# Picard rank jumps for K3 surfaces with bad reduction

Salim Tayou  Department of Mathematics, Harvard University, 1 Oxford St,
Cambridge, MA 02138, USA<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely
many specializations where its Picard rank jumps, hence extending our previous
work with Shankar–Shankar–Tang to the case where $X$ has bad reduction. We
prove a similar result for generically ordinary non-isotrivial families of K3
surfaces over curves over $\overline{\mathbb{F}}_{p}$ which extends previous
work of Maulik–Shankar–Tang. As a consequence, we give a new proof of the
ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.

###### Contents

  1. 1 Introduction
  2. 2 GSpin Shimura varieties: integral models and compactifications
  3. 3 Arithmetic intersection theory and modularity
  4. 4 The main estimates and proof of the main theorems
  5. 5 Bounding the contribution from bad reduction places
  6. 6 Applications

##  1\. Introduction

Let $X$ be a K3 surface over a number field $K$. Let
$\mathscr{X}\rightarrow\mathscr{S}$ be a smooth projective model, where
$\mathscr{S}\hookrightarrow\mathrm{Spec}(\mathcal{O}_{K})$ is an open subset
of the spectrum of the ring of integers $\mathcal{O}_{K}$. For every place
$\mathfrak{P}$ of $\mathcal{O}_{K}$ with finite residual field
$k(\mathfrak{P})$, we have an injective specialization map:

$\mathrm{Pic}(X_{\overline{K}})\hookrightarrow\mathrm{Pic}(\mathscr{X}_{\overline{k(\mathfrak{P})}}),$

and both groups have finite rank, _the Picard rank_ , denoted
$\rho(X_{\overline{K}})$ and $\rho(\mathscr{X}_{\overline{k(\mathfrak{P})}})$
respectively.

Inspired by the classical density result of Noether–Lefschetz loci for weight
$2$ polarized variations of Hodge structures, see [Voi02, Ogu03], Charles
asked in [Cha14] what can be said about the _arithmetic Noether–Lefschetz
locus_ :

$NL=\\{\mathfrak{P}\in\mathscr{S}|\,\rho(X_{\overline{K}})<\rho(\mathscr{X}_{\overline{k(\mathfrak{P})}})\\}.$

In a prior work [SSTT19, Theorem 1.1], we proved that the set $NL$ is infinite
under the additional assumption that $X$ has potentially everywhere good
reduction, i.e., up to taking a finite extension of $K$, we assumed that
$\mathscr{S}=\mathrm{Spec}(\mathcal{O}_{K})$. The first main result of this
paper is the following unconditional result.

######  Theorem 1.1.

Let $X$ be a K3 surface over a number field $K$. Then the set $NL$ is
infinite.

This theorem is a particular instance of Theorem 4.1 which is formulated for
GSpin Shimura varieties and which has many other applications. As a
consequence, Theorems 1.4, 1.6, and Corollary 1.7 in [SSTT19] hold with no
assumptions on the reduction type. In particular, we have the following
theorem.

######  Theorem 1.2.

Let $K$ be a number field and $A$ an abelian surface over $K$. Then there
exists infinitely many places where $A$ has good reduction, and the reduction
is geometrically non-simple.

###  1.1. Picard rank jumps over function fields

Let $p\geq 5$ be a prime number. Let $\mathscr{X}\rightarrow\mathscr{S}$ be a
family of K3 surfaces over a curve $\mathscr{S}$ over
$\overline{\mathbb{F}}_{p}$. Let $\eta$ be the generic point of $\mathscr{S}$.
For every $s\in\mathscr{S}(\overline{\mathbb{F}}_{p})$, we have similarly an
inequality of Picard ranks

$\rho(\mathscr{X}_{\overline{\eta}})\leq\rho(\mathscr{X}_{s}),$

and one can introduce similarly the Noether–Lefschetz locus as the subset of
$\mathscr{S}$ where the above inequality is strict:

$NL=\\{s\in\mathscr{S}(\overline{\mathbb{F}}_{p})|\rho(\mathscr{X}_{\overline{\eta}})<\rho(\mathscr{X}_{s})\\}.$

In [MST22, Theorem 1.1], Maulik, Shankar and Tang proved that if $\mathscr{S}$
is proper and the family $\mathscr{X}\rightarrow\mathscr{S}$ is generically
ordinary and not isotrivial then the set $NL$ is infinite. Our second main
theorem in this paper is to remove the properness assumption in their result.

######  Theorem 1.3.

Let $\mathscr{X}\rightarrow\mathscr{S}$ be a generically ordinary non-
isotrivial family of K3 surfaces over a smooth curve $\mathscr{S}$ over
$\overline{\mathbb{F}}_{p}$ with $p\geq 5$. Suppose that the discriminant of
the generic geometric Picard lattice is prime to $p$. Then the locus $NL$ is
infinite.

The theorem is also a particular instance of Theorem 4.8 for GSpin Shimura
varieties, which has several other applications and also has an analogue for
unitary Shimura varieties, see Theorem 6.1. In particular, we have the
following theorem which extends [MST18, Theorem 1 (1)] to the quasi-projective
case.

######  Theorem 1.4.

Let $A$ be a non-isotrivial ordinary abelian surface over the function field
of a curve over $\overline{\mathbb{F}}_{p}$. Then $A$ has infinitely many
smooth and non-simple specializations.

Both Theorem 1.1 and Theorem 1.3 are motivated by the density of Hodge loci in
polarized variations of Hodge structure of weight $2$ of K3 type, see for
example [Voi02, Ogu03, Tay20]. Recent density results for general polarized
variations of Hodge structures of level less than $2$ as in [TT23, BKU21]
suggest that density of Hodge loci in arithmetic and function field settings
are natural problems to investigate, and we hope to address these questions in
future work.

###  1.2. Hecke orbit conjecture

As an application of Theorem 1.3, we give a new proof of the Hecke orbit
conjecture for orthogonal and certain unitary Shimura varieties. We refer to
[MST22, Section 1.2] for the context and prior results on this conjecture.

######  Theorem 1.5.

Let $\mathcal{M}_{\mathbb{F}_{p}}$ be the reduction at $p\geq 5$ of the
integral model of a Shimura variety of either:

  1. (1)

Orthogonal type associated to a lattice of signature $(b,2)$ having
discriminant prime to p.

  2. (2)

Unitary type associated to an imaginary quadratic field $K$ split at $p$ and
to a Hermitian lattice over $\mathcal{O}_{K}$ of signature $(n,1)$ with
discriminant prime to $p$.

Then the prime-to-p Hecke orbit of an ordinary point is Zariski dense in
$\mathcal{M}_{\mathbb{F}_{p}}$.

The density of Hecke orbits in characteristic zero is a consequence of the
work of Clozel–Oh–Ullmo [CU05], see also [EO06] for a dynamical approach using
Ratner theory. Chai first proved the Hecke orbit conjecture for the ordinary
locus of the moduli space of principally polarized abelian varieties in
[Cha95]. For orthogonal and some unitary Shimura varieties, a first proof of
the Hecke orbit conjecture in the ordinary case has been obtained by
Maulik–Shankar–Tang in [MST22] and our approach is inspired from theirs. Very
recently, Pol Van Hoften [van21] proved this conjecture for the ordinary locus
of Shimura varieties of abelian type under certain conditions on the reflex
field and using completely different methods.

###  1.3. Strategy of the proof

Theorem 1.1 and Theorem 1.3 are proved using a strategy initiated by Chai-Oort
[CO06] and Charles [Cha18] for the product of two modular curves and
subsequently used in [MST18, ST20] for Hilbert modular surfaces over number
fields and Siegel threefolds over $\overline{\mathbb{F}}_{p}$. Here we follow
the set-up in [SSTT19] and [MST22] to which we refer for more details. For
Theorem 1.1, we first translate it into an intersection theory type statement
between a curve and a sequence of divisors in the integral model of a toroidal
compactification of a Shimura variety of GSpin type. For this matter, we use
the Arakelov intersection theory with prelog forms developed in [BBK07]. We
follow a similar approach for Theorem 1.3, using the usual intersection theory
on the reduction modulo $p$ of the aforementioned compactification of the
integral model of a GSpin Shimura variety. The new ingredients which were
missing in both [SSTT19] and [MST22] are the local estimates on multiplicities
of intersection with special divisors at points of bad reduction and the
estimates of extra terms coming from the boundary divisors in the global
intersection numbers coming from the work of [BZ21], see also [EGT23] for a
recent approach. These are the main contributions of this paper. To obtain the
first estimates, we use an explicit description of the special divisors in the
formal completions along toroidal boundary components. This allows us to
define in each case a decreasing sequence of positive definite lattices
$(L_{n},Q)$ which computes the local intersection number. We give an estimate
on the growth of the successive minima of these lattices, then a geometry-of-
numbers type argument allows us to derive the desired estimates. To obtain the
bounds on the extra terms in the global intersection number, we use the
explicit expressions from [BZ21] and [Bru02] combined with an equidistribution
result from [Duk88, EO06].

###  1.4. Organization of the paper

The key input of this paper is the description of the special divisors in
terms of local coordinates of integral models of toroidal compactifications of
Shimura varieties of GSpin type. In Section 2, we explain these constructions
following [HP20] and [MP16], and the section culminates with a description of
the special divisors in formal completions along locally closed boundary
divisors. In Section 3, we recall briefly Arakelov arithmetic intersection
theory with prelog forms following [BBK07], and we assemble different
ingredients from the literature ([BZ21, HP20, Bor99]) to state the modularity
of the generating series of special divisors in the integral models of
toroidal compactifications of Shimura varieties of GSpin type. In Section 4,
we state the archimidean and finite place estimates needed to prove our main
theorems, and then we prove the archimidean estimates. Section 5 is devoted to
estimating contributions from bad reduction places. Finally, we prove the
application to Hecke orbit conjecture in Section 6.

###  1.5. Acknowledgments

I am very grateful to François Charles, François Greer, Keerthi Madapusi Pera,
Ananth Shankar and Yunqing Tang for many helpful discussions and insights.
This project started while I was invited by Yunqing Tang to Princeton
University, and I also benefited from support from the Institute for Advanced
Study in Princeton. I thank both institutions for their hospitality. I thank
the referee for the very valuable comments and suggestions.

##  2\. GSpin Shimura varieties: integral models and compactifications

This section summarizes the construction of the GSpin Shimura variety, its
toroidal compactifications and their integral models following [BZ21, HP20,
Per19, AGHMP18], see also [Kis10, MP16, Pin90] for earlier work. The ultimate
goal is to describe the special divisors in formal completions along the
toroidal boundary strata. The familiar reader may wish to skip directly to
Section 2.4 for these results.

###  2.1. The GSpin Shimura variety

Let $(L,Q)$ be an even quadratic lattice of signature $(b,2)$ with $b\geq 1$
and with associated even bilinear form

$(\,.\,):L\times L\rightarrow\mathbb{Z}$

such that $Q(x)=\frac{(x.x)}{2}\in\mathbb{Z}$ for all $x\in L$.

Let $G=\mathrm{GSpin}(L_{\mathbb{Q}})$ be the algebraic group over
$\mathbb{Q}$ of spinor similitudes defined as in [MP16, Section 1.2]. The
group $G(\mathbb{R})$ acts on the Hermitian symmetric space

$\mathcal{D}=:\\{z\in\mathbb{P}(L_{\mathbb{C}})|\,(z.z)=0,\,(z.\overline{z})<0\\}.$

The pair $(G,\mathcal{D})$ is the GSpin Shimura datum. Its reflex field is
$\mathbb{Q}$ by [MP16, Section 3.1].

For $K\subset G(\mathbb{A}_{f})$ a compact open subgroup, the GSpin Shimura
variety

$M(\mathbb{C})=G(\mathbb{Q})\backslash\mathcal{D}\times G(\mathbb{A}_{f})/K$

is the set of complex points of a Deligne–Mumford stack $M$ defined over
$\mathbb{Q}$. In what follows, we choose the compact open group $K\subset
G(\mathbb{A}_{f})$ as in [AGHMP18, Equation (4.1.2)]. Its image in
$\mathrm{SO}(L_{\mathbb{Q}})(\mathbb{A}_{f})$ stabilizes
$L\otimes\widehat{\mathbb{Z}}\subseteq L\otimes\mathbb{A}_{f}$ and is equal to
the subgroup that acts trivially on the quotient
$\widehat{L}^{\vee}/\widehat{L}=L^{\vee}/L$, where the dual lattice $L^{\vee}$
is defined as

$L^{\vee}=\\{x\in L_{\mathbb{Q}}|\forall y\in L,\,(x.y)\in\mathbb{Z}\\}.$

The Shimura variety $M$ carries a line bundle of weight $1$ modular forms that
we denote by $\mathcal{L}_{\mathbb{Q}}$ and we refer to [AGHMP18, Section 4.1]
for a definition. The Shimura datum $(G,\mathcal{D})$ is of Hodge type by
[AGHMP17, Section 2.2]: there exists a Shimura datum of Siegel type
$(G^{Sg},\mathcal{D}^{Sg})$ and a compact open subgroup $K^{sg}\subset
G^{sg}(\mathbb{A}_{f})$ such that we have an embedding of Shimura varieties
over $\mathbb{Q}$

$M\hookrightarrow M^{Sg}.$

This is the Kuga–Satake embedding. The pull-back of the universal abelian
scheme on $M^{Sg}$ yields the Kuga–Satake abelian scheme $A\rightarrow M$.

###  2.2. Toroidal compactifications over $\mathbb{C}$

In this section, we describe the toroidal compactifcations of $M$ as well as
the structure of the boundary components following [BZ21] and [HP20]. See also
[AMRT10] for the general theory of toroidal compactifications over
$\mathbb{C}$.

Recall from [HP20, Section 2.2] that an admissible parabolic subgroup
$P\subseteq G$ is either a maximal proper parabolic subgroup of $G$ or $G$
itself. 111$G^{ad}$ is simple in our case. A cusp label representative
$\Phi=(P,\mathcal{D}^{\circ},h)$ is a triple constituted from an admissible
parabolic subgroup $P$, a connected component
$\mathcal{D}^{\circ}\subset\mathcal{D}$ and an element $h\in
G(\mathbb{A}_{f})$.

Attached to a cusp label representative $\Phi=(P,\mathcal{D}^{\circ},h)$,
there exists a mixed Shimura variety that we now describe. Let $U_{\Phi}$ be
the unipotent radical of $P$ and let $W_{\Phi}$ be the center of $U_{\Phi}$
222We follow the notations of [Per19] which differ from other references.. Let
$Q_{\Phi}$ be the normal subgroup of $P$ defined as in [Pin90, §4.7], see also
[HP20, 2.2]. Define as in loc. cit.
$\mathcal{D}_{\Phi}=Q_{\Phi}(\mathbb{R})W_{\Phi}(\mathbb{C})\mathcal{D}^{\circ}$
and let $K_{\Phi}=hKh^{-1}\cap Q_{\Phi}(\mathbb{A}_{f})$. We define then the
mixed Shimura variety

$\displaystyle
M_{\Phi}(\mathbb{C})=Q_{\Phi}(\mathbb{Q})\backslash\mathcal{D}_{\Phi}\times
Q_{\Phi}(\mathbb{A}_{f})/K_{\Phi}.$ (2.2.1)

By [Pin90, Prop. 12.1], $M^{\Phi}(\mathbb{C})$ has a canonical model
$M^{\Phi}$ also defined over $\mathbb{Q}$. Let
$\overline{Q}_{\Phi}=Q_{\Phi}/W_{\Phi}$ and
$\overline{\mathcal{D}}_{\Phi}=W_{\Phi}(\mathbb{C})\backslash\mathcal{D}_{\Phi}$.
Let $\overline{K}_{\Phi}$ be the image of $K_{\Phi}$ under the quotient map
$Q_{\Phi}(\mathbb{A}_{f})\rightarrow\overline{Q}_{\Phi}(\mathbb{A}_{f})$. Then
from the data
$(\overline{Q}_{\Phi},\overline{\mathcal{D}}_{\Phi},\overline{K}_{\Phi})$ we
define similarly to 2.2.1 a mixed Shimura variety $\overline{M}_{\phi}$ and we
have a canonical morphism

$\displaystyle M_{\Phi}\rightarrow\overline{M}_{\Phi}.$ (2.2.2)

This map has a torsor structure that we now describe. Let
$\Gamma_{\Phi}=K_{\Phi}\cap W_{\Phi}(\mathbb{Q})$. It is a
$\mathbb{Z}$-lattice in $W_{\Phi}(\mathbb{Q})$. By [HP20, Proposition 2.3.1],
the map 2.2.2 is canonically a torsor under the torus $T_{\Phi,\mathbb{Q}}$
whose cocharacter group is $\Gamma_{\Phi}$.

The mixed Shimura variety $\overline{M}_{\Phi}$ has itself a fibration
structure over a pure Shimura variety constructed as follows, see [Per19,
2.1.7] for more details.

Let $G^{h}_{\Phi}=Q_{\Phi}/U_{\Phi}$ be the Levi quotient of $Q_{\Phi}$,
$V_{\Phi}=U_{\Phi}/W_{\Phi}$ the unipotent radical of $\overline{Q}_{\Phi}$
and let
$\mathcal{D}^{h}_{\Phi}=V_{\Phi}(\mathbb{R})\backslash\overline{D}_{\Phi}$.
Then the pair $(G^{h}_{\Phi},\mathcal{D}^{h}_{\Phi})$ is a pure Shimura datum
with reflex field equal to $\mathbb{Q}$. Let $K_{\Phi}^{h}\subset
G_{\Phi}^{h}(\mathbb{A}_{f})$ be the image of $K_{\Phi}$. Then the quotient

$M_{\Phi}^{h}(\mathbb{C})=G^{h}_{\Phi}(\mathbb{Q})\backslash(\mathcal{D}^{h}_{\Phi}\times
G^{h}_{\Phi}(\mathbb{A}_{f}))/K_{\Phi}^{h}$

is the set of complex points of a Shimura variety which admits a canonical
model $M_{\Phi}^{h}$ defined over $\mathbb{Q}$ and we have a canonical map

$\displaystyle\overline{M}_{\Phi}\rightarrow M_{\Phi}^{h}.$ (2.2.3)

By [Per19, 2.1.12], there exists a natural abelian scheme
$A_{K}(\Phi)\rightarrow M_{\Phi}^{h}$ such that the map 2.2.3 is a torsor
under $A_{K}(\Phi)$.

In what follows, we will describe the above data for the GSpin Shimura variety
introduced in Section 2.1 following [HP20, Section 4] and [BZ21, Section 3].
Let $\Phi$ be a cusp label representative. The admissible parabolic subgroup
$P$ is the stabilizer of a totally isotropic subspace $I_{\Phi}$ of
$L_{\mathbb{Q}}$ of dimension at most $2$. The dimension $0$ case corresponds
to $P=G$. If $P$ is the stabilizer of a primitive isotropic line
$I_{\mathbb{Q}}\subset L_{\mathbb{Q}}$, then the cusp label representative is
said to be of type III. If $P$ is the stabilizer of a primitive isotropic
plane $J_{\mathbb{Q}}\subset L_{\mathbb{Q}}$, then $\Phi$ is said to be of
type III. We will follow the notations of [BZ21] and denote by $\Upsilon$,
resp. $\Xi$, a cusp label representative of type II, resp. of type III.

Given two cusp label representatives $\Phi_{1}$ and $\Phi_{2}$, there is a
notion of a $K$-morphism $\Phi_{1}\xrightarrow{(\gamma,q_{2})_{K}}\Phi_{2}$
given by $\gamma\in G(\mathbb{Q})$ and $q_{2}\in Q_{\Phi_{2}}(\mathbb{A}_{f})$
which we don’t define here and refer to [Per19, 2.1.14] for the definition,
see also [HP20, Definition 2.4.1].

Let $\Phi$ be a cusp label representative. By the general theory of toroidal
compactifications, see [Pin90, 4.15] or [AMRT10, Ch. II.§1.1] for the
definitions, there exists a canonical open non-degenerate self-adjoint convex
cone $C_{\Phi}\subset W_{\Phi}(\mathbb{R})$ homogeneous under $P(\mathbb{R})$
and which allows to realize $\mathcal{D}^{\circ}$ as a tube domain inside an
affine space, see [Per19, 2.1.5]. We define the extended cone $C^{*}_{\Phi}$
as in [Per19, 2.1.22]: for any map
$\Phi^{\prime}\xrightarrow{(\gamma,q)_{K}}\Phi$, the conjugation by
$\gamma^{-1}$ induces an embedding

$\mathrm{int}(\gamma^{-1}):W_{\Phi^{\prime}}(\mathbb{R})\hookrightarrow
W_{\Phi}(\mathbb{R})$

and we define then

$C_{\Phi}^{*}=\bigcup_{\Phi^{\prime}\rightarrow\Phi}\mathrm{int}(C_{\Phi^{\prime}}).$

This cone lies between $C_{\Phi}$ and its topological closure in
$W_{\Phi}(\mathbb{R})$ but in general, it is neither open nor closed. See also
[Pin90, Definition-Proposition 4.22] for more details.

Recall from [HP20, Definition 2.4.3] that a rational polyhedral cone
decomposition (rpcd for short) of $C^{*}_{\Phi}$ is a collection
$\Sigma_{\Phi}=\\{\sigma\\}$ of rational polyhedral cones $\sigma\subset
W_{\Phi}(\mathbb{R})$ satisfying natural compatibility conditions (we don’t
recall these conditions here and invite the reader to consult the reference
above for more information). The rpcd $\Sigma_{\Phi}$ is said to be smooth if
it is smooth in the sense of [Pin90, §5.2] with respect to the lattice
$\Gamma_{\Phi}$. It is complete if

$C^{*}_{\Phi}=\bigcup_{\sigma\in\Sigma_{\Phi}}\sigma.$

####  2.2.1. Boundary components of type II

Let $\Upsilon$ be a cusp label representative of type II. Then $P$ is the
stabilizer of a primitive isotropic plane $J_{\mathbb{Q}}\subseteq
L_{\mathbb{Q}}$ and let $J=J_{\mathbb{Q}}\cap h.L$, where $h\in
G(\mathbb{A}_{f})$ acts on $L$ via the map
$G(\mathbb{A}_{f})\rightarrow\mathrm{SO}(L_{\mathbb{Q}})(\mathbb{A}_{f})$.
Then by [HP20, Page 31], the group $W_{\Upsilon}$ is identified with
$\bigwedge^{2}J_{\mathbb{Q}}$, hence it is one dimensional. The lattice
$\Gamma_{\Upsilon}\subset W_{\Upsilon}$ is also of rank $1$. The open convex
cone $C_{\Upsilon}$ is given by a half line $\mathbb{R}^{+}\backslash\\{0\\}$
and the extended cone is $C^{*}_{\Upsilon}=\\{0\\}\cup C_{\Upsilon}$.

Let $M_{\Upsilon}$ and $\overline{M}_{\Upsilon}$ be the mixed Shimura
varieties associated to $\Upsilon$. Then
$M_{\Upsilon}\rightarrow\overline{M}_{\Upsilon}$ is a torsor under the $1$
dimensional torus $T_{\Upsilon}$ with cocharacter group $\Gamma_{\Upsilon}$.
The group $G^{h}_{\Upsilon}$ is equal to $\mathrm{SL}_{2}$ and
$\mathcal{D}^{h}_{\Upsilon}$ is equal to the Poincaré upper half-plane. The
Shimura variety $M^{h}_{\Upsilon}$ is a modular curve and the abelian scheme
$A_{\Upsilon}$ is equal to the Kuga–Sato variety $D\otimes E$ where
$E\rightarrow M^{h}_{\Upsilon}$ is the universal elliptic curve over over
$M^{h}_{\Upsilon}$ and $D$ is the positive definite plane $J^{\bot}/J$, see
[BZ21, Corollary 3.17] and [Zem20, Proposition 4.3] for details and proofs.
Notice that our choice for the compact open subgroup $K$ gives exactly the
stable orthogonal group used in [BZ21] and [Zem20].

The only possible cone decomposition of $C^{*}_{\Upsilon}$ in this situation
is $\Sigma_{\Upsilon}=\\{\\{0\\},C_{\Upsilon}\cup\\{0\\}\\}$ and this
determines a partial compactification $M_{\Upsilon}\hookrightarrow
M_{\Upsilon,\Sigma}$ which is a fibration by $\mathbb{A}^{1}_{\mathbb{C}}$
over $\overline{M}_{\Upsilon}$. Finally, there is only one boundary divisor
denoted by $B_{\Upsilon}$ associated to the ray $C_{\Upsilon}$.

####  2.2.2. Boundary components of type III

Let $\Xi$ be a cusp label representative of type III. Then $P$ is the
stabilizer of a primitive isotropic line $I_{\mathbb{Q}}\subset
L_{\mathbb{Q}}$ and let $I=I_{\mathbb{Q}}\cap h.L$. Set $K_{I}=I^{\bot}/I$.
Then by [HP20, Equation (4.4.2)], we have $U_{\Xi}=W_{\Xi}$ and we have an
isomorphism of vector spaces

$K_{I,\mathbb{Q}}\otimes I_{\mathbb{Q}}\simeq W_{\Xi}(\mathbb{Q}).$

The lattice $(K_{I},Q)$ is a Lorentzian lattice of signature $(b-1,1)$. Under
the above isomorphism, and assuming we have chosen a primitive generator of
$I$, the open convex cone $C_{\Xi}\subset W_{\Xi}(\mathbb{R})$, see [HP20,
Section 2.4] is identified with a connected component of the light cone

$\\{x\in K_{I,\mathbb{R}},\,Q(x)<0\\}.$

The spaces $M^{h}_{\Xi}$ and $\overline{M}_{\Xi}$ are equal and are Shimura
varieties of dimension zero that we can describe as follows. Let
$(\mathbb{G}_{m},\mathcal{H}_{0})$ be the Shimura data given by

$\mathcal{H}_{0}:=\\{2\pi\epsilon:\epsilon^{2}=-1\\},$

on which $\mathbb{R}^{\times}$ acts naturally through the quotient
$\mathbb{R}^{\times}/\mathbb{R}^{\times}_{+}$. There is a morphism of mixed
Shimura data
$(Q_{\Xi},\mathcal{D}_{\Xi})\rightarrow(\mathbb{G}_{m},\mathcal{H}_{0})$ given
by a canonical character $v_{\Xi}:Q_{\Xi}\rightarrow\mathbb{G}_{m}$ defined as
in [HP20, Equation (4.4.1)] and a map
$\mathcal{D}_{\Xi}\rightarrow\mathcal{H}_{0}$ given as in [HP20, Equation
(4.6.3)]. Then the Shimura variety
$\mathrm{Sh}_{\nu_{\Xi}(K_{\Xi})}(\mathbb{G}_{m},\mathcal{H}_{0})$ is zero
dimensional and the canonical map
$M_{\Xi}\rightarrow\mathrm{Sh}_{\nu_{\Xi}(K_{\Xi})}(\mathbb{G}_{m},\mathcal{H}_{0})$
is a torsor under the torus
$T_{\Xi}=\mathrm{Spec}\left(\mathbb{Q}[q_{\alpha}]_{\alpha\in\Gamma_{\Xi}^{\vee}}\right)$
with cocharacter group $\Gamma_{\Xi}=K_{I}$ by [BZ21, Proposition 3.7].

The intermediate cone $C_{\Phi}^{*}$ can be described explicitly as follows,
see also [BZ21, page 23]: for any type II boundary component $\Upsilon$ with
corresponding isotropic plane $J$ containing $I$, the quotient $J/I$ has a
generator $\omega_{\Xi,\Upsilon}$ lying on the boundary of the $C_{\Xi}$.
Hence

$C_{\Xi}^{*}=C_{\Xi}\cup\bigcup_{\Upsilon}\mathbb{R}\omega_{\Xi,\Upsilon}.$

The rays $\mathbb{R}\omega_{\Xi,\Upsilon}$ will be referred to as the external
rays and the rays in $C_{\Xi}$ are the inner rays.

####  2.2.3. Toroidal compactifications

Recall from [HP20, Definition 2.4.4] that a $K$-admissible rational polyhedral
cone decomposition for $(G,\mathcal{D})$ is a collection
$\Sigma=\\{\Sigma_{\Xi},\Sigma_{\Upsilon}\\}$ such that $\Sigma_{\Xi}$ and
$\Sigma_{\Upsilon}$ are rpcd for any cusp label representative $\Xi$ and
$\Upsilon$ respectively satisfying the compatibility conditions of [HP20,
Definitions 2.4.3, 2.4.4]. It is said smooth (resp. complete) if every
$\Sigma_{\Phi}$ is smooth (resp. complete).

A toroidal stratum representative is a pair $(\Phi,\sigma)$ where $\Phi$ is a
cusp label representative and $\sigma\subset C_{\Phi}^{*}$ is a rational
polyhedral cone whose interior is contained in $C_{\Phi}$. There is similarly
a notion of $K$-morphism between stratum representatives, see [HP20,
Definition 2.4.6] and the set of $K$-isomorphism classes of toroidal stratum
representatives will be denoted $\mathrm{Start}_{K}(G,\mathcal{D},\Sigma)$. We
say that $\Sigma$ is finite if

$|\mathrm{Start}_{K}(G,\mathcal{D},\Sigma)|<\infty.$

Let $\Sigma$ be a finite $K$-admissible complete cone decomposition. The main
result of [Pin90, §12], see also [Per19, Theorem 2.1.27], ensures that there
exists a proper toroidal compactification

$M\hookrightarrow M^{\Sigma}$

in the category of Deligne-Mumford stacks over $\mathbb{Q}$ such that
$M^{\Sigma}$ is proper over $\mathbb{Q}$ and has a stratification

$\displaystyle
M^{\Sigma}=\bigsqcup_{(\Phi,\sigma)\in\mathrm{Start}_{K}(G,\mathcal{D},\Sigma)}B^{\Phi,\sigma}$
(2.2.4)

by locally closed subspaces indexed by the finite set of strata
$\mathrm{Start}_{K}(G,\mathcal{D},\Sigma)$. The stratum indexed by
$(\Phi,\sigma)$ lies in the closure of the stratum index by
$(\Phi^{\prime},\sigma^{\prime})$ if and only if there is a $K$-morphism of
strata representatives
$(\Phi,\sigma)\rightarrow(\Phi^{\prime},\sigma^{\prime})$. Then the closure of
the stratum $B^{\Phi,\sigma}_{K}$ is given by

$\overline{B^{\Phi,\sigma}}=\bigcup_{(\Phi^{\prime},\sigma^{\prime})\rightarrow(\Phi,\sigma)}B^{\Phi^{\prime},\sigma^{\prime}}.$

Moreover, by [HP20, Theorem 3.4.1] following the work of Harris and
Harris–Zucker [HZ01], the line bundle of weight $1$ modular forms
$\mathcal{L}$ extends to a line bundle on $M^{\Sigma}$ which we still denote
$\mathcal{L}$ by abuse of notation.

Let $(\Phi,\sigma)$ be a toroidal stratum representative. Then $(\Phi,\sigma)$
determines a partial compactification of the mixed Shimura variety
$M_{\Phi}\hookrightarrow M_{\Phi}(\sigma)$ with boundary component index by
$\sigma$ denoted by $Z^{\Phi}(\sigma)$. Pink proved that there is a canonical
isomorphism [Pin90, Corollary 7.17, Theorem 12.4], see also [Per19, Theorem
2.1.27], of Deligne–Mumford stacks:

$\Delta_{K}(\Phi,\sigma)\backslash Z^{\sigma}(\sigma)\simeq B^{\Phi,\sigma}$

where $\Delta_{K}(\Phi,\sigma)$ is the finite group defined in [Per19,
2.1.19]. The latter induces an isomorphism of formal Deligne–Mumford stacks:

$\displaystyle\Delta_{K}(\Phi,\sigma)\backslash\widehat{M}_{\Phi}(\sigma)\simeq\widehat{M}^{\Sigma}$
(2.2.5)

where $\widehat{M}_{\Phi}(\sigma)$ is the completion of $M_{\Phi}(\sigma)$
along the locally closed subspace $Z^{\Phi}(\sigma)$ and
$\widehat{M}^{\Sigma}$ is the formal completion of $M^{\Sigma}$ along the
locally closed stratum $B^{\Phi,\sigma}$.

Our goal in the next two sections is to make the above isomorphisms explicit
for type II and type III boundary strata.

####  2.2.4. Formal completion along type II boundary strata

Let $\Upsilon$ be a cusp label representative of type II. By the discussion in
Section 2.2.1, there is a unique choice of a 1 dimensional ray $\sigma$ and
hence a unique choice of boundary stratum representative $(\Upsilon,\sigma)$
which corresponds to a locally closed divisor $B^{\Upsilon,\sigma}$.

The morphism $M_{\Upsilon}\rightarrow\overline{M}_{\Upsilon}$ is then a torsor
under a $1$-dimensional torus $T_{\Upsilon}$ with cocharacter group
$\Gamma_{\Upsilon}\simeq\mathbb{Z}$, i.e.,
$T_{\Upsilon}\simeq\mathrm{Spec}(\mathbb{Q}[q,q^{-1}])$. The partial
compactification $T_{\Upsilon}(\sigma)$ is then isomorphic to
$\mathrm{Spec}\left(\mathbb{Q}[q]\right)$ and the partial toroidal
compactification of $M_{\Upsilon}$ is given as a twisted torus embedding over
$\overline{M}_{\Upsilon}$ with fiber
$\mathrm{Spec}\left(\mathbb{Q}[q]\right)$. Hence we have the following
description of $\widehat{M}_{\Upsilon}(\sigma)$

$\widehat{M}_{\Upsilon}(\Sigma)\xrightarrow{\mathrm{Spf}(\mathbb{Q}[[X]])}\overline{M}_{\Upsilon}\xrightarrow{D\otimes
E}M^{h}_{\Upsilon}.$

####  2.2.5. Formal completion along type III boundary strata

Let $(\Xi,\sigma)$ be a toroidal stratum representative of type III such that
$\sigma$ is a one dimensional inner ray. The corresponding boundary component
is denoted by $B^{\Phi,\sigma}$ and is a locally closed divisor. Write
$\sigma=\mathbb{R}\omega$ where $\omega\in C_{\Xi}\cap K$ is an integral
primitive generator that satisfies $(\omega.\omega)<0$.

The morphism
$M_{\Xi}\rightarrow\mathrm{Sh}_{\nu_{\Xi}(K_{\Xi})}(\mathbb{G}_{m},\mathcal{H}_{0})$
is a torsor under the torus

$T_{\Xi}=\mathrm{Spec}\left(\mathbb{Q}[q_{\alpha}]_{\alpha\in\Gamma_{\Xi}^{\vee}}\right).$

The partial compactification $T_{\Xi}(\sigma)$ is equal to

$T_{\Xi}(\sigma)=\mathrm{Spec}\left(\mathbb{Q}[q_{\alpha}]_{\underset{(\alpha.\omega)\geq
0}{\alpha\in\Gamma_{\Xi}^{\vee}}}\right)$

and the ideal defining the boundary divisor is given by
$I_{\sigma}=(q_{\alpha},(\alpha,\omega)>0)$. It is generated by
$q_{\omega^{\prime}}$ for any $\omega^{\prime}\in\Gamma_{\Xi}^{\vee}$ for
which $(\omega,\omega^{\prime})=1$. We fix such $\omega^{\prime}$.

The formal completion along the boundary divisor is then given by:

$\widehat{T}_{\Xi}(\sigma)=\mathrm{Spec}\left(\mathbb{Q}[q_{\alpha},\,\alpha\in\Gamma_{\Xi}^{\vee}\cap\omega^{\bot}][[q_{\omega^{\prime}}]]\right),$

and the map
$M_{\Xi}(\sigma)\rightarrow\mathrm{Sh}_{\nu_{\Xi}(K_{\Xi})}(\mathbb{G}_{m},\mathcal{H}_{0})$
is a twisted torus embedding with fibers $\widehat{T}_{\Xi}(\sigma)$. We will
trivialize this fibration following an approach similar to [HP20, Page 34].

First choose an auxiliary isotropic line $I_{*}\subset L_{\mathbb{Q}}$ such
that $(I.I_{*})\neq 0$. Then by [HP20, Equation (4.6.6)] and the discussion
that follows, this determines a section

$(\mathbb{G}_{m},\mathcal{H}_{0})\xrightarrow{s}(Q_{\Xi},\mathcal{D}_{\Xi}).$

The section $s$ determines a Levi decomposition $Q_{\Xi}=\mathbb{G}_{m}\ltimes
U_{\Xi}$. Let $K_{0}\subset\mathbb{G}_{m}(\mathbb{A}_{f})$ be a compact open
subgroup small enough such that the image under the section $s$ is contained
in $K_{\Xi}$ and let

$K_{\Xi,0}=K_{0}\ltimes(U_{\Xi}(\mathbb{A}_{f}\cap K_{\Xi}))\subset K_{\Xi}.$

Then by reasoning similarly to [HP20, Proposition 4.6.2], we have the
following.

######  Proposition 2.1.

We have an isomorphism of formal algebraic spaces:

$\underset{{a\in\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}}}{\bigsqcup}\widehat{T}_{\Xi}(\sigma)_{/\mathbb{C}}\xrightarrow{\simeq}\widehat{M}_{K_{\Xi,0}}(\sigma)_{/\mathbb{C}},$

and the map

$\widehat{M}_{K_{\Xi,0}}(\sigma)_{/\mathbb{C}}\rightarrow\widehat{M}_{K_{\Xi}}(\sigma)_{/\mathbb{C}}$

is a formally étale map of formal Deligne–Mumford stacks given by the quotient
by $K_{\Xi}/K_{\Xi,0}$. In particular, if $K$ is neat, then the above map is a
formally étale surjection of algebraic spaces.

###### Proof.

The same proof as in [HP20, Proposition 4.6.2] works with no change in our
setting. ∎

###  2.3. Integral models

We recall in this section the construction of integral models of GSpin Shimura
varieties and their compactifications following [HP20, AGHMP18, Per19]. We
assume henceforth that the lattice $(L,Q)$ is a maximal lattice, i.e., there
is no strict superlattice in $L_{\mathbb{Q}}$ containing $L$ over which $Q$ is
$\mathbb{Z}$-valued.

By [AGHMP18, Section 4.4], there exists a flat and normal integral model
$\mathcal{M}\rightarrow\mathrm{Spec}(\mathbb{Z})$ which is a Deligne–Mumford
stack of finite type over $\mathbb{Z}$. It enjoys the following properties:

  1. (1)

If the lattice $(L,Q)$ is almost self dual333See [HP20, Definition 6.1.1]. at
a prime $p$ then the restriction of the integral model to
$\mathrm{Spec}(\mathbb{Z}_{(p)})$ is smooth.

  2. (2)

If $p$ is odd and $p^{2}$ does not divide the discriminant of $(L,Q)$, the
restriction of $\mathcal{M}$ to $\mathrm{Spec}(\mathbb{Z}_{(p)})$ is regular.

  3. (3)

If $n\geq 6$, the reduction mod $p$ is geometrically normal.

  4. (4)

The line bundle of modular forms of weight $1$ extends to a line bundle on
$\mathcal{M}$ that we denote by $\mathcal{L}$.

Furthermore, given a $K$-admissible polyhedral complete cone decomposition,
$\mathcal{M}$ admits by [Per19, Theorem 4.1.5] a toroidal compactification
$\mathcal{M}^{\Sigma}$ proper over $\mathrm{Spec}(\mathbb{Z})$ and which
extends the compactification $M^{\Sigma}$ previously defined over
$\mathbb{Q}$. Moreover, it has a stratification

$\displaystyle\mathcal{M}^{\Sigma}=\bigsqcup_{(\Phi,\sigma)\in\mathrm{Start}_{K}(G,\mathcal{D},\Sigma)}\mathcal{B}^{\Phi,\sigma}$
(2.3.1)

which extends the stratification in Equation 2.2.4 and such every stratum is
flat over $\mathbb{Z}$. The unique open stratum is $\mathcal{M}$ and its
complement is a Cartier divisor. Moreover, for any cusp label representative
$(\Phi,\sigma)$, the tower of maps

$M_{\Phi}(\sigma)\rightarrow\overline{M}_{\phi}\rightarrow M^{h}_{\Phi}$

has an integral model

$\mathcal{M}_{\Phi}(\sigma)\rightarrow\overline{\mathcal{M}}_{\phi}\rightarrow\mathcal{M}^{h}_{\Phi}$

which satisfies the following: the abelian scheme $A_{\Phi}$ has an extension
$\mathcal{A}_{\Phi}\rightarrow\mathcal{M}^{h}_{\Phi}$ such that the map
$\overline{\mathcal{M}}_{\phi}\rightarrow\mathcal{M}^{h}_{\Phi}$ is a torsor
under $\mathcal{A}_{\Phi}$ and the map
$\mathcal{M}_{\Phi}(\sigma)\rightarrow\overline{\mathcal{M}}_{\phi}$ is a
twisted torus embedding with structure group the torsor $\mathcal{T}_{\Xi}$
extending $T_{\Xi}$. Finally, the boundary component $Z_{\Phi}(\sigma)$ has a
flat extension $\mathcal{Z}_{\Phi}(\sigma)$ such that we have an isomorphism
of completions:

$\displaystyle\Delta_{K}(\Phi,\sigma)\backslash\widehat{\mathcal{M}}_{\Phi}(\sigma)\simeq\widehat{\mathcal{M}}^{\Sigma}$
(2.3.2)

extending the isomorphism in Equation 2.2.5. See [Per19, Theorem 4.1.5] and
[HP20, Section 8.1] for more details.

Fix a prime $p$. The goal of the next two subsections is to describe the
formal completions of $\mathcal{M}^{\Sigma}$ along the boundary divisors of
these compactifications explicitly over $\mathbb{Z}_{(p)}$ in the type II and
the type III case.

####  2.3.1. Type II

Let $(\Upsilon,\sigma)$ be a toroidal stratum representative of type II where
$\sigma$ is the unique one dimensional ray.

Let
$\mathcal{T}_{\Upsilon}=\mathrm{Spec}\left(\mathbb{Z}_{(p)}[q,q^{-1}]\right)$
with partial compactification
$\mathcal{T}_{\Upsilon}(\sigma)=\mathrm{Spec}\left(\mathbb{Z}_{(p)}[q]\right)$.
By Equation 2.3.1 and [Per19, Theorem 4.1.5 (2-4)], the morphism
$\mathcal{M}_{\Upsilon}\rightarrow\overline{\mathcal{M}}_{\Upsilon}$ is a
torsor under $\mathcal{T}_{\Upsilon}$ and the morphism
$\overline{\mathcal{M}}_{\Upsilon}\rightarrow\mathcal{M}^{h}_{\Xi}$ is a
torsor under $D\otimes\mathcal{E}$ where
$\mathcal{E}\rightarrow\mathcal{M}^{h}_{\Xi}$ is the universal elliptic curve.
Moreover, the partial toroidal compactification of $\mathcal{M}_{\Xi}$ is
given as a twisted torus embedding over $\overline{\mathcal{M}}_{\Upsilon}$
with fibers isomorphic to $\mathcal{T}_{\Upsilon}(\sigma)$. In particular, the
formal completion of $\mathcal{M}_{\Upsilon}$ along the boundary component is
describe by the following diagram:

$\displaystyle\widehat{\mathcal{M}}_{\Upsilon}(\sigma)\xrightarrow{\widehat{\mathcal{T}}_{\Upsilon}(\sigma)}\overline{\mathcal{M}}_{\Upsilon}\xrightarrow{D\otimes\mathcal{E}}\mathcal{M}^{h}_{\Upsilon},$
(2.3.3)

where
$\widehat{\mathcal{T}}_{\Upsilon}(\sigma)=\mathrm{Spf}\left(\mathbb{Z}_{(p)}[[q]]\right)$.

####  2.3.2. Type III

Let $(\Xi,\sigma)$ be a toroidal stratum representative of type III such that
$\sigma$ is one dimensional and generated by a primitive integral element
$\omega\in C_{\Xi}$ with $(\omega.\omega)=-2N$. Let
$\mathcal{T}_{\Xi}=\mathrm{Spec}\left(\mathbb{Z}_{(p)}[q_{\alpha}]_{\alpha\in\Gamma_{\Xi}^{\vee}}\right)$
and recall that we have a $T_{\Xi}$ torsor structure

$M_{\Xi}\rightarrow\mathrm{Sh}_{\nu_{\Xi}(K_{\Xi})}(\mathbb{G}_{m},\mathcal{H}_{0}).$

The cone $\sigma$ determines a partial compactification
$\mathcal{T}_{\Xi}(\sigma)=\mathrm{Spec}\left(\mathbb{Z}_{(p)}[q_{\alpha}]_{\underset{(\alpha,\omega)\geq
0}{\alpha\in\Gamma_{\Xi}^{\vee}}}\right)$ and also a partial compactification
$\mathcal{M}_{\Xi}\hookrightarrow\mathcal{M}_{\Xi}(\sigma)$ which is a twisted
torus embedding with fibers $\mathcal{T}_{\Xi}(\sigma)$.

The boundary divisor in $\mathcal{T}_{\Xi}(\sigma)$ is defined by the ideal
$I_{\sigma}=(q_{\alpha},(\alpha,\omega)>0)$. If
$\omega^{\prime}\in\Gamma^{\vee}_{\Xi}$ is as before an element such that
$(\omega^{\prime}.\omega)=1$, then $I_{\sigma}=(q_{\omega^{\prime}})$. The
formal completion of $\mathcal{T}_{\Xi}(\sigma)$ along $I_{\sigma}$ is then
given by

$\widehat{\mathcal{T}}_{\Xi}=\mathrm{Spf}\left(\mathbb{Z}_{(p)}[q_{\alpha},\,\alpha\in\Gamma_{\Xi}^{\vee}\cap\omega^{\bot}][[q_{\omega^{\prime}}]]\right).$

Recall that we have a morphism of Shimura data

$(Q_{\Xi},\mathcal{D}_{\Xi})\xrightarrow{v_{\Xi}}(\mathbb{G}_{m},\mathcal{H}_{0}),$

and let $s$ be the section of $v_{\Xi}$ defined in Section 2.2.5. Let
$K_{0}\subset\mathbb{A}_{f}^{\times}$ be a compact open subgroup such that
$s(K_{0})\subset K_{\Xi}$. We can furthermore assume that $K_{0}$ factors as

$K_{0}=\mathbb{Z}_{p}^{\times}.K_{0}^{p}.$

Let $F$ be the abelian extension of $\mathbb{Q}$ determined by the reciprocity
morphism in global class field theory:

$\mathrm{rec}:\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}\simeq\mathrm{Gal}(F/\mathbb{Q}).$

Fix a prime $\mathfrak{P}\subset\mathcal{O}_{F}$ above $p$ and let $R$ be the
localization of $\mathcal{O}_{F}$ at $\mathfrak{P}$. Then using similar
arguments as in [HP20, Proposition 8.2.3], we have the following proposition.

######  Proposition 2.2.

There is an isomorphism

$\bigsqcup_{\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}}\widehat{\mathcal{T}}_{\Xi}(\sigma)_{/R}\rightarrow\widehat{\mathcal{M}}_{\Xi,0}(\sigma)/R$

of formal Deligne–Mumford stacks over $R$ whose base change to $\mathbb{C}$
agrees with Proposition 2.1. Moreover, the map

$\widehat{\mathcal{M}}_{\Xi,0}(\sigma)/R\rightarrow\widehat{\mathcal{M}}_{\Xi}(\sigma)/R$

is an étale map of Deligne–Mumford stacks given as the quotient by
$K_{\Xi}/K_{\Xi,0}$.

The proof follows from the description given over $\mathbb{C}$ 2.1, the
flatness of both sides over $\mathbb{Z}_{(p)}$ and the fact the normalization
of $\mathrm{Spec}(\mathbb{Z}_{(p)})$ in
$\mathrm{Sh}_{K_{0}}(\mathbb{G}_{m},\mathcal{H}_{0})$ is isomorphic to
$\bigsqcup_{a\in\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}}\mathrm{Spec}(R)$,
see [HP20, Proposition 8.2.3] for a proof and more details.

###  2.4. Special divisors

We continue to assume in this section that the lattice $(L,Q)$ is maximal and
let $\Sigma$ be a smooth $K$-admissible cone decomposition.

For every $\beta\in L^{\vee}/L$, $m\in Q(\beta)+\mathbb{Z}$ such that $m>0$,
one can define a special divisor $\mathcal{Z}(\beta,m)\rightarrow\mathcal{M}$
following [AGHMP18, Definition 4.5.6]. We recall briefly the definition and
refer to loc. cit. for more details.

The Shimura variety $\mathcal{M}$ carries the family of Kuga-Satake abelian
varieties $\mathcal{A}\rightarrow\mathcal{M}$. For any scheme
$S\rightarrow\mathcal{M}$, a group of special quasi-endomorphisms
$V_{\beta}(\mathcal{A}_{S})$ is defined in [AGHMP18, Section 4.5]. Then the
functor sending a scheme $S$ to

$\mathcal{Z}(\beta,m)(S)=\\{x\in V_{\beta}(\mathcal{A}_{S})|Q(x)=m\\}$

is representable by a Deligne–Mumford stack which is étale locally an
effective Cartier divisor on $\mathcal{M}$. We will rather consider its image
in $\mathcal{M}$ by a procedure described in [HP20] after Proposition 6.5.2.
By abuse of notation, we also denote by $\mathcal{Z}(\beta,m)$ its closure in
$\mathcal{M}^{\Sigma}$, which is again a Cartier divisor.

In what follows, we will give an explicit description of
$\mathcal{Z}(\beta,m)$ in the formal completions of $\mathcal{M}^{\Sigma}$
along its boundary components. Since for our purposes we only need $\beta=0$
and $m$ coprime to $p$, we will only describe what happens in this situation
and we abbreviate for short $\mathcal{Z}(\beta,m)=\mathcal{Z}(m)$. We assume
that $m\geq 1$ is coprime to $p$ for the rest of this section.

By [AGHMP18, Page 434], $\mathcal{Z}(m)(\mathbb{C})$ has a complex
uniformization as follows: for any $g\in G(\mathbb{A}_{f})$, let
$L_{g}=g.\widehat{L}\cap L_{\mathbb{Q}}$ and consider the sub-Hermtian domain
of $\mathcal{D}$

$\mathcal{D}^{\circ}(\lambda)=\\{x\in\mathcal{D}^{\circ}|(x,\lambda)=0\\},$

where $\lambda\in L_{g}$, $Q(\lambda)=m$. Then $\mathcal{Z}(m)(\mathbb{C})$ is
equal to the union of $\mathcal{D}^{\circ}(\lambda)$ for $g\in
G(\mathbb{A}_{f})$ and $\lambda\in L_{g}$ with $Q(\lambda)=m$.

For any $\lambda\in L_{g}$ with $Q(\lambda)=m$, let $G_{\lambda}$ be the
fixator of $\lambda$, $L_{\lambda}$ the orthogonal lattice to $\lambda$ in
$L_{\mathbb{Q}}$, and let $\mathcal{D}_{\lambda}\subset\mathcal{D}$ be the
orthogonal to $\lambda$. Notice that $\mathcal{D}_{\lambda}$ does not depend
on $g$ but only on $\lambda\in L_{\mathbb{C}}$. Notice that since $m$ is
coprime to $p$, the lattice $L_{\lambda}$ is also maximal at $p$. Then
$(G_{\lambda},D_{\lambda})$ is again a Shimura datum of GSpin associated to
the lattice $(L_{\lambda},Q)$ which is of signature $(b-1,2)$ and has reflex
field equal to $\mathbb{Q}$. If we choose $K_{\lambda}\subset
G_{\lambda}(\mathbb{A}_{f})$ a compact open subgroup as in [AGHMP18, Equation
(4.1.2)], then $K_{\lambda}\subset K\cap G_{\lambda}(\mathbb{A}_{f})$ and we
obtain a morphism of complex Shimura varieties

$M_{\lambda}(\mathbb{C})\rightarrow M(\mathbb{C}).$

By the description [AGHMP18, Equation (2.4)], the union over $g\in
G(\mathbb{A}_{f})$, $\lambda\in L_{g}$ with $Q(\lambda)=m$ of the images of
$M_{\lambda}(\mathbb{C})$ is equal to $\mathcal{Z}(m)(\mathbb{C})$.

Now since $(G_{\lambda},D_{\lambda})$ is again a Shimura variety of GSpin type
associated to a lattice maximal at $p$, the discussion in the previous
sections applies verbatim to the Shimura variety $M_{\lambda}$ and yields
similar description for the compactification and the integral model over
$\mathbb{Z}_{(p)}$. In particular, we have a map between integral models
$\mathcal{M}_{\lambda}\rightarrow\mathcal{M}$ over $\mathbb{Z}_{(p)}$ which
factors through $\mathcal{Z}(m)$ by [HP20, Page 82].

$\mathcal{M}_{\lambda}\rightarrow\mathcal{Z}(m)\hookrightarrow\mathcal{M}$

and the union over of images444This union is in fact finite. of such maps for
$g\in G(\mathbb{A}_{f})$ and $\lambda\in L_{g}$ with $Q(\lambda)=m$ is equal
to $\mathcal{Z}(m)$.

Let $(\Phi,\sigma)$ be a toroidal stratum representative for $\mathcal{M}$.
From the description of the parabolic subgroups of $\mathrm{GSpin}(b,2)$, we
have the following lemma.

######  Lemma 2.3.

The group $P\cap G_{\lambda}$ is an admissible parabolic subgroup of
$G_{\lambda}$ if and only if $\lambda\in I_{\Phi}^{\bot}$.

Notice also that if $\lambda\notin I_{\Phi}^{\bot}$, then the image of
$\mathcal{D}_{\lambda}$ in $M^{\Sigma}(\mathbb{C})$ will not intersect the
boundary components parameterized by $\Phi$, as its projection to the Baily-
Borel compactification will not do so. Hence they will not appear in the
formal completions of $\mathcal{M}^{\Sigma}$ along these boundary components.

We can write $\Phi=(P,\mathcal{D}^{\circ},h)$ and let $\lambda\in L_{g}$ with
$Q(\lambda)=m$ such that $\lambda\in I_{\Phi}^{\bot}$. 2.3 shows that
$(\Phi,\sigma)$ can also be seen as a toroidal stratum representative with
respect to $(G_{\lambda},\mathcal{D}_{\lambda})$ by considering $P\cap
G_{\lambda}$, see [MP16, Section 2.1.28] for more details. Let
$\mathcal{M}_{\lambda,\Upsilon}$ be the integral model over $\mathbb{Z}_{(p)}$
of the mixed Shimura variety associated to $\Phi$. We get then a morphism of
mixed Shimura varieties

$\mathcal{M}_{\lambda,\Phi}\rightarrow\mathcal{M}_{\Phi},$

as well as a morphism of partial compactifications respecting the strata

$\mathcal{M}_{\lambda,\Phi}(\sigma)\rightarrow\mathcal{M}_{\Phi}(\sigma).$

By [Per19, Proposition 2.1.29], the morphism induced at the level of formal
completions along the boundary strata given by $\sigma$ is compatible with the
toroidal compacitifications of $\mathcal{M}_{\lambda}$ and $\mathcal{M}$. In
particular, we get a commutative diagram

where the right vertical map is an étale cover of Deligne–Mumford stacks, the
left vertical map is an étale cover of an open and closed subset by [HP20,
Page 82]. Finally, the union over $g\in G(\mathbb{A}_{f})$, $\lambda\in L_{g}$
with $Q(\lambda)=m$ of the images of the left map covers the whole
$\widehat{\mathcal{Z}}(m)$.

####  2.4.1. Special divisors along type II boundary components

Let $(\Upsilon,\sigma)$ be a toroidal stratum representative of type II.

Let $\lambda\in L$ with $Q(\lambda)=m$ such that $\lambda\in
I_{\Upsilon}^{\bot}$ and $m$ is coprime to $p$. We have a morphism of formal
completions of the partial compactifications of mixed Shimura varieties

$\widehat{\mathcal{M}}_{\lambda,\Upsilon}(\sigma)\rightarrow\widehat{\mathcal{M}}_{\Upsilon}(\sigma).$

Let
$x\in\mathcal{B}^{\Upsilon,\sigma}(\overline{\mathbb{F}}_{p})\subset\mathcal{M}_{\Upsilon}(\sigma)(\overline{\mathbb{F}}_{p})$
and let $\mathcal{O}_{\mathcal{M}_{\Upsilon}(\sigma),x}$ be the local ring at
$x$. Let $\overline{x}$ be the image of $x$ in
$\overline{\mathcal{M}}_{\Upsilon}(\overline{\mathbb{F}}_{p})$ and let $z$ the
image in $\mathcal{M}^{h}_{\Upsilon}(\overline{\mathbb{F}}_{p})$. If follows
from Equation 2.3.3 that the formal completion
$\widehat{\mathcal{O}}_{\mathcal{M}_{\Upsilon}(\sigma),x}$ is isomorphic to

$\widehat{\mathcal{O}}_{\mathcal{M}_{\Upsilon}(\sigma),x}\simeq\mathbb{Z}_{p}[[X]]\widehat{\otimes}\widehat{\mathcal{O}}_{\overline{\mathcal{M}}_{\Phi},\overline{x}}.$

Moreover, the pull-back of the torsor
$\overline{\mathcal{M}}_{\Upsilon}\rightarrow\mathcal{M}^{h}_{\Upsilon}$ to
$\mathrm{Spf}(\widehat{\mathcal{O}}_{\mathcal{M}^{h}_{\Upsilon},z})$ is
trivial, as it is trivial by reduction to $\overline{\mathbb{F}}_{p}$ and we
can lift formally any section. Hence

$\widehat{\mathcal{O}}_{\overline{\mathcal{M}}_{\Phi},\overline{x}}\simeq\widehat{\mathcal{O}}_{D\otimes\mathcal{E},\overline{x}}$

For $\lambda\in D$, consider the map over $\mathcal{M}^{h}_{\Upsilon}$

$\displaystyle
D\otimes\mathcal{E}\xrightarrow{(\lambda.\,)\otimes\mathrm{Id}}\mathcal{E}.$
(2.4.1)

Its kernel is flat over $\mathcal{M}^{h}_{\Upsilon}$. Let
$I_{\lambda}\subset\lambda$ be the ideal defining it. Then
$\widehat{I}_{\lambda}\hookrightarrow\widehat{\mathcal{O}}_{D\otimes\mathcal{E},\overline{x}}$
is flat over $\widehat{\mathcal{O}}_{\mathcal{M}^{h}_{\Upsilon},z}$.

######  Proposition 2.4.

The formal completion $\widehat{\mathcal{Z}}(m)$ along $x$ is the union over
$\lambda\in D$ with $Q(\lambda)=m$ of the vanishing loci inside
$\widehat{\mathcal{O}}_{\mathcal{M}_{\Upsilon}(\sigma),x}$ of the ideals
$\mathbb{Z}_{p}[[X]]\widehat{\otimes}\widehat{I}_{\lambda}$.

###### Proof.

Let $\lambda\in L$ such that $\lambda\in J^{\bot}$ and $Q(\lambda)=m$. Then we
have a description of the mixed Shimura variety $M_{\Upsilon,\lambda}$ similar
to Equation 2.3.3, namely, it has a fibration structure which fits into the
following diagram:

One can check that $D_{\lambda}=\overline{\lambda}^{\bot}$ where
$\overline{\lambda}$ is the image of $\lambda$ in $D=J^{\bot}/J$. Moreover,
the right vertical map in the above diagram is an étale cover and the vertical
middle map is equivariant with respect to the inclusion

$\overline{\lambda}^{\bot}\otimes\mathcal{E}\hookrightarrow
D\otimes\mathcal{E},$

and the left vertical map has image given by an open and closed subset of
$\widehat{\mathcal{Z}}(m)$.

Let
$z^{\prime}\in\mathcal{M}^{h}_{\lambda,\Upsilon}(\overline{\mathbb{F}}_{p})$
be a point mapping to $z$, then
$\widehat{\mathcal{O}}_{\mathcal{M}^{h}_{\lambda,\Upsilon,z^{\prime}}}\simeq\widehat{\mathcal{O}}_{\mathcal{M}^{h}_{\Upsilon,z}}$.
Hence the above diagram becomes at the level of completed local rings:

where the vertical map is contained in the kernel of the map 2.4.1. By
considering all the $\lambda\in J^{\bot}$ that map to a given class
$\overline{\lambda}\in D$, we get that the image is exactly the kernel of the
map 2.4.1 and hence the image of left vertical map is defined by the ideal
$\mathbb{Z}_{p}[[X]]\widehat{\otimes}\widehat{I}_{\lambda}$, see [Zem20,
Equation (26)] for a description over $\mathbb{C}$. Finally, since
$\widehat{\mathcal{Z}}(m)$ is equal to the union of such images, the
conclusion follows. ∎

####  2.4.2. Special divisors along type III boundary components

Let $(\Xi,\sigma)$ be a stratum representative of type III. Let
$K_{I}=I^{\bot}/I$ be the Lorentzian lattice as introduced in Section 2.2.2
and we continue to assume that $\sigma$ is a one dimensional inner ray. Let
$\omega\in K_{I}\cap C_{\Xi}$ be a generator of $\sigma$ with
$(\omega.\omega)=-2N$, $N\geq 1$. Let $\omega^{\prime}\in K_{I}^{\vee}$ be an
element such that $(\omega.\omega^{\prime})=1$.

Let $\lambda\in L$ with $Q(\lambda)=m$ and such that $\lambda\in I^{\bot}$.
The projection $\overline{\lambda}\in K_{I}$ defines a divisor in the torus
$\mathcal{T}_{\Xi}=\mathrm{Spec}(\mathbb{Z}_{(p)}[q^{\alpha}]_{\alpha\in{\Gamma_{\Xi}^{\vee}}})$
given by the equation $q^{\overline{\lambda}}=1$.

In the partial compactification
$\mathcal{T}_{\Xi}\hookrightarrow\mathcal{T}_{\Xi}(\sigma)$, the equation of
this divisor becomes $q^{\overline{\lambda}}-1=0$ if
$(\overline{\lambda}.\omega)\geq 0$ or $q^{-\overline{\lambda}}-1=0$
otherwise. Notice also that this divisor intersects the toric boundary divisor
defined by $\sigma$ if and only if $(\omega.\overline{\lambda})=0$. We will
hence restrict ourselves to this latter situation and we denote by

$\mathcal{T}_{\Xi}(\lambda,\sigma)\hookrightarrow\mathcal{T}_{\Xi}(\sigma)$

the divisor defined by $\lambda$. By construction, it only depends on the
class of $\lambda$ in $K_{I}$.

######  Proposition 2.5.

Let $\widehat{\mathcal{Z}}(m)$ be the formal completion of $\mathcal{Z}(m)$
along the boundary component of $\mathcal{M}^{\Xi}$ index by $(\Xi,\sigma)$.
Then the following diagram is commutative and compatible with Proposition 2.1.

$\textstyle{\underset{{a\in\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}}}{\bigsqcup}\underset{{\underset{Q(\lambda)=m,(\lambda.\omega)=0}{\lambda\in
K}}}{\bigsqcup}\widehat{\mathcal{T}}_{\Xi,0}(\lambda,\sigma)_{/\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{{a\in\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}}}{\bigsqcup}\widehat{\mathcal{T}}_{\Xi,0}(\sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{Z}}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{M}}^{\Sigma}}$

The vertical maps are étale coverings of formal Deligne–Mumford stacks and the
union over $\lambda\in I^{\bot}$ covers $\widehat{\mathcal{Z}}(m)$.

###### Proof.

Let $\lambda\in L\cap I^{\bot}$ with $Q(\lambda)=m$ and such that projection
$\overline{\lambda}\in I^{\bot}/I$ is orthogonal to $\omega$. Then we have
similarly a description of the mixed Shimura variety
$\mathcal{M}_{\lambda,\Xi}$ associated to the Shimura datum
$(G_{\lambda},\mathcal{D}_{\lambda})$ as a torus fibration and such that the
following diagram is commutative

The left vertical map is equivariant with respect to the inclusion
$\widehat{\mathcal{T}}_{\Xi}(\lambda,\sigma)\hookrightarrow\widehat{\mathcal{T}}_{\Xi}(\sigma)$
and its image only depends on $\overline{\lambda}\in I^{\bot}/I$. Since the
formal completion $\widehat{\mathcal{Z}}(m)$ is the union over $\lambda\in L$
of the images of the left vertical maps, we get the desired result. ∎

##  3\. Arithmetic intersection theory and modularity

We recall in this section the Arakelov arithmetic intersection theory on
$\mathcal{M}^{\Sigma}$ following [BBK07], the modularity results of the
special divisors from [HP20, Bor99] and its extension to complex toroidal
compactification by [BZ21]. Then we derive a further extension to the integral
model of the toroidal compactifications of GSpin Shimura varieties.

###  3.1. Modularity of special divisors

Let $(L,Q)$ be a maximal quadratic lattice with signature $(b,3)$ and assume
that $b\geq 3$.

Let $K\subset G(\mathbb{A}_{f})$ be the compact open subgroup from Section 2.1
and let $\Sigma$ be a $K$-admissible smooth polyhedral cone decomposition.
Denote by $\mathcal{M}^{\Sigma}$ the toroidal compactification of the integral
model of the GSpin Shimura variety constructed in Section 2.3. Let
$\widehat{\operatorname{CH}}^{1}(\mathcal{M}^{\Sigma},\mathcal{D}_{pre})_{\mathbb{Q}}$
be the first Chow group of prelog forms as defined in [BBK07, Definition
1.15].

Let $\Upsilon$ be a cusp label representative of type II. Then there is a
unique one dimensional ray in the cone decomposition associated to $\Upsilon$
and we denote by abuse of notation $\mathcal{B}^{\Upsilon}$ the closure of the
boundary divisor associated to $\Upsilon$.

Consider now $(\Xi,\sigma)$ a toroidal stratum representative of type III such
that $\sigma$ is a $1$-dimensional inner ray in the cone decomposition
$\Sigma$. Then we denote by $\mathcal{B}^{\Xi,\sigma}$ the closed boundary
divisor in $\mathcal{M}^{\Sigma}$ associated to $(\Xi,\sigma)$.

Let $\beta\in L^{\vee}/L$ and $m\in Q(\beta)+\mathbb{Z}$ with $m>0$. For every
toroidal stratum representative $\Upsilon$ and $(\Xi,\sigma)$, let
$\mu_{\Upsilon}(\beta,m)$ and $\mu_{\Xi,\omega}(\beta,m)$ be the real numbers
defined by Equation 4.5.1 and Equation 4.6.1, see also [BZ21]. Consider then
the following divisor on $\mathcal{M}^{\Sigma}$:

$\displaystyle\mathcal{Z}^{tor}(\beta,m)=\mathcal{Z}(\beta,m)+\sum_{\Upsilon}\mu_{\Upsilon}(\mu,m)\cdot\mathcal{B}^{\Upsilon}+\sum_{(\Xi,\omega)}\mu_{\Xi,\omega}(\mu,m)\cdot\mathcal{B}^{\Xi,\omega},$
(3.1.1)

where the two last sums are over toroidal stratum representatives of type II
and type III respectively. Then by [BZ21], the Cartier divisor
$\mathcal{Z}^{tor}(\beta,m)$ can be endowed with a Green function
$\Phi_{\beta,m}$ such that the resulting pair

$\widehat{\mathcal{Z}}^{tor}(\beta,m)=(\mathcal{Z}^{tor}(\beta,m),\Phi_{\beta,m})$

is an element of the first Chow group of prelog forms
$\widehat{\operatorname{CH}}^{1}(\mathcal{M}^{\Sigma},\mathcal{D}_{pre})_{\mathbb{Q}}$.
For $m=0$ and $\beta=0$, we define $\widehat{\mathcal{Z}}(0,0)$ to be any
arithmetic divisor whose is class is the dual of the hermitian line bundle
$\widehat{\mathcal{L}}=(\mathcal{L},||.||_{pet})$ endowed with the Petersson
metric $||z||^{2}=[z,\overline{z}]$.

Consider then the following generating series

$\Phi_{L}:=\sum_{\beta\in L^{\vee}/L}\sum_{m\in
Q(\beta)+\mathbb{Z}}\widehat{\mathcal{Z}}^{tor}(\beta,m)q^{m}e_{\beta}\in\mathbb{C}[L^{\vee}/L][[q^{\frac{1}{D_{L}}}]]\otimes\widehat{\operatorname{CH}}^{1}(\mathcal{M}^{\Sigma},\mathcal{D}_{pre})_{\mathbb{Q}},$

where $(e_{\beta})_{\beta\in L^{\vee}/L}$ is a basis of the
$\mathbb{C}$-vector space $\mathbb{C}[L^{\vee}/L]$, $D_{L}$ is the
discriminant of $L$, and $q=e^{2i\pi\tau}$, where $\tau\in\mathbb{H}$ is in
the the upper-half plane.

Let

$\rho_{L}:\mathrm{Mp}_{2}(\mathbb{Z})\rightarrow\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}[L^{\vee}/L])$

be the Weil representation associated to the quadratic lattice $(L,Q)$, where
$\mathrm{Mp}_{2}(\mathbb{R})$ is the metaplectic double cover if
$\mathrm{Mp}_{2}(\mathbb{R})$. For $k\in\frac{1}{2}\mathbb{Z}$, let
$\mathrm{Mod}_{k}(\rho_{L})$ denote the vector space of vector valued modular
forms of weight $k$ with respect to $\rho_{L}$. We then have the following
theorem.

######  Theorem 3.1.

The generating series $\Phi_{L}$ is the Fourier development of a vector-valued
modular forms of weight $1+\frac{b}{2}$ and representation $\rho_{L}$, i.e.,

$\Phi_{L}\in\mathrm{Mod}_{1+\frac{b}{2}}(\rho_{L})\otimes\widehat{\operatorname{CH}}^{1}(\mathcal{M}^{\Sigma},\mathcal{D}_{pre})_{\mathbb{Q}}.$

###### Proof.

Let $F\in M^{!}_{1-\frac{b}{2}}(\overline{\rho_{L}})$ be a weakly holomorphic
modular form of weight $1-\frac{b}{2}$ with respect to the complex conjugate
Weil representation of $\rho_{L}$ such that that $F$ has integral principal
part, and let $\Psi$ be the associated Borcherds product. Then by [BZ21,
Theorem 5.5], the divisor in $\mathcal{M}^{\Sigma}(\mathbb{C})$ of
$\Psi(F)_{\mathbb{C}}$ is equal to

$\sum_{\beta\in L^{\vee}/L}\sum_{m\in
Q(\beta)+\mathbb{Z}}c_{\beta}(-m)\mathcal{Z}^{tor}(\beta,m)(\mathbb{C}).$

Since Borcherds products are defined rationally by [HP20, Theorem A], we only
need to check that the divisor of the Borcherds products has the expected form
over $\mathbb{Z}$ and this will be automatic if all the special divisors and
the boundary divisors are flat. By [Per19, Theorem 4.1.5], the boundary
divisors are flat and by [HP20, Proposition 7.2.2], the special divisors are
flat over $\mathbb{Z}[\frac{1}{2}]$ and over $\mathbb{Z}$ if $b\geq 4$. For
$b=3$, one can use the algebraic version of the Borcherds embedding trick as
in [HP20, Section 9.2] to prove that no further components appear at $2$ and
hence the divisor of the Borcherds product has the correct form. Hence we
conclude by the criterion in [BZ21, Proposition 5.4]. ∎

##  4\. The main estimates and proof of the main theorems

We state in this section the local and global estimates that will allow us to
prove Theorem 1.1 and Theorem 1.3. Then we will prove the global estimates and
we postpone the proof of local estimates to the next section.

###  4.1. Number field setting

Let $X$ be K3 surface over a number field $K$. Given an embedding
$\tau:K\hookrightarrow\mathbb{C}$, let $(L,Q)$ be a maximal lattice containing
the transcendental lattice of $X^{\tau}(\mathbb{C})$. It is an even lattice of
signature $(b,2)$ whose genus is independent from $\tau$. We can assume
furthermore that $b\geq 3$, as the case $b\leq 2$ has already been treated,
see [Cha18, ST20].

Let $\mathcal{M}$ be the integral model of the GSpin Shimura variety
associated to the lattice $(L,Q)$ and, given an admissible polyhedral cone
decomposition $\Sigma$, let $\mathcal{M}^{\Sigma}$ be its toroidal
compactitfication as in Section 2. By [MP15], the K3 surface has an associated
Kuga-Satake abelian variety which we can also assume to be defined over the
number field $K$, up to taking a finite extension. Hence it defines a
$K$-point of $\mathcal{M}^{\Sigma}$. By the extension property of the integral
model, there exists $N\geq 1$ such that, up to taking a finite extension of
$K$, we have a flat morphism over $\mathbb{Z}$:

$\mathrm{Spec}(\mathcal{O}_{K}\left[\frac{1}{N}\right])\rightarrow\mathcal{M},$

and by properness, this map extends to

$\rho:\mathcal{Y}=\mathrm{Spec}(\mathcal{O}_{K})\rightarrow\mathcal{M}^{\Sigma}.$

By construction, the image of this map is not contained in any special
divisor. A prime over $N$ is said to be a prime of bad reduction and otherwise
of good reduction.

As in [SSTT19, Theorem 2.4], we will rather prove the following more general
version, which is easily seen to imply Theorem 1.1.

######  Theorem 4.1.

Let $\mathcal{Y}\in\mathcal{M}^{\Sigma}(\mathcal{O}_{K})$ with smooth
reduction outside $N$. Let $D\in\mathbb{Z}_{>0}$ be a fixed integer
represented by $(L,Q)$ and coprime to $N$. Assume that $\mathcal{Y}_{K}\in
M(K)$ is not contained in any special divisor $\mathcal{Z}(m)(K)$. Then there
are infinitely many places $\mathfrak{P}$ of $K$ of good reduction such that
$\mathcal{Y}_{\overline{\mathfrak{P}}}$ lies in the image of
$\mathcal{Z}(Dm^{2})\rightarrow\mathcal{M}$ for some $m\in\mathbb{Z}_{>0}$
coprime to $N$.

Let $\rho:\mathcal{Y}\rightarrow\mathcal{M}^{\Sigma}$ be as in the previous
theorem. We first begin by the following proposition.

######  Proposition 4.2.

There exists a refinement of the cone decomposition $\Sigma$, such that the
map $\rho:\mathcal{Y}\rightarrow\mathcal{M}^{\Sigma}$ satisfies the following
property: for any prime $\mathfrak{P}$ of bad reduction, the image of the
closed point $\\{\mathfrak{P}\\}$ under $\rho$ is contained in a stratum which
is a locally closed divisor of $\mathcal{M}^{\Sigma}$.

###### Proof.

Let $s_{\mathfrak{P}}\in\mathcal{Y}$ be the closed point $\mathfrak{P}$ of
$\mathcal{Y}$. By Equation 2.3.1, the image of $s_{\mathfrak{P}}$ lies in a
stratum indexed either by either a type II boundary component $\Upsilon$ or a
type III $(\Xi,\sigma)$ toroidal stratum representative. In the type II case,
the boundary is already a divisor and there is nothing to prove. In the type
III case, let $r$ be the dimension of the cone $\sigma$. Then we get a
morphism:

$\displaystyle\mathrm{Spf}(W(\overline{\mathbb{F}}_{p}))\rightarrow\widehat{\mathcal{M}}^{\Sigma},$
(4.1.1)

where $\widehat{\mathcal{M}}^{\Sigma}$ is the formal completion along the
boundary component defined by $(\Xi,\sigma)$. By a similar analysis to Section
2.3.2, we have an étale cover of formal Deligne–Mumford stacks:

$\widehat{\mathcal{T}}_{\Xi}(\sigma)\rightarrow\widehat{\mathcal{M}}^{\Sigma}.$

Hence the map Equation 4.1.1 lifts to a morphism

$\displaystyle\mathrm{Spf}(W(\overline{\mathbb{F}}_{p}))\rightarrow\widehat{\mathcal{T}}_{\Xi}(\sigma),$
(4.1.2)

where

$\widehat{\mathcal{T}}_{\Xi}(\sigma)=\mathrm{Spf}\left(\mathbb{Z}_{p}[q^{\alpha}|(\alpha,\sigma)=0]\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}[[q^{\alpha}|(\alpha,\sigma)>0]]\right).$

Hence this corresponds to a map

$\mathbb{Z}_{p}[[q^{\alpha}|(\alpha,C)>0]]\otimes\mathbb{Z}_{p}[q^{\alpha}|(\alpha,C)=0]\rightarrow
W(\overline{\mathbb{F}}_{p}).$

The linear form on $\Gamma_{\Xi}^{\vee}$ given by sending an element $\alpha$
to the $p$-adic valuation of the image of $q^{\alpha}$ under the above map is
represented by an element $\omega\in\Gamma_{\Xi}$ which satisfies
$(\omega.\alpha)>0$ whenever $(\alpha.\sigma)>0$, hence $\omega$ is in
$\sigma$. The cocharacter defined by $\omega$ is in fact tangent to the map
given in Equation 4.1.2. Let $\sigma^{\prime}$ in $\sigma$ be the ray defined
by $\omega$ and let $\Sigma^{\prime}$ be the new cone decomposition obtained
by refining $\Sigma$ and which contains $\sigma^{\prime}$ as a one dimensional
ray. Then $\mathcal{M}^{\Sigma^{\prime}}$ is a blow-up of
$\mathcal{M}^{\Sigma}$ and by the preceding discussion, the point
$s_{\mathfrak{P}}$ belongs to the boundary divisor parameterized by
$(\Xi,\sigma^{\prime})$. Since there are only finitely many primes of bad
reduction, then by repeating this procedure finitely many times, we get the
desired cone decomposition. ∎

We will work from now on with the toroidal compactification given by the above
proposition. For $m\geq 1$ an integer, let $\mathcal{Z}(m)$ be the closed
special divisor $\mathcal{Z}(0,m)\hookrightarrow\mathcal{M}^{\Sigma}$ and
$\widehat{\mathcal{Z}}^{tor}(m)$ the arithmetic divisor associated to
$\mathcal{Z}^{tor}(m)$ by Equation 3.1.1. The pullback via the period map
$\rho:\mathcal{Y}\rightarrow\mathcal{M}^{\Sigma}$ allows us to define the
height $h_{\widehat{\mathcal{Z}}^{tor}(m)}(\mathcal{Y})$ of $\mathcal{Y}$ with
respect to the arithmetic divisor $\widehat{\mathcal{Z}}^{tor}(m)$ as its
image under the composition:

$\displaystyle\widehat{\operatorname{CH}}^{1}(\mathcal{M}^{\Sigma},\mathcal{D}_{pre})_{\mathbb{Q}}\rightarrow$
$\displaystyle\widehat{\operatorname{CH}}^{1}(\mathcal{Y},\mathcal{D}_{pre})_{\mathbb{Q}}\xrightarrow{\widehat{\deg}}\mathbb{R},$
$\displaystyle\widehat{\mathcal{Z}}^{tor}(m)$ $\displaystyle\longrightarrow
h_{\widehat{\mathcal{Z}}^{tor}(m)}(\mathcal{Y}).$

By choice of the lattice $(L,Q)$, the arithmetic curve $\mathcal{Y}$
intersects properly the divisors $\mathcal{Z}(m)$, $\mathcal{B}^{\Xi,\omega}$
and $\mathcal{B}^{\Upsilon}$ for every $\Upsilon$ and $(\Xi,\omega)$. Hence we
have

$\displaystyle
h_{\widehat{\mathcal{Z}}^{tor}(m)}(\mathcal{Y})=\sum_{\tau:K\hookrightarrow\mathbb{C}}\Phi_{m}(\mathcal{Y}^{\tau})+\sum_{\mathfrak{P}}(\mathcal{Y}.\mathcal{Z}^{tor}(m))_{\mathfrak{P}}\log|\mathcal{O}_{K}/\mathfrak{P}|,$
(4.1.3)

where for $\tau:K\hookrightarrow\mathbb{C}$, we use $\mathcal{Y}^{\tau}$ to
denote the point in $M(\mathbb{C})$ induced by

$\operatorname{Spec}(\mathbb{C})\xrightarrow{\tau}\operatorname{Spec}({\mathcal{O}_{K}})=\mathcal{Y}\rightarrow\mathcal{M}^{\Sigma}.$

We have

$\displaystyle(\mathcal{Y}.\mathcal{Z}^{tor}(m))_{\mathfrak{P}}=(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}+\sum_{\Upsilon}\mu_{\Upsilon}(m)(\mathcal{Y}.\mathcal{B}^{\Upsilon})_{\mathfrak{P}}+\sum_{(\Xi,\omega)}\mu_{\Xi,\omega}(m)\cdot(\mathcal{Y}.\mathcal{B}^{\Xi,\omega})_{\mathfrak{P}}.$
(4.1.4)

Let us denote by
$\mathcal{O}_{\mathcal{Y}\times_{\mathcal{M}^{\Sigma}}\mathcal{Z}(m),v}$ the
étale local ring of $\mathcal{Y}\times_{\mathcal{M}}\mathcal{Z}(m)$ at $v$,
then

$\displaystyle(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=\sum_{v\in\mathcal{Y}\times_{\mathcal{M}}\mathcal{Z}(m)(\overline{\mathbb{F}}_{\mathfrak{P}})}\operatorname{length}(\mathcal{O}_{\mathcal{Y}\times_{\mathcal{M}^{\Sigma}}\mathcal{Z}(m),v}),$
(4.1.5)

where $\mathbb{F}_{\mathfrak{P}}$ denotes the residue field of $\mathfrak{P}$.

Let

$(\mathcal{Y}.\mathcal{Z}(m))=\sum_{\mathfrak{P}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}\log|\mathcal{O}_{K}/\mathfrak{P}|.$

The first new contribution of this paper is to prove the following estimate
which results from Borcherds modularity and ad hoc bounds on the
multiplicities $\mu_{\Upsilon}(m)$ and $\mu_{\Xi,\omega}(m)$.

######  Proposition 4.3.

As $m\rightarrow\infty$, we have

$(\mathcal{Y}.\mathcal{Z}(m))+\sum_{\tau:K\hookrightarrow\mathbb{C}}\Phi_{m}(\mathcal{Y}^{\tau})=O(m^{\frac{b}{2}}).$

As a corollary, we get the following bound, which is referred to as the
diophantine bound in [SSTT19, Equation (5.2)].

######  Corollary 4.4.

For any finite place $\mathfrak{P}$, we have the following bound.

$(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=O(m^{\frac{b}{2}}\log
m),\quad\Phi_{m}(\mathcal{Y}^{\tau})=O(m^{\frac{b}{2}}\log m).$

For our next estimate, we recall the notion of asymptotic density from
[SSTT19]: for a subset $S\subset\mathbb{Z}_{>0}$, the _logarithmic asymptotic
density_ of $S$ is defined to be

$\displaystyle\limsup_{X\rightarrow\infty}\frac{\log|S_{X}|}{\log X},$

where $S_{X}:=\\{a\in S\mid X\leq a<2X\\}$.

Recall from Theorems 5.7 and 6.1 in [SSTT19] that we have the following
estimate:

######  Proposition 4.5.

There exists a subset $S_{bad}\subset\mathbb{Z}_{>0}$ of zero logarithmic
asymptotic density such that

$\sum_{\tau:K\hookrightarrow\mathbb{C}}\Phi_{m}(\mathcal{Y}^{\tau})=c(m)\log(m)+o(m^{\frac{b}{2}}\log(m)),$

where $-c(m)\asymp m^{\frac{b}{2}}$ and is defined in [SSTT19, Section 3.3].

For a prime $\mathfrak{P}$ of good reduction, i.e., where the intersection of
$\mathcal{Y}$ and $\mathcal{Z}(m)$ above $\mathfrak{P}$ is supported in
$\mathcal{M}$, we have the following estimate which follows easily from
[SSTT19, Theorem 7.1].

######  Proposition 4.6.

Let $\mathfrak{P}$ be a finite place of good reduction. Let
$D\in\mathbb{Z}_{\geq 1}$ coprime to $N$. For $X\in\mathbb{Z}_{>0}$, let
$S_{D,X}$ denote the set

$\\{m\in\mathbb{Z}_{>0}\mid X\leq
m<2X,\,\frac{m}{D}\in\mathbb{Z}\cap(\mathbb{Q}^{\times})^{2},\,(m,N)=1\\}.$

Then we have

$\sum_{m\in
S_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=o(X^{\frac{b+1}{2}}\log
X).$

Finally, for a prime $\mathfrak{P}$ of bad reduction, we prove the following
proposition which is the second new contribution of this paper.

######  Proposition 4.7.

Let $\mathfrak{P}$ a finite place of bad reduction. Let $D\in\mathbb{Z}_{\geq
1}$ coprime to $N$. For $X\in\mathbb{Z}_{>0}$, let $S_{D,X}$ be the set
defined in the previous proposition. Then we have

$\sum_{m\in
S_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=o(X^{\frac{b+1}{2}}\log
X).$

###  4.2. Function field setting

We assume in this section that the lattice $(L,Q)$ is self-dual at $p$. Then
the Shimura variety $\mathcal{M}$ has smooth reduction at $p$ and we denote
its reduction by $\mathcal{M}_{\mathbb{F}_{p}}$. Given an admissible cone
decomposition $\Sigma$, we denote by $\mathcal{M}_{\mathbb{F}_{p}}^{\Sigma}$
the reduction of the toroidal compactification $\mathcal{M}^{\Sigma}$. We
first give a new formulation of Theorem 1.3, see Theorem 4.8, then we will
give the main estimates that will allow us to prove the latter.

Let $\mathcal{X}\rightarrow\mathscr{S}$ be a generically ordinary non-
isotrivial family of K3 surfaces over a smooth curve $\mathscr{S}$ over
$\overline{\mathbb{F}}_{p}$. The quadratic lattice $(L,Q)$ in this case
corresponds to a maximal quadratic lattice orthogonal to the generic geometric
Picard group in the K3 lattice. Hence $(L,Q)$ has discriminant coprime to $p$
by assumption and we get a period map by [MP15, section 4]

$\rho:\mathscr{S}\rightarrow\mathcal{M}_{\mathbb{F}_{p}},$

which is a finite map and the image of the generic point is in the ordinary
locus. The locus in $\mathscr{S}$ where the Picard rank jumps corresponds then
exactly to the union over $m\geq 1$ of the intersections
$\mathscr{S}\cap\mathcal{Z}(m)_{\mathbb{F}_{p}}$. Hence Theorem 1.3 follows
from the following theorem.

######  Theorem 4.8.

Let $\mathscr{S}\rightarrow\mathcal{M}_{\mathbb{F}_{p}}$ be a finite map with
generically ordinary image and not contained in any special divisor. Then
there exists infinitely many closed points in $\mathscr{S}$ in the union of
$\mathcal{Z}(m)_{\mathbb{F}_{p}}$ for integers $m$ coprime with $p$.

Let $\mathscr{S}$ be a smooth curve as in the theorem above. By properness, we
can extend the map

$\rho:\overline{\mathscr{S}}\rightarrow\mathcal{M}_{\mathbb{F}_{p}}^{\Sigma},$

where $\overline{\mathscr{S}}$ is the smooth compactification of
$\mathscr{S}$. We have the following proposition whose proof is similar to
Proposition 4.2 and hence we omit it.

######  Proposition 4.9.

There exists a refinement of the cone decomposition $\Sigma$ such that the
image of $\overline{\mathscr{S}}$ in $\mathcal{M}_{\overline{\mathbb{F}}_{p}}$
intersects the boundary only in strata corresponding to locally closed
divisors.

Let $\Sigma$ be a polyhedral cone decomposition which satisfies the conditions
of the previous proposition. By abuse of notations, if
$D\subset\mathcal{M}^{\Sigma}_{\mathbb{F}_{p}}$ is a Cartier divisor, we write

$(D.\overline{\mathscr{S}})=\deg_{\overline{\mathscr{S}}}\rho^{*}D.$

We have then the following global estimate.

######  Proposition 4.10.

As $m\rightarrow\infty$, we have

$(\mathcal{Z}(m)_{\overline{\mathbb{F}}_{p}}.\overline{\mathscr{S}})=|c(m)|(\overline{\mathscr{S}}.\mathcal{L}_{\mathbb{F}_{p}})+o(m^{\frac{b}{2}}).$

For any integer $m$, we can decompose:

$(\mathcal{Z}(m)_{\overline{\mathbb{F}}_{p}}.\overline{\mathscr{S}})=\sum_{P\in\overline{\mathbb{F}}_{p}}m_{P}(\mathcal{Z}(m)_{\mathbb{F}_{p}},\overline{\mathscr{S}}),$

where $m_{P}(\mathcal{Z}(m)_{\mathbb{F}_{p}},\overline{\mathscr{S}})$ is the
multiplicity of intersection at $P$. Our next goal is to estimate in average
these local multiplicities and we start by the good reduction case already
treated in [MST22, Prop. 7.11, Th. 7.18 ].

Let $S$ be as in [MST22, Section 7.1], i.e., a set of integers of positive
density such that every $m\in S$ is coprime to $p$ and is representable by the
quadratic lattice $(L,Q)$.

For $P\in(\mathscr{S}\cap\mathcal{M})(\mathbb{F}_{p})$, we define as in
[MST22, Definition 7.6]

$g_{P}(m)=\frac{h_{p}}{p-1}|c(m)|,$

where $h_{p}$ is the order of vanishing of the Hasse invariant at $P$, see
loc. cit. The following proposition is the combination of Proposition 7.11 and
Theorem 7.18 from [MST22].

######  Proposition 4.11.

Let $P\in\mathscr{S}(\overline{\mathbb{F}}_{p})$. Then:

  1. (1)

If $P$ is not supersingular then

$\sum_{m\in
S_{X}}m_{P}(\mathcal{Z}(m)_{\mathbb{F}_{p}}.\mathscr{S})=O(X^{\frac{b}{2}}\log
X).$

  2. (2)

There exists an absolute constant $0<\alpha<1$ such that for any supersingular
point $P$ we have

$\sum_{m\in
S_{X}}m_{P}(\mathcal{Z}(m)_{\mathbb{F}_{p}}.\mathscr{S})=\alpha\sum_{m\in
S_{X}}g_{p}(m)+O(X^{\frac{b+1}{2}}).$

Our new contribution in this setting is the following theorem which gives an
estimate on intersection multiplicities at points where $\overline{S}$
intersects the boundary of $\mathcal{M}^{\Sigma}_{p}$.

######  Proposition 4.12.

Let $P\in\overline{\mathscr{S}}(\overline{\mathbb{F}}_{p})$ a point mapping to
the boundary of $\mathcal{M}^{\Sigma}_{\mathbb{F}_{p}}$. Then we have the
following estimate:

$\sum_{m\in
S_{X}}m_{P}(\mathcal{Z}(m)_{\mathbb{F}_{p}}.\overline{\mathscr{S}})=O(X^{\frac{b}{2}}\log
X).$

###  4.3. Proof of the main theorems

Assuming the estimates in the previous section we now indicate how to prove
Theorem 1.1 and Theorem 1.3.

###### Proof of Theorem 1.1.

It is enough to prove Theorem 4.1 in a similar way to [SSTT19, Section 8]. For
convenience of the reader, we will sketch the proof. Assume for the sake of
contradiction that there are only finitely many primes of good reduction such
that $\mathcal{Y}$ intersects a special divisor of the form
$\mathcal{Z}(Dm^{2})$ where $Dm^{2}$ is coprime with $N$ and is represented by
$(L,Q)$. By Proposition 4.3 and Proposition 4.5, there exists a subset
$S_{bad}\subset\mathbb{Z}_{>0}$ of logarithmic asymptotic density zero such
that:

$(\mathcal{Y}.\mathcal{Z}(m))=-c(m)\log(m)+o(m^{\frac{b}{2}}\log(m))\asymp
m^{\frac{b}{2}}\log(m).$

Let $S^{good}_{D,X}=\\{m\in S_{D,X},m\notin S_{bad},\,(m,N)=1\\}$, then one
can easily check that $|S^{good}_{D,X}|\asymp X^{\frac{1}{2}}$ and $c(m)\gg
X^{\frac{b}{2}}\log X$ for $m\in S^{good}_{D,X}$. Hence we get

$\displaystyle\sum_{m\in S^{good}_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))\asymp
X^{\frac{b+1}{2}}\log X.$ (4.3.1)

On the other hand, by Proposition 4.5 and Proposition 4.7, we get by summing
over the finitely many places where either $\mathcal{Y}$ intersects a
$\mathcal{Z}(Dm^{2})$ or which are of bad reduction

$\sum_{m\in
S^{good}_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))=o(X^{\frac{b+1}{2}}\log X),$

which contradicts Equation 4.3.1. ∎

###### Proof Theorem 1.3.

The proof is similar: assume that there are only finitely many points in the
union $\left(\cup_{m,m\wedge
p=1}\mathcal{Z}(m)\cap\mathscr{S}\right)(\overline{\mathbb{F}}_{p})$ and let
$S$ be a set as in Section 4.2. Then by Proposition 4.10, we have

$\sum_{m\in
S_{X}}(\mathcal{Z}(m)_{\mathbb{F}_{p}}.\overline{\mathscr{S}})=\sum_{m\in
S_{X}}|c(m)|(\overline{\mathscr{S}}.\mathcal{L}_{\mathbb{F}_{p}})+o(X^{b/2+1}).$

On the other hand, by Proposition 4.11 and Proposition 4.12 we have

$\displaystyle\sum_{m\in
S_{X}}(\mathcal{Z}(m)_{\mathbb{F}_{p}}.\overline{\mathscr{S}})$
$\displaystyle=\sum_{m\in S_{X}}\sum_{P\in\left(\cup_{m,m\wedge
p=1}\mathcal{Z}(m)\cap\mathscr{S}\right)(\overline{\mathbb{F}}_{p})}m_{P}(\mathcal{Z}(m)_{\mathbb{F}_{p}}.\overline{\mathscr{S}})$
$\displaystyle=\alpha\sum_{m\in S_{X}}|g_{P}(m)|+O(X^{\frac{b+1}{2}}).$
$\displaystyle\leq\alpha\sum_{m\in
S_{X}}|c(m)|(\overline{\mathscr{S}}.\mathcal{L}_{\mathbb{F}_{p}})+O(X^{\frac{b+1}{2}}),$

where the last equality results from the fact that the Hasse invariant is a
section of $\mathcal{L}_{\mathbb{F}_{p}}^{\otimes p-1}$. These two estimates
contradict each other, hence the result. ∎

###  4.4. Global estimate

We prove in this section simultaneously Propositions 4.3 and 4.10.

By Theorem 3.1, the following generating series

$\sum_{\beta\in L^{\vee}/L}\sum_{m\in
Q(\beta)+\mathbb{Z}}h_{\widehat{\mathcal{Z}}^{tor}(\beta,m)}(\mathcal{Y})q^{m}e_{\beta},$

and

$\sum_{\beta\in L^{\vee}/L}\sum_{m\in
Q(\beta)+\mathbb{Z}}(\mathcal{Z}^{tor}(\beta,m)_{\mathbb{F}_{p}}.\overline{\mathscr{S}})q^{m}e_{\beta}$

are elements of $\mathrm{Mod}_{1+\frac{b}{2}}(\rho_{L})$. Classical estimates
on the growth of coefficients of modular forms imply that (see [Tay20, Example
2.3] for more details):

$h_{\widehat{\mathcal{Z}}^{tor}(m)}(\mathcal{Y})=O(m^{\frac{b}{2}})$

and

$(\mathcal{Z}(m)^{tor}_{\mathbb{F}_{p}}.\overline{\mathscr{S}})=|c(m)|(\overline{\mathscr{S}}.\mathcal{L}_{\mathbb{F}_{p}})+o(m^{\frac{b}{2}}).$

By Equation 4.1.3 and Equation 4.1.4, we can write

$\displaystyle(\mathcal{Y}.\mathcal{Z}(m))+\sum_{\tau:K\hookrightarrow\mathbb{C}}\Phi_{m}(\mathcal{Y}^{\tau})=h_{\widehat{\mathcal{Z}}^{tor}(m)}(\mathcal{Y})-\sum_{\Upsilon}\mu_{\Upsilon}(m)(\mathcal{Y}.\mathcal{B}^{\Upsilon})_{\mathfrak{P}}\log|\mathcal{O}_{K}/\mathfrak{P}|$
(4.4.1)
$\displaystyle-\sum_{\Xi}\mu_{\Xi,\sigma}(m)\cdot(\mathcal{Y}.\mathcal{B}^{\Xi,\sigma})_{\mathfrak{P}}\log|\mathcal{O}_{K}/\mathfrak{P}|$
(4.4.2)

and similarly, we can write:

$\displaystyle(\overline{\mathscr{S}}.\mathcal{Z}(m)_{\mathbb{F}_{p}})=(\mathcal{Z}^{tor}(m)_{\mathbb{F}_{p}}.\overline{\mathscr{S}})-\sum_{\Upsilon}\mu_{\Upsilon}(m)(\overline{\mathscr{S}}.\mathcal{B}^{\Upsilon,\mathbb{F}_{p}})$
(4.4.3)
$\displaystyle-\sum_{\Xi,\sigma}\mu_{\Xi,\omega}(m)\cdot(\overline{\mathscr{S}}.\mathcal{B}^{\Xi,\sigma}_{\mathbb{F}_{p}}).$
(4.4.4)

Hence we only have to bound the growth of the multiplicities
$\mu_{\Upsilon}(m)$ and $\mu_{\Xi,\omega}(m)$.555$\omega$ is the unique
integral generator of $\sigma$. This is given by the following lemma.

######  Proposition 4.13.

As $m\rightarrow\infty$, we have the following estimates.

  1. (1)

For any type II cusp label representative $\Upsilon$, we have

$\mu_{\Upsilon}(m)\ll_{\epsilon}m^{\frac{b}{2}-1+\epsilon}.$

  2. (2)

For any type III toroidal stratum representative $(\Xi,\sigma)$ such that
$\sigma$ is a ray, we have

$\mu_{\Xi,\omega}(m)\ll_{\epsilon}m^{\frac{b-1}{2}+\epsilon}.$

This proposition will be proved in the following two sections.

###  4.5. Estimates on type II multiplicities

The goal of this section is to prove the type II estimate in Proposition 4.13.
First we recall some notations associated to isotropic planes introduced in
[BZ21, Section 3.2].

Let $\Upsilon=(P,\mathcal{D}^{\circ},h)$ be a cusp label representative
corresponding to a boundary component of type II. Recall from Section 2.2.1
that $P$ is the stabilizer of an isotropic plane $J_{\mathbb{Q}}$ and
$J=J_{\mathbb{Q}}\cap h.L$ is a primitive isotropic plane of $h.L\cap
L_{\mathbb{Q}}$.

To simplify the notations, assume that $h.L\cap L_{\mathbb{Q}}=L$, the reader
may otherwise replace $L$ by $L_{h}=h.L\cap L_{\mathbb{Q}}$ in what follows.
Define then:

$J_{L^{\vee}}=J_{\mathbb{R}}\cap L^{\vee},\quad J^{\bot}_{L}=J^{\bot}\cap
L,\quad J^{\bot}_{L^{\vee}}=J^{\bot}\cap L^{\vee},\quad\textrm{and}\quad
D=J^{\bot}_{L}/J.$

The lattice $D$ is positive definite lattice of rank $b-2$. Its dual lattice
can be described as

$D^{\vee}=J^{\bot}_{L^{\vee}}/J_{L^{\vee}}.$

and the discriminant lattice is given by:

$\Delta_{D}=D^{\vee}/D=J^{\bot}_{L^{\vee}}/(J^{\bot}_{L}+J_{L^{\vee}}=L_{J}^{\vee}/(L+J_{L^{\vee}}),$

where $L^{\vee}_{J}$ is the subgroup of $L^{\vee}$

$L+J^{\bot}_{L^{\vee}}=\\{\mu\in L^{\vee}|\exists\nu\in L\,\textrm{such
that}\,(\mu,\lambda)=(\nu,\lambda)\,\forall\lambda\in J\\}.$

Let $\Theta_{D}$ denote the vector-valued Theta function associated to $D$
defined by

$\Theta_{D}(\tau)=\sum_{\beta\in
D^{\vee}}q^{Q(\beta)}e_{\beta+D}\in\mathbb{C}[\Delta_{D}][[q^{\frac{1}{|\Delta_{D}|}}]].$

It is an element of $M_{\frac{b}{2}-1}(\rho_{D})$, which is the space of
vector-valued modular forms of weight $\frac{b}{2}-1$ with respect to the Weil
representation $\rho_{D}$ associated to the positive definite lattice $(D,Q)$.
We can can also write

$\displaystyle\Theta_{D}(\tau)=\sum_{\beta\in D^{\vee}/D}\sum_{m\geq
0}c(D,\beta,m)q^{m}e_{\beta},$

where for $\beta\in D^{\vee}/D$, $m\in Q(\beta)+\mathbb{Z}$, $m\geq 0$, we
have

$c(D,\beta,m)=|\\{\lambda\in\beta+D,Q(\lambda)=m\\}|.$

Following Bruinier–Zemel’s notations [BZ21, Section 4.4], define

$\displaystyle\uparrow^{L}_{D}(\Theta_{D})(\tau)$
$\displaystyle=\sum_{\beta\in
J^{\bot}_{L^{\vee}}/J}q^{\frac{Q(\beta)}{2}}e_{\beta+L}$
$\displaystyle=\sum_{\beta\in L^{\vee}/L}\sum_{m\in
Q(\beta)+L}c(D,\beta,m)q^{m}e_{\beta}\in M_{\frac{b}{2}-1}(\rho_{L}),$

where $c(D,\beta,m)=0$ if $\beta\notin J^{\bot}_{L^{\vee}}/J^{\bot}$ or
$m\notin Q(\beta)+\mathbb{Z}$, and otherwise
$c(D,\beta,m)=c(D,\overline{\beta},m)$ where $\overline{\beta}$ is the image
of $\beta$ under the reduction map $J^{\bot}_{L\vee}\rightarrow D^{\vee}/D$.

In particular, we have

$\displaystyle q\frac{d}{dq}\uparrow^{L}_{D}(\Theta_{D})(\tau)=\sum_{\beta\in
L^{\vee}/L}\sum_{m\in Q(\beta)+L}mc(D,\beta,m)q^{m}e_{\beta},$

which is a quasi-modular form in the sense of [IRR14, Definition 1].

Then by [BZ21, Definition 4.18, Proposition 4.21 4.15], we can define:

$\displaystyle\mu_{\Upsilon}(m)=\frac{1}{b-2}\mathrm{CT}\left(\langle
q\frac{d}{dq}\uparrow^{L}_{D}(\Theta_{D}),F^{+}_{m}\rangle_{L}\right),$
(4.5.1)

where $F^{+}_{m}$ is the holomorphic part of the Harmonic Mass form $F_{m,0}$
from [BZ21, Proposition 4.2]. A direct computation shows then (see also the
second formula in [Bru02, Theorem 2.14])

$\mu_{\Upsilon}(m)=\frac{2}{b-2}mc(D,0,m).$

Classical estimates on coefficients of modular forms, see for example [Sar90,
Prop.1.5.5], show that

$\displaystyle|c(D,\beta,m)|\ll_{\epsilon}m^{\frac{b}{2}-2+\epsilon}$ (4.5.2)

for all $\epsilon>0$. Hence we get that

$|\mu_{\Upsilon}(m)|\ll_{\epsilon}m^{\frac{b}{2}-1+\epsilon},$

which proves the first part of Proposition 4.13.

###  4.6. Estimates on type III multiplicities

In this section, we prove the estimates on the type III multiplicities in
Proposition 4.13.

Let $(\Xi,\sigma)$ be a toroidal stratum representative of type III such that
$\sigma$ is a ray. Keeping the notations from Section 2.2.2, let
$I_{\mathbb{Q}}$ be the isotropic line of $L_{\mathbb{Q}}$ whose stabilizer is
the parabolic subgroup attached to $\Xi$ and let $I=I_{\mathbb{Q}}\cap h.L$.
To simplify notations, we assume that $h.L=L$, the reader may notice that this
is harmless, up to replacing $L$ by $h.L$ in what follows.

The line $I$ is an isotropic line of $L$ and the lattice $K_{I}=I^{\bot}/I$ is
Lorentzian. Let $C_{\mathbb{R}}$ be the cone of negative elements of the
Lorentzian space $K_{I,\mathbb{R}}$ and let $C=C_{\mathbb{R}}\cap K$. As is
explained in Section 2.2.2, the ray $\omega$ is generated by an element
$\omega\in K_{I}\cap C$ which is primitive and such that $Q(\omega)=-N$.
Following [BZ21, Definition 4.18], we define

$\displaystyle\mu_{\Xi,\omega}(m)=\frac{\sqrt{N}}{8\sqrt{2}\pi}\Phi^{K}_{m}\left(\frac{\omega}{\sqrt{N}}\right).$
(4.6.1)

Let $v=\frac{\omega}{\sqrt{N}}$. By [Bru02, Proposition 2.11] and [Bru02,
Theorem 2.14], we have

$\displaystyle\Phi^{K}_{m}(v)$
$\displaystyle=\Phi^{K}_{m}(v,\frac{1}{2}+\frac{b}{4})$
$\displaystyle=\frac{2\Gamma(\frac{b-1}{2})(4\pi
m)^{\frac{b}{2}}}{1+\frac{b}{2}}\sum_{\underset{Q(\lambda)=m}{\lambda\in
K_{I}}}\frac{F(\frac{b-1}{2},1,1+\frac{b}{2};\frac{m}{q(Q(\lambda_{v^{\bot}})})}{(4\pi|Q(\lambda_{v^{\bot}})^{\frac{b-1}{2}}|)},$

where $F(a,b,c;z)$ is the usual Gauss hypergeometric function given by:

$F(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!},$

and $(a)_{n}=\Gamma(a+n)/\Gamma(a)$. Recall that the above series has $1$ as a
radius of convergence and converges absolutely in the unit circle $|z|=1$ if
$\mathcal{R}(c-a-b)>0$. In our situation, the latter quantity is equal to
$1+b/2-1-\frac{b-1}{2}=1/2>0$. Hence the series
$F(\frac{b-1}{2},1,1+\frac{b}{2};z)$ is globally bounded over the unit disc.
For $\lambda\in K$ such that $Q(\lambda)=m$, we have
$m=Q(\lambda_{v})+Q(\lambda_{v^{\bot}})$ and $Q(\lambda_{v})\leq 0$, hence
$0<m\leq Q(\lambda_{v^{\bot}})$. Hence we get

$\displaystyle|\Phi^{K}_{m}(v)|$
$\displaystyle\ll\sqrt{m}.\sum_{\underset{Q(\lambda)=1}{\sqrt{m}\lambda\in
K_{I}}}\frac{1}{Q(\lambda_{v^{\bot}})^{\frac{b-1}{2}}}$
$\displaystyle\ll\sqrt{m}\sum_{N\geq
1}\sum_{\underset{Q(\lambda)=1}{\underset{\sqrt{m}\lambda\in
K_{I}}{Q(\lambda_{v}^{\bot})\in[N,N+1[}}}\frac{1}{N^{\frac{b-1}{2}}}.$

By Proposition 4.14 below, we have

$|\\{\lambda\in K_{I,\mathbb{R}},Q(\lambda)=1,\sqrt{m}\lambda\in
K,Q(\lambda_{v}^{\bot})\in[N,N+1[\\}|\ll_{\epsilon}m^{\frac{b}{2}-1+\epsilon}N^{\frac{b}{2}-2}.$

Hence

$\displaystyle|\Phi^{K}_{m}(v)|$
$\displaystyle\ll_{\epsilon}m^{\frac{b-1}{2}+\epsilon}\sum_{N\geq
1}\frac{N^{\frac{b}{2}-2}}{{N^{\frac{b-1}{2}}}}$
$\displaystyle\ll_{\epsilon}m^{\frac{b-1}{2}+\epsilon}\sum_{N\geq
1}\frac{1}{N^{3/2}}.$ $\displaystyle\ll_{\epsilon}m^{\frac{b-1}{2}+\epsilon}.$

which proves the second part of proposition 4.13.

######  Proposition 4.14.

Let $m\geq 1$ be an integer and $X>0$ a positive real number. Then

$|\\{\lambda\in K_{I,\mathbb{R}},Q(\lambda)=1,\sqrt{m}\lambda\in
K_{I},Q(\lambda_{v}^{\bot})\in[N,N+1[\\}|\ll_{\epsilon}m^{\frac{b}{2}-1+\epsilon}N^{\frac{b}{2}-2}$

###### Proof.

Recall that $(K_{I},Q)$ is a quadratic lattice of signature $(b-1,1)$ and we
have a canonical measure $\mu_{\infty}$ on the quadric $K_{1}:=\\{x\in
K_{I,\mathbb{R}}|Q(x)=1\\}$ defined as follows: for $W$ an open subset of
$K_{\mathbb{R}}$, let

$\mu_{\infty}(W\cap K_{1})=\lim_{\epsilon\rightarrow
0}\frac{\mathrm{Leb}\left(\\{x\in
W,\,|Q(x)-1|<\epsilon\\}\right)}{2\epsilon}.$

Here $\mathrm{Leb}$ is the Lebesgue measure on $K_{\mathbb{R}}$ for which the
lattice $K$ is of covolume $1$. One can then prove that (see for example the
proof of [SSTT19, Corollary 4.12]):

$\mu_{\infty}(\\{\lambda\in K_{1},Q(\lambda_{v^{\bot}})\in[X,X+1[\\})\ll
X^{\frac{b}{2}-2}.$

On the other hand, by the equidistribution of integral points in quadrics, see
[EO06, Duk88], 666Or the circle method. we have:

$|\\{\lambda\in K_{1},\sqrt{m}\lambda\in
K,Q(\lambda_{v}^{\bot})\in[N,N+1[\\}|\ll_{\epsilon}m^{\frac{b}{2}-1+\epsilon}\mu_{\infty}(\\{\lambda\in
K_{1},Q(\lambda_{v^{\bot}})\in[X,X+1[\\}),$

which yields the desired result. ∎

##  5\. Bounding the contribution from bad reduction places

In this section we prove Proposition 4.7 and Proposition 4.12. Let
$\mathcal{M}^{\Sigma}$ be as before the toroidal compactification of the GSpin
Shimura variety associated to a quadratic lattice $(L,Q)$ and a $K$-admissible
polyhedral cone decomposition $\Sigma$. The lattice $(L,Q)$ is assumed to be
maximal in the number field case and moreover self-dual at $p$ in the function
field case.

###  5.1. Bad reduction in the number field setting

In this section, we prove Proposition 4.7. We assume hence that the lattice
$(L,Q)$ is maximal and that the polyhedral cone decomposition $\Sigma$ is
chosen in such way that Proposition 4.2 is satisfied.

By the choice of the cone decomposition $\Sigma$, the intersection points of
$\mathcal{Y}$ and $\mathcal{M}^{\Sigma}$ lie either in a boundary divisor of
type II or a boundary divisor of type III associated to a toroidal stratum
representative $(\Xi,\sigma)$ of type III where $\sigma$ is a ray.

Let $\mathfrak{P}$ be a prime of bad reduction, i.e., where $\mathcal{Y}$
intersects the boundary of $\mathcal{M}^{\Sigma}$. Let $K_{\mathfrak{P}}$ be
the completion at $\mathfrak{P}$ of the number field $K$ and
$v_{\mathfrak{P}}$ its normalized valuation. Let $k_{\mathfrak{P}}$ be the
residue field of $\mathfrak{P}$ and $\overline{k}_{\mathfrak{P}}$ an algebraic
closure.

####  5.1.1. Type II degeneration

Assume in this section that the boundary point lies in
$\mathcal{B}^{\Upsilon}_{\mathbb{F}_{p}}$ where $\Upsilon$ is a cusp label
representative of type II.

Let $J$ be the primitive isotropic plane associated to $\Upsilon$ and let
$D=J_{L}^{\bot}/J$, see Section 4.5 for notations.

Recall from Equation 2.2.5 and Equation 2.3.3 that the completion of
$\mathcal{M}^{\Sigma}$ along the boundary divisor $\mathcal{B}^{\Upsilon}$
fits into the following commutative diagram

$\textstyle{\widehat{\mathcal{M}}_{\Upsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{=}$$\textstyle{\widehat{\mathcal{M}^{\Sigma}}}$$\textstyle{\widehat{\mathcal{M}}_{\Upsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Spf}(\mathbb{Z}_{p}[[X]])}$$\textstyle{\overline{\mathcal{M}}_{\Upsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D\otimes\mathcal{E}}$$\textstyle{\mathcal{M}^{h}_{\Upsilon}}$

where the map $\pi$ is an étale map of formal Deligne–Mumford stacks.

The formal completion of $\mathcal{Y}$ along $\mathfrak{P}$ induces a map

$\mathrm{Spf}(\mathcal{O}_{K_{\mathfrak{P}}})\rightarrow\widehat{\mathcal{M}^{\Sigma}},$

which lifts by étaleness of $\pi$ to a map

$\mathrm{Spf}(\mathcal{O}_{K_{\mathfrak{P}}})\rightarrow\widehat{\mathcal{M}}_{\Upsilon}.$

Denoting by $x$ the image of the closed point $s_{\mathfrak{P}}$, then we get
a map of local rings

$\Psi:\widehat{\mathcal{O}}_{\widehat{\mathcal{M}}_{\Upsilon},x}\rightarrow\mathcal{O}_{K_{\mathfrak{P}}}.$

Let $m\geq 1$ be an integer coprime to $N$. By Proposition 2.4, the formal
completion of the divisor $\mathcal{Z}(m)$ is described as the union over
$\lambda\in D$ with $Q(\lambda)=m$, of the vanishing set of the ideals
$\mathbb{Z}_{p}[X]]\otimes\widehat{I}_{\lambda}$. If $f_{\lambda}$ is a
generator of $\widehat{I}_{\lambda}$777Recall that $\mathcal{Z}(m)$ is
Cartier., then the multiplicity of intersection of the branch parameterized by
$\lambda$ at $\mathfrak{P}$ is equal to

$v(\lambda)=v_{\mathfrak{P}}(\Psi(f_{\lambda})).$

Hence the multiplicity of intersection of $\mathcal{Y}$ and $\mathcal{Z}(m)$
at $\mathfrak{P}$ is given by:

$(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=\frac{1}{d}\sum_{\underset{Q(\lambda)=m}{\lambda\in
D}}v(\lambda),$

where $d$ is the degree of $\pi$ at $\rho(s_{\mathfrak{P}})$.

For an integer $n$, define the set:

$L_{n}=\\{\lambda\in D|v(\lambda)\geq n\\},$

and notice that $(L_{n})$ is a decreasing chain of sets. It follows then

$\displaystyle(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}$
$\displaystyle\leq\sum_{\underset{Q(\lambda)=m}{\lambda\in D}}v(\lambda)$
(5.1.1) $\displaystyle\leq$ $\displaystyle\sum_{n\geq 1}|\\{\lambda\in
L_{n}|Q(\lambda)=m\\}|.$ (5.1.2)

The Proposition below should be compared to what happens in the good reduction
case in [SSTT19, Section 7]. For a definition of the successive minima used,
we refer to [EK95, Definition 2.2].

######  Proposition 5.1.

The sequence $(L_{n},Q)_{n}$ is a decreasing sequence of positive definite
lattices which all have the same rank $r\leq b-2$. Moreover, the following
holds:

  1. (1)

$\cap_{n}L_{n}=\\{0\\}$.

  2. (2)

For every $n\geq 1$, $pL_{n}\subseteq L_{n+1}$.

  3. (3)

For $1\leq r\leq b-2$, let $\mu_{i}(L_{n})$ be the $i^{th}$ successive minima
of $L_{n}$ and let $a_{i}(L_{n})=\prod_{1\leq k\leq i}\mu_{i}(L_{n})$. Then we
have

$a_{i}(L_{n})\gg_{\epsilon}n^{\frac{i}{b+\epsilon}}.$

###### Proof.

Let $\lambda,\lambda^{\prime}\in L_{n}$. From Equation 2.4.1, we see that
$\mathrm{ker}(p_{\lambda})\cap\mathrm{ker}(p_{\lambda^{\prime}})$ and thus

$\widehat{I}_{\lambda+\lambda}\subset\widehat{I_{\lambda}}+\widehat{I_{\lambda^{\prime}}}.$

It follows that

$v(\lambda+\lambda^{\prime})\geq\min\\{v(\lambda),v(\lambda^{\prime})\\}\geq
n.$

We conclude that $L_{n}\subseteq D$ is a subgroup and $(L_{n},Q)$ is obviously
positive definite. Moreover, since the curve $\mathcal{Y}$ is not contained in
any special divisor, $(1)$ follows immediately.

For (2), let $\lambda\in L_{n}$ with $v(\lambda)\geq n\geq 1$. Then
$\widehat{I}_{p\lambda}$ is the ideal defining the kernel of the composition

$D\otimes\widehat{\mathcal{E}}\rightarrow\widehat{\mathcal{E}}\rightarrow\widehat{\mathcal{E}},$

over $\mathrm{Spf}(\widehat{\mathcal{O}}_{\mathcal{M}^{h},z})$.

Since the multiplication by $p$ map is ramified at $0$ with ramification
degree equal to $p$, we conclude that

$\displaystyle v(p\lambda)$ $\displaystyle\geq pv(\lambda)$ $\displaystyle\geq
n+1.$

This also proves that the lattices $L_{n}$ have the same rank.

For (3), let $n\geq 1$ and let $w_{0}$ be a vector in $L_{n}$ such that
$Q(w_{0})=\mu_{1}(L_{n})^{2}$. By choosing $m_{0}=\mu_{1}(L_{n})^{2}$, the
height bound 4.4 implies

$n\leq(\mathcal{Y}.\mathcal{Z}(m_{0}))_{\mathfrak{P}}\ll_{\epsilon}m^{\frac{b+\epsilon}{2}}.$

Hence $\mu_{1}(L_{n})\gg_{\epsilon}n^{\frac{1}{b+\epsilon}}$. Since
$a_{i}(n)\geq\mu_{1}(n)^{r}$, this concludes the proof. ∎

######  Proposition 5.2.

Let $D\in\mathbb{Z}_{\geq 1}$. For $X\in\mathbb{Z}_{>0}$, let $S_{D,X}$ denote
the set

$\\{m\in\mathbb{Z}_{>0}\mid X\leq
m<2X,\,\frac{m}{D}\in\mathbb{Z}\cap(\mathbb{Q}^{\times})^{2},\,(m,N)=1\\}.$

Then we have

$\sum_{m\in
S_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=o(X^{\frac{b+1}{2}}\log
X).$

###### Proof.

We have

$\displaystyle\sum_{m\in S_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}$
$\displaystyle\leq\sum_{m\in S_{D,X}}\sum_{n\geq 1}|\\{\lambda\in
L_{n}|Q(\lambda)=m\\}|$ $\displaystyle=\sum_{n\geq 1}\sum_{m\in
S_{D,X}}|\\{\lambda\in L_{n}|Q(\lambda)=m\\}|.$

By [EK95, Lemma 2.4], we have the following estimate which only depends on the
rank $r$ of the lattices $L_{n}$ and hence not on $n$

$\sum_{m\in S_{D,X}}|\\{\lambda\in
L_{n}|Q(\lambda)=m\\}|\ll\sum_{j=0}^{r}\frac{X^{j}}{a_{j}(L_{n})}.$

On the other hand, if $\lambda\in L_{n}$ with $Q(\lambda)=m\in S_{D,X}$, then
$\mu_{1}(L_{n})^{2}\leq m\leq X$, hence $n\ll X^{\frac{b+\epsilon}{2}}$ and

$\displaystyle\sum_{m\in S_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}$
$\displaystyle\ll\sum_{m\in
S_{D,X}}\sum^{O_{\epsilon}(X^{\frac{b+\epsilon}{2}})}_{n\geq 1}|\\{\lambda\in
L_{n}|Q(\lambda)=m\\}|$ $\displaystyle\ll\sum_{j=0}^{r}\sum_{n\geq
1}^{X^{\frac{b+\epsilon}{2}}}\frac{X^{\frac{j}{2}}}{n^{\frac{j}{b+\epsilon}}}$
$\displaystyle\ll\sum_{j=0}^{r}X^{\frac{j}{2}+(1-\frac{j}{b+\epsilon})\frac{b+\epsilon}{2}}$
$\displaystyle=O(X^{\frac{b+\epsilon}{2}}).$

Hence the result. ∎

####  5.1.2. Type III degeneration

Let $(\Xi,\sigma)$ be a toroidal stratum representative of type III such that
$\sigma$ is a ray. We use notations from Section 2.2.2.

By our choice of $\Sigma$, the curve $\mathcal{Y}$ touches the boundary of
$\mathcal{M}^{\Sigma}$ at a locally closed boundary divisor
$\mathcal{B}^{\Xi,\sigma}$. Let $\widehat{M}^{\Sigma}$ be the formal
completion of $\mathcal{M}^{\Sigma}$ along $\mathcal{B}^{\Xi,\sigma}$ and
hence we get a map

$\displaystyle\widehat{\mathcal{Y}}\rightarrow\widehat{M}^{\Sigma}.$ (5.1.3)

By Section 2.2.5, the following maps of formal Deligne–Mumford stacks are
finite étale:

$\bigsqcup_{\mathbb{Q}^{\times}_{>0}\backslash\mathbb{A}_{f}^{\times}/K_{0}}\widehat{\mathcal{T}}_{\Xi/R}\rightarrow\widehat{\mathcal{M}}_{\Xi,\sigma}\rightarrow\widehat{\mathcal{M}^{\Sigma}}.$

Hence map 5.1.3 lifts to map

$\widehat{\mathcal{Y}}\rightarrow\mathrm{Spf}\left(\mathbb{Z}_{p}[q_{\alpha}|\alpha\in\Gamma_{\Xi}^{\vee}\cap\omega^{\bot}][[q_{\omega^{\prime}}]]\right).$

This corresponds to a morphism:

$\displaystyle\mathbb{Z}_{(p)}[[q_{\omega^{\prime}}]][q_{\alpha}]_{\underset{(\alpha.\omega)=0}{\alpha\in\Gamma_{\Xi}^{\vee}}}\rightarrow\mathcal{O}_{K_{\mathfrak{P}}}.$
(5.1.4)

Let $\lambda\in K_{I}=\Gamma_{\Xi}$ with $Q(\lambda)=m$. By Section 2.4.2 the
branch of the special divisor $\mathcal{Z}(m)$ parameterized by $\lambda$
intersects the boundary only if $(\lambda.\omega)=0$. In the latter case, by
Proposition 2.5, its equation is given by $q^{\lambda}-1$ and the multiplicity
of intersection of $\mathcal{Y}$ with the branch given by $\lambda$ is the
$p$-adic valuation of the element $q^{\lambda}-1$ under the map Equation
5.1.4.

Let $x\in\mathcal{B}^{\Xi,\sigma}(\overline{\mathbb{F}}_{p})$ be the image of
$\mathfrak{P}$. Then by the previous discussion, we conclude that

$(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=\frac{1}{d}\sum_{\underset{Q(\lambda)=m}{\lambda\in
K_{I}\cap\omega^{\bot}}}v_{p}(q^{\lambda}-1),$

where $d$ is the degree of the map Equation 5.1.3 at $x$.

For $n\geq 1$, let

$L_{n}=\\{\lambda\in K_{I}\cap\omega^{\bot}|v_{p}(q^{\lambda}-1)\geq n\\}.$

Then we can rewrite the multiplicity intersection at $\mathfrak{P}$ as:

$(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=\frac{1}{d}\sum_{n\geq
1}\\{\lambda\in L_{n}|Q(\lambda)=m\\}.$

######  Proposition 5.3.

The lattices $(L_{n},Q)$ are positive definite lattices of rank $r\leq b-1$
independent from $n$ and they satisfy the following properties:

  1. (1)

$\cap_{n}L_{n}=\\{0\\}$.

  2. (2)

For every $n\geq 1$, $pL_{n}\subseteq L_{n+1}$.

  3. (3)

For $1\leq r\leq b-1$, let $\mu_{i}(L_{n})$ be the $i^{th}$ successive minima
and let $a_{i}(L_{n})=\prod_{1\leq k\leq i}\mu_{i}(L_{n})$. Then we have

$a_{i}(L_{n})\gg_{\epsilon}n^{\frac{i}{b+\epsilon}}.$

###### Proof.

The proof is similar to the proof of Proposition 5.1. Let
$\lambda,\lambda^{\prime}\in K\cap\omega^{\bot}$. By writing

$q^{\lambda+\lambda^{\prime}}-1=q^{\lambda}(q^{\lambda^{\prime}}-1)+q^{\lambda}-1,$

we get that $L_{n}$ is a lattice and it is obviously positive definite as
$K_{I}$ is Lorentzian and $\omega$ is a negative normed vector.

Let $\pi$ be a uniformizer of $\mathcal{O}_{K_{\mathfrak{P}}}$ and let
$\lambda\in L_{n}$. Then $q^{\lambda}=1+\pi^{n}.u$ for some
$u\in\mathcal{O}_{K_{\mathfrak{P}}}$. Hence

$\displaystyle q^{p\lambda}-1$ $\displaystyle=(1+\pi^{n}.u)^{p}-1$
$\displaystyle=\sum_{i\geq 1}\binom{p}{i}\pi^{ni}u^{i}$
$\displaystyle=\pi^{n+1}u^{\prime}.$

Hence (2). The rest of the proof is similar to Proposition 5.1. ∎

As a consequence, we get the following proposition, whose proof is identical
to the proof of Proposition 5.2 and we omit it.

######  Proposition 5.4.

Let $D\in\mathbb{Z}_{\geq 1}$ be coprime to $N$. For $X\in\mathbb{Z}_{>0}$,
let $S_{D,X}$ denote the set

$\\{m\in\mathbb{Z}_{>0}\mid X\leq
m<2X,\,\frac{m}{D}\in\mathbb{Z}\cap(\mathbb{Q}^{\times})^{2},\,(m,N)=1\\}.$

Then we have

$\sum_{m\in
S_{D,X}}(\mathcal{Y}.\mathcal{Z}(m))_{\mathfrak{P}}=o(X^{\frac{b+1}{2}}\log
X).$

###  5.2. Function field setting

In this section, we prove Proposition 4.11. We assume here that the lattice
$(L,Q)$ is self-dual at $p$ and we let $\mathcal{M}_{\mathbb{F}_{p}}$ be the
mod $p$ GSpin Shimura variety associated to $(L,Q)$. Let $\Sigma$ be a
polyhedral cone decomposition which satisfies Proposition 4.9.

Let $\overline{\mathscr{S}}\rightarrow\mathcal{M}^{\Sigma}_{\mathbb{F}_{p}}$
be a finite map as before and let
$P\in\overline{\mathscr{S}}(\overline{\mathbb{F}}_{p})$ be a point mapping to
the boundary of $\mathcal{M}^{\Sigma}_{\mathbb{F}_{p}}$. Let denote
$k=\overline{\mathbb{F}}_{p}$. The point $P$ lies either in a boundary stratum
of of type II or type III. We treat each case separately.

####  5.2.1. Type II degeneration

Assume that the image of $P$ is in
$\mathcal{B}^{\Upsilon}_{\mathbb{F}_{p}}(k)$ where $\Upsilon$ is a cusp label
representative of type II.

Let $\widehat{\overline{\mathscr{S}}}\simeq\mathrm{Spf}(k[[t]])$ be the formal
completion of $\overline{\mathscr{S}}$ along $s$. Then by reasoning similarly
to Section 5.1.1, specifically using the reduction mod $p$ of Equation 2.3.3,
we get for every $\lambda\in D$ with $Q(\lambda)=m\geq 1$, $m$ coprime to $N$
a map

$\Phi_{p}:\widehat{\mathcal{O}}_{\mathcal{M}_{\Upsilon,\mathbb{F}_{p},x}}\rightarrow
k[[t]],$

Let $v(\lambda)$ denote the $t$-adic valuation of the generator $f_{\lambda}$
of $I_{\lambda,p}$. Then similarly to the number field case, we have:

######  Lemma 5.5.

The multiplicity of intersection of $\overline{\mathscr{S}}$ and
$\mathcal{Z}(m)_{\mathbb{F}_{p}}$ at $P$ satisfies:

$m_{P}(\overline{\mathscr{S}},\mathcal{Z}(m)_{\mathbb{F}_{p}})\ll\sum_{n\geq
1}|\\{\lambda\in L_{n}|\,Q(\lambda)=m\\}|.$

Now we are ready to prove Proposition 4.12.

######  Proposition 5.6.

The sequence of lattices $(L_{n},Q)$ satisfy the same properties as in
Proposition 5.1 and letting $S$ be as in Section 4.2, we have the following
estimate for $X>0$:

$\sum_{m\in
S_{X}}m_{P}(\overline{\mathscr{S}},\mathcal{Z}(m)_{\mathbb{F}_{p}})=O_{\epsilon}(X^{\frac{b+\epsilon}{2}})$

###### Proof.

The same proof as in Proposition 5.1 shows that the lattices $(L_{n},Q)$ enjoy
the same properties of the aforementioned proposition. For the second part, we
have

$\displaystyle\sum_{m\in
S_{X}}m_{P}(\overline{\mathscr{S}},\mathcal{Z}(m)_{\mathbb{F}_{p}})$
$\displaystyle\ll\sum_{m\in S_{X}}\sum_{n\geq 1}|\\{\lambda\in
L_{n}|\,Q(\lambda)=m\\}|$
$\displaystyle\ll\sum^{O(X^{\frac{b+\epsilon}{2}})}_{n=1}|\\{\lambda\in
L_{n}|Q(\lambda)\leq m\\}|$
$\displaystyle\ll\sum^{O(X^{\frac{b+\epsilon}{2}})}_{n=1}\sum_{j=0}^{r}\frac{X^{\frac{j}{2}}}{a_{j}(L_{n})}$
$\displaystyle\ll\sum_{j=0}^{r}\sum^{O(X^{\frac{b+\epsilon}{2}})}_{n=1}\frac{X^{\frac{j}{2}}}{n^{\frac{j}{b+\epsilon}}}$
$\displaystyle=O(X^{\frac{b+\epsilon}{2}}).$

∎

####  5.2.2. Type III degeneration

Assume now that there exists a toroidal stratum representative $(\Xi,\sigma)$
such that $\sigma$ is a ray and such that $P$ lies in
$\mathcal{B}^{\Xi,\sigma}_{\mathbb{F}_{p}}(k)$. Using a similar approach to
Section 5.1.2 by taking reduction mod $p$, we get a map

$k[q_{\alpha}|\,\alpha\in\Gamma_{\Xi}^{\vee}\cap\omega^{\bot}][[q_{\omega^{\prime}}]]\rightarrow
k[[t]],$

sending $q_{\omega^{\prime}}$ to an element of the ideal $(t)$. Let $v$ denote
the $t$-adic valuation on $k[[t]]$. Then, for $m$ coprime to $N$, the
multiplicity of intersection of $\overline{\mathscr{S}}$ and
$\mathcal{Z}(m)_{\mathbb{F}_{p}}$ at $P$ satisfies:

$m_{P}(\overline{\mathscr{S}},\mathcal{Z}(m)_{\mathbb{F}_{p}})\leq\sum_{\underset{Q(\lambda)=m}{\lambda\in
K_{I}\cap\omega^{\bot}}}v(q^{\lambda}-1).$

If we define the sequence lattices $L_{n}$ as

$L_{n}=\\{\lambda\in K\cap\omega^{\bot}|v(q^{\lambda}-1)\geq n,\\}$

then

$m_{P}(\overline{\mathscr{S}},\mathcal{Z}(m)_{\mathbb{F}_{p}})\leq\sum_{n\geq
1}|\\{\lambda\in L_{n},\,Q(\lambda)=m\\}|.$

Now the rest of the proof is similar to Section 5.1.2. This proves Proposition
4.12 in the remaining type III case.

##  6\. Applications

In this section, we present a proof of Theorem 1.5. This approach is inspired
from [MST22].

###  6.1. Hecke orbit conjecture

####  6.1.1. The orthogonal case

Since GSpin Shimura varieties are finite covers of orthogonal ones, it is
enough to prove the result for GSpin Shimura varieties.

Let $\mathcal{M}_{\mathbb{F}_{p}}$ be the reduction mod $p\geq 5$ of a GSpin
type Shimura variety with hyperspecial level at $p$ associated to a lattice
$(L,Q)$, which is assumed to be self-dual at $p$ and of signature $(b,2)$. We
will prove Theorem 1.5 by induction on $b$, which is also the dimension of
$\mathcal{M}_{\mathbb{F}_{p}}$.

The case $b=1$ is immediate: the prime-to-$p$ Hecke orbit of $x$ is infinite,
hence Zariski dense.

Assume now that $n\geq 2$ and the result of Theorem 1.5 holds for all ordinary
points in GSpin Shimura varieties of dimension less than $b-1$ with
hyperspecial level at $p$. Let $x$ be an ordinary point in
$\mathcal{M}(\overline{\mathbb{F}}_{p})$ and let $\overline{T_{x}}$ be the
Zariski closure of its prime-to-$p$ Hecke orbit. Then $\overline{T_{x}}$ has
positive dimension and intersects the ordinary locus non-trivially. Hence we
can find a smooth quasi-projective curve $\mathscr{S}$ and a finite map

$\mathscr{S}\rightarrow\mathcal{M}_{\mathbb{F}_{p}}$

whose image is contained in $\overline{T_{x}}$ and which is contained in the
ordinary locus. Moreover, we can assume that this image is not contained in
any special divisor. Indeed, the same argument used for proper curves in
[MST22, Lemma 8.11] works in our setting with no change. By Theorem 1.3, the
curve $\mathscr{S}$ intersects infinitely many divisors
$\mathcal{Z}(m)_{\mathbb{F}_{p}}$ with $(m,p)=1$. The special divisors
$\mathcal{Z}(m)_{\mathbb{F}_{p}}$ are themselves the union of GSpin Shimura
varieties of dimension $b-1$ with hyperspecial level at $p$ since $m$ is
coprime to $p$. Let
$y\in\mathscr{S}(\overline{\mathbb{F}}_{p})\cap\mathcal{Z}^{\prime}(m)(\overline{\mathbb{F}}_{p})$
for some irreducible component $\mathcal{Z}^{\prime}(m)$ of $\mathcal{Z}(m)$.
Then $y$ is ordinary and the prime-to-$p$ Hecke orbit of $y$ in
$\mathcal{Z}^{\prime}(m)_{\mathbb{F}_{p}}$ is Zariski dense by the induction
hypothesis. Since this orbit is a sub-orbit of the Hecke orbit in
$\mathcal{M}_{\mathbb{F}_{p}}$, we conclude that
$\mathcal{Z}^{\prime}(m)_{\mathbb{F}_{p}}\subset\overline{T_{x}}$.
Furthermore, it is straightforward to check that the collection of the
divisors $\mathcal{Z}^{\prime}(m)_{\mathbb{F}_{p}}$ must be infinite by
Theorem 1.3. Hence we conclude that
$\overline{T_{x}}=\mathcal{M}_{\mathbb{F}_{p}}$ which is the desired result.

####  6.1.2. The unitary case

We prove in this section the Hecke orbit conjecture in the unitary case using
the reduction to the orthogonal case already used in [MST22, Remark 8.12] and
in [SSTT19, Section 9.3].

Let $\mathcal{M}_{\mathbb{F}_{p}}$ be the mod $p$ points of the canonical
model of a unitary Shimura variety associated to an imaginary quadratic field
$\mathbf{k}$, a unitary group of signature $(r,1)$ with hyperspecial level at
$p$ as described in [BHK+20, Section 2.1] such that $p$ is split in
$\mathbf{k}$. Consider the family of special divisors $\mathcal{Z}_{Kra}(m)$
as described in Section 2.5 loc. cit. which are themselves unitary Shimura
varieties associated to unitary groups of signature $(r-1,1)$ and hyperspecial
at $p$ when $p$ does not divide $m$. Then using a similar argument to [SSTT19,
Section 9.3] and further explained in [MST22, Remark 8.12], we have the
following theorem which is a consequence of Theorem 1.3.

######  Theorem 6.1.

Assume that $p\geq 5$ and let
$\mathscr{S}\rightarrow\mathcal{M}_{\mathbb{F}_{p}}$ be a finite map from a
smooth quasi-projective curve $\mathscr{S}$ over $\overline{\mathbb{F}}_{p}$
and with generically ordinary image. Then the union over $m$ prime to $p$ of
the intersections $\mathscr{S}\cap\mathcal{Z}_{Kra}(m)$ is infinite.

Now the Hecke orbit conjecture in the unitary case is an easy consequence of
the above theorem and the induction method explained in the previous
paragraph.

## References

  * [AGHMP17] Fabrizio Andreatta, Eyal Z. Goren, Benjamin Howard, and Keerthi Madapusi Pera.  Height pairings on orthogonal Shimura varieties.  Compos. Math., 153(3):474–534, 2017.
  * [AGHMP18] Fabrizio Andreatta, Eyal Z. Goren, Benjamin Howard, and Keerthi Madapusi Pera.  Faltings heights of abelian varieties with complex multiplication.  Ann. of Math. (2), 187(2):391–531, 2018.
  * [AMRT10] Avner Ash, David Mumford, Michael Rapoport, and Yung-Sheng Tai.  Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze.  Cambridge: Cambridge University Press, 2010.
  * [BBK07] Jan H. Bruinier, José I. Burgos Gil, and Ulf Kühn.  Borcherds products and arithmetic intersection theory on Hilbert modular surfaces.  Duke Math. J., 139(1):1–88, 2007.
  * [BHK+20] Jan H. Bruinier, Benjamin Howard, Stephen S. Kudla, Michael Rapoport, and Tonghai Yang.  Modularity of generating series of divisors on unitary Shimura varieties.  Astérisque, (421, Diviseurs arithmétiques sur les variétés orthogonales et unitaires de Shimura):7–125, 2020.
  * [BKU21] Gregorio Baldi, Bruno Klingler, and Emmanuel Ullmo.  On the distribution of the Hodge locus.  arXiv e-prints, page arXiv:2107.08838, July 2021.
  * [Bor99] Richard E. Borcherds.  The Gross-Kohnen-Zagier theorem in higher dimensions.  Duke Math. J., 97(2):219–233, 1999.
  * [Bru02] Jan H. Bruinier.  Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, volume 1780 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2002.
  * [BZ21] Jan Hendrik Bruinier and Shaul Zemel.  Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties.  Math. Z., 2021.
  * [Cha95] Ching-Li Chai.  Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli.  Invent. Math., 121(3):439–479, 1995.
  * [Cha14] François Charles.  On the Picard number of K3 surfaces over number fields.  Algebra Number Theory, 8(1):1–17, 2014.
  * [Cha18] François Charles.  Exceptional isogenies between reductions of pairs of elliptic curves.  Duke Math. J., 167(11):2039–2072, 08 2018.
  * [CO06] Ching-Li Chai and Frans Oort.  Hypersymmetric abelian varieties.  Pure and Applied Mathematics Quarterly, 2(1):1–27, 2006.
  * [CU05] Laurent Clozel and Emmanuel Ullmo.  Équidistribution de sous-variétés spéciales.  Ann. of Math. (2), 161(3):1571–1588, 2005.
  * [Duk88] W. Duke.  Hyperbolic distribution problems and half-integral weight Maass forms.  Invent. Math., 92(1):73–90, 1988.
  * [EGT23] Philip Engel, François Greer, and Salim Tayou.  Mixed mock modularity of special divisors.  arXiv e-prints, page arXiv:2301.05982, January 2023.
  * [EK95] Alex Eskin and Yonatan R. Katznelson.  Singular symmetric matrices.  Duke Math. J., 79(2):515–547, 1995.
  * [EO06] Alex Eskin and Hee Oh.  Representations of integers by an invariant polynomial and unipotent flows.  Duke Math. J., 135(3):481–506, 2006.
  * [HP20] Benjamin Howard and Keerthi Madapusi Pera.  Arithmetic of Borcherds products.  Astérisque, (421, Diviseurs arithmétiques sur les variétés orthogonales et unitaires de Shimura):187–297, 2020.
  * [HZ01] Michael Harris and Steven Zucker.  Boundary cohomology of Shimura varieties. III. Coherent cohomology on higher-rank boundary strata and applications to Hodge theory.  Mém. Soc. Math. Fr. (N.S.), (85):vi+116, 2001.
  * [IRR14] Özlem Imamoglu, Martin Raum, and Olav K. Richter.  Holomorphic projections and Ramanujan’s mock theta functions.  Proc. Natl. Acad. Sci. USA, 111(11):3961–3967, 2014.
  * [Kis10] Mark Kisin.  Integral models for Shimura varieties of abelian type.  J. Amer. Math. Soc., 23(4):967–1012, 2010.
  * [MP15] Keerthi Madapusi Pera.  The Tate conjecture for K3 surfaces in odd characteristic.  Invent. Math., 201(2):625–668, 2015.
  * [MP16] Keerthi Madapusi Pera.  Integral canonical models for spin Shimura varieties.  Compos. Math., 152(4):769–824, 2016.
  * [MST18] Davesh Maulik, Ananth N. Shankar, and Yunqing Tang.  Reductions of abelian surfaces over global function fields.  arXiv e-prints, page arXiv:1812.11679, December 2018.
  * [MST22] Davesh Maulik, Ananth N. Shankar, and Yunqing Tang.  Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture.  Inventiones Mathematicae, January 2022.
  * [Ogu03] Keiji Oguiso.  Local families of K3 surfaces and applications.  J. Algebraic Geom., 12(3):405–433, 2003.
  * [Per19] Keerthi Madapusi Pera.  Toroidal compactifications of integral models of Shimura varieties of Hodge type.  Ann. Sci. Éc. Norm. Supér. (4), 52(2):393–514, 2019.
  * [Pin90] Richard Pink.  Arithmetical compactification of mixed Shimura varieties, volume 209 of Bonner Mathematische Schriften [Bonn Mathematical Publications].  Universität Bonn, Mathematisches Institut, Bonn, 1990.  Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1989.
  * [Sar90] Peter Sarnak.  Some applications of modular forms, volume 99 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 1990.
  * [SSTT19] Ananth N. Shankar, Arul Shankar, Yunqing Tang, and Salim Tayou.  Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields.  arXiv e-prints, page arXiv:1909.07473, September 2019.
  * [ST20] Ananth N. Shankar and Yunqing Tang.  Exceptional splitting of reductions of abelian surfaces.  Duke Math. J., 169(3):397–434, 2020.
  * [Tay20] Salim Tayou.  On the equidistribution of some Hodge loci.  J. Reine Angew. Math., 762:167–194, 2020.
  * [TT23] Salim Tayou and Nicolas Tholozan.  Equidistribution of Hodge loci. II.  Compos. Math., 159(1):1–52, 2023.
  * [van21] Pol van Hoften.  On the ordinary Hecke orbit conjecture.  arXiv e-prints, page arXiv:2112.12422, December 2021.
  * [Voi02] C. Voisin.  Théorie de Hodge et géométrie algébrique complexe.  Collection SMF. Société Mathématique de France, 2002.
  * [Zem20] Shaul Zemel.  The structure of integral parabolic subgroups of orthogonal groups.  Journal of Algebra, 559:95–128, 2020.