda _{v_0}=\Lambda _{w_0}\oplus \Lambda _{\overline {w}_0}$
denote the natural decomposition of the lattice
$\Lambda _{v_0}$
attached to the split place
$v_0$
. For a point
$(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline {\eta }^p)\in \mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G})(S)$
, we have a decomposition of p-divisible groups

where w ranges over the places of F lying over p. Moreover, we further decompose the
$v_0$
-term in (B.2.21) and consider

where, when p is locally nilpotent on S, the p-divisible group
$A[w_0^{\infty }]$
satisfies the Kottwitz condition of type
$r|_{w_0}$
for the action of
$\mathcal {O}_{F,w_0}$
on its Lie algebra, in the sense of [Reference Rapoport and ZinkRZ17,
$\S$
8]. Here,
$r|_{w_0}$
denotes the restriction of the function r on
${\mathrm {Hom}}_{\mathbb {Q}}(F,\overline {\mathbb {Q}})$
to
${\mathrm {Hom}}_{\mathbb {Q}_p}(F_{w_0},\overline {\mathbb {Q}}_p)$
under (B.2.19).
Likewise, we have the same decomposition as (B.2.22) for
$A_0$
; that is, we have

Let
$\pi _{w_0}$
be a uniformizer of
$F_{0,w_0}$
. In addition to the moduli functor
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G})$
which classifies tuples
$(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline {\eta }^p)$
, we impose the following additional Drinfeld level structure as in [Reference Harris and TaylorHT01,
$\S$
II.2]; that is,
-
• an
$\mathcal {O}_{F,w_0}$ -linear homomorphism of finite flat group schemes
(B.2.24)$$ \begin{align} \eta:\pi_{w_0}^{-m}\Lambda_{w_0}/\Lambda_{w_0}\to \underline{{\mathrm{Hom}}}_{\mathcal{O}_{F,w_0}}(A_0[w_0^m],A[w_0^m]). \end{align} $$
We denote the resulting moduli problem by
$\mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
, which is relatively representable by a finite flat morphism to
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G})$
. In fact,
$\mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
is regular and flat over
$\operatorname {\mathrm {Spec}}\mathcal {O}_{E,(\nu )}$
by [Reference Harris and TaylorHT01, Lemma III.4.1].
B.2.25
Recall the integral model
$\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
(resp.
$\mathscr {S}_{K_G}(G,\{h_G\})$
) defined in (B.2.4) (resp. (B.2.11)). We define
$\mathscr {S}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\})$
(resp.
$\mathscr {S}_{K_G^m}(G,\{h_G\})_{\mathcal {O}_{L,(\nu )}}$
) as the normalization of
$\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
(resp.
$\mathscr {S}_{K_G}(G,\{h_G\})_{\mathcal {O}_{L, (\nu )}}$
) inside
${\mathrm {Sh}}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ } M_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
${\mathrm {Sh}}_{{K_G^m}}(G,\{h_G\})_L{\ \simeq \ } M_{{K_G^m}}(G)_L$
).
Corollary B.2.26.
$\mathscr {S}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ } \mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
as
$\operatorname {\mathrm {Spec}}\mathcal {O}_{E,(\nu )}$
-schemes, and

as
$\operatorname {\mathrm {Spec}}\mathcal {O}_{L,(\nu )}$
-schemes.
Proof. By Lemma B.2.5 (resp. B.2.12),
$\mathscr {S}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\})$
(resp.
$\mathscr {S}_{K_G^m}(G,\{h_G\})_{\mathcal {O}_L}$
) is the normalization of
$\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ } \mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
(resp.
$\mathscr {S}_{K_G}(G,\{h_G\})_{\mathcal {O}_L}{\ \simeq \ } \mathcal {M}_{K_G}(G,\{h_G\})$
) inside
${\mathrm {Sh}}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ } M_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
${\mathrm {Sh}}_{{K_G^m}}(G)_L{\ \simeq \ } M_{{K_G^m}}(G)_L$
). Since
$\mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
is regular and flat, in particular it is normal. Thus, by [Reference GrothendieckGro67, IV-2, 6.14.1],
$\mathcal {M}_{K_G^m}(G)$
is normal (even though it may not necessarily be regular). By [Reference ProjectSta18, 035I] applied to the scheme
$\mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
$\mathcal {M}_{K_G^m}(G)$
), there exists a unique morphism
$\mathscr {S}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\})\to \mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
$\mathscr {S}_{K_{G}^m}(G,\{h_{G}\})\to \mathcal {M}_{K_{G}^m}(G)$
), which is the normalization of
$\mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
$\mathcal {M}_{K_G^m}(G)$
) in
${\mathrm {Sh}}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\})$
(resp.
${\mathrm {Sh}}_{K_G^m}(G,\{h_G\})$
). Since
$\mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
$\mathcal {M}_{K_G^m}(G)$
) is already normal, we have an isomorphism
$\mathscr {S}_{K_{\widetilde {G}}^m}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ } \mathcal {M}_{K_{\widetilde {G}}^m}(\widetilde {G})$
(resp.
$\mathcal {M}_{K_G^m}(G,\{h_G\}){\ \simeq \ }\mathscr {S}_{K_G^m}(G,\{h_G\})_{\mathcal {O}_{L,(\nu )}}$
).
B.2.28
In this last section, we recall the construction of semi-global integral models with AT parahoric level as in [Reference Rapoport, Smithling and ZhangRSZ20,
$\S$
4.4]. Recall the notion of vertex lattice from §B.1.26. We say that a vertex lattice
$\Lambda $
is almost self-dual if it is a vertex lattice of type
$1$
. We say that a vertex lattice
$\Lambda $
is
$\pi _v$
-modular (resp. almost
$\pi _v$
-modular) if
$\Lambda ^{*}=\pi _v^{-1}\Lambda $
(resp.
$\Lambda \subset \Lambda ^{*}\subset ^1 \pi _v^{-1}\Lambda $
).
Suppose
$p\neq 2$
and
$v_0$
is unramified over p. As in §B.1.26, we take a vertex lattice
$\Lambda _v\subset W_v$
for each prime v of
$F_0$
above p. Unlike in §B.1.26, let
$(v_0,\Lambda _{v_0})$
be of one of the following types:
-
1.
$v_0$ is inert in F and
$\Lambda _{v_0}$ is almost self-dual as an
$\mathcal {O}_{F,v_0}$ -lattice;
-
2. n is even,
$v_0$ ramifies in F and
$\Lambda _{v_0}$ is
$\pi _{v_0}$ -modular;
-
3. n is odd,
$v_0$ ramifies in F and
$\Lambda _{v_0}$ is almost
$\pi _{v_0}$ -modular;
-
4.
$n=2$ ,
$v_0$ ramifies in F and
$\Lambda _{v_0}$ is self-dual.
To the moduli functor
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G})$
which classifies tuples
$(A_0,\iota _0,\lambda _0,A,\iota ,\lambda ,\overline {\eta }^p)$
as in §B.1.26 (except that the condition on
$(v_0,\Lambda _{v_0})$
is different), we impose the following additional condition:
-
• When the pair
$(v_0,\Lambda _{v_0})$ is of AT type (2), (3) or (4), we impose the Eisenstein condition on the summand
${\mathrm {Lie}}_{\psi }A[v_0^{\infty }]$ [Reference Rapoport, Smithling and ZhangRSZ20, 4.10];
-
• When the pair
$(v_0,\Lambda _{v_0})$ is of AT type (2), we impose additionally the wedge condition [Reference Rapoport, Smithling and ZhangRSZ20, 4.27] and the spin condition [Reference Rapoport, Smithling and ZhangRSZ20, 4.28]
-
• When the pair
$(v_0,\Lambda _{v_0})$ is of AT type (3), we impose additionally the refined spin condition [Reference Rapoport, Smithling and ZhangRSZ18, (7.9)] on
${\mathrm {Lie}}_{\psi _0}A[v_0^{\infty }]$ .
By [Reference Rapoport, Smithling and ZhangRSZ20, Theorem 4.7], the moduli functor above is representable by a Deligne-Mumford stack flat over
$\operatorname {\mathrm {Spec}}\mathcal {O}_{E,(\nu )}$
and relatively representable over
$\mathcal {M}_0^{\mathcal {O}_F,\xi }$
, i.e. (B.1.30) still holds in this case. To see that
$\mathcal {M}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\}){\ \simeq \ }\mathscr {S}_{K_{\widetilde {G}}}(\widetilde {G},\{h_{\widetilde {G}}\})$
as
$\operatorname {\mathrm {Spec}}\mathcal {O}_{E,(\nu )}$
-schemes, one simply proceeds as in Lemma B.2.5.
Acknowledgements
The author thanks Yifeng Liu for his many useful comments and suggestions. He also thanks Xinyi Yuan for sharing his manuscript and answering the author’s questions. He is grateful to Shouwu Zhang for proposing the arithmetic mixed Siegel-Weil formula and for his help on the revision of the paper. He thanks Jan Hendrik Bruinier, Stephan Ehlen, Ziqi Guo, Chao Li, and Siddarth Sankaran, Wei Zhang, Yihang Zhu and Jialiang Zou for their help. He also thanks Yujie Xu for providing Appendix B. He also thanks the anonymous referees for their advice to improve the presentation of this paper. The research is partially supported by the NSF grant DMS-2000533.
Competing interest
The authors have no competing interests to declare.