We shall use the same notation $\overline{h}^{\nabla}$ to denote the induced
$\mathcal{O}_{U^{(N-1)}}$-linear morphism
$\mathcal{O}_{\mathcal{E}}^{[N-1]}|_{U}\rightarrow\Omega^{(N-1)}\otimes\mathcal{O}_{\mathcal{E}}^{[N-1]}|_{U}$
(where $\Omega^{(N-1)}:=\mathcal{T}^{(N-1)\vee}$) via the natural morphism
$\mathfrak{g}_{\mathcal{E}}^{[N-1]}\rightarrow\mathcal{E}nd_{\mathcal{O}_{X^{(N-1)}}}(\mathcal{O}_{\mathcal{E}}^{[N-1]})$.
If $\nabla_{R}$ (resp., $R\times_{k}\nabla$) denotes the
$S^{\mathrm{log}}$-connection on the $\mathcal{O}_{U^{(N-1)}}$-module
$\mathcal{O}_{\mathcal{E}}|_{U}$ induced by $\phi_{R}^{(N-1)}|_{U}$ (resp.,
$R\times_{k}\phi^{(N-1)}|_{U}$), then (493) implies
$\nabla_{R}=R\times_{k}\nabla+\varepsilon\cdot\overline{h}^{\nabla}$. Since
both $\nabla_{R}$ and $R\times_{k}\nabla$ have vanishing $p$-curvature (cf.
Proposition 1.6.1, (ii)), it follows from Lemma 6.3.3 described below that we
can find, after possibly shrinking $U$, an element
$v\in\Gamma(U,\mathfrak{g}_{\mathcal{E}}^{[N-1]})$ satisfying

(497)
$\displaystyle(\nabla_{\phi}^{\mathrm{ad}})^{[N-1]}(v)=\overline{h}^{\nabla}.$

Denote by $h_{v}$ the automorphism of $R\times_{k}\mathcal{E}$ corresponding
to the automorphism of $\mathcal{O}_{R\times_{k}\mathcal{E}}$ described as
$\mathrm{id}_{\mathcal{O}_{R\times_{k}\mathcal{E}}}-\varepsilon\cdot v$. Also,
denote by $h_{v}^{*}(R\times_{k}\phi^{\natural\natural}|_{U})^{(N-1)}$ the
$\mathcal{D}^{(N-1)}$-module structure on
$\mathcal{O}_{R\times_{k}\mathcal{E}|_{U}}$ defined as the pull-back of
$R\times_{k}\phi^{(N-1)\natural\natural}|_{U}$ by $h_{v}$. Since $v$ belongs
to $\mathfrak{g}_{\mathcal{E}}^{[N-1]}$, we have

(498) $\displaystyle
h_{v}^{*}(R\times_{k}\phi^{\natural\natural}|_{U})^{(N-2)}=\left(R\times_{k}\phi^{\natural\natural(N-2)}|_{U}=\right)\phi_{R}^{\natural\natural(N-2)}|_{U}.$

On the other hand, the equality (497) together with the same argument as
[Wak6, Remark 6.2] shows that

(499) $\displaystyle
h_{v}^{*}(R\times_{k}\phi^{\natural\natural}|_{U})^{[N-1]}=\phi_{R}^{\natural\natural[N-1]}|_{U}.$

By (498) and (499), we can apply Lemma 2.1.2 to obtain the equality
$h_{v}^{*}(R\times_{k}\phi^{\natural\natural}|_{U})^{(N-1)}=\phi_{R}^{\natural\natural(N-1)}|_{U}$.
This means that $\phi_{R}^{\natural\natural(N-1)}|_{U}$ can be transformed
into the trivial deformation of $\phi^{\natural\natural(N-1)}$ via a suitable
trivialization
$\mathcal{E}_{R}|_{U}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}R\times_{k}\mathcal{E}|_{U}$.
By applying this argument to various points $q$ of $X$, we finish the proof of
this proposition. ∎

The following lemma was used in the proof of the above proposition.

######  Lemma 6.3.3.

Let us keep the notation in the proof of Proposition 6.3.2. Then, the
$p$-curvature of $\nabla_{R}$ satisfies the equality

(500) $\displaystyle\psi(\nabla_{R})=-\varepsilon\cdot
C_{(\mathfrak{g}_{\mathcal{E}}^{[N-1]},(\nabla_{\phi}^{\mathrm{ad}})^{[N-1]})}(\overline{h}^{\nabla})$

under the composite of natural inclusions

(501)
$\displaystyle\Gamma(U^{(N)},\Omega^{(N)}\otimes\mathfrak{g}_{\mathcal{E}}^{[N]})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Hom}_{\mathcal{O}_{U^{(N)}}}(\mathcal{T}^{[N]}|_{U^{(N)}},\mathfrak{g}_{\mathcal{E}}^{[N]}|_{U^{(N)}})$
$\displaystyle\hookrightarrow\mathrm{Hom}_{\mathcal{O}_{U^{(N)}}}(\mathcal{T}^{[N]}|_{U^{(N)}},\mathcal{E}nd_{k}(\mathcal{O}_{\mathcal{E}})|_{U^{(N)}})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Hom}_{\mathcal{O}_{U^{(N-1)}}}(F_{X^{(N-1)}/k}^{*}(\mathcal{T}^{[N]})|_{U^{(N-1)}},\mathcal{E}nd_{k}(\mathcal{O}_{\mathcal{E}})|_{U^{(N-1)}}).$

In particular, if the $p$-curvature of $\nabla_{R}$ vanishes, then
$\overline{h}^{\nabla}$ may be expressed, Zariski locally on $X$, as
$\overline{h}^{\nabla}=(\nabla_{\phi}^{\mathrm{ad}})^{[N-1]}(v)$ for some
local section $v$ of $\mathfrak{g}_{\mathcal{E}}^{[N-1]}$.

###### Proof.

The assertion follows from computations similar to those carried out in the
proof of [Wak6, Proposition 6.11]. So we omit the details of the proof. ∎

######  Remark 6.3.4 (Higher rank cases).

The same assertion as Proposition 6.3.2 also holds for a general rank
$n\left(<p\right)$ not only the case of $n=2$. In fact, it can be proved by a
similar argument because the result of Proposition 4.2.4, which was
essentially used in the proof, can be applied to
$\mathrm{PGL}_{n}^{(N)}$-opers.

For each $R\in\mathrm{Ob}(\mathcal{A}rt_{/k}$), we shall set

(502)
$\displaystyle\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R):=\left(\begin{matrix}\text{the
set of isomorphism classes}\\\ \text{of deformations of
$(\mathscr{X},\mathscr{E}^{\spadesuit})$ over $R$}\end{matrix}\right).$

If $\sigma:R^{\prime}\rightarrow R$ is a morphism in $\mathcal{A}rt_{/k}$,
then the base-change along the induced morphism
$\mathrm{Spec}(R)\rightarrow\mathrm{Spec}(R^{\prime})$ gives a map of sets
$\sigma^{*}:\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R^{\prime})\rightarrow\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R)$.
Thus, the assignments
$R\mapsto\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R)$ and
$\sigma\mapsto\sigma^{*}$ determines a functor

(503)
$\displaystyle\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}:\mathcal{A}rt_{/k}\rightarrow\mathcal{S}et.$

######  Theorem 6.3.5.

The functor $\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}$ is a rigid
deformation functor with good deformation theory, in the sense of [Wak5, §
6.1]. Moreover, the triple

(504)
$\displaystyle(\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})},\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1},\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{2})$

forms a deformation triple, in the sense of [Wak5, Definition 6.1.1], i.e.,
$\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}$ has a tangent-
obstruction theory by putting $\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1}$ as
the tangent space and $\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{2}$ as the
obstruction space. In particular,
$\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}$ is smooth and pro-
representable by
$k[\\![t_{1},\cdots,t_{\mathrm{dim}(\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1})}]\\!]$
(cf. [Wak5, Remark 6.1.3, (i)]).

###### Proof.

First, we shall consider the tangent space of
$\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}$. Let
$\mathfrak{e}:=(0\rightarrow M\rightarrow R\rightarrow R_{0}\rightarrow 0)$ be
a small extension in $\mathcal{A}rt_{/k}$ and
$(\mathscr{X}_{R},\mathscr{E}^{\spadesuit}_{R})$ (where
$\mathscr{X}_{R}:=(X_{R}/R,\\{\sigma_{R,i}\\}_{i})$ and
$\mathscr{E}^{\spadesuit}_{R}:=((\mathcal{E}_{B})_{R},\phi_{R})$) a
deformation of $(\mathscr{X},\mathscr{E}^{\spadesuit})$ over $R$. Set
$\mathcal{E}_{R}:=(\mathcal{E}_{B})_{R}\times^{B}\mathrm{PGL}_{2}$. Also, let
us take an element $v$ of
$M\otimes_{k}\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1}$, which may be
described as $v=(v_{1},v_{2})$ for
$v_{1}\in\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\mathrm{id}_{M}\otimes\widetilde{\nabla}_{\phi,B}^{\mathrm{ad}(0)}])$,
$v_{2}\in
H^{1}(X,M\otimes_{k}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/k^{\mathrm{log}},\phi})$,
where $\mathrm{id}_{M}\otimes\widetilde{\nabla}_{\phi,B}^{\mathrm{ad}(0)}$
denotes the tensor product
$M\otimes_{k}\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}/k^{\mathrm{log}}}\rightarrow
M\otimes_{k}(\Omega\otimes\mathfrak{g}_{\mathcal{E}})$ of $\mathrm{id}_{M}$
and $\widetilde{\nabla}_{\phi,B}^{\mathrm{ad}(0)}$. Note here that
$H^{1}(X,M\otimes_{k}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/k^{\mathrm{log}},\phi})$
can be realized as the cohomology of the Čech complex of
$M\otimes_{k}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/k^{\mathrm{log}},\phi}$
associated with an affine covering. That is to say, we can find an affine open
covering $\mathcal{U}:=\\{U_{\alpha}\\}_{\alpha\in I}$ (where $I$ is a finite
index set) of $X$, and $v$ may be represented by a Čech $1$-cocycle

(505) $\displaystyle
v_{2}=\\{\widetilde{\partial}_{\alpha\beta}\\}_{(\alpha,\beta)\in
I_{2}}\in\check{C}^{1}(\mathcal{U},M\otimes\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/k^{\mathrm{log}},\phi})$

where $I_{2}$ denotes the subset of $I\times I$ consisting of pairs
$(\alpha,\beta)$ with $U_{\alpha\beta}:=U_{\alpha}\cap
U_{\beta}\neq\emptyset$, and
$\widetilde{\partial}_{\alpha\beta}\in\Gamma(U_{\alpha\beta},M\otimes_{k}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/k^{\mathrm{log}},\phi})$.
For each $(\alpha,\beta)\in I_{2}$, we set
$\partial_{\alpha\beta}:=d_{\mathcal{E}_{B}}(\widetilde{\partial}_{\alpha\beta})\in\Gamma(U_{\alpha\beta},M\otimes_{k}\mathcal{T})$.
Also, we shall set
$U_{R,\alpha}^{\mathrm{log}}:=X_{R}^{\mathrm{log}}|_{U_{\alpha}}$ for each
$\alpha\in I$ and set
$U_{R,\alpha\beta}^{\mathrm{log}}:=X_{R}^{\mathrm{log}}|_{U_{\alpha\beta}}$
for each $(\alpha,\beta)\in I_{2}$. The element $\partial_{\alpha\beta}$
(resp., $\widetilde{\partial}_{\alpha\beta}$) may be regarded as a $k$-linear
morphism

(506)
$\displaystyle\partial_{\alpha\beta}:\mathcal{O}_{U_{\alpha\beta}}\rightarrow
M\otimes_{k}\mathcal{O}_{U_{\alpha\beta}}\ \left(\text{resp.,}\
\widetilde{\partial}_{\alpha\beta}:\mathcal{O}_{\mathcal{E}|_{U_{\alpha\beta}}}\rightarrow
M\otimes_{k}\mathcal{O}_{\mathcal{E}|_{U_{\alpha\beta}}}\right)$

satisfying the Leibniz rule. Then,
$\mathrm{id}_{\mathcal{O}_{U_{R,\alpha\beta}}}+\partial_{\alpha\beta}$ (resp.,
$\mathrm{id}_{\mathcal{O}_{\mathcal{E}_{R}|_{U_{R,\alpha\beta}}}}+\widetilde{\partial}_{\alpha\beta}$)
defines a well-defined automorphism $\partial^{\sharp}_{\alpha\beta}$ (resp.,
$\widetilde{\partial}^{\sharp}_{\alpha\beta}$) of
$U_{R,\alpha\beta}^{\mathrm{log}}$ (resp.,
$\mathcal{E}_{R}^{\mathrm{log}}|_{U_{R,\alpha\beta}}$). By means of the
$\partial_{\alpha\beta}^{\sharp}$’s ($(\alpha,\beta)\in I_{2}$), we can glue
together the $U_{R,\alpha}^{\mathrm{log}}$’s ($\alpha\in I$) to obtain a log
curve $X^{\mathrm{log}}_{R,v}$ over $R\times_{k}S^{\mathrm{log}}$. It
determines a deformation $\mathscr{X}_{R,v}$ of $\mathscr{X}$ over $R$ (cf.
[FKa, Theorem 4.1]). Also, the automorphism
$\widetilde{\partial}^{\sharp}_{\alpha\beta}$ is
$\mathrm{PGL}_{2}$-equivariant, and the following square diagram is
commutative and Cartesian:

(511)

For each local section
$D\in\mathcal{D}^{(N-1)}_{U_{R,\alpha\beta}^{\mathrm{log}}/R^{\mathrm{log}}}$,
the following sequence of equalities holds:

(512) $\displaystyle\hskip
14.22636pt(\widetilde{\partial}^{\sharp}_{\alpha\beta})^{*}(\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}})(D)$
$\displaystyle=(\mathrm{id}+\widetilde{\partial}_{\alpha\beta})\circ\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}}(D-[\partial_{\alpha\beta},D])\circ(\mathrm{id}+\widetilde{\partial}_{\alpha\beta})^{-1}$
$\displaystyle=(\mathrm{id}+\widetilde{\partial}_{\alpha\beta})\circ\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}}(D-[\partial_{\alpha\beta},D])\circ(\mathrm{id}-\widetilde{\partial}_{\alpha\beta})$
$\displaystyle=\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}}(D)-\left([\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}}(D),\widetilde{\partial}_{\alpha\beta}]-\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}}([D,\partial_{\alpha\beta}])\right)$
$\displaystyle=\phi_{R}^{\natural\natural}|_{U_{R,\alpha\beta}}(D)$

(cf. [Wak6, Remark 6.2] for the first equality), where the last equality
follows from $\widetilde{\partial}_{\alpha\beta}\in
M\otimes_{k}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/k^{\mathrm{log}},\phi}$.
This means that $\widetilde{\partial}^{\sharp}_{\alpha\beta}$ preserves the
$(N-1)$-PD stratification $\phi_{R}|_{U_{R,\alpha\beta}}$. Hence, the pairs
$(\mathcal{E}_{R}|_{U_{R,\alpha}},\phi_{R}|_{U_{R,\alpha\beta}})$ may be glued
together by means of the $\widetilde{\partial}^{\sharp}_{\alpha\beta}$’s; we
denote the resulting $(N-1)$-PD stratified $\mathrm{PGL}_{2}$-bundle by
$(\mathcal{E}_{R,v},\phi_{R,v})$. By construction, $\phi_{R,v}$ has vanishing
$p^{N}$-curvature.

On the other hand, according to [Wak6, Eq. (792)], the element $v_{1}$
determines a deformation $((\mathcal{E}_{B})_{R,v},(\nabla_{\phi})_{R,v})$
over $R$ of the $\mathrm{PGL}_{2}$-oper $(\mathcal{E}_{B},\nabla_{\phi})$.
Since $v_{1}$ and $v_{2}$ agree under the maps $\mathbb{H}^{1}(\gamma_{1})$
and $\mathbb{H}^{1}(\gamma_{2})$, we see that
$(\mathcal{E}_{B})_{R,v}\times^{B}\mathrm{PGL}_{2}\cong\mathcal{E}_{R,v}$ and
that the $k^{\mathrm{log}}$-connection on $\mathcal{E}_{R,v}$ induced by
$\phi_{R,v}$ coincides with $(\nabla_{\phi})_{R,v}$. Hence,
$\mathscr{E}^{\spadesuit}_{R,v}:=((\mathcal{E}_{B})_{R,v},\phi_{R,v})$ forms a
dormant $\mathrm{PGL}_{2}^{(N)}$-oper on $\mathscr{X}_{R,v}$, and the pair
$(\mathscr{X}_{R,v},\mathscr{E}_{R,v}^{\spadesuit})$ specifies an element of
$\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R)$. By Proposition
6.3.2 and the bijection displayed in [Wak6, Eq. (792)], the assignment
$(\mathscr{X}_{R},\mathscr{E}^{\spadesuit}_{R})\mapsto(\mathscr{X}_{R,v},\mathscr{E}_{R,v}^{\spadesuit})$
(for every $v\in M\otimes_{k}\mathcal{G}^{1}_{\mathscr{E}^{\spadesuit}}$)
defines a transitive action of
$M\otimes_{k}\mathcal{G}^{1}_{\mathscr{E}^{\spadesuit}}$ on the fibers of
$\mathrm{Def}_{(X,\mathscr{E}^{\spadesuit})}(\sigma):\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R)\rightarrow\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}(R_{0})$.
This action is, by construction, verified to be free if $R=k$. Consequently,
$\mathcal{G}^{1}_{\mathscr{E}^{\spadesuit}}$ is isomorphic to the tangent
space of $\mathrm{Def}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}$.

The assertion concerning an obstruction space can be proved by a routine
argument using the Čech cohomological descriptions of relevant deformations as
discussed above, so the proof is left to the reader. (Similar discussions can
be found in the proofs of [Wak5, Propositions 6.4.3 and 6.8.2].) Finally, the
last assertion follows from [Sch, Theorem 2.11] together with the fact that
$\mathrm{dim}(\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1})<\infty$ and
$\mathcal{G}^{2}_{\mathscr{E}^{\spadesuit}}=0$ (cf. Proposition 6.2.4). ∎

As a consequence of the above theorem, we obtain the following assertion.

######  Corollary 6.3.6.

  * (i)

The Deligne-Mumford stack $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ is
smooth over $\mathbb{F}_{p}$.

  * (ii)

Let $k$ be an algebraically closed field over $\mathbb{F}_{p}$, $\mathscr{X}$
an $r$-pointed stable curve of genus $g$ over $k$, and
$\mathscr{E}^{\spadesuit}$ a dormant $\mathrm{PGL}_{2}^{(N)}$-oper on
$\mathscr{X}$. Denote by
$s:\mathrm{Spec}(k)\rightarrow\overline{\mathcal{M}}_{g,r}$ (resp.,
$\widetilde{s}:\mathrm{Spec}(k)\rightarrow\mathcal{O}p_{g,r}^{{}^{\mathrm{Zzz...}}}$)
the morphism classifying $\mathscr{X}$ (resp.,
$(\mathscr{X},\mathscr{E}^{\spadesuit})$). Also, denote by
$T_{s}\overline{\mathcal{M}}_{g,r}$ (resp.,
$T_{\widetilde{s}}\mathcal{O}p_{g,r}^{{}^{\mathrm{Zzz...}}}$) the $k$-vector
space defined as the tangent space of $\overline{\mathcal{M}}_{g,r}$ (resp.,
$\mathcal{O}p_{g,r}^{{}^{\mathrm{Zzz...}}}$) at $s$ (resp., $\widetilde{s}$).
Then, there exists a canonical isomorphism of $k$-vector spaces

(513)
$\displaystyle\mathrm{KS}_{(\mathscr{X},\mathscr{E}^{\spadesuit})}:T_{\widetilde{s}}\mathcal{O}p_{g,r}^{{}^{\mathrm{Zzz...}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1}$

which makes the following diagram commute:

(518)

where $\mathrm{KS}_{\mathscr{X}}$ denotes the Kodaira-Spencer morphism (cf.,
e.g., [Wak6, §6̇.1.1]) and the upper horizontal arrow $d\Pi_{g,r}$ denotes the
differential of the projection $\Pi_{g,r}$ (cf. (306)).

###### Proof.

The assertion follows from Theorem 6.3.5 and the definition of
$D_{\mathscr{E}^{\spadesuit}}$. ∎

###  6.4. Dimension of the moduli space

Let $S$, $\mathscr{X}$, and $\mathscr{E}^{\spadesuit}$ be as introduced at the
beginning of § 6.1. Suppose further that the $\mathrm{PGL}_{2}$-oper induced
from $\mathscr{E}^{\spadesuit}$ is normal, in the sense of [Wak6, Definition
4.53]. (According to [Wak6, Proposition 4.55], any $\mathrm{PGL}_{2}$-oper is
isomorphic to a normal one.) Consider the composite surjection

(519) $\displaystyle
d^{\prime}_{\mathcal{E}}:\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}\twoheadrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}/\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}/S^{\mathrm{log}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}_{\mathcal{E}}/\mathfrak{b}_{\mathcal{E}_{B}}\left(=\mathfrak{g}_{\mathcal{E}}^{-1}/\mathfrak{g}_{\mathcal{E}}^{0}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{T},$

where the second and third arrows are the isomorphisms defined in [Wak6, Eq.
(163)] and [Wak6, Eq. (617)], respectively. By applying the functor
$\mathbb{R}^{1}f_{*}(-)$ to the composite

(520)
$\displaystyle\left(\mathfrak{g}_{\mathcal{E},\phi}=\right)\mathcal{S}ol(\nabla_{\phi}^{\mathrm{ad}})\xrightarrow{\mathrm{inclusion}}\left(\mathfrak{g}_{\mathcal{E}}\hookrightarrow\right)\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}\xrightarrow{d^{\prime}_{\mathcal{E}}}\mathcal{T},$

we obtain an $\mathcal{O}_{S}$-linear morphism

(521) $\displaystyle
D^{\prime}_{\mathscr{E}^{\spadesuit}}:\mathbb{R}^{1}f_{*}(\mathcal{S}ol(\nabla_{\phi}^{\mathrm{ad}}))\rightarrow\mathbb{R}^{1}f_{*}(\mathcal{T}).$

Next, let us consider the $\mathcal{O}_{X}$-linear endomorphism of
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}$
defined as

(522)
$\displaystyle\chi:=\mathrm{id}_{\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}}-\nabla_{\phi}^{(0)}\circ
d_{\mathcal{E}}-\nabla^{(0)}_{\phi}\circ
d^{\prime}_{\mathcal{E}}:\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}\rightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}.$

######  Lemma 6.4.1.

The automorphism $\chi$ of
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}$ is
involutive (i.e., $\chi\circ\chi=\mathrm{id}$), and the following equalities
hold:

(523) $\displaystyle d_{\mathcal{E}}\circ\chi=-d^{\prime}_{\mathcal{E}},\hskip
28.45274ptd^{\prime}_{\mathcal{E}}\circ\chi=-d_{\mathcal{E}}.$

###### Proof.

Since we have assumed that the $\mathrm{PGL}_{2}$-oper induced from
$\mathscr{E}^{\spadesuit}$ is normal, the equality
$d^{\prime}_{\mathcal{E}}\circ\nabla_{\phi}=\mathrm{id}_{\mathcal{T}}$ holds
(cf. [Wak6, Example 2.7 and Remark 2.16]). Hence, the following sequence of
equalities holds:

(524) $\displaystyle\chi\circ\chi$
$\displaystyle=\left(\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}}\right)\circ\left(\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}}\right)$
$\displaystyle=\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}}$
$\displaystyle\hskip 14.22636pt-\nabla_{\phi}\circ
d_{\mathcal{E}}\circ\left(\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}}\right)$
$\displaystyle\hskip 14.22636pt-\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}}\circ\left(\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}}\right)$
$\displaystyle=\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}}$
$\displaystyle\hskip 14.22636pt-\nabla_{\phi}\circ
d_{\mathcal{E}}+\nabla_{\phi}\circ d_{\mathcal{E}}+\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}}$ $\displaystyle\hskip 14.22636pt-\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}}+\nabla_{\phi}\circ d_{\mathcal{E}}+\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}}$ $\displaystyle=\mathrm{id}.$

Moreover, we have

(525) $\displaystyle d_{\mathcal{E}}\circ\chi$
$\displaystyle=d_{\mathcal{E}}\circ(\mathrm{id}-\nabla_{\phi}\circ
d_{\mathcal{E}}-\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}})$
$\displaystyle=d_{\mathcal{E}}-d_{\mathcal{E}}\circ\nabla_{\phi}\circ
d_{\mathcal{E}}-d_{\mathcal{E}}\circ\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}}$
$\displaystyle=d_{\mathcal{E}}-d_{\mathcal{E}}-d^{\prime}_{\mathcal{E}}$
$\displaystyle=-d^{\prime}_{\mathcal{E}}.$

A similar calculation shows
$d^{\prime}_{\mathcal{E}}\circ\chi=-d_{\mathcal{E}}$. This completes the proof
of this lemma. ∎

Moreover, we can prove the following assertion.

######  Proposition 6.4.2.

The automorphism $\chi$ restricts to an automorphism
$\chi_{\phi}:\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}$
of
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}$,
and restricts to an isomorphism
$\chi_{B}:\mathfrak{g}_{\mathcal{E}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}/S^{\mathrm{log}}}$.
That is to say, we obtain the following commutative square diagrams:

(534)

###### Proof.

By the definition of $\nabla_{\phi}$, the equality

(535)
$\displaystyle\phi^{\natural\natural}(D)=\nabla_{\phi}(D)\left(\in\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{\mathcal{E}})\right)$

holds for every local section
$D\in\mathcal{T}\left(\subseteq\mathcal{D}^{(N-1)}\right)$. For local sections
$v\in\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}$
and $D\in\mathcal{D}^{(N-1)}$, the following sequence of equalities holds:

(536) $\displaystyle\hskip
14.22636pt[\phi^{\natural\natural}(D),\chi(v)]-\phi^{\natural\natural}([D,d_{\mathcal{E}}(\chi(v))])$
$\displaystyle=[\phi^{\natural\natural}(D),v-(\nabla_{\phi}\circ
d_{\mathcal{E}})(v)-(\nabla\circ d^{\prime}_{\mathcal{E}})(v)]$
$\displaystyle\hskip
14.22636pt-\phi^{\natural\natural}([D,d_{\mathcal{E}}(v-(\nabla_{\phi}\circ
d_{\mathcal{E}})(v)-(\nabla_{\phi}\circ d^{\prime}_{\mathcal{E}})(v))])$
$\displaystyle=\cancel{[\phi^{\natural\natural}(D),v]}-[\phi^{\natural\natural}(D),(\nabla_{\phi}\circ
d_{\mathcal{E}})(v)]-[\phi^{\natural\natural}(D),(\nabla_{\phi}\circ
d^{\prime}_{\mathcal{E}})(v)]$ $\displaystyle\hskip
14.22636pt-\cancel{\phi^{\natural\natural}([D,d_{\mathcal{E}}(v)])}+\phi^{\natural\natural}([D,d_{\mathcal{E}}(v)])+\phi^{\natural\natural}([D,d^{\prime}_{\mathcal{E}}(v)])$
$\displaystyle=-[\phi^{\natural\natural}(D),\phi^{\natural\natural}(d_{\mathcal{E}}(v))]-[\phi^{\natural\natural}(D),\phi^{\natural\natural}(d^{\prime}_{\mathcal{E}}(v))]+\phi^{\natural\natural}([D,d_{\mathcal{E}}(v)])+\phi^{\natural\natural}([D,d^{\prime}_{\mathcal{E}}(v)])$
$\displaystyle=0,$

where the third equality follows from (535) together with the equality
$[\phi^{\natural\natural}(D),v]-\phi^{\natural\natural}([D,d_{\mathcal{E}}(v)])=0$,
and the fourth equality follows from the fact that $\phi^{\natural\natural}$
preserves the Lie bracket operator $[-,-]$. This implies that
$\chi(v)\in\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}$,
and hence $\chi$ restricts to an automorphism of
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}$.

Moreover, since $\mathrm{Ker}(d_{\mathcal{E}})=\mathfrak{g}_{\mathcal{E}}$ and
$\mathrm{Ker}(d^{\prime}_{\mathcal{E}})=\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{B}/S^{\mathrm{log}}}$,
the equality $d^{\prime}_{\mathcal{E}}\circ\chi=-d_{\mathcal{E}}$ asserted in
Lemma 6.4.1 implies that $\chi$ restricts to an isomorphism
$\mathfrak{g}_{\mathcal{E}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}/S^{\mathrm{log}}}$.
This completes the proof of the assertion. ∎

######  Proposition 6.4.3.

There exists an isomorphism of $\mathcal{O}_{S}$-modules

(537)
$\displaystyle\Xi:\mathbb{R}^{1}f_{*}(\mathcal{S}ol(\nabla_{\phi}^{\mathrm{ad}}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1},$

which makes the following diagram commute:

(542)

###### Proof.

By Proposition 6.4.2, we obtain the following commutative square diagram:

(547)

where $\delta_{1}$ and $\delta_{2}$ denote the natural inclusions
$\mathfrak{g}_{\mathcal{E}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}$
and
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}}}$,
respectively. This induces an isomorphism of $\mathcal{O}_{S}$-modules

(548)
$\displaystyle\mathrm{Ker}(\mathbb{R}^{1}f_{*}(\delta_{1}\oplus(-\delta_{2})))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1}.$

On the other hand, it follows from Lemma 6.2.1, (i), that the commutative
square diagram

(553)

is Cartesian, where the upper horizontal arrow (resp., the left-hand vertical
arrow) arises from the natural inclusion
$\mathfrak{g}_{\mathcal{E},\phi}\hookrightarrow\mathfrak{g}_{\mathcal{E}}$
(resp.,
$\mathfrak{g}_{\mathcal{E},\phi}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}/S^{\mathrm{log}},\phi}$).
Hence, this diagram yields an isomorphism of $\mathcal{O}_{S}$-modules

(554)
$\displaystyle\mathbb{R}^{1}f_{*}(\mathfrak{g}_{\mathcal{E},\phi})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}(\mathbb{R}^{1}f_{*}(\delta_{1}\oplus(-\delta_{2}))).$

By composing (548) and (554), we obtain an isomorphism
$\mathbb{R}^{1}f_{*}(\mathfrak{g}_{\mathcal{E},\phi})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{G}_{\mathscr{E}^{\spadesuit}}^{1}$,
as desired. ∎

######  Corollary 6.4.4.

The Deligne-Mumford stack $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ is
smooth over $\mathbb{F}_{p}$ and equidimensional of dimension $3g-3+r$.
Moreover, the projection $\Pi_{g,r}$ is faithfully flat. In particular,
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}\times_{\overline{\mathcal{M}}_{g,r}}\mathcal{M}_{g,r}$
is nonempty.

###### Proof.

The former assertion follows from Corollary 6.3.6, Propositions 6.1.2, (ii),
and 6.4.3.

Next, suppose that the projection $\Pi_{g,r}$ is not surjective. Since
$\Pi_{g,r}$ is proper (cf. Corollary 4.6.2) and $\overline{\mathcal{M}}_{g,r}$
is irreducible, the image $\mathrm{Im}(\Pi_{g,r})$ of $\Pi_{g,r}$ does not
coincides with the entire space $\overline{\mathcal{M}}_{g,r}$ and has
dimension $<3g-3+r$. In particular, we have
$\mathrm{dim}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r})-\mathrm{dim}(\overline{\mathcal{M}}_{g,r})>0$.
This implies that each fiber of $\Pi_{g,r}$ has positive dimension (cf. [Har,
Chap. II, Exercise 3.22]); it contradicts the finiteness of $\Pi_{g,r}$
asserted in Corollary 4.6.2. Thus, $\Pi_{g,r}$ turns out to be surjective.

The flatness of $\Pi_{g,r}$ follows immediately from [Har, Chap. III, Exercise
10.9] because both $\overline{\mathcal{M}}_{g,r}$,
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ are smooth over $\mathbb{F}_{p}$
and $\Pi_{g,r}$ is finite. Thus, we have finished the proof of the assertion.
∎

###  6.5. Ordinariness

Let $S$, $\mathscr{X}$, and $\mathscr{E}^{\spadesuit}$ be as before.

######  Definition 6.5.1.

We shall say that $\mathscr{E}^{\spadesuit}$ is ordinary if the morphism
$D_{\mathscr{E}^{\spadesuit}}$ (cf. (458)) is an isomorphism.

The following assertion is a direct consequence of Proposition 6.4.3.

######  Proposition 6.5.2.

$\mathscr{E}^{\spadesuit}$ is ordinary if and only if the morphism
$D^{\prime}_{\mathscr{E}^{\spadesuit}}$ (cf. (521)) is an isomorphism.

######  Remark 6.5.3 (Previous definition of ordinariness).

When $\mathscr{X}$ is unpointed and smooth, it follows from Proposition 6.5.2
that the definition of ordinariness introduced here is equivalent to that of
the corresponding indigenous $(\mathrm{PGL}_{2},B)$-bundle, in the sense of
[Wak5, Definition 6.7.1].

Denote by

(555) $\displaystyle\mathcal{O}p_{N,g,r,\mathrm{ord}}^{{}^{\mathrm{Zzz...}}},\
\text{or simply},\mathcal{O}p_{g,r,\mathrm{ord}}^{{}^{\mathrm{Zzz...}}},$

the open substack of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ (i.e.,
$\mathcal{O}p_{N,g,r}^{{}^{\mathrm{Zzz...}}}$) classifying ordinary dormant
$\mathrm{PGL}_{2}^{(N)}$-opers. According to Corollary 6.3.6, this stack
coincides with the étale locus of $\mathcal{O}p_{g,r}^{{}^{\mathrm{Zzz...}}}$
relative to $\overline{\mathcal{M}}_{g,r}$.

######  Proposition 6.5.4.

The projection $\Pi_{N\Rightarrow
1}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}\rightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,g,r}$
obtained by reducing the level of dormant $\mathrm{PGL}_{2}$-opers to $1$
restricts to a morphism
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r,\mathrm{ord}}\rightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,g,r,\mathrm{ord}}$.

###### Proof.

As shown in [Moc2, Chap. II, Theorem 2.8],
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,g,r}$ is irreducible. Hence, by an
argument similar to the proof of Corollary 6.4.4, the projection
$\Pi_{N\Rightarrow 1}$ turns out to be flat. If the projection
$\Pi_{N,g,r,\mathbb{F}_{p}}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}\rightarrow\overline{\mathcal{M}}_{g,r}$
is unramified (or equivalently, étale) at a geometric point $q$ of
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$, then $\Pi_{N\Rightarrow 1}$ is
unramified (or equivalently, étale) at the same point. It follows that the
differential of the projection
$\Pi_{1,g,r,\mathbb{F}_{p}}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,g,r}\rightarrow\overline{\mathcal{M}}_{g,r}$
at $\Pi_{N\Rightarrow 1}(q)$ yields an isomorphism between the respective
tangent spaces. This means that $\Pi_{1,g,r,\mathbb{F}_{p}}$ is étale at
$\Pi_{N\Rightarrow 1}(q)$. So the proof of the assertion is completed. ∎

Next, suppose that $\mathscr{X}$ is unpointed (i.e., $r=0$) and smooth (which
implies $X^{\mathrm{log}}=X$ and $S^{\mathrm{log}}=S$). We shall write
$\mathscr{G}:=(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}}_{\phi})$ for
simplicity. Since the equivalence of categories (104) commutes the formation
of duals, the identification
$(\mathfrak{g}_{\mathcal{E}},\nabla_{\phi}^{\mathrm{ad}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathfrak{g}_{\mathcal{E}}^{\vee},\nabla_{\phi}^{\mathrm{ad}\vee})$
induced from the Killing form on $\mathfrak{g}$ gives an identification
$\mathfrak{g}_{\mathcal{E},\phi}=\mathfrak{g}_{\mathcal{E},\phi}^{\vee}$.
Under these identifications, the diagram (140) where “$(\mathcal{F},\nabla)$”
is taken to be $\mathscr{G}$ defines a commutative square diagram

(560)

Let us consider the composite

(561) $\displaystyle\Omega^{\otimes
2}\hookrightarrow\Omega\otimes\mathfrak{g}_{\mathcal{E}}\xrightarrow{C_{\mathscr{G}}}\Omega^{(N)}\otimes\mathfrak{g}_{\mathcal{E},\phi},$

where the first arrow denotes the morphism induced by the inclusion
$\mathfrak{g}^{1}_{\mathcal{E}}\hookrightarrow\mathfrak{g}_{\mathcal{E}}$
under the natural identification
$\left(\mathfrak{g}^{1}_{\mathcal{E}}/\mathfrak{g}^{2}_{\mathcal{E}}=\right)\mathfrak{g}_{\mathcal{E}}^{1}=\Omega$
(cf. [Wak6, Eq. (617)]). By applying the functor $\mathbb{R}^{1}f_{*}(-)$ to
this composite, we obtain an $\mathcal{O}_{S}$-linear morphism

(562) $\displaystyle
D^{\prime\prime}_{\mathscr{E}^{\spadesuit}}:f_{*}(\Omega^{\otimes
2})\rightarrow f_{*}(\Omega^{(N)}\otimes\mathfrak{g}_{\mathcal{E},\phi})).$

######  Proposition 6.5.5.

The following square diagram is commutative up to multiplication by an
invertible factor in $\mathbb{F}_{p}$:

(567)

where $\int$ denotes the isomorphism obtained by applying Grothendieck-Serre
duality to $\Omega$. In particular, $\mathscr{E}^{\spadesuit}$ is ordinary if
and only if $D^{\prime\prime}_{\mathscr{E}^{\spadesuit}}$ is an isomorphism.

###### Proof.

Consider the diagram

(572)

where the upper and lower horizontal arrows arise from the isomorphisms
$\Omega\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}_{\mathcal{E}}^{1}\left(\hookrightarrow\mathfrak{g}_{\mathcal{E}}\right)$
and
$\left(\mathfrak{g}_{\mathcal{E}}\twoheadrightarrow\right)\mathfrak{g}_{\mathcal{E}}/\mathfrak{g}_{\mathcal{E}}^{0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{T}$,
respectively, defined in [Wak6, Eq. (617)]. By the definition of the Killing
form on $\mathfrak{g}$ (i.e.,
$(a,b)\mapsto\frac{1}{4}\cdot\mathrm{tr}(\mathrm{ad}(a)\cdot\mathrm{ad}(b))\left(=\mathrm{tr}(ab)\right)$
for $a,b\in\mathfrak{g}$), this diagram is commutative up to multiplication by
an invertible factor. Hence, the assertion follows from the commutativity (in
this sense) of this diagram and that of (560). The latter assertion follows
from the former assertion and Proposition 6.5.2. ∎

###  6.6. Dormant $\mathrm{PGL}_{2}^{(N)}$-opers on a $3$-pointed projective
line

The goal of this subsection is to show that the stack
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,0,3}$ defined for each
$\rho\in(\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm\\}$ is, if it is
nonempty, isomorphic to $\mathrm{Spec}(\mathbb{F}_{p})$ (cf. Theorem 6.6.5).
Since we have already obtained the finiteness and smoothness of the moduli
space (cf. Theorem 5.3.6 and Corollary 6.3.6, (i)), the problem is reduced to
proving that $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,0,3}$ consists of
exactly one point in the set-theoretic sense. So the desired assertion follows
directly from [Wak7, Proposition 6.4.1]. However, in this subsection, we give
an alternative (somewhat simpler) proof of
“$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,0,3}\cong\mathrm{Spec}(\mathbb{F}_{p})$”
without applying the finiteness and smoothness, which were actually proved by
technically complicated arguments. We do so by slightly generalizing the
discussion in [Wak7].

We shall consider the case where $\mathscr{X}$ is the $3$-pointed projective
line $\mathscr{P}:=(\mathbb{P}/S,\\{[0],[1],[\infty]\\})$, introduced in
(390). Let $k$ be an algebraically closed field over $\mathbb{F}_{p}$, and
suppose that $S=\mathrm{Spec}(R)$ for $R\in\mathrm{Ob}(\mathcal{A}rt_{/k})$.
To clarify the base space “$\mathrm{Spec}(R)$”, we occasionally write
$\mathbb{P}_{R}$ and $\mathscr{P}_{R}$ instead of $\mathbb{P}$ and
$\mathscr{P}$, respectively.

Let us take a triple $(\rho_{0},\rho_{1},\rho_{\infty})$ of elements of
$(\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm 1\\}$. Then, there exists a
triple of integers $(\lambda_{0},\lambda_{1},\lambda_{\infty})$ satisfying the
following conditions:

  * •

$2\cdot\rho_{x}=\lambda_{x}$ as elements of
$(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm 1\\}$ and $0<\lambda_{x}<p^{N}$ for every
$x=0,1,\infty$;

  * •

The sum $\lambda_{0}+\lambda_{1}+\lambda_{\infty}$ is odd $<2\cdot p^{N}$.

Set
$\mathcal{O}_{\mathbb{P}}^{+}:=\mathcal{O}_{\mathbb{P}}(\lambda_{0}\cdot[0]+\lambda_{1}\cdot[1]+\lambda_{\infty}\cdot[\infty])$.
Note that we can find a unique $\mathcal{D}^{(N-1)}$-module structure
$\nabla^{+}$ on $\mathcal{O}_{\mathbb{P}}^{+}$ whose restriction to
$U:=\mathbb{P}\setminus\\{[0],[1],[\infty]\\}$ coincides with the trivial
$\mathcal{D}_{U/S}^{(N-1)}$-module structure on
$\mathcal{O}_{\mathbb{P}}^{+}|_{U}=\mathcal{O}_{U}$. Also, let $\mathcal{L}$
be a line bundle on $\mathbb{P}$ of relative degree
$\frac{\lambda_{0}+\lambda_{1}+\lambda_{\infty}+1}{2}$, which is uniquely
determined up to isomorphism because $R$ is a local ring. Let us fix an
identification $\mathcal{L}^{\otimes
2}\otimes\mathcal{T}=\mathcal{O}_{\mathbb{P}}^{+}$. Under this identification,
the pair $\vartheta:=(\mathcal{L},\nabla^{+})$ form a dormant $2^{(N)}$-theta
characteristic of $\mathbb{P}^{\mathrm{log}}/S$.

Now, let $\mathscr{E}^{\spadesuit}$ be a dormant $\mathrm{PGL}_{2}^{(N)}$-oper
on $\mathscr{P}$ of radii $(\rho_{0},\rho_{1},\rho_{\infty})$, i.e., whose
radius at $[x]$ ($x\in\\{0,1,\infty\\}$) coincides with $\rho_{x}$. We shall
write

(573)
$\displaystyle\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{j=0}^{2})$

for the dormant $\mathrm{GL}_{2}^{(N)}$-oper determined by a unique (up to
isomorphism) dormant $(\mathrm{GL}_{2}^{(N)},\vartheta)$-oper
$\nabla^{\diamondsuit}$ on $\mathscr{P}$ corresponding to
$\mathscr{E}^{\spadesuit}$ via the isomorphism
$\Lambda_{\diamondsuit\Rightarrow\spadesuit}^{{}^{\mathrm{Zzz...}}}$ (cf.
Theorem 4.5.1). That is to say, we set
$\mathscr{F}^{\heartsuit}:=\nabla^{\diamondsuit\Rightarrow\heartsuit}$ (hence
$\mathcal{F}=\mathcal{F}_{\mathcal{L}}$ and $\nabla=\nabla^{\diamondsuit}$).
In particular, since $\mathcal{F}^{1}=\mathcal{L}$, we can regard
$\mathcal{L}$ as a line subbundle of $\mathcal{F}$.

######  Definition 6.6.1.

We shall refer to $\mathscr{F}^{\heartsuit}$ as the canonical dormant
$\mathrm{GL}_{2}^{(N)}$-oper associated to $\mathscr{E}^{\spadesuit}$. (Note
that the formation of $\mathscr{F}^{\heartsuit}$ depends on the choice of
$(\lambda_{0},\lambda_{1},\lambda_{\infty})$, but its isomorphism class does
not depend on the choice of the identification $\mathcal{L}^{\otimes
2}\otimes\mathcal{T}=\mathcal{O}_{\mathbb{P}}^{+}$.)

######  Remark 6.6.2 (Previous works on canonical
$\mathrm{GL}_{2}^{(N)}$-opers).

The construction of the canonical $\mathrm{GL}_{2}$-opers was discussed in
[Moc2, Chap. IV, § 2.1] (for $S=\mathrm{Spec}(k)$ and $N=1$) and [Wak7, § 6.3]
(for $S=\mathrm{Spec}(k)$) to establish a bijective correspondence between
dormant $\mathrm{PGL}_{2}$-opers on $\mathscr{P}$ and certain tamely ramified
coverings on $\mathscr{P}$ (i.e., dynamical Belyi maps).

We shall write
$\mathcal{F}^{\flat}:=F^{(N)*}_{\mathbb{P}/S}(\mathcal{S}ol(\nabla))$ and
write

(574) $\displaystyle\tau:\mathcal{F}^{\flat}\hookrightarrow\mathcal{F}$

for the $\mathcal{O}_{\mathbb{P}}$-linear injection extending the
$\mathcal{O}_{X^{(N)}}$-linear inclusion
$\mathcal{S}ol(\nabla)\hookrightarrow\mathcal{F}$; this morphism commutes with
the $\mathcal{D}^{(N-1)}$-module structures
$\nabla^{(N-1)}_{\mathcal{S}ol(\nabla),\mathrm{can}}$ (cf. (103)) and
$\nabla$. Next, we shall set
$\mathcal{L}^{\flat}:=\mathcal{L}\cap\mathrm{Im}(\tau)$. Since the restriction
of $\tau$ to $U:=\mathbb{P}\setminus\\{[0],[1],[\infty]\\}$ is an isomorphism
(cf. (104)), the quotient sheaf $\mathcal{L}/\mathcal{L}^{\flat}$ is a torsion
sheaf supported on $\\{[0],[1],[\infty]\\}$.

######  Lemma 6.6.3.

Let us keep the above notation.

  * (i)

$\mathcal{L}/\mathcal{L}^{\flat}$ is flat over $S$, and the stalk of
$\mathcal{L}/\mathcal{L}^{\flat}$ at the marked point
$x\in\\{[0],[1],[\infty]\\}$ is of rank $\lambda_{x}$ (as a free
$\mathcal{O}_{S}$-module). Moreover, the injection

(575)
$\displaystyle\iota:\mathcal{L}/\mathcal{L}^{\flat}\hookrightarrow\mathrm{Coker}(\tau)$

induced from the inclusion $\mathcal{L}\hookrightarrow\mathcal{F}$ is an
isomorphism.

  * (ii)

There exists an $\mathcal{O}_{\mathbb{P}^{(N)}}$-linear isomorphism
$\gamma:\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{S}ol(\nabla)$.

###### Proof.

In what follows, we shall use the notation $(-)_{k}$ to denote the result of
reducing objects and morphisms over $R$ modulo $\mathfrak{m}_{R}$.

First, let us consider assertion (i). By Proposition 6.3.2 (and the
isomorphism
$\Lambda_{\diamondsuit\Rightarrow\spadesuit}^{{}^{\mathrm{Zzz...}}}$),
$(\mathcal{F},\nabla)$ is locally isomorphic to the base-change of
$(\mathcal{F}_{k},\nabla_{k})$ over $R$. Hence, $\mathrm{Coker}(\tau)$ can be
locally identified with the base-change of $\mathrm{Coker}(\tau_{k})$. This
implies that $\mathrm{Coker}(\tau)$ is flat over $S$. Recall from [Wak7, Lemma
6.3.2] that the reduction modulo $\mathfrak{m}_{R}$ of (575) is an
isomorphism. By Nakayama’s lemma, the morphism (575) turns out to be an
isomorphism. In particular, $\mathcal{L}/\mathcal{L}^{\flat}$ is flat over
$S$, and the rank of its stalk at $x$ (as a free $\mathcal{O}_{S}$-module)
coincides with the rank of its reduction modulo $\mathfrak{m}_{R}$, i.e.,
$\lambda_{x}$. This completes the proof of assertion (i).

Next, we shall consider assertion (ii). Since $(\mathcal{F},\nabla)$ is
locally isomorphic to the base-change of $(\mathcal{F}_{k},\nabla_{k})$ (as
mentioned above), the vector bundle $\mathcal{S}ol(\nabla)$ on
$\mathbb{P}^{(N)}$ is locally isomorphic to the base-change of
$\mathcal{S}ol(\nabla_{k})$. Hence, $\mathcal{S}ol(\nabla)$ specifies a
deformation of $\mathcal{S}ol(\nabla_{k})$ over $R$. Here, let us observe the
following fact from well-known generalities of deformation theory: if we are
given a small extension in $\mathcal{A}rt_{/k}$ of the form $0\rightarrow
k\rightarrow R_{1}\rightarrow R_{0}\rightarrow 0$ and a deformation
$\mathcal{S}ol(\nabla_{k})_{R_{0}}$ of $\mathcal{S}ol(\nabla_{k})$ over
$R_{0}$, then the set of deformations of $\mathcal{S}ol(\nabla_{k})_{R_{0}}$
over $R_{1}$ has a structure of torsor modeled on
$H^{1}(\mathbb{P}_{k}^{(N)},\mathcal{E}nd(\mathcal{S}ol(\nabla_{k})))$. On the
other hand, $\mathcal{S}ol(\nabla_{k})$ is isomorphic to
$\mathcal{O}_{\mathbb{P}_{k}^{(N)}}^{\oplus 2}$ as proved in [Wak7, Lemma
6.3.3], so we have

(576) $\displaystyle
H^{1}(\mathbb{P}_{k}^{(N)},\mathcal{E}nd(\mathcal{S}ol(\nabla_{k})))\cong
H^{1}(\mathbb{P}^{(N)}_{k},\mathcal{E}nd(\mathcal{O}_{\mathbb{P}_{k}^{(N)}}^{\oplus
2}))\cong
H^{1}(\mathbb{P}_{k}^{(N)},\mathcal{O}_{\mathbb{P}_{k}^{(N)}})^{\oplus 4}=0.$

This implies that any deformation of $\mathcal{S}ol(\nabla_{k})$ over $R$ is
trivial. In particular, there exists an isomorphism
$\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{S}ol(\nabla)$. ∎

The following assertion is a slight generalization of [Wak7, Proposition
6.4.1]. (The proof is entirely similar.)

######  Proposition 6.6.4.

Let $\rho:=(\rho_{0},\rho_{1},\rho_{\infty})$ be an element of
$((\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm 1\\})^{\times 3}$. Then, any
two dormant $\mathrm{PGL}_{2}^{(N)}$-opers on $\mathscr{P}$ of radii $\rho$
are isomorphic.

###### Proof.

Suppose that we are given two dormant $\mathrm{PGL}_{2}^{(N)}$-opers
$\mathscr{E}_{\circ}^{\spadesuit}$, $\mathscr{E}^{\spadesuit}_{\bullet}$ on
$\mathscr{P}$ of radii $\rho$. For each $\star\in\\{\circ,\bullet\\}$, denote
by
$\mathscr{F}^{\heartsuit}_{\star}:=(\mathcal{F}_{\star},\nabla_{\star},\\{\mathcal{F}^{j}_{\star}\\}_{j})$
the canonical dormant $\mathrm{GL}_{n}^{(N)}$-oper associated to
$\mathscr{E}^{\spadesuit}_{\star}$. As mentioned previously, the inclusion
$\mathcal{S}ol(\nabla_{\star})\hookrightarrow\mathcal{F}_{\star}$ extends to
an $\mathcal{O}_{\mathbb{P}}$-linear injection
$\tau_{\star}:F^{(N)*}_{\mathbb{P}/S}(\mathcal{S}ol(\nabla_{\star}))\hookrightarrow\mathcal{F}_{\star}$,
and we obtain
$\mathcal{L}_{\star}^{\flat}:=\mathcal{L}_{\star}\cap\mathrm{Im}(\tau_{\star})$
by regarding $\mathcal{L}_{\star}:=\mathcal{L}$ as a line subbundle of
$\mathcal{F}_{\star}$. The morphism
$\iota_{\star}:\mathcal{L}_{\star}/\mathcal{L}_{\star}^{\flat}\rightarrow\mathrm{Coker}(\tau_{\star})$
induced by $\mathcal{L}_{\star}\hookrightarrow\mathcal{F}_{\star}$ is an
isomorphism (cf. Lemma 6.6.3, (i)). Write $\nabla_{\star}^{(0)}$ for the
$\mathcal{D}^{(0)}$-module structure on $\mathcal{F}_{\star}$ induced from
$\nabla_{\star}$. Then, the collection
$\mathscr{F}_{\star}^{\heartsuit(1)}:=(\mathcal{F}_{\star},\nabla_{\star}^{(0)},\\{\mathcal{F}_{\star}^{j}\\}_{j})$
defines a dormant $(\mathrm{GL}_{2}^{(1)},\vartheta^{(1)})$-oper, where
$\vartheta^{(1)}$ denotes the $2^{(1)}$-theta characteristic obtained by
reducing the level of $\vartheta$ to $1$. Recall from [Moc2, Chap. I, Theorem
4.4] that dormant $\mathrm{PGL}_{2}$-opers on $\mathscr{P}$ are completely
determined by their radii. In particular, since
$(\mathscr{F}^{\heartsuit(1)}_{\circ})^{\Rightarrow\spadesuit}$ and
$(\mathscr{F}^{\heartsuit(1)}_{\bullet})^{\Rightarrow\spadesuit}$ have the
same radii (cf. Proposition 5.3.3, (i)), we have
$(\mathscr{F}^{\heartsuit(1)}_{\circ})^{\Rightarrow\spadesuit}\cong(\mathscr{F}^{\heartsuit(1)}_{\bullet})^{\Rightarrow\spadesuit}$.
It follows that there exists an isomorphism

(577)
$\displaystyle\alpha:\mathscr{F}^{(1)\heartsuit}_{\circ}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{F}^{(1)\heartsuit}_{\bullet}$

(cf. Theorem 4.5.1). This isomorphism restricts to an isomorphism
$\alpha|_{\mathcal{L}_{\circ}}:\mathcal{L}_{\circ}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}_{\bullet}$,
and hence induces an isomorphism
$\alpha|_{\mathcal{L}_{\circ}/\mathcal{L}^{\flat}_{\circ}}:\mathcal{L}_{\circ}/\mathcal{L}_{\circ}^{\flat}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}_{\bullet}/\mathcal{L}^{\flat}_{\bullet}$.
Thus, we obtain the composite isomorphism

(578)
$\displaystyle\alpha|_{\mathrm{Coker}(\tau_{\circ})}:=\iota_{\bullet}\circ\alpha|_{\mathcal{L}_{\circ}/\mathcal{L}^{\flat}_{\circ}}\circ\iota_{\circ}^{-1}:\mathrm{Coker}(\tau_{\circ})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Coker}(\tau_{\bullet}).$

In what follows, we shall prove the commutativity of the following square
diagram:

(583)

where $\pi_{\star}$ ($\star\in\\{\circ,\bullet\\}$) denotes the natural
projection
$\mathcal{F}_{\star}\twoheadrightarrow\mathrm{Coker}(\tau_{\star})$. Let us
take $\star\in\\{\circ,\bullet\\}$ and $x\in\\{[0],[1],[\infty]\\}$. Also,
choose a local function $t$ on $\mathbb{P}$ defining $x$. The formal
neighborhood $\widehat{U}_{x}$ of $x$ in $\mathbb{P}$ may be identified with
$U_{\oslash}$ (cf. (195)). By Proposition 3.3.1 and Lemma 6.6.3, one may
verify that the restriction
$(\mathcal{F}_{\star},\nabla_{\star})|_{\widehat{U}_{x}}$ of
$(\mathcal{F}_{\star},\nabla_{\star})$ to $\widehat{U}_{x}$ is isomorphic to
$\mathscr{O}_{\oslash,\overline{\lambda}_{x}}^{(N-1)}\oplus\mathscr{O}_{\oslash,0}^{(N-1)}$,
where $\overline{\lambda}_{x}$ denotes the image of $\lambda_{x}$ via the
natural quotient $\mathbb{Z}\twoheadrightarrow\mathbb{Z}/p^{N}\mathbb{Z}$. Let
us fix an identification
$(\mathcal{F}_{\star},\nabla_{\star})|_{\widehat{U}_{x}}=\mathscr{O}_{\oslash,\overline{\lambda}_{x}}^{(N-1)}\oplus\mathscr{O}_{\oslash,0}^{(N-1)}$.
Under this identification, the restriction of $\alpha$ to $\widehat{U}_{x}$
determines an automorphism $\alpha_{\widehat{U}_{x}}$ of
$\mathscr{O}_{\oslash,\overline{\lambda}_{x}}^{(N-1)}\oplus\mathscr{O}_{\oslash,0}^{(N-1)}$.
Here, we shall use the notation $\mathrm{mult}_{(-)}$ to denote the
endomorphism of $\mathcal{O}_{\oslash}$ given by multiplication by $(-)$.
Since $\overline{\lambda}_{x}\neq 0$, the automorphism
$\alpha|_{\widehat{U}_{x}}$ may be expressed as
$\mathrm{mult}_{v}\oplus\mathrm{mult}_{w}$ for some
$v,w\in\mathcal{O}_{\oslash}^{\times}$ after possibly replacing the fixed
identification
$(\mathcal{F}_{\star},\nabla_{\star})|_{\widehat{U}_{x}}=\mathscr{O}_{\oslash,\overline{\lambda}_{x}}^{(N-1)}\oplus\mathscr{O}_{\oslash,0}^{(N-1)}$
with another (cf. Proposition 3.3.4, (ii)). By Proposition 5.1.3, (i), the
inclusion $\mathcal{L}_{\star}\hookrightarrow\mathcal{F}_{\star}$ corresponds,
after choosing a suitable trivialization
$\Gamma(\widehat{U}_{x},\mathcal{L}_{\star}|_{\widehat{U}_{x}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{\oslash}$,
to the $\mathcal{O}_{\oslash}$-linear morphism
$\mathcal{O}_{\oslash}\rightarrow\mathcal{O}_{\oslash}^{\oplus 2}$ given by
$1\mapsto(u_{\star},1)$ for some $u_{\star}\in\mathcal{O}_{\oslash}^{\times}$.
Then, the restriction of $\alpha|_{\mathcal{L}_{\star}}$ to $\widehat{U}_{x}$
may be expressed as $\mathrm{mult}_{w}$, and the equality $v\cdot
u_{\circ}=w\cdot u_{\bullet}$ holds. Hence, for each
$(g,h)\in\mathcal{O}_{\oslash}^{\oplus
2}\left(=\mathcal{F}_{\circ}|_{\widehat{U}_{x}}\right)$, we have

(584) $\displaystyle(\pi_{\bullet}\circ\alpha)((g,h))$
$\displaystyle=\pi_{\bullet}((v\cdot g,w\cdot h))$
$\displaystyle=\left(\frac{v\cdot g}{u_{\bullet}}\cdot
u_{\bullet},\frac{v\cdot g}{u_{\bullet}}\cdot 1\right)\text{mod}\
\mathrm{Im}(\tau_{\bullet})$ $\displaystyle=\iota_{\bullet}\left(\frac{v\cdot
g}{u_{\bullet}}\ \mathrm{mod}\ \mathcal{L}^{\flat}_{\bullet}\right)$
$\displaystyle=(\iota_{\bullet}\circ\alpha|_{\mathcal{L}_{\circ}/\mathcal{L}^{\flat}_{\circ}})\left(\frac{v\cdot
g}{wu_{\bullet}}\ \mathrm{mod}\ \mathcal{L}^{\flat}_{\circ}\right)$
$\displaystyle=(\alpha|_{\mathrm{Coker}(\tau_{\circ})}\circ\iota_{\circ})\left(\frac{g}{u_{\circ}}\
\mathrm{mod}\ \mathcal{L}^{\flat}_{\circ}\right)$
$\displaystyle=\alpha|_{\mathrm{Coker}(\tau_{\circ})}\left(\left(\frac{g}{u_{\circ}}\cdot
u_{\circ},\frac{g}{u_{\circ}}\cdot 1\right)\text{mod}\
\mathrm{Im}(\tau_{\circ})\right)$
$\displaystyle=(\alpha|_{\mathrm{Coker}(\tau_{\circ})}\circ\pi_{\circ})((g,h)).$

This shows the desired commutativity of (583).

Moreover, the commutativity of (583) just proved implies that $\alpha$ is
restricted, via $\iota_{\circ}$ and $\iota_{\bullet}$, to an isomorphism

(585)
$\displaystyle\alpha^{\prime}:F^{(N)*}_{\mathbb{P}/S}(\mathcal{S}ol(\nabla_{\circ}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{(N)*}_{\mathbb{P}/S}(\mathcal{S}ol(\nabla_{\bullet})).$

Since
$\mathcal{S}ol(\nabla_{\circ})\cong\mathcal{S}ol(\nabla_{\bullet})\cong\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2}$ (cf. Lemma 6.6.3, (ii)), the morphism

(586)
$\displaystyle\mathrm{Hom}_{\mathcal{O}_{\mathbb{P}^{(N)}}}(\mathcal{S}ol(\nabla_{\circ}),\mathcal{S}ol(\nabla_{\bullet}))\rightarrow\mathrm{End}_{\mathcal{O}_{\mathbb{P}}}(F^{(N)*}_{\mathbb{P}/S}(\mathcal{S}ol(\nabla_{\circ})),F^{(N)*}_{\mathbb{P}/S}(\mathcal{S}ol(\nabla_{\bullet})))$

arising from pull-back by $F^{(N)}_{\mathbb{P}/S}$ is bijective. In
particular, $\alpha^{\prime}$ comes from an isomorphism
$\mathcal{S}ol(\nabla_{\circ})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{S}ol(\nabla_{\bullet})$,
and hence $\alpha^{\prime}$ is compatible with the respective
$\mathcal{D}^{(N-1)}$-actions
$\nabla^{(N-1)}_{\mathcal{S}ol(\nabla_{\circ}),\mathrm{can}}$,
$\nabla^{(N-1)}_{\mathcal{S}ol(\nabla_{\bullet}),\mathrm{can}}$. Since
$\nabla_{\star}$ ($\star\in\\{\circ,\bullet\\}$) is the unique
$\mathcal{D}^{(N-1)}$-module structure on $\mathcal{F}_{\star}$ extending
$\nabla^{(N-1)}_{\mathcal{S}ol(\nabla_{\star}),\mathrm{can}}$ via
$\tau_{\star}$, the isomorphism $\alpha$, being an extension of
$\alpha^{\prime}$, preserves the $\mathcal{D}^{(N-1)}$-module structure. It
follows that $\alpha$ defines an isomorphism of $\mathrm{GL}_{2}^{(N)}$-opers
$\mathscr{F}^{\heartsuit}_{\circ}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{F}^{\heartsuit}_{\bullet}$.
Thus, we obtain
$\mathscr{E}_{\circ}^{\spadesuit}\cong\mathscr{E}_{\bullet}^{\spadesuit}$, as
desired. ∎

######  Theorem 6.6.5.

Let $\rho\in((\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm 1\\})^{\times 3}$.
Then, the stack $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,0,3}$ is either
empty or isomorphic to $\mathrm{Spec}(\mathbb{F}_{p})$. In particular, any
dormant $\mathrm{PGL}_{2}^{(N)}$-oper on $\mathscr{P}$ is ordinary, i.e., we
have
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{0,3,\mathrm{ord}}=\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{0,3}$.

###### Proof.

The assertion follows from Proposition 6.6.4. ∎

######  Remark 6.6.6 (When does
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,0,3}$ become nonempty?).

It is an important problem to determines for what elements
$\rho\in((\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm 1\\})^{\times 3}$ the
associated moduli stacks $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,0,3}$ are
nonempty. The complete answer to this problem for $N=1$ was given in [Moc2,
Introduction, Theorem 1.3] (see also [Wak6, § 7.8.1]). But, little is known
about the general case at the time of writing the present paper. In a
subsequent study in this series of papers, we will investigate that problem
(cf. [Wak9]).

######  Corollary 6.6.7.

For each $N\in\mathbb{Z}_{>0}$, denote by
$\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r})$ the set of connected
components of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$. Let us consider
the projective system of sets

(587)
$\displaystyle\cdots\rightarrow\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r})\rightarrow\cdots\rightarrow\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,g,r})\rightarrow\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,g,r})$

arising from (307). Then, the projective limit

(588)
$\displaystyle\varprojlim_{N\in\mathbb{Z}_{>0}}\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r})$

of this system is nonempty.

###### Proof.

Let us take an $r$-pointed totally degenerate curve $\mathscr{X}$ of genus $g$
over an algebraically closed field $k$ in characteristic $p$. Choose clutching
data
$\mathbb{G}:=({{}^{\rotatebox[origin={c}]{-20.0}{$\vartriangle$}}}G,\\{(g_{j},r_{j})\\}_{j=1}^{J},\\{\lambda_{j}\\}_{j=1}^{J})$
that is compatible with $\mathscr{X}$ (cf. Definition 5.5.1). In particular,
$\mathscr{X}$ may be obtained by gluing together $J$ copies of $\mathscr{P}$
by means of $\mathbb{G}$. Denote by
$\rho_{\mathbb{G}}:=\\{\rho^{j}\\}_{j=1}^{J}$ the set of
$\mathbb{G}$-$\Xi_{2,N}$-radii defined by
$\rho^{j}:=\rho_{N,\circledcirc}^{\times 3}$ (cf. (402)). By Theorem 6.6.5 and
the equivalence of categories (377), there exists a unique (up to isomorphism)
dormant $\mathbb{G}\text{-}\mathrm{PGL}_{2}^{(N)}$-oper of radii
$\rho_{\mathbb{G}}$ on $\mathscr{X}$; we shall denote by
$\mathscr{E}^{\spadesuit}_{\mathscr{X},N}$ the corresponding dormant
$\mathrm{PGL}_{2}^{(N)}$-oper on $\mathscr{X}$ via the equivalence of
categories (376). Also, denote by $\mathfrak{I}_{N}$ the connected component
of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$ containing the point $s_{N}$
classifying $(\mathscr{X},\mathscr{E}^{\spadesuit}_{\mathscr{X},N})$. For each
$N^{\prime}\leq N$, the equality $q_{N\Rightarrow
N^{\prime}}(\rho_{N,\circledcirc}^{\times
3})=\rho_{N^{\prime},\circledcirc}^{\times 3}$ holds (cf. (366) for the
definition of $q_{N\Rightarrow N^{\prime}}$). This implies that the dormant
$\mathrm{PGL}_{2}^{(N^{\prime})}$-oper obtained from
$\mathscr{E}^{\spadesuit}_{\mathscr{X},N}$ by reducing its level is isomorphic
to $\mathscr{E}^{\spadesuit}_{\mathscr{X},N^{\prime}}$. In other words, the
projection
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}\rightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N^{\prime},g,r}$
sends $s_{N}$ to $s_{N^{\prime}}$. Since the stacks $\mathfrak{I}_{N}$ are
irreducible because of their smoothness proved in Corollary 6.3.6, (i), the
projective system (307) restricts to a projective system

(589)
$\displaystyle\cdots\rightarrow\mathfrak{I}_{N}\rightarrow\cdots\rightarrow\mathfrak{I}_{2}\rightarrow\mathfrak{I}_{1}.$

In particular, this system defines an element of the set
$\varprojlim_{N}\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r})$, so we
conclude that
$\varprojlim_{N}\pi_{0}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r})\neq\emptyset$,
as desired. ∎

###  6.7. Generic étaleness of the moduli space

When $N=1$, the generic étaleness of
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}/\overline{\mathcal{M}}_{g,r}$ was
already shown in [Moc2, Chap. II, Theorem 2.8]. By using Theorem 6.6.5, we can
prove the following higher-level generalization.

######  Theorem 6.7.1.

The stack $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r,\mathrm{ord}}$ is a dense
open substack of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ containing all
the points lying over the points in $\overline{\mathcal{M}}_{g,r}$ classifying
totally degenerate curves. In particular, the projection
$\Pi_{g,r}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}\\!\rightarrow\overline{\mathcal{M}}_{g,r}$
is generically étale.

###### Proof.

First, we shall prove the former assertion. Let $k$ be an algebraically closed
field over $\mathbb{F}_{p}$ and $\rho$ an element of $\Xi_{2,N}^{\times r}$.
Also, let us take an arbitrary $r$-pointed totally degenerate curve
$\mathscr{X}$ of genus $g$ over $k$; there exists clutching data
$\mathbb{G}=({{}^{\rotatebox[origin={c}]{-20.0}{$\vartriangle$}}}G,\\{(g_{j},r_{j})\\}_{j=1}^{J},\\{\lambda_{j}\\}_{j=1}^{J})$
that is compatible with $\mathscr{X}$. In particular, the equality
$(g_{j},r_{j})=(0,3)$ holds for every $j=1,\cdots,J$, and the point
$q_{\mathbb{G}}$ of $\overline{\mathcal{M}}_{g,r}$ determined by the morphism

(590)
$\displaystyle\mathrm{Clut}_{\mathbb{G}}:\left(\mathrm{Spec}(\mathbb{F}_{p})=\right)\prod_{j=1}^{J}\overline{\mathcal{M}}_{g_{j},r_{j}}\rightarrow\overline{\mathcal{M}}_{g,r}$

(cf. (382)) classifies $\mathscr{X}$. By applying Theorem 5.4.7, we obtain the
Cartesian square diagram (389) for $\mathbb{G}$ and $\rho$. It follows from
Theorem 6.6.5 that the left-hand vertical arrow in this diagram is unramified.
Hence, the projection
$\Pi_{\rho,g,r}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,g,r}\rightarrow\overline{\mathcal{M}}_{g,r}$,
i.e., the right-hand vertical arrow in (389), is unramified over the point
$q_{\mathbb{G}}$. By the flatness of $\Pi_{g,r}$ obtained in Corollary 6.4.4,
$\Pi_{\rho,g,r}$ is moreover étale over the same point. This completes the
proof of the former assertion.

Also, since the projection $\Pi_{g,r}$ is finite and faithfully flat, one may
verify that any irreducible component of
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ contains a point lying over the
point of $\overline{\mathcal{M}}_{g,r}$ classifying a totally degenerate
curve. Thus, the latter assertion follows from the former assertion together
with the open nature of étaleness. ∎

######  Remark 6.7.2.

(Divisor defined by the nonordinary locus) The complement of the ordinary
locus forms a divisor on $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$.
Indeed, let us consider the morphism $D_{\mathscr{E}^{\spadesuit}}$ (cf.
(458)) in the case where the pair “$(\mathscr{X},\mathscr{E}^{\spadesuit})$”
is taken to be the universal object over
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$. This morphism determines a
morphism

(591)
$\displaystyle\bigwedge^{3g-3+r}D_{\mathscr{E}^{\spadesuit}}:\bigwedge^{3g-3}\mathbb{R}^{1}f_{*}(\mathcal{G}^{1}_{\mathscr{E}^{\spadesuit}})\rightarrow\bigwedge^{3g-3+r}\mathbb{R}^{1}f_{*}(\mathcal{T})$

between line bundles on $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ (cf.
Propositions 6.1.2, (ii), 6.2.4, and 6.4.3), which corresponds to a global
section of the line bundle

(592)
$\displaystyle\left(\bigwedge^{3g-3+r}\mathbb{R}^{1}f_{*}(\mathcal{T})\right)\otimes\left(\bigwedge^{3g-3+r}\mathbb{R}^{1}f_{*}(\mathcal{G}^{1}_{\mathscr{E}^{\spadesuit}})\right)^{\vee}.$

The morphism $D_{\mathscr{E}^{\spadesuit}}$ is an isomorphism exactly on the
complement of the divisor
$\mathcal{D}_{\cancel{\mathrm{ord}}}\left(\subseteq\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}\right)$
associated to this section. Thus, the nonordinary locus in
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{g,r}$ coincides with the support of
$\mathcal{D}_{\cancel{\mathrm{ord}}}$.

######  Remark 6.7.3 (Generic étaleness for higher rank cases).

For a (sufficiently small) general $n>1$, the projection
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,g,r,\mathbb{F}_{p}}\rightarrow\overline{\mathcal{M}}_{g,r,\mathbb{F}_{p}}$
is generically étale (cf. [Wak6, Theorem G]). The generic étaleness is
essential in the proofs of Joshi’s conjecture described in [Wak1] and [Wak6].
Indeed, we used this geometric property to lift relevant moduli spaces to
characteristic $0$ and then compared them with certain Quot schemes over
$\mathbb{C}$; this argument enables us to compute explicitly the number of all
possible dormant $\mathrm{PGL}_{n}$-opers on a general curve. To develop the
enumerative geometry of higher-level dormant opers in such a way that this
argument works, we expect that the following conjecture is true:

> Suppose that $p$ is sufficiently large relative to $n$, $g$, and $r$. Then,
> the stack $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r,\mathbb{F}_{p}}$
> associated to such a collection $(p,n,g,r)$ is generically étale over
> $\overline{\mathcal{M}}_{g,r,\mathbb{F}_{p}}$, i.e., any irreducible
> component of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r,\mathbb{F}_{p}}$
> that dominates $\overline{\mathcal{M}}_{g,r,\mathbb{F}_{p}}$ admits a dense
> open subscheme which is étale over
> $\overline{\mathcal{M}}_{g,r,\mathbb{F}_{p}}$.

For each positive integer $N^{\prime}<N$, we shall write $\Pi_{N\Rightarrow
N^{\prime}}$ for the morphism
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}\rightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N^{\prime},g,r}$
obtained by reducing the level of dormant $\mathrm{PGL}_{2}$-opers to
$N^{\prime}$. We shall set

(593) $\displaystyle\mathcal{J}:=\bigcap_{N^{\prime}\leq N}\Pi_{N\Rightarrow
N^{\prime}}^{-1}(\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N^{\prime},g,r,\mathrm{ord}}),$

which is a dense open substack of
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$ (cf. Theorem 6.7.1). That is to
say, $\mathcal{J}$ classifies dormant $\mathrm{PGL}_{2}^{(N)}$-opers
$\mathscr{E}^{\spadesuit}$ such that the morphisms
$D_{\mathscr{E}^{\spadesuit(N^{\prime})}}$ are isomorphisms for all
$N^{\prime}\leq N$, where $\mathscr{E}^{\spadesuit(N^{\prime})}$ denotes the
dormant $\mathrm{PGL}_{2}^{(N^{\prime})}$-oper induced from
$\mathscr{E}^{\spadesuit}$. Hence, we obtain the dense open substack

(594) $\displaystyle\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}$

of $\overline{\mathcal{M}}_{g,r}$ defined as the complement of the image of
the natural projection
$\mathcal{O}p_{N,g,r}^{{}^{\mathrm{Zzz...}}}\setminus\mathcal{J}\rightarrow\overline{\mathcal{M}}_{g,r}$.

If $\mathscr{X}$ is a pointed stable curve over a geometric point classified
by $\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}$, then all the dormant
$\mathrm{PGL}_{2}^{(N)}$-opers on $\mathscr{X}$ and their reductions to level
$N^{\prime}\leq N$ are ordinary. In particular, the set of isomorphism classes
of dormant $\mathrm{PGL}_{2}^{(N)}$-opers on $\mathscr{X}$ has cardinality
equal to the degree of the projection $\Pi_{g,r}$.

######  Definition 6.7.4.

An $r$-pointed stable curve of genus $g$ is called (dormant-)$N$-ordinary if
it is classified by the open substack
$\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}$. (Since
$\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}$ is dense in
$\overline{\mathcal{M}}_{g,r}$, a general pointed stable curve is
$N$-ordinary.)

The following assertion describes the relationship between $N$-ordinariness
and $1$-ordinariness.

######  Proposition 6.7.5.

The inclusion relation
$\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}\subseteq\overline{\mathcal{M}}_{g,r,1\text{-}\mathrm{ord}}$
holds. That is to say, if $\mathscr{X}$ is $N$-ordinary, then it is also
$1$-ordinary.

###### Proof.

The assertion follows from Proposition 6.5.4. ∎

######  Remark 6.7.6 ($N$-ordinariness vs. $N^{\prime}$-ordinariness).

For the same reason as above, the inclusion relation
$\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}\subseteq\overline{\mathcal{M}}_{g,r,N^{\prime}\text{-}\mathrm{ord}}$
(where $N^{\prime}\leq N$) holds if the projection $\Pi_{N\Rightarrow
N^{\prime}}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}\rightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N^{\prime},g,r}$
is surjective. In particular, if we can show the connectedness of the stacks
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$ (which implies the surjectivity
of the $\Pi_{N\Rightarrow N^{\prime}}$’s), there exists a sequence of open
immersions

(595)
$\displaystyle\cdots\subseteq\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}\subseteq\cdots\subseteq\overline{\mathcal{M}}_{g,r,2\text{-}\mathrm{ord}}\subseteq\overline{\mathcal{M}}_{g,r,1\text{-}\mathrm{ord}}\subseteq\overline{\mathcal{M}}_{g,r}.$

It will be natural to ask what the intersection
$\bigcap_{N\in\mathbb{Z}_{>0}}\overline{\mathcal{M}}_{g,r,N\text{-}\mathrm{ord}}$
is. However, at the time of writing the present paper, the author does not
know the existence of a closed point of $\overline{\mathcal{M}}_{g,r}$
classifying a pointed curve that is $N$-ordinary for every $N$.

##  7\. Canonical diagonal liftings

In this final section, we construct the canonical diagonal lifting of a
dormant $\mathrm{PGL}_{2}^{(N)}$-oper, as well as a Frobenius-projective
structure, on a general hyperbolic curve. A key observation for doing this is
that the ordinariness introduced in the previous section enables us to connect
various deformation spaces involved. As a consequence, we obtain an effective
way to discuss the counting problem of dormant $\mathrm{PGL}_{2}$-opers in
characteristic $p^{N}$ via reduction modulo $p$.

Let us fix an odd prime $p$ and a positive integer $N$.

###  7.1. Canonical diagonal liftings of dormant
$(\mathrm{GL}_{2}^{(N)},\vartheta)$-opers

Let $R$ be a flat $\mathbb{Z}/p^{N}\mathbb{Z}$-algebra and $X$ a geometrically
connected, proper, and smooth curve over $S:=\mathrm{Spec}(R)$ of genus $g>1$.
For each positive integer $N^{\prime}\leq N$, we shall denote by a subscript
“$N^{\prime}-1$” the result of reducing an object over
$\mathbb{Z}/p^{N}\mathbb{Z}$ modulo $p^{N^{\prime}}$. For simplicity, we write
$\Omega_{0}:=\Omega_{X_{0}/S_{0}}$ and
$\Omega^{(N^{\prime})}_{0}:=\Omega_{X_{0}^{(N^{\prime})}/S_{0}}$.

We shall fix a dormant $2^{(1)}$-theta characteristic
$\vartheta:=(\varTheta,\nabla_{\vartheta})$ of $X/S$ (cf. Definition 4.4.1 and
Proposition 4.4.3) and fix an integer $N^{\prime}$ with $1\leq N^{\prime}<N$.
The diagonal reduction of $\vartheta_{N^{\prime}-1}$ defines a
$2^{(N^{\prime})}$-theta characteristic $\vartheta_{0}^{(N^{\prime})}$ of
$X_{0}/S_{0}$.

Also, let $\nabla^{\diamondsuit}_{N^{\prime}-1}$ be a dormant
$(\mathrm{GL}_{2}^{(1)},\vartheta_{N^{\prime}-1})$-oper on
$X_{N^{\prime}-1}/S_{N^{\prime}-1}$, and write
$\mathscr{F}:=(\mathcal{F}_{N^{\prime}-1},\nabla_{N^{\prime}-1}^{\diamondsuit})$,
where $\mathcal{F}:=\mathcal{F}_{\varTheta}$. The diagonal reduction of
$\nabla^{\diamondsuit}_{N^{\prime}-1}$ defines a
$(\mathrm{GL}_{2}^{(N^{\prime})},\vartheta_{0}^{(N^{\prime})})$-oper
$\nabla_{0}^{\diamondsuit(N^{\prime})}$ on $X_{0}/S_{0}$. The
$(\mathrm{GL}_{2}^{(1)},\vartheta_{0})$-oper on $X_{0}/S_{0}$ induced by
$\nabla_{0}^{\diamondsuit(N^{\prime})}$ via level reduction will be denoted by
$\nabla_{0}^{\diamondsuit(1)}$.

We shall write

(596)
$\displaystyle\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}$

for the set of isomorphism classes of
$(\mathrm{GL}_{2}^{(1)},\vartheta_{N^{\prime}})$-opers
$\nabla^{\diamondsuit}_{N^{\prime}}$ on $X_{N^{\prime}}/S_{N^{\prime}}$ with
$(\nabla^{\diamondsuit}_{N^{\prime}})_{N^{\prime}-1}=\nabla^{\diamondsuit}_{N^{\prime}-1}$.
Then, $\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}$ is
nonempty and has a canonical structure of torsor modeled on the $R$-module
$H^{0}(X_{0},\Omega_{0}^{\otimes 2})$ (cf. [Moc1, Chap. I, Corollary 2.9]).

Now, let us consider the composite

(597) $\displaystyle H^{0}(X_{0},\Omega_{0}^{\otimes 2})\hookrightarrow
H^{0}(\Omega_{0}\otimes\mathcal{E}nd^{0}(\mathcal{F}_{0}))\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[(\nabla^{\diamondsuit(1)}_{0})_{\mathcal{E}nd^{0}}])$

(cf. (176) for the definition of $(-)_{\mathcal{E}nd^{0}}$), where the first
arrow denotes the canonical inclusion (cf. [Wak6, § 4.9.5]) and the second
arrow denotes the morphism induced by the natural inclusion of complexes
$(\Omega_{0}\otimes\mathcal{E}nd^{0}(\mathcal{F}_{0}))[-1]\rightarrow\mathcal{K}^{\bullet}[(\nabla_{0}^{\diamondsuit(1)})_{\mathcal{E}nd^{0}}]$.
Then, the assignment
$\nabla_{N^{\prime}}^{\diamondsuit}\mapsto(\mathcal{F}_{N^{\prime}},\nabla_{N^{\prime}}^{\diamondsuit})$
defines a map of sets

(598)
$\displaystyle\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}\rightarrow\mathrm{Lift}(\mathscr{F})_{\mathscr{L}}$

where $\mathscr{L}:=(\mathrm{det}(\mathcal{F}),\nabla_{\vartheta})$ (cf. (177)
for the definition of $\mathrm{Lift}(\mathscr{F})_{\mathscr{L}}$). Recall from
the discussion preceding Proposition 2.4.2 that
$\mathrm{Lift}(\mathscr{F})_{\mathscr{L}}$ has a structure of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[(\nabla_{0}^{\diamondsuit(1)})_{\mathcal{E}nd^{0}}])$-torsor.
The map (598) is verified to be compatible with the respective torsor
structures via the morphism (597) (cf. [Wak6, § 6.3.2]). By combining this
fact and Proposition 2.4.3, we obtain the following assertion.

######  Proposition 7.1.1.

Taking the diagonal reductions
$\nabla_{N^{\prime}}^{\diamondsuit}\mapsto{{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\\!\\!\nabla_{N^{\prime}}^{\diamondsuit}$
yields a map of sets

(599)
$\displaystyle\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}\rightarrow\mathrm{Lift}(\nabla_{0}^{\diamondsuit(N^{\prime})})_{\mathscr{L}}$

(cf. (172) for the definition of the codomain of this map), and this map
commutes with the respective torsor structures via the composite

(600) $\displaystyle
D^{\prime\prime}_{\nabla_{0}^{\diamondsuit(N^{\prime})}}:H^{0}(X_{0},\Omega_{0}^{\otimes
2})$
$\displaystyle\xrightarrow{(\ref{fE14})}\mathbb{H}^{1}(X_{0},\mathcal{K}^{\bullet}[(\nabla_{0}^{\diamondsuit(1)})_{\mathcal{E}nd^{0}}])$
$\displaystyle\xrightarrow{(\ref{fE8})}H^{0}(X_{0}^{(N^{\prime})},\Omega^{(N^{\prime})}_{0}\otimes\mathcal{E}nd^{0}(\mathcal{S}ol(\nabla_{0}^{\diamondsuit(N^{\prime})}))).$

In particular, the map (599) becomes bijective when
$D^{\prime\prime}_{\nabla_{0}^{\diamondsuit(N^{\prime})}}$ is bijective.

Denote by

(601)
$\displaystyle\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}^{\psi}$

the subset of
$\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}$ consisting
of dormant $(\mathrm{GL}_{2}^{(1)},\vartheta_{N^{\prime}})$-opers.
Propositions 2.4.4 and 7.1.1 together imply the following proposition.

######  Proposition 7.1.2.

  * (i)

Taking the diagonal reductions
$\nabla_{N^{\prime}}^{\diamondsuit}\mapsto{{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\\!\\!\nabla_{N^{\prime}}^{\diamondsuit}$
yields a map of sets

(602)
$\displaystyle\mathrm{Lift}(\nabla_{N^{\prime}-1}^{\diamondsuit})_{\vartheta}^{\psi}\rightarrow\mathrm{Lift}(\nabla_{0}^{\diamondsuit(N^{\prime})})^{\psi},$

and the following commutative square diagram is Cartesian:

(607)

  * (ii)

Suppose that $D^{\prime\prime}_{\nabla_{0}^{\diamondsuit(N^{\prime})}}$ is
bijective. Then, the map (602) is bijective. In particular, for each dormant
$(\mathrm{GL}_{2}^{(N^{\prime}+1)},\vartheta^{(N^{\prime}+1)}_{0})$-oper
$\nabla_{0}^{\diamondsuit(N^{\prime}+1)}$ of $X_{0}/S_{0}$ that induces
$\nabla_{0}^{\diamondsuit(N^{\prime})}$ by reducing its level, there exists a
unique (up to isomorphism) dormant
$(\mathrm{GL}_{2}^{(1)},\vartheta_{N^{\prime}})$-oper
$\nabla_{N^{\prime}}^{\diamondsuit}$ on $X_{N^{\prime}}/S_{N^{\prime}}$ with
$(\nabla_{N^{\prime}}^{\diamondsuit})_{N^{\prime}-1}=\nabla_{N^{\prime}-1}^{\diamondsuit}$
and
${{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\\!\\!\nabla_{N^{\prime}}^{\diamondsuit}=\nabla_{0}^{\diamondsuit(N^{\prime}+1)}$.

######  Corollary 7.1.3.

Let $\nabla^{\diamondsuit(N)}_{0}$ be a dormant
$(\mathrm{GL}_{2}^{(N)},\vartheta_{0}^{(N)})$-oper on $X_{0}/S_{0}$. Suppose
that the morphism $D^{\prime\prime}_{\nabla_{0}^{\diamondsuit(N^{\prime})}}$
is bijective for every $N^{\prime}\leq N$, where
$\nabla_{0}^{\diamondsuit(N^{\prime})}$ denotes the dormant
$(\mathrm{GL}_{2}^{(N^{\prime})},\vartheta^{(N^{\prime})}_{0})$-oper obtained
by reducing the level of $\nabla^{\diamondsuit(N)}_{0}$ to $N^{\prime}$. Then,
there exists a unique (up to isomorphism) dormant
$(\mathrm{GL}_{n}^{(1)},\vartheta)$-oper $\nabla^{\diamondsuit}$ on $X/S$
whose diagonal reduction coincides with $\nabla^{\diamondsuit(N)}_{0}$.

###### Proof.

The assertion can be proved by applying assertion (ii) of Proposition 7.1.2
successively with respect to $N^{\prime}$ ($1\leq N^{\prime}<N$). ∎

###  7.2. Canonical diagonal liftings of dormant
$\mathrm{PGL}_{2}^{(N)}$-opers

Denote by

(608) $\displaystyle\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\
\left(\text{resp.,}\ \mathrm{Op}_{N,X_{0}}^{{}^{\mathrm{Zzz...}}}\right)$

the set of isomorphism classes of dormant $\mathrm{PGL}_{2}$-opers on $X/S$
(resp., dormant $\mathrm{PGL}_{2}^{(N)}$-opers on $X_{0}/S_{0}$). By regarding
it as a (discrete) category, we obtain, from Proposition 4.2.4, an equivalence
of categories

(609)
$\displaystyle\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\cong\mathcal{O}p_{1,g,0,R}^{{}^{\mathrm{Zzz...}}}\\!\times_{\overline{\mathcal{M}}_{g,0,R},\sigma_{X}}S\
\left(\text{resp.,}\
\mathrm{Op}_{N,X_{0}}^{{}^{\mathrm{Zzz...}}}\cong\mathcal{O}p_{N,g,0,\mathbb{F}_{p}}^{{}^{\mathrm{Zzz...}}}\\!\times_{\overline{\mathcal{M}}_{g,0,\mathbb{F}_{p}},\sigma_{X_{0}}}S\right),$

where $\sigma_{X}$ (resp., $\sigma_{X_{0}}$) denotes the classifying morphism
$S\rightarrow\overline{\mathcal{M}}_{g,r,R}$ (resp.,
$S_{0}\rightarrow\overline{\mathcal{M}}_{g,r,\mathbb{F}_{p}}$) of $X$ (resp.,
$X_{0}$). Taking the diagonal reductions of dormant $\mathrm{PGL}_{2}$-opers
yields a map of sets

(610)
$\displaystyle\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\\!\\!\spadesuit}:\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\rightarrow\mathrm{Op}_{N,X_{0}}^{{}^{\mathrm{Zzz...}}}$

(cf. (309)). Then, we have the following assertion.

######  Theorem-Definition 7.2.1.

Suppose that $X_{0}/S_{0}$ is $N$-ordinary (cf. Definition 6.7.4). Then, the
map
$\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\\!\\!\spadesuit}:\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\rightarrow\mathrm{Op}_{N,X_{0}}^{{}^{\mathrm{Zzz...}}}$
is bijective. In particular, for each dormant $\mathrm{PGL}_{2}^{(N)}$-oper
$\mathscr{E}^{\spadesuit}_{0}$ on $X_{0}/S_{0}$, there exists a unique (up to
isomorphism) dormant $\mathrm{PGL}_{2}$-oper $\mathscr{E}^{\spadesuit}$ on
$X/S$ with
${{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\\!\\!\mathscr{E}^{\spadesuit}=\mathscr{E}^{\spadesuit}_{0}$.
We shall refer to $\mathscr{E}^{\spadesuit}$ as the canonical diagonal lifting
of $\mathscr{E}^{\spadesuit}_{0}$.

###### Proof.

According to Proposition 4.4.3, we can find a dormant $2^{(1)}$-theta
characteristic $\vartheta:=(\varTheta,\nabla_{\vartheta})$ of $X/S$. Denote by
$\vartheta_{0}^{(N)}$ the $2^{(N)}$-theta characteristic of $X_{0}/S_{0}$
obtained as the diagonal reduction of $\vartheta$. Now, let
$\mathscr{E}_{0}^{\spadesuit}:=(\mathcal{E}_{B,0},\phi_{0}^{(N)})$ be a
dormant $\mathrm{PGL}_{2}^{(N)}$-oper on $X_{0}/S_{0}$. For each positive
integer $N^{\prime}\leq N$, we denote by $\phi^{(N^{\prime})}_{0}$ the
$(N^{\prime}-1)$-PD stratification on $\mathcal{E}$ induced by
$\phi^{(N)}_{0}$. The resulting dormant $\mathrm{PGL}_{2}^{(N^{\prime})}$-oper
$\mathscr{E}_{0}^{\spadesuit(N^{\prime})}:=(\mathcal{E}_{B,0},\phi_{0}^{(N^{\prime})})$
corresponds, via the bijection
$\Lambda_{\diamondsuit\Rightarrow\spadesuit,\vartheta}^{{}^{\mathrm{Zzz...}}}$
(cf. Theorem 4.5.1), to a dormant
$(\mathrm{GL}_{2}^{(N^{\prime})},\vartheta_{0}^{(N^{\prime})})$-oper
$\nabla^{\diamondsuit(N^{\prime})}_{0}$ on $X_{0}/S_{0}$. Note that there
exists a canonical identification
$\mathcal{E}nd^{0}(\mathcal{F}_{\varTheta_{0}})=\mathfrak{g}_{\mathcal{E}_{0}}$
(where $\mathcal{E}_{0}:=\mathcal{E}_{B,0}\times^{B}\mathrm{PGL}_{2}$), which
is compatible with the respective
$\mathcal{D}_{X_{0}/S_{0}}^{(N^{\prime}-1)}$-module structures, i.e.,
$(\nabla_{0}^{\diamondsuit(N^{\prime})})_{\mathcal{E}nd^{0}}$ and
$\nabla_{\phi_{0}^{(N^{\prime})}}^{\mathrm{ad}}$. In particular, we have

(611)
$\displaystyle\mathcal{E}nd^{0}(\mathcal{S}ol(\nabla_{0}^{\diamondsuit(N^{\prime})}))\left(=\mathcal{S}ol((\nabla_{0}^{\diamondsuit(N^{\prime})})_{\mathcal{E}nd^{0}})\right)=\mathfrak{g}_{\mathcal{E}_{0},\phi_{0}^{(N^{\prime})}}.$

Under this identification, the morphism
$D^{\prime\prime}_{\nabla_{0}^{\diamondsuit(N^{\prime})}}$ coincides with
$D^{\prime\prime}_{\mathscr{E}_{0}^{\spadesuit(N^{\prime})}}$ (cf. (562)).
Hence, since $X_{0}/S_{0}$ is $N$-ordinary, the morphisms
$D^{\prime\prime}_{\nabla_{0}^{\diamondsuit(N^{\prime})}}$ are bijective for
all $N^{\prime}$’s. By Corollary 7.1.3, there exists a unique (up to
isomorphism) dormant $(\mathrm{GL}_{2}^{(1)},\vartheta)$-oper
$\nabla_{N}^{\diamondsuit}$ on $X/S$ whose diagonal reduction coincides with
$\nabla^{\diamondsuit(N)}_{0}$. The dormant $\mathrm{PGL}_{2}$-oper
$\mathscr{E}^{\spadesuit}$ corresponding to $\nabla_{N}^{\diamondsuit}$
specifies a diagonal lifting of $\mathscr{E}^{\spadesuit}_{0}$. The resulting
assignment $\mathscr{E}_{0}^{\spadesuit}\mapsto\mathscr{E}^{\spadesuit}$
specifies a well-defined map of sets
$\mathrm{Op}_{N,X_{0}}^{{}^{\mathrm{Zzz...}}}\rightarrow\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}$,
and it gives an inverse map of
$\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\\!\\!\spadesuit}$. This
completes the proof of the assertion. ∎

######  Remark 7.2.2 (Affineness assumption).

By the uniqueness assertion in the above theorem, the formation of the
canonical diagonal liftings commutes with base-change over $S$-schemes. Hence,
Theorem-Definition 7.2.1 is verified to be true even when we remove the
affineness assumption on $S$.

######  Remark 7.2.3 (Representability of an ordinary locus of
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,0,R}$).

Denote by $\mathcal{M}_{\mathbb{F}_{p}}$ (resp., $\mathcal{M}_{R}$) the open
substack of $\mathcal{M}_{g,0,\mathbb{F}_{p}}$ (resp., $\mathcal{M}_{g,0,R}$)
consisting of points lying on
$\overline{\mathcal{M}}_{g,0,N\text{-}\mathrm{ord}}$. Since the projection
$\mathcal{O}p_{N,g,0,\mathbb{F}_{p}}^{{}^{\mathrm{Zzz...}}}\times_{\overline{\mathcal{M}}_{g,0,\mathbb{F}_{p}}}\mathcal{M}_{\mathbb{F}_{p}}\rightarrow\mathcal{M}_{\mathbb{F}_{p}}$
is étale, there exists a unique (up to isomorphism) étale stack $\mathcal{U}$
over $\mathcal{M}_{R}$ whose reduction modulo $p$ is isomorphic to
$\mathcal{O}p_{N,g,0,\mathrm{ord}}^{{}^{\mathrm{Zzz...}}}\times_{\overline{\mathcal{M}}_{g,0,\mathbb{F}_{p}}}\mathcal{M}_{\mathbb{F}_{p}}$.

Now, let us take an étale scheme $S$ over $\mathcal{M}_{g,0,R}$ and a section
$u$ of $\mathcal{U}$ over $S$. The morphism $S\rightarrow\mathcal{M}_{g,0,R}$
classifies a smooth curve $X/S$ whose reduction $X_{0}/S_{0}$ modulo $p$ is
$N$-ordinary, and the reduction modulo $p$ of $u$ defines a dormant
$\mathrm{PGL}_{2}^{(N)}$-oper $\mathscr{E}^{\spadesuit}_{0}$ on $X_{0}/S_{0}$.
According to Theorem-Definition 7.2.1, we obtain the canonical diagonal
lifting $\mathscr{E}^{\spadesuit}$ of $\mathscr{E}^{\spadesuit}_{0}$ defined
on $X/S$. In particular, the pair $(X/S,\mathscr{E}^{\spadesuit})$ specifies
an object of
$\mathcal{O}p_{N,g,0,R}^{{}^{\mathrm{Zzz...}}}\times_{\overline{\mathcal{M}}_{g,0,R}}\mathcal{M}_{R}$
over $S$. One may verify that the functor

(612)
$\displaystyle\mathcal{U}\rightarrow\mathcal{O}p_{N,g,0,R}^{{}^{\mathrm{Zzz...}}}\times_{\overline{\mathcal{M}}_{g,0,R}}\mathcal{M}_{R}$

given by assigning $u\mapsto(X/S,\mathscr{E}^{\spadesuit})$ becomes an
isomorphism between the étale sheaves on $\mathcal{M}_{R}$ represented by the
respective fibered categories. In particular, by using this isomorphism, we
can construct a universal family of ordinary dormant
$\mathrm{PGL}_{2}^{(N)}$-opers parametrized by $\mathcal{U}$.

######  Remark 7.2.4 (Comparison with $p$-adic Teichmüller theory).

One of the main achievements in $p$-adic Teichmüller theory asserts the
existence of a canonical $p$-adic lifting of a curve equipped with a certain
additional structure involving a $\mathrm{PGL}_{2}$-oper (= a torally
indigenous bundle). Each object classified by the VF-stack, or the shifted VF-
stack, of pure tone (cf. [Moc2, Chap. IV, Definition 2.6]) is related to the
notion of a dormant $\mathrm{PGL}_{2}^{(N)}$-oper because both are
characterized by the vanishing of the $p$-curvature of the associated flat
bundles. However, despite the similarity in concept, there seems to be no (at
least direct) correspondence with the canonical diagonal liftings asserted in
the above theorem.

Also, other lifting constructions of $\mathrm{PGL}_{2}$-opers can be found in
[Moc1] (which is also in the context of $p$-adic Teichmüller theory, but in a
more classical setting) and [LSYZ]. These constructions are entirely different
from ours because the $\mathrm{PGL}_{2}$-opers treated there never have
vanishing $p$-curvature.

######  Corollary 7.2.5.

Let $N$ be a positive integer and $k$ an algebraically closed field over
$\mathbb{F}_{p}$. Denote by $W_{N}$ the ring of Witt vectors of length $N$
over $k$. Also, let $X$ be a geometrically connected, proper, and smooth curve
of genus $g>1$ over $W_{N}$. Denote by $X_{0}$ the reduction modulo $p$ of
$X$. Then, there are only finitely many isomorphism classes of dormant
$\mathrm{PGL}_{2}$-opers on $X$. If, moreover, $X_{0}$ is general in
$\mathcal{M}_{g,0,}$, then the cardinality
$\sharp(\mathrm{Op}^{{}^{\mathrm{Zzz...}}}_{1,X})$ of
$\mathrm{Op}^{{}^{\mathrm{Zzz...}}}_{1,X}$ satisfies the following equalities:

(613)
$\displaystyle\sharp(\mathrm{Op}^{{}^{\mathrm{Zzz...}}}_{1,X})=\sharp(\mathrm{Op}^{{}^{\mathrm{Zzz...}}}_{N,X_{0}})=\mathrm{deg}(\Pi_{N,g,0,\mathbb{F}_{p}}).$

###### Proof.

The assertion follows from Theorem 5.3.6 and Theorem-Definition 7.2.1. ∎

######  Remark 7.2.6 ($p$-adic liftings of dormant
$\mathrm{PGL}_{2}^{(\infty)}$-opers).

Let $K$ be an algebraically closed field over $\mathbb{F}_{p}$, and let
$X_{0}$ be a connected proper smooth curve of genus $g$ over $K$ classified by
a $K$-rational generic point of $\mathcal{M}_{g,0}$. Denote by $W$ the ring of
Witt vectors over $K$. Also, choose a deformation $X$ of $X_{0}$ over $W$.

According to Corollary 6.6.7, we can find a collection of data

(614) $\displaystyle\\{s_{N}\\}_{N\in\mathbb{Z}_{>0}},$

where

  * •

each $s_{N}$ ($N\in\mathbb{Z}_{>0}$) denotes a $K$-valued generic point of an
irreducible component of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}$;

  * •

the points $\\{s_{N}\\}_{N\in\mathbb{Z}_{>0}}$ are compatible with respect to
the projective system (307), i.e., $s_{N}$ is mapped to $s_{N^{\prime}}$ via
the natural projection $\Pi_{N\Rightarrow
N^{\prime}}:\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N,g,r}\rightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{N^{\prime},g,r}$
for every $N^{\prime}\leq N$.

The point $s_{N}$ determines a dormant $\mathrm{PGL}_{2}^{(N)}$-oper
$\mathscr{E}_{N}^{\spadesuit}$ on $X_{0}/K$. The resulting compatible
collection $\\{\mathscr{E}^{\spadesuit}_{N}\\}_{N\in\mathbb{Z}_{>0}}$ should
be called a dormant $\mathrm{PGL}_{2}$-oper of level $\infty$, or dormant
$\mathrm{PGL}_{2}^{(\infty)}$-oper. Since $X_{0}$ is $N$-ordinary for every
$N$, we can apply Corollary 7.2.5 to the $\mathscr{E}_{N}^{\spadesuit}$’s, and
obtain a dormant $\mathrm{PGL}_{2}$-oper $\mathscr{E}_{\infty}^{\spadesuit}$
defined on a curve $X/W$ via algebraization. This construction would partially
realize the expected correspondence (at least in the generic situation)
between dormant opers of level $\infty$ on a curve in characteristic $p$ and
dormant opers of level $1$ on its $p$-adic lifting.

###  7.3. Canonical diagonal liftings of $F^{N}$-projective structures

The notion of an $F^{N}$-projective structures (i.e., a Frobenius-projective
structures of level $N$) on a curve in characteristic $p$ was originally
introduced by Y. Hoshi (cf. [Hos, Definition 2.1]) and later investigated by
the author (cf. [Wak4], [Wak5]). In this final subsection, we extend the
definition of an $F^{N}$-projective structure to the case where the underlying
curve has characteristic $p^{N}$ (cf. Definition 7.3.2 below). After proving a
bijective correspondence with dormant $\mathrm{PGL}_{2}^{(N)}$-opers, we
formulate the canonical diagonal liftings of $F^{N}$-projective structures.

Let $l$ be a nonnegative integer, $S$ a flat
$\mathbb{Z}/p^{l+1}\mathbb{Z}$-scheme, and $X$ a geometrically connected,
proper, and smooth curve of genus $g>1$ over $S$. Suppose that $S$ is equipped
with an $(N-1)$-PD structure extending $X$. Denote by $(\mathrm{PGL}_{2})_{X}$
the Zariski sheaf on $X$ represented by $\mathrm{PGL}_{2}$; this may be
identified with the sheaf of automorphisms of the trivial
$\mathrm{PGL}_{2}$-bundle $X\times\mathrm{PGL}_{2}$. Also, denote by

(615) $\displaystyle(\mathrm{PGL}_{2}^{(N)})_{X}$

the subsheaf of $(\mathrm{PGL}_{2})_{X}$ consisting of (locally defined)
automorphisms of $X\times\mathrm{PGL}_{2}$ preserving the $(N-1)$-PD
stratification $\phi_{\mathrm{triv}}$ (cf. (60) for the definition of
$\phi_{\mathrm{triv}}$).

Next, let us write

(616) $\displaystyle\mathcal{P}^{\text{\'{e}t}}$

for the Zariski sheaf of sets on $X$ that assigns, to each open subscheme $U$
of $X$, the set of étale $S$-morphisms from $U$ to the projective line
$\mathbb{P}$ over $S$.

######  Lemma 7.3.1.

Let $U$ be an open subscheme of $X$, $\varphi:U\rightarrow\mathbb{P}$ an
element of $\mathcal{P}^{\text{\'{e}t}}(U)$. Also, let $h$ be an element of
$(\mathrm{PGL}_{2}^{(N)})_{X}(U)$, regarded as an $S$-morphism
$U\rightarrow\mathrm{PGL}_{2}$. Then, the composite

(617)
$\displaystyle{{}^{h}}\varphi:U\xrightarrow{(h,\varphi)}\mathrm{PGL}_{2}\times\mathbb{P}\xrightarrow{\Psi}\mathbb{P}$

belongs to $\mathcal{P}^{\text{\'{e}t}}(U)$, where the second arrow $\Psi$
denotes the usual $\mathrm{PGL}_{2}$-action on $\mathbb{P}$.

###### Proof.

After possibly replacing $U$ with its covering with respect to the étale
topology, we may assume that there exists an automorphism $\widetilde{h}$ of
$(\mathcal{O}_{X}^{\oplus 2},(\nabla_{X,\mathrm{triv}}^{(N-1)})^{\oplus 2})$
inducing $h$ via projectivization; we will use the same notation
$\widetilde{h}$ to denote the associated morphism
$U\rightarrow\mathrm{GL}_{2}$. Let us choose a point $q\in U(k)$, where $k$
denotes a field. The differential of
$(h,\varphi):U\rightarrow\mathrm{PGL}_{2}\times\mathbb{P}$ at $q$ yields a
morphism of $k$-vector spaces

(618)
$\displaystyle(dh,d\varphi):\mathcal{T}_{U/S}|_{q}\rightarrow\mathcal{T}_{\mathrm{PGL}_{2}}|_{h(q)}\oplus\mathcal{T}_{\mathbb{P}/S}|_{\varphi(q)}\left(=\mathcal{T}_{(\mathrm{PGL}_{2}\times\mathbb{P})/S}|_{(h(q),\varphi(q))}\right).$

Note that the differential of $\widetilde{h}$ coincides with the zero map
because $\widetilde{h}$ lies in $\mathrm{Aut}(\mathcal{O}_{X}^{\oplus
2},(\nabla_{X,\mathrm{triv}}^{(N-1)})^{\oplus 2})$. Hence, since $h$ factors
through $\widetilde{h}$, the morphism
$dh:\mathcal{T}_{U/S}|_{q}\rightarrow\mathcal{T}_{\mathrm{PGL}_{2}}|_{h(q)}$
coincides with the zero map. On the other hand, by the étaleness assumption on
$\varphi$, the morphism
$d\varphi:\mathcal{T}_{U/S}|_{q}\rightarrow\mathcal{T}_{\mathbb{P}/S}|_{\varphi(q)}$
is an isomorphism. Also, the differential
$d\Psi:\mathcal{T}_{\mathrm{PGL}_{2}}|_{h(q)}\oplus\mathcal{T}_{\mathbb{P}/S}|_{\varphi(q)}\rightarrow\mathcal{T}_{\mathbb{P}/S}|_{h(\varphi)(q)}$
of the $\mathrm{PGL}_{2}$-action $\Psi$ restricts to an isomorphism
$\mathcal{T}_{\mathbb{P}/S}|_{\varphi(q)}\left(=0\oplus\mathcal{T}_{\mathbb{P}/S}|_{\varphi(q)}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{T}_{\mathbb{P}/S}|_{{{}^{h}}\varphi(q)}$.
It follows that the differential
$d({{}^{h}}\varphi)\left(=d\Psi\circ(dh,d\varphi)\right)$ of ${{}^{h}}\varphi$
at $q$ is an isomorphism. Since both $U$ and $\mathbb{P}$ are smooth over $S$,
the morphism ${{}^{h}}\varphi$ turns out to be étale. This completes the proof
of this lemma. ∎

By the above lemma, the assignment $(h,\varphi)\mapsto{{}^{h}}\varphi$ defines
a $(\mathrm{PGL}_{2}^{(N)})_{X}$-action

(619)
$\displaystyle\Psi^{(N)}:(\mathrm{PGL}_{2}^{(N)})_{X}\times\mathcal{P}^{\text{\'{e}t}}\rightarrow\mathcal{P}^{\text{\'{e}t}}$

on the sheaf $\mathcal{P}^{\text{\'{e}t}}$.

######  Definition 7.3.2.

Let $\mathcal{S}^{\blacklozenge}$ be a subsheaf of
$\mathcal{P}^{\text{\'{e}t}}$. We shall say that $\mathcal{S}^{\blacklozenge}$
is a Frobenius-projective structure of level $N$ (or simply, an
$F^{N}$-projective structure) on $X/S$ if it is closed under the
$(\mathrm{PGL}_{2}^{(N)})_{X}$-action $\Psi^{(N)}$ and forms a
$(\mathrm{PGL}_{2}^{(N)})_{X}$-torsor on $X$ with respect to the resulting
$(\mathrm{PGL}_{2}^{(N)})_{X}$-action on $\mathcal{S}^{\blacklozenge}$.

######  Remark 7.3.3 (Comparison with the classical definition).

Suppose that $l=0$. Then, since $(\mathrm{PGL}_{2}^{(N)})_{X}$ coincides with
the subsheaf $(F_{X/S}^{(N)})^{-1}((\mathrm{PGL}_{2})_{X^{(N)}})$ of
$(\mathrm{PGL}_{2})_{X}$, the definition of an $F^{N}$-projective structure
described above is the same as the classical definition discussed in [Hos,
Definition 2.1] and [Wak4, Definition 1.2.1].

Denote by

(620) $\displaystyle F^{N}\text{-}\mathrm{Pr}_{X}$

the set of $F^{N}$-projective structures on $X/S$. The following assertion
generalizes [Hos, Theorem A] and [Wak5, Theorem B] (for
$(G,H)=(\mathrm{PGL}_{2},B)$) to characteristic $p^{N}$.

######  Theorem 7.3.4.

Suppose that either $l=0$ or $N=1$ is satisfied (hence the set
$\mathrm{Op}_{N,X}^{{}^{\mathrm{Zzz...}}}$ can be defined as in (608)). Then,
there exists a canonical bijection of sets

(621)
$\displaystyle\mathrm{Op}_{N,X}^{{}^{\mathrm{Zzz...}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{N}\text{-}\mathrm{Pr}_{X}.$

In particular, there are only finitely many $F^{N}$-projective structures on
$X/S$.

###### Proof.

Since the assertion for $l=0$ was already proved in [Hos, Theorem A], it
suffices to consider the case of $N=1$. First, we shall construct a map of
sets $\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\rightarrow
F^{1}\text{-}\mathrm{Pr}_{X}$. Let
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a dormant
$\mathrm{PGL}_{2}$-oper on $X/S$, and write
$\mathcal{E}:=\mathcal{E}_{B}\times^{B}\mathrm{PGL}_{2}$. Denote by
$\mathbb{P}(\mathcal{E})$ the projective line bundle associated to
$\mathcal{E}$. The $B$-reduction $\mathcal{E}_{B}$ of $\mathcal{E}$ determines
a global section $\sigma:X\rightarrow\mathbb{P}(\mathcal{E})$. Next, let us
take an open subscheme $U$ of $X$ on which $(\mathcal{E},\nabla)$ is
trivialized (cf. Corollary 4.5.2, (ii)). Choose a trivialization

(622)
$\displaystyle\alpha:(\mathcal{E}|_{U},\nabla|_{U})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(U\times\mathrm{PGL}_{2},\nabla_{\mathrm{triv}}),$

where $\nabla_{\mathrm{triv}}$ denotes the trivial $S$-connection on
$U\times\mathrm{PGL}_{2}$ (cf. [Wak6, Eq. (78)]); it induces an isomorphism
$\alpha_{\mathbb{P}}:\mathbb{P}(\mathcal{E})|_{U}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}U\times_{S}\mathbb{P}$.
Thus, we obtain the composite

(623)
$\displaystyle\varphi_{\alpha}:U\xrightarrow{\sigma|_{U}}\mathbb{P}(\mathcal{E})|_{U}\xrightarrow{\alpha_{\mathbb{P}}}U\times_{S}\mathbb{P}\xrightarrow{\mathrm{pr}_{2}}\mathbb{P}.$

Similarly to the proof of [Wak4, Corollary 1.6.2], we see that
$\varphi_{\alpha}$ is étale. Let
$\mathscr{E}^{\spadesuit\Rightarrow\blacklozenge}(U)$ denote the set of étale
morphisms $\varphi_{\alpha}$ constructed in this way for all possible
trivializations $\alpha$ as in (622). The set
$\mathscr{E}^{\spadesuit\Rightarrow\blacklozenge}(U)$ has a structure of
$(\mathrm{PGL}_{2}^{(1)})_{X}(U)$-torsor with respect to the
$(\mathrm{PGL}_{2}^{(1)})_{X}(U)$-action defined by
$\varphi_{\alpha}\mapsto\varphi_{h\circ\alpha}$ for each
$h\in(\mathrm{PGL}_{2}^{(1)})_{X}(U)$. Hence, the sheaf associated to the
assignment $U\mapsto\mathscr{E}^{\spadesuit\Rightarrow\blacklozenge}(U)$
specifies a subsheaf

(624) $\displaystyle\mathscr{E}^{\spadesuit\Rightarrow\blacklozenge}$

of $\mathcal{P}^{\text{\'{e}t}}$. One may verify that
$\mathscr{E}^{\spadesuit\Rightarrow\blacklozenge}$ forms an $F^{1}$-projective
structure on $X/S$, and the resulting assignment
$\mathscr{E}^{\spadesuit}\mapsto\mathscr{E}^{\spadesuit\Rightarrow\blacklozenge}$
defines a map of sets $\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\rightarrow
F^{1}\text{-}\mathrm{Pr}_{X}$. Moreover, we can reverse the steps in the above
construction, so the map $\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\rightarrow
F^{1}\text{-}\mathrm{Pr}_{X}$ is verified to be bijective.

Finally, the latter assertion follows from the former assertion together with
Proposition 4.6.1, (ii). This completes the proof of the theorem. ∎

Hereinafter, we suppose that $l+1=N$. Denote by $X_{0}/S_{0}$ the curve
obtained as the reduction modulo $p$ of $X/S$. Let us define a map

(625)
$\displaystyle\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\\!\\!\blacklozenge}:F^{1}\text{-}\mathrm{Pr}_{X}\rightarrow
F^{N}\text{-}\mathrm{Pr}_{X_{0}}$

to be the unique map that makes the following square diagram commute:

(630)

######  Theorem-Definition 7.3.5.

Suppose that $X_{0}/S_{0}$ is $N$-ordinary. Then, the map
$\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\\!\\!\blacklozenge}$ is
bijective. That is to say, for any $F^{N}$-projective structure
$\mathcal{S}^{\blacklozenge}_{0}$ on $X_{0}/S_{0}$, there exists a unique
$F^{1}$-projective structure $\mathcal{S}^{\blacklozenge}$ on $X/S$ which is
mapped to $\mathcal{S}^{\blacklozenge}_{0}$ via
$\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\\!\\!\blacklozenge}$. We shall
refer to $\mathcal{S}^{\blacklozenge}$ as the canonical diagonal lifting of
$\mathcal{S}^{\blacklozenge}_{0}$.

###### Proof.

The assertion follows from Theorem-Definition 7.2.1, Remark 7.2.2, and Theorem
7.3.4. ∎

######  Corollary 7.3.6.

Suppose that $X_{0}/S_{0}$ is $N$-ordinary. Then, there are only finitely many
$F^{1}$-projective structures on $X/S$ and the following equality holds:

(631)
$\displaystyle\sharp(F^{1}\text{-}\mathrm{Pr}_{X/S})=\sharp(F^{N}\text{-}\mathrm{Pr}_{X_{0}/S_{0}}).$

###### Proof.

The assertion follows Theorem-Definition 7.3.5. ∎

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