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# Geometric weight-shifting operators on Hilbert modular forms in
characteristic $p$

Fred Diamond<EMAIL_ADDRESS>Department of Mathematics, King’s College
London, WC2R 2LS, UK

###### Abstract.

We carry out a thorough study of weight-shifting operators on Hilbert modular
forms in characteristic $p$, generalizing the author’s prior work with Sasaki
to the case where $p$ is ramified in the totally real field. In particular we
use the partial Hasse invariants and Kodaira–Spencer filtrations defined by
Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial
$\Theta$-operators, obtaining ones whose effect on weights is optimal from the
point of view of geometric Serre weight conjectures. Furthermore we describe
the kernels of partial $\Theta$-operators in terms of images of geometrically
constructed partial Frobenius operators. Finally we apply our results to prove
a partial positivity result for minimal weights of mod $p$ Hilbert modular
forms.

November 2020

##  1\. Introduction

The study of weight-shifting operations on modular forms has a rich and
fruitful history. Besides those naively obtained from the graded algebra
structure on the space of classical modular forms of all weights, there is a
deeper construction due to Ramanujan [30] which shifts the weight by two using
differentiation, leading to a more general theory of Maass–Shimura operators.
Analogous weight-shifting operations in characteristic $p$, first studied by
Swinnerton-Dyer and Serre [34], take on special significance in the context of
congruences between modular forms and the implications for associated Galois
representations. In particular one has the following linear maps from the
space of mod $p$ modular forms of weight $k$ and some fixed level $N$ prime to
$p$:

  * •

multiplication by a Hasse invariant $H$, to forms of weight $k+p-1$;

  * •

a differential operator $\Theta$, to forms of weight $k+p+1$;

  * •

a linearized $p$-power map $V$, to forms of weight $pk$.

These maps all have simple descriptions in terms of associated $q$-expansions:
if $f$ has $q$-expansion $\sum a_{n}q^{n}$, then that of $Hf$ (resp. $\Theta
f$, $Vf$) is $\sum a_{n}q^{n}$ (resp. $\sum na_{n}q^{n}$, $\sum a_{n}q^{pn}$).
Following the work of Swinnerton-Dyer and Serre, there were further
significant developments to the theory due to Katz [23, 24] (interpreting the
constructions more geometrically), Jochnowitz [21, 22] (on the weight
filtration and Tate’s $\Theta$-cycles) and Gross [20] (in the study of
companion forms), providing crucial ingredients for Edixhoven’s proof of the
weight part of Serre’s Conjecture in [14].

Suppose now that $F$ is a totally real field of degree $d=[F:{\mathbb{Q}}]$
and consider spaces of Hilbert modular forms of weight
$\mathbf{k}\in\mathbb{Z}^{\Sigma}$ and fixed level prime to $p$, where
$\Sigma$ denotes the set of embeddings
$\\{\sigma:F\hookrightarrow\overline{{\mathbb{Q}}}_{p}\\}$. For such spaces of
$p$-adic modular forms, Katz [25] constructed a family of commuting
differential operators $\Theta_{\sigma}$, indexed by the $d$-embeddings
$\sigma\in\Sigma$. The theory was further developed by Andreatta and Goren [1]
who, building on Katz’s work and Goren’s definition of partial Hasse
invariants in [18, 19] (if $p$ is unramified in $F$), defined partial
$\Theta$-operators on spaces of mod $p$ Hilbert modular forms.

Under the assumption that $p$ is unramified in $F$, some aspects of the
construction of partial $\Theta$-operators in [1] were simplified in [12],
which also went on to define partial Frobenius operators (generalizing $V$)
geometrically and use their image to describe kernels of partial
$\Theta$-operators. When $p$ is ramified in $F$, the effectiveness of the
approach in [1] was limited by the singularities of the available
(Deligne–Pappas) model for the Hilbert modular variety. Since then however, a
smooth integral model was constructed by Pappas and Rapoport [29], and the
theory of partial Hasse invariants was further developed in this context by
Reduzzi and Xiao in [32]. The theory of partial $\Theta$-operators was
revisited in that light by Deo, Dimitrov and Wiese in [8], where they closely
follow [1]. Here we instead exploit the observations and techniques introduced
in [12], applying them directly to the special fibre of the Pappas–Rapoport
model to construct and relate partial $\Theta$ and Frobenius operators. In
particular this eliminates extraneous multiples of partial Hasse invariants
that appear in [8], and yields results whose implications for minimal weights
are motivated by the forthcoming generalization to the ramified case of the
geometric Serre weight conjectures of [12]. The main contributions of this
paper may be summarized as follows:

  * •

a construction of operators $\Theta_{\tau}$ with optimal effect on weight
(Theorem 5.2.1);

  * •

a geometric construction of partial Frobenius operators $V_{\mathfrak{p}}$
(see §6.2);

  * •

a description of the kernel of $\Theta_{\tau}$ in terms of the image of
$V_{\mathfrak{p}}$ (see §9.1);

  * •

an application to positivity of minimal weights (Theorem 9.2.1).

We should emphasize that the focus of this paper is entirely on Hilbert
modular forms in characteristic $p$. There is also a rich theory of
$\Theta$-operators on $p$-adic automorphic forms which has seen major progress
recently in the work of de Shalit and Goren [7], and Eischen, Fintzen,
Mantovan and Varma [15], which in turn has implications in the characteristic
$p$ setting [6, 7, 16, 17]. Another advance in characteristic $p$ has been
Yamauchi’s construction [36] of $\Theta$-operators for mod $p$ Siegel modular
forms of degree two. We remark however that all of the work just mentioned
only considers automorphic forms on reductive groups which are unramified at
$p$; the novelty of this paper is largely in the treatment of ramification at
$p$.

We now describe the contents in more detail.

We first set out some basic notation and constructions in §2.1. In particular
we fix a prime $p$, a totally real field $F$ of degree111Including the case
$F=\mathbb{Q}$ would introduce different complications in the treatment of
cusps and provide no new results. $d=[F:\mathbb{Q}]>1$, and let
${\mathcal{O}}_{F}$ denote the ring of integers of $F$ and $S_{p}$ the set of
prime ideals of ${\mathcal{O}}_{F}$ dividing $p$. For each ${\mathfrak{p}}\in
S_{p}$, let $\Sigma_{{\mathfrak{p}},0}$ denote the set of $f_{\mathfrak{p}}$
embeddings ${\mathcal{O}}_{F}/{\mathfrak{p}}\to\overline{{\mathbb{F}}}_{p}$
and $\Sigma_{\mathfrak{p}}$ the set of $e_{\mathfrak{p}}f_{\mathfrak{p}}$
embeddings $F_{\mathfrak{p}}\to\overline{{\mathbb{Q}}}_{p}$, where
$f_{\mathfrak{p}}$ (resp. $e_{\mathfrak{p}}$) is the residual (resp.
ramification) degree of ${\mathfrak{p}}$. We let

$\begin{array}[]{rcccl}\Sigma_{0}&=&\coprod_{{\mathfrak{p}}\in
S_{p}}\Sigma_{{\mathfrak{p}},0}&=&\\{\,\tau_{{\mathfrak{p}},i}\,|\,{\mathfrak{p}}\in
S_{p},i\in\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}\,\\}\\\
\mbox{and}\qquad\Sigma&=&\coprod_{{\mathfrak{p}}\in
S_{p}}\Sigma_{\mathfrak{p}}&=&\\{\,\theta_{{\mathfrak{p}},i,j}\,|\,{\mathfrak{p}}\in
S_{p},i\in\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z},j=1,\ldots,e_{\mathfrak{p}}\,\\},\end{array}$

where each $\tau_{{\mathfrak{p}},0}\in\Sigma_{{\mathfrak{p}},0}$ is chosen
arbitrarily, $\tau_{{\mathfrak{p}},i}=\tau_{{\mathfrak{p}},0}^{p^{i}}$ and
$\theta_{{\mathfrak{p}},i,1},\ldots,\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$
is any ordering of the lifts of $\tau_{{\mathfrak{p}},i}$ to
$\Sigma_{\mathfrak{p}}$. We also define a “right-shift” permutation $\sigma$
of $\Sigma$ by

$\sigma(\theta_{{\mathfrak{p}},i,j})=\left\\{\begin{array}[]{ll}\theta_{{\mathfrak{p}},i,j-1},&\mbox{if
$j>1$};\\\ \theta_{{\mathfrak{p}},i-1,e_{\mathfrak{p}}},&\mbox{if
$j=1$.}\end{array}\right.$

In §2.2 we recall the definition of the Pappas–Rapoport model $Y_{U}$ for the
Hilbert modular variety of level $U$, where $U$ is any sufficiently small open
compact subgroup of $\operatorname{GL}_{2}(\mathbb{A}_{F,\mathbf{f}})$ of
level prime to $p$. This may be viewed as a coarse moduli space for Hilbert-
Blumenthal abelian varieties with additional structure, where this additional
structure includes a suitable collection of filtrations on direct summands of
its sheaf of invariant differentials. The scheme $Y_{U}$ is then smooth of
relative dimension $d$ over ${\mathcal{O}}$, where ${\mathcal{O}}$ is the ring
of integers of a finite extension of ${\mathbb{Q}}_{p}$ in
$\overline{{\mathbb{Q}}}_{p}$. Since the main results of the paper concern
Hilbert modular forms in characteristic $p$, we will restrict our attention to
this setting for the remainder of the Introduction, and let
$\overline{Y}_{U}=Y_{U,\mathbb{F}}$ where $\mathbb{F}$ is the residue field of
${\mathcal{O}}$.

In §§3.1–3.2 we construct the automorphic line bundles
$\overline{{\mathcal{A}}}_{\mathbf{k},\mathbf{l}}$ on $\overline{Y}_{U}$ for
all $\mathbf{k}$, $\mathbf{l}\in\mathbb{Z}^{\Sigma}$ and sufficiently small
$U$ (of level prime to $p$), and define the space of Hilbert modular forms of
weight $(\mathbf{k},\mathbf{l})$ and level $U$ over $\mathbb{F}$ to be

$M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})=H^{0}(\overline{Y}_{U},\overline{{\mathcal{A}}}_{\mathbf{k},\mathbf{l}}).$

The spaces are equipped with a natural Hecke action making

$M_{\mathbf{k},\mathbf{l}}(\mathbb{F}):=\varinjlim_{U}M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$

a smooth admissible representation of
$\operatorname{GL}_{2}(\mathbb{A}_{F,\mathbf{f}})$ over $\mathbb{F}$. A key
point, as already observed in [12] in the unramified case, is that the parity
condition on $\mathbf{k}$ imposed in the definition of Hilbert modular forms
in characteristic zero (for the group
${\operatorname{Res}}_{F/\mathbb{Q}}\operatorname{GL}_{2}$) disappears in
characteristic $p$. We remark also that the effect of the weight parameter
$\mathbf{l}$ (in characteristic $p$) is to introduce twists by torsion bundles
that make various constructions, in particular that of partial
$\Theta$-operators, compatible with the natural Hecke action.

In §§3.3–4.2 we recall results of Reduzzi and Xiao [32] that will underpin our
construction of partial $\Theta$-operators. Firstly there is a natural
Kodaira–Spencer filtration on direct summands of
$\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}}$ whose graded pieces are isomorphic
to the automorphic line bundles
$\overline{{\mathcal{A}}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}$
(where ${\mathbf{e}}_{\theta}$ denotes the basis element of
$\mathbb{Z}^{\Sigma}$ indexed by $\theta$). Secondly for each
$\theta=\theta_{{\mathfrak{p}},i,j}\in\Sigma$, there is a partial Hasse
invariant

$H_{\theta}\in M_{\mathbf{h}_{\theta},\bf{0}}(U;\mathbb{F}),\qquad\mbox{where
\,\,$\mathbf{h}_{\theta}=n_{\theta}{\mathbf{e}}_{\sigma^{-1}\theta}-{\mathbf{e}}_{\theta}$}$

with $n_{\theta}=p$ if $\theta_{{\mathfrak{p}},i,1}$ for some
${\mathfrak{p}},i$, and $n_{\theta}=1$ otherwise. Multiplication by the
partial Hasse invariant $H_{\theta}$ thus defines a map

$\cdot H_{\theta}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\longrightarrow
M_{\mathbf{k}+\mathbf{h}_{\theta},\mathbf{l}}(U;\mathbb{F})$

which is easily seen to be Hecke-equivariant. We also analogously define
invariants $G_{\theta}\in M_{\bf{0},\mathbf{h}_{\theta}}(U;\mathbb{F})$ which
trivialize the bundles
$\overline{{\mathcal{A}}}_{\bf{0},\mathbf{h}_{\theta}}$. The partial Hasse
invariants $H_{\theta}$ refine the ones defined by Andreatta and Goren in [1]
and give rise to a natural stratification on $\overline{Y}_{U}$ and a notion
of minimal weight $\mathbf{k}_{\min}(f)$ for non-zero $f\in
M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$, which the main result of [10] shows
lies in a certain cone $\Xi^{\min}\subset\mathbb{Z}_{\geq 0}^{\Sigma}$ (see
(10)).

We then follow the approach of [12, §8] to define partial $\Theta$-operators
in §5. For each $\tau=\tau_{{\mathfrak{p}},i}\in\Sigma_{0}$, this gives a
Hecke-equivariant operator

$\Theta_{\tau}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\longrightarrow
M_{\mathbf{k}+\mathbf{h}_{\theta}+2{\mathbf{e}}_{\theta},\mathbf{l}-{\mathbf{e}}_{\theta}}(U;\mathbb{F})$

where $\theta=\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$. Note in particular
that if ${\mathfrak{p}}$ is ramified, then the shift222This precise shift is
predictable from the point of view of forthcoming work with Sasaki
generalizing the geometric Serre weight conjectures of [12] to the ramified
case. in the weight parameter $\mathbf{k}$ is by
${\mathbf{e}}_{\sigma^{-1}\theta}+{\mathbf{e}}_{\theta}$. The idea of the
construction, inspired by the one in [1, §12], is to divide by fundamental
Hasse invariants to get a rational function on the Igusa cover of
$\overline{Y}_{U}$, differentiate, project to the top graded piece of the
$\tau$-component of the Kodaira–Spencer filtration, and finally multiply by
fundamental Hasse invariants to descend to $\overline{Y}_{U}$ and eliminate
poles. The argument also gives a direct (albeit local) definition of the
$\Theta$-operator without reference to the Igusa cover in (12), and
establishes the following result (Theorem 5.2.1) generalizing [12, Thm.
8.2.2]:

######  Theorem A.

Let $\tau=\tau_{{\mathfrak{p}},i}$ and
$\theta=\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$. Then $\Theta_{\tau}(f)$
is divisible by $H_{\theta}$ if and only if either $f$ is divisible by
$H_{\theta}$ or $p|k_{\theta}$.

We turn to the construction of partial Frobenius operators $V_{\mathfrak{p}}$
in §6. This essentially generalizes a definition in [12, §9.8], but requires
significantly more work to actualize if ${\mathfrak{p}}$ is ramified. We do
this using Dieudonné theory to define a partial Frobenius endomorphism
$\Phi_{\mathfrak{p}}$ of $\overline{Y}_{U}$ and an isomorphism
$\Phi_{\mathfrak{p}}^{*}\overline{{\mathcal{A}}}_{\mathbf{k},\mathbf{l}}\cong\overline{{\mathcal{A}}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}$,
where

$\mathbf{k}^{\prime\prime}=\mathbf{k}+\sum_{\theta\in\Sigma_{\mathfrak{p}}}k_{\theta}\mathbf{h}_{\theta}\quad\mbox{and}\quad\mathbf{l}^{\prime\prime}=\mathbf{l}+\sum_{\theta\in\Sigma_{\mathfrak{p}}}l_{\theta}\mathbf{h}_{\theta},$

in order to obtain, for ${\mathfrak{p}}\in S_{p}$, commuting Hecke-equivariant
operators

$V_{\mathfrak{p}}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\longrightarrow
M_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}(U;\mathbb{F}).$

We will use $q$-expansions to relate the kernel of $\Theta_{\tau}$ to the
image of $V_{\mathfrak{p}}$ for $\tau\in\Sigma_{{\mathfrak{p}},0}$, so we
recall the theory in §7. This is a straightforward adaptation to our setting
of results and methods developed in [31, 4, 13, 5]. In §8 we compute the
(constant) $q$-expansions of the invariants $H_{\theta}$ and $G_{\theta}$ at
each cusp of $\overline{Y}_{U}$, and we obtain formulas generalizing the
classical ones for the effect of the operators $\Theta_{\tau}$ and
$V_{\mathfrak{p}}$ on all $q$-expansions. In particular this shows that the
operators $\Theta_{\tau}$ for varying $\tau$ commute.

In §9.1 we turn our attention to the description of the kernel of
$\Theta_{\tau}$. The $q$-expansion formulas also show that $\Theta_{\tau}\circ
V_{\mathfrak{p}}=0$ if $\tau\in\Sigma_{{\mathfrak{p}},0}$, and that
$\ker(\Theta_{\tau})$ is the same for all $\tau\in\Sigma_{{\mathfrak{p}},0}$.
Theorem A then reduces the study of the kernel to the case of weights of the
form $(\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime})$ where
$\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}$ are as in the definition
of $V_{\mathfrak{p}}$, for which the argument proving [12, Thm. 9.8.2] gives
the following:333This is a slight reformulation of Theorem 9.1.1.

######  Theorem B.

If $\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}$ and
$\tau=\tau_{{\mathfrak{p}},i}$ and
$\theta=\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$, then the sequence

$0\longrightarrow
M_{\mathbf{k},\mathbf{l}}(U,\mathbb{F})\stackrel{{\scriptstyle
V_{\mathfrak{p}}}}{{\longrightarrow}}M_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}(U;\mathbb{F})\stackrel{{\scriptstyle\Theta_{\tau}}}{{\longrightarrow}}M_{\mathbf{k}^{\prime\prime}+\mathbf{h}_{\theta}+2{\mathbf{e}}_{\theta},\mathbf{l}^{\prime\prime}-{\mathbf{e}}_{\theta}}(U;\mathbb{F})$

is exact.

Before discussing the application to positivity of minimal weights, we remark
that a less precise relation among the weight-shifting operations can be
neatly encapsulated in terms of the algebra of modular forms of all weights

$M_{\operatorname{tot}}(U;\mathbb{F}):=\bigoplus_{\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}}M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F}),$

or even its direct limit
$M_{\operatorname{tot}}(\mathbb{F}):=\varinjlim_{U}M_{\operatorname{tot}}(U;\mathbb{F})$
(over all sufficiently small levels prime to $p$). It follows from its
definition that the operator $V_{\mathfrak{p}}$ (resp. $\Theta_{\tau}$) on the
direct sum is an $\mathbb{F}$-algebra endomorphism (resp.
$\mathbb{F}$-derivation) of $M_{\operatorname{tot}}(\mathbb{F})$. One also
finds that $V_{\mathfrak{p}}$ maps the ideal

${\mathfrak{I}}=\langle
H_{\theta}^{\prime}-1,G_{\theta}^{\prime}-1\rangle_{\theta\in\Sigma}$

to itself.444The $H_{\theta}^{\prime}$ and $G_{\theta}^{\prime}$ are slight
modifications of the $H_{\theta}$ and $G_{\theta}$ obtained by rescaling those
for which $j=1$; see §9.3. Furthermore
$\Theta_{\tau}(H_{\theta}^{\prime})=\Theta_{\tau}(G_{\theta}^{\prime})=0$ for
all $\theta\in\Sigma$, so $V_{\mathfrak{p}}$ (resp. $\Theta_{\tau}$) induces
an $\mathbb{F}$-algebra endomorphism (resp. $\mathbb{F}$-derivation) of the
quotient $M_{\operatorname{tot}}(\mathbb{F})/{\mathfrak{I}}$. We then have the
following consequence of Theorem B (see Theorem 9.3.4):

######  Theorem C.

If $\tau\in\Sigma_{{\mathfrak{p}},0}$, then the sequence

$0\longrightarrow
M_{\operatorname{tot}}(\mathbb{F})/{\mathfrak{I}}\stackrel{{\scriptstyle
V_{\mathfrak{p}}}}{{\longrightarrow}}M_{\operatorname{tot}}(\mathbb{F})/{\mathfrak{I}}\stackrel{{\scriptstyle\Theta_{\tau}}}{{\longrightarrow}}M_{\operatorname{tot}}(\mathbb{F})/{\mathfrak{I}}$

is exact.

In §9.2 we apply our results to refine the main result of [10], which we
recall states that minimal weights of non-zero forms always lie in
$\Xi^{\min}$. The geometric Serre weight conjectures of [12] (and its
forthcoming generalization to the ramified case) predict that if $f$ is a mod
$p$ Hecke eigenform which is non-Eisenstein (in the sense that the associated
Galois representation is irreducible), then $\mathbf{k}_{\min}(f)$ should be
totally positive. We use Theorems A and B to prove a partial result in this
direction (Theorem 9.2.1):

######  Theorem D.

Suppose that ${\mathfrak{p}}\in S_{p}$ is such that
$F_{\mathfrak{p}}\neq{\mathbb{Q}}_{p}$ and $p^{f_{\mathfrak{p}}}>3$. Suppose
that $f\in M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})$ is non-zero and
$\mathbf{k}=\mathbf{k}_{\min}(f)$. If $k_{\theta}=0$ for some
$\theta\in\Sigma_{\mathfrak{p}}$, then $\mathbf{k}=\bf{0}$.

Since the Hecke action on forms of weight $(\bf{0},\mathbf{l})$ is Eisenstein
(see Proposition 3.2.2), the theorem implies the total positivity of minimal
weights of non-Eisenstein eigenforms in many situations, for example if $p>3$
and there are no primes ${\mathfrak{p}}\in S_{p}$ such that
$F_{\mathfrak{p}}={\mathbb{Q}}_{p}$. We remark that the hypothesis
$F_{\mathfrak{p}}\neq{\mathbb{Q}}_{p}$ cannot be removed from Theorem D: if
$\Sigma_{\mathfrak{p}}=\\{\theta\\}$, then there are non-zero forms whose
minimal weight $\mathbf{k}$ satisfies $k_{\theta}=0$ and
$k_{\theta^{\prime}}>0$ for some $\theta^{\prime}\neq\theta$. However
forthcoming work with Kassaei will show that the Hecke action on such forms is
Eisenstein; like in [9, 10], the case of split primes seems to require a
completely different method. Unfortunately the case of $p\leq 3$,
$f_{\mathfrak{p}}=1$, $e_{\mathfrak{p}}>1$ slips through the crack between the
two methods. We do not know whether Theorem D should hold in this case, but we
still at least conjecture the failure is Eisenstein.

Acknowledgements: The author would like to thank Payman Kassaei and Shu Sasaki
for numerous helpful conversations, particularly in the context of
collaboration on forthcoming work that motivates the results in this paper. It
is also a pleasure to acknowledge the evident debt this work owes to that of
Reduzzi and Xiao in [32]. The author is also grateful to Deo, Dimitrov and
Wiese, whose interest in partial $\Theta$-operators (and their kernels)
encouraged him to explore the ramifications of the approach in [12], and to
the referee for pointing out a number of typos and useful references.

##  2\. Preliminaries

###  2.1. Embeddings and decompositions

We first set out notation and conventions for various constructions associated
to the set of embeddings of a totally real field $F$, which together with a
prime $p$, will be fixed throughout the paper.

We assume that $F$ has degree $d=[F:\mathbb{Q}]>1$, let ${\mathcal{O}}_{F}$
denote its ring of integers, ${\mathfrak{d}}$ its different, and $\Sigma$ the
set of embeddings $F\to{\overline{{\mathbb{Q}}}}$, where
${\overline{{\mathbb{Q}}}}$ is the algebraic closure of ${\mathbb{Q}}$ in
${\mathbb{C}}$.

We also fix an embedding
${\overline{{\mathbb{Q}}}}\to\overline{{\mathbb{Q}}}_{p}$. We let $S_{p}$
denote the set of primes of ${\mathcal{O}}_{F}$ dividing $p$, and identify
$\Sigma$ with $\coprod_{{\mathfrak{p}}\in S_{p}}\Sigma_{{\mathfrak{p}}}$ under
the natural bijection, where $\Sigma_{{\mathfrak{p}}}$ denotes the set of
embeddings $F_{{\mathfrak{p}}}\to\overline{{\mathbb{Q}}}_{p}$.

For each ${\mathfrak{p}}\in S_{p}$, we let $F_{{\mathfrak{p}},0}$ denote the
maximal unramified subextension of $F_{\mathfrak{p}}$, which we identify with
the field of fractions of $W({\mathcal{O}}_{F}/{\mathfrak{p}})$. We also let
$f_{\mathfrak{p}}$ denote the residue degree
$[F_{{\mathfrak{p}},0}:{\mathbb{Q}}_{p}]$, $e_{\mathfrak{p}}$ the ramification
index $[F_{\mathfrak{p}}:F_{{\mathfrak{p}},0}]$, and
$\Sigma_{{\mathfrak{p}},0}$ the set of embeddings
$F_{{\mathfrak{p}},0}\to\overline{{\mathbb{Q}}}_{p}$, which we may identify
with the set of embeddings
${\mathcal{O}}_{F}/{\mathfrak{p}}\to\overline{{\mathbb{F}}}_{p}$, or
homomorphisms $W({\mathcal{O}}_{F}/{\mathfrak{p}})\to
W(\overline{{\mathbb{F}}}_{p})$. For each ${\mathfrak{p}}\in S_{p}$, we fix a
choice of embedding $\tau_{{\mathfrak{p}},0}\in\Sigma_{{\mathfrak{p}},0}$, and
for $i\in{\mathbb{Z}}/f_{\mathfrak{p}}{\mathbb{Z}}$, we let
$\tau_{{\mathfrak{p}},i}=\phi^{i}\circ\tau_{{\mathfrak{p}},0}$ where $\phi$ is
the Frobenius automorphism of $\overline{{\mathbb{F}}}_{p}$ (or
$W(\overline{{\mathbb{F}}}_{p})$ or its field of fractions), so that
$\Sigma_{{\mathfrak{p}},0}=\\{\tau_{{\mathfrak{p}},1},\tau_{{\mathfrak{p}},2},\ldots,\tau_{{\mathfrak{p}},f_{{\mathfrak{p}}}}\\}$.
We also let $\Sigma_{0}=\coprod_{{\mathfrak{p}}\in
S_{p}}\Sigma_{{\mathfrak{p}},0}$. Letting
${\mathfrak{q}}=\prod_{{\mathfrak{p}}\in S_{p}}{\mathfrak{p}}$ denote the
radical of $p$ in ${\mathcal{O}}_{F}$, note that $\Sigma_{0}$ may also be
identified with the set of ring homomorphisms
${\mathcal{O}}_{F}/{\mathfrak{q}}\to\overline{{\mathbb{F}}}_{p}$ (or indeed
${\mathcal{O}}_{F}\to\overline{{\mathbb{F}}}_{p}$).

For each $\tau=\tau_{{\mathfrak{p}},i}\in\Sigma_{0}$, we let
$\Sigma_{\tau}\subset\Sigma_{\mathfrak{p}}$ denote the set of embeddings
restricting to $\tau$, for which we choose an ordering
$\theta_{{\mathfrak{p}},i,1},\theta_{{\mathfrak{p}},i,2},\ldots,\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$,
so that

$\Sigma=\coprod_{\tau\in\Sigma_{0}}\Sigma_{\tau}=\\{\,\theta_{{\mathfrak{p}},i,j}\,|\,{\mathfrak{p}}\in
S_{p},i\in\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z},1\leq j\leq
e_{\mathfrak{p}}\,\\}.$

We also define a permutation $\sigma$ of $\Sigma$ whose restriction to each
$\Sigma_{\mathfrak{p}}$ is the $e_{\mathfrak{p}}f_{\mathfrak{p}}$-cycle
corresponding to the right shift of indices with respect to the lexicographic
ordering, i.e.,

$\begin{array}[]{ccccccccc}(1,1)&\mapsto&(1,2)&\mapsto&\cdots&\mapsto&(1,e_{\mathfrak{p}})&\mapsto&\\\
(2,1)&\mapsto&(2,2)&\mapsto&\cdots&\mapsto&(2,e_{\mathfrak{p}})&\mapsto&\\\
&&&&\vdots&&&&\\\
(f_{\mathfrak{p}},1)&\mapsto&(f_{\mathfrak{p}},2)&\mapsto&\cdots&\mapsto&(f_{\mathfrak{p}},e_{\mathfrak{p}})&\mapsto&(1,1).\end{array}$

Let $E\subset{\overline{{\mathbb{Q}}}}$ be a number field containing the image
of $\theta$ for all $\theta\in\Sigma$, let ${\mathcal{O}}$ be the completion
of ${\mathcal{O}}_{E}$ at the prime determined by the choice of
${\overline{{\mathbb{Q}}}}\to\overline{{\mathbb{Q}}}_{p}$, and let
$\mathbb{F}$ be its residue field. For any
${\mathcal{O}}_{F,p}={\mathcal{O}}_{F}\otimes\mathbb{Z}_{p}$-module $M$, we
write $M=\bigoplus_{{\mathfrak{p}}\in S_{p}}M_{\mathfrak{p}}$ for the
decomposition obtained from that of

${\mathcal{O}}_{F,p}\cong\prod_{{\mathfrak{p}}\in
S_{p}}{\mathcal{O}}_{F,{\mathfrak{p}}}.$

Similarly for any
$W({\mathcal{O}}_{F}/{\mathfrak{p}})\otimes_{\mathbb{Z}_{p}}{\mathcal{O}}$-module
$N$, we have a decomposition
$N=\bigoplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}N_{\tau}$ obtained from

$W({\mathcal{O}}_{F}/{\mathfrak{p}})\otimes_{\mathbb{Z}_{p}}{\mathcal{O}}\cong\prod_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\mathcal{O}}.$

In particular, for any ${\mathcal{O}}_{F}\otimes{\mathcal{O}}$-module $M$, we
have the decomposition

$M=\bigoplus_{{\mathfrak{p}}\in
S_{p}}M_{\mathfrak{p}}=\bigoplus_{\tau\in\Sigma_{0}}M_{\tau},$

where we simply write $M_{\tau}$ for $M_{{\mathfrak{p}},\tau}$. We also write
$M_{{\mathfrak{p}},i}$ for $M_{\tau}$ if $\tau=\tau_{{\mathfrak{p}},i}$; thus
$M_{{\mathfrak{p}},i}$ is the summand of the
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{\mathbb{Z}_{p}}{\mathcal{O}}$-module
$M_{\mathfrak{p}}$ on which $W({\mathcal{O}}_{F}/{\mathfrak{p}})$ acts via
$\tau_{{\mathfrak{p}},i}$.

We also fix a choice of uniformizer $\varpi_{\mathfrak{p}}$ for each
${\mathfrak{p}}\in S_{p}$. We let $f_{\mathfrak{p}}(u)$ denote the minimal
polynomial of $\varpi_{\mathfrak{p}}$ over
$W({\mathcal{O}}_{F}/{\mathfrak{p}})$, and let $f_{\tau}$ denote its image in
${\mathcal{O}}[u]$ for each $\tau\in\Sigma_{{\mathfrak{p}},0}$; thus
$u\mapsto\varpi_{\mathfrak{p}}\otimes 1$ induces an isomorphism

${\mathcal{O}}[u]/(f_{\tau}(u))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}.$

Furthermore we have
$f_{\tau}(u)=\prod_{\theta\in\Sigma_{\tau}}(u-\theta(\varpi_{\mathfrak{p}}))$,
and we define elements

(1)
$\begin{array}[]{rcc}s_{\tau,j}&=&(u-\theta_{{\mathfrak{p}},i,1}(\varpi_{\mathfrak{p}}))\cdots(u-\theta_{{\mathfrak{p}},i,j}(\varpi_{\mathfrak{p}}))\\\
\mbox{and}\quad
t_{\tau,j}&=&(u-\theta_{{\mathfrak{p}},i,j+1}(\varpi_{\mathfrak{p}}))\cdots(u-\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}(\varpi_{\mathfrak{p}}))\end{array}$

of ${\mathcal{O}}[u]/(f_{\tau}(u))$ for $j=0,\ldots,e_{\mathfrak{p}}$ (with
the obvious convention that $s_{\tau,0}=t_{\tau,e_{\mathfrak{p}}}=1$). Note
that each of the ideals $(s_{\tau,j})$ and $(t_{\tau,j})$ is the other’s
annihilator; furthermore the quotients of ${\mathcal{O}}[u]/(f_{\tau}(u))$ by
these ideals are free over ${\mathcal{O}}$, and the corresponding ideals in
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}$
may be described as kernels of projection maps to products of copies of
${\mathcal{O}}$, hence depend only on $j$ and the ordering of embeddings, and
not on the choice of uniformizer $\varpi_{\mathfrak{p}}$.

For an invertible ${\mathcal{O}}_{F}$-module $L$ and an embedding
$\theta=\theta_{{\mathfrak{p}},i,j}\in\Sigma_{\tau}$, we define $L_{\theta}$
to be the free rank one ${\mathcal{O}}$-module

(2)
$L_{\theta}=t_{\tau,j}(L\otimes{\mathcal{O}})_{\tau}\otimes_{{\mathcal{O}}[u],\theta}{\mathcal{O}}.$

Note that $L_{\theta}$ is not to be identified with
$L\otimes_{{\mathcal{O}}_{F},\theta}{\mathcal{O}}$; rather there is a
canonical map $L_{\theta}\to L\otimes_{{\mathcal{O}}_{F},\theta}{\mathcal{O}}$
which is an isomorphism if and only if $j=e_{\mathfrak{p}}$. If $L$ and
$L^{\prime}$ are invertible ${\mathcal{O}}_{F}$-modules, we will write
$LL^{\prime}$ for $L\otimes_{{\mathcal{O}}_{F}}L^{\prime}$ and $L^{-1}$ for
${\operatorname{Hom}\,}_{{\mathcal{O}}_{F}}(L,{\mathcal{O}}_{F})$. Note that
there are natural maps
$L_{\theta}\otimes_{{\mathcal{O}}}L^{\prime}_{\theta}\to(LL^{\prime})_{\theta}$
and
$(L^{-1})_{\theta}\to{\operatorname{Hom}\,}_{{\mathcal{O}}}(L_{\theta},{\mathcal{O}})$,
but again these are isomorphisms if and only if $j=e_{\mathfrak{p}}$.

###  2.2. Pappas–Rapoport models

In this section we recall the description of the Hilbert modular variety as a
coarse moduli space for abelian varieties with additional structure, along
with the construction by Pappas and Rapoport of a smooth integral model (see
[29] and [33]).

Let $G={\operatorname{Res}}_{F/\mathbb{Q}}\operatorname{GL}_{2}$ and let $U$
be an open compact subgroup of
$\operatorname{GL}_{2}(\widehat{{\mathcal{O}}}_{F})\subset\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}})=G({\mathbb{A}}_{\mathbf{f}})$
of the form $U_{p}U^{p}$, where
$U_{p}=\operatorname{GL}_{2}({\mathcal{O}}_{F,p})$ and
$U^{p}\subset\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ is
sufficiently small, in a sense to be specified below.

We consider the functor which associates, to a locally Noetherian
${\mathcal{O}}$-scheme $S$, the set of isomorphism classes of data
$(A,\iota,\lambda,\eta,{\mathcal{F}}^{\bullet})$, where:

  * •

$s:A\to S$ is an abelian scheme of relative dimension $d$;

  * •

$\iota:{\mathcal{O}}_{F}\to{\operatorname{End}\,}_{S}(A)$ is an embedding such
that $(s_{*}\Omega^{1}_{A/S})_{\mathfrak{p}}$ is, locally on $S$, free of rank
$e_{\mathfrak{p}}$ over
$W({\mathcal{O}}_{F}/{\mathfrak{p}})\otimes_{\mathbb{Z}_{p}}{\mathcal{O}}_{S}$
for each ${\mathfrak{p}}\in S_{p}$;

  * •

$\lambda$ is an ${\mathcal{O}}_{F}$-linear quasi-polarization of $A$ such that
for each connected component $S_{i}$ of $S$, $\lambda$ induces an isomorphism
${\mathfrak{c}}_{i}{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}A_{S_{i}}\to
A_{S_{i}}^{\vee}$ for some fractional ideal ${\mathfrak{c}}_{i}$ of $F$ prime
to $p$;

  * •

$\eta$ is a level $U^{p}$ structure on $A$, i.e., for a choice of geometric
point $\overline{s}_{i}$ on each connected component $S_{i}$ of $S$, the data
of a $\pi_{1}(S_{i},\overline{s}_{i})$-invariant $U^{p}$-orbit of
$\widehat{{\mathcal{O}}}_{F}^{(p)}={\mathcal{O}}_{F}\otimes\widehat{{\mathbb{Z}}}^{(p)}$-linear
isomorphisms555Note the conventions in place with respect to the different,
which are motivated by the point of view that we wish to systematically
trivialize modules defined by cohomological constructions.

$\eta_{i}:(\widehat{{\mathcal{O}}}_{F}^{(p)})^{2}\to{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}T^{(p)}(A_{\overline{s}_{i}}),$

where $T^{(p)}$ denotes the product over $\ell\neq p$ of the $\ell$-adic Tate
modules, and $g\in U^{p}$ acts on $\eta_{i}$ by pre-composing with right
multiplication by $g^{-1}$;

  * •

${\mathcal{F}}^{\bullet}$ is a collection of Pappas–Rapoport filtrations,
i.e., for each $\tau=\tau_{{\mathfrak{p}},i}\in\Sigma_{0}$, an increasing
filtration of
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}_{S}$-modules

$0={\mathcal{F}}_{\tau}^{(0)}\subset{\mathcal{F}}_{\tau}^{(1)}\subset\cdots\subset{\mathcal{F}}_{\tau}^{(e_{\mathfrak{p}}-1)}\subset{\mathcal{F}}_{\tau}^{(e_{\mathfrak{p}})}=(s_{*}\Omega_{A/S}^{1})_{\tau}$

such that for $j=1,\ldots,e_{{\mathfrak{p}}}$, the quotient

${{\mathcal{L}}}_{{\mathfrak{p}},i,j}:={\mathcal{F}}_{\tau}^{(j)}/{\mathcal{F}}_{\tau}^{(j-1)}$

is a line bundle on $S$ on which ${\mathcal{O}}_{F}$ acts via
$\theta_{{\mathfrak{p}},i,j}$.

The proof of [12, Lemma 2.4.1] does not assume $p$ is unramified in $F$, and
shows that if $U^{p}$ is sufficiently small and $\alpha$ is an automorphism of
a triple $(A,\iota,\eta)$ over a connected scheme $S$, then
$\alpha=\iota(\mu)$ for some $\mu\in U\cap{\mathcal{O}}_{F}^{\times}$. If we
assume further that $-1\not\in U\cap{\mathcal{O}}_{F}^{\times}$, then it
follows from standard arguments that the functor above is representable by an
infinite disjoint union of quasi-projective schemes over ${\mathcal{O}}$,
which we denote by $\widetilde{Y}_{U}$, and the argument in the proof of [33,
Prop. 6] shows that $\widetilde{Y}_{U}$ is smooth of relative dimension $d$
over ${\mathcal{O}}$. Furthermore defining an action of
${\mathcal{O}}_{F,(p),+}^{\times}$ on $\widetilde{Y}_{U}$ by

$\nu\cdot(A,\iota,\lambda,\eta,{\mathcal{F}}^{\bullet})=(A,\iota,\nu\lambda,\eta,{\mathcal{F}}^{\bullet})$

(as in [11, §2.1.3]), we see that the resulting action of
${\mathcal{O}}_{F,(p),+}^{\times}/(U\cap{\mathcal{O}}_{F}^{\times})^{2}$ is
free and the quotient is representable by a smooth quasi-projective scheme
over ${\mathcal{O}}$, which we denote by $Y_{U}$.

We also have a natural right action of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ on the inverse
system of schemes $Y_{U}$ induced by pre-composing the level structure $\eta$
with right-multiplication by $g^{-1}$. More precisely suppose that $U_{1}$ and
$U_{2}$ are as above (with $U_{1}^{p}$ and $U_{2}^{p}$ sufficiently small) and
$g\in\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ is such that
$g^{-1}U_{1}g\subset U_{2}$. Letting
$(A,\iota,\lambda,\eta,{\mathcal{F}}^{\bullet})$ denote the universal object
over $\widetilde{Y}_{U_{1}}$, there is a prime-to-$p$ quasi-isogeny $A\to
A^{\prime}$ of abelian varieties with ${\mathcal{O}}_{F}$-action inducing
isomorphisms
${\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}T^{(p)}(A^{\prime}_{\overline{s}_{i}})\cong\eta_{i}((\widehat{{\mathcal{O}}}_{F}^{(p)})^{2}g^{-1})$
for each $i$, from which we obtain a level $U_{2}$-structure
$\eta^{\prime}=\eta\circ r_{g^{-1}}$ on $A^{\prime}$ (where $r_{g^{-1}}$
denotes right multiplication by $g^{-1}$). Together with the other data
inherited from $A$, we obtain an object
$(A^{\prime},\iota^{\prime},\lambda^{\prime},\eta^{\prime},{\mathcal{F}}^{\prime\bullet})$
corresponding to a morphism
$\widetilde{\rho}_{g}:\widetilde{Y}_{U_{1}}\to\widetilde{Y}_{U_{2}}$ and
descending to a morphism $Y_{U_{1}}\to Y_{U_{2}}$ which we denote $\rho_{g}$.
These morphisms satisfy the evident compatibility
$\rho_{g_{2}}\circ\rho_{g_{1}}=\rho_{g_{1}g_{2}}$ whenever
$g_{1}^{-1}U_{1}g_{1}\subset U_{2}$ and $g_{2}^{-1}U_{2}g_{2}\subset U_{3}$.

Finally we remark that the schemes $Y_{U}$ define smooth integral models over
${\mathcal{O}}$ for the Hilbert modular varieties associated to the group $G$
(with the usual choice of Shimura datum), and their generic fibres and
resulting $\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$-action
may be identified with those obtained from a system of canonical models. In
particular for any ${\mathcal{O}}\to\mathbb{C}$, we have isomorphisms

$Y_{U}(\mathbb{C})\cong\operatorname{GL}_{2}(F)_{+}\backslash(\mathfrak{H}^{\Sigma}\times\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}})/U)\cong\operatorname{GL}_{2}({\mathcal{O}}_{F,(p)})_{+}\backslash(\mathfrak{H}^{\Sigma}\times\operatorname{GL}_{2}({\mathbb{A}}^{(p)}_{F,\mathbf{f}})/U^{p})$

compatible with the right action of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ on the inverse
system, and inducing a bijection between the set of geometric components of
$Y_{U}$ and

${\mathbb{A}}_{F,\mathbf{f}}^{\times}/F_{+}^{\times}\det(U)\cong({\mathbb{A}}_{F,\mathbf{f}}^{(p)})^{\times}/{\mathcal{O}}_{F,(p),+}^{\times}\det(U^{p})$

These isomorphisms arise in turn from ones of the form

$\widetilde{Y}_{U}(\mathbb{C})\cong\operatorname{SL}_{2}({\mathcal{O}}_{F,(p)})\backslash(\mathfrak{H}^{\Sigma}\times\operatorname{GL}_{2}({\mathbb{A}}^{(p)}_{F,\mathbf{f}})/U^{p}),$

under which the set of geometric components of $\widetilde{Y}_{U}$ is
described by $({\mathbb{A}}_{F,\mathbf{f}}^{(p)})^{\times}/\det(U^{p})$.

##  3\. Automorphic bundles

###  3.1. Pairings and duality

Before introducing the line bundles whose sections define the automorphic
forms of interest in the paper, we present a plethora of perfect pairings
provided by Poincaré duality.

We fix a sufficiently small $U$ as in §2.2 and consider the de Rham cohomology
sheaves
${\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)=\mathbb{R}^{1}s_{*}\Omega^{\bullet}_{A/S}$
on the universal abelian scheme $A$ over $S=\widetilde{Y}_{U}$. Recall that
these sheaves are locally free of rank two over
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}$. Furthermore Poincaré duality and
the polarization $\lambda$ induce an
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}$-linear isomorphism

$\begin{array}[]{ccccc}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}&{\mathcal{H}om}_{{\mathcal{O}}_{S}}({\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\vee}/S),{\mathcal{O}}_{S})&\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}&{\mathcal{H}om}_{{\mathcal{O}}_{S}}({\mathcal{H}}^{1}_{\operatorname{dR}}({\mathfrak{c}}{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}A)/S),{\mathcal{O}}_{S})\\\
&&&&\wr\parallel\\\
&&&&{\mathcal{H}om}_{{\mathcal{O}}_{S}}({\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S),{\mathcal{O}}_{S})\end{array}$

(where ${\mathfrak{c}}$ depends on the connected component of $S$ and
disappears from the last expression since it is prime to $p$). This in turn
induces
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}_{S}$-linear
isomorphisms

(3)
${\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)_{\tau}\cong{\mathcal{H}om}_{{\mathcal{O}}_{S}}({\mathfrak{d}}_{\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F,{\mathfrak{p}}}}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)_{\tau},{\mathcal{O}}_{S})$

which we view as defining a perfect ${\mathcal{O}}_{S}$-bilinear pairing
$\langle\cdot,\cdot\rangle_{\tau}^{0}$ between
${\mathcal{H}}_{\tau}:={\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)_{\tau}$ and
${\mathfrak{d}}_{{\mathfrak{p}}}^{-1}\otimes_{{\mathcal{O}}_{F,{\mathfrak{p}}}}{\mathcal{H}}_{\tau}$
for $\tau\in\Sigma_{{\mathfrak{p}},0}$. Furthermore the pairing is alternating
in the sense that $\langle x,c\otimes y\rangle_{\tau}^{0}=-\langle y,c\otimes
x\rangle_{\tau}^{0}$ on sections. Alternatively, we may apply the canonical
${\mathcal{O}}_{F}\otimes R$-linear isomorphism

${\operatorname{Hom}\,}_{{\mathcal{O}}_{F}\otimes
R}(M,{\mathcal{O}}_{F}\otimes
R)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\operatorname{Hom}\,}_{R}({\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}M,R)$

induced by the trace for any ${\mathcal{O}}_{F}\otimes R$-module $M$ to obtain
an ${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}$-linear isomorphism

(4)
${\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{H}om}_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}}({\mathfrak{c}}^{-1}\otimes_{{\mathcal{O}}_{F}}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S),{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}),$

and hence a perfect alternating
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}_{S}$-bilinear
pairing $\langle\cdot,\cdot\rangle_{\tau}$ on ${\mathcal{H}}_{\tau}$.

Note that ${\mathcal{H}}_{\tau}$ is locally free of rank two over
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}_{S}$,
hence a vector bundle of rank $2e_{\mathfrak{p}}$ over ${\mathcal{O}}_{S}$.
Furthermore $(s_{*}\Omega^{1}_{A/S})_{\tau}$ is a subbundle of
${\mathcal{H}}_{\tau}$ of rank $e_{\mathfrak{p}}$, but is not locally free
over
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}_{S}$
if $e_{\mathfrak{p}}>1$ (in which case the failure is on a closed subscheme of
codimension one), and more generally ${\mathcal{F}}_{\tau}^{(j)}$ is a
subbundle of rank $j$ for $j=0,1,\ldots,e_{\mathfrak{p}}$.

Recall that for $j=0,1,\ldots,e_{\mathfrak{p}}$, we defined (see (1)) elements
$s_{\tau,j}$ and $t_{\tau,j}$ of the ring
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}}\cong{\mathcal{O}}[u]/(f_{\tau}(u))$,
where $f_{\tau}$ is the image under $\tau$ of the Eisenstein polynomial
associated to our choice of uniformizer $\varpi_{\mathfrak{p}}$. In the
following, we shall fix $\tau$ and omit the subscripts $\tau$,
${\mathfrak{p}}$ and $i$ to disencumber the notation; we also write simply $W$
for $W({\mathcal{O}}_{F}/{\mathfrak{p}})$. Note that since ${\mathcal{H}}$ is
locally free over
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}$ and
${\mathcal{F}}^{(j)}$ is annihilated by $s_{j}$, we have that
${\mathcal{F}}^{(j)}$ is in fact a subbundle of $t_{j}{\mathcal{H}}$.

For a subsheaf ${\mathcal{E}}\subset{\mathcal{H}}$ of
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}$-submodules, we
define ${\mathcal{E}}^{\perp}$ to be its orthogonal complement under the
pairing $\langle\cdot,\cdot\rangle$, i.e., the kernel of the morphism:

${\mathcal{H}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({\mathcal{H}},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})\longrightarrow{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({\mathcal{E}},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}),$

or equivalently the orthogonal complement of
${\mathfrak{d}}_{\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F,{\mathfrak{p}}}}{\mathcal{E}}$
under the pairing $\langle\cdot,\cdot\rangle^{0}$. Note from the latter
description that if ${\mathcal{E}}$ is an ${\mathcal{O}}_{S}$-subbundle of
${\mathcal{H}}$, then so is ${\mathcal{E}}^{\perp}$.

######  Lemma 3.1.1.

We have the equality
$({\mathcal{F}}^{(j)})^{\perp}=t_{j}^{-1}{\mathcal{F}}^{(j)}$ for
$j=0,1,\ldots,e$, where
$t_{j}^{-1}{\mathcal{F}}^{(j)}=\ker({\mathcal{H}}\stackrel{{\scriptstyle
t_{j}}}{{\longrightarrow}}t_{j}{\mathcal{H}}\longrightarrow
t_{j}{\mathcal{H}}/{\mathcal{F}}^{(j)})$ is the preimage sheaf of
${\mathcal{F}}^{(j)}$ under $t_{j}$.

Proof. We prove the lemma by induction on $j$, the case of $j=0$ being
obvious.

Suppose then that $1\leq j\leq e$ and that
$({\mathcal{F}}^{(j-1)})^{\perp}=t_{j-1}^{-1}{\mathcal{F}}^{(j-1)}$. Note that
$({\mathcal{F}}^{(j)})^{\perp}$ and $t_{j}^{-1}{\mathcal{F}}^{(j)}$ are both
kernels of surjective morphisms from ${\mathcal{H}}$ to vector bundles of rank
$j$ on $S$, so each is a subbundle of rank $2e-j$, and hence it suffices to
prove the inclusion
$t_{j}^{-1}{\mathcal{F}}^{(j)}\subset({\mathcal{F}}^{(j)})^{\perp}$. To do so,
we may work locally on $S$, and assume that
${\mathcal{F}}^{(j)}(V)={\mathcal{F}}^{(j-1)}(V)\oplus Rx_{j}$ where
$V={\operatorname{Spec}\,}R$ is a Noetherian open subscheme of
$\widetilde{Y}_{U}$ such that ${\mathcal{H}}|_{V}$ is free over
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{V}$ and
$x_{j}\in{\mathcal{H}}(V)$ satisfies
$(u-\theta_{j}(\varpi))x_{j}\in{\mathcal{F}}^{(j-1)}(V)$. In particular
$x_{j}=t_{j}y_{j}$ for some $y_{j}\in t_{j-1}^{-1}{\mathcal{F}}^{(j-1)}(V)$,
so that

$t_{j}^{-1}{\mathcal{F}}^{(j)}(V)=t_{j}^{-1}{\mathcal{F}}^{(j-1)}(V)\oplus
Ry_{j}\quad\mbox{and}\quad({\mathcal{F}}^{(j)}(V))^{\perp}=({\mathcal{F}}^{(j-1)}(V))^{\perp}\cap(Rx_{j})^{\perp}.$

Note that if $w\in t_{j}^{-1}{\mathcal{F}}^{(j)}(V)$, then

$t_{j-1}w=(u-\theta_{j}(\varpi))t_{j}w\in(u-\theta_{j}(\varpi)){\mathcal{F}}^{(j)}(V)\subset{\mathcal{F}}^{(j-1)}(V),$

so that $w\in
t_{j-1}^{-1}{\mathcal{F}}^{(j-1)}(V)=({\mathcal{F}}^{(j-1)})(V))^{\perp}$.
Furthermore if $w\in t_{j}^{-1}{\mathcal{F}}^{(j-1)}(V)$, then

$\langle w,x_{j}\rangle=\langle w,t_{j}y_{j}\rangle=\langle
t_{j}w,y_{j}\rangle$

since $t_{j}w\in{\mathcal{F}}^{(j-1)}$ and $y_{j}\in
t_{j-1}^{-1}{\mathcal{F}}^{(j-1)}(V)=({\mathcal{F}}^{(j-1)}(V))^{\perp}$.
Finally since the pairing is alternating, we have

$\langle y_{j},x_{j}\rangle=\langle y_{j},t_{j}y_{j}\rangle=\langle
t_{j}y_{j},y_{j}\rangle=\langle x_{j},y_{j}\rangle=-\langle
y_{j},x_{j}\rangle,$

which implies that $\langle y_{j},x_{j}\rangle=0$ (since $\widetilde{Y}_{U}$
is flat over $\mathbb{Z}_{2}$ if $p=2$).

We have now shown that
$t_{j}^{-1}{\mathcal{F}}^{(j)}(V)\subset({\mathcal{F}}^{(j-1)}(V))^{\perp}\cap(Rx_{j})^{\perp}=({\mathcal{F}}^{(j)}(V))^{\perp}$,
as required.  $\square$

We now define
${\mathcal{G}}^{(j)}=(u-\theta_{j}(\varpi))^{-1}{\mathcal{F}}^{(j-1)}$ for
$j=1,\ldots,e$. Thus ${\mathcal{G}}^{(j)}$ is a rank $j+1$ subbundle of
${\mathcal{H}}$, and we have inclusions of subbundles
${\mathcal{F}}^{(j-1)}\subset{\mathcal{F}}^{(j)}\subset{\mathcal{G}}^{(j)}$,
so that ${{\mathcal{L}}}_{j}:={\mathcal{F}}^{(j)}/{\mathcal{F}}^{(j-1)}$ is a
rank one subbundle of the rank two vector bundle
${{\mathcal{P}}}_{j}:={\mathcal{G}}^{(j)}/{\mathcal{F}}^{(j-1)}$. Furthermore
all the inclusions are morphisms of
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}$-modules, and
${\mathcal{O}}_{F}$ acts on ${{\mathcal{P}}}_{j}$ via $\theta_{j}$.

Note that ${\mathcal{G}}^{(j)}$ is annihilated by $s_{j}$, so that
${\mathcal{G}}^{(j)}\subset t_{j}{\mathcal{H}}$, and we have
$t_{j}^{-1}{\mathcal{G}}^{(j)}=t_{j-1}^{-1}{\mathcal{F}}^{(j-1)}=({\mathcal{F}}^{j-1})^{\perp}$
by Lemma 3.1.1, from which it follows also that

$t_{j}^{-1}{\mathcal{F}}^{(j-1)}=t_{j}^{-1}(t_{j}^{-1}{\mathcal{G}}^{(j)})^{\perp}=({\mathcal{G}}^{(j)})^{\perp}.$

(The last equality can be seen by arguing locally on sections, or by noting
that the diagram

$\textstyle{{\mathcal{H}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({\mathcal{H}},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({\mathcal{G}}^{(j)},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({\mathcal{H}},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}(t_{j}^{-1}{\mathcal{G}}^{(j)},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})}$

commutes, where the left horizontal morphisms are defined by the pairing, the
right by restriction, and all the vertical morphisms by $t_{j}$. The kernel of
the composite along the top is $({\mathcal{G}}^{(j)})^{\perp}$, whereas
$t_{j}^{-1}(t_{j}^{-1}{\mathcal{G}}^{(j)})^{\perp}$ is the kernel of the
composite along the left and bottom. Since
$t_{j}:t_{j}^{-1}{\mathcal{G}}^{(j)}\to{\mathcal{G}}^{(j)}$ is a surjective
morphism of vector bundles, the leftmost vertical arrow is injective, so these
kernels coincide.) Therefore multiplication by $t_{j}$ defines an isomorphism
$({\mathcal{F}}^{(j-1)})^{\perp}/({\mathcal{G}}^{(j)})^{\perp}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{G}}^{(j)}/{\mathcal{F}}^{(j-1)}$,
and composing its inverse with the isomorphism

$({\mathcal{F}}^{(j-1)})^{\perp}/({\mathcal{G}}^{(j)})^{\perp}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({\mathcal{G}}^{(j)}/{\mathcal{F}}^{(j-1)},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})$

induced by the pairing on ${\mathcal{H}}$, we obtain an alternating
$({\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})$-valued
pairing $\langle\cdot,\cdot\rangle_{j}$ on
${{\mathcal{P}}}_{j}:={\mathcal{G}}^{(j)}/{\mathcal{F}}^{(j-1)}$, whose
description on sections is given in terms of (4) by

$\langle t_{j}x,t_{j}y\rangle_{j}=\langle x,t_{j}y\rangle=\langle
t_{j}x,y\rangle.$

Note that since ${\mathcal{O}}_{F,{\mathfrak{p}}}$ acts via $\tau_{j}$ on
${{\mathcal{P}}}_{j}$, we in fact have the identification

${\mathcal{H}om}_{{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}}({{\mathcal{P}}}_{j},{\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S})={\mathcal{H}om}_{{\mathcal{O}}_{S}}({{\mathcal{P}}}_{j},{\mathcal{I}}_{j})$

where ${\mathcal{I}}_{j}$ is the sheaf of ideals, and trivial rank one
${\mathcal{O}}_{S}$-subbundle, of
${\mathcal{O}}_{F,{\mathfrak{p}}}\otimes_{W}{\mathcal{O}}_{S}$ generated by
the global section $\prod_{j^{\prime}\neq
j}(u-\tau_{j^{\prime}}(\varpi_{\mathfrak{p}}))$. We thus obtain a
trivialization of $\wedge^{2}_{{\mathcal{O}}_{S}}{{\mathcal{P}}}_{j}$
corresponding to a perfect ${\mathcal{O}}_{S}$-valued pairing
$\langle\cdot,\cdot\rangle_{j}^{0}$, which an unravelling of definitions shows
is given in terms of the original pairing $\langle\cdot,\cdot\rangle^{0}$ of
(3) by the formula

$\langle t_{j}x,y\rangle_{j}^{0}=\langle
x,f^{\prime}(\varpi_{\pi_{\mathfrak{p}}})^{-1}\otimes y\rangle^{0}$

on sections (where $f^{\prime}$ is the derivative of the Eisenstein polynomial
$f$).

###  3.2. Automorphic line bundles

Recall that the Pappas–Rapoport model $\widetilde{Y}_{U}$ is equipped with
line bundles666For the moment, we continue to suppress the fixed
$\tau=\tau_{{\mathfrak{p}},i}\in\Sigma_{0}$ from the notation.
${\mathcal{L}}_{j}$, which we described in §3.1 as sub-bundles of the rank two
vector bundles ${\mathcal{P}}_{j}$. It is natural and convenient to consider
also the twists of ${\mathcal{L}}_{j}$ by powers of the determinant bundle of
${\mathcal{P}}_{j}$:

${{\mathcal{N}}}_{j}=\wedge^{2}_{{\mathcal{O}}_{S}}{{\mathcal{P}}}_{j}\cong{{\mathcal{L}}}_{j}\otimes_{{\mathcal{O}}_{S}}{{\mathcal{M}}}_{j},$

where ${{\mathcal{M}}}_{j}$ is the line bundle
${{\mathcal{P}}}_{j}/{{\mathcal{L}}}_{j}$. Note that the pairing
$\langle\cdot,\cdot\rangle^{0}_{j}$ defines an isomorphism
${{\mathcal{M}}}_{j}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{{\mathcal{L}}}_{j}^{-1}$
and a trivialization
${\mathcal{O}}_{S}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{{\mathcal{N}}}_{j}$
(which depends on the choice of $\varpi$).

As we will now consider these bundles for varying $\tau$, we resume writing
the indicative subscripts; thus for $\tau=\tau_{{\mathfrak{p}},i}$, we will
denote ${\mathcal{G}}^{(j)}$ by ${\mathcal{G}}_{\tau}^{(j)}$,
${{\mathcal{P}}}_{j}$ by ${{\mathcal{P}}}_{{\mathfrak{p}},i,j}$, and similarly
for ${{\mathcal{M}}}_{j}$ and ${{\mathcal{N}}}_{j}$. We also freely replace
the subscript “${\mathfrak{p}},i,j$” by $\theta$, where
$\theta=\theta_{{\mathfrak{p}},i,j}$, so that for each $\theta\in\Sigma$, we
have now defined a rank two vector bundle ${{\mathcal{P}}}_{\theta}$ and line
bundles ${{\mathcal{L}}}_{\theta}$, ${{\mathcal{M}}}_{\theta}$,
${{\mathcal{N}}}_{\theta}$ on $S=\widetilde{Y}_{U}$, along with exact
sequences

(5)
$0\longrightarrow{{\mathcal{L}}}_{\theta}\longrightarrow{{\mathcal{P}}}_{\theta}\longrightarrow{{\mathcal{M}}}_{\theta}\longrightarrow
0$

and a trivialization of
${{\mathcal{N}}}_{\theta}=\wedge^{2}_{{\mathcal{O}}_{S}}{{\mathcal{P}}}_{\theta}\cong{\mathcal{L}}_{\theta}\otimes_{{\mathcal{O}}_{S}}{{\mathcal{M}}}_{\theta}$.
Furthermore the bundles ${{\mathcal{P}}}_{\theta}$, ${{\mathcal{L}}}_{\theta}$
and ${{\mathcal{M}}}_{\theta}$ are
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}$-subquotients of
${\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)$ on which ${\mathcal{O}}_{F}$ acts
via $\theta$.

Recall that we have an action of ${\mathcal{O}}_{F,(p),+}^{\times}$ on
$\widetilde{Y}_{U}$ defined by multiplication on the quasi-polarization. In
particular if $\nu\in{\mathcal{O}}_{F,(p),+}^{\times}$, then the
identification of $\nu^{*}A$ with $A$ induces an
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}$-linear isomorphism
$\nu^{*}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)\cong{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)$
under which $\nu^{*}{\mathcal{F}}^{\bullet}$ corresponds to
${\mathcal{F}}^{\bullet}$, and we thus obtain isomorphisms
$\alpha_{\nu}:\nu^{*}{{\mathcal{P}}}_{\theta}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{{\mathcal{P}}}_{\theta}$
compatible with (5) and satisfying
$\alpha_{\nu^{\prime}\nu}=\alpha_{\nu}\circ\nu^{*}(\alpha_{\nu^{\prime}})$
(for $\nu,\nu^{\prime}\in{\mathcal{O}}_{F,(p),+}^{\times}$). Recall also that
the action of ${\mathcal{O}}_{F,(p),+}^{\times}$ on $\widetilde{Y}_{U}$
factors through
${\mathcal{O}}_{F,(p),+}^{\times}/(U\cap{\mathcal{O}}_{F}^{\times})^{2}$, the
isomorphism
${\underline{A}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\nu^{*}{\underline{A}}$
being defined by $\iota(\mu^{-1})$ if $\nu=\mu^{2}$ for $\mu\in
U\cap{\mathcal{O}}_{F}^{\times}$, and one finds that the automorphism of
${{\mathcal{P}}}_{\theta}$ obtained from $\alpha_{\nu}$ is multiplication by
$\theta(\mu)$, so the natural action of ${\mathcal{O}}_{F,(p),+}^{\times}$ on
the bundles fails to define descent data with respect to the cover
$\widetilde{Y}_{U}\to Y_{U}$. We do however obtain descent data after taking
suitable tensor products or base-changes of these bundles, which we now
consider.

For any ${\mathcal{O}}$-algebra $R$, we will use $\cdot_{R}$ to denote the
base-change to $R$ of an ${\mathcal{O}}$-scheme $X$, as well as the pull-back
to $X_{R}$ of a quasi-coherent sheaf on $X$. Let
$\\{\,{\mathbf{e}}_{\theta}\,|\,\theta\in\Sigma\,\\}$ denote the standard
basis of $\mathbb{Z}^{\Sigma}$. For $\mathbf{k}=\sum
k_{\theta}{\mathbf{e}}_{\theta}$ and $\mathbf{l}=\sum
l_{\theta}{\mathbf{e}}_{\theta}\in\mathbb{Z}^{\Sigma}$, we define the line
bundle

$\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l}}=\bigotimes_{\theta\in\Sigma}\left({{\mathcal{L}}}_{\theta}^{\otimes
k_{\theta}}\otimes{{\mathcal{N}}}_{\theta}^{\otimes
l_{\theta}}\right)\cong\bigotimes_{\theta\in\Sigma}\left({{\mathcal{L}}}_{\theta}^{\otimes
k_{\theta}+l_{\theta}}\otimes{{\mathcal{M}}}_{\theta}^{\otimes
l_{\theta}}\right)$

on $S=\widetilde{Y}_{U}$, where all tensor products are over
${\mathcal{O}}_{S}$. For $\mathbf{n}=\sum
n_{\theta}{\mathbf{e}}_{\theta}\in\mathbb{Z}^{\Sigma}$, we let
$\chi_{\mathbf{n}}:{\mathcal{O}}_{F}^{\times}\to{\mathcal{O}}^{\times}$ denote
the character defined by
$\chi_{\mathbf{n}}(\mu)=\prod_{\theta}\theta(\mu)^{n_{\theta}}$, and we let
$\chi_{\mathbf{n},R}$ denote the associated $R^{\times}$-valued character. If
$\mathbf{k}$, $\mathbf{l}$, $R$ and $U$ are such that
$\chi_{\mathbf{k}+2\mathbf{l},R}$ is trivial on
${\mathcal{O}}_{F}^{\times}\cap U$, then the action of
${\mathcal{O}}_{F,(p),+}^{\times}$ on
$\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},R}$ (over its action on
$\widetilde{Y}_{U,R}$) factors through
${\mathcal{O}}_{F,(p),+}^{\times}/(U\cap{\mathcal{O}}_{F}^{\times})^{2}$ and
hence defines descent data, in which case we denote the resulting line bundle
on $Y_{U,R}$ by ${\mathcal{A}}_{\mathbf{k},\mathbf{l},R}$.

######  Definition 3.2.1.

For $\mathbf{k}$, $\mathbf{l}$, $U$ and $R$ as above, we call
${\mathcal{A}}_{\mathbf{k},\mathbf{l},R}$ the automorphic line bundle of
weight $(\mathbf{k},\mathbf{l})$ on $Y_{U,R}$, and we define the space of
Hilbert modular forms of weight $(\mathbf{k},\mathbf{l})$ and level $U$ with
coefficients in $R$ to be

$M_{\mathbf{k},\mathbf{l}}(U;R):=H^{0}(Y_{U,R},{\mathcal{A}}_{\mathbf{k},\mathbf{l},R}).$

We note some general situations in which this space is defined:

  * •

The paritious setting: if $w=k_{\theta}+2l_{\theta}$ is independent of
$\theta$, then
$\chi_{\mathbf{k}+2\mathbf{l}}(\mu)=\mathrm{Nm}_{F/\mathbb{Q}}(\mu)^{w}=1$ for
all $\mu\in U\cap{\mathcal{O}}_{F}^{\times}$ (assuming only $U$ is small
enough that $U\cap{\mathcal{O}}_{F}^{\times}$ has no elements of norm $-1$ if
$w$ is odd).

  * •

The mod $p$ setting: if $R$ is any
$\mathbb{F}={\mathcal{O}}/{\mathfrak{m}}_{\mathcal{O}}$-algebra and $U$ is
sufficiently small that $\mu\equiv 1\bmod{\mathfrak{p}}$ for all $\mu\in
U\cap{\mathcal{O}}_{F}^{\times}$ and ${\mathfrak{p}}\in S_{p}$, then
$\theta(\mu)\equiv 1\bmod{\mathfrak{m}}_{\mathcal{O}}$ for all
$\theta\in\Sigma$, so $\chi_{\mathbf{k}+2\mathbf{l},\mathbb{F}}$ is trivial on
$U\cap{\mathcal{O}}_{F}^{\times}$, and hence so is
$\chi_{\mathbf{k}+2\mathbf{l},R}$.

  * •

The torsion setting: if $R$ is an ${\mathcal{O}}/p^{N}{\mathcal{O}}$-algebra
and $U$ is sufficiently small that $\mu\equiv 1\bmod p^{N}{\mathcal{O}}_{F}$
for all $\mu\in U\cap{\mathcal{O}}_{F}^{\times}$, so
$\chi_{\mathbf{k}+2\mathbf{l},{\mathcal{O}}/p^{N}{\mathcal{O}}}$ is trivial on
$U\cap{\mathcal{O}}_{F}^{\times}$, and hence so is
$\chi_{\mathbf{k}+2\mathbf{l},R}$.

We also have a natural left action of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ on the direct limit
over $U$ of the spaces $M_{\mathbf{k},\mathbf{l}}(U;R)$. More precisely
suppose that $U_{1}$ and $U_{2}$ are as above and
$g\in\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ is such that
$g^{-1}U_{1}g\subset U_{2}$, in which case recall that in §2.2 we defined a
morphism $\widetilde{\rho}_{g}:\widetilde{Y}_{U_{1}}\to\widetilde{Y}_{U_{2}}$
descending to a morphism $\rho_{g}:Y_{U_{1}}\to Y_{U_{2}}$. Furthermore the
morphism $\widetilde{\rho}_{g}$ is obtained from a prime-to-$p$ quasi-isogeny
$A\to A^{\prime}$ where $A$ is the universal abelian scheme over
$\widetilde{Y}_{U_{1}}$ and $A^{\prime}$ is the pull-back of the universal
abelian scheme. We thus obtain isomorphisms
$\widetilde{\rho}_{g}^{*}{{\mathcal{P}}}_{\theta,2}\to{{\mathcal{P}}}_{\theta,1}$
compatible with (5) and the action of ${\mathcal{O}}_{F,(p),+}^{\times}$
(augmenting the notation for the automorphic bundles on
$\widetilde{Y}_{U_{i}}$ and $Y_{U_{i},R}$ with the subscript $i$). Note that
$U_{1}\cap{\mathcal{O}}_{F}^{\times}\subset
U_{2}\cap{\mathcal{O}}_{F}^{\times}$, so if
$\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},R,2}$ descends to
$Y_{U_{2},R}$, then $\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},R,1}$
descends to $Y_{U_{1},R}$, and we obtain an isomorphism
$\rho_{g}^{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},R,2}\cong{\mathcal{A}}_{\mathbf{k},\mathbf{l},R,1}$.
We then define $[g]=[g]_{U_{1},U_{2}}:M_{\mathbf{k},\mathbf{l}}(U_{2};R)\to
M_{\mathbf{k},\mathbf{l}}(U_{1};R)$ as the composite

$H^{0}(Y_{U_{2}},{\mathcal{A}}_{\mathbf{k},\mathbf{l},R,2})\stackrel{{\scriptstyle\rho_{g}^{*}}}{{\longrightarrow}}H^{0}(Y_{U_{1}},\rho_{g}^{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},R,2})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(Y_{U_{1}},{\mathcal{A}}_{\mathbf{k},\mathbf{l},R,1}).$

These maps satisfy the obvious compatibility, namely that

$[g_{1}]_{U_{1},U_{2}}\circ[g_{2}]_{U_{2},U_{3}}=[g_{1}g_{2}]_{U_{1},U_{3}}$

whenever $g_{1}^{-1}U_{1}g_{1}\subset U_{2}$ and $g_{2}^{-1}U_{2}g_{2}\subset
U_{3}$, and hence define an action of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ on

(6)
$M_{\mathbf{k},\mathbf{l}}(R):=\varinjlim_{U}M_{\mathbf{k},\mathbf{l}}(U;R)$

(where the limit is over sufficiently small open compact $U$ with respect to
the maps $[1]_{U_{1},U_{2}}$). For paritious $\mathbf{k}$, $\mathbf{l}$ and
any choice of ${\mathcal{O}}\to\mathbb{C}$, we may identify the spaces
$M_{\mathbf{k},\mathbf{l}}(U;\mathbb{C})$ with those of holomorphic Hilbert
modular forms, compatibly with the usual action (up to normalization by a
factor of $||\det g||$ depending on conventions).

Finally we remark that the action of $\nu\in{\mathcal{O}}_{F,(p),+}^{\times}$
on the trivialization of ${{\mathcal{N}}}_{\theta}$ is given by multiplication
by $\theta(\nu)$, so their products do not descend to trivializations of line
bundles on $Y_{U,R}$. However since the stabilizer of each geometric connected
component of $\widetilde{Y}_{U}$ is ${\mathcal{O}}_{F,+}^{\times}\cap\det(U)$,
we can obtain a (non-canonical) trivialization of
${\mathcal{A}}_{{\mathbf{0}},\mathbf{l},R}$ as in [12, Prop. 3.6.1], provided
$\chi_{\mathbf{l},R}$ is trivial on ${\mathcal{O}}_{F,+}^{\times}\cap\det(U)$
and the geometric connected components of $Y_{U}$ are defined over $R$.
Furthermore the same argument as in the proof of [12, Lemma 4.5.1] shows the
following:

######  Proposition 3.2.2.

If $p^{N}R=0$, then the action of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ on
$M_{\bf{0},\mathbf{l}}(R)$ factors through
$\det:\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})\to({\mathbb{A}}_{F,\mathbf{f}}^{(p)})^{\times}$;
furthermore as a representation of
$({\mathbb{A}}_{F,\mathbf{f}}^{(p)})^{\times}$, $M_{\bf{0},\mathbf{l}}(R)$ is
isomorphic to the smooth induction of the character
${\mathcal{O}}_{F,(p),+}^{\times}\to R^{\times}$ defined by
$\alpha\mapsto\prod_{\theta}\theta(\alpha)^{l_{\theta}}$.

###  3.3. The Kodaira–Spencer filtration

In this section we define a filtration on $\Omega^{1}_{Y_{U}/{\mathcal{O}}}$
whose pieces are described by automorphic bundles with weight components
$k_{\theta}=2$, $l_{\theta}=-1$. The construction of the filtration is due to
Reduzzi and Xiao (see [32, §2.8]), but their presentation is complicated by
the fact they wish to prove smoothness simultaneously, and it obscures the
fact that the bundles we denoted ${\mathcal{G}}^{(j)}$ automatically satisfy
the orthogonality condition appearing in the definition of their counterparts
in [32]. We will show below that, with smoothness already established, one can
give a more direct conceptual description of the filtration and its
properties.777The simultaneous treatment in [32] seems natural in view of the
inherent overlap in the analysis of deformations needed for both results.
However the decision not to appeal to the results in [35] and [33] also makes
reference to a perceived minor gap in the proof of [35, Prop 2.11]; we found
no such gap, nor is that result even needed, but we remark that we made
implicit use of [35, Cor. 2.10] when invoking the proof of [33, Prop. 6] to
conclude that $\widetilde{Y}_{U}$ is smooth over ${\mathcal{O}}$. Furthermore
in the case $p=2$, the argument in [32] appeals to a very general flatness
assertion for divided power envelopes for which we could not find a proof or
reference, so it is not used here.

######  Theorem 3.3.1 (Reduzzi–Xiao).

There exists a decomposition

$\Omega^{1}_{Y_{U}/{\mathcal{O}}}=\bigoplus_{{\mathfrak{p}}\in
S_{p}}\bigoplus_{i=1}^{f_{\mathfrak{p}}}\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i},$

together with an increasing filtration

$\begin{array}[]{rcccl}0={\operatorname{Fil}\,}^{0}(\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i})&\subset&{\operatorname{Fil}\,}^{1}(\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i})&\subset&\cdots\\\
&\subset&{\operatorname{Fil}\,}^{e_{\mathfrak{p}}-1}(\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i})&\subset&{\operatorname{Fil}\,}^{e_{\mathfrak{p}}}(\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i})=\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i}\end{array}$

for each ${\mathfrak{p}}\in S_{p}$ and $i=1,\ldots,f_{\mathfrak{p}}$, such
that for each $j=1,\ldots,e_{\mathfrak{p}}$,
${\operatorname{gr}\,}^{j}(\Omega^{1}_{Y_{U}/{\mathcal{O}},{\mathfrak{p}},i})$
is isomorphic to the automorphic bundle
${\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta},{\mathcal{O}}}$,
where $\theta=\theta_{{\mathfrak{p}},i,j}$.

Proof. As usual, we first prove the analogous result for
$S:=\widetilde{Y}_{U}$ and then descend to $Y_{U}$.

We let
$\delta_{0}:S\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Delta\hookrightarrow
Z_{0}$ denote the first infinitesimal thickening of the diagonal embedding,
and we view $\Omega^{1}_{S/{\mathcal{O}}}$ as $\delta_{0}^{*}{\mathcal{I}}$,
where ${\mathcal{I}}$ denotes the sheaf of ideals defining $\Delta$ in
$Z_{0}$. Letting $s:A\to S$ denote the universal abelian scheme, the
transition maps for the crystal
$R^{1}s_{{\operatorname{cris}},*}{\mathcal{O}}_{A/{\mathbb{Z}}_{p}}$ and
canonical isomorphisms with de Rham cohomology yield an
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{Z_{0}}$-linear isomorphism

$\alpha:p_{0}^{*}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q_{0}^{*}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)$

extending the identity on $S\cong\Delta$, where $p_{0},q_{0}:Z_{0}\to S$ are
the two projection maps $Z_{0}\to S$. Since $\alpha$ is
${\mathcal{O}}_{F}$-linear, it follows from the definition of
${\mathcal{P}}_{\tau,1}={\mathcal{G}}_{\tau}^{(1)}$ that $\alpha$ restricts to
an isomorphism

$\alpha_{\tau,1}:p_{0}^{*}{\mathcal{P}}_{\tau,1}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q_{0}^{*}{\mathcal{P}}_{\tau,1}$

for each ${\mathfrak{p}}\in S_{p}$ and $\tau\in\Sigma_{{\mathfrak{p}},0}$.
Furthermore since the composite

$p_{0}^{*}{\mathcal{L}}_{\tau,1}\hookrightarrow
p_{0}^{*}{\mathcal{P}}_{\tau,1}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q_{0}^{*}{\mathcal{P}}_{\tau,1}\twoheadrightarrow
q_{0}^{*}{\mathcal{M}}_{\tau,1}$

has trivial pull-back to $S\cong\Delta$, it factors through a morphism

$\delta_{0,*}{\mathcal{L}}_{\tau,1}=p_{0}^{*}{\mathcal{L}}_{\tau,1}\otimes_{{\mathcal{O}}_{Z_{0}}}({\mathcal{O}}_{Z_{0}}/{\mathcal{I}})\longrightarrow
q_{0}^{*}{\mathcal{M}}_{\tau,1}\otimes_{{\mathcal{O}}_{Z_{0}}}{\mathcal{I}}=\delta_{0,*}{\mathcal{M}}_{\tau,1}\otimes_{{\mathcal{O}}_{Z_{0}}}{\mathcal{I}},$

and hence induces a morphism

$\beta_{\tau,1}:\delta_{0,*}({\mathcal{L}}_{\tau,1}\otimes_{{\mathcal{O}}_{S}}{\mathcal{M}}_{\tau,1}^{-1})\longrightarrow{\mathcal{I}}.$

We then define the sheaf of ideals ${\mathcal{I}}_{\tau,1}$ on $Z_{0}$ to be
the image of $\beta_{\tau,1}$, and we let $Z_{\tau,1}$ denote the subscheme of
$Z_{0}$ defined by ${\mathcal{I}}_{\tau,1}$, and $p_{\tau,1}$ and $q_{\tau,1}$
the resulting projection maps $Z_{\tau,1}\to S$. By construction the pull-back
of $\beta_{\tau,1}$ to $Z_{\tau,1}$ is trivial, and hence so is that of the
morphism $p_{0}^{*}{\mathcal{L}}_{\tau,1}\to q_{0}^{*}{\mathcal{M}}_{\tau,1}$,
which implies that the pull-back of $\alpha$ maps
$p_{\tau,1}^{*}{\mathcal{L}}_{\tau,1}=p_{\tau,1}^{*}{\mathcal{F}}_{\tau}^{(1)}$
isomorphically to
$q_{\tau,1}^{*}{\mathcal{L}}_{\tau,1}=q_{\tau,1}^{*}{\mathcal{F}}_{\tau}^{(1)}$.
It follows from ${\mathcal{O}}_{F}$-linearity that $\alpha$ induces an
isomorphism
$p_{\tau,1}^{*}{\mathcal{G}}_{\tau}^{(2)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q_{\tau,1}^{*}{\mathcal{G}}_{\tau}^{(2)}$
(if $e_{\mathfrak{p}}>1$), and hence an isomorphism

$\alpha_{\tau,2}:p_{\tau,1}^{*}{\mathcal{P}}_{\tau,2}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q_{\tau,1}^{*}{\mathcal{P}}_{\tau,2}.$

The same argument as above now yields a morphism

$\beta_{\tau,2}:\delta_{0,*}({\mathcal{L}}_{\tau,2}\otimes_{{\mathcal{O}}_{S}}{\mathcal{M}}_{\tau,2}^{-1})\longrightarrow{\mathcal{I}}/{\mathcal{I}}_{\tau,1},$

whose image is that of a sheaf of ideals on $Z_{0}$ we denote by
${\mathcal{I}}_{\tau,2}$.

Iterating the above construction thus yields, for each
$\tau\in\Sigma_{{\mathfrak{p}},0}$, a chain of sheaves of ideals

$0={\mathcal{I}}_{\tau,0}\subset{\mathcal{I}}_{\tau,1}\subset\cdots\subset{\mathcal{I}}_{\tau,e_{\mathfrak{p}}}$

on $Z_{0}$ such that $\alpha$ induces

  * •

isomorphisms
$p_{\tau,j}^{*}{\mathcal{F}}_{\tau}^{(j)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q_{\tau,j}^{*}{\mathcal{F}}_{\tau}^{(j)}$,

  * •

and surjections
$\delta_{0,*}({\mathcal{L}}_{\tau,j}\otimes_{{\mathcal{O}}_{S}}{\mathcal{M}}_{\tau,j}^{-1})\twoheadrightarrow{\mathcal{I}}_{\tau,j}/{\mathcal{I}}_{\tau,j-1}$,

for $j=1,\ldots,e_{\mathfrak{p}}$, where $Z_{\tau,j}$ denotes the closed
subscheme of $Z_{0}$ defined by ${\mathcal{I}}_{\tau,j}$ and $p_{\tau,j}$,
$q_{\tau,j}$ are the projections $Z_{\tau,j}\to S$.

Furthermore we claim that the map

$\bigoplus_{{\mathfrak{p}}\in
S_{p}}\bigoplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\mathcal{I}}_{\tau,e_{\mathfrak{p}}}\to{\mathcal{I}}$

is surjective. Indeed let ${\mathcal{J}}$ denote the image and let $T$ denote
the corresponding closed subscheme of $Z_{0}$, so $T$ is the scheme-theoretic
intersection of the $Z_{\tau,e_{\mathfrak{p}}}$, and let888With apologies for
the temporary dual use of $p$. $p$, $q:T\to S$ denote the projection maps. By
construction $\alpha$ pulls back to an isomorphism
$p^{*}{\mathcal{H}}^{1}_{{\operatorname{dR}}}(A/S)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q^{*}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)$
under which
$p^{*}{\mathcal{F}}_{\tau}^{(j)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q^{*}{\mathcal{F}}_{\tau}^{(j)}$
for all $\tau$ and $j$. In particular
$t_{*}\Omega^{1}_{p^{*}A/T}=p^{*}(s_{*}\Omega^{1}_{A/S})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}q^{*}(s_{*}\Omega^{1}_{A/S})=u_{*}\Omega^{1}_{q^{*}A/T}$
(where $t:p^{*}A\to T$ and $u:q^{*}A\to T$ are the structure morphisms), which
the Grothendieck–Messing Theorem implies is induced by an isomorphism
$p^{*}A\cong q^{*}A$ of abelian schemes lifting the identity over $S$. Since
the isomorphism respects the filtrations ${\mathcal{F}}^{\bullet}$, and the
lifts of the universal auxiliary structures $\iota$, $\lambda$ and $\eta$ over
$T$ are unique, it follows that $p^{*}\underline{A}\cong q^{*}\underline{A}$,
which means that $p=q\in S(T)$, so $T=\Delta$.

Now defining
$\Omega^{1}_{S/{\mathcal{O}},{\mathfrak{p}},i}=\delta_{0}^{*}{\mathcal{I}}_{\tau_{{\mathfrak{p}},i},e_{\mathfrak{p}}}$
and
${\operatorname{Fil}\,}^{j}(\Omega^{1}_{S/{\mathcal{O}},{\mathfrak{p}},i})=\delta_{0}^{*}{\mathcal{I}}_{\tau_{{\mathfrak{p}},i},j}$
for ${\mathfrak{p}}\in S_{p}$, $1\leq i\leq f_{\mathfrak{p}}$, $1\leq j\leq
e_{\mathfrak{p}}$, we obtain surjective morphisms

${\mathcal{L}}_{\tau_{{\mathfrak{p}},i},j}\otimes_{{\mathcal{O}}_{S}}{\mathcal{M}}_{\tau_{{\mathfrak{p}},i},j}^{-1}\twoheadrightarrow{\operatorname{gr}\,}^{j}(\Omega^{1}_{S/{\mathcal{O}},{\mathfrak{p}},i})\qquad\mbox{and}\qquad\bigoplus_{{\mathfrak{p}},i}\Omega^{1}_{S/{\mathcal{O}},{\mathfrak{p}},i}\twoheadrightarrow\Omega^{1}_{S/{\mathcal{O}}}.$

Since the
${\mathcal{L}}_{\tau_{{\mathfrak{p}},i},j}\otimes_{{\mathcal{O}}_{S}}{\mathcal{M}}_{\tau_{{\mathfrak{p}},i},j}^{-1}$
are line bundles and $\Omega^{1}_{S/{\mathcal{O}}}$ is locally free of rank
$d$, it follows that all the maps are isomorphisms.

Finally the constructions above are independent of the polarization $\lambda$,
hence are compatible with the action of
$\nu\in{\mathcal{O}}_{F,(p),+}^{\times}$ on $S=\widetilde{Y}_{U}$. More
precisely, the pull-back of $\alpha$ via the diagonal map $(\nu,\nu)$ is
compatible with the canonical isomorphism
$\nu^{*}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)\cong{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)$
induced by the identification of $\nu^{*}A$ with $A$, from which it follows
easily that the morphisms in the construction of the filtration are invariant
under the action of ${\mathcal{O}}_{F,(p),+}^{\times}$, hence descend to give
the decomposition, filtrations and isomorphisms in the statement of the
theorem.  $\square$

Let us also note the interpretation of the Kodaira–Spencer filtration in terms
of tangent spaces. For a closed point $y$ of $S$ corresponding to the data
$\underline{A}_{0}=(A_{0},\iota_{0},\lambda_{0},\eta_{0},{\mathcal{F}}_{0}^{\bullet})$
over a finite extension $k$ of the residue field of ${\mathcal{O}}$, the fibre
$T_{y}(S)$ of
${\mathcal{H}om}_{{\mathcal{O}}_{S}}(\Omega^{1}_{S/{\mathcal{O}}},{\mathcal{O}}_{S})$
is canonically identified with the set of isomorphism classes of data
$\underline{A}_{1}$ over $k[\epsilon]$ lifting $\underline{A}_{0}$, and the
decomposition and filtrations of the theorem yield dual decompositions of
$T_{y}(S)$ into components $T_{y}(S)_{\tau}$ with decreasing filtrations
${\operatorname{Fil}\,}^{j}(T_{y}(S)_{\tau})$. From the proof of the theorem
one sees immediately that
$\bigoplus_{\tau}{\operatorname{Fil}\,}^{j_{\tau}}(T_{y}(S)_{\tau})$
corresponds to the set of
$(A_{1},\iota_{1},\lambda_{1},\eta_{1},{\mathcal{F}}_{1}^{\bullet})$ such that
${\mathcal{F}}_{1,\tau}^{(j)}$ is the image of
${\mathcal{F}}_{0,\tau}^{(j)}\otimes_{k}k[\epsilon]$ for all $\tau$ and $j\leq
j_{\tau}$ under the canonical isomorphism

$H^{1}_{\operatorname{dR}}(A_{1}/k[\epsilon])\cong
H^{1}_{\operatorname{cris}}(A_{0}/k[\epsilon])\cong
H^{1}_{\operatorname{dR}}(A_{0}/k)\otimes_{k}k[\epsilon].$

We note also that the theorem yields a canonical (Kodaira–Spencer) isomorphism

$\Omega_{Y_{U}/{\mathcal{O}}}^{d}\cong\wedge_{{\mathcal{O}}_{Y_{U}}}^{d}\Omega^{1}_{Y_{U}/{\mathcal{O}}}\cong{\mathcal{A}}_{{\bf{2}},{\bf{-1}},{\mathcal{O}}}$

(writing $\sum m{\mathbf{e}}_{\theta}$ as $\mathbf{m}$ for $m\in\mathbb{Z}$).
Furthermore the decomposition, filtrations and isomorphisms of the theorem
(and hence also the Kodaira–Spencer isomorphism) are Hecke-equivariant in the
obvious sense. More precisely the same argument as for the compatibility with
the ${\mathcal{O}}_{F,(p),+}^{\times}$-action, but using the quasi-isogeny in
the construction of $\rho_{g}$, shows that if $U_{1}$, $U_{2}$ and
$g\in\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ are such that
$g^{-1}U_{1}g\subset U_{2}$, then
$\rho_{g}^{*}{\operatorname{Fil}\,}^{j}(\Omega^{1}_{Y_{U_{2}}/{\mathcal{O}},{\mathfrak{p}},i})$
corresponds to
${\operatorname{Fil}\,}^{j}(\Omega^{1}_{Y_{U_{1}}/{\mathcal{O}},{\mathfrak{p}},i})$
for all ${\mathfrak{p}},i,j$ under the canonical isomorphism
$\rho_{g}^{*}\Omega^{1}_{Y_{U_{2}}/{\mathcal{O}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Omega^{1}_{Y_{U_{1}}/{\mathcal{O}}}$,
and the resulting diagrams

$\textstyle{\rho_{g}^{*}{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta},{\mathcal{O}},2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\wr}$$\textstyle{{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta},{\mathcal{O}},1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{\rho_{g}^{*}{\operatorname{gr}\,}^{j}(\Omega^{1}_{Y_{U_{2}}/{\mathcal{O}},{\mathfrak{p}},i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{\operatorname{gr}\,}^{j}(\Omega^{1}_{Y_{U_{1}}/{\mathcal{O}},{\mathfrak{p}},i})}$

commute (where the top arrow is defined in the discussion preceding (6)).

##  4\. Partial Hasse invariants

###  4.1. Construction of $H_{\theta}$ and $G_{\theta}$

We now recall the definition, due to Reduzzi and Xiao [32], of generalized
partial Hasse invariants on Pappas–Rapoport models. These will be, for each
$\theta=\theta_{{\mathfrak{p}},i,j}\in\Sigma$, a Hilbert modular form
$H_{\theta}$ of weight $(\mathbf{h}_{\theta},\bf{0})$ with coefficients in
$\mathbb{F}={\mathcal{O}}/{\mathfrak{m}}_{\mathcal{O}}$, where
$\mathbf{h}_{\theta}:=n_{\theta}{\mathbf{e}}_{\sigma^{-1}\theta}-{\mathbf{e}}_{\theta}$,
with $n_{\theta}=p$ if $j=1$ and $n_{\theta}=1$ if $j>1$. We also define below
a(n in)variant $G_{\theta}$ of weight $(\bf{0},\mathbf{h}_{\theta})$.

We will now be working in the mod $p$ setting, so until further notice $S$
will denote $\widetilde{Y}_{U,\mathbb{F}}$, and $s:A\to S$ the universal
abelian scheme over it. Thus ${\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)$ is a
locally free sheaf of rank two over

${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}\cong\bigoplus_{{\mathfrak{p}}\in
S_{p}}\bigoplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\mathcal{O}}_{S}[u]/(u^{e_{\mathfrak{p}}}),$

where $u$ acts via $\iota(\varpi_{\mathfrak{p}})^{*}$ on the
${\mathfrak{p}}$-component of

${\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)=\bigoplus_{{\mathfrak{p}}\in
S_{p}}{\mathcal{H}}_{\mathfrak{p}}=\bigoplus_{{\mathfrak{p}}\in
S_{p}}\bigoplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\mathcal{H}}_{\tau}.$

We will also now be working with a fixed ${\mathfrak{p}}$ and omit the
subscript from the notation, so that

${\mathcal{H}}={\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)_{\mathfrak{p}}=\bigoplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\mathcal{H}}_{\tau}=\bigoplus_{i\in\mathbb{Z}/f\mathbb{Z}}{\mathcal{H}}_{i}$

with each ${\mathcal{H}}_{i}$ locally free of rank two over
${\mathcal{O}}_{S}[u]/(u^{e})$ (where we have also abbreviated the subscript
$\tau_{{\mathfrak{p}},i}$ by $i$). Furthermore for each
$i\in\mathbb{Z}/f\mathbb{Z}$, we have a filtration

$0={\mathcal{F}}_{i}^{(0)}\subset{\mathcal{F}}_{i}^{(1)}\subset\cdots\subset{\mathcal{F}}_{i}^{(e-1)}\subset{\mathcal{F}}_{i}^{(e)}=(s_{*}\Omega^{1}_{A/S})_{i}$

by sheaves of ${\mathcal{O}}_{S}[u]/(u^{e})$-modules such that the quotients
${\mathcal{L}}_{i,j}={\mathcal{F}}_{i}^{(j)}/{\mathcal{F}}_{i}^{(j-1)}$ are
line bundles annihilated by $u$.

Firstly note that if $j>1$, then
$u:{\mathcal{F}}_{i}^{(j)}\to{\mathcal{F}}_{i}^{(j-1)}$ induces a morphism
${\mathcal{L}}_{i,j}\to{\mathcal{L}}_{i,j-1}$. On the other hand if $j=1$,
then the Verschiebung morphism $\phi_{S}^{*}A\to A$ over $S$ induces
${\mathcal{O}}_{S}[u]/(u^{e})$-linear morphisms

$\operatorname{Ver}_{i}^{*}:{\mathcal{H}}_{i}={\mathcal{H}}_{\tau_{i}}\longrightarrow{\mathcal{H}}^{1}_{\operatorname{dR}}(\phi_{S}^{*}(A)/S)_{\tau_{i}}\cong\phi_{S}^{*}({\mathcal{H}}_{\tau_{i-1}})=\phi_{S}^{*}({\mathcal{H}}_{i-1})$

with image $\phi_{S}^{*}({\mathcal{F}}_{i-1}^{(e)})$ (where $\phi_{S}$ denotes
the absolute Frobenius on $S$). Note that
${\mathcal{L}}_{i,1}={\mathcal{F}}_{i}^{(1)}\subset u^{e-1}{\mathcal{H}}_{i}$,
so that $u^{e-1}$ defines an isomorphism

$u^{1-e}{\mathcal{L}}_{i,1}/u{\mathcal{H}}_{i}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{F}}_{i}^{(1)},$

and that
$\operatorname{Ver}_{i}^{*}(u{\mathcal{H}}_{i})=u\phi_{S}^{*}({\mathcal{F}}_{i-1}^{(e)})\subset\phi_{S}^{*}({\mathcal{F}}_{i-1}^{(e-1)})$,
so we obtain a well-defined ${\mathcal{O}}_{S}$-linear morphism
“$\operatorname{Ver}_{i}^{*}\circ u^{1-e}$”

${\mathcal{L}}_{i,1}\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}u^{1-e}{\mathcal{L}}_{i,1}/u{\mathcal{H}}_{i}\longrightarrow\phi_{S}^{*}({\mathcal{L}}_{i-1,e})\cong{\mathcal{L}}_{i-1,e}^{\otimes
p}.$

We have now defined a morphism
${\mathcal{L}}_{\theta}\longrightarrow{\mathcal{L}}_{\sigma^{-1}\theta}^{\otimes
n_{\theta}}$ for all $\theta$, and hence a section of
$\widetilde{{\mathcal{A}}}_{\mathbf{h}_{\theta},\bf{0},\mathbb{F}}$ over $S$.
Furthermore it is straightforward to check that the section is invariant under
the action of ${\mathcal{O}}_{F,(p),+}^{\times}$ and therefore descends to an
element

$H_{\theta}\in
M_{\mathbf{h}_{\theta},\bf{0}}(U;\mathbb{F})=H^{0}(Y_{U,\mathbb{F}},{\mathcal{A}}_{\mathbf{h}_{\theta},\bf{0},\mathbb{F}}),$

which we call the partial Hasse invariant (indexed by $\theta$). Furthermore
the partial Hasse invariants are stable under the Hecke action, in the sense
that if $U_{1}$, $U_{2}$ and
$g\in\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ are such that
$g^{-1}U_{1}g\subset U_{2}$, then $[g]H_{\theta,2}=H_{\theta,1}$. (Note also
that the partial Hasse invariants are dependent on the choice of uniformizer
$\varpi=\varpi_{\mathfrak{p}}$ only up to a scalar in $\mathbb{F}^{\times}$:
if $\varpi$ is replaced by $a\varpi$ for some
$a\in{\mathcal{O}}_{F,{\mathfrak{p}}}^{\times}$, then $H_{\theta}$ is replaced
by $\tau(a)H_{\theta}$ if $j>1$ and by $\tau(a)^{1-e}H_{\theta}$ if $j=1$.)

We remark also that the line bundles
${\mathcal{A}}_{\bf{0},\mathbf{h}_{\theta},\mathbb{F}}$ have canonical
trivializations. Indeed for each $i\in\mathbb{Z}/f\mathbb{Z}$ and
$j=2,\ldots,e$, we have the exact sequence

$0\to{\mathcal{G}}_{i}^{(j-1)}/{\mathcal{F}}_{i}^{(j-1)}\longrightarrow{\mathcal{G}}_{i}^{(j)}/{\mathcal{F}}_{i}^{(j-1)}\stackrel{{\scriptstyle
u}}{{\longrightarrow}}{\mathcal{F}}_{i}^{(j-1)}/{\mathcal{F}}_{i}^{(j-2)}\to
0$

over $S$, i.e.,
$0\to{\mathcal{M}}_{i,j-1}\to{\mathcal{P}}_{i,j}\to{\mathcal{N}}_{i,j-1}\to
0$, inducing an isomorphism

${\mathcal{N}}_{i,j-1}={\mathcal{L}}_{i,j-1}\otimes_{{\mathcal{O}}_{S}}{\mathcal{M}}_{i,j-1}\cong\wedge^{2}_{{\mathcal{O}}_{S}}{\mathcal{P}}_{i,j}={\mathcal{N}}_{i,j}$

and hence
${\mathcal{O}}_{S}\simeq{\mathcal{N}}_{i,j-1}^{-1}\otimes_{{\mathcal{O}}_{S}}{\mathcal{N}}_{i,j}=\widetilde{{\mathcal{A}}}_{\bf{0},\mathbf{h}_{\theta},\mathbb{F}}$
for $\theta=\theta_{{\mathfrak{p}},i,j}$, which it is straightforward to check
descends to $Y_{U,\mathbb{F}}$. Similarly we have the exact sequence

$0\to\phi_{S}^{*}({\mathcal{G}}_{i-1}^{(e)}/{\mathcal{F}}_{i-1}^{(e)})\stackrel{{\scriptstyle{\operatorname{Frob}}_{A}^{*}}}{{\longrightarrow}}{\mathcal{G}}_{i}^{(1)}\stackrel{{\scriptstyle\operatorname{Ver}_{A}^{*}u^{1-e}}}{{\longrightarrow}}\phi_{S}^{*}({\mathcal{F}}_{i-1}^{(e)}/{\mathcal{F}}_{i-1}^{(e-1)})\to
0$

inducing an isomorphism
$\phi_{S}^{*}({\mathcal{N}}_{i-1,e})\cong{\mathcal{N}}_{i,1}$ and hence
${\mathcal{O}}_{S}\simeq\widetilde{{\mathcal{A}}}_{\bf{0},\mathbf{h}_{\theta},\mathbb{F}}$
for $\theta=\theta_{{\mathfrak{p}},i,1}$ descending to $Y_{U,\mathbb{F}}$.
Furthermore these isomorphisms are Hecke-equivariant in the usual sense, but
note that they depend via $u$ on the choice of $\varpi_{\mathfrak{p}}$. For
each $\theta$, we let $G_{\theta}\in
M_{\bf{0},\mathbf{h}_{\theta}}(U;\mathbb{F})$ denote the canonical
trivializing section.

###  4.2. Stratification

We also recall how the partial Hasse invariants define a stratification of the
Hilbert modular variety in characteristic $p$. For any $\theta\in\Theta$, we
define $\widetilde{Z}_{\theta}$ (resp. $Z_{\theta}$) to be the closed
subscheme of $S=\widetilde{Y}_{U,\mathbb{F}}$ (resp. $Y_{U,\mathbb{F}}$)
defined by the vanishing of $H_{\theta}$, and for any subset $T\subset\Sigma$,
we let

$\widetilde{Z}_{T}=\bigcap_{\theta\in
T}\widetilde{Z}_{\theta}\quad\mbox{and}\quad Z_{T}=\bigcap_{\theta\in
T}Z_{\theta}.$

Note that the schemes $Z_{T}$ are stable under the Hecke action, in the strong
sense that $Z_{T,1}$ is the pull-back of $Z_{T,2}$ under
$\rho_{g}:Y_{U_{1}}\to Y_{U_{2}}$.

We then have the following consequence ([32, Thm. 3.10]) of the description of
the Kodaira–Spencer filtration on tangent spaces at closed points; we give a
proof here as some of the details are relevant to the construction of
$\Theta$-operators in §5.2.

######  Proposition 4.2.1.

The schemes $\widetilde{Z}_{T}$ and $Z_{T}$ are smooth over $\mathbb{F}$ of
dimension $|\Sigma-T|$.

Proof. We prove the result for $\widetilde{Z}_{T}$, from which the result for
$Z_{T}$ is immediate.

Let $y$ be a closed point of $S$ with local ring $R={\mathcal{O}}_{S,y}$,
maximal ideal ${\mathfrak{m}}$ and residue field $k=R/{\mathfrak{m}}$. For
each $\theta\in\Sigma$, choose a basis $b_{\theta}$ for
${{\mathcal{L}}}_{\theta,y}$ over $R$ and write
$H_{\theta,y}b_{\theta}=x_{\theta}b_{\sigma^{-1}\theta}^{n_{\theta}}$. Thus if
$y\in\widetilde{Z}_{\theta}$, then $x_{\theta}\in{\mathfrak{m}}$, and we let
$\overline{x}_{\theta}$ denote its image in
${\mathfrak{m}}/{\mathfrak{m}}^{2}$.

Identifying ${\mathfrak{m}}/{\mathfrak{m}}^{2}$ with the fibre of
$\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}}$ at $y$ and writing
${\operatorname{Fil}\,}^{j}({\mathfrak{m}}/{\mathfrak{m}}^{2})_{\tau}$ for the
subspaces obtained from the Kodaira–Spencer filtration, we claim that if $y\in
Z_{\theta}$, then

(7)
${\operatorname{Fil}\,}^{j}({\mathfrak{m}}/{\mathfrak{m}}^{2})_{\tau}=k\overline{x}_{\theta}+{\operatorname{Fil}\,}^{j-1}({\mathfrak{m}}/{\mathfrak{m}}^{2})_{\tau},$

where $\tau=\tau_{{\mathfrak{p}},i}$ and $\theta=\theta_{{\mathfrak{p}},i,j}$.
Comparing dimensions, we see it suffices to prove the inclusion of the left-
hand side in the right, or equivalently that if

$v\in T_{y}(S)=\bigoplus_{\tau^{\prime}\in\Sigma_{0}}T_{y}(S)_{\tau^{\prime}}$

is such that its $\tau$-component $v_{\tau}$ lies in
${\operatorname{Fil}\,}^{j-1}T_{y}(S)_{\tau}$ and $v$ is orthogonal to
$\overline{x}_{\theta}$, then in fact
$v_{\tau}\in{\operatorname{Fil}\,}^{j}(T_{y}(S)_{\tau})$ (using the notation
of the discussion following the proof of Theorem 3.3.1).

Let
$\underline{A}_{0}=(A_{0},\iota_{0},\lambda_{0},\eta_{0},{\mathcal{F}}_{0}^{\bullet})$
denote the data corresponding to the point $y\in S(k)$ and
$\underline{A}_{1}=(A_{1},\iota_{1},\lambda_{1},\eta_{1},{\mathcal{F}}_{1}^{\bullet})$
that of its lift $v\in S(k[\epsilon])$. With $\tau=\tau_{{\mathfrak{p}},i}$
fixed for now, we will suppress ${\mathfrak{p}}$ from the notation and replace
the subscript $\tau_{p,i^{\prime}}$ by $i^{\prime}$ (for $i^{\prime}=i,i-1$).
Recall the assumption that
$v_{i}\in{\operatorname{Fil}\,}^{j-1}(T_{y}(S)_{i})$ means that
${\mathcal{F}}_{1,i}^{(j^{\prime})}$ corresponds to
${\mathcal{F}}_{0,i}^{(j^{\prime})}\otimes_{k}k[\epsilon]$ for
$j^{\prime}=1,\ldots,j-1$ under the canonical isomorphism

(8) $H^{1}_{\operatorname{dR}}(A_{1}/k[\epsilon])\cong
H^{1}_{\operatorname{cris}}(A_{0}/k[\epsilon])\cong
H^{1}_{\operatorname{dR}}(A_{0}/k)\otimes_{k}k[\epsilon].$

For $v_{i}$ to be orthogonal to $\overline{x}_{\theta}$ means that the
morphism

(9)
${\mathcal{L}}_{1,\theta}\longrightarrow{\mathcal{L}}_{1,\sigma^{-1}\theta}^{\otimes
n_{\theta}}$

induced by $H_{\theta}$ vanishes, and we need to show this implies that
${\mathcal{F}}_{1,i}^{(j)}$ is the image of
${\mathcal{F}}_{0,i}^{(j)}\otimes_{k}k[\epsilon]$.

Suppose first that $j>1$. Then (9) is simply

$u:{\mathcal{F}}_{1,i}^{(j)}/{\mathcal{F}}_{1,i}^{(j-1)}\longrightarrow{\mathcal{F}}_{1,i}^{(j-1)}/{\mathcal{F}}_{1,i}^{(j-2)},$

whose vanishing means
${\mathcal{F}}_{1,i}^{(j)}=u^{-1}{\mathcal{F}}_{1,i}^{(j-2)}$. Since (8) sends
${\mathcal{F}}_{1,i}^{(j-2)}$ to
${\mathcal{F}}_{0,i}^{(j-2)}\otimes_{k}k[\epsilon]$ and is compatible with
$u$, it follows that it also sends ${\mathcal{F}}_{1,i}^{(j)}$ to
${\mathcal{F}}_{0,i}^{(j)}\otimes_{k}k[\epsilon]$.

On the other hand if $j=1$, then the vanishing of (9) means that
$u^{1-e}{\mathcal{F}}_{1,i}^{(1)}$ is the preimage of
$\phi_{1}^{*}({\mathcal{F}}_{1,i-1}^{(e-1)})$ under
$\operatorname{Ver}_{i}^{*}$ (where $\phi_{1}$ is the absolute Frobenius on
$k[\epsilon]$, and $\phi_{0}$ will denote the absolute Frobenius on $k$).
Since the diagram

$\textstyle{H^{1}_{\operatorname{dR}}(A_{1}/k[\epsilon])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{dR}}(A_{0}/k)\otimes_{k}k[\epsilon]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi_{1}^{*}H^{1}_{\operatorname{dR}}(A_{1}/k[\epsilon])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi_{0}^{*}H^{1}_{\operatorname{dR}}(A_{0}/k)\otimes_{k}k[\epsilon]}$

commutes, where the vertical maps are induced by Verschiebung, the top arrow
is (8) and the bottom one is given by the identification of
$\phi_{1}^{*}A_{1}$ with $\phi_{0}^{*}A_{0}\otimes_{k}k[\epsilon]$, so in
particular identifies $\phi_{1}^{*}({\mathcal{F}}_{1,i-1}^{(e-1)})$ with
$\phi_{0}^{*}({\mathcal{F}}_{0,i-1}^{(e-1)})\otimes_{k}k[\epsilon]$, it
follows that the top arrow sends $u^{1-e}{\mathcal{F}}_{1,i}^{(1)}$ to
$u^{1-e}{\mathcal{F}}_{0,i}^{(1)}\otimes_{k}k[\epsilon]$, and hence
${\mathcal{F}}_{1,i}^{(1)}$ to
${\mathcal{F}}_{0,i}^{(1)}\otimes_{k}k[\epsilon]$. This completes the proof of
the claim.

Now note that if $y\in\widetilde{Z}_{T}$, then (7) implies that the elements
$\overline{x}_{\theta}$ for $\theta\in T$ can be extended to a basis for
${\mathfrak{m}}/{\mathfrak{m}}^{2}$ over $k$, hence are linearly independent.
Since $R$ is regular of dimension $d=|\Sigma|$, it follows that
${\mathcal{O}}_{\widetilde{Z}_{T},y}=R/\langle x_{\theta}\rangle_{\theta\in
T}$ is regular of dimension of $d-|T|$, and hence that $\widetilde{Z}_{T}$ is
smooth over $\mathbb{F}$ of dimension $d-|T|$.  $\square$

Finally we recall the definition of the minimal weight of a non-zero mod $p$
Hilbert modular form. If $f\in M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$, then
$\mathbf{k}_{\min}(f)$ is defined to be
$\mathbf{k}-\sum_{\theta}m_{\theta}\mathbf{h}_{\theta}$ where
$\sum_{\theta}m_{\theta}{\mathbf{e}}_{\theta}$ is the unique maximal element
of the set

$\left\\{\,\left.\sum_{\theta}m_{\theta}{\mathbf{e}}_{\theta}\in\mathbb{Z}_{\geq
0}^{\Sigma}\,\right|\,f=g\prod_{\theta\in\Sigma}H_{\theta}^{m_{\theta}}\mbox{\,for
some\,}g\in
M_{\mathbf{k}-\sum_{\theta}m_{\theta}\mathbf{h}_{\theta},\mathbf{l}}(U;\mathbb{F})\,\right\\}.$

By the main result of [10], the minimal weight of $f$ always lies in the cone:

(10)
$\Xi^{\min}:=\left\\{\,\left.\sum_{\theta}k_{\theta}{\mathbf{e}}_{\theta}\,\right|\,\mbox{$n_{\theta}k_{\theta}\geq
k_{\sigma^{-1}\theta}$ for all $\theta\in\Sigma$}\,\right\\}.$

Note that the result stated in [10] applies to forms on a finite étale cover
of $Y_{U,\mathbb{F}}$, from which the analogous result for forms on
$Y_{U,\mathbb{F}}$ is immediate.

##  5\. Partial Theta operators

###  5.1. Fundamental Hasse invariants

In order to define the partial $\Theta$-operators (in §5.2 below), we first
define a canonical factorization of the partial Hasse invariants over a finite
flat (Igusa) cover of the Hilbert modular variety over $\mathbb{F}$.

We fix a sufficiently small $U$ that the line bundles
${\mathcal{L}}_{\theta}$, ${\mathcal{M}}_{\theta}$, ${\mathcal{N}}_{\theta}$
(and hence $\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},\mathbb{F}}$) on
$\widetilde{Y}_{U,\mathbb{F}}$ descend to $Y_{U,\mathbb{F}}$ for all
$\theta\in\Sigma$ (and all $\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}$), and
we write simply $\overline{Y}$ for $Y_{U,\mathbb{F}}$, and
$\overline{{\mathcal{L}}}_{\tau,j}$, $\overline{{\mathcal{M}}}_{\tau,j}$ and
$\overline{{\mathcal{N}}}_{\tau,j}$ for the line bundles on $\overline{Y}$.
For each ${\mathfrak{p}}\in S_{p}$ and $\tau\in\Sigma_{{\mathfrak{p}},0}$, we
let

$H_{\tau}=\prod_{j=1}^{e_{\mathfrak{p}}}H_{\tau,j}\in
H^{0}(\overline{Y},\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}\otimes_{{\mathcal{O}}_{\overline{Y}}}\overline{{\mathcal{L}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}}^{\otimes
p}).$

Viewing each $H_{\tau}$ as a morphism
$(\overline{{\mathcal{L}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}}^{-1})^{\otimes
p}\to\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}$ and
$H_{\mathfrak{p}}:=\prod_{\tau\in\Sigma_{{\mathfrak{p}},0}}H_{\tau}$ as a
morphism
$\otimes_{\tau\in\Sigma_{{\mathfrak{p}},0}}(\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1})^{\otimes(p-1)}\to{\mathcal{O}}_{\overline{Y}}$,
we define the Igusa cover999The cover has a natural moduli-theoretic
interpretation in terms of $A[{\mathfrak{q}}]$, but we will not need this
here. of $\overline{Y}$ (of level ${\mathfrak{q}}=\prod{\mathfrak{p}}$) to be

${Y}^{\mathrm{Ig}}=\mathbf{Spec}\left(\mathrm{Sym}_{\mathcal{O}_{\overline{Y}}}(\bigoplus_{\tau\in\Sigma_{0}}\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1})/\mathcal{I}\right),$

where ${\mathcal{I}}$ is the sheaf of ideals generated by the
${\mathcal{O}}_{\overline{Y}}$-submodules

$(H_{\tau}-1)\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}$ for
$\tau\in\Sigma_{0}$, and
$(H_{\mathfrak{p}}-1)\left(\bigotimes_{\tau\in\Sigma_{{\mathfrak{p}},0}}(\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1})^{\otimes(p-1)}\right)$
for ${\mathfrak{p}}\in S_{p}$

(where all tensor products are over ${\mathcal{O}}_{\overline{Y}}$). We then
define an action of $({\mathcal{O}}_{F}/{\mathfrak{q}})^{\times}$ on
${Y}^{{\mathrm{Ig}}}$ over $\overline{Y}$ by letting
$\alpha\in({\mathcal{O}}_{F}/{\mathfrak{q}})^{\times}$ act on the structure
sheaf by the automorphism of sheaves of
${\mathcal{O}}_{\overline{Y}}$-algebras induced by multiplication by
$\tau({\alpha})^{-1}$ on
$\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}$ for each $\tau$. We
then see, exactly as in the proof of parts (1) and (2) of [12, Prop. 8.1.1],
that the projection $\pi:{Y}^{{\mathrm{Ig}}}\to\overline{Y}$ is finite flat,
generically étale, and identifies $\overline{Y}$ with the quotient of
${Y}^{{\mathrm{Ig}}}$ by the action of
$({\mathcal{O}}_{F}/{\mathfrak{q}})^{\times}$.

For each $\tau\in\Sigma_{{\mathfrak{p}},0}$, we let
$h_{\tau,e_{\mathfrak{p}}}$ denote the tautological section of
$\pi^{*}\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}$ induced by the
inclusion
$\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}^{-1}\hookrightarrow\pi_{*}{\mathcal{O}}_{{Y}^{\mathrm{Ig}}}$.
We also define the section

$h_{\tau,j}=\pi^{*}(H_{\tau,j+1}\cdots
H_{\tau,e_{\mathfrak{p}}})h_{\tau,e_{\mathfrak{p}}}$

of $\pi^{*}\overline{{\mathcal{L}}}_{\tau,j}$ for
$j=1,\ldots,e_{{\mathfrak{p}}}-1$. Note that since ${Y}^{\mathrm{Ig}}$ is
reduced (or since
$\prod_{\tau\in\Sigma_{{\mathfrak{p}},0}}h_{\tau,e_{\mathfrak{p}}}^{p-1}=\pi^{*}H_{\mathfrak{p}}$
by construction), the sections $h_{\tau,e_{\mathfrak{p}}}$ are injective, and
hence so are the $h_{\tau,j}$ for all $\tau$ and $j$. We write simply
$h_{\theta}$ for the section $h_{\tau,j}$ of
$\pi^{*}\overline{{\mathcal{L}}}_{\theta}=\pi^{*}\overline{{\mathcal{L}}}_{\tau,j}$,
and we call the $h_{\theta}$ the fundamental Hasse invariant (indexed by
$\theta$).

###  5.2. Construction of $\Theta_{\tau}$

We now explain how the construction of $\Theta$-operators in [12] directly
generalizes to the case where $p$ is ramified in $F$, yielding an operator
that shifts the weight $\mathbf{k}$ by $(1,1)$ in the final two components
corresponding to embeddings with the same reduction, i.e.,
$\theta_{{\mathfrak{p}},i,e_{{\mathfrak{p}}}-1}$,
$\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$ (and hence, by composing with
multiplication by partial Hasse invariants, one can shift weights by $+1$ for
any pair of embeddings with the same reduction).

Indeed for each $\tau\in\Sigma_{0}$, we define the operator $\Theta_{\tau}$
exactly as in [12, §8], but using the morphism

$KS_{\tau}:\Omega^{1}_{\overline{Y}/{\mathbb{F}}}\longrightarrow{\operatorname{gr}\,}^{e_{{\mathfrak{p}}}}(\Omega^{1}_{\overline{Y}/{\mathbb{F}}})_{\tau}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\overline{{\mathcal{L}}}_{\tau,e_{\mathfrak{p}}}\otimes_{{\mathcal{O}}_{\overline{Y}}}\overline{{\mathcal{M}}}^{-1}_{\tau,e_{\mathfrak{p}}}$

provided by Theorem 3.3.1 via projection to the top graded piece of the
filtration of the $\tau$-component of $\Omega^{1}_{Y/{\mathbb{F}}}$. More
precisely, fix ${\mathfrak{p}}_{0}\in S_{p}$ and
$\tau_{0}=\tau_{{\mathfrak{p}}_{0},i}$, let
$\theta_{0}=\theta_{{\mathfrak{p}}_{0},i,e_{{\mathfrak{p}}_{0}}}$, and
consider the morphism

$KS_{\tau_{0}}^{{\mathrm{Ig}}}:\Omega^{1}_{{Y}^{\mathrm{Ig}}/{\mathbb{F}}}\otimes_{\mathcal{O}_{{Y}^{\mathrm{Ig}}}}\mathcal{F}^{\mathrm{Ig}}\cong\pi^{*}\Omega^{1}_{\overline{Y}/{\mathbb{F}}}\otimes_{\mathcal{O}_{{Y}^{\mathrm{Ig}}}}\mathcal{F}^{\mathrm{Ig}}\longrightarrow\pi^{*}({\mathcal{A}}_{2{\mathbf{e}}_{\theta_{0}},-{\mathbf{e}}_{\theta_{0}},\mathbb{F}})\otimes_{\mathcal{O}_{Y^{\mathrm{Ig}}}}\mathcal{F}^{\mathrm{Ig}}$

induced by $KS_{\tau_{0}}$, where ${\mathcal{F}}^{\mathrm{Ig}}$ is the sheaf
of total fractions on ${Y}^{\mathrm{Ig}}$. Suppose now that $f\in
M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})$, and write
$h^{\mathbf{k}}=\prod_{\theta\in\Sigma}h_{\theta}^{k_{\theta}}$ and
$g^{\mathbf{l}}=\prod_{\theta\in\Sigma}g_{\theta}^{l_{\theta}}$ for any choice
of trivializations $g_{\theta}$ of the line bundles
$\overline{{\mathcal{N}}}_{\theta}$ on $\overline{Y}$. We then define the
section

$\Theta_{\tau_{0}}^{\mathrm{Ig}}(f):=h^{\mathbf{k}}\pi^{*}(g^{\mathbf{l}}H_{\theta_{0}})KS_{\tau_{0}}^{\mathrm{Ig}}(d(h^{-\mathbf{k}}\pi^{*}(g^{-\mathbf{l}}f))),$

where

(11)
$\mathbf{k}^{\prime}=\mathbf{k}+n_{\theta_{0}}{\mathbf{e}}_{\sigma^{-1}\theta_{0}}+{\mathbf{e}}_{\theta_{0}}\quad\mbox{and}\quad\mathbf{l}^{\prime}=\mathbf{l}+{\mathbf{e}}_{\theta_{0}}.$

Furthermore, the section is independent of the choices of $g_{\theta}$ and
invariant under the action of $({\mathcal{O}}_{F}/{\mathfrak{q}})^{\times}$,
hence descends to a section of
${\mathcal{A}}_{\mathbf{k}^{\prime},\mathbf{l}^{\prime},{\mathbb{F}}}\otimes_{{\mathcal{O}}_{\overline{Y}}}{\mathcal{F}}$,
where ${\mathcal{F}}$ is the sheaf of total fractions on $\overline{Y}$.
Denoting the section $\Theta_{\tau_{0}}(f)$, we have the following
generalization of [12, Thm. 8.2.2]:

######  Theorem 5.2.1.

If $f\in M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})$, then
$\Theta_{\tau_{0}}(f)\in
M_{\mathbf{k}^{\prime},\mathbf{l}^{\prime}}(U;{\mathbb{F}})$. Moreover
$\Theta_{\tau_{0}}(f)$ is divisible by $H_{\theta_{0}}$ if and only if either
$f$ is divisible by $H_{\theta_{0}}$ or $p|k_{\theta_{0}}$.

Proof. We see exactly as in [12] that $\Theta_{\tau_{0}}(f)$ is regular on the
ordinary locus of $\overline{Y}$, i.e., the complement of the divisor
$\cup_{\theta\in\Sigma}Z_{\theta}$ (where $Z_{\theta}$ was defined in §4.2),
so the theorem reduces to proving that if $z$ is the generic point of an
irreducible component of $Z_{\theta_{1}}$ for some $\theta_{1}\in\Sigma$, then

  * •

${\operatorname{ord}}_{z}(\Theta_{\tau_{0}}(f))\geq 0$,

  * •

if $\theta_{1}=\theta_{0}$, then
${\operatorname{ord}}_{z}(\Theta_{\tau_{0}}(f))>0$ if and only if
$p|k_{\theta_{0}}$ or ${\operatorname{ord}}_{z}(f)>0$.

Let $R$ denote the discrete valuation ring ${\mathcal{O}}_{\overline{Y},z}$,
and for each $\tau\in\Sigma_{{\mathfrak{p}},0}$ and $\theta\in\Sigma_{\tau}$,
let $y_{\theta}=y_{\tau,j}$ be a basis for the stalk
$\overline{{\mathcal{L}}}_{\theta,z}=\overline{{\mathcal{L}}}_{\tau,j,z}$ over
$R$ (for $j=1,\ldots,e_{\mathfrak{p}}$). For each $\theta\in\Sigma$, we may
then write

$H_{\theta}y_{\theta}=r_{\theta}y_{\sigma^{-1}\theta}^{n_{\theta}}$

for some $r_{\theta}=r_{\tau,j}\in R$, and we let
$r_{\tau}=\prod_{j=1}^{e_{\mathfrak{p}}}r_{\tau,j}$. By construction, we have
$T:=(\pi_{*}{\mathcal{O}}_{{Y}^{{\mathrm{Ig}}}})_{z}=R[x_{\tau}]_{\tau\in\Sigma_{0}}/I$,
where $I$ is the ideal generated by

$x_{\phi^{-1}\circ\tau}^{p}-r_{\tau}x_{\tau}\mbox{\,\,for
$\tau\in\Sigma_{0}$,}\quad\mbox{and}\quad\prod_{\tau\in\Sigma_{{\mathfrak{p}},0}}x_{\tau}^{p-1}-\prod_{\tau\in\Sigma_{{\mathfrak{p}},0}}r_{\tau}^{p-1}\mbox{\\\
,,for ${\mathfrak{p}}\in S_{p}$,}$

where each $x_{\tau}$ is the dual basis of $y_{\tau,e_{\mathfrak{p}}}$. We
then have $h_{\tau,e_{\mathfrak{p}}}=x_{\tau}y_{\tau,e_{\mathfrak{p}}}$ (in
$(\pi_{*}\pi^{*}{\mathcal{L}}_{\tau,e_{\mathfrak{p}}})_{z}$), from which it
follows that

$h_{\tau,j}=r_{\tau,j+1}r_{\tau,j+2}\cdots
r_{\tau,e_{\mathfrak{p}}}x_{\tau}y_{\tau,j}$

for $j=1,\ldots,e_{{\mathfrak{p}}}-1$, and hence that
$h^{\mathbf{k}}=\varphi_{\mathbf{k}}y^{\mathbf{k}}$, where
$y^{\mathbf{k}}=\prod_{\theta\in\Sigma}y_{\theta}^{k_{\theta}}$ and

$\varphi_{\mathbf{k}}=\prod_{{\mathfrak{p}}\in
S_{p}}\prod_{\tau\in\Sigma_{{\mathfrak{p}},0}}\left((r_{\tau}x_{\tau})^{\sum_{\theta\in\Sigma_{\tau}}k_{\theta}}\prod_{j=1}^{e_{\mathfrak{p}}}r_{\tau,j}^{-\sum_{j^{\prime}=j}^{e_{\mathfrak{p}}}k_{\tau,j^{\prime}}}\right)$

(writing $k_{\tau_{{\mathfrak{p}},i},j}$ for $k_{\theta_{{\mathfrak{p}},i,j}}$
as usual, and working over the field of fractions of $T$).

Writing $f=\varphi_{f}y^{\mathbf{k}}g^{\mathbf{l}}$, we see that

$\Theta_{\tau_{0}}^{\mathrm{Ig}}(f)=KS_{\tau_{0}}^{\mathrm{Ig}}(r_{\theta_{0}}\varphi_{\mathbf{k}}d(\varphi_{f}\varphi_{\mathbf{k}}^{-1}))y_{\theta_{0}}^{-1}y_{\sigma^{-1}\theta_{0}}^{n_{\theta_{0}}}y^{\mathbf{k}}g^{\mathbf{l}}.$

Since $r_{\tau}x_{\tau}=x_{\phi^{-1}\circ\tau}^{p}$, we have
$d(r_{\tau}x_{\tau})=0$ and

(12)
$\Theta_{\tau_{0}}(f)=KS_{\tau_{0}}\left(r_{\theta_{0}}d\varphi_{f}+r_{\theta_{0}}\varphi_{f}\sum_{\theta\in\Sigma}k_{\theta}^{\prime}\frac{dr_{\theta}}{r_{\theta}}\right)y_{\theta_{0}}^{-1}y_{\sigma^{-1}\theta_{0}}^{n_{\theta_{0}}}y^{\mathbf{k}}g^{\mathbf{l}},$

where $k^{\prime}_{\theta}=k_{\tau,j}+k_{\tau,j+1}+\cdots
k_{\tau,e_{\mathfrak{p}}}$ if $\tau=\tau_{{\mathfrak{p}},i}$ and
$\theta=\theta_{{\mathfrak{p}},i,j}$. We are therefore reduced to showing that
${\operatorname{ord}}_{z}KS_{\tau_{0}}(dr_{\theta_{1}})>0$ if and only if
$\theta_{1}=\theta_{0}$. However the proof of Proposition 4.2.1 shows that if
$y$ is a closed point of $Z_{\theta_{1}}$, then
$KS_{\tau_{0}}(dr_{\theta_{1}})$ vanishes at $y$ if and only if
$\theta_{1}=\theta_{0}$.  $\square$

######  Remark 5.2.2.

The Kodaira–Spencer isomorphism is defined in [12] using the Gauss–Manin
connection. Much of the work in [12, §8.2] amounts to an explicit translation
of this to the context of deformation theory. Here however we defined the
morphism $KS_{\tau_{0}}$ more directly using deformation theory, so the
analogue of [12, Lem. 8.2.3] was not needed here.

######  Remark 5.2.3.

It is straightforward to check directly that the right-hand side of (12) is
independent of the choice of local trivializations $y_{\tau}$ and $g_{\tau}$,
and can therefore be used to define the partial $\Theta$-operator without
reference to the Igusa cover ${Y}^{\mathrm{Ig}}$.

We call $\Theta_{\tau_{0}}$ the partial $\Theta$-operator (indexed by
$\tau_{0}$). It is immediate from its definition that the resulting map on
$\mathbb{F}$-algebras

$\bigoplus_{\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}}M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})\longrightarrow\bigoplus_{\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}}M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}}),$

given by the direct sum over all weights of the operators $\Theta_{\tau_{0}}$,
is an $\mathbb{F}$-linear derivation, i.e. that

$\Theta_{\tau_{0}}(f_{1}f_{2})=f_{1}\Theta_{\tau_{0}}(f_{2})+\Theta_{\tau_{0}}(f_{1})f_{2}$

for all $f_{1},f_{2}$ in $\oplus M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$. It
is also clear that $\Theta_{\tau_{0}}(H_{\theta})=0$ for all
$\theta\in\Sigma$, and hence that $\Theta_{\tau_{0}}$ commutes with
multiplication by partial Hasse invariants.

It is also straightforward to check that the operator $\Theta_{\tau_{0}}$
commutes with the Hecke action in the obvious sense, and hence induces a
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$-equivariant map

$\Theta_{\tau}:M_{\mathbf{k},\mathbf{l}}({\mathbb{F}})\longrightarrow
M_{\mathbf{k}^{\prime},\mathbf{l}^{\prime}}({\mathbb{F}})$

where $M_{\mathbf{k},\mathbf{l}}({\mathbb{F}})$ (and
$M_{\mathbf{k}^{\prime},\mathbf{l}^{\prime}}({\mathbb{F}})$, with their
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$ actions) are
defined in (6) as direct limits over suitable open compact $U$.

Let us also make the effect of $\Theta_{\tau_{0}}$ on the weight $\mathbf{k}$
more explicit. Note that if
$\mathbf{k}=\sum_{\theta}k_{\theta}{\mathbf{e}}_{\theta}$ and
$\tau_{0}=\tau_{{\mathfrak{p}}_{0},i_{0}}$, then
$\mathbf{k}^{\prime}=\sum_{\theta}k^{\prime}_{\theta}{\mathbf{e}}_{\theta}$,
where

  * •

if $e_{{\mathfrak{p}}_{0}}=f_{{\mathfrak{p}}_{0}}=1$, then
$k^{\prime}_{\theta}=\left\\{\begin{array}[]{ll}k_{\theta}+p+1,&\mbox{if
$\theta=\theta_{0}=\theta_{{\mathfrak{p}}_{0},1,1}$,}\\\
k_{\theta},&{otherwise;}\end{array}\right.$

  * •

if $e_{{\mathfrak{p}}_{0}}=1$ and $f_{{\mathfrak{p}}_{0}}>1$, then
$k^{\prime}_{\theta}=\left\\{\begin{array}[]{ll}k_{\theta}+1,&\mbox{if
$\theta=\theta_{0}=\theta_{{\mathfrak{p}}_{0},i_{0},1},$}\\\
k_{\theta}+p,&\mbox{if
$\theta=\sigma^{-1}\theta_{0}=\theta_{{\mathfrak{p}}_{0},i_{0}-1,1}$,}\\\
k_{\theta},&{otherwise;}\end{array}\right.$

  * •

if $e_{{\mathfrak{p}}_{0}}>1$, then
$k^{\prime}_{\theta}=\left\\{\begin{array}[]{ll}k_{\theta}+1,&\mbox{if
$\theta=\theta_{0}=\theta_{{\mathfrak{p}}_{0},i_{0},e_{{\mathfrak{p}}_{0}}}$
or
$\theta=\sigma^{-1}\theta_{0}=\theta_{{\mathfrak{p}}_{0},i_{0},e_{{\mathfrak{p}}_{0}}-1}$,}\\\
k_{\theta},&{otherwise.}\end{array}\right.$

######  Remark 5.2.4.

Considerations from the theory of Serre weights from the point of [12] suggest
that the above weight shifts are in a certain sense optimal. One can also
define cruder partial $\Theta$-operators by composing with multiplication by
(products of) partial Hasse invariants. For example, the operator
$H_{\tau_{0},1}H_{\tau_{0},2}\cdots
H_{\tau_{0},e_{{\mathfrak{p}}_{0}}-1}\Theta_{\tau_{0}}$ is the one constructed
in [8], and for any $j=1,\ldots,e_{{\mathfrak{p}}_{0}}$, the operator

$H_{\tau_{0},j}H^{2}_{\tau_{0},j+1}\cdots
H^{2}_{\tau_{0},e_{{\mathfrak{p}}_{0}}-1}H_{\tau_{0},e_{{\mathfrak{p}}_{0}}}\Theta_{\tau_{0}}$

shifts the weight $\mathbf{k}$ by
${\mathbf{e}}_{\theta}+n_{\sigma^{-1}\theta}{\mathbf{e}}_{\sigma^{-1}\theta}$,
where $\theta=\theta_{{\mathfrak{p}}_{0},i_{0},j}$.

##  6\. Partial Frobenius operators

###  6.1. Partial Frobenius endomorphisms

In order to define partial Frobenius operators on Hilbert modular forms (in
§6.2 below), we first need to define partial Frobenius endomorphisms of
Hilbert modular varieties over $\mathbb{F}$.

Fix a prime ${\mathfrak{p}}$ dividing $p$, and a level $U$, assumed as usual
to be sufficiently small and prime to $p$. We will draw on ideas from [11,
§7.1] to construct an isogeny on the universal abelian variety $s:A\to S$,
where $S=\widetilde{Y}_{U,\mathbb{F}}$.

We begin by associating Raynaud data to the line bundles
${{\mathcal{L}}}_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$ over $S$, which we write
simply as ${{\mathcal{L}}}_{i}$ for
$i\in\mathbb{Z}/f\mathbb{Z}=\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}$ (omitting
the subscripts for the fixed ${\mathfrak{p}}$ and $j=e=e_{\mathfrak{p}}$). We
define $f_{i}:{{\mathcal{L}}}^{\otimes p}_{i}\to{{\mathcal{L}}}_{i+1}$ to be
zero, and we define $v_{i}:{{\mathcal{L}}}_{i+1}\to{{\mathcal{L}}}^{\otimes
p}_{i}$ to be the morphism induced by the restriction of

$\operatorname{Ver}^{*}_{A}:{\mathcal{H}}={\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)_{\mathfrak{p}}\to{\mathcal{H}}^{1}_{\operatorname{dR}}((\phi_{S}^{*}A)/S)_{\mathfrak{p}}=\phi_{S}^{*}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)_{\mathfrak{p}}=\phi_{S}^{*}{\mathcal{H}}$

to ${\mathcal{F}}_{i}^{(e)}=(s_{*}\Omega^{1}_{A/S})_{i}$ (abbreviating
subscripts $\tau_{{\mathfrak{p}},i}$ by $i$). Note that since the image of
${\mathcal{H}}_{i+1}$ under $\operatorname{Ver}_{A}^{*}$ is
$\phi_{S}^{*}({\mathcal{F}}_{i}^{(e)})$, the inclusions
${\mathcal{F}}_{i+1}^{(e-1)}\subset u{\mathcal{H}}_{i+1}$ and
$u{\mathcal{F}}_{i}^{(e)}\subset{\mathcal{F}}_{i}^{(e-1)}$ ensure that
$\operatorname{Ver}_{A}^{*}({\mathcal{F}}_{i+1}^{(e-1)})\subset\phi_{S}^{*}({\mathcal{F}}_{i}^{(e-1)})$,
so the morphism $v_{i}$ is well-defined. We then let $H$ denote the finite
flat $({\mathcal{O}}_{F}/{\mathfrak{p}})$-vector space scheme over $S$
associated to the Raynaud data
$({{\mathcal{L}}}_{i},f_{i},v_{i})_{i\in\mathbb{Z}/f\mathbb{Z}}$.

Recall that the Dieudonné crystal of $\ker({\operatorname{Frob}}_{A})$ is
canonically isomorphic to $\Phi^{*}(s_{*}\Omega^{1}_{A/S})$, with $F=0$ and
$V$ induced by $\Phi^{*}(\operatorname{Ver}_{A}^{*})$ (in the notation of [2,
§4.4.3]). On the other hand the Dieudonné crystal of $H$ is identified with
$\Phi^{*}(\oplus_{i}{{\mathcal{L}}}_{i})$ with $F=\Phi^{*}(\oplus_{i}f_{i})=0$
and $V=\Phi^{*}(\oplus_{i}v_{i})$ (as a simple special case of [11, Prop.
7.1.3]). Therefore the canonical projection
$s_{*}\Omega^{1}_{A/S}\to\oplus_{i}{{\mathcal{L}}}_{i}$ induces a surjective
morphism of Dieudonné crystals
${\mathbb{D}}(\ker({\operatorname{Frob}}_{A}))\to{\mathbb{D}}(H)$. As the base
$S$ is smooth over $\mathbb{F}$, the exact contravariant functor
${\mathbb{D}}$ is fully faithful on finite flat $p$-group schemes over $S$
([3, Thm. 4.1.1]), so the surjection arises from a closed immersion
$H\hookrightarrow\ker({\operatorname{Frob}}_{A})$, and we let

$\alpha:A\longrightarrow A^{\prime}:=A/H$

denote the resulting isogeny of abelian varieties over $S$. Note that
$A^{\prime}$ naturally inherits an ${\mathcal{O}}_{F}$-action $\iota^{\prime}$
from the action $\iota$ on $A$.

Let ${\mathcal{I}}$ denote the image of the morphism
$\alpha^{*}:{\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\prime}/\widetilde{Y}_{U,\mathbb{F}})_{\mathfrak{p}}\to{\mathcal{H}}$.
By construction, we have the exact sequence

$\begin{array}[]{ccccccc}{\mathbb{D}}(A^{\prime}[p])_{S}&\longrightarrow&{\mathbb{D}}(A[p])_{S}&\longrightarrow&{\mathbb{D}}(H)_{S}&\longrightarrow&0\\\
\parallel\wr&&\parallel\wr&&\parallel\wr&&\\\
{\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\prime}/S)&\stackrel{{\scriptstyle\alpha^{*}}}{{\longrightarrow}}&{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)&\stackrel{{\scriptstyle\operatorname{Ver}_{A}^{*}}}{{\longrightarrow}}&\bigoplus_{i}\phi_{S}^{*}({\mathcal{F}}_{\tau}^{(e)}/{\mathcal{F}}_{\tau}^{(e-1)})&\longrightarrow&0,\end{array}$

showing that ${\mathcal{I}}=\oplus_{i}{\mathcal{I}}_{i}$, where
${\mathcal{I}}_{i}$ is the preimage of
$\phi_{S}^{*}({\mathcal{F}}_{i-1}^{(e)})$ under
$\operatorname{Ver}_{A,i}^{*}:{\mathcal{H}}_{i}\to\phi_{S}^{*}({\mathcal{H}}_{i-1})$.
Note in particular that $u{\mathcal{H}}_{i}\subset{\mathcal{I}}_{i}$ for all
$i$, so that $H\subset A[{\mathfrak{p}}]$ and there is a unique isogeny
$\beta:{\mathfrak{p}}\otimes_{{\mathcal{O}}_{F}}A^{\prime}\longrightarrow A$
such that $\alpha\circ\beta$ is the canonical isogeny
${\mathfrak{p}}\otimes_{{\mathcal{O}}_{F}}A^{\prime}\to A^{\prime}$.

We now equip $A^{\prime}$ with auxiliary data corresponding to an element of
$\widetilde{Y}_{U,\mathbb{F}}(S)$.

Since $\alpha$ induces isomorphisms
$T^{(p)}(A_{\overline{s}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T^{(p)}(A^{\prime}_{\overline{s}})$
for all geometric points $\overline{s}$ of $S$, we immediately have a level
$U^{p}$ structure $\eta^{\prime}$ on $A^{\prime}$ inherited from $A$.

Next we claim that the quasi-polarization $\lambda$ on $A$ induces an
isomorphism101010Here ${\mathfrak{c}}$ depends on the connected component of
$S$.
${\mathfrak{p}}{\mathfrak{c}}{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}A^{\prime}\to(A^{\prime})^{\vee}$,
or equivalently
$A^{\prime}\to{\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}(A^{\prime})^{\vee}$,
which amounts to the claim that $H$ corresponds to the kernel of

${\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes\beta^{\vee}:{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}A^{\vee}\longrightarrow{\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}(A^{\prime})^{\vee}$

under the isomorphism induced by $\lambda$. Denoting this kernel by $I$, we
have that $H$ and $I$ are finite flat group schemes over $S$ of the same rank,
so it suffices to prove that the composite

$I\longrightarrow{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}A^{\vee}[p]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}A[p]\longrightarrow
A^{\prime}[p]$

is trivial. Taking Dieudonné modules, this in turn amounts to the vanishing of
the composite

${\mathbb{D}}(A^{\prime}[p])_{S}\longrightarrow{\mathbb{D}}(A[p])_{S}\longrightarrow{\mathbb{D}}({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}A^{\vee}[p])_{S}\longrightarrow{\mathbb{D}}(I)_{S}.$

We have already noted that the image of the first map has
${\mathfrak{p}}$-component $\oplus_{i}{\mathcal{I}}_{i}$; on the other hand
the kernel of the last map is the image of the map

${\mathbb{D}}({\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}(A^{\prime})^{\vee}[p])_{S}\longrightarrow{\mathbb{D}}({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}A^{\vee}[p])_{S}$

corresponding to the adjoint of

$\beta^{*}:{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S)\to{\mathcal{H}}^{1}_{\operatorname{dR}}(({\mathfrak{p}}\otimes_{{\mathcal{O}}_{F}}A^{\prime})/S)\cong{\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F}}{\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\prime}/S)$

under the canonical isomorphisms

$\begin{array}[]{rl}{\mathbb{D}}({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}A^{\vee}[p])_{S}\cong{\mathcal{H}}^{1}_{\operatorname{dR}}(({\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}A^{\vee}/S)&\\\
\cong{\mathcal{H}om}_{{\mathcal{O}}_{S}}({\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}{\mathcal{H}}^{1}_{\operatorname{dR}}(A/S),{\mathcal{O}}_{S})&\cong{\mathcal{H}om}_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}}({\mathcal{H}}^{1}_{\operatorname{dR}}(A/S),{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S})\end{array}$

and similarly

${\mathbb{D}}({\mathfrak{p}}^{-1}{\mathfrak{c}}^{-1}{\mathfrak{d}}^{-1}\otimes_{{\mathcal{O}}_{F}}(A^{\prime})^{\vee}[p])_{S}\cong{\mathcal{H}om}_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S}}({\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F}}{\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\prime}/S),{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{S})$

obtained from duality. We are therefore reduced to proving that
${\mathcal{I}}_{i}$ is orthogonal to the kernel of $\beta_{i}^{*}$ for each
$i\in\mathbb{Z}/f\mathbb{Z}$ under the pairing $\langle\cdot,\cdot\rangle_{i}$
defined by (4). Note however that the kernel of $\beta_{i}^{*}$ is
$u^{e-1}{\mathcal{I}}_{i}$, as can be seen for example from the commutative
diagram

$\textstyle{H^{1}_{\mathrm{crys}}(A_{{\overline{s}}}/W(\overline{{\mathbb{F}}}_{p}))_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F}}H^{1}_{\mathrm{crys}}(A^{\prime}_{{\overline{s}}}/W(\overline{{\mathbb{F}}}_{p})))_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}_{\operatorname{dR}}(A_{{\overline{s}}}/\overline{{\mathbb{F}}}_{p}))_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F}}H^{1}_{\operatorname{dR}}(A^{\prime}_{{\overline{s}}}/\overline{{\mathbb{F}}}_{p}))_{i}}$

of $W(\overline{{\mathbb{F}}}_{p})[u]$-modules for ${\overline{s}}\in
S(\overline{{\mathbb{F}}}_{p})$. Finally the orthogonality of
${\mathcal{I}}_{i}$ and $u^{e-1}{\mathcal{I}}_{i}$ is immediate from that of
${\mathcal{F}}_{i-1}^{(e-1)}$ and $u^{-1}{\mathcal{F}}_{i-1}^{(e-1)}$ provided
by Lemma 3.1.1, completing the proof of the claim. We may then define the
quasi-polarization on $A^{\prime}$ by
$\alpha^{*}(\lambda^{\prime})=\delta\lambda$ for any totally positive
generator $\delta=\delta_{\mathfrak{p}}$ of
${\mathfrak{p}}{\mathcal{O}}_{F,(p)}$, so that $\lambda^{\prime}$ induces an
isomorphism
${\mathfrak{c}}^{\prime}{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}A^{\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(A^{\prime})^{\vee}$
where ${\mathfrak{c}}^{\prime}=\delta^{-1}{\mathfrak{p}}{\mathfrak{c}}$.

Finally we define a Pappas–Rapoport filtration on
${\mathcal{F}}^{\prime}_{\tau}:=(s^{\prime}_{*}\Omega^{1}_{A^{\prime}/S})_{\tau}$
for all $\tau\in\Sigma_{0}$.

First note that if $\tau\not\in\Sigma_{{\mathfrak{p}},0}$, then $\alpha^{*}$
induces an isomorphism
${\mathcal{F}}^{\prime}_{\tau}\simeq{\mathcal{F}}_{\tau}^{(e)}$, and we define
$({\mathcal{F}}_{\tau}^{\prime})^{(j)}$ as the pre-image of
${\mathcal{F}}_{\tau}^{(j)}$.

Suppose now that $\tau=\tau_{{\mathfrak{p}},i}$. Recall from the construction
of $A^{\prime}=A/H$ that
$\operatorname{Ver}_{A}^{*}({\mathcal{F}}_{i}^{(e-1)})\subset\phi_{S}^{*}({\mathcal{F}}_{i-1}^{(e-1)})$,
so we have ${\mathcal{F}}_{i}^{(e-1)}\subset{\mathcal{I}}_{i}$. It follows
that $(\alpha_{i}^{*})^{-1}({\mathcal{F}}_{i}^{(e-1)})$ is a subbundle of
${\mathcal{H}}^{\prime}_{i}:={\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\prime})_{i}$
of the same rank as ${\mathcal{F}}^{\prime}_{i}$, namely $e$. Furthermore we
have

$\begin{array}[]{rl}\phi_{S}^{*}(\alpha^{*}({\mathcal{F}}_{i}^{\prime}))=(\phi_{S}^{*}(\alpha))^{*}({\mathcal{F}}_{i}^{\prime})=&(\phi_{S}^{*}(\alpha))^{*}(\operatorname{Ver}_{A^{\prime}}^{*}({\mathcal{H}}_{i+1}^{\prime}))\\\
=&\operatorname{Ver}_{A}^{*}(\alpha^{*}({\mathcal{H}}_{i+1}^{\prime}))=\operatorname{Ver}_{A}^{*}({\mathcal{I}}_{i+1})\subset\phi_{S}^{*}({\mathcal{F}}_{i}^{(e+1)}),\end{array}$

so in fact
${\mathcal{F}}_{i}^{\prime}\subset(\alpha_{i}^{*})^{-1}({\mathcal{F}}_{i}^{(e-1)})$,
and hence equality holds. We thus obtain an exact sequence

$0\to\ker(\alpha_{i}^{*})\longrightarrow{\mathcal{F}}^{\prime}_{i}\stackrel{{\scriptstyle\alpha_{i}^{*}}}{{\longrightarrow}}{\mathcal{F}}_{i}^{(e-1)}\to
0.$

We may thus define a Pappas–Rapoport filtration on
${\mathcal{F}}_{i}^{\prime}$ by setting

$({\mathcal{F}}_{i}^{\prime})^{(j)}=(\alpha_{i}^{*})^{-1}({\mathcal{F}}_{i}^{(j-1)})$

for $j=1,\ldots,e$, so in particular
$({\mathcal{F}}_{i}^{\prime})^{(1)}=\ker(\alpha_{i}^{*})$.

We now define
$\widetilde{\Phi}_{\mathfrak{p}}:\widetilde{Y}_{U,\mathbb{F}}=S\to\widetilde{Y}_{U,\mathbb{F}}$
to be the endomorphism corresponding to the data
$(A^{\prime},\iota^{\prime},\lambda^{\prime},\eta^{\prime},({\mathcal{F}}^{\prime})^{\bullet})$.
Note that $\widetilde{\Phi}_{\mathfrak{p}}$ depends on the choice of $\delta$
in the definition of $\lambda^{\prime}$; however it is straightforward to
check that $\widetilde{\Phi}_{\mathfrak{p}}$ is compatible with the
${\mathcal{O}}_{F,(p),+}^{\times}$-action on $S$ and descends to an
endomorphism $\Phi_{\mathfrak{p}}$ of $\overline{Y}_{U}$ which is independent
of this choice. We call $\widetilde{\Phi}_{\mathfrak{p}}$ (resp.
$\Phi_{\mathfrak{p}}$) the partial Frobenius endomorphism (indexed by
${\mathfrak{p}}$) of $\widetilde{Y}_{U,\mathbb{F}}$ (resp.
$\overline{Y}_{U}$); the terminology is justified by the next proposition.

For the statement of the proposition, we also define the endomorphism
$\widetilde{\Phi}$ of $S=\widetilde{Y}_{U,\mathbb{F}}$ corresponding to the
data
$\phi_{S}^{*}({\underline{A}})=(\phi_{S}^{*}A,\phi_{S}^{*}\iota,\phi_{S}^{*}\lambda,\phi_{S}^{*}\eta,(\phi_{S}^{*}{\mathcal{F}})^{\bullet})$,
where $(\phi_{S}^{*}{\mathcal{F}})^{\bullet}$ is the collection of filtrations
on the vector bundles

$((\phi_{S}^{*}s)_{*}(\Omega^{1}_{(\phi_{S}^{*}A)/S}))_{\tau}=(\phi_{S}^{*}(s_{*}\Omega^{1}_{A/S}))_{\tau}=\phi_{S}^{*}((s_{*}\Omega^{1}_{A/S})_{\phi^{-1}\circ\tau})$

given by
$(\phi_{S}^{*}{\mathcal{F}})_{\tau}^{(j)}=\phi_{S}^{*}({\mathcal{F}}_{\phi^{-1}\circ\tau}^{(j)})$.
Note that $\widetilde{\Phi}$ is not the absolute Frobenius $\phi_{S}$ on $S$
(unless $\mathbb{F}=\mathbb{F}_{p}$), but we may write
$\phi_{S}=\widetilde{\Phi}\circ\widetilde{\epsilon}$ where
$\widetilde{\epsilon}$ is the isomorphism defined by the commutative diagram

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0.0pt\hbox{$\textstyle{{\operatorname{Spec}\,}\mathbb{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
71.10416pt\raise-73.71312pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
71.74097pt\raise-84.10828pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\phi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
88.82639pt\raise-77.99718pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
88.82639pt\raise-77.99718pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\operatorname{Spec}\,}\mathbb{F}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where the square is Cartesian and $\varepsilon$ is the inverse of the
isomorphism associated to $\phi^{*}A=A\times_{\mathbb{F},\phi}\mathbb{F}$ with
the evident auxiliary data. We thus have an isomorphism
$\widetilde{\epsilon}^{*}A\cong A$ compatible with $\iota$, $\lambda$ and
$\eta$, and inducing
$\widetilde{\epsilon}^{*}{\mathcal{F}}_{\tau}^{(j)}\cong{\mathcal{F}}_{\phi\circ\tau}^{(j)}$
for all $\tau$ and $j$. (Note also that $\widetilde{\Phi}$ may be viewed as
the base-change of the absolute Frobenius on the descent of $S$ to
$\mathbb{F}_{p}$ defined by the diagram.)

The endomorphism $\widetilde{\Phi}$ is compatible with the
${\mathcal{O}}_{F,(p),+}$-action on $S=\widetilde{Y}_{U,\mathbb{F}}$, and we
let $\Phi$ denote the resulting endomorphism of $\overline{Y}_{U}$. Similarly
$\widetilde{\epsilon}$ descends to a $\phi$-linear automorphism $\epsilon$ of
$\overline{Y}_{U}$ such that the absolute Frobenius on $\overline{Y}_{U}$ is
$\Phi\circ\epsilon$.

######  Proposition 6.1.1.

The morphisms $\Phi_{\mathfrak{p}}$ are finite flat of (constant) degree
$\mathrm{Nm}_{F/\mathbb{Q}}({\mathfrak{p}})$, commute with each other, and
satisfy the formula

$\prod_{{\mathfrak{p}}\in S_{p}}\Phi_{\mathfrak{p}}^{e_{\mathfrak{p}}}=\Phi.$

Proof. We first prove the commutativity and analogous formula for the maps
$\widetilde{\Phi}_{\mathfrak{p}}$ on $S=\widetilde{Y}_{U,\mathbb{F}}$, from
which the corresponding assertions for $\Phi_{\mathfrak{p}}$ follow. To that
end it suffices to consider the maps on geometric closed points
$\overline{s}\in S(\overline{{\mathbb{F}}}_{p})$, which we will do in order to
facilitate computations on Dieudonné modules.

Let ${\underline{A}}_{0}$ denote the data corresponding to $\overline{s}\in
S(\overline{{\mathbb{F}}}_{p})$ and
${\underline{A}}_{0}^{\prime}={\underline{A}}_{0,{\mathfrak{p}}}^{\prime}$
denote the data corresponding to
$\widetilde{\Phi}_{\mathfrak{p}}(\overline{s})$. Let
$D=H^{1}_{\mathrm{crys}}(A_{0}/W(\overline{{\mathbb{F}}}_{p}))$ and
$D^{\prime}=H^{1}_{\mathrm{crys}}(A_{0}^{\prime}/W(\overline{{\mathbb{F}}}_{p}))$,
so we may decompose the ${\mathcal{O}}_{F}\otimes
W(\overline{{\mathbb{F}}}_{p})$-modules $D=\oplus_{\tau\in\Sigma}D_{\tau}$ and
$D^{\prime}=\oplus_{\tau\in\Sigma^{\prime}}D_{\tau}^{\prime}$ where $D_{\tau}$
and $D_{\tau}^{\prime}$ are free
$W(\overline{{\mathbb{F}}}_{p})[u]/(f^{\tau}(u))$-modules of rank two.
Furthermore the canonical isogeny $\alpha:A_{0}\to A_{0}^{\prime}$ induces an
injective $W(\overline{{\mathbb{F}}}_{p})[u]/(f^{\tau}(u))$-linear map
$\alpha_{\tau}^{*}:D_{\tau}^{\prime}\to D_{\tau}$ for each $\tau$, compatible
in the obvious sense with the maps

${\operatorname{Frob}}_{A_{0},\tau}^{*}:\phi^{*}(D_{\phi^{-1}\circ\tau})\to
D_{\tau}\quad\mbox{and}\quad{\operatorname{Frob}}_{A_{0}^{\prime},\tau}^{*}:\phi^{*}(D_{\phi^{-1}\circ\tau}^{\prime})\to
D^{\prime}_{\tau},$

as well as
$\operatorname{Ver}_{A_{0},\tau}^{*}=p({\operatorname{Frob}}_{A_{0},\tau}^{*})^{-1}$
and
$\operatorname{Ver}_{A_{0}^{\prime},\tau}^{*}=p({\operatorname{Frob}}_{A_{0}^{\prime},\tau}^{*})^{-1}$.
Letting $F_{\tau}^{(j)}\subset D_{\tau}$ denote the pre-image of
${\mathcal{F}}_{\tau}^{(j)}$ under the canonical surjection

$D_{\tau}\longrightarrow(D/pD)_{\tau}\cong
H^{1}_{\operatorname{dR}}(A_{0}/\overline{{\mathbb{F}}}_{p})_{\tau},$

we have $\alpha_{\tau}^{*}(D_{\tau}^{\prime})=D_{\tau}$ if
$\tau\not\in\Sigma_{{\mathfrak{p}},0}$ and

$\alpha_{\tau}^{*}(D_{\tau}^{\prime})=(\operatorname{Ver}_{A_{0},\tau}^{*})^{-1}(\phi^{*}(F_{\phi^{-1}\circ\tau}^{(e_{\mathfrak{p}}-1)}))$

if $\tau\in\Sigma_{{\mathfrak{p}},0}$ (by the construction of
$\Phi_{\mathfrak{p}}$). Furthermore writing $F^{\prime(j)}_{\tau}$ for the
submodules of $D_{\tau}^{\prime}$ similarly defined by the Pappas–Rapoport
filtration on
$H^{0}(A_{0}^{\prime},\Omega^{1}_{A_{0}^{\prime}/\overline{{\mathbb{F}}}_{p}})_{\tau}$,
we have

  * •

$\alpha_{\tau}^{*}(F^{\prime(j)}_{\tau})=F^{(j)}_{\tau}$ if
$\tau\not\in\Sigma_{{\mathfrak{p}},0}$,

  * •

$\alpha_{\tau}^{*}(F^{\prime(j)}_{\tau})=F^{(j-1)}_{\tau}$ for
$j=2,\ldots,e_{\mathfrak{p}}$ if $\tau\in\Sigma_{{\mathfrak{p}},0}$,

  * •

and
$\alpha_{\tau}^{*}(F^{\prime(1)}_{\tau})=pD_{\tau}={\operatorname{Frob}}_{A_{0},\tau}^{*}(\phi^{*}(F^{(e_{\mathfrak{p}})}_{\phi^{-1}\circ\tau}))$
if $\tau\in\Sigma_{{\mathfrak{p}},0}$.

Thus if ${\mathfrak{p}}_{1}$ and ${\mathfrak{p}}_{2}$ are distinct elements of
$S_{p}$, then
$\widetilde{\Phi}_{{\mathfrak{p}}_{1}}(\widetilde{\Phi}_{{\mathfrak{p}}_{2}}(\overline{s}))$
corresponds to the data ${\underline{A}}_{0}^{\prime\prime}$ for which we have
an isogeny $\alpha^{\prime}:A_{0}\to A_{0}^{\prime\prime}$ such that if
$\tau\not\in\Sigma_{{\mathfrak{p}}_{1},0}\cup\Sigma_{{\mathfrak{p}}_{2},0}$
then $(\alpha^{\prime})_{\tau}^{*}(D_{\tau}^{\prime\prime})=D_{\tau}$ and
$(\alpha^{\prime})_{\tau}^{*}(F^{\prime\prime(j)}_{\tau})=F^{(j)}_{\tau}$
(with the obvious notation), but if $\tau\in\Sigma_{{\mathfrak{p}}_{i},0}$ for
$i=1$ or $2$, then