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

$(\alpha^{\prime})_{\tau}^{*}(F^{\prime\prime(1)}_{\tau})=pD_{\tau}$ and
$(\alpha^{\prime})_{\tau}^{*}(F^{\prime\prime(j)}_{\tau})=F^{(j-1)}_{\tau}$
for $j=2,\ldots,e_{{\mathfrak{p}}_{i}}$. Furthermore we have
$\eta^{\prime\prime}=\alpha^{\prime}\circ\eta$ and
$(\alpha^{\prime})^{*}(\lambda^{\prime\prime})=\delta_{{\mathfrak{p}}_{2}}\delta_{{\mathfrak{p}}_{1}}\lambda$,
from which it follows that the isomorphism class of the data
${\underline{A}}_{0}^{\prime\prime}$ also corresponds to
$\widetilde{\Phi}_{{\mathfrak{p}}_{2}}(\widetilde{\Phi}_{{\mathfrak{p}}_{1}}(\overline{s}))$.

Now consider the data ${\underline{A}}_{0}^{(r)}$ associated to
$\widetilde{\Phi}_{\mathfrak{p}}^{r}(\overline{s})$ for
$r=1,\ldots,e_{\mathfrak{p}}$, and write $D_{r}=\oplus D_{r,\tau}$ for
$H^{1}_{\mathrm{crys}}(A_{0}^{(r)}/W(\overline{{\mathbb{F}}}_{p}))$,
$F_{r,\tau}$ for the submodule of $D_{r,\tau}$ determined as above by the
Pappas–Rapoport filtration, and $\alpha_{r}$ for the composite isogeny
$A_{0}\to A_{0}^{(1)}\to\cdots\to A_{0}^{(r)}$. By induction on $r$, we find
that if $\tau\not\in\Sigma_{{\mathfrak{p}},0}$, then
$\alpha_{r,\tau}^{*}(D_{r,\tau})=D_{\tau}$ and
$\alpha_{r,\tau}^{*}(F_{r,\tau}^{(j)})=F^{(j)}_{\tau}$, but if
$\tau\in\Sigma_{{\mathfrak{p}},0}$, then

  * •

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

  * •

$\alpha_{r,\tau}^{*}(F^{(j)}_{r,\tau})=F^{(j-r)}_{\tau}$ for
$j=r+1,\ldots,e_{\mathfrak{p}}$,

  * •

$\alpha_{r,\tau}^{*}(F^{(j)}_{r,\tau})={\operatorname{Frob}}_{A_{0},\tau}^{*}(\phi^{*}(F^{(e_{\mathfrak{p}}+j-r)}_{\phi^{-1}\circ\tau}))$
for $j=1,\ldots,r$.

In particular taking $r=e_{\mathfrak{p}}$ gives

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

and
$\alpha_{e_{\mathfrak{p}},\tau}^{*}(F^{(j)}_{e_{\mathfrak{p}},\tau})={\operatorname{Frob}}_{A_{0},\tau}^{*}(\phi^{*}(F^{(j)}_{\phi^{-1}\circ\tau}))$
for $j=1,\ldots,e_{\mathfrak{p}}$ and $\tau\in\Sigma_{{\mathfrak{p}},0}$. It
then follows that $\prod_{{\mathfrak{p}}\in
S_{p}}\widetilde{\Phi}_{\mathfrak{p}}^{e_{\mathfrak{p}}}(\overline{s})$
corresponds to ${\underline{A}}_{0}^{\prime\prime}$ with
$\alpha^{\prime}:A_{0}\to A_{0}^{\prime\prime}$ satisfying
$(\alpha^{\prime})_{\tau}^{*}(D_{\tau}^{\prime\prime})={\operatorname{Frob}}_{A_{0},\tau}^{*}(\phi^{*}(D_{\phi^{-1}\circ\tau}))$
and
$(\alpha^{\prime})_{\tau}^{*}(F^{\prime\prime(j)}_{\tau})={\operatorname{Frob}}_{A_{0},\tau}^{*}(\phi^{*}(F^{(j)}_{\phi^{-1}\circ\tau}))$
for all $\tau$ and $j$. Furthermore we have
$\eta^{\prime\prime}=\alpha^{\prime}\circ\eta$ and
$(\alpha^{\prime})^{*}(\lambda^{\prime\prime})=\prod_{{\mathfrak{p}}}\delta_{{\mathfrak{p}}}^{e_{\mathfrak{p}}}\lambda$,
from which it follows that ${\underline{A}}_{0}^{\prime\prime}$ is isomorphic
to
$(\phi^{*}A_{0},\phi^{*}\iota,\nu\phi^{*}\lambda,(\phi^{*}{\mathcal{F}})^{\bullet})$
with
$\nu=p^{-1}\prod_{{\mathfrak{p}}}\delta_{{\mathfrak{p}}}^{e_{\mathfrak{p}}}\in{\mathcal{O}}_{F,(p),+}^{\times}$.
This proves that

$\prod_{{\mathfrak{p}}\in
S_{p}}\widetilde{\Phi}_{\mathfrak{p}}^{e_{\mathfrak{p}}}=\nu\cdot\widetilde{\Phi},$

which in turn implies the desired formula.

Since $\Phi$ is finite (and $\overline{Y}_{U}$ is separated), it follows that
$\Phi_{\mathfrak{p}}$ is finite, and therefore also flat since
$\overline{Y}_{U}$ regular. Note furthermore that $\Phi_{\mathfrak{p}}$ is
therefore bijective on closed points and induces isomorphisms on their residue
fields, so the degree of $\Phi_{\mathfrak{p}}$ in a neighborhood of any closed
point $x$ of $\overline{Y}_{U}$ is that of the extension of completed regular
local rings
$\Phi_{{\mathfrak{p}},x}^{*}:{\mathcal{O}}_{\overline{Y}_{U},y}^{\wedge}\to{\mathcal{O}}_{\overline{Y}_{U},x}^{\wedge}$,
where $y=\Phi_{\mathfrak{p}}(x)$. Since $\Phi$ factors through
$\Phi_{\mathfrak{p}}$, so does the absolute Frobenius on $\overline{Y}_{U}$,
and hence the absolute Frobenius on
${\mathcal{O}}_{\overline{Y}_{U},x}^{\wedge}$ factors through
$\Phi_{{\mathfrak{p}},x}^{*}$. Therefore it follows from [26, Cor. 2] that
$\deg(\Phi_{{\mathfrak{p}},x}^{*})=p^{n}$ where $n=n_{\mathfrak{p}}$ is the
dimension of the kernel of the induced map on tangent spaces
$T_{x}(\overline{Y}_{U})\to T_{y}(\overline{Y}_{U})$. Furthermore since
$\prod_{{\mathfrak{p}}\in S_{p}}\Phi_{\mathfrak{p}}^{e_{\mathfrak{p}}}=\Phi$
has degree $p^{[F:\mathbb{Q}]}=\prod_{{\mathfrak{p}}\in
S_{p}}p^{e_{\mathfrak{p}}f_{\mathfrak{p}}}$, it suffices to prove that
$n_{\mathfrak{p}}\geq f_{\mathfrak{p}}=Nm_{F/\mathbb{Q}}({\mathfrak{p}})$ for
each ${\mathfrak{p}}$. Note also that we may replace $\overline{Y}_{U}$ by
$S=\widetilde{Y}_{U,\mathbb{F}}$, $x$ by any point in its pre-image in $S$ and
$\Phi_{\mathfrak{p}}$ by $\widetilde{\Phi}_{{\mathfrak{p}}}$.

Suppose then that $x$ corresponds to the data
$(A_{0},\iota_{0},\lambda_{0},\eta_{0},{\mathcal{F}}_{0}^{\bullet})$ over the
residue field $k$, and its image $y=\widetilde{\Phi}_{\mathfrak{p}}(x)$
corresponds to the data
$(A^{\prime}_{0},\iota^{\prime}_{0},\lambda^{\prime}_{0},\eta^{\prime}_{0},{\mathcal{F}}^{\prime\bullet}_{0})$.
Recall that the Kodaira–Spencer filtration on the fibre of
$\Omega^{1}_{S/\overline{{\mathbb{F}}}_{p}}$ at $x$ is dual to one on
$T_{x}(S)$ which was described using Grothendieck–Messing deformation theory
(see the discussion following the proof of Theorem 3.3.1). In particular, we
have a decomposition $T_{x}(S)=\oplus_{\tau\in\Sigma_{0}}T_{x}(S)_{\tau}$ and
a decreasing filtration of length $e_{{\mathfrak{p}}^{\prime}}$ on
$T_{x}(S)_{\tau}$ for each $\tau\in\Sigma_{{\mathfrak{p}}^{\prime},0}$ (where
${\mathfrak{p}}^{\prime}\in S_{p}$) such that

$\bigoplus_{\tau\in\Sigma_{0}}{\operatorname{Fil}\,}^{j_{\tau}}T_{x}(S)_{\tau}$

corresponds to the set of lifts of ${\underline{A}}_{0}$ to
${\underline{A}}_{1}=(A_{1},\iota_{1},\lambda_{1},\eta_{1},{\mathcal{F}}_{1}^{\bullet})\in
S(k[\epsilon])$ 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 claim that the $f_{\mathfrak{p}}$-dimensional subspace
$\oplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\operatorname{Fil}\,}^{e_{\mathfrak{p}}-1}T_{x}(S)_{\tau}$
is contained in the kernel of $T_{x}(S)\to T_{y}(S)$. Indeed if
${\underline{A}}_{1}$ is a lift corresponding to an element of this subspace
and ${\underline{A}}_{1}^{\prime}$ is its image in $T_{y}(S)$ and
$\alpha_{i}:A_{i}\to A_{i}^{\prime}$ are the specializations of the universal
isogeny $\alpha:A\to A^{\prime}$, then the commutativity of the diagram

$\textstyle{H^{1}_{\operatorname{dR}}(A_{1}^{\prime}/k[\epsilon])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}^{*}}$$\scriptstyle{\wr}$$\textstyle{H^{1}_{\operatorname{dR}}(A_{1}/k[\epsilon])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{1}_{\operatorname{dR}}(A^{\prime}_{0}/k)\otimes_{k}k[\epsilon]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{0}^{*}\otimes
1}$$\textstyle{H^{1}_{\operatorname{dR}}(A_{0}/k)\otimes_{k}k[\epsilon]}$

and the definition of $\widetilde{\Phi}_{\mathfrak{p}}$ imply that
${\mathcal{F}}^{\prime(j)}_{1,\tau}$ corresponds to
${\mathcal{F}}^{\prime(j)}_{0,\tau}\otimes_{k}k[\epsilon]$ for all $\tau$ and
$j$. (Note in particular that
${\mathcal{F}}_{1,\tau}^{(1)}=\ker(\alpha_{1,\tau}^{*})$ for all
$\tau\in\Sigma_{{\mathfrak{p}},0}$, and that
$H^{0}(A_{1}^{\prime},\Omega_{A_{1}^{\prime}/k[\epsilon]})_{\tau}$ corresponds
to $H^{0}(A_{0},\Omega_{A_{0}^{\prime}/k})_{\tau}\otimes_{k}k[\epsilon]$ for
all $\tau\in\Sigma_{0}$.) It follows that ${\underline{A}}_{1}^{\prime}$ is
the trivial deformation of ${\underline{A}}_{0}^{\prime}$, so the kernel of
$T_{x}(S)\to T_{y}(S)$ has dimension $n\geq f_{\mathfrak{p}}$ as required.
$\square$

######  Remark 6.1.2.

The final part of the proof of the proposition shows that the kernel of the
pull-back map
$\Phi_{\mathfrak{p}}^{*}\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}}\to\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}}$
is precisely

$\Phi_{\mathfrak{p}}^{*}\left(\oplus_{\tau\in\Sigma_{{\mathfrak{p}},0}}{\operatorname{Fil}\,}^{1}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau}\right).$

Furthermore a similar argument shows that the map preserves the
Kodaira–Spencer decomposition and filtration, in the obvious sense, and
induces isomorphisms

$\begin{array}[]{cc}\Phi_{\mathfrak{p}}^{*}\left({\operatorname{Fil}\,}^{j}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau}/{\operatorname{Fil}\,}^{1}(\Omega^{!}_{\overline{Y}_{U}/\mathbb{F}})_{\tau}\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\operatorname{Fil}\,}^{j-1}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau},&\mbox{if
$\tau\in\Sigma_{{\mathfrak{p}},0},j=1,\ldots,e_{\mathfrak{p}}$,}\\\
\Phi_{\mathfrak{p}}^{*}\left({\operatorname{Fil}\,}^{j}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau}\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\operatorname{Fil}\,}^{j}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau},&\mbox{if
$\tau\not\in\Sigma_{{\mathfrak{p}},0}$.}\end{array}$

###  6.2. Construction of $V_{\mathfrak{p}}$

In this section we generalize the construction111111The operators defined here
differ slightly from the ones defined in [12] in the unramified case. The
construction there is tailored to be compatible with the classical case and to
be simply interpreted on $q$-expansions at cusps at $\infty$. Doing this in
the general ramified case would introduce complications that make it seem not
worthwhile. of [12, §9.8] to define partial Frobenius operators, similar to
the $V_{p}$-operator on classical modular forms.

We maintain the notation of §6.1, so that $\widetilde{\Phi}_{\mathfrak{p}}$ is
an endomorphism of $S=\widetilde{Y}_{U,\mathbb{F}}$ corresponding to the data
$(A^{\prime},\iota^{\prime},\lambda^{\prime},\eta^{\prime},({\mathcal{F}}^{\prime})^{\bullet})$,
where $A^{\prime}=A/H$ for a certain finite flat subgroup scheme $H\subset
A[{\mathfrak{p}}]$, and $\alpha$ is the projection $A\to A^{\prime}$.

It is immediate from the definition of ${\mathcal{F}}^{\prime(j)}_{\tau}$ that
$\alpha_{\tau}^{*}$ induces an isomorphism
${\mathcal{L}}_{\tau,j}^{\prime}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{L}}_{\tau,j}$
for all $j$ if $\tau\not\in\Sigma_{{\mathfrak{p}},0}$, as well as

${\mathcal{L}}_{\tau,j}^{\prime}={\mathcal{F}}^{\prime(j)}_{\tau}/{\mathcal{F}}^{\prime(j-1)}_{\tau}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{F}}^{(j-1)}_{\tau}/{\mathcal{F}}^{(j-2)}_{\tau}={\mathcal{L}}_{\tau,j-1}$

for $j=2,\ldots,e_{\mathfrak{p}}$ if $\tau\in\Sigma_{{\mathfrak{p}},0}$. To
describe ${\mathcal{L}}_{\tau,1}^{\prime}$ for
$\tau\in\Sigma_{{\mathfrak{p}},0}$, note that since
$H\subset\ker({\operatorname{Frob}}_{A})$, there is an isogeny
$\gamma:A^{\prime}\to\phi_{S}^{*}A$ such that
$\gamma\circ\alpha={\operatorname{Frob}}_{A}$. We thus obtain a surjection

$\gamma_{\tau}^{*}:\phi_{S}^{*}({\mathcal{F}}_{\phi^{-1}\circ\tau}^{(e_{\mathfrak{p}})})=\ker({\operatorname{Frob}}_{A})_{\tau}^{*}\longrightarrow\ker(\alpha_{\tau}^{*})={\mathcal{L}}^{\prime}_{\tau,1}.$

Furthermore
$\phi_{S}^{*}({\mathcal{F}}_{\phi^{-1}\circ\tau}^{(e_{\mathfrak{p}}-1)})=\ker(\gamma_{\tau}^{*})$,
as can be seen on closed points, so we obtain a surjection, hence isomorphism,
of line bundles

${\mathcal{L}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}}^{\otimes
p}\cong\phi_{S}^{*}({\mathcal{L}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}})=\phi_{S}^{*}({\mathcal{F}}_{\phi^{-1}\circ\tau}^{(e_{\mathfrak{p}})})/\phi_{S}^{*}({\mathcal{F}}_{\phi^{-1}\circ\tau}^{(e_{\mathfrak{p}}-1)})\stackrel{{\scriptstyle\gamma_{\tau}^{*}}}{{\longrightarrow}}{\mathcal{L}}^{\prime}_{\tau,1}.$

By construction, we have
$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{L}}_{\tau,j}={\mathcal{L}}^{\prime}_{\tau,j}$,
so we have now defined isomorphisms

$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{L}}_{\theta}\cong\left\\{\begin{array}[]{ll}{\mathcal{L}}_{\sigma^{-1}\theta}^{\otimes
n_{\theta}},&\mbox{if $\theta\in\Sigma_{{\mathfrak{p}}}$;}\\\
{\mathcal{L}}_{\theta},&\mbox{if
$\theta\not\in\Sigma_{{\mathfrak{p}}}$.}\end{array}\right.$

Similarly we find that if $\tau\in\Sigma_{{\mathfrak{p}},0}$, then
$\gamma^{*}$ induces
$\phi_{S}^{*}({\mathcal{P}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}})\cong{\mathcal{P}}^{\prime}_{\tau,1}$
and $\alpha^{*}$ induces
${\mathcal{P}}^{\prime}_{\tau,j}\cong{\mathcal{P}}_{\tau,j-1}$ for
$j=2,\ldots,e_{\mathfrak{p}}$, so that
$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{N}}_{\theta}\cong{\mathcal{N}}_{\sigma^{-1}\theta}^{\otimes
n_{\theta}}$ if $\theta\in\Sigma_{\mathfrak{p}}$, while
$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{N}}_{\theta}\cong{\mathcal{N}}_{\theta}$
if $\theta\not\in\Sigma_{\mathfrak{p}}$. Taking tensor products, we thus
obtain isomorphisms

$\Phi_{\mathfrak{p}}^{*}\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},\mathbb{F}}\cong\widetilde{{\mathcal{A}}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}}$

for all $\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}$, where

  * •

$k^{\prime\prime}_{\theta}=k_{\theta}$ and
$l_{\theta}^{\prime\prime}=l_{\theta}$ if
$\theta\not\in\Sigma_{\mathfrak{p}}$, and

  * •

$k^{\prime\prime}_{\theta}=n_{\sigma\theta}k_{\sigma\theta}$ and
$l^{\prime\prime}_{\theta}=n_{\sigma\theta}l_{\sigma\theta}$ if
$\theta\in\Sigma_{\mathfrak{p}}$.

Furthermore it is straightforward to check that the isomorphisms are
compatible with the descent data relative to
$S=\widetilde{Y}_{U,\mathbb{F}}\to Y_{U,\mathbb{F}}$, so we obtain
isomorphisms

(13)
$\Phi_{\mathfrak{p}}^{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}}\cong{\mathcal{A}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}}$

for all $\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}$ and sufficiently small
$U$. Note that

(14)
$\mathbf{k}^{\prime\prime}=\mathbf{k}+\sum_{\theta\in\Sigma_{\mathfrak{p}}}k_{\theta}\mathbf{h}_{\theta}\quad\mbox{and}\quad\mathbf{l}^{\prime\prime}=\mathbf{l}+\sum_{\theta\in\Sigma_{\mathfrak{p}}}l_{\theta}\mathbf{h}_{\theta}.$

######  Remark 6.2.1.

One can check that the resulting isomorphisms
$\Phi_{\mathfrak{p}}^{*}{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta},\mathbb{F}}\cong{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta},\mathbb{F}}$
(for $\theta\not\in\Sigma_{\mathfrak{p}}$), and

$\Phi_{\mathfrak{p}}^{*}{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\tau},\mathbb{F}}\cong{\mathcal{A}}_{2{\mathbf{e}}_{\sigma^{-1}\theta},-{\mathbf{e}}_{\sigma^{-1}\theta},\mathbb{F}}$

(for $\theta=\theta_{{\mathfrak{p}},i,j}$, $j=2,\ldots,e_{\mathfrak{p}}$) are
compatible via the Kodaira–Spencer isomorphisms of Theorem 3.3.1 with the
corresponding isomorphisms

$\Phi_{\mathfrak{p}}^{*}({\operatorname{gr}\,}^{j}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau})\cong{\operatorname{gr}\,}^{j^{\prime}}(\Omega^{1}_{\overline{Y}_{U}/\mathbb{F}})_{\tau}$

given by Remark 6.1.2, where $j^{\prime}=j$ if
$\tau\not\in\Sigma_{{\mathfrak{p}},0}$ and $j^{\prime}=j-1$ if
$\tau\in\Sigma_{{\mathfrak{p}},0}$.

We are now ready to define the partial Frobenius operator (indexed by
${\mathfrak{p}}$)

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

as the composite

$H^{0}(Y_{U,\mathbb{F}},{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}})\stackrel{{\scriptstyle\Phi_{\mathfrak{p}}^{*}}}{{\longrightarrow}}H^{0}(Y_{U,\mathbb{F}},\Phi_{\mathfrak{p}}^{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(Y_{U,\mathbb{F}},{\mathcal{A}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}}),$

where the second map is the isomorphism (13). It is immediate from the
definition that $V_{\mathfrak{p}}$ is injective, and that taking the direct
sum over all weights yields an $\mathbb{F}$-algebra homomorphism

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

for all sufficiently small $U$ containing
$\operatorname{GL}_{2}({\mathcal{O}}_{F,p})$. It is also straightforward to
check that $V_{\mathfrak{p}}$ is compatible with the Hecke action in the usual
sense, and hence defines a
$\operatorname{GL}_{2}(\mathbb{A}_{F,\mathbf{f}}^{(p)})$-equivariant map

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

where the spaces are defined in (6) as direct limits over sufficiently small
$U$ containing $\operatorname{GL}_{2}({\mathcal{O}}_{F,p})$.

It will also be convenient at times to consider instead the operator

(15)
$V^{0}_{\mathfrak{p}}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\longrightarrow
M_{\mathbf{k}^{\prime\prime},\mathbf{l}}(U,\mathbb{F})$

defined by
$V^{0}_{\mathfrak{p}}(f)=V_{\mathfrak{p}}(f)\prod_{\theta\in\Sigma_{\mathfrak{p}}}G_{\theta}^{-l_{\theta}}$,
where $G_{\theta}$ is the trivialization of
${\mathcal{A}}_{\bf{0},\mathbf{h}_{\theta},\mathbb{F}}$ defined at the end of
§4.1. Thus $V_{\mathfrak{p}}^{0}$ is also Hecke-equivariant, but depends on
the choice of uniformizer $\varpi_{\mathfrak{p}}$.

We also record the relation between the partial Frobenius operators and the
$p$-power map. First note that the identification
$\widetilde{\Phi}^{*}({\mathcal{F}}_{\tau}^{(j)})=\phi_{S}^{*}({\mathcal{F}}_{\phi^{-1}\circ\tau}^{(j)})$
arising from the definition of $\widetilde{\Phi}$ yields isomorphisms

$\widetilde{\Phi}^{*}{\mathcal{L}}_{\tau,j}\cong\phi_{S}^{*}({\mathcal{L}}_{\phi^{-1}\circ\tau,j})\cong{\mathcal{L}}_{\phi^{-1}\circ\tau,j}^{\otimes
p}$

for all $\tau$ and $j$. We similarly have
$\widetilde{\Phi}^{*}{\mathcal{N}}_{\tau,j}\cong{\mathcal{N}}_{\phi^{-1}\circ\tau,j}^{\otimes
p}$, and taking tensor products and descending to $\overline{Y}$ yields
isomorphisms
$\Phi^{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}}\cong{\mathcal{A}}_{p\mathbf{k}^{\phi},p\mathbf{l}^{\phi},\mathbb{F}}$
for all $\mathbf{k},\mathbf{l}$, where
$k^{\phi}_{\theta_{{\mathfrak{p}},i,j}}=k_{\theta_{{\mathfrak{p}},i+1,j}}$,
and hence an operator $V_{p}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\to
M_{p\mathbf{k}^{\phi},p\mathbf{l}^{\phi}}(U;\mathbb{F})$. Similarly the
isomorphisms
$\widetilde{\epsilon}^{*}{\mathcal{F}}_{\phi^{-1}\circ\tau}^{(j)}\cong{\mathcal{F}}_{\tau}^{(j)}$
yield
$\widetilde{\epsilon}^{*}{\mathcal{L}}_{\phi^{-1}\circ\tau,j}\cong{\mathcal{L}}_{\tau,j}$
and
$\widetilde{\epsilon}^{*}{\mathcal{N}}_{\phi^{-1}\circ\tau,j}\cong{\mathcal{N}}_{\tau,j}$
whose tensor products descend to isomorphisms
$\epsilon^{*}{\mathcal{A}}_{\mathbf{k}^{\phi},\mathbf{l}^{\phi},\mathbb{F}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}}$,
yielding a $\phi$-linear isomorphism
$M_{\mathbf{k}^{\phi},\mathbf{l}^{\phi}}(U;\mathbb{F})\to
M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$ which we denote
$\epsilon_{\mathbf{k},\mathbf{l}}$. Furthermore the above isomorphisms of line
bundles on $S$ are compatible in the sense that the resulting diagram

$\textstyle{\widetilde{\epsilon}^{*}\widetilde{\Phi}^{*}{\mathcal{L}}_{\tau,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\wr}$$\textstyle{\widetilde{\epsilon}^{*}{\mathcal{L}}_{\phi^{-1}\circ\tau,j}^{\otimes
p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{\phi_{S}^{*}{\mathcal{L}}_{\tau,j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{{\mathcal{L}}_{\tau,j}^{\otimes
p}}$

commutes, as does its analogue for the ${\mathcal{N}}_{\theta}$, from which it
follows that the composite

$M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\stackrel{{\scriptstyle
V_{p}}}{{\longrightarrow}}M_{p\mathbf{k}^{\phi},p\mathbf{l}^{\phi}}(U;\mathbb{F})\stackrel{{\scriptstyle\epsilon_{p\mathbf{k},p\mathbf{l}}}}{{\longrightarrow}}M_{p\mathbf{k},p\mathbf{l}}(U;\mathbb{F})$

is the $p$-power map.

Returning to the partial Frobenius operators, the isomorphisms between
$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{L}}_{\theta}$ and
${\mathcal{L}}_{\sigma^{-1}\theta}^{\otimes n_{\theta}}$ (resp.
${\mathcal{L}}_{\theta}$) for $\theta\in\Sigma_{\mathfrak{p}}$ (resp.
$\theta\not\in\Sigma_{\mathfrak{p}}$) for different ${\mathfrak{p}}\in S_{p}$
are compatible with each other in the obvious sense, and taken together with
the formula $\prod_{{\mathfrak{p}}\in
S_{p}}\widetilde{\Phi}_{\mathfrak{p}}^{e_{\mathfrak{p}}}=\nu\cdot\widetilde{\Phi}$
and the canonical isomorphism
$\nu^{*}{\mathcal{L}}_{\theta}\cong{\mathcal{L}}_{\theta}$ yield the
isomorphisms
$\widetilde{\Phi}^{*}{\mathcal{L}}_{\tau,j}\cong{\mathcal{L}}_{\phi^{-1}\circ\tau,j}^{\otimes
p}$ defined above. A similar assertion holds for the line bundles
${\mathcal{N}}_{\theta}$, and it follows that the operators $V_{\mathfrak{p}}$
for ${\mathfrak{p}}\in S_{p}$ commute with each other and that
$\prod_{\mathfrak{p}}V_{\mathfrak{p}}^{e_{\mathfrak{p}}}=V_{p}$, so that

(16) $\left(\epsilon_{p\mathbf{k},p\mathbf{l}}\prod_{{\mathfrak{p}}\in
S_{p}}V_{\mathfrak{p}}^{e_{\mathfrak{p}}}\right)(f)=f^{p}.$

##  7\. Compactifications and $q$-expansions

###  7.1. Toroidal compactifications

We next recall how $q$-expansions of Hilbert modular forms are obtained using
compactifications of Hilbert modular varieties. In this section we review
properties of the toroidal compactification constructed by Rapoport [31] (see
also [4] and [13]). We will consider toroidal compactifications only in the
case $U=U(N)$, but we first describe the set of cusps adelically for any $U$
of level prime to $p$.

For an arbitrary open compact subgroup $U$ of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}})$ containing
$\operatorname{GL}_{2}({\mathcal{O}}_{F,p})$, we define the set of cusps of
$Y_{U}$ to be

$Y_{U}^{\infty}=B({\mathcal{O}}_{F,(p)})_{+}\backslash\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})/U^{p}=B(F)_{+}\backslash\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}})/U,$

where $B$ denotes the subgroup of $\operatorname{GL}_{2}$ consisting of upper-
triangular matrices. Similarly we define the set of cusps of
$\widetilde{Y}_{U}$ to be

$\widetilde{Y}_{U}^{\infty}=B_{1}({\mathcal{O}}_{F,(p)})_{+}\backslash\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})/U^{p}$

Note that the natural surjection $\widetilde{Y}^{\infty}_{U}\to
Y^{\infty}_{U}$ identifies $Y^{\infty}_{U}$ with the quotient of
$\widetilde{Y}^{\infty}_{U}$ by the left action of
$B({\mathcal{O}}_{F,(p)})_{+}/B_{1}({\mathcal{O}}_{F,(p)})\cong{\mathcal{O}}_{F,(p),+}^{\times}$.
Furthermore the subgroup $({\mathcal{O}}_{F}^{\times}\cap U)^{2}$ acts
trivially on $\widetilde{Y}^{\infty}_{U}$, but the quotient
${\mathcal{O}}_{F,(p),+}^{\times}/({\mathcal{O}}_{F}^{\times}\cap U)^{2}$ need
not act freely; more precisely, the stabilizer of the cusp
$B_{1}({\mathcal{O}}_{F,(p)})gU$ of $\widetilde{Y}_{U}$ is the group
$\det(gUg^{-1}\cap B(F))_{+}\subset{\mathcal{O}}_{F,+}^{\times}$, in which
$({\mathcal{O}}_{F}^{\times}\cap U)^{2}$ has finite index.

We also have a natural bijection between $Y^{\infty}_{U}$ and the set of
isomorphism classes of data $(H,I,[\lambda],[\eta])$ where

  * •

$H$ is a projective ${\mathcal{O}}_{F}$-module of rank two;

  * •

$I$ is an invertible submodule of $H$ such that $J:=H/I$ is invertible;

  * •

$[\lambda]$ is a prime-to-$p$ orientation of
$\wedge^{2}_{{\mathcal{O}}_{F}}H\cong I\otimes_{{\mathcal{O}}_{F}}J=IJ$, by
which we mean an ${\mathcal{O}}_{F,(p),+}^{\times}$-orbit of isomorphisms

$\lambda:\wedge^{2}_{{\mathcal{O}}_{F,(p)}}H_{(p)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}_{F,(p)}.$

  * •

$[\eta]$ is a level $U^{p}$-structure on $H$, i.e., a $U^{p}$-orbit of
$\widehat{{\mathcal{O}}}_{F}^{(p)}$-linear isomorphisms

$\eta:(\widehat{{\mathcal{O}}}_{F}^{(p)})^{2}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widehat{{\mathcal{O}}}_{F}^{(p)}\otimes_{{\mathcal{O}}_{F}}H.$

The bijection is defined by associating the data
$(H_{g},I_{g},[\lambda_{g}],[\eta_{g}])$ to the coset
$B({\mathcal{O}}_{F,(p)})_{+}gU^{p}$, where
$H_{g}=\widehat{{\mathcal{O}}}_{F}^{2}g^{-1}\cap F^{2}$, $I_{g}$ is its
intersection with the subspace $\\{0\\}\times F$, $\lambda_{g}$ is induced by
the determinant, and $\eta_{g}$ is induced by right pre-multiplication by
$g^{-1}$.

Note that to give a prime-to-$p$ orientation of
$\wedge^{2}_{{\mathcal{O}}_{F}}H$ is equivalent to giving an
$F_{+}^{\times}$-orbit of isomorphisms $\wedge_{F}^{2}(\mathbb{Q}\otimes
H)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}F$, but the integrality
condition is imposed for consistency with the fact that we have a bijection
between $\widetilde{Y}^{\infty}_{U}$ and the set of isomorphism classes of
data of the form $(H,I,\lambda,[\eta])$, which is similarly defined,
${\mathcal{O}}_{F,(p),+}^{\times}$-equivariant, and compatible in the obvious
sense with the bijection describing $Y^{\infty}_{U}$. In particular if
$U=U(1)=\operatorname{GL}_{2}(\widehat{{\mathcal{O}}}_{F})$, then the map
sending $(H,I,[\lambda],[\eta])$ to the pair
$(\wedge^{2}_{{\mathcal{O}}_{F}}H,I)$ defines a bijection between
$Y^{\infty}_{U}$ and $C_{F}^{+}\times C_{F}$, where $C_{F}^{(+)}$ denotes the
(strict) class group of $F$. For each such cusp we choose a polyhedral cone
decomposition as in [31, Lemme 4.2] (with $U^{\prime}=U_{+}$ in the notation
there) for $(M^{*}\otimes\mathbb{R})_{+}\cup\\{0\\}$, where

(17)
$\begin{array}[]{l}M={\operatorname{Hom}\,}_{{\mathcal{O}}_{F}}(I,{\mathfrak{d}}^{-1}J)={\mathfrak{d}}^{-1}I^{-1}J,\\\
M^{*}={\operatorname{Hom}\,}(M,\mathbb{Z})\cong{\operatorname{Hom}\,}_{{\mathcal{O}}_{F}}(J,I)=J^{-1}I,\end{array}$

and the positivity is induced by the orientation of
$I\otimes_{{\mathcal{O}}_{F}}J$.

Suppose now that $U=U(N)$ for some $N\geq 3$ (not divisible by $p$) and that
${\mathcal{O}}$ contains the $N$th roots of unity. The above choice of cone
decomposition (for the image of each cusp of $\widetilde{Y}^{\infty}_{U}$ in
$Y^{\infty}_{U(1)}$) yields a toroidal compactification121212Compactified in
the sense that its (infinitely many) connected components are proper over
${\mathcal{O}}$.
$\widetilde{Y}_{U}\hookrightarrow\widetilde{Y}_{U}^{\operatorname{tor}}$ such
that the set of (geometrically) connected components of its (reduced) closed
subscheme
$\widetilde{Z}_{U}^{\operatorname{tor}}:=\widetilde{Y}_{U}^{\operatorname{tor}}-\widetilde{Y}_{U}$
is identified with $\widetilde{Y}^{\infty}_{U}$. The construction of
$\widetilde{Y}_{U}^{\operatorname{tor}}$ identifies its completion along the
component corresponding to a cusp $\widetilde{{\mathcal{C}}}$ represented by
$(H,I,\lambda,[\eta])$ with the quotient of a formal scheme131313The formal
scheme depends on the chosen cone decomposition
$\\{\sigma_{\alpha}^{\widetilde{{\mathcal{C}}}}\\}$ and is denoted
$S_{N}(\\{\sigma_{\alpha}^{\widetilde{{\mathcal{C}}}}\\})^{\wedge}$ in [4,
3.4.2]. $\widehat{S}_{\widetilde{{\mathcal{C}}}}$ by an action of $V_{N}^{2}$,
where
$V_{N}=\ker({\mathcal{O}}_{F}^{\times}\to({\mathcal{O}}_{F}/N{\mathcal{O}}_{F})^{\times})$.
Furthermore this extends to an action of ${\mathcal{O}}_{F,+}^{\times}$ on
$\widehat{S}=\widehat{S}_{\widetilde{{\mathcal{C}}}}$, and we have an
isomorphism

(18)
$\Gamma(\widehat{S},{\mathcal{O}}_{\widehat{S}})\cong{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}$

compatible with the obvious action of ${\mathcal{O}}_{F,+}^{\times}$ on the
target. (The isomorphism depends on a choice of splitting of the exact
sequence

$0\to I\to H\to J\to 0$

of ${\mathcal{O}}_{F}$-modules; modifying the splitting by an element
$\beta\in J^{-1}I\cong M^{*}$ alters it by composition with the automorphism
defined by $q^{m}\mapsto\zeta_{N}^{-\beta(m)}q^{m}$ for $m\in N^{-1}M$.) We
let $\xi:{\widehat{S}}\to\widetilde{Y}_{U}$ denote the natural morphism of
formal schemes, and we write $F_{\widehat{S}}$ for the field of fractions of
$\Gamma({\widehat{S}},{\mathcal{O}}_{\widehat{S}})$ and $\mu_{\nu}$ for the
automorphism of ${\widehat{S}}$ defined by
$\nu\in{\mathcal{O}}_{F,+}^{\times}$.

The construction of the toroidal compactification also extends the universal
abelian scheme $A$ to a semi-abelian scheme $A^{\operatorname{tor}}$ whose
pull-back to $\widehat{S}$ is identified with that of the Tate semi-abelian
scheme141414More precisely, the formal scheme $\widehat{S}$ has an open cover
by affine formal subschemes ${\operatorname{Spf}\,}R_{\sigma}$ (indexed by
cones $\sigma$) such that
${\operatorname{Spec}\,}R_{\sigma}\times_{\widetilde{Y}_{U}^{\operatorname{tor}}}A^{\operatorname{tor}}$
is identified with the semi-abelian scheme $T_{I,J}$ over
${\operatorname{Spec}\,}R_{\sigma}$. The compatibilities in the discussion
that follows are then systematically checked by verifying them over the open
subschemes
${\operatorname{Spec}\,}R_{\sigma}^{0}={\operatorname{Spec}\,}R_{\sigma}\times_{\widetilde{Y}_{U}^{\operatorname{tor}}}\widetilde{Y}_{U}$.
associated to a quotient of the form

(19)
$T_{I,J}:=({\mathfrak{d}}^{-1}I\otimes{\mathbb{G}}_{m})/\widetilde{q}^{{\mathfrak{d}}^{-1}J},$

where $\widetilde{q}^{\cdot}:{\mathfrak{d}}^{-1}J\to
F_{\widehat{S}}^{\times}\otimes{\mathfrak{d}}^{-1}I$ is the homomorphism
corresponding to the tautological element under the canonical isomorphism

${\operatorname{Hom}\,}(M,F_{\widehat{S}}^{\times})={\operatorname{Hom}\,}_{{\mathcal{O}}_{F}}({\mathfrak{d}}^{-1}J,{\mathfrak{d}}^{-1}I\otimes
F_{\widehat{S}}^{\times}).$

Similarly its dual $A^{\vee}$ extends to a Tate semi-abelian scheme
$(A^{\vee})^{\operatorname{tor}}$ whose pull-back via $\xi$ is associated to
$T_{{\mathfrak{d}}J^{-1},{\mathfrak{d}}I^{-1}}$, with the isomorphism
${\mathfrak{c}}{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}A^{\operatorname{tor}}\to(A^{\vee})^{\operatorname{tor}}$
defined by the quasi-polarization pulling back to the composite

${\mathfrak{c}}{\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}T_{I,J}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathfrak{d}}(IJ)^{-1}\otimes_{{\mathcal{O}}_{F}}T_{I,J}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T_{{\mathfrak{d}}J^{-1},{\mathfrak{d}}I^{-1}},$

where ${\mathfrak{c}}=\\{\,\alpha\in
F\,|\,\alpha\lambda(IJ)\subset{\mathcal{O}}_{F}\,\\}$, the first morphism is
the isomorphism induced by $\lambda$, and the second is the canonical one.

The subschemes $Z_{\theta}$ of $\widetilde{Y}_{U}$ (defined in §4.2 by the
vanishing of the partial Hasse invariants $H_{\theta}$) are closed in
$\widetilde{Y}_{U}^{\operatorname{tor}}$, and we let
$\widetilde{Y}_{U}^{\operatorname{ord}}$ (resp.
$\widetilde{Y}_{U}^{\operatorname{tord}}$) denote the complement of their
union, i.e., the ordinary locus, in $\widetilde{Y}_{U}$ (resp.
$\widetilde{Y}_{U}^{\operatorname{tor}}$), and we use similar notation for the
restrictions of $A^{\operatorname{tor}}$ and
$(A^{\vee})^{\operatorname{tor}}$. Since the sheaf
${\mathcal{L}ie}(A^{\operatorname{tord}}/\widetilde{Y}_{U}^{\operatorname{tord}})$
is locally free over
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tord}}}$,
the universal filtration ${\mathcal{F}}_{\tau}^{(j)}$ on

$(s_{*}\Omega_{A/\widetilde{Y}_{U}}^{1})_{\tau}\cong{\mathcal{H}om}_{{\mathcal{O}}_{\widetilde{Y}_{U}}}({\mathcal{L}ie}(A/\widetilde{Y}_{U})_{\tau},{\mathcal{O}}_{\widetilde{Y}_{U}})$

extends canonically to one on
${\mathcal{H}om}_{{\mathcal{O}}_{\widetilde{Y}^{\operatorname{tor}}_{U}}}({\mathcal{L}ie}(A^{\operatorname{tor}}/\widetilde{Y}^{\operatorname{tor}}_{U})_{\tau},{\mathcal{O}}_{\widetilde{Y}^{\operatorname{tor}}_{U}})$
for each $\tau\in\Sigma_{0}$. Furthermore its pull-back to ${\widehat{S}}$ is
identified (in the notation of (1)) with

$0\subset t_{\tau,1}(I^{-1}\otimes{\mathcal{O}}_{\widehat{S}})_{\tau}\subset
t_{\tau,2}(I^{-1}\otimes{\mathcal{O}}_{\widehat{S}})\subset\cdots\subset
t_{\tau,e_{\mathfrak{p}}}(I^{-1}\otimes{\mathcal{O}}_{\widehat{S}})_{\tau}=(I^{-1}\otimes{\mathcal{O}}_{\widehat{S}})_{\tau}$

under the canonical isomorphism

${\mathcal{H}om}_{{\mathcal{O}}_{{\widehat{S}}}}({\mathcal{L}ie}(T_{I,J}/{\widehat{S}}),{\mathcal{O}}_{\widehat{S}})\cong{\operatorname{Hom}\,}({\mathfrak{d}}^{-1}I,{\mathcal{O}}_{\widehat{S}})\cong
I^{-1}\otimes{\mathcal{O}}_{\widehat{S}}.$

We thus obtain extensions
${\widetilde{{\mathcal{L}}}}_{\theta}^{\operatorname{tor}}$ of the line
bundles
${\widetilde{{\mathcal{L}}}}_{\theta}={\widetilde{{\mathcal{L}}}}_{{\mathfrak{p}},i,j}$
to $\widetilde{Y}_{U}^{\operatorname{tor}}$ whose pull-back to ${\widehat{S}}$
is identified with
$(I^{-1})_{\theta}\otimes_{{\mathcal{O}}}{\mathcal{O}}_{\widehat{S}}$, where
$(I^{-1})_{\theta}$ is defined by (2).

Similarly
${\mathcal{L}ie}((A^{\vee})^{\operatorname{tord}}/\widetilde{Y}_{U}^{\operatorname{tord}})$
is locally free over
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tord}}}$,
but the line bundles
${\widetilde{{\mathcal{M}}}}_{\theta}={\widetilde{{\mathcal{M}}}}_{{\mathfrak{p}},i,j}$
over $\widetilde{Y}_{U}^{\operatorname{ord}}$ are canonically identified with

$(R^{1}s_{*}{\mathcal{O}}_{A^{\operatorname{ord}}})_{\tau}[u-\theta(\varpi_{\mathfrak{p}})]\cong{\mathcal{L}ie}((A^{\vee})^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}})_{\tau}[u-\theta(\varpi_{\mathfrak{p}})].$

It follows that each ${\widetilde{{\mathcal{M}}}}_{\theta}$ extends to a line
bundle ${\widetilde{{\mathcal{M}}}}_{\theta}^{\operatorname{tor}}$ on
$\widetilde{Y}_{U}^{\operatorname{tor}}$ such that the identification

${\mathcal{L}ie}(T_{{\mathfrak{d}}J^{-1},{\mathfrak{d}}I^{-1}}/{\widehat{S}})=J^{-1}\otimes{\mathcal{O}}_{\widehat{S}}$

induces an isomorphism

$\xi^{*}{\widetilde{{\mathcal{M}}}}_{\theta}^{\operatorname{tor}}\cong(J^{-1}\otimes{\mathcal{O}}_{{\widehat{S}}})_{\tau}[u-\theta(\varpi_{\mathfrak{p}})]\cong({\mathfrak{d}}J^{-1}\otimes{\mathcal{O}}_{\widehat{S}})_{\tau}\otimes_{{\mathcal{O}}[u],\theta}{\mathcal{O}}.$

We can thus identify the pull-back
$\xi^{*}{\widetilde{{\mathcal{N}}}}_{\theta}^{\operatorname{tor}}$ of the line
bundle
${\widetilde{{\mathcal{N}}}}_{\theta}^{\operatorname{tor}}={\widetilde{{\mathcal{L}}}}_{\theta}^{\operatorname{tor}}\otimes_{{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tor}}}}{\widetilde{{\mathcal{M}}}}_{\theta}^{\operatorname{tor}}$
with
$({\mathfrak{d}}(IJ)^{-1})_{\theta}\otimes_{{\mathcal{O}}}{\mathcal{O}}_{\widehat{S}}$,
which the polarization in turn identifies with
$({\mathfrak{c}}{\mathfrak{d}})_{\theta}\otimes_{{\mathcal{O}}}{\mathcal{O}}_{\widehat{S}}$
in the notation of (2). Finally it follows that the automorphic bundles
${\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}$ extend to line bundles
${\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}^{\operatorname{tor}}$ on
$\widetilde{Y}_{U}^{\operatorname{tor}}$ such that

(20)
$\xi^{*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}^{\operatorname{tor}}\cong
D_{\mathbf{k},\mathbf{l}}\otimes_{{\mathcal{O}}}{\mathcal{O}}_{\widehat{S}},\quad\mbox{where\,\,}D_{\mathbf{k},\mathbf{l}}:=\bigotimes_{\theta\in\Sigma}\left((I^{-1})^{\otimes
k_{\theta}}_{\theta}\otimes({\mathfrak{d}}(IJ)^{-1})^{\otimes
l_{\theta}}_{\theta}\right)$

(the tensor products being over ${\mathcal{O}}$). We refer to this isomorphism
as the canonical trivialization of
$\xi^{*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}^{\operatorname{tor}}$.

Next we consider the completion of $\widetilde{Y}_{U}^{\operatorname{tor}}$
along the component corresponding to the cusp $\widetilde{{\mathcal{C}}}$
represented by $(H,I,\lambda,[\eta])$, which we denote
$(\widetilde{Y}_{U^{\operatorname{tor}}})_{\widetilde{{\mathcal{C}}}}^{\wedge}$.
We now describe the global sections of the completions of the line bundles
${\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}^{\operatorname{tor}}$
using the identification
$(\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}}^{\wedge}={\widehat{S}}/V_{N}^{2}$
and taking invariants under the action of $V_{N}^{2}$ on their trivializations
over ${\widehat{S}}$. Note firstly that
$\Gamma((\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}}^{\wedge},{\mathcal{O}}_{(\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}}^{\wedge}})=\Gamma({\widehat{S}},{\mathcal{O}}_{\widehat{S}})^{V_{N}^{2}}$
corresponds to

(21) $\left\\{\,\left.\sum r_{m}q^{m}\in{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\,\right|\,r_{\alpha^{2}m}=r_{m}\,\,\forall\alpha\in
V_{N},m\in N^{-1}M_{+}\,\right\\}$

under the isomorphism of (18). One then finds that the descent data for
$\xi^{*}A^{\operatorname{tor}}$ is provided by the isomorphisms
$T_{I,J}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mu_{\alpha^{2}}^{*}T_{I,J}$
induced by $\alpha\otimes 1$ on ${\mathfrak{d}}^{-1}I\otimes{\mathbb{G}}_{m}$,
from which it follows that the descent data for
$\xi^{*}{\widetilde{{\mathcal{L}}}}_{\theta}^{\operatorname{tor}}$ is provided
on the trivialization by the isomorphisms

$(I^{-1})_{\theta}\otimes_{\mathcal{O}}\mu^{*}_{\alpha^{2}}{\mathcal{O}}_{{\widehat{S}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(I^{-1})_{\theta}\otimes_{\mathcal{O}}{\mathcal{O}}_{{\widehat{S}}}$

induced by $\theta(\alpha)$ on $(I^{-1})_{\theta}$. On the other hand the
descent data for
$\xi^{*}{\widetilde{{\mathcal{M}}}}_{\theta}^{\operatorname{tor}}$ is
similarly induced on the canonical trivialization by $\theta(\alpha)^{-1}$, so
that the resulting trivialization of
$\xi^{*}{\widetilde{{\mathcal{N}}}}_{\theta}^{\operatorname{tor}}$ descends to
$(\widetilde{Y}_{U^{\operatorname{tor}}})_{\widetilde{{\mathcal{C}}}}^{\wedge}$
(in fact extending the one already defined over $\widetilde{Y}_{U}$ via the
choice of generator $t_{\tau,j}(f^{\prime}(\varpi_{\mathfrak{p}})\otimes 1)$
of $({\mathfrak{c}}{\mathfrak{d}})_{\theta}$). Since
$\Gamma((\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}},({\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l}})_{\widetilde{{\mathcal{C}}}}^{\wedge})=\Gamma({\widehat{S}},\xi^{*}{\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l}})^{V_{N}^{2}}$,
we conclude the following:

######  Proposition 7.1.1.

Suppose that $U=U(N)$ and ${\mathcal{O}}$ contains the $N$th roots of unity.
Then the isomorphism (18) and the canonical trivialization (20) identify
$\Gamma((\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}},({\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l}})_{\widetilde{{\mathcal{C}}}}^{\wedge})$
with

$\left\\{\,\left.\sum_{m\in N^{-1}M_{+}\cup\\{0\\}}b\otimes
r_{m}q^{m}\,\right|\,r_{\alpha^{2}m}=\chi_{\mathbf{k}}(\alpha)r_{m}\,\,\forall\alpha\in
V_{N},m\in N^{-1}M_{+}\,\right\\},$

where $b$ is any choice of basis for $D_{\mathbf{k},\mathbf{l}}$.

###  7.2. Minimal compactifications

We now recall the construction due to Chai [4] of minimal compactifications of
Hilbert modular varieties. The presentation in [4] is very concise with
numerous typos, but a more detailed treatment of the construction can be found
in [13] in the case of $U_{1}({\mathfrak{n}})$ (with different conventions
than ours), and of the descriptions of $q$-expansions in that case in [5].

We continue to assume for the moment that $U=U(N)$ for some sufficiently large
$N$ prime to $p$. The minimal compactification
$\widetilde{Y}_{U}\hookrightarrow\widetilde{Y}_{U}^{\min}$ is then constructed
as in [4, §4] or [13, §8]. More precisely, letting
$\mathbf{t}=\sum{\mathbf{e}}_{\theta}$ and taking the global sections of
$\oplus_{k\geq
0}{\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{k\mathbf{t},\bf{0}}$ over
each component of $\widetilde{Y}_{U}^{\operatorname{tor}}$ yields a projective
scheme over ${\mathcal{O}}$ containing the corresponding component of the
Deligne-Pappas model as an open subscheme. Gluing their ordinary loci to
$\widetilde{Y}_{U}$ along $\widetilde{Y}_{U}^{\operatorname{ord}}$ yields a
scheme $\widetilde{Y}_{U}^{\min}$ and a proper morphism
$\pi:\widetilde{Y}_{U}^{\operatorname{tor}}\to\widetilde{Y}_{U}^{\min}$ such
that ${\widetilde{\iota}}:\widetilde{Y}_{U}\to\widetilde{Y}_{U}^{\min}$ is an
open immersion. Furthermore the (reduced) complement
$\widetilde{Y}_{U}^{\min}-\widetilde{Y}_{U}$ is an infinite disjoint union of
copies of ${\operatorname{Spec}\,}{\mathcal{O}}$ indexed by
$\widetilde{Y}^{\infty}_{U}$, the preimage of each in
$\widetilde{Y}_{U}^{\operatorname{tor}}$ being the corresponding connected
component of $\widetilde{Z}_{U}^{\operatorname{tor}}$, and the scheme
$\widetilde{Y}_{U}^{\min}$ is independent of the choice of cone decomposition
in the construction of $\widetilde{Y}_{U}^{\operatorname{tor}}$.

Now recall that the Koecher Principle implies that
${\mathcal{O}}_{\widetilde{Y}_{U}^{\min}}=\pi_{*}{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tor}}}={\widetilde{\iota}}_{*}{\mathcal{O}}_{\widetilde{Y}_{U}}$,
so that
${\mathcal{O}}^{\wedge}_{\widetilde{Y}_{U}^{\min},\widetilde{{\mathcal{C}}}}=\Gamma((\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}}^{\wedge},{\mathcal{O}}_{(\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}}^{\wedge}})$
is the ring described by (21), where we have written
$\widetilde{{\mathcal{C}}}$ for the corresponding point of
$\widetilde{Y}_{U}^{\min}$. Furthermore the argument of [31, Prop. 4.9] shows
that
${\widetilde{\iota}}_{*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}=\pi_{*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}^{\operatorname{tor}}$
(see the discussion following [31, Def. 6.10], or view $\widetilde{Y}_{U}$ as
a disjoint union of PEL Shimura varieties and apply [28, Thm. 2.5]), so the
Theorem on Formal Functions gives that
$(\iota_{*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}})^{\wedge}_{\widetilde{{\mathcal{C}}}}=\Gamma((\widetilde{Y}_{U^{\operatorname{tor}}})_{\widetilde{{\mathcal{C}}}}^{\wedge},({\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l}})_{\widetilde{{\mathcal{C}}}}^{\wedge}))$
is the
${\mathcal{O}}^{\wedge}_{\widetilde{Y}_{U}^{\min},\widetilde{{\mathcal{C}}}}$-module
described in Proposition 7.1.1. (Note that
${\widetilde{\iota}}_{*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l}}$
is coherent, but not necessarily invertible.)

Similarly for any ${\mathcal{O}}$-algebra $R$, we may identify
$({\widetilde{\iota}}_{R,*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l},R})^{\wedge}_{\widetilde{{\mathcal{C}}}}$
with

(22) $\left\\{\,\left.\sum_{m\in N^{-1}M_{+}\cup\\{0\\}}b\otimes
r_{m}q^{m}\,\right|\,r_{\alpha^{2}m}=\chi_{\mathbf{k},R}(\alpha)r_{m}\,\,\forall\alpha\in
V_{N},m\in N^{-1}M_{+}\,\right\\}$

as a module over
${\mathcal{O}}^{\wedge}_{\widetilde{Y}_{U,R}^{\min},\widetilde{{\mathcal{C}}}}$,
which the Koecher Principle and (18) identify with

(23) $\left\\{\,\left.\sum r_{m}q^{m}\in R[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\,\right|\,r_{\alpha^{2}m}=r_{m}\,\,\forall\alpha\in
V_{N},m\in N^{-1}M_{+}\,\right\\},$

where
${\widetilde{\iota}}_{R}:\widetilde{Y}_{U,R}\to\widetilde{Y}_{U,R}^{\min}$ is
the base-change of ${\widetilde{\iota}}$ to $R$, the completions are at the
fibre over $\widetilde{{\mathcal{C}}}$, and $b$ is any basis for
$D_{\mathbf{k},\mathbf{l}}$.

The compatibility of the choices of polyhedral cone decompositions ensures
that the natural action of ${\mathcal{O}}_{F,(p),+}$ on $\widetilde{Y}_{U}$
extends (uniquely) to one on $\widetilde{Y}_{U}^{\operatorname{tor}}$.
Furthermore the stabilizer of each component of
$\widetilde{Z}_{U}^{\operatorname{tor}}$ is $V_{N,+}$, and the action of
$V_{N,+}$ on each completion
$(\widetilde{Y}_{U^{\operatorname{tor}}})_{\widetilde{{\mathcal{C}}}}^{\wedge}={\widehat{S}}/V_{N}^{2}$
is induced by an action of $V_{N,+}$ on ${\widehat{S}}$ such that the effect
of $\nu\in V_{N,+}$ on global sections of ${\mathcal{O}}_{{\widehat{S}}}$ is
induced by multiplication by $\nu^{-1}$ on $M$. We see also that the canonical
isomorphism $A\to\nu^{*}A$ extends to an isomorphism
$A^{\operatorname{tor}}\to\nu^{*}A^{\operatorname{tor}}$ whose pull-back via
$\xi$ is induced by the identity on
${\mathfrak{d}}^{-1}I\otimes{\mathbb{G}}_{m}$, from which it follows that the
action of $\nu$ is compatible with the canonical trivialization of the line
bundle ${\widetilde{{\mathcal{L}}}}_{\theta}^{\operatorname{tor}}$ over
${\widehat{S}}$. On the other hand the induced isomorphisms
$\nu^{*}{\widetilde{{\mathcal{M}}}}_{\theta}^{\operatorname{tor}}\to{\mathcal{M}}_{\theta}^{\operatorname{tor}}$
and
$\nu^{*}{\widetilde{{\mathcal{N}}}}_{\theta}^{\operatorname{tor}}\to{\widetilde{{\mathcal{N}}}}_{\theta}^{\operatorname{tor}}$
pull back to ones corresponding to multiplication by $\theta(\nu)$.

Since $\widetilde{Y}_{U}^{\min}$ is a disjoint union of projective schemes
over ${\mathcal{O}}$ on which ${\mathcal{O}}_{F,(p),+}^{\times}/V_{N}^{2}$
acts with finite stabilizers, the quotient scheme exists, and we define this
to be the minimal compactification $Y_{U}^{\min}$ of $Y_{U}$. We thus obtain
an open immersion $\iota:Y_{U}\to Y_{U}^{\min}$ such that $Y_{U}^{\min}$ is
projective over ${\mathcal{O}}$ and the (reduced) complement of $Y_{U}$ is a
disjoint union of copies of ${\operatorname{Spec}\,}{\mathcal{O}}$ in
canonical bijection with the set of cusps $Y_{U}^{\infty}$. Furthermore we
again have that $\iota_{*}{\mathcal{O}}_{Y_{U}}={\mathcal{O}}_{Y_{U}^{\min}}$,
and its completion ${\mathcal{O}}^{\wedge}_{Y_{U}^{\min},{\mathcal{C}}}$ at
the cusp ${\mathcal{C}}$ represented by $(H,I,[\lambda],[\eta])$ is identified
under (21) with

$\left\\{\,\left.\sum r_{m}q^{m}\in{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\,\right|\,r_{\nu m}=r_{m}\,\,\forall\nu\in
V_{N,+},m\in N^{-1}M_{+}\,\right\\}$

(where the identification depends as in (18) on a choice of splitting of the
exact sequence $0\to I\to H\to J\to 0$). Now suppose that $R$ is an
${\mathcal{O}}$-algebra such that $\chi_{\mathbf{k}+2\mathbf{l},R}$ is trivial
on $V_{N}$, so that the line bundle
$\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},R}$ descends to one over
$Y_{U,R}$ which we denote by ${\mathcal{A}}_{\mathbf{k},\mathbf{l},R}$. We
then see that $\iota_{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},R}$ is a coherent
sheaf on $Y_{U,R}^{\min}$ whose completion at the (base-change to $R$ of the)
cusp ${\mathcal{C}}$ is identified under (22) with

$\left\\{\,\left.\sum_{m\in N^{-1}M_{+}\cup\\{0\\}}b\otimes
r_{m}q^{m}\,\right|\,r_{\nu^{-1}m}=\chi_{\mathbf{l},R}(\nu)r_{m}\,\,\forall\nu\in
V_{N,+},m\in N^{-1}M_{+}\,\right\\}.$

In particular $\iota_{*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},R}$ is a line
bundle if $\chi_{\mathbf{l},R}$ is trivial on $V_{N,+}$.

Suppose now that $U^{\prime}$ is any sufficiently small open compact subgroup
of $\operatorname{GL}_{2}(\widehat{{\mathcal{O}}}_{F})$ containing
$\operatorname{GL}_{2}({\mathcal{O}}_{F,p})$. One can then carry out a
construction similar to the one above to obtain the minimal compactification,
or choose an $N$ prime to $p$ such that $U(N)\subset U^{\prime}$, extend the
natural (right) action of $U^{\prime}/U(N)$ on $Y_{U(N)}$ to $Y_{U(N)}^{\min}$
and take the quotient; we do the latter (see [13] for the former in the case
of $U^{\prime}=U_{1}({\mathfrak{n}})$). Firstly our choice of polyhedral cone
decompositions ensures that the natural right action of $U^{\prime}/U$ on
$\widetilde{Y}_{U}$ extends to $\widetilde{Y}_{U}^{\operatorname{tor}}$, where
$U=U(N)$ for some choice of $N$ as above. Denoting the resulting automorphism
of $\widetilde{Y}_{U}^{\operatorname{tor}}$ by $\widetilde{\rho}_{g}$ for
$g\in U^{\prime}$, the canonical identification of the universal $A$ over
$\widetilde{Y}_{U}$ with its pull-back extends to an identification
$A^{\operatorname{tor}}=\widetilde{\rho}_{g}^{*}A^{\operatorname{tor}}$,
giving rise to canonical isomorphisms
$\widetilde{\rho}_{g}^{*}{\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{k\mathbf{t},\bf{0}}={\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{k\mathbf{t},\bf{0}}$,
and hence to an action of $U^{\prime}/U$ on $\widetilde{Y}_{U}^{\min}$
extending its action on $\widetilde{Y}_{U}$. Moreover the action commutes with
the natural action of ${\mathcal{O}}_{F,(p),+}$, so it descends to an action
on $Y_{U}^{\min}$ extending the action on $Y_{U}$. We denote the resulting
automorphisms of $Y_{U}^{\min}$ by $\rho_{g}$, and define
$Y_{U^{\prime}}^{\min}$ to be the quotient of $Y_{U}^{\min}$ by the action of
$U^{\prime}/U$ (which we will shortly see is independent of the choice of $N$
in its definition).

Identifying the set of components of $Y_{U}^{\min}-Y_{U}$ with
$Y^{\infty}_{U}$, the resulting action of $g\in U^{\prime}$ is given by pre-
composing $\eta$ with right multiplication by $g^{-1}$, so the set of
components of $Y_{U^{\prime}}^{\min}-Y_{U^{\prime}}$ may be identified with
$Y^{\infty}_{U^{\prime}}$. For each cusp ${\mathcal{C}}^{\prime}\in
Y^{\infty}_{U^{\prime}}$, the completion
${\mathcal{O}}^{\wedge}_{Y_{U^{\prime}}^{\min},{\mathcal{C}}^{\prime}}$ is
identified with the subring of
${\mathcal{O}}^{\wedge}_{Y_{U}^{\min},{\mathcal{C}}}$ invariant under the
stabilizer of ${\mathcal{C}}$ in $U^{\prime}/U$, where ${\mathcal{C}}$ is any
cusp of $Y_{U}$ in the preimage of ${\mathcal{C}}^{\prime}$. Choose such an
$(H,I,[\lambda],[\eta])$ representing ${\mathcal{C}}$ and a splitting
$\sigma:H\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J\times I$, and let

(24)
$\Gamma_{{\mathcal{C}}}=\left\\{\,\left.\left(\begin{array}[]{cc}\alpha&\beta\\\
0&\delta\end{array}\right)\,\right|\,\alpha\in{\mathcal{O}}_{F}^{\times},\beta\in
J^{-1}I,\delta\in\alpha{\mathcal{O}}_{F,+}^{\times}\,\right\\},$

which we view as acting on $J\times I$ by right multiplication. The stabilizer
of ${\mathcal{C}}$ is then the set of classes $Ug=gU\in U^{\prime}/U$ such
that

$g\equiv\eta^{-1}\sigma^{-1}\gamma\sigma\eta\bmod
N\widehat{{\mathcal{O}}}_{F}\mbox{\,\,for
some\,\,}\gamma\in\Gamma_{{\mathcal{C}}},$

and we let
$\Gamma_{{\mathcal{C}},U^{\prime}}=\Gamma_{{\mathcal{C}}}\cap\sigma\eta
U^{\prime}\eta^{-1}\sigma^{-1}$. Thus the stabilizer of ${\mathcal{C}}$ is the
image of the homomorphism $s:\Gamma_{{\mathcal{C}},U^{\prime}}\longrightarrow
U^{\prime}/U$ defined by $\gamma\mapsto\eta^{-1}\sigma^{-1}\gamma\sigma\eta
U$.

We claim that if $g=s(\gamma)$, then $\rho_{g}^{*}$ on
${\mathcal{O}}^{\wedge}_{Y_{U}^{\min},{\mathcal{C}}}=H^{0}({\widehat{S}},{\mathcal{O}}_{{\widehat{S}}})^{V_{N,+}}$
is induced by an automorphism $\psi_{\gamma}$ of $\widehat{S}$ whose effect on
global sections corresponds to the map defined by

(25)
$\psi_{\gamma}^{*}:q^{m}\mapsto\zeta_{N}^{-\beta(\alpha^{-1}Nm)}q^{\alpha^{-1}\delta
m}$

under (18) and the identification $M^{*}=J^{-1}I$ of (17). Indeed letting
$\nu$ denote $\alpha\delta$ (as well as the automorphism of
$\widetilde{Y}_{U}^{\operatorname{tor}}$ defined by its effect on the
universal polarization), we see that $\delta\otimes 1$ on $T_{I,J}$ defines an
isomorphism
$\xi^{*}\rho_{g}^{*}\nu^{*}A^{\operatorname{tor}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\psi_{\gamma}^{*}\xi^{*}A^{\operatorname{tor}}$
compatible with all auxiliary data, from which one deduces that
$\nu\circ\rho_{g}\circ\xi=\xi\circ\psi_{\gamma}$. Note also that (25) defines
an action of $\Gamma_{{\mathcal{C}}}$ on
$\Gamma({\widehat{S}},{\mathcal{O}}_{{\widehat{S}}})$ which factors through
the surjection

$\begin{array}[]{ccc}\Gamma_{{\mathcal{C}}}&\longrightarrow&(J^{-1}I\otimes\mathbb{Z}/N\mathbb{Z})\rtimes{\mathcal{O}}_{F,+}^{\times}\\\
\left(\begin{array}[]{cc}\alpha&\beta\\\
0&\delta\end{array}\right)&\mapsto&(-\alpha^{-1}\beta,\alpha^{-1}\delta),\end{array}$

and the latter group acts on
$\Gamma({\widehat{S}},{\mathcal{O}}_{{\widehat{S}}})^{V_{N,+}}$ via its
quotient

$(J^{-1}I\otimes\mathbb{Z}/N\mathbb{Z})\rtimes({\mathcal{O}}_{F,+}^{\times}/V_{N,+}).$

We conclude that
${\mathcal{O}}^{\wedge}_{Y_{U^{\prime}}^{\min},{\mathcal{C}}^{\prime}}=H^{0}({\widehat{S}},{\mathcal{O}}_{{\widehat{S}}})^{\Gamma_{{\mathcal{C}},U^{\prime}}}$
is identified with

$\left\\{\,\sum_{m\in
N^{-1}M_{+}\cup\\{0\\}}r_{m}q^{m}\,\left|\,r_{\alpha^{-1}\delta
m}=\zeta_{N}^{-\beta(\alpha^{-1}Nm)}r_{m}\,\,\forall m\in
N^{-1}M_{+},\bigl{(}\begin{smallmatrix}{\alpha}&{\beta}\\\
{0}&{\delta}\end{smallmatrix}\bigr{)}\in\Gamma_{{\mathcal{C}},U^{\prime}}\,\right.\right\\},$

where we recall that the isomorphism may depend on the choice of the splitting
$\sigma$ and that we view $\beta$ as an element of $M^{*}$. We note in
particular if $U^{\prime}=U(N^{\prime})$ for some $N^{\prime}|N$, then the
resulting description of
${\mathcal{O}}^{\wedge}_{Y_{U^{\prime}}^{\min},{\mathcal{C}}^{\prime}}$
coincides with the one previously obtained, from which it follows that the
same holds for the scheme $Y_{U^{\prime}}^{\min}$, and hence that
$Y_{U^{\prime}}^{\min}$ is independent of the choice of $N$ in its definition
(for any sufficiently small $U^{\prime}$ containing
$\operatorname{GL}_{2}({\mathcal{O}}_{F,p})$.)

Suppose now that $\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}$ and $R$ is an
${\mathcal{O}}$-algebra such that $\chi_{\mathbf{k}+2\mathbf{l},R}$ is trivial
on $U^{\prime}\cap{\mathcal{O}}_{F}^{\times}$, and consider the automorphic
bundle ${\mathcal{A}}^{\prime}_{\mathbf{k},\mathbf{l},R}$ on
$Y_{U^{\prime},R}$. Letting $\iota_{R}^{\prime}$ denote the open immersion of
$Y_{U^{\prime},R}$ in $Y_{U^{\prime},R}^{\min}$, similar considerations to
those above show that
$\iota^{\prime}_{R,*}{\mathcal{A}}^{\prime}_{\mathbf{k},\mathbf{l},R}$ is a
coherent sheaf on $Y_{U,R}^{\min}$ whose completion at
${\mathcal{C}}^{\prime}$ is identified with the
${\mathcal{O}}^{\wedge}_{Y_{U^{\prime},R}^{\min},{\mathcal{C}}^{\prime}}$-module
of $\Gamma_{{\mathcal{C}},U^{\prime}}$-invariants in
$({\widetilde{\iota}}_{R,*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l},R})^{\wedge}_{{\mathcal{C}}}$.
Using that the isomorphism
$\xi^{*}A^{\operatorname{tor}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\psi_{\gamma}^{*}\xi^{*}A^{\operatorname{tor}}$
is induced by $\delta\otimes 1$ on the Tate semi-abelian scheme, we find that
the resulting automorphism multiplies the canonical trivialization (20) of
$\xi^{*}\widetilde{{\mathcal{A}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l},R}$
by $\chi_{\mathbf{l},R}(\alpha)\chi_{\mathbf{k}+\mathbf{l},R}(\delta)$. We
therefore conclude:

######  Proposition 7.2.1.

If $\chi_{\mathbf{k}+2\mathbf{l},R}$ is trivial on
$U^{\prime}\cap{\mathcal{O}}_{F}^{\times}$, then
$\iota^{\prime}_{R,*}{\mathcal{A}}^{\prime}_{\mathbf{k},\mathbf{l},R}$ is a
coherent sheaf on $Y_{U^{\prime},R}^{\min}$ whose completion at (the fibre
over) ${\mathcal{C}}^{\prime}$ is identified by the Koecher Principle and
Proposition 7.1.1 with

$\left\\{\,\sum_{m\in N^{-1}M_{+}\cup\\{0\\}}b\otimes
r_{m}q^{m}\,\left|\,\begin{array}[]{c}r_{\alpha^{-1}\delta
m}=\zeta_{N}^{-\beta(\alpha^{-1}Nm)}\chi_{\mathbf{l},R}(\alpha)\chi_{\mathbf{k}+\mathbf{l},R}(\delta)r_{m}\\\
\mbox{\rm for all $m\in
N^{-1}M_{+},\bigl{(}\begin{smallmatrix}{\alpha}&{\beta}\\\
{0}&{\delta}\end{smallmatrix}\bigr{)}\in\Gamma_{{\mathcal{C}},U^{\prime}}$}\end{array}\,\right.\right\\}.$

Note that the description of
${\mathcal{O}}^{\wedge}_{Y_{U^{\prime},R}^{\min},{\mathcal{C}}^{\prime}}=(\iota^{\prime}_{R,*}{\mathcal{O}}_{Y_{U^{\prime},R}^{\min}})^{\wedge}_{{\mathcal{C}}^{\prime}}$
may be viewed as a special case (with $\mathbf{k}=\mathbf{l}=\bf{0}$), as can
the prior formula for $U=U(N)$. Furthermore the identifications are compatible
in the obvious senses with base changes $R\to R^{\prime}$, inclusions
$U^{\prime\prime}\subset U^{\prime}$ (provided the splittings $\sigma$ are
chosen compatibly), and the natural algebra structure on
$\bigoplus_{\mathbf{k},\mathbf{l}}{\mathcal{A}}^{\prime}_{\mathbf{k},\mathbf{l},R}$
(taking the direct sum over $\mathbf{k},\mathbf{l}$ as in the statement).

Recall that the $q$-expansion Principle allows one to characterize Hilbert
modular forms in terms of their $q$-expansions:

######  Proposition 7.2.2.

If $C\subset Y^{\infty}_{U^{\prime}}$ is any set of cusps containing at least
one on each component of $Y_{U^{\prime}}$, then the natural map

$M_{\mathbf{k},\mathbf{l}}(U^{\prime};R)=H^{0}(Y_{U^{\prime}}^{\min},\iota^{\prime}_{R,*}{\mathcal{A}}^{\prime}_{\mathbf{k},\mathbf{l},R})\longrightarrow\bigoplus_{{\mathcal{C}}^{\prime}\in
C}(\iota^{\prime}_{R,*}{{\mathcal{A}}^{\prime}}_{\mathbf{k},\mathbf{l},R})^{\wedge}_{{\mathcal{C}}^{\prime}}$

is injective.

Note also that we may replace
$D_{\mathbf{k},\mathbf{l}}\otimes_{{\mathcal{O}}}\cdot$ with
$D_{\mathbf{k},\mathbf{l},R}\otimes_{R}\cdot$ in the description of
$q$-expansions over $R$. In particular if $R$ is an $\mathbb{F}$-algebra, the
identification

$\begin{array}[]{rl}(I^{-1})_{\theta}\otimes_{{\mathcal{O}}}\mathbb{F}&=(I^{-1}\otimes{\mathcal{O}})_{\tau}\otimes_{{\mathcal{O}}[u],\theta}\mathbb{F}=t_{\tau,j}(I_{{\mathfrak{p}}}^{-1}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}})\otimes_{{\mathcal{O}}[u],\theta}\mathbb{F}\\\
&=t_{\tau,j}(I_{{\mathfrak{p}}}^{-1}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}})/(u,{\mathfrak{m}}_{\mathcal{O}})t_{\tau,j}(I_{{\mathfrak{p}}}^{-1}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}{\mathcal{O}})\\\
&=u^{e_{\mathfrak{p}}-j}(I_{{\mathfrak{p}}}^{-1}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}\mathbb{F})/u^{e_{\mathfrak{p}}-j+1}(I_{{\mathfrak{p}}}^{-1}\otimes_{W({\mathcal{O}}_{F}/{\mathfrak{p}}),\tau}\mathbb{F})\end{array}$

yields a canonical isomorphism

(26)
$(I^{-1})_{\theta}\otimes_{{\mathcal{O}}}R=({\mathfrak{p}}^{e_{\mathfrak{p}}-j}I^{-1}/{\mathfrak{p}}^{e_{\mathfrak{p}}-j+1}I^{-1})\otimes_{{\mathcal{O}}_{F}/{\mathfrak{p}},\tau}R.$

The analogous formula holds for the factors
$({\mathfrak{d}}(IJ)^{-1})_{\theta}$ appearing in the definition of
$D_{\mathbf{k},\mathbf{l}}$.

The condition on the $q$-expansion coefficients in the description of the
completions in Proposition 7.2.1 simplifies for certain standard level
structures and cusps, as in [12, Prop. 9.1.2]. Suppose that ${\mathfrak{n}}$
is an ideal of ${\mathcal{O}}_{F}$ such that $\chi_{\mathbf{k}+2\mathbf{l},R}$
is trivial on
$V_{{\mathfrak{n}}}=\ker({\mathcal{O}}_{F}^{\times}\to({\mathcal{O}}_{F}/{\mathfrak{n}})^{\times})$.
Letting $U^{\prime}=U({\mathfrak{n}})$, we have

$\Gamma_{{\mathcal{C}},U^{\prime}}=\left\\{\,\left.\left(\begin{array}[]{cc}\alpha&\beta\\\
0&\delta\end{array}\right)\,\right|\,\alpha\in
V_{\mathfrak{n}},\beta\in{\mathfrak{n}}M^{*},\delta\in\alpha
V_{{\mathfrak{n}},+}\,\right\\}$

for every cusp ${\mathcal{C}}$ of $Y_{U}$. Note that
$m\in{\mathfrak{n}}^{-1}M$ if and only if $\beta(Nm)\in N\mathbb{Z}$ for all
$\beta\in{\mathfrak{n}}M^{*}$, and that $\alpha,\delta\in V_{\mathfrak{n}}$
implies that
$\chi_{\mathbf{l},R}(\alpha)\chi_{\mathbf{k}+2\mathbf{l}}(\delta)=\chi_{\mathbf{l},R}(\alpha\delta^{-1})$,
so we see that

$(\iota^{\prime}_{R,*}{{\mathcal{A}}^{\prime}}_{\mathbf{k},\mathbf{l},R})^{\wedge}_{{\mathcal{C}}^{\prime}}\simeq\left\\{\left.\,\sum_{m\in(\mathfrak{n}^{-1}M)_{+}\cup\\{0\\}}b\otimes
r_{m}q^{m}\,\right|\,r_{\nu^{-1}m}=\chi_{\mathbf{l},R}(\nu)r_{m}\ \mbox{for
all $\nu\in V_{\mathfrak{n},+}$}\,\right\\}$

for every cusp ${\mathcal{C}}^{\prime}$ of $Y_{U^{\prime}}$.

Keep the same assumption on ${\mathfrak{n}}$, but now let
$U^{\prime}=U_{1}({\mathfrak{n}})$ and suppose that ${\mathcal{C}}^{\prime}$
is a cusp of $Y_{U^{\prime}}$ “at $\infty$” in the sense that $\eta(0,1)\in
I+{\mathfrak{n}}\widehat{H}^{(p)}$. We then find that

$\Gamma_{{\mathcal{C}},U^{\prime}}=\left\\{\,\left.\left(\begin{array}[]{cc}\alpha&\beta\\\
0&\delta\end{array}\right)\,\right|\,\alpha\in\delta{\mathcal{O}}_{F,+}^{\times},\beta\in
M^{*},\delta\in V_{{\mathfrak{n}}}\,\right\\},$

and we similarly conclude that

$(\iota^{\prime}_{R,*}{{\mathcal{A}}^{\prime}}_{\mathbf{k},\mathbf{l},R})^{\wedge}_{{\mathcal{C}}^{\prime}}\simeq\left\\{\left.\,\sum_{m\in
M_{+}\cup\\{0\\}}b\otimes
r_{m}q^{m}\,\right|\,r_{\nu^{-1}m}=\chi_{\mathbf{l},R}(\nu)r_{m}\ \mbox{for
all $\nu\in{\mathcal{O}}_{F,+}$}\,\right\\}.$

We remark that every component of $Y_{U^{\prime}}$ contains such cusps (in the
obvious sense), and that in this case the isomorphism is independent of the
choice of splitting $\sigma$.

###  7.3. Kodaira–Spencer filtration

We next explain how the Kodaira–Spencer filtration on differentials extends to
compactifications.

We maintain the notation from the preceding section. In particular, we first
assume $U=U(N)$ for some $N$ prime to $p$ before deducing results for more
general level structures. The construction of
$\widetilde{Y}_{U}^{\operatorname{tor}}$ via torus embeddings then yields a
canonical isomorphism

(27)
$\xi^{*}(\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tor}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}}))\cong
N^{-1}M\otimes{\mathcal{O}}_{{\widehat{S}}}$

for each cusp ${\widetilde{{\mathcal{C}}}}$ of $\widetilde{Y}_{U}$ under which
the descent data relative to the quotient map
${\widehat{S}}={\widehat{S}}_{{\widetilde{{\mathcal{C}}}}}\to(\widetilde{Y}_{U}^{\operatorname{tor}})_{\widetilde{{\mathcal{C}}}}^{\wedge}$
corresponds to that induced by the obvious action of $V_{N}^{2}$ on $N^{-1}M$,
and the completion of the canonical derivation

$d:{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tor}}}\to\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tor}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})$

pulls back to a derivation ${\mathcal{O}}_{{\widehat{S}}}\to
N^{-1}M\otimes{\mathcal{O}}_{{\widehat{S}}}$ whose effect on global sections
corresponds under (18) to the map defined by

$\sum_{m\in N^{-1}M_{+}\cup\\{0\\}}r_{m}q^{m}\mapsto\sum_{m\in
N^{-1}M_{+}\cup\\{0\\}}m\otimes r_{m}q^{m}.$

Recall also that

$\begin{array}[]{rcl}s_{*}\Omega^{1}_{A^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}}}&\cong&{\mathcal{H}om}_{{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{ord}}}}({\mathcal{L}ie}(A^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}}),{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{ord}}})\\\
\mbox{and}\quad
R^{1}s_{*}{\mathcal{O}}_{A^{\operatorname{ord}}}&\cong&{\mathcal{L}ie}((A^{\operatorname{ord}})^{\vee}/\widetilde{Y}_{U}^{\operatorname{ord}})\end{array}$

are locally free sheaves of
${\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{ord}}}$-modules
over $\widetilde{Y}_{U}^{\operatorname{ord}}$, and therefore so is

$s_{*}\Omega^{1}_{A^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}}}\otimes_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{ord}}}}s_{*}\Omega^{1}_{(A^{\operatorname{ord}})^{\vee}/\widetilde{Y}_{U}^{\operatorname{ord}}}$

Decomposing this sheaf over embeddings $\tau\in\Sigma_{0}$ and equipping it
with the filtration defined by the images of the endomorphisms $t_{\tau,j}$
defined by (1), we see that the successive quotients

$\begin{array}[]{l}t_{j}\left(s_{*}\Omega^{1}_{A^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}}}\otimes_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{ord}}}}s_{*}\Omega^{1}_{(A^{\operatorname{ord}})^{\vee}/\widetilde{Y}_{U}^{\operatorname{ord}}}\right)_{\tau}\otimes_{{\mathcal{O}}[u],\theta}{\mathcal{O}}\\\
\qquad\cong{\mathcal{H}om}_{{\mathcal{O}}_{S}}(R^{1}s_{*}{\mathcal{O}}_{A^{\operatorname{ord}}}[u-\theta(\varpi)],t_{j}(s_{*}\Omega^{1}_{(A^{\operatorname{ord}})^{\vee}/\widetilde{Y}_{U}^{\operatorname{ord}}})_{\tau}\otimes_{{\mathcal{O}}[u],\theta}{\mathcal{O}})\end{array}$

(where $\tau=\tau_{i}$, $t_{j}=t_{\tau,j}$ and
$\theta=\theta_{{\mathfrak{p}},i,j}$) are canonically identified with the
automorphic bundles
$\widetilde{{\mathcal{A}}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}$
over $\widetilde{Y}_{U}^{\operatorname{ord}}$. Furthermore the proof of
Theorem 3.3.1 shows that the natural map

(28)
$s_{*}\Omega^{1}_{A^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}}}\otimes_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{ord}}}}s_{*}\Omega^{1}_{(A^{\operatorname{ord}})^{\vee}/\widetilde{Y}_{U}^{\operatorname{ord}}}\longrightarrow\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{ord}}/{\mathcal{O}}}$

arising from Grothendieck–Messing theory, or equivalently the Gauss–Manin
connection on
${\mathcal{H}}^{1}_{\operatorname{dR}}(A^{\operatorname{ord}}/\widetilde{Y}_{U}^{\operatorname{ord}})$
(see [27, §2.1.7]), is an isomorphism. In particular the Kodaira–Spencer
filtration on
$\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{ord}}/{\mathcal{O}}}$
corresponds under (28) to the one defined by the images of the endomorphisms
$t_{\tau,j}$. Furthermore (28) extends over
$\widetilde{Y}_{U}^{\operatorname{tord}}$ to an isomorphism

$\begin{array}[]{c}{\mathcal{H}om}_{{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tord}}}}({\mathfrak{d}}\otimes_{{\mathcal{O}}_{F}}{\mathcal{L}ie}(A^{\operatorname{tord}}/\widetilde{Y}_{U}^{\operatorname{tord}})\otimes_{{\mathcal{O}}_{F}\otimes{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tord}}}}{\mathcal{L}ie}((A^{\operatorname{tord}})^{\vee}/\widetilde{Y}_{U}^{\operatorname{tord}}),{\mathcal{O}}_{\widetilde{Y}_{U}^{\operatorname{tord}}})\\\
\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\,\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tord}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})\end{array}$

whose pull-back via $\xi=\xi_{\widetilde{{\mathcal{C}}}}$ for each cusp
${\widetilde{{\mathcal{C}}}}$ of $\widetilde{Y}_{U}$ is compatible with the
canonical isomorphisms of the pull-back of each with
$M\otimes{\mathcal{O}}_{{\widehat{S}}}=N^{-1}M\otimes{\mathcal{O}}_{{\widehat{S}}}$
(the latter via (27)). Indeed the existence of the extension and the claimed
compatibility follow from the analogous well-known result after base-change to
$\mathbb{C}$. We therefore conclude that the Kodaira–Spencer filtration on
$\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}}$ extends over
$\widetilde{Y}_{U}^{\operatorname{tor}}$ in the form of a decomposition

$\Omega^{1}_{\widetilde{Y}^{\operatorname{tor}}_{U}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})=\bigoplus_{{\mathfrak{p}}\in
S_{p}}\bigoplus_{i\in\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}}\left(\Omega^{1}_{\widetilde{Y}^{\operatorname{tor}}_{U}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})\right)_{{\mathfrak{p}},i},$

together with an increasing filtration of length $e_{\mathfrak{p}}$ on each
component
$(\Omega^{1}_{\widetilde{Y}^{\operatorname{tor}}_{U}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}}))_{{\mathfrak{p}},i}$,
and isomorphisms

$\widetilde{{\mathcal{A}}}^{\operatorname{tor}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\operatorname{gr}\,}^{j}\left(\Omega^{1}_{\widetilde{Y}^{\operatorname{tor}}_{U}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})\right)_{{\mathfrak{p}},i}.$

Furthermore for each cusp ${\widetilde{{\mathcal{C}}}}$ of $\widetilde{Y}_{U}$
and embeddings $\tau=\tau_{{\mathfrak{p}},i}$ and
$\theta=\theta_{{\mathfrak{p}},i,j}$, the pull-back via
$\xi=\xi_{{\widetilde{{\mathcal{C}}}}}$ of
${\operatorname{Fil}\,}^{j}(\Omega^{1}_{\widetilde{Y}^{\operatorname{tor}}_{U}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}}))_{{\mathfrak{p}},i}$
corresponds to
$t_{j}(N^{-1}M\otimes{\mathcal{O}})_{\tau}\otimes_{\mathcal{O}}{\mathcal{O}}_{{\widehat{S}}}=t_{j}({\mathfrak{d}}^{-1}I^{-1}J\otimes{\mathcal{O}})_{\tau}\otimes_{\mathcal{O}}{\mathcal{O}}_{{\widehat{S}}}$
under (27), and the resulting isomorphism

$\xi^{*}\widetilde{{\mathcal{A}}}^{\operatorname{tor}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\operatorname{gr}\,}^{j}\left(\xi^{*}(\Omega^{1}_{\widetilde{Y}^{\operatorname{tor}}_{U}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}}))_{{\mathfrak{p}},i}\right)\cong({\mathfrak{d}}^{-1}I^{-1}J\otimes{\mathcal{O}})_{\theta}\otimes_{\mathcal{O}}{\mathcal{O}}_{{\widehat{S}}}$

coincides with the canonical trivialization of (20).

We now interpret this in the context of minimal compactifications. First we
note that the argument of [31, Prop. 4.9] yields a Koecher Principle for
$\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tor}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})$,
so that

$\pi_{*}(\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tor}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}}))\longrightarrow\widetilde{\iota}_{*}(\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}})$

is an isomorphism of coherent sheaves on $\widetilde{Y}_{U}^{\min}$ whose
completion at the cusp ${\widetilde{{\mathcal{C}}}}$ is identified with
$\left(M\otimes{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\right)^{V_{N}^{2}}$

$=\left\\{\left.\,\sum_{m\in N^{-1}M_{+}}c_{m}\otimes q^{m}\,\right|\,c_{\nu
m}=\nu c_{m}\,\,\forall\nu\in V_{N}^{2},m\in N^{-1}M_{+}\,\right\\}.$

(Note that $c_{0}\in M^{V_{N}^{2}}=0$.) Furthermore the completion at
${\widetilde{{\mathcal{C}}}}$ of the canonical derivation
${\mathcal{O}}_{\widetilde{Y}_{U}^{\min}}\to\widetilde{\iota}_{*}\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}}$
is given by $\sum r_{m}q^{m}\mapsto\sum m\otimes r_{m}q^{m}$.

We see also from the description of the extension of the Kodaira–Spencer
filtration to
$\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tor}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}})$
in terms of $q$-expansions that

$\pi_{*}({\operatorname{Fil}\,}^{j}(\Omega^{1}_{\widetilde{Y}_{U}^{\operatorname{tor}}/{\mathcal{O}}}(\log\widetilde{Z}_{U}^{\operatorname{tor}}))_{{\mathfrak{p}},i})={\widetilde{\iota}}_{*}({\operatorname{Fil}\,}^{j}(\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}})_{{\mathfrak{p}},i}),$

with completion at ${\widetilde{{\mathcal{C}}}}$ given by
$\left(t_{j}(M\otimes{\mathcal{O}})_{\tau}\otimes_{\mathcal{O}}{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\right)^{V_{N}^{2}}$

$=\left\\{\left.\,\sum_{m\in N^{-1}M_{+}}c_{m}\otimes q^{m}\,\right|\,c_{\nu
m}=(\nu\otimes 1)c_{m}\,\,\forall\nu\in V_{N}^{2},m\in
N^{-1}M_{+}\,\right\\}.$

Furthermore since $V_{N}^{2}$ acts freely on $N^{-1}M_{+}$ (and
$M_{\theta}^{V_{N}^{2}}=0$), the morphisms

$\widetilde{\iota}_{*}({\operatorname{Fil}\,}^{j}(\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}})_{{\mathfrak{p}},i})\to\widetilde{\iota}_{*}\widetilde{{\mathcal{A}}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}$

are surjective on completions at cusps, and hence surjective. It follows that
the graded pieces of (the obvious extension of) the Kodaira–Spencer filtration
on $\widetilde{\iota}_{*}\Omega^{1}_{\widetilde{Y}_{U}/{\mathcal{O}}}$ are
canonically isomorphic to
$\widetilde{\iota}_{*}\widetilde{{\mathcal{A}}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}$.

The constructions above are compatible with the natural actions of
${\mathcal{O}}_{F,(p),+}^{\times}$ on
$\widetilde{Y}_{U}^{\operatorname{tor}}$, $\widetilde{Y}_{U}^{\min}$ and $M$,
so that the resulting descriptions carry over to $Y_{U}^{\min}$. More
precisely, $\iota_{*}(\Omega^{1}_{Y_{U}/{\mathcal{O}}})$ is a coherent sheaf
on ${Y}_{U}^{\min}$ whose completion at the cusp ${\mathcal{C}}$ is identified
with $\left(M\otimes{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\right)^{V_{N,+}}$

$=\left\\{\left.\,\sum_{m\in N^{-1}M_{+}}c_{m}\otimes q^{m}\,\right|\,c_{\nu
m}=\nu c_{m}\,\,\forall\nu\in V_{N,+},m\in N^{-1}M_{+}\,\right\\},$

in terms of which the canonical derivation is $\sum r_{m}q^{m}\mapsto\sum
m\otimes r_{m}q^{m}$. Furthermore the completion at ${\mathcal{C}}$ of
${\operatorname{Fil}\,}^{j}(\iota_{*}(\Omega^{1}_{Y_{U}/{\mathcal{O}}})_{{\mathfrak{p}},i}):=\iota_{*}({\operatorname{Fil}\,}^{j}(\Omega^{1}_{Y_{U}/{\mathcal{O}}})_{{\mathfrak{p}},i})$
is identified with
$\left(t_{j}(M\otimes{\mathcal{O}})_{\tau}\otimes_{\mathcal{O}}{\mathcal{O}}[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\right)^{V_{N,+}}$, and the natural maps

(29)
${\operatorname{gr}\,}^{j}(\iota_{*}(\Omega^{1}_{Y_{U}/{\mathcal{O}}})_{{\mathfrak{p}},i})\hookrightarrow\iota_{*}({\operatorname{gr}\,}^{j}(\Omega^{1}_{Y_{U}/{\mathcal{O}}})_{{\mathfrak{p}},i})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\iota_{*}{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}$

are isomorphisms whose completions at the cusps are induced by the surjections
$t_{j}(M\otimes{\mathcal{O}})_{\tau}\to M_{\theta}$.

Suppose now that $U^{\prime}$ is an arbitrary sufficiently small open compact
subgroup of $\operatorname{GL}_{2}(\mathbb{A}_{F,\mathbf{f}})$ of level prime
to $p$, and choose $N$ so that $U(N)\subset U^{\prime}$. The constructions
above are then also compatible with the natural actions of $U^{\prime}$, so we
arrive at similar conclusions with minor modifications to the descriptions of
completions that result from taking invariants under
$\Gamma_{{\mathcal{C}},U^{\prime}}$. We omit the details, but we remark that
letting $L$ (resp. $V$) denote the kernel (resp. image) of the homomorphism

$\begin{array}[]{ccc}\Gamma_{{\mathcal{C}},U^{\prime}}/({\mathcal{O}}_{F}^{\times}\cap
U^{\prime})&\to&{\mathcal{O}}_{F,+}^{\times}\\\
\left(\begin{array}[]{cc}\alpha&\beta\\\
0&\delta\end{array}\right)\cdot({\mathcal{O}}_{F}^{\times}\cap
U^{\prime})&\mapsto&\alpha^{-1}\delta,\end{array}$

the coefficients $r_{m}$ of $q$-expansions in
${\mathcal{O}}_{Y_{U^{\prime}}^{\min},{\mathcal{C}}^{\prime}}^{\wedge}$ are
indexed by $m\in(L^{*})_{+}\cup\\{0\\}$ (where $L$ is identified with a finite
index subgroup of $M$). Since $V$ acts freely on $L^{*}$ (twisting
$q$-expansion coefficients by a possibly non-trivial cocycle valued in
$L\otimes\mu_{N}({\mathcal{O}})$), we still obtain the isomorphism of (29)
with $U$ replaced by $U^{\prime}$, identifying the graded pieces of the
Kodaira–Spencer filtration on
$\iota_{*}\Omega^{1}_{Y_{U^{\prime}}/{\mathcal{O}}}$ with the sheaves
$\iota_{*}{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta}}$.

The description of the extension of the Kodaira–Spencer filtration over
compactifications also applies after base-change to an arbitrary
${\mathcal{O}}$-algebra $R$, with one significant difference. If $R$ is not
flat over ${\mathcal{O}}$, then the modules $M\otimes R$ (and their
subquotients) may have invariants under the action of the unit groups
$V_{N,+}$ (or more generally the isotropy groups
$\Gamma_{{\mathcal{C}},U^{\prime}}$), so that $q$-expansions of meromorphic
differentials on $Y_{U,R}$ (and forms of weight
$(2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta})$) may have non-zero constant
terms, and the morphism analogous to (29) may fail to be an isomorphism. (Note
that in this case the relevant base-change morphisms
$(\iota_{*}{\mathcal{F}})_{R}\to\iota_{R,*}({\mathcal{F}}_{R})$ fail to be
surjective at the cusps.)

We can however simplify matters by placing ourselves in the situation when
this fails in the extreme. Suppose then that $p^{n}R=0$ for some $n>0$, and
that $N$ is sufficiently large that $\nu\equiv 1\bmod p^{n}{\mathcal{O}}_{F}$
for all $\nu\in V_{N,+}$. Arguing exactly as above, we find that
$\iota_{R,*}(\Omega^{1}_{Y_{U,R}/R})$ is now a vector bundle over
${Y}_{U,R}^{\min}$ whose completion at ${\mathcal{C}}$ is identified with
$\left(M\otimes R[[q^{m}]]_{m\in
N^{-1}M_{+}\cup\\{0\\}}\right)^{V_{N,+}}=M\otimes{\mathcal{O}}_{Y^{\min}_{U,R},{\mathcal{C}}}^{\wedge}$

$=\left\\{\left.\,\sum_{m\in N^{-1}M_{+}\cup\\{0\\}}c_{m}\otimes
q^{m}\,\right|\,c_{\nu m}=\nu c_{m}\,\,\forall\nu\in V_{N,+},m\in
N^{-1}M_{+}\,\right\\},$

with the canonical derivation given by $\sum r_{m}q^{m}\mapsto\sum m\otimes
r_{m}q^{m}$. Furthermore each

${\operatorname{Fil}\,}^{j}(\iota_{R,*}(\Omega^{1}_{Y_{U,R}/R})_{{\mathfrak{p}},i})=\iota_{R,*}({\operatorname{Fil}\,}^{j}(\Omega^{1}_{Y_{U,R}/R})_{{\mathfrak{p}},i})$

is a sub-bundle whose completion at ${\mathcal{C}}$ is identified with
$t_{j}(M\otimes{\mathcal{O}})_{\tau}\otimes_{\mathcal{O}}{\mathcal{O}}_{Y^{\min}_{U,R},{\mathcal{C}}}^{\wedge}$
and the natural maps

${\operatorname{gr}\,}^{j}(\iota_{R,*}(\Omega^{1}_{Y_{U,R}/R})_{{\mathfrak{p}},i})\hookrightarrow\iota_{R,*}({\operatorname{gr}\,}^{j}(\Omega^{1}_{Y_{U,R}/R})_{{\mathfrak{p}},i})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\iota_{R,*}{\mathcal{A}}_{2{\mathbf{e}}_{\theta},-{\mathbf{e}}_{\theta},R}$

are isomorphisms of line bundles over $Y_{U,R}^{\min}$ whose completions at
the cusps are induced by the surjections
$t_{j}(M\otimes{\mathcal{O}})_{\tau}\to M_{\theta}$. We remark also that this
carries over with $U$ replaced by arbitrary $U^{\prime}$, provided
$U^{\prime}$ is sufficiently small that (in additional to the usual
hypotheses) $\alpha\equiv\delta\mod p^{n}{\mathcal{O}}_{F}$ for all
$\left(\begin{array}[]{cc}\alpha&\beta\\\
0&\delta\end{array}\right)\in\Gamma_{{\mathcal{C}},U^{\prime}}$ and cusps
${\mathcal{C}}^{\prime}$ of $Y_{U^{\prime}}$ (the condition being independent
of the choice of $N$ and ${\mathcal{C}}$ in the definition of
$\Gamma_{{\mathcal{C}},U^{\prime}}$).

##  8\. Operators on $q$-expansions

###  8.1. Partial Hasse invariants

We next describe the effect of the various weight-shifting operators on
$q$-expansions, beginning with the simplest case of (multiplication by)
partial Hasse invariants. We will now only be working in the setting of
$R=\mathbb{F}$, and we will use $\overline{\cdot}$ to denote base-changes from
${\mathcal{O}}$ to $\mathbb{F}$. Since the formation of $q$-expansions is
compatible in the obvious sense with pull-back under the projections
$\overline{Y}_{U}\to\overline{Y}_{U^{\prime}}$, it will suffice to consider
the case $U=U(N)$.

Recall that in §4.1 we defined the partial Hasse invariants as certain
elements

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

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 $j>1$. In particular if
$j>1$, then $H_{\theta}$ is defined by the morphism
$u:{{\mathcal{L}}}_{\tau,j}\to{{\mathcal{L}}}_{\tau,j-1}$ induced by
$\varpi_{\mathfrak{p}}$ on the universal abelian variety over
$\widetilde{Y}_{U,\mathbb{F}}$, which evidently extends to the endomorphism
$\varpi_{\mathfrak{p}}$ of $A^{\operatorname{tor}}$ over
$S:=\widetilde{Y}_{U,\mathbb{F}}^{\operatorname{tor}}$. Since its pull-back
via $\overline{\xi}$ is defined by $\varpi_{\mathfrak{p}}$ on
$\overline{T}_{I,J}$, the resulting morphism of line bundles

$\overline{\xi}^{*}{{{\mathcal{L}}}}_{\tau,j}^{\operatorname{tor}}\longrightarrow\overline{\xi}^{*}{{{\mathcal{L}}}}_{\tau,j-1}^{\operatorname{tor}}$

is compatible with their canonical trivializations, and more precisely with
the morphism

$(I^{-1})_{\theta}\otimes_{{\mathcal{O}}}\mathbb{F}\to(I^{-1})_{\sigma^{-1}\theta}\otimes_{{\mathcal{O}}}\mathbb{F}$

induced by $u=\varpi\otimes 1$ on $(I^{-1}\otimes{\mathcal{O}})_{\tau}$. It
follows that $H_{\theta}$ has constant $q$-expansion, where the constant
corresponds to the basis element $\varpi_{\mathfrak{p}}$ under the
identification

$\overline{D}_{\mathbf{h}_{\theta},\bf{0}}\cong{\mathfrak{p}}\otimes_{{\mathcal{O}}_{F},\theta}\mathbb{F}.$

provided by (26).

For $j=1$ we use also that the morphism of line bundles

$\overline{\xi}^{*}{{{\mathcal{L}}}}_{\tau,1}^{\operatorname{tor}}\longrightarrow\overline{\xi}^{*}\phi_{S}^{*}({{{\mathcal{L}}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}}^{\operatorname{tor}})$

induced by the Verschiebung
$\phi_{S}^{*}\overline{T}_{I,J}\to\overline{T}_{I,J}$ is compatible via the
canonical trivializations with the canonical isomorphism

$(I^{-1}\otimes\mathbb{F})_{\tau}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\phi_{S}^{*}(I^{-1}\otimes\mathbb{F})_{\phi^{-1}\circ\tau}.$

So in this case we again find that $H_{\theta}$ has constant $q$-expansion,
the constant now corresponding to the basis element
$\varpi_{\mathfrak{p}}^{1-e_{\mathfrak{p}}}$ under the identification

$\overline{D}_{\mathbf{h}_{\theta},\bf{0}}\cong{\mathfrak{p}}^{1-e_{\mathfrak{p}}}\otimes_{{\mathcal{O}}_{F},\theta}\mathbb{F}$

given by (26).

The $q$-expansions of the canonical sections $G_{\theta}\in
M_{\bf{0},\mathbf{h}_{\theta}}(U;\mathbb{F})$ may be described similarly.
Indeed for $j>1$, the composites

$\begin{array}[]{cccccc}&{{\mathcal{L}}}_{\tau,j}&\longrightarrow&{{\mathcal{P}}}_{\tau,j}&\stackrel{{\scriptstyle
u}}{{\longrightarrow}}&{{\mathcal{L}}}_{\tau,j-1}\\\
\mbox{and}&{{\mathcal{M}}}_{\tau,j-1}&\longrightarrow&{{\mathcal{P}}}_{\tau,j}&\stackrel{{\scriptstyle
u}}{{\longrightarrow}}&{{\mathcal{M}}}_{\tau,j}\end{array}$

on $\widetilde{Y}^{\operatorname{ord}}_{U,\mathbb{F}}$ are isomorphisms whose
tensor product defines
${{\mathcal{N}}}_{\theta}\cong{{\mathcal{N}}}_{\sigma^{-1}\theta}$. Its unique
extension to $\widetilde{Y}^{\operatorname{tord}}_{U,\mathbb{F}}$, and hence
to $\widetilde{Y}^{\operatorname{tor}}_{U,\mathbb{F}}$, is therefore the
isomorphism whose pull-back via $\overline{\xi}$ is the tensor product of the
isomorphisms defined on canonical trivializations by
$(I^{-1})_{\theta}\otimes_{{\mathcal{O}}}\mathbb{F}\stackrel{{\scriptstyle
u}}{{\to}}(I^{-1})_{\sigma^{-1}\theta}\otimes_{{\mathcal{O}}}\mathbb{F}$ and
by the identity on
$({\mathfrak{d}}J^{-1}\otimes\mathbb{F})_{\tau}/u({\mathfrak{d}}J^{-1}\otimes\mathbb{F})_{\tau}$,
and hence corresponds to

$({\mathfrak{d}}(IJ)^{-1})_{\theta}\otimes_{{\mathcal{O}}}\mathbb{F}\stackrel{{\scriptstyle
u}}{{\to}}({\mathfrak{d}}(IJ)^{-1})_{\sigma^{-1}\theta}\otimes_{{\mathcal{O}}}\mathbb{F}.$

Therefore $G_{\theta}$ has constant $q$-expansion, with constant corresponding
to the basis element $\varpi_{\mathfrak{p}}$ under the identification

$\overline{D}_{\bf{0},\mathbf{h}_{\theta}}\cong{\mathfrak{p}}\otimes_{{\mathcal{O}}_{F},\theta}\mathbb{F}$

provided by the analogue of (26) for $({\mathfrak{d}}(IJ)^{-1})_{\theta}$.

Similarly if $j=1$, we find that
${{\mathcal{N}}}_{\tau,1}\cong\phi_{S}^{*}({{\mathcal{N}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}})$
over $S=\widetilde{Y}^{\operatorname{ord}}_{U,\mathbb{F}}$ is the tensor
product of the isomorphism

${{\mathcal{L}}}_{\tau,1}\to\phi_{S}^{*}({{\mathcal{L}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}})$

defining the Hasse invariant and the isomorphism

$\phi_{S}^{*}({{\mathcal{M}}}_{\phi^{-1}\circ\tau,e_{\mathfrak{p}}})\to{{\mathcal{M}}}_{\tau,1}$

induced by ${\operatorname{Frob}}_{A}$. The extensions to
$\widetilde{Y}^{\operatorname{tord}}_{U,\mathbb{F}}$ are again compatible with
the canonical trivializations, now corresponding to maps whose tensor product
is the inverse of the isomorphism

$(({\mathfrak{d}}(IJ)^{-1})_{\sigma^{-1}\theta}\otimes_{{\mathcal{O}}}\mathbb{F})^{\otimes
p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}({\mathfrak{d}}(IJ)^{-1})_{\theta}\otimes_{{\mathcal{O}}}\mathbb{F}$

induced by $u^{e_{\mathfrak{p}}-1}$. So again $G_{\theta}$ has constant
$q$-expansion, with constant corresponding to
$\varpi_{\mathfrak{p}}^{1-e_{\mathfrak{p}}}$ under the identification

$\overline{D}_{\bf{0},\mathbf{h}_{\theta}}\cong{\mathfrak{p}}^{1-e_{\mathfrak{p}}}\otimes_{{\mathcal{O}}_{F},\theta}\mathbb{F}$

given by (26) for $({\mathfrak{d}}(IJ)^{-1})_{\theta}$.

###  8.2. Partial $\Theta$-operators

We now compute the effect of $\Theta$-operators on $q$-expansions exactly as
in [12]. Recall from §5.2 that for each
$\tau_{0}=\tau_{{\mathfrak{p}},i}\in\Sigma_{0}$, the associated partial
$\Theta$-operator is a map

$\Theta_{\tau_{0}}:M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})\to
M_{\mathbf{k}^{\prime},\mathbf{l}^{\prime}}(U;{\mathbb{F}})$

where
$\mathbf{k}^{\prime}=\mathbf{k}+n_{\theta_{0}}{\mathbf{e}}_{\sigma^{-1}\theta_{0}}+{\mathbf{e}}_{\theta_{0}}$,
$\mathbf{l}^{\prime}=\mathbf{l}+{\mathbf{e}}_{\theta_{0}}$ and
$\theta_{0}=\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}$. It is defined for all
sufficiently small $U$ of level prime to $p$, and is Hecke-equivariant. In
particular it is compatible with restriction for $U\subset U^{\prime}$, so we
may assume $U=U(N)$ for some $N$ sufficiently large that $\nu\equiv 1\mod
p{\mathcal{O}}_{F}$ for all $\nu\in V_{N,+}$.

Recall from the proof of Theorem 5.2.1 that $\Theta_{\tau_{0}}$ is defined by
a morphism
${\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}}\to{\mathcal{A}}_{\mathbf{k}^{\prime},\mathbf{l}^{\prime},\mathbb{F}}$
given locally on sections by formula (12). Our assumptions on $U$ imply that

$(\iota_{\mathbb{F},*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}})_{{\mathcal{C}}}^{\wedge}=D_{\mathbf{k},\mathbf{l},\mathbb{F}}\otimes_{\mathbb{F}}\cdot{\mathcal{O}}_{Y_{U,\mathbb{F}}^{\min},{\mathcal{C}}}^{\wedge}$

is free of rank one over
${\mathcal{O}}_{Y_{U,\mathbb{F}}^{\min},{\mathcal{C}}}^{\wedge}$ for all
weights $\mathbf{k},\mathbf{l}$ and cusps ${\mathcal{C}}$, so the completion
at ${\mathcal{C}}$ of $\iota_{\mathbb{F},*}\Theta_{\tau_{0}}$ is the map

$(\iota_{\mathbb{F},*}{\mathcal{A}}_{\mathbf{k},\mathbf{l},\mathbb{F}})_{{\mathcal{C}}}^{\wedge}\to(\iota_{\mathbb{F},*}{\mathcal{A}}_{\mathbf{k}^{\prime},\mathbf{l}^{\prime},\mathbb{F}})_{{\mathcal{C}}}^{\wedge}$

defined by (12), where $y_{\theta}$ is any basis for
$(\iota_{\mathbb{F},*}{{\mathcal{L}}}_{\theta})_{{\mathcal{C}}}^{\wedge}$ and
$y^{\mathbf{k}}=\prod_{\theta}y_{\theta}^{k_{\theta}}$. In particular we may
choose $y_{\theta}=b_{\theta}\otimes 1$ where $b_{\theta}$ is a basis for
$D_{{\mathbf{e}}_{\theta},\bf{0},\mathbb{F}}=(I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}$.
The fact that $H_{\theta}$ has (non-zero) constant $q$-expansion at
${\mathcal{C}}$ then means the same holds for the element
$r_{\theta}\in{\mathcal{O}}_{Y_{U,\mathbb{F}}^{\min},{\mathcal{C}}}^{\wedge}$,
i.e., $r_{\theta}\in\mathbb{F}^{\times}$. We can even select the bases
$b_{\theta}$ so that $r_{\theta}=1$ for all $\theta$ by choosing any basis
$b_{\mathfrak{p}}$ for
$(I_{\mathfrak{p}}\otimes\mathbb{F})_{{\mathfrak{p}},0}$ over
$\mathbb{F}[u]/u^{e_{\mathfrak{p}}}$ for each ${\mathfrak{p}}$, letting
$b_{{\mathfrak{p}},i}$ correspond to $(\phi^{i})^{*}(b_{\mathfrak{p}})$ under
the canonical isomorphism
$(I_{\mathfrak{p}}\otimes\mathbb{F})_{{\mathfrak{p}},i}\cong(\phi^{i})^{*}(I_{\mathfrak{p}}\otimes\mathbb{F})_{{\mathfrak{p}},0}$,
and defining $b_{{\mathfrak{p}},i,j}$ to be the image of
$u^{e_{\mathfrak{p}}-j}b_{{\mathfrak{p}},i}$ in
$(I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}$

Recall also that $g^{\mathbf{l}}=\prod_{\theta}g_{\theta}^{l_{\theta}}$ in
(12), where each $g_{\theta}$ is a trivialization of
${{\mathcal{N}}}_{\theta}$ over $Y_{U,\mathbb{F}}$. Therefore $g_{\theta}$
trivializes $\iota_{\mathbb{F},*}{{\mathcal{N}}}_{\theta}$ over
$Y^{\min}_{U,\mathbb{F}}$, from which it follows that
$g_{\theta}=c_{\theta}\otimes 1$ for some basis $c_{\theta}$ of
$D_{\bf{0},{\mathbf{e}}_{\theta},\mathbb{F}}=({\mathfrak{d}}(IJ)^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}$.
The formula (12) therefore takes the form

$\Theta_{\tau_{0}}(b^{\mathbf{k}}c^{\mathbf{l}}\otimes\varphi_{f})=H_{\theta_{0}}b^{\mathbf{k}}c^{\mathbf{l}}\otimes
KS_{\tau_{0}}(d\varphi_{f})=b^{\mathbf{k}+\mathbf{h}_{\theta_{0}}}c^{\mathbf{l}}\otimes
KS_{\tau_{0}}(d\varphi_{f}),$

for
$\varphi_{f}\in{\mathcal{O}}_{Y_{U,\mathbb{F}}^{\min},{\mathcal{C}}}^{\wedge}$.
Finally the descriptions in §7.3 of the canonical derivation, the
Kodaira–Spencer filtration and the isomorphism (29) in terms of $q$-expansions
yield the formula

(30) $\Theta_{\tau_{0}}\left(\sum_{m\in
N^{-1}M_{+}\cup\\{0\\}}(b^{\mathbf{k}}c^{\mathbf{l}}\otimes
r_{m})q^{m}\right)=\sum_{m\in
N^{-1}M_{+}}(b^{\mathbf{k}+\mathbf{h}_{\theta_{0}}}c^{\mathbf{l}}\overline{\tau}_{0}(m)\otimes
r_{m})q^{m}$

where $\overline{\tau}_{0}$ is the canonical projection

$\begin{array}[]{rcl}N^{-1}M&\to&(N^{-1}M\otimes\mathbb{F})_{\tau_{0}}=(M\otimes\mathbb{F})_{\tau_{0}}\\\
&\to&M\otimes_{{\mathcal{O}}_{F},\overline{\theta}_{0}}\mathbb{F}=({\mathfrak{d}}^{-1}I^{-1}J)\otimes_{{\mathcal{O}}_{F},\overline{\theta}_{0}}\mathbb{F}=D_{2{\mathbf{e}}_{\theta_{0}},-{\mathbf{e}}_{\theta_{0}},\mathbb{F}}\end{array}$

(writing $\overline{\theta}_{0}$ for the composite of
${\mathcal{O}}_{F}\stackrel{{\scriptstyle\theta_{0}}}{{\to}}{\mathcal{O}}\to\mathbb{F}$).
As noted above, it follows that (30) holds with $U=U(N)$ replaced by any
sufficiently small open compact $U^{\prime}$ of level prime to $p$ and
${\mathcal{C}}$ replaced by any cusp of $Y_{U^{\prime}}$. In this case the
$q$-expansions are necessarily invariant under the natural action of
$\Gamma_{{\mathcal{C}},U^{\prime}}$ (whose compatibility with (30) is a
consequence of the construction, but is straightforward to check directly).

We see immediately from (30) that the operators $\Theta_{\tau}$ for varying
$\tau$ commute. We see also that

$\Theta_{\tau_{0}}^{p}(f)=\Theta_{\tau_{1}}(f)H_{\theta_{0}}^{p}H_{\theta_{1}}G_{\theta_{1}}^{-1}\prod_{j=1}^{e_{\mathfrak{p}}-1}\left(H_{\sigma^{j}\theta_{0}}G^{-1}_{\sigma^{j}\theta_{0}}\right),$

where $\tau_{1}=\tau_{0}\circ\phi=\tau_{{\mathfrak{p}},i+1}$ and
$\theta_{1}=\sigma^{e_{\mathfrak{p}}}\theta_{0}=\theta_{{\mathfrak{p}},i+1,e_{\mathfrak{p}}}$.
Indeed this follows from (30) together with the fact that the $q$-expansions
of

$H_{\tau_{1}}=\prod_{j=1}^{e_{\mathfrak{p}}}H_{\sigma^{j}\theta_{0}}\quad\mbox{and}\quad
G_{\tau_{1}}=\prod_{j=1}^{e_{\mathfrak{p}}}G_{\sigma^{j}\theta_{0}}$

are constants given by the canonical isomorphisms
$((I^{-1})_{\theta_{0}}\otimes_{{\mathcal{O}}}\mathbb{F})^{\otimes
p}\cong(I^{-1})_{\theta_{1}}\otimes_{{\mathcal{O}}}\mathbb{F}$ and
$(({\mathfrak{d}}(IJ)^{-1})_{\theta_{0}}\otimes_{{\mathcal{O}}}\mathbb{F})^{\otimes
p}\cong({\mathfrak{d}}(IJ)^{-1})_{\theta_{1}}\otimes_{{\mathcal{O}}}\mathbb{F}$,
so we get a commutative diagram

$\textstyle{D_{2{\mathbf{e}}_{\theta_{1}},-{\mathbf{e}}_{\theta_{1}},\mathbb{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N^{-1}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\tau}_{1}}$$\scriptstyle{\overline{\tau_{0}}^{\otimes
p}}$$\textstyle{D_{2{\mathbf{e}}_{\theta_{0}},-{\mathbf{e}}_{\theta_{0}},\mathbb{F}}^{\otimes
p}}$

where the downward arrow is multiplication by the (constant) $q$-expansion of
$H_{\tau_{1}}^{2}G_{\tau_{1}}^{-1}$.

###  8.3. Partial Frobenius operators

Finally we compute the effect on $q$-expansions of the partial Frobenius
operators $V_{\mathfrak{p}}$ defined in §6.2. We must first extend the partial
Frobenius endomorphisms $\widetilde{\Phi}_{\mathfrak{p}}$ (and
$\Phi_{\mathfrak{p}}$) defined in §6.1 to compactifications. To that end let
$h_{\mathfrak{p}}$ denote the matrix $\bigl{(}\begin{smallmatrix}{1}&{0}\\\
{0}&{\delta}\end{smallmatrix}\bigr{)}$, viewed as an element of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}}^{(p)})$, where $\delta\in
F^{\times}$ is as in the definition fo $\widetilde{\Phi}_{\mathfrak{p}}$. We
let
$\widetilde{\Phi}_{{\mathfrak{p}}}^{\infty}:\widetilde{Y}^{\infty}_{U}\to\widetilde{Y}^{\infty}_{U}$
be the permutation defined on double cosets by
$B_{1}({\mathcal{O}}_{F,(p)})gU^{p}\mapsto
B_{1}({\mathcal{O}}_{F,(p)})h_{\mathfrak{p}}^{-1}gU^{p}$, and similarly let
$\Phi_{{\mathfrak{p}}}^{\infty}$ denote the induced permutation of
$Y^{\infty}_{U}$. Then $\widetilde{\Phi}^{\infty}_{\mathfrak{p}}$ translates
to the map on corresponding data sending $(H,I,\lambda,[\eta])$ to
$(H^{\prime},I^{\prime},\lambda^{\prime},[\eta^{\prime}])$, where

  * •

$I^{\prime}={\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F}}I={\mathfrak{p}}^{-1}I$,

  * •

$H^{\prime}$ is the push-out of $H$ with respect to $I\to I^{\prime}$;

  * •

$\lambda^{\prime}=\delta\otimes\lambda$ (identifying
$\wedge^{2}_{{\mathcal{O}}_{F}}H^{\prime}={\mathfrak{p}}^{-1}\otimes_{{\mathcal{O}}_{F}}\wedge^{2}_{{\mathcal{O}}_{F}}H$),

  * •

and $\eta^{\prime}=\eta$ (identifying
$\widehat{{\mathcal{O}}}_{F}^{(p)}\otimes_{{\mathcal{O}}_{F}}H^{\prime}=\widehat{{\mathcal{O}}}_{F}^{(p)}\otimes_{{\mathcal{O}}_{F}}H$).

Suppose now that $U=U(N)$ for some sufficiently large $N$. One then checks
that the morphism
$\widetilde{\Phi}_{\mathfrak{p}}:\widetilde{Y}_{U,\mathbb{F}}\to\widetilde{Y}_{U,\mathbb{F}}$
extends to a morphism
$\widetilde{\Phi}_{\mathfrak{p}}^{\operatorname{tor}}:\widetilde{Y}_{U,\mathbb{F}}^{{\operatorname{tor}}^{\prime}}\to\widetilde{Y}_{U,\mathbb{F}}^{\operatorname{tor}}$
(where the ′ indicates the choice of cone decomposition need not be the same)
under which the component corresponding to a cusp
${\widetilde{{\mathcal{C}}}}$ represented by $(H,I,\lambda,[\eta])$ is sent to
the one corresponding to
${\widetilde{{\mathcal{C}}}}^{\prime}=\widetilde{\Phi}_{\mathfrak{p}}^{\infty}({\widetilde{{\mathcal{C}}}})$
represented by $(H^{\prime},I^{\prime},\lambda^{\prime},[\eta^{\prime}])$, and
the resulting map on completions pulls back to a morphism
${\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}}\to{\widehat{S}}_{{\widetilde{{\mathcal{C}}}}^{\prime},\mathbb{F}}$
whose effect on global sections corresponds under the isomorphisms of (18) to
the homomorphism induced by the canonical inclusion

$M^{\prime}:={\mathfrak{d}}^{-1}(I^{\prime})^{-1}J={\mathfrak{p}}{\mathfrak{d}}^{-1}I^{-1}J\hookrightarrow{\mathfrak{d}}^{-1}I^{-1}J.$

Furthermore the pull-back of
$A^{\prime{\operatorname{tor}}}=(\widetilde{\Phi}^{\operatorname{tor}}_{\mathfrak{p}})^{*}A^{\operatorname{tor}}$
to ${\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}}$ is
identified with the Tate semi-abelian variety $T_{{\mathfrak{p}}^{-1}I,J}$,
and the isomorphisms defined in §6.2 relating the line bundles
$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{L}}_{\theta}$ to
${\mathcal{L}}_{\theta}$ or ${\mathcal{L}}_{\sigma^{-1}\theta}$ extend to
isomorphisms

$(\widetilde{\Phi}^{\operatorname{tor}}_{\mathfrak{p}})^{*}{\mathcal{L}}^{\operatorname{tor}}_{\theta}\cong{\mathcal{L}}^{{\operatorname{tor}}^{\prime}}_{\theta}$
for $\theta\not\in\Sigma_{\mathfrak{p}}$, and
$(\widetilde{\Phi}^{\operatorname{tor}}_{\mathfrak{p}})^{*}{\mathcal{L}}_{\theta}^{\operatorname{tor}}\cong({\mathcal{L}}^{{\operatorname{tor}}^{\prime}}_{\sigma^{-1}\theta})^{\otimes
n_{\theta}}$ for $\theta\in\Sigma_{\mathfrak{p}}$

over $\widetilde{Y}_{U,\mathbb{F}}^{{\operatorname{tor}}^{\prime}}$ whose
pull-backs are compatible via their canonical trivializations with
isomorphisms induced by the canonical
$({\mathcal{O}}_{F}\otimes\mathbb{F})_{\tau}$-equivariant maps

$(pI^{-1}\otimes\mathbb{F})\longrightarrow({\mathfrak{p}}I^{-1}\otimes\mathbb{F})_{\tau}\longrightarrow(I^{-1}\otimes\mathbb{F})_{\tau}.$

More precisely if $\tau\not\in\Sigma_{{\mathfrak{p}},0}$, then the second map
is an isomorphism identifying
$({\mathfrak{p}}I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}$ with
$(I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}$, and if
$\tau=\tau_{{\mathfrak{p}},i}$ and $\theta=\theta_{{\mathfrak{p}},i,j}$, then
this map also induces the desired isomorphisms

$\begin{array}[]{rcl}({\mathfrak{p}}I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}&=&u^{e_{\mathfrak{p}}-j}({\mathfrak{p}}I^{-1}\otimes\mathbb{F})_{\tau}\otimes_{\mathbb{F}[u]/u^{e_{\mathfrak{p}}}}\mathbb{F}\\\
&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}&u^{e_{\mathfrak{p}}-j+1}(I^{-1}\otimes\mathbb{F})_{\tau}\otimes_{\mathbb{F}[u]/u^{e_{\mathfrak{p}}}}\mathbb{F}=(I^{-1})_{\sigma^{-1}\theta}\otimes_{\mathcal{O}}\mathbb{F}\end{array}$

for $j=2,\ldots,e_{\mathfrak{p}}$. On the other hand if
$\tau=\tau_{{\mathfrak{p}},i}$ and $\theta=\theta_{{\mathfrak{p}},i,1}$, then
the first map induces an isomorphism

$(pI^{-1}\otimes\mathbb{F})_{\tau}\otimes_{\mathbb{F}[u]/u^{e_{\mathfrak{p}}}}\mathbb{F}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}u^{e_{\mathfrak{p}}-j}({\mathfrak{p}}I^{-1}\otimes\mathbb{F})_{\tau}\otimes_{\mathbb{F}[u]/u^{e_{\mathfrak{p}}}}\mathbb{F}=({\mathfrak{p}}I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}$

whose composite with the ones induced by

$\phi^{*}(I^{-1}\otimes\mathbb{F})_{\phi^{-1}\circ\tau}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(I^{-1}\otimes\mathbb{F})_{\tau}\stackrel{{\scriptstyle
p\otimes 1}}{{\longrightarrow}}(pI^{-1}\otimes\mathbb{F})_{\tau}$

yields the desired isomorphism

$((I^{-1})_{\sigma^{-1}\theta}\otimes_{\mathcal{O}}\mathbb{F})^{\otimes
p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(I^{-1})_{\theta}\otimes_{\mathcal{O}}\mathbb{F}.$

The relations between the line bundles
$\widetilde{\Phi}_{\mathfrak{p}}^{*}{\mathcal{N}}_{\theta}$ and
${\mathcal{N}}_{\theta}$ or ${\mathcal{N}}_{\sigma^{-1}\theta}$ extend
similarly over $\widetilde{Y}_{U,\mathbb{F}}^{{\operatorname{tor}}^{\prime}}$,
so for $\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}$ as in (14) we
obtain isomorphisms

$(\widetilde{\Phi}^{\operatorname{tor}}_{\mathfrak{p}})^{*}\widetilde{{\mathcal{A}}}_{\mathbf{k},\mathbf{l},\mathbb{F}}^{\operatorname{tor}}\cong\widetilde{{\mathcal{A}}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}}^{{\operatorname{tor}}^{\prime}}$

whose pull-backs to
${\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}}$ are
compatible via their canonical trivializations with the isomorphisms

(31)
$\overline{D}^{\prime}_{\mathbf{k},\mathbf{l}}\cong\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}$

obtained as the tensor products of the ones just defined (where
$\overline{D}^{\prime}_{\mathbf{k},\mathbf{l}}$ is associated to the data for
the cusp ${\widetilde{{\mathcal{C}}}}^{\prime}$, and
$\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}$ to the
data for ${\widetilde{{\mathcal{C}}}}$).

It follows from the above description of
$\widetilde{\Phi}_{\mathfrak{p}}^{\operatorname{tor}}$ that
$\widetilde{\Phi}_{\mathfrak{p}}$ extends to the morphism
$\widetilde{\Phi}_{\mathfrak{p}}^{\min}:\widetilde{Y}_{U,\mathbb{F}}^{\min}\to\widetilde{Y}_{U,\mathbb{F}}^{\min}$
restricting to $\widetilde{\Phi}^{\infty}_{{\mathfrak{p}}}$ on the set of
cusps, with the induced maps on completed local rings151515Note that we have
implicitly chosen different representatives $(H,I,\lambda,[\eta])$ for each
cusp ${\widetilde{{\mathcal{C}}}}$ according to whether
$\widetilde{Y}_{U,\mathbb{F}}$ is viewed as the source or target of
$\widetilde{\Phi}_{\mathfrak{p}}$, but the rings and modules arising from the
two different descriptions of completions at ${\widetilde{{\mathcal{C}}}}$ are
canonically isomorphic. being the restriction to $V_{N}^{2}$-invariants of the
canonical inclusion

(32)
$\mathbb{F}[[q^{m}]]_{m\in(N^{-1}M^{\prime})_{+}\cup\\{0\\}}\longrightarrow\mathbb{F}[[q^{m}]]_{m\in(N^{-1}M)_{+}\cup\\{0\\}},$

where $M={\mathfrak{d}}^{-1}I^{-1}J$ and $M^{\prime}={\mathfrak{p}}M$.
Furthermore the commutativity of the diagram

$\textstyle{({\widetilde{\iota}}_{\mathbb{F},*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l},\mathbb{F}})^{\wedge}_{{\widetilde{{\mathcal{C}}}}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma((\widetilde{Y}_{U,\mathbb{F}}^{\operatorname{tor}})_{{\widetilde{{\mathcal{C}}}}^{\prime}}^{\wedge},({\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l},\mathbb{F}})_{{\widetilde{{\mathcal{C}}}}^{\prime}}^{\wedge})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\textstyle{(\overline{D}^{\prime}_{\mathbf{k},\mathbf{l}}\otimes_{\mathbb{F}}\Gamma({\widehat{S}}_{{\widetilde{{\mathcal{C}}}}^{\prime},\mathbb{F}},{\mathcal{O}}_{{\widehat{S}}_{{\widetilde{{\mathcal{C}}}}^{\prime},\mathbb{F}}}))^{V_{N}^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\widetilde{\iota}}_{\mathbb{F},*}{\widetilde{\Phi}_{\mathfrak{p}}^{*}\widetilde{{\mathcal{A}}}}_{\mathbf{k},\mathbf{l},\mathbb{F}})^{\wedge}_{\widetilde{{\mathcal{C}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{\Gamma((\widetilde{Y}_{U,\mathbb{F}}^{{\operatorname{tor}}^{\prime}})_{{\widetilde{{\mathcal{C}}}}}^{\wedge},(\widetilde{\Phi}_{\mathfrak{p}}^{{\operatorname{tor}},*}({\widetilde{{\mathcal{A}}}}^{\operatorname{tor}}_{\mathbf{k},\mathbf{l},\mathbb{F}}))_{{\widetilde{{\mathcal{C}}}}}^{\wedge})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\textstyle{(\overline{D}^{\prime}_{\mathbf{k},\mathbf{l}}\otimes_{\mathbb{F}}\Gamma({\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}},{\mathcal{O}}_{{\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}}}))^{V_{N}^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{({\widetilde{\iota}}_{\mathbb{F},*}{\widetilde{{\mathcal{A}}}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}})^{\wedge}_{\widetilde{{\mathcal{C}}}}}$$\textstyle{\Gamma((\widetilde{Y}_{U,\mathbb{F}}^{{\operatorname{tor}}^{\prime}})_{{\widetilde{{\mathcal{C}}}}}^{\wedge},({\widetilde{{\mathcal{A}}}}^{{\operatorname{tor}}^{\prime}}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}})_{{\widetilde{{\mathcal{C}}}}}^{\wedge})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\textstyle{(\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}\otimes_{\mathbb{F}}\Gamma({\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}},{\mathcal{O}}_{{\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}}}))^{V_{N}^{2}}}$

(where the top vertical arrows are defined by pulling back via
$\widetilde{\Phi}_{\mathfrak{p}}^{\min}$,
$\widetilde{\Phi}_{\mathfrak{p}}^{{\operatorname{tor}}}$ and the map
${\widehat{S}}^{\prime}_{{\widetilde{{\mathcal{C}}}},\mathbb{F}}\to{\widehat{S}}_{{\widetilde{{\mathcal{C}}}}^{\prime},\mathbb{F}}$)
shows that the resulting map on $q$-expansions is the restriction to
$V_{N}^{2}$-invariants of the map

$\overline{D}^{\prime}_{\mathbf{k},\mathbf{l}}\otimes_{\mathbb{F}}\mathbb{F}[[q^{m}]]_{m\in(N^{-1}M^{\prime})_{+}\cup\\{0\\}}\longrightarrow\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}\otimes_{\mathbb{F}}\mathbb{F}[[q^{m}]]_{m\in(N^{-1}M)_{+}\cup\\{0\\}},$

obtained as the tensor product of (31) and (32). (Note that the isomorphism
(31) is $V_{N}^{2}$-equivariant, but we can also choose $N$ sufficiently large
that the action is trivial.)

The constructions above are all compatible with the natural action of
${\mathcal{O}}_{F,(p),+}^{\times}$, so the morphism
$\widetilde{\Phi}_{\mathfrak{p}}^{\min}$ induces a morphism
$\Phi_{{\mathfrak{p}}}^{\min}:\overline{Y}_{U}^{\min}\to\overline{Y}_{U}^{\min}$
extending $\Phi_{\mathfrak{p}}$ by the map $\Phi^{\infty}_{{\mathfrak{p}}}$ on
cusps, and its effect on completed local rings is given by the
$V_{N,+}$-invariants of (32). Furthermore the map
$V_{\mathfrak{p}}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\longrightarrow
M_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}(U,\mathbb{F})$ is
described on $q$-expansions by taking the $V_{N,+}$-invariants of the tensor
product of (31) and (32). (Note that $\Phi_{\mathfrak{p}}^{\min}$ is proper
and quasi-finite, hence finite, but not necessarily flat at the cusps.)

Similarly for any sufficiently small $U^{\prime}$, we may choose $N$ so
$U=U(N)\subset U^{\prime}$ and take invariants under the natural action of
$U^{\prime}/U$, with which the above constructions are also easily seen to be
compatible. We thus obtain the description of $V_{\mathfrak{p}}$ on
$q$-expansions (under the identifications of Proposition 7.2.1) as the
resulting map

(33)
$\left(\overline{D}^{\prime}_{\mathbf{k},\mathbf{l}}\otimes_{\mathbb{F}}\mathbb{F}[[q^{m}]]_{m\in(N^{-1}M^{\prime})_{+}\cup\\{0\\}}\right)^{\Gamma_{{\mathcal{C}}^{\prime},U^{\prime}}}\longrightarrow\left(\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}\otimes_{\mathbb{F}}\mathbb{F}[[q^{m}]]_{m\in(N^{-1}M)_{+}\cup\\{0\\}}\right)^{\Gamma_{{\mathcal{C}},U^{\prime}}}.$

(Note that the maps (31) and (32) are in fact
$\Gamma_{{\mathcal{C}}}$-equivariant, where $\Gamma_{{\mathcal{C}}}$ is
defined in (24) and its action on the target is via the natural inclusion in
$\Gamma_{{\mathcal{C}}^{\prime}}$.)

Finally we note that the effect of the operator
$V_{\mathfrak{p}}^{0}:M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})\longrightarrow
M_{\mathbf{k}^{\prime\prime},\mathbf{l}}(U,\mathbb{F})$ on $q$-expansions has
the same description, but with (31) replaced by its composite with the
isomorphism

$\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}\cong\overline{D}_{\bf{0},\mathbf{l}-\mathbf{l}^{\prime\prime}}\otimes_{\mathbb{F}}\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}^{\prime\prime}}=\overline{D}_{\mathbf{k}^{\prime\prime},\mathbf{l}}$

given by choosing the basis element of
$\overline{D}_{\bf{0},\mathbf{l}-\mathbf{l}^{\prime\prime}}$ to be the
(constant) $q$-expansion of $\prod_{\theta\in\Sigma}G_{\theta}^{-l_{\theta}}$.

##  9\. The kernel of $\Theta$

###  9.1. Determination of the kernel

In this section we analyze the kernel of the partial $\Theta$-operator
$\Theta_{\tau}:M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})\to
M_{\mathbf{k}^{\prime},\mathbf{l}^{\prime}}(U;{\mathbb{F}})$ for
$\tau\in\Sigma_{{\mathfrak{p}},0}$ and relate it to the image of a partial
Frobenius operator.

We allow $U$ to be any sufficiently small open compact subgroup of
$\operatorname{GL}_{2}({\mathbb{A}}_{F,\mathbf{f}})$ of level prime to $p$,
and $(\mathbf{k},\mathbf{l})$ any weight such that
$\chi_{\mathbf{k}+2\mathbf{l},\mathbb{F}}$ is trivial on
$U\cap{\mathcal{O}}_{F}^{\times}$. First note that by (30) and the
$q$-expansion Principle (Proposition 7.2.2), the kernel of $\Theta_{\tau}$
consists precisely of those $f$ whose $q$-expansions $\sum_{m\in
N^{-1}M_{+}\cup\\{0\\}}(b^{\mathbf{k}}c^{\mathbf{l}}\otimes r_{m})q^{m}$ at
all cusps, or even a cusp on each connected component, satisfy $r_{m}=0$ for
all $m\not\in{\mathfrak{p}}N^{-1}M$. (Recall that
$M={\mathfrak{d}}^{-1}I^{-1}J$ if the cusp ${\mathcal{C}}$ if represented by
$(H,I,[\lambda],[\eta])$, and note that the condition depends only on the
isomorphism class of $(H,I,[\lambda],[\eta])$ and in particular is independent
of $N$ prime to $p$ such that $U(N)\subset U$.) Note that the condition is the
same for all $\tau\in\Sigma_{{\mathfrak{p}},0}$, so that

$\ker(\Theta_{\tau})=\ker(\Theta_{\tau^{\prime}})\quad\mbox{for all
$\tau,\tau^{\prime}\in\Sigma_{{\mathfrak{p}},0}$}.$

Note also that the condition is invariant under multiplication by the Hasse
invariants $H_{\theta}$ (and of course the forms $G_{\theta}$) for all
$\theta$, so that $\Theta_{\tau}(f)=0$ if and only if

$\Theta_{\tau}(f\prod_{\theta\in\Sigma}G_{\theta}^{m_{\theta}}H_{\theta}^{n_{\theta}})=0$

for all $\mathbf{m}\in\mathbb{Z}^{\Sigma}$, $\mathbf{n}\in\mathbb{Z}_{\geq
0}^{\Sigma}$, if and only if
$\Theta_{\tau}(f\prod_{\theta\in\Sigma}G_{\theta}^{m_{\theta}}H_{\theta}^{n_{\theta}})=0$
for some $\mathbf{m}\in\mathbb{Z}^{\Sigma}$, $\mathbf{n}\in\mathbb{Z}_{\geq
0}^{\Sigma}$. (Alternatively note that this follows from the fact the partial
$\Theta$-operators commute with multiplication by the $G_{\theta}$ and
$H_{\theta}$, as can be seen directly from their definition.)

Suppose now that $\mathbf{k}=\mathbf{k}_{\min}(f)$, so that $f$ is not
divisible by any partial Hasse invariants (see §4.2). Then if
$f\in\ker(\Theta_{\tau})$, and hence
$f\in\ker(\Theta_{\tau_{{\mathfrak{p}},i}})$ for all
$i\in\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}$, then Theorem 5.2.1 implies that
$p|k_{\theta_{{\mathfrak{p}},i,e_{\mathfrak{p}}}}$ for all
$i\in\mathbb{Z}/f_{\mathfrak{p}}\mathbb{Z}$. Therefore $\mathbf{k}$ is of the
form $\mathbf{k}_{0}^{\prime\prime}$ for some $\mathbf{k}_{0}$, where
$\mathbf{k}_{0}^{\prime\prime}$ is as in the definition of $V_{\mathfrak{p}}$
in §6.2, or equivalently $V_{\mathfrak{p}}^{0}$. Furthermore it is immediate
from the description of the effect of $V_{\mathfrak{p}}^{0}$ on $q$-expansions
in (33) that its image is contained in the kernel of $\Theta_{\tau}$. We now
use the method of [12, Thm. 9.8.2] to prove the kernel is precisely the image
of $V_{\mathfrak{p}}^{0}$.

######  Theorem 9.1.1.

Suppose that $f\in
M_{\mathbf{k}_{0}^{\prime\prime},\mathbf{l}}(U;{\mathbb{F}})$ and
$\tau\in\Sigma_{{\mathfrak{p}},0}$. If $\Theta_{\tau}(f)=0$, then
$f=V_{\mathfrak{p}}^{0}(g)$ for some $g\in
M_{\mathbf{k}_{0},\mathbf{l}}(U;{\mathbb{F}})$.

Proof. Let $\overline{\iota}$ denote the embedding
$\overline{Y}_{U}\hookrightarrow\overline{Y}_{U}^{\min}$, and choose a set of
cusps ${\mathcal{S}}\subset Y^{\infty}_{U}$ consisting of precisely one on
each connected component of $\overline{Y}_{U}$. Note that since
$\Phi_{\mathfrak{p}}^{\min}$ (defined in §8.3) is bijective on cusps as well
as connected components, the set
${\mathcal{S}}^{\prime}:=\Phi^{\infty}_{{\mathfrak{p}}}({\mathcal{S}})$ also
includes exactly one cusp on each connected component.

Recall from Proposition 7.2.1 that the sheaves
$\overline{\iota}_{*}{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}}$ and
$\overline{\iota}_{*}{\mathcal{A}}_{\mathbf{k}^{\prime\prime}_{0},\mathbf{l},\mathbb{F}}$
are coherent, as is
$\overline{\iota}_{*}\Phi_{{\mathfrak{p}},*}{\mathcal{A}}_{\mathbf{k}_{0}^{\prime\prime},\mathbf{l}^{\prime\prime},\mathbb{F}}=\Phi_{{\mathfrak{p}},*}^{\min}\overline{\iota}_{*}{\mathcal{A}}_{\mathbf{k}_{0}^{\prime\prime},\mathbf{l},\mathbb{F}}$
since $\Phi_{{\mathfrak{p}}}^{\min}$ is finite. For each
${\mathcal{C}}\in{\mathcal{S}}$, let
${\mathcal{C}}^{\prime}=\Phi_{{\mathfrak{p}}}^{\infty}({\mathcal{C}})$, so
that $\Phi_{{\mathfrak{p}}}^{{\min},*}$ defines a finite extension
${\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}^{\prime}}\hookrightarrow{\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}}$
of local rings. We let
$N_{{\mathcal{C}}^{\prime}}=(\overline{\iota}_{*}{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}})_{{\mathcal{C}}^{\prime}}$
denote the stalk at ${\mathcal{C}}^{\prime}$ of
$\overline{\iota}_{*}{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}}$,
and similarly let
$N^{\prime\prime}_{{\mathcal{C}}}=(\overline{\iota}_{*}{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}})_{{\mathcal{C}}}=(\overline{\iota}_{*}\Phi_{{\mathfrak{p}},*}{\mathcal{A}}_{\mathbf{k}_{0}^{\prime\prime},\mathbf{l},\mathbb{F}})_{{\mathcal{C}}^{\prime}}$.
The stalk at ${\mathcal{C}}^{\prime}$ of $\overline{\iota}_{*}$ of the adjoint
of
$\Phi_{{\mathfrak{p}}}^{*}{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}}_{\mathbf{k}_{0}^{\prime\prime},\mathbf{l},\mathbb{F}}$
then defines an injective homomorphism $N_{{\mathcal{C}}^{\prime}}\to
N^{\prime\prime}_{{\mathcal{C}}}$ of finitely generated
${\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}^{\prime}}$-modules,
extending $V_{\mathfrak{p}}^{0}$ to a map

$\bigoplus_{{\mathcal{C}}^{\prime}\in{\mathcal{S}}^{\prime}}N_{{\mathcal{C}}^{\prime}}\to\bigoplus_{{\mathcal{C}}\in{\mathcal{S}}}N^{\prime\prime}_{{\mathcal{C}}}.$

Similarly localizing at the generic points of $\overline{Y}_{U}$ (or
equivalently $\overline{Y}_{U}^{\min}$) extends $V_{\mathfrak{p}}^{0}$ to a
map
$H^{0}(\overline{Y}_{U},{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}}\otimes_{\overline{Y}_{U}}{\mathcal{F}}_{U})\to
H^{0}(\overline{Y}_{U},{\mathcal{A}}_{\mathbf{k}^{\prime\prime}_{0},\mathbf{l},\mathbb{F}}\otimes_{\overline{Y}_{U}}{\mathcal{F}}_{U})$,
so we obtain a commutative diagram of injective maps

(34)
$\textstyle{M_{\mathbf{k}_{0},\mathbf{l}}(U;\mathbb{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{{\mathcal{C}}^{\prime}\in{\mathcal{S}}^{\prime}}N_{{\mathcal{C}}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\overline{Y}_{U},{\mathcal{A}}_{\mathbf{k}_{0},\mathbf{l},\mathbb{F}}\otimes_{\overline{Y}_{U}}{\mathcal{F}}_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{\mathbf{k}^{\prime\prime}_{0},\mathbf{l}}(U;\mathbb{F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{{\mathcal{C}}\in{\mathcal{S}}}N^{\prime\prime}_{{\mathcal{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\overline{Y}_{U},{\mathcal{A}}_{\mathbf{k}^{\prime\prime}_{0},\mathbf{l},\mathbb{F}}\otimes_{\overline{Y}_{U}}{\mathcal{F}}_{U}).}$

(Note that the horizontal maps, defined by localization, are injective since
${\mathcal{S}}$ and ${\mathcal{S}}^{\prime}$ each contain a unique cusp on
each component of $\overline{Y}_{U}$.)

Let $N_{{\mathcal{C}}^{\prime}}^{\wedge}$ denote the completion of
$N_{{\mathcal{C}}^{\prime}}$ with respect to the maximal ideal of
${\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}^{\prime}}$, and
similarly let $N^{\prime\prime\wedge}_{{\mathcal{C}}}$ denote the completion
of $N^{\prime\prime}_{{\mathcal{C}}}$ with respect to the maximal ideal of
${\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}}$, or equivalently
${\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}^{\prime}}$. Note that
the map $N_{{\mathcal{C}}^{\prime}}^{\wedge}\to
N^{\prime\prime\wedge}_{{\mathcal{C}}}$ is the one described by (33), or more
precisely its variant for $V_{\mathfrak{p}}^{0}$.

Recall from (30) that if $f\in\ker\Theta_{\tau}$, then for each
${\mathcal{C}}\in{\mathcal{S}}$, the $q$-expansion of $f$:

$\sum_{m\in
N^{-1}M_{+}\cup\\{0\\}}(b^{\mathbf{k}_{0}^{\prime\prime}}c^{\mathbf{l}}\otimes
r_{m})q^{m}\in N^{\prime\prime\wedge}_{{\mathcal{C}}}$

satisfies $r_{m}=0$ for all
$m\not\in{\mathfrak{p}}N^{-1}M=N^{-1}M^{\prime}_{+}$, where
$M^{\prime}={\mathfrak{p}}M$, and is therefore in the image of
$N_{{\mathcal{C}}^{\prime}}^{\wedge}$. Since the completion
${\mathcal{O}}^{\wedge}_{\overline{Y}_{U}^{\min},{\mathcal{C}}^{\prime}}$ is
faithfully flat over
${\mathcal{O}}_{\overline{Y}_{U}^{\min},{\mathcal{C}}^{\prime}}$, it follows
that the image of $f$ in
$\bigoplus_{{\mathcal{C}}\in{\mathcal{S}}}N^{\prime\prime}_{{\mathcal{C}}}$ is
of the form $V_{\mathfrak{p}}^{0}(g)$ for some
$g\in\bigoplus_{{\mathcal{C}}^{\prime}\in{\mathcal{S}}^{\prime}}N_{{\mathcal{C}}^{\prime}}$,
and hence that its image in
$H^{0}(\overline{Y}_{U},{\mathcal{A}}_{\mathbf{k}^{\prime\prime}_{0},\mathbf{l},\mathbb{F}}\otimes_{\overline{Y}_{U}}{\mathcal{F}}_{U})$
is of the form $V_{\mathfrak{p}}^{0}(g)$ for some $g\in
H^{0}(\overline{Y}_{U},{\mathcal{A}}_{\mathbf{k}^{\prime\prime}_{0},\mathbf{l},\mathbb{F}}\otimes_{\overline{Y}_{U}}{\mathcal{F}}_{U})$.

It just remains to prove that $g\in
M_{\mathbf{k}_{0},\mathbf{l}}(U;\mathbb{F})$, or equivalently that
${\operatorname{ord}}_{z}(g)\geq 0$ for all prime divisors $z$ on
$\overline{Y}_{U}$. To that end, note that the operators
$V_{{\mathfrak{p}}^{\prime}}$ for all ${\mathfrak{p}}^{\prime}\in S_{p}$, and
$\epsilon_{\mathbf{k},\mathbf{l}}$ for all
$\mathbf{k},\mathbf{l}\in\mathbb{Z}^{\Sigma}$ (see §6.2)) similarly extend to
maps on stalks at generic points satisfying (16), so that
$V_{\mathfrak{p}}(g)=f\prod_{\theta\in\Sigma}G_{\theta}^{l_{\theta}}$ and

$g^{p}=\left(\epsilon_{p\mathbf{k},p\mathbf{l}}\prod_{{\mathfrak{p}}^{\prime}\in
S_{p}}V_{\mathfrak{q}}^{e_{\mathfrak{q}}}\right)(g)=\left(\delta_{p\mathbf{k},p\mathbf{l}}\prod_{{\mathfrak{p}}^{\prime}\neq{\mathfrak{p}}}V_{{\mathfrak{p}}^{\prime}}^{e_{\mathfrak{p}}}\right)V_{\mathfrak{p}}^{e_{\mathfrak{p}}-1}(f).$

Therefore $p\cdot{\operatorname{ord}}_{z}(g)\geq 0$, and hence
${\operatorname{ord}}_{z}(g)\geq 0$.  $\square$

For the following corollary, recall that $\Xi^{\min}$ is defined by (10) and
that the main result of [10] states that if $f$ is a non-zero form in
$M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})$, then
$\mathbf{k}_{\min}(f)\in\Xi^{\min}$.

######  Corollary 9.1.2.

Suppose that $f\in M_{\mathbf{k},\mathbf{l}}(U;{\mathbb{F}})$ and
$\tau\in\Sigma_{{\mathfrak{p}},0}$. Then $\Theta_{\tau}(f)=0$ if and only if
there exist $\mathbf{k}_{0}\in\Xi^{\min}$,
$\mathbf{n}\in\mathbb{Z}^{\Sigma}_{\geq 0}$ and $g\in
M_{\mathbf{k}_{0},\mathbf{l}}(U;{\mathbb{F}})$ such that
$\mathbf{k}=\mathbf{k}_{0}^{\prime\prime}+\sum_{\theta}n_{\theta}\mathbf{h}_{\theta}$
and

$f=V_{\mathfrak{p}}^{0}(g)\prod_{\theta}H_{\theta}^{n_{\theta}}.$

Proof. We have already seen that if
$f=V_{\mathfrak{p}}^{0}(g)\prod_{\theta}H_{\theta}^{n_{\theta}}$, then
$\Theta_{\tau}(f)=0$.

For the converse, note that we may assume $\mathbf{k}=\mathbf{k}_{\min}(f)$,
so that $\mathbf{k}\in\Xi^{\min}$ and
$\mathbf{k}=\mathbf{k}_{0}^{\prime\prime}$ for some
$\mathbf{k}_{0}\in\mathbb{Z}^{\Sigma}$. Therefore the theorem implies that
$f=V_{\mathfrak{p}}^{0}(g)$ for some $g\in
M_{\mathbf{k}_{0},\mathbf{l}}(U;{\mathbb{F}})$. Finally it is immediate from
the definitions of $\Xi^{\min}$ and $\mathbf{k}_{0}^{\prime\prime}$ that
$\mathbf{k}_{0}\in\Xi^{\min}$ if (and only if)
$\mathbf{k}_{0}^{\prime\prime}\in\Xi^{\min}$.  $\square$

###  9.2. Forms of partial weight $0$

We now apply our results on partial $\Theta$-operators to prove a partial
positivity result for minimal weights of Hilbert modular forms. Recall the
main result of [10] proves that minimal weights $\mathbf{k}=\sum
k_{\theta}{\mathbf{e}}_{\theta}$ of Hilbert modular forms necessarily lie in
the cone $\Xi^{\min}$, and hence satisfy $k_{\theta}\geq 0$ for all $\theta$,
and that forms with $\mathbf{k}=\bf{0}$ are easily described by Proposition
3.2.2. We prove the following restriction on possible minimal weights
$\mathbf{k}$ with $k_{\theta}=0$ for some (but not all) $\theta\in\Sigma$.

######  Theorem 9.2.1.

Suppose that ${\mathfrak{p}}\in S_{p}$ is such that $f_{\mathfrak{p}}>1$, or
$e_{\mathfrak{p}}>1$ and $p>3$. 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 $\mathbf{k}=\bf{0}$.

Proof. Writing simply $f=f_{\mathfrak{p}}$ and $e=e_{\mathfrak{p}}$, note that
the hypotheses mean that $ef>1$ and $p^{f}>3$. Choose any
$\tau=\tau_{{\mathfrak{p}},i}\in\Sigma_{{\mathfrak{p}},0}$ and let
$\theta_{0}=\theta_{{\mathfrak{p}},i,e}$. We will first prove that
$\Theta_{\tau}(f)=0$.

Note that since $\mathbf{k}\in\Xi^{\min}$ and $k_{\theta}=0$ for some
$\theta\in\Sigma_{\mathfrak{p}}$, we in fact have $k_{\theta}=0$ for all
$\theta\in\Sigma_{\mathfrak{p}}$. In particular $p|k_{\theta_{0}}$, so Theorem
5.2.1 implies that $H_{\theta_{0}}|\Theta_{\tau}(f)$. Therefore if
$\Theta_{\tau}(f)\neq 0$, then
$\mathbf{k}_{\min}(\Theta_{\tau}(f))\leq_{{\mathrm{Ha}}}\mathbf{k}+2{\mathbf{e}}_{\theta_{0}}$,
i.e.,

$\mathbf{k}+2{\mathbf{e}}_{\theta_{0}}-\sum_{\theta\in\Sigma}m_{\theta}\mathbf{h}_{\theta}\in\Xi^{\min}$

for some integers $m_{\theta}\geq 0$. Letting $m_{r}=m_{\sigma^{r}\theta_{0}}$
for $r=1,\ldots,ef$, this implies that

(35) $\begin{array}[]{rl}&m_{1}-m_{2}\leq m_{2}-m_{3}\leq\cdots\leq
m_{e-1}-m_{e}\leq m_{e}-pm_{e+1}\\\ \leq&p(m_{e+1}-m_{e+2})\leq
p(m_{e+2}-m_{e+3})\leq\cdots\leq p(m_{2e-1}-m_{2e})\leq p(m_{2e}-pm_{2e+1})\\\
&\vdots\\\ \leq&p^{f-1}(m_{(f-1)e+1}-m_{(f-1)e+2})\leq\cdots\leq
p^{f-1}(m_{ef-1}-m_{ef})\leq p^{f-1}(2+m_{ef}-pm_{1})\\\
\leq&p^{f}(m_{1}-m_{2}),\end{array}$

(with the obvious collapsing here and in subsequent inequalities if $e$ or
$f=1$). In particular all the expressions in (35) are non-negative, so we have

$m_{1}\geq m_{2}\geq\cdots m_{e}\geq pm_{e+1}\geq pm_{e+2}\geq\cdots\geq
p^{f-1}m_{ef-1}\geq p^{f-1}m_{ef}$

and $2+m_{ef}-pm_{1}\geq 0$, which implies that $(p^{f}-1)m_{ef}\leq 2$. Since
$p^{f}>3$, it follows that $m_{ef}=0$, so $pm_{1}\leq 2$, which implies that
either $m_{1}=0$, or $m_{1}=1$ and $p=2$. If $m_{1}=0$, then $m_{r}=0$ for all
$r$, which contradicts the final inequality in (35). On the other hand if
$m_{1}=1$ and $p=2$, then all the expressions in (35) are zero, which in turn
implies that $m_{1}=p^{f-1}m_{ef}$, which again yields a contradiction.

We have now shown that $\Theta_{\tau}(f)=0$. Note that
$\mathbf{k}^{\prime\prime}=\mathbf{k}$ since $k_{\theta}=0$ for all
$\theta\in\Sigma_{\mathfrak{p}}$, so Theorem 5.2.1 implies that
$f=V_{\mathfrak{p}}^{0}(f_{1})$ for some $f_{1}\in
M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$. We may therefore iterate the above
argument to conclude that $f_{1}=V_{\mathfrak{p}}^{0}(f_{2})$ for some
$f_{2}\in M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$, and by induction that for
all $n\geq 1$, we have $f=(V_{\mathfrak{p}}^{0})^{n}(f_{n})$ for some
$f_{n}\in M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$. It follows that for all
$n\geq 1$, the $q$-expansion of $f$ at every cusp of $\overline{Y}_{U}$
satisfies $r_{m}=0$ for all $m\not\in{\mathfrak{p}}^{n}M$, so in fact the
$q$-expansion of $f$ at every cusp is constant.

To prove that $\mathbf{k}=\bf{0}$, recall that $\Xi^{\min}$ is contained in
the cone spanned by the partial Hasse invariants, so
$\mathbf{k}=\sum_{\theta\in\Sigma}s_{\theta}\mathbf{h}_{\theta}$ for some
$s_{\theta}\in\mathbb{Q}_{\geq 0}$. Furthermore the denominators are divisors
of $M=\operatorname{lcm}\\{\,p^{f_{\mathfrak{q}}}-1\,|\,{\mathfrak{q}}\in
S_{p}\,\\}$, so that $M\mathbf{k}=\sum m_{\theta}\mathbf{h}_{\theta}$ for some
$m_{\theta}\in\mathbb{Z}_{\geq 0}$. Similarly $M\mathbf{l}=\sum
n_{\theta}\mathbf{h}_{\theta}$ for some $n_{\theta}\in\mathbb{Z}$. Since $f$
has constant $q$-expansions, so does $f^{M}$, and therefore

$f^{M}=h\prod_{\theta\in\Sigma}(H_{\theta}^{m_{\theta}}G_{\theta}^{n_{\theta}})$

for some $h\in H^{0}(\overline{Y}_{U},{\mathcal{O}}_{\overline{Y}_{U}})$.

For each $\theta\in\Sigma$, the assumption that
$\mathbf{k}=\mathbf{k}_{\min}(f)$ means that $f$ is not divisible by
$H_{\theta}$, so ${\operatorname{ord}}_{z}(f)=0$ for some irreducible
component $z$ of $Z_{\theta}$. On the other hand we have
$M{\operatorname{ord}}_{z}(f)={\operatorname{ord}}_{z}(f^{M})\geq m_{\theta}$,
so $m_{\theta}=0$. As this holds for all $\theta\in\Sigma$, we conclude that
$\mathbf{k}=\bf{0}$.  $\square$

###  9.3. The kernel revisited

Finally we present a cleaner, but less explicit, variant of Corollary 9.1.2
describing the kernels of partial $\Theta$-operators.

We first record the effect of $V_{\mathfrak{p}}$ on the partial Hasse
invariants $H_{\theta}$. For each prime ${\mathfrak{p}}\in S_{p}$, we let
$\beta_{\mathfrak{p}}=p^{-1}\varpi_{\mathfrak{p}}^{e_{\mathfrak{p}}}\in{\mathcal{O}}_{F,{\mathfrak{p}}}^{\times}$.
It is straightforward to check, directly from the definition of
$V_{\mathfrak{p}}$ or from the description (33) of its effect on
$q$-expansions (and those of the $H_{\theta}$ in §8.1), that if
$\theta\not\in\Sigma_{\mathfrak{p}}$ then
$V_{\mathfrak{p}}(H_{\theta})=H_{\theta}$ , but if
$\theta=\theta_{{\mathfrak{p}},i,j}$ then

$V_{\mathfrak{p}}(H_{\theta})=\left\\{\begin{array}[]{ll}H_{\sigma^{-1}\theta}^{p},&\mbox{if
$e_{\mathfrak{p}}=1$;}\\\
\overline{\theta}(\beta_{\mathfrak{p}})^{-1}H_{\sigma^{-1}\theta}^{p},&\mbox{if
$e_{\mathfrak{p}}>1$ and $j=1$;}\\\
\overline{\theta}(\beta_{\mathfrak{p}})H_{\sigma^{-1}\theta},&\mbox{if
$e_{\mathfrak{p}}>1$ and $j=2$;}\\\
H_{\sigma^{-1}\theta},&\mbox{otherwise.}\end{array}\right.$

Therefore we define the modified partial Hasse invariant to be
$H^{\prime}_{\theta}=\overline{\theta}(\beta_{{\mathfrak{p}}^{\prime}})H_{\theta}$
if $\theta=\theta_{{\mathfrak{p}}^{\prime},i,1}$ for some
${\mathfrak{p}}^{\prime}\in S_{p}$ and
$i\in\mathbb{Z}/f_{{\mathfrak{p}}^{\prime}}\mathbb{Z}$, and
$H^{\prime}_{\theta}=H_{\theta}$ otherwise, so that

$V_{\mathfrak{p}}(H^{\prime}_{\theta})=\left\\{\begin{array}[]{ll}H^{\prime
n_{\theta}}_{\sigma^{-1}\theta}&\mbox{if
$\theta\in\Sigma_{{\mathfrak{p}},0}$;}\\\ H^{\prime}_{\theta}&\mbox{if
$\theta\not\in\Sigma_{{\mathfrak{p}},0}$.}\end{array}\right.$

Similarly letting
$G^{\prime}_{\theta}=\overline{\theta}(\beta_{{\mathfrak{p}}^{\prime}})G_{\theta}$
if $\theta=\theta_{{\mathfrak{p}}^{\prime},i,1}$ and
$G^{\prime}_{\theta}=G_{\theta}$ otherwise, we have
$V_{\mathfrak{p}}(G^{\prime}_{\theta})=G^{\prime
n_{\theta}}_{\sigma^{-1}\theta}$ if $\theta\in\Sigma_{{\mathfrak{p}}}$ and
$V_{\mathfrak{p}}(G^{\prime}_{\theta})=G_{\theta}$ if
$\theta\not\in\Sigma_{\mathfrak{p}}$.

Now for any sufficiently small $U$ of level prime to $p$, consider the
$\mathbb{F}$-algebra

$M_{{\operatorname{tot}}}(U;\mathbb{F})=\bigoplus_{\mathbf{k},\mathbf{l}\in(\mathbb{Z}^{\Sigma})^{2}}M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})$

of Hilbert modular forms of all weights and level $U$ (where we let
$M_{\mathbf{k},\mathbf{l}}(U;\mathbb{F})=0$ if
$\chi_{\mathbf{k}+2\mathbf{l},\mathbb{F}}$ is non-trivial on
$U\cap{\mathcal{O}}_{F}^{\times}$). We may then consider $V_{\mathfrak{p}}$
(resp. $\Theta_{\tau}$) as an $\mathbb{F}$-algebra homomorphism (resp.
$\mathbb{F}$-linear derivation) $M_{{\operatorname{tot}}}(U;\mathbb{F})\to
M_{{\operatorname{tot}}}(U;\mathbb{F})$ for any ${\mathfrak{p}}\in S_{p}$ and
$\tau\in\Sigma_{{\mathfrak{p}},0}$. Furthermore letting ${\mathfrak{I}}$
denote the ideal
$\langle\,H_{\theta}^{\prime}-1,G_{\theta}^{\prime}-1\,\rangle_{\theta\in\Sigma}$
in $M_{{\operatorname{tot}}}(U;\mathbb{F})$ and
$R_{U}=M^{{\operatorname{tot}}}(U;\mathbb{F})/{\mathfrak{I}}$, we see that
$V_{\mathfrak{p}}({\mathfrak{I}})\subset{\mathfrak{I}}$ and
$\Theta_{\tau}({\mathfrak{I}})\subset{\mathfrak{I}}$, so we obtain an
$\mathbb{F}$-algebra homomorphism $V_{\mathfrak{p}}$ and derivation
$\Theta_{\tau}$ such that the composite

$R_{U}\stackrel{{\scriptstyle
V_{\mathfrak{p}}}}{{\longrightarrow}}R_{U}\stackrel{{\scriptstyle\Theta_{\tau}}}{{\longrightarrow}}R_{U}$

is zero for any ${\mathfrak{p}}\in S_{p}$, $\tau\in\Sigma_{{\mathfrak{p}},0}$.

Let $\Lambda$ denote the subgroup
$\bigoplus_{\theta\in\Sigma}\mathbb{Z}\mathbf{h}_{\theta}$ of
$\mathbb{Z}^{\Sigma}=\bigoplus_{\theta\in\Sigma}\mathbb{Z}{\mathbf{e}}_{\theta}$,
so $\Lambda$ is the image of the image of the endomorphism of
$\mathbb{Z}^{\Sigma}$ defined by
$\sum_{\theta}m_{\theta}{\mathbf{e}}_{\theta}\mapsto\sum_{\theta}m_{\theta}\mathbf{h}_{\theta}$.
Writing
$\mathbf{h}_{\theta}=\sum_{\theta^{\prime}}n_{\theta,\theta^{\prime}}{\mathbf{e}}_{\theta^{\prime}}$,
it is straightforward to check that the matrix $(n_{\theta,\theta^{\prime}})$
has determinant $\prod_{{\mathfrak{p}}\in S_{p}}(p^{f_{\mathfrak{p}}}-1)$, so
this is the index of $\Lambda$ in $\mathbb{Z}^{\Sigma}$. On the other hand,
let $\Psi$ denote the group of characters
$\psi:({\mathcal{O}}_{F}/{\mathfrak{q}})^{\times}=\bigoplus_{{\mathfrak{p}}\in
S_{p}}({\mathcal{O}}_{F}/{\mathfrak{p}})^{\times}\longrightarrow\mathbb{F}^{\times},$
and consider the surjective homomorphism $\varrho:\mathbb{Z}^{\Sigma}\to\Psi$
defined by

$\varrho(\mathbf{k})=\prod_{\theta\in\Sigma}\overline{\theta}^{k_{\theta}}=\bigoplus_{{\mathfrak{p}}\in
S_{p}}(\prod_{\theta\in\Sigma_{\mathfrak{p}}}\overline{\theta}^{k_{\theta}}).$

Note that $\varrho(\mathbf{h}_{\theta})$ is trivial for all $\theta\in\Sigma$,
so $\Lambda\subset\ker(\varrho)$. Since $\mathbb{Z}^{\Sigma}/\Lambda$ and
$\Psi$ each have order $\prod_{{\mathfrak{p}}\in
S_{p}}(p^{f_{\mathfrak{p}}}-1)$, it follows that

$\Lambda=\ker(\varrho)=\left\\{\,\sum