Abelian \'{e
Yasuhiro Wakabayashi
In the present paper, we introduce and study a certain ordinariness of $\mathrm{GL_n$-do'pers (= dormant $\mathrm{GL_n$-opers) on a pointed proper smooth algebraic curve, which may be thought of as a generalization of the classical ordinariness of algebraic curves in positive characteristic. We consider an analogy of a type of assertion discussed and proved by S. Nakajima and M. Raynaud, i.e., ordinariness of $\mathrm{GL_n$-do'pers pulled-back via abelian coverings of the underlying curve. % pulled back via an \'{etale abelian covering map between pointed (prover) smooth curves. The main results of the present paper consist of the following two assertions: the first is that if the $\mathrm{GL_n$-do'per pulled-back via an \'{etale covering is ordinary, then the original $\mathrm{GL_n$-do'per is ordinary; the second is that if a given $\mathrm{GL_n$-do'per is ordinary and its underlying curve is sufficiently general, then the pull-back of this $\mathrm{GL_n$-do'per via % $\mr{GL_n$-do'per on an abelian covering is ordinary whenever its Galois group has the order prime to the characteristic of the base field.
1602.07061.tex
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\begin{document}

\title[Abelian coverings and ordinariness of dormant opers]{Abelian \'{e}tale coverings of generic curves\\ and ordinariness of dormant opers}
\author{Yasuhiro Wakabayashi}
\date{}
\markboth{Yasuhiro Wakabayashi}{}
\maketitle
\footnotetext{Y. Wakabayashi: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo,  153-8914, Japan;}
\footnotetext{e-mail: {\tt wkbysh@ms.u-tokyo.ac.jp};}
\footnotetext{2010 {\it Mathematical Subject Classification}: Primary 14H10, Secondary 14H60;}
\footnotetext{Key words and phrases: $p$-adic Teichm\"{u}ller theory, $p$-curvature, pointed stable curves,  opers, dormant opers, ordinariness.}
\begin{abstract}
In the present paper, we introduce and study a certain ordinariness of  $\mathrm{GL}_n$-do'pers (= dormant $\mathrm{GL}_n$-opers) on a pointed proper smooth algebraic curve, which may be thought of as a generalization of the classical ordinariness of algebraic curves in positive characteristic.
We consider  an analogy of a type of assertion   discussed and proved by S. Nakajima and  M. Raynaud, i.e., 
ordinariness of  $\mathrm{GL}_n$-do'pers pulled-back  via  abelian coverings of the underlying curve.
% pulled back via an  \'{e}tale abelian covering map between  pointed (prover) smooth curves. 
The main  results of the present paper consist of  the following two assertions:
the first is that  if the $\mathrm{GL}_n$-do'per pulled-back via an \'{e}tale covering is ordinary, then the original $\mathrm{GL}_n$-do'per is ordinary;
the second is that  if a given  $\mathrm{GL}_n$-do'per is ordinary and  its  underlying curve is sufficiently general, 
 then  the pull-back of this $\mathrm{GL}_n$-do'per via 
% $\mr{GL}_n$-do'per on 
 an abelian covering is ordinary whenever  its  Galois group has the order prime to the characteristic of the base field.
 
\end{abstract}
\tableofcontents 
\section*{Introduction}

\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection*{0.1} \label{S01}
In the present paper, we introduce and study a certain {\it ordinariness of  $\mr{GL}_n$-do'pers} (= {\it dormant $\mr{GL}_n$-opers}) on a pointed proper smooth  (or more generally, stable) algebraic  curve, which may be thought of as a generalization of the classical ordinariness of algebraic curves in positive characteristic  (i.e.,  the  $p$-rank of its jacobian is maximal).
%In the present paper, we study {\it ordinariness of $\mr{GL}_n$-do'pers (= dormant $\mr{GL}_n$-opers)} on 
%algebraic  curves of positive characteristic  and their \'{e}tale coverings.

Recall (cf. Definition \ref{y0108} (i)) that a $\mr{GL}_n$-oper on  an algebraic curve is, by definition, a triple $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j=0}^n)$ consisting of  a rank $n$  vector bundle $\mcF$ on the curve,  a(n) (integrable) connection $\nabla_\mcF$ on $\mcF$,   and a complete flag $\{ \mcF^j \}_{j=0}^n$ on $\mcF$ satisfying certain conditions.
 $\mr{GL}_n$-opers on a family of pointed  stable curves of arbitrary characteristic  were   defined and discussed  in ~\cite{Wak5} (or, in ~\cite{Wak6}).
But,  various properties of opers  in  the case where $n=2$ were originally  discussed, under the name of indigenous bundles,  in the context of the $p$-adic Teichm\"{u}ller theory developed by S. Mochizuki (cf. ~\cite{Mzk1} and ~\cite{Mzk2}). Also,  opers  defined  on a proper smooth curve over the field of complex numbers $\mbC$ or a formal disc over $\mbC$ were introduced and studied in the work of geometric Langlands program by A. Beilinson and V. Drinfeld (cf.  ~\cite{BD1} and ~\cite{BD2}). Thus, opers play a central role in  integrable systems  and representation theory of loop algebras.
If the underlying curve has characteristic $p>0$,
then it makes sense to speak of $\mr{GL}_n$-opers with {\it vanishing $p$-curvature}, which we refer to  as {\it dormant $\mr{GL}_n$-opers} or {\it $\mr{GL}_n$-do'pers} for short (cf. Definition \ref{y01019}).
$\mr{GL}_n$-do'pers and their moduli occur naturally in mathematics, as discussed in, e.g., ~\cite{JRXY},   ~\cite{Wak2}, and ~\cite{Wak5}.
Thus, a much more understanding of them will be of use in various areas relevant to the theory of opers in positive characteristic.

\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection*{0.2} \label{S021}

As  declared at the beginning of Introduction, the property  on $\mr{GL}_n$-do'pers which  we want to focus on  is ordinariness (cf. Definition \ref{y05019}).
The  condition of being ``{\it ordinary}"  may be   described  in terms  of  certain complexes of sheave obtained from the adjoint bundles associated with $\mr{GL}_n$-do'pers. 
%According to
The precise definition  of  ordinariness, as well as $\circledast$-ordinariness,  of  $\mr{GL}_n$-do'pers  will be given in Definition \ref{y05019}.
%of being ``{\it ordinary}", as well as ``{\it $\circledast$-ordinary}",    will be given.
In the case where $n=1$,
%If $n=1$, then
 the ordinariness of    $\mr{GL}_1$-do'pers is equivalent to the classical ordinariness of their underlying curves  (cf. the discussion in \S\,\ref{S25}, especially, Proposition \ref{g00078}).
Hence,  from this point of view, the ordinariness of $\mr{GL}_n$-do'pers may be thought of as a higher rank analogue of the classical ordinariness.


Here, we focus  our attention on the relation between the ordinariness and coverings of the curve.
Let us review an assertion   proved by S. Nakajima and M. Raynaud (cf. ~\cite{Nak}, ~\cite{Ray1}, and ~\cite{Bou}) as follows.
Let $Y \migi X$ be a Galois covering of connected proper smooth (hyperbolic) curves with Galois group $G$.
Suppose that  $X$ is  general (in the sense described in Theorem B below) and  the finite group  $G$ is either abelian or a central extension of two abelian groups with $(\sharp (G), g !) =1$.
Then, it follows from ~\cite{Nak}, Theorem 2, and ~\cite{Ray1}, TH\'{E}OR\`{E}ME 14, that $Y$ is ordinary.
(See also ~\cite{Ray1}, TH\'{E}OR\`{E}ME 2 and ~\cite{Bou} for the other assertions concerning the relation between the ordinariness and Galois coverings.) 
%If $X$ is ordinary and $G$ is either abelian 
 In the present paper, we consider  an analogy of such a type of  assertion
%   discussed and proved by S. Nakajima (cf. ~\cite{Nak}, Theorem 2) and  M. Raynaud (cf. ~\cite{Ray1}, TH\'{E}OR\`{E}ME 2 and TH\'{E}OR\`{E}ME 14),
%and COROLLAIRE 8),
for ordinariness of  $\mr{GL}_n$-do'pers pulled-back  via  abelian coverings.

\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection*{0.3} \label{S02}

Let us describe the main results of the present paper.
Let $p$ be a prime, $n$ a positive integer  with $n <p$,  $(g,r)$ a pair of nonnegative integers with $2g-2+r >0$,  and  $k$  an algebraically closed field of characteristic $p$.
Also, let $\mfX := (X, \{ \sigma_i \}_{i=1}^r)$ be a pointed proper smooth  curve of type $(g,r)$ over  $k$, that is to say, a geometrically connected, proper, and  smooth curve $X$ of genus $g$ over $k$ together with  $r$ marked points $\{ \sigma_i \}_{ i=1}^r$ in $X$.
Suppose that we are given a $\mr{GL}_n$-do'per $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{i=1}^j)$ on $\mfX$ and an \'{e}tale covering 
$(\mfY, w : Y \migi X)$ of $\mfX$ over $k$
%$\mfw : \mfY \migi \mfX$ 
(cf. Definition \ref{Y50199} (i) for the definition of an \'{e}tale covering of a curve), where $\mfY$ denotes  another pointed proper smooth curve over $k$ whose underlying curve is $Y$.
By pulling-back the data in $\mcF^\heartsuit$ via $w$, one obtains a $\mr{GL}_n$-oper $w^* (\mcF)^\heartsuit$ on $\mfY$.
%According to Definition \ref{y05019}, the notion of ``{\it ordinary}" of $\mr{GL}_n$-do'pers is defined.
The goal of the present paper is to prove the following two assertions concerning the relation between the ordinariness of $\mcF^\heartsuit$ and the ordinariness of its pull-back $w^*(\mcF)^\heartsuit$. 

%---------------------------------------------------------------------[begin theorem]----------------------
\vspace{3mm}
 %\vspace{5mm}
\begin{intthm} 
% [{\bf Generic \'{e}taleness of $\mfO \mfp_{\mfs \mfl_n, \hslash, \hslash \star \rho, g,r}^\ZZZ/\overline{\mfM}_{g,r}$}]
  \label{y019}
\leavevmode\\
\ \ \ 
Suppose that  the pull-back $w^*(\mcF)^\heartsuit$ is ordinary.
Then $\mcF^\heartsuit$ is  ordinary.
%If the $\mr{GL}_n$-do'per on $Y$ obtained by pulling back $\mcF^\heartsuit$ is ordinary, then $\mcF^\heartsuit$ is ordinary.
%If $w^*(\mcF^\heartsuit)$ is ordinary, then $\mcF^\heartsuit$ is ordinary.
\end{intthm}
%\vspace{5mm}
%------------------------------------------------------------------------------[end theorem]-------------

%Conversely, we shall prove the following assertion.
 
%---------------------------------------------------------------------[begin theorem]----------------------
\vspace{1mm}
 %\vspace{5mm}
\begin{intthm} 
% [{\bf Generic \'{e}taleness of $\mfO \mfp_{\mfs \mfl_n, \hslash, \hslash \star \rho, g,r}^\ZZZ/\overline{\mfM}_{g,r}$}]
  \label{y019}
\leavevmode\\
\ \ \ 
Suppose the following conditions:
\begin{itemize}
\item[(i)]
$\mcF^\heartsuit$ is ordinary;
\item[(ii)]
 $(\mfY, w)$
% $\mfw : \mfY \migi \mfX$ is
 is  an abelian covering with  $p \nmid \sharp (\mr{Gal} (\mfY/\mfX))$ (cf. Definition \ref{Y50199} (i) and (ii));
\item[(iii)]
$\mfX$ is general in the moduli stack $\mfM_{g,r}$ classifying pointed proper smooth curves of type $(g,r)$ over $k$.
%\item[(iii)]
%$\mcF^\heartsuit$ is ordinary.
\end{itemize}
(Here,  we recall that $\mfM_{g,r}$ is irreducible (cf. ~\cite{DM}, \S\,5); thus, it makes sense to speak of a ``{\it general}" $\mfX$, i.e., an $\mfX$  that determines  a point of $\mfM_{g,r}$ lying  outside some fixed closed substack.)
Then, the pull-back $w^* (\mcF)^\heartsuit$ is ordinary.
\end{intthm}
\vspace{5mm}
%------------------------------------------------------------------------------[end theorem]-------------



%----------------------------------------------------------------------[begin subsection]-------------
\subsection*{0.4} \label{S03}
In the final section of the present paper, we shall also consider, as a generalization of the ordinariness of $\mr{GL}_n$-do'pers,  a certain ordinariness of $\mfg$-do'pers (= dormant $\mfg$-oper) for a general  semisimple Lie algebra $\mfg$.
It is a key observation  (cf. Proposition \ref{ghu}) that 
the $\circledast$-ordinariness of $\mr{GL}_n$-do'pers corresponds exactly  
the unramifiedness  of the moduli stack $\mfO \mfp_{\mr{GL}_n, g,r}^{^\text{Zzz...}}$ (cf. \S\,\ref{S26}, (\ref{DD077})) of  $\mr{GL}_n$-do'pers (which is isomorphic to the moduli stack $\mfO \mfp_{\mfs \mfl_n, g,r}^{^\text{Zzz...}}$ of $\mfs \mfl_n$-do'pers)  at the classifying points. 
Thus, by taking account of this observation, one   may generalize the notion of ordinariness  to the case of  $\mfg$-do'pers (i.e., the {\it $\circledast_\mfg$-ordinariness} defined in  Definition \ref{D0001}) 
in terms  of such a local property applied to  the moduli stack $\mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}}$ of $\mfg$-do'pers.
If $\mfg$ satisfies a certain condition, which we refer to as being  {\it  admissibly of classical type $A$} (cf. Definition \ref{D00012}), then there exists a closed immersion
$\mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}} \migi \mfO \mfp_{\mfs \mfl_m, g,r}^{^\text{Zzz...}}$ for a sufficiently large $m$.
%For an arbitrary $\mfg$, a choice of $\mfs \mfl_2$-triple of $\mfg$ yields
%a closed immersion $\mfO \mfp_{\mfs \mfl_2, g,r}^{^\text{Zzz...}} \migi \mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}}$.
This morphism allow us to  relate    the ($\circledast$-)ordinariness of $\mr{GL}_m$-do'pers  with the $\circledast_\mfg$-ordinariness (cf. Proposition \ref{PPP2} and Theorem 5.2.2).
Moreover, for such a $\mfg$, we obtain (cf. Theorem 5.1.4)  an assertion which may be thought of as 
% IN the last section, we consider
  a generalization of Theorem B.
  % to the ordinariness of $\mfg$-ordinarimess for some what general $\mfg$.
% By means of Theorem B, one may prove an assertion similar to Theorem B.
% (See also Proposition ? for a relation with $\mfs \mfl_2$-ordinariness.)
 
 
\vspace{5mm}
\hspace{-4mm}{\bf Acknowledgement} \leavevmode\\
 \ \ \  
 The author cannot express enough his sincere and  deep gratitude to  all those who give  the opportunity or   impart  the great joy of studying 
 %getting in touch with
    mathematics to him.
The author means the present paper for a gratitude letter to them.
 The author would  also like to thank Professor Yuichiro Hoshi for helpful discussions and advice.
The author was partially  supported by the Grant-in-Aid for Scientific Research (KAKENHI No.\,15J02721) and the FMSP program at the Graduate School of Mathematical Sciences of the University of Tokyo.
%The author is partially supported by a 


 
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%--[ begin  section1]---%%%%%%
\vspace{10mm}
\section{Preliminaries} \vspace{3mm}


%----------------------------------------------------------------------[begin subsection]-------------
%\subsection{}\label{y0}

First, we shall introduce our notation and review some topics involved (e.g., integrable vector bundles on log-curves, Cartier operators,  and ordinariness of algebraic curves in positive characteristic).
% in the present paper.
%In this section, we 
%, and a pair of nonnegative integers $(g, r)$ with $2g -2+r >0$.
%Let $g$ be an integer with $g > 1$, and
%denote by $\mfM_{g}$ the moduli space classifying proper smooth curve over $k$ of genus $g$
A {\it log scheme} means, in the present paper,  a scheme equipped with a logarithmic structure in the sense of Fontaine-Illusie (cf.  ~\cite{KaKa}, (1.2)).
For a log scheme (resp., a morphism of log schemes) indicated, say, by the notation ``$X^\mr{log}$" (resp., ``$f^\mr{log}$"), we shall use the notation ``$X$" (resp., ``$f$")  for indicating the underlying scheme (resp., the underlying morphism of schemes).
Throughout the present paper, {\it we fix a prime $p$ and  
%a positive integer  $n$ with $n <p$,  
an algebraically closed field $k$ of characteristic $p>0$}.





\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S12}

Recall from ~\cite{ACGH},  Definition 4.5, the definition of a log-curve as follows.
%-----------------------------------------------------------------------[begin definition]------------------
\vspace{0mm}
\bde \label{Y5019}\leavevmode\\
 \ \ \  
A {\bf log-curve} is a log smooth integrable morphism $f^\mr{log} : X^\mr{log} \migi S^\mr{log}$ of fs log schemes
% (cf. ~\cite{KaKa}, \S\,3;  ~\cite{KaFu}, Theorem 2.6)
%(or its base-change via a morphism $T^\mr{log} \migi S^\mr{log}$ of log schemes)
 such that the geometric fibers of the underlying morphism $f : X \migi S$  are reduced connected $1$-dimensional schemes.
If $T^\mr{log}$ is an arbitrary log scheme, then we shall mean, by {\bf a log-curve over $T^\mr{log}$}, a log scheme $Y^\mr{log}$ over $T^\mr{log}$ which is isomorphic to $X^\mr{log} \times_{f^\mr{log}, S^\mr{log}}T^\mr{log}$ for some log-curve $f^\mr{log} : X^\mr{log} \migi S^\mr{log}$ and some morphism $T^\mr{log} \migi S^\mr{log}$ of log schemes.
(If $\Omega_{Y^\mr{log}/T^\mr{log}}$ denotes  the sheaf of logarithmic $1$-forms on $Y^\mr{log}$ relative to $T^\mr{log}$,  then it  is a line bundle and admits a universal log derivation $d : \mcO_Y \migi \Omega_{Y^\mr{log}/T^\mr{log}}$.)
   \ede
%-----------------------------------------------------------------------[end definition]-------------------
\vspace{3mm}

Also, let us recall (families of) pointed stable curves and  log-curves associated with them, as follows.
Let $(g,r)$ be  a pair of nonnegative integers with $2g -2+r >0$.
Denote by $\overline{\mfM}_{g,r}$ the moduli stack of $r$-pointed stable curves (cf. ~\cite{Kn2}, Definition 1.1) over $k$ of genus $g$ (where we shall refer to such pointed stable curves as being {\bf of type $(g,r)$}),
%with $r$ marked points
% (i.e., of type $(g,r)$),
  and by $f_\mr{\mft \mfa \mfu} : \mfC_{g,r} \migi \overline{\mfM}_{g,r}$ the tautological curve over $\overline{\mfM}_{g,r}$, with its $r$ marked points $\sigma_{\mft \mfa \mfu, 1} , \cdots, \sigma_{\mft \mfa \mfu, r} : \overline{\mfM}_{g,r} \migi \mfC_{g,r}$.
Recall (cf. ~\cite{Kn2}, Corollary 2.6 and Theorem 2.7; ~\cite{DM}, \S\,5) that $\overline{\mfM}_{g,r}$ may be represented by a geometrically connected, proper, and smooth Deligne-Mumford stack over $k$ of dimension $3g-3+r$.
Also, recall (cf. ~\cite{KaFu}, Theorem 4.5) that $\overline{\mfM}_{g,r}$ has a natural log structure given by the divisor at infinity,
where we shall denote the resulting log stack  by $\overline{\mfM}_{g,r}^{\mr{log}}$.
Moreover, we obtain a log structure on $\mfC_{g,r}$ by taking   the divisor which is the union of the $\sigma_{\mft \mfa \mfu, i}$'s and the pull-back of the divisor at infinity of $\overline{\mfM}_{g,r}$ (resp., the divisor defined as the pull-back of the divisor at infinity of $\overline{\mfM}_{g,r}$); let us  denote  the resulting log stack by   $\mfC^{\mr{log}}_{g,r}$ (resp., $\mfC^{\mr{log}'}_{g,r}$).
$f_\mr{\mft \mfa \mfu} : \mfC_{g,r} \migi \overline{\mfM}_{g,r}$ extends naturally to a morphism 
$f_\mr{\mft \mfa \mfu}^\mr{log} : \mfC^\mr{log}_{g,r} \migi \overline{\mfM}^\mr{log}_{g,r}$ (resp., $f_\mr{\mft \mfa \mfu}^{\mr{log}'} : \mfC^{\mr{log}'}_{g,r} \migi \overline{\mfM}^\mr{log}_{g,r}$) of log stacks.
%If, moreover,
Let $\mfM_{g,r}$ denotes the substack of $\overline{\mfM}_{g,r}$ classifying pointed proper smooth curves; it is a dense open substack of $\overline{\mfM}_{g,r}$ and coincides  with  the locus  in which the log structure of $\overline{\mfM}_{g,r}^\mr{log}$  becomes trivial.

%In the rest of this section,  w
Let us  fix a scheme $S$ over $k$ and
% be a scheme (or, more generally, a stack ) over $k$ and 
 a pointed stable curve
\begin{equation}
\label{X}
 \mfX : =(f :X \migi S, \{ \sigma_i : S \migi X\}_{i=1}^r)\end{equation}
 over $S$ of type $(g,r)$, which consists of a (proper) semistable curve $f : X \migi S$ over $S$ of genus $g$ and $r$ marked points $\sigma_i : S \migi X$ ($i = 1, \cdots, r$).
%$\mfX_{/S}$ determines its classifying morphism $s : S \migi \overline{\mfM}_{g,r}$ and an isomorphism $X \isom S \times_{s, \overline{\mfM}_{g,r}} \mfC_{g,r}$ over $S$.
By pulling-back the  log structures of $\overline{\mfM}^{\mr{log}}_{g,r}$ and $\mfC^{\mr{log}}_{g,r}$ (resp.,  $\mfC^{\mr{log}'}_{g,r}$) via its classifying morphism,
we obtain log structures on $S$ and $X$ respectively; we denote  the resulting log stacks by 
\begin{align}
S^{\mfX\text{-}\mr{log}} \ \ \text{and}  \ \ X^{\mfX\text{-}\mr{log}} \ \  (\text{resp.},  \ X^{\mfX\text{-}\mr{log}'}).
\end{align} 
If there is no fear of confusion, we shall abbreviate 
them
%$S^{\mfX_{/S}\text{-}\mr{log}}$ and $X^{\mfX_{/S}\text{-}\mr{log}}$
 to $S^\mr{log}$ and $X^\mr{log}$ (resp., $X^{\mr{log}'}$) respectively.
The structure morphism $f : X \migi S$ extends to a morphism $f^\mr{log} : X^{\mr{log}} \migi S^{\mr{log}}$ (resp., $f^{\mr{log}'} : X^{\mr{log}'} \migi S^{\mr{log}}$) of log schemes, 
by  which the log scheme $X^\mr{log}$ (resp., $X^{\mr{log}'}$)  becomes   a log-curve over $S^\mr{log}$.
%which is log smooth 
%(cf. ~\cite{KaKa}, \S\,3;  ~\cite{KaFu}, Theorem 2.6). 
%In particular, $\Omega_{X^\mr{log}/S^\mr{log}}$  is a line bundle.
If, moreover, the underlying scheme $X$ is smooth over $S$, then $S^{\mfX \text{-} \mr{log}} = S$.

Denote by $D_{\mfX}$ the \'{e}tale effective  relative divisor on $X$ relative to $S$ defined to be the union of the image of the marked points $\sigma_i$ ($i =1, \cdots, r$).
If $\omega_{X/S}$ denotes  the dualizing sheaf of $X$ over $S$, then $\omega_{X/S}$ is  isomorphic to $\Omega_{X^\mr{log}/S^\mr{log}}(-D_{\mfX_{}})$ ($= \Omega_{X^{\mr{log}'}/S^\mr{log}}$).

\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S11}

%Let $\mfX_{/S}$ be as above and 
Let $S^\mr{log}$ be a log scheme and $f^\mr{log} : X^\mr{log} \migi S^\mr{log}$ a log-curve over $S^\mr{log}$,
%Let  $T^\mr{log} = (T, \alpha_T : M_T \migi \mcO_T)$, $Y^\mr{log} := (Y, \alpha_Y : M_Y \migi \mcO_Y)$ be fine log schemes and $f^\mr{log} : Y^\mr{log} \migi T^\mr{log}$ a log smooth morphism of relative dimension $1$ (cf. ~\cite{KaKa}, \S\,3;  ~\cite{KaFu}, Theorem 2.6), 
and $\mcF$  a rank $n$ ($\geq 0$) vector bundle (i.e., a locally free coherent sheaf of finite rank) on $X$.
%Denote by $\Omega_{Y^\mr{log}/T^\mr{log}}$  the sheaf of logarithmic $1$-forms on $Y^\mr{log}$ relative to $T^\mr{log}$, which  is a line bundle and admits a universal log derivation $d : \mcO_Y \migi \Omega_{Y^\mr{log}/T^\mr{log}}$.
%Also, let 
By an {\bf $S$-log connection} on  $\mcF$,  we mean
% (cf. ~\cite{Tra}, \S\,2.2)
 an
$f^{-1}(\mcO_S)$-linear morphism
\begin{equation}
  \nabla : \mcF \migi \Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcF
  \end{equation}
 satisfying the condition that
 \begin{equation}  \nabla (a \cdot m) =  d (a) \otimes m + a \cdot \nabla (m)\end{equation}
 for any local sections $a \in \mcO_X$, $m \in \mcF$.


An {\bf  integrable vector bundle} on $X^\mr{log}/S^\mr{log}$ (of rank $n$) is a pair $(\mcF, \nabla_\mcF)$ consisting of a vector bundle $\mcF$ on $X$ (of rank $n$) and an $S$-log connection $\nabla_\mcF$ on $\mcF$. 
(Note that since $\Omega_{X^\mr{log}/S^\mr{log}}$ is a line bundle, any $S$-log connection on a vector bundle  is automatically  integrable, i.e., has vanishing curvature.)
If, moreover, $\mcF$ is of rank $1$, then we shall refer to $(\mcF, \nabla_\mcF)$ as an {\bf  integrable line bundle} on $X^\mr{log}/S^\mr{log}$.  
For simplicity, we shall  refer  to an integral  vector bundle on $X^{\mfX \text{-}\mr{log}}/S^{\mfX \text{-}\mr{log}}$ (where $\mfX$ denotes a  pointed stable curve whose underlying curve coincides with $X$) as an {\bf integrable vector bundle on $\mfX_{}$}. 


Let $(\mcF, \nabla_\mcF)$ and $(\mcG, \nabla_\mcG)$  be an integrable vector bundles
%If, moreover, $(\mcG, \nabla_\mcG)$ is a log integrable vector bundle
 on  $X^\mr{log}/S^\mr{log}$.
 We shall write $\nabla_\mcF \otimes \nabla_\mcG$ (by abuse of notation) for the $S$-log connection on the tensor product $\mcF \otimes \mcG$
 determined by
 \begin{equation} \label{DD091}
 (\nabla_\mcF \otimes \nabla_\mcG) (a \otimes b) = \nabla_\mcF (a )\otimes b + a \otimes \nabla_\mcG (b).
 \end{equation}
 for any local sections $a \in \mcF$ and $b \in \mcG$.
 %  induced by $\nabla_\mcF$ and $\nabla_\mcG$.
 Also, we define an {\bf isomorphism of integrable vector bundles} from $(\mcF, \nabla_\mcF)$ to $(\mcG, \nabla_\mcG)$ is an isomorphism $\mcF \isom \mcG$ of vector bundles that is compatible with the respective connections $\nabla_\mcF$ and $\nabla_\mcG$.



%Next, we shall consider the pull-back of an integrable vector bundle via an \'{e}tale covering of pointed stable curves.
Next, suppose that we are given 
an integrable vector bundle $(\mcF, \nabla_\mcF)$ on $X^\mr{log}/S^\mr{log}$ and a commutative square diagram
\begin{align}
\xymatrix{
Y^\mr{log} \ar[r]^{w^\mr{log}} \ar[d] & X^\mr{log} \ar[d]
\\
T^\mr{log} \ar[r] & S^\mr{log}
}
\end{align} 
of log schemes, where  $T^\mr{log}$ denotes a log scheme over $k$ and $Y^\mr{log}$ denotes  a log-curve over   $T^\mr{log}$.
(Hence, we have a morphism $w^\sharp : w^{-1} (\Omega_{X^\mr{log}/S^\mr{log}}) \migi \Omega_{Y^\mr{log}/T^\mr{log}}$ induced by $w$.)
One may obtain a $T$-log connection 
\begin{align}
\nabla_{w^*(\mcF)} : w^*(\mcF) \migi \Omega_{Y^\mr{log}/T^\mr{log}}\otimes w^*(\mcF)
\end{align}
 on the pull-back $w^*(\mcF)$ ($= \mcO_Y \otimes_{w^{-1}(\mcO_X)} w^{-1}(\mcF)$) given by 
assigning   
\begin{align}
a \otimes v \mapsto d (a) \otimes v + a \cdot (w^\sharp \otimes \mr{id}_{w^{-1}(\mcF)}) ( w^{-1}(\nabla_\mcF) (v))
\end{align}
 for any local sections $a \in \mcO_Y$,  $v \in w^{-1}(\mcF)$.
Thus, we have an integrable vector bundle 
\begin{align}
(w^*(\mcF), \nabla_{w^* (\mcF)})
\end{align}
 on $Y^\mr{log}/T^\mr{log}$. 




\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S13}


%Let $T$ be a scheme  over $k$ ($\supseteq \mbF_p$) and $f : Y \migi T$ a scheme over $T$.
In the  rest of this section, we fix a scheme $S$ over $k$ and a pointed stable curve
$\mfX := (f : X \migi S, \{ \sigma_i\}_{i=1}^r)$  over $S$ of type $(g,r)$ (hence, we have a log-curve $X^\mr{log}/S^\mr{log} := X^{\mfX\text{-}\mr{log}} / S^{\mfX\text{-}\mr{log}}$).
%Let $S^\mr{log}$ be a log scheme over $k$ and 
%$f^\mr{log} : X^\mr{log} \migi S^\mr{log}$ a log-curve over $S^\mr{log}$.
%$\mfX : = (X/S, \{ \sigma_i \}_{i=1}^r)$  a pointed stable curve over $S^\mr{log}$.
Denote by $F_S : S \migi S$ (resp., $F_X : X \migi X$) the absolute Frobenius morphism of $S$ (resp., $X$).
The  {\it Frobenius twist of $X$ over $S$} is, by definition,  the base-change $X^{(1)}_S$ ($:= X \times_{S, F_S} S$) of $f : X\migi S$ via
 $F_S : S \migi S$.
Denote by $f^{(1)} : X^{(1)}_S \migi S$ the structure morphism of the Frobenius twist of $X$ over $S$. 
The {\it relative Frobenius morphism of $X$ over $S$} is  the unique morphism $F_{X/S} : X \migi X^{(1)}_S$ over $S$ that fits into a commutative diagram of the form 
\begin{align}
\xymatrix{
X \ar[rrrrd]^{F_X}\ar[ddrr]_{f} \ar[rrd]_{ \ \ \ \ \ \ \  \  \   F_{X/S}} & & &   &   \\
& & X_{S}^{(1)}  \ar[rr]_{\mr{id}_X \times F_S} \ar[d]^{f^{(1)}}  \ar@{}[rrd]|{\Box}  &  &  X \ar[d]_{f} \\
&  & S \ar[rr]_{F_S} & &  S.
}
\end{align}
%\begin{equation} \begin{CD}
%X @> F_{X/S} >> X^{(1)}_S @> \mr{id}_X \times F_S >> X
%\\
%@V f VV @V f^{(1)} VV @V f VV
%\\
%S @> \mr{id}_S >> S @> F_S >> S.
%\end{CD} \end{equation}
%Here,  the upper  composite in this diagram coincides with $F_X$
% and the right-hand square is, by the definition of $X^{(1)}_S$, cartesian.

%If  $Y^\mr{log}$ is a log-curve over $S^\mr{log}$ and , then 
%$\mfY := (Y/S, \{ \delta_{i'} \}_{i'})$ is a pointed stable curve over $S$ (of type $(g',r')$ for some pair $(g', r')$) and  $w : Y \migi X$ is a morphism over $S$, 
%then we shall denote by $w^{(1)} : Y^{(1)}_S \migi X^{(1)}_S$ the base-change of  $w$  via $F_S : \mr{Spec}(S) \migi \mr{Spec}(S)$. 


 If we write 
 \begin{equation}
 \sigma_i^{(1)} := (\sigma_i \circ F_S, \mr{id}_S)  : S \migi X \times_{S, F_S} S \ (= X_S^{(1)})
 \end{equation}
  (for each $i \in \{ 1, \cdots, r\}$), then the collection of data
 \begin{equation}
 \mfX^{(1)}_{} := (f^{(1)} : X^{(1)}_S \migi S, \{ \sigma^{(1)}_i \}_{i=1}^r) 
 \end{equation}
forms a pointed stable curve over $S$ of type $(g,r)$.
In particular, we obtain canonically a log structure on $X^{(1)}_S$.

If $\mfY := (Y/S, \{ \sigma'_{i} \}_{i}^{r'})$ is a pointed stable curve over $S$ (of type $(g',r')$ for some pair $(g', r')$) and  $w : Y \migi X$ is a morphism over $S$, then we shall denote by $w^{(1)} : Y^{(1)}_S \migi X^{(1)}_S$ the base-change of  $w$  via $F_S : \mr{Spec}(S) \migi \mr{Spec}(S)$. 




\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S14}

%Before preceding the discussion, 
In this subsection, we shall recall  the  Cartier operator associated with an integrable vector bundle.
%For the rest of this section, we suppose that $k$ is of characteristic $p > 0$ (i.e., the condition  $(\text{Char})_{p}^{\mbW}$ is satisfied) and  the pinning $\ \ooalign{$\mbG$ \cr $\hspace{-1mm}\swarrow$}$  may be defined, in a natural sense,  over $\mbF_p$.
 Let $(\mcF, \nabla_\mcF)$ be an integrable vector bundle on $\mfX$ with {\it vanishing $p$-curvature}   (cf. ~\cite{Wak5}, Definition 3.2.1,  for the definition of $p$-curvature).
Although $\nabla_\mcF$ is not $\mcO_X$-linear, but it
may be thought, via the underlying homeomorphism of $F_{X/S}$,   of as an {\it $\mcO_{X_S^{(1)}}$-linear} morphism
\begin{equation} F_{X/S*}(\nabla_\mcF) : F_{X/S*}(\mcF) \migi F_{X/S*}(\Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcF). \label{F}\end{equation}
In particular, both $ F_{X/S*}(\mr{Ker}(\nabla_\mcF))$ ($= \mr{Ker}(F_{X/S*}(\nabla_\mcF))$) and $ F_{X/S*}(\mr{Coker}(\nabla_\mcF))$ ($= \mr{Coker}(F_{X/S*}(\nabla_\mcF))$) may be thought of as $\mcO_{X^{(1)}_S}$-modules.

The {\it Cartier operator} (cf. ~\cite{Og}, Proposition 1.2.4)  associated with 
$(\mcF, \nabla_\mcF)$ is, by definition, a unique $\mcO_{X^{(1)}_S}$-linear morphism
\begin{equation}
C^{(\mcF, \nabla_\mcF)} : F_{X/S*}(\Omega_{X^{\mr{log}}/S^\mr{log}} \otimes \mcF) \migi \Omega_{X^{(1)\mr{log}}_S/S^\mr{log}} \otimes F_{X/S*}(\mcF)
\label{CartierF}
\end{equation}
satisfying the following condition:
 for any locally defined logarithmic derivation $\partial \in \mcT_{X^\mr{log}/S^\mr{log}}$ and any local section $a \in \Omega_{X^{\mr{log}}/S^\mr{log}} \otimes \mcF$, the equality 
\begin{equation}
\langle C^{(\mcF, \nabla_\mcF)} (a), (\mr{id}_X \times F_S)^*(\partial)\rangle = \langle a, \partial^{[p]}\rangle - \nabla_{\mcF} (\partial)^{\circ (p-1)}(\langle a, \partial\rangle)
\label{Cartier2F}
\end{equation}
is satisfied, where $\nabla_\mcF (\partial)^{\circ (p-1)}$ denotes the $(p-1)$-st iterate of the endomorphism $\nabla_\mcF (\partial)$ of $\mcF$,
and $\langle -, -\rangle$ in the both sides  denote  the pairings induced by the natural pairing  $\Omega_{X^{\mr{log}}/S^\mr{log}}  \times \mcT_{X^{\mr{log}}/S^\mr{log}} \migi \mcO_X$.
Moreover, since $(\mcF, \nabla_\mcF)$ has vanishing $p$-curvature,
there exists (cf. ~\cite{Og}, the discussion following  Proposition 1.2.4) a commutative square
\begin{equation} \label{Cartier3F}  \begin{CD}
F_{X/S*}(\Omega_{X^{\mr{log}}/S^\mr{log}} \otimes \mcF) @> C^{(\mcF, \nabla_\mcF)} >>\Omega_{X^{(1)\mr{log}}_S/S^\mr{log}} \otimes F_{X/S*}(\mcF)
\\
@V VV @AA A
\\
F_{X/S*}(\mr{Coker}(\nabla_\mcF)) @> \overline{C}^{(\mcF, \nabla_\mcF)} >> \Omega_{X^{(1)\mr{log}}_S/S^\mr{log}} \otimes F_{X/S*}(\mr{Ker}(\nabla_\mcF)),
\end{CD} \end{equation} 
where
%$\pi^{(\mcF, \nabla_\mcF)}$ and $\iota^{(\mcF, \nabla_\mcF)}$ denote
 the left-hand  and  right-hand vertical arrows denote
  the $\mcO_{X^{(1)}_S}$-linear morphisms induced by the quotient  $\Omega_{X^{\mr{log}}/S^\mr{log}} \otimes \mcF \migisurj \mr{Coker}(\nabla_\mcF)$ and the inclusion $\mr{Ker}(\nabla_\mcF) \migiincl \mcF$ respectively.
%natural surjection and injection respectively.
Finally, it follows from ~\cite{Og3}, Theorem 3.1.1, that the morphism $\overline{C}^{(\mcF, \nabla_\mcF)}$ (i.e., the lower horizontal arrow in (\ref{Cartier3F})) is an isomorphism.
(Here, we note that since $X^\mr{log}/S^\mr{log}$ is of Cartier type in the sense of ~\cite{KaKa}, Definition (4.8), the notion of the exact relative Frobenius map in the statement of 
 ~\cite{Og}
%{\it loc.\,cit.}
 coincides with $F_{X/S}$).
Denote by $d' : \mcO_X \migi \omega_{X/S} \ (= \Omega_{X^{\mr{log}'}/S^\mr{log}})$ the universal log derivation of $X^{\mr{log}'}/S^\mr{log}$.
By
% considering the case where $(\mcF, \nabla)$ is taken to be $(\mcO_X, d)$ (resp., $(\mcO_X, d')$) and 
 composing $\overline{C}^{(\mcO_X, d)}$ (resp., $\overline{C}^{(\mcO_X, d')}$) with the quotient  $\Omega_{X^\mr{log}/S^\mr{log}} \migi \mr{Coker} (d)$ (resp., $\omega_{X/S} \migisurj \mr{Coker}(d')$), we obtain an $\mcO_{X_S^{(1)}}$-linear  morphism
\begin{align} \label{E1002}
C_{X/S} & : F_{X/S*} (\Omega_{X^\mr{log}/S^\mr{log}}) \migi \Omega_{X^{(1)\mr{log}}_S/S^\mr{log}}  \\ (\text{resp.,} \ C'_{X/S} & :  F_{X/S*} (\omega_{X/S}) \migi \omega_{X_S^{(1)}/S}). \notag
\end{align}





\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S15}

Let us recall the ordinariness of algebraic curves,  as follows.
%-----------------------------------------------------------------------[begin definition]------------------
\vspace{0mm}
\bde \label{Y5019}\leavevmode\\
 \ \ \  
 We shall say  that $\mfX$ is {\bf ordinary} if the $\mcO_S$-linear morphism
\begin{equation}
\mbR^1 f^{(1)}_* (F^*_{X/S}) : \mbR^1f^{(1)}_* (\mcO_{X_S^{(1)}}) \migi \mbR^1 f_* (\mcO_X) \ (=\mbR^1 f^{(1)}_* (F_{X/S*}(\mcO_X)))
\end{equation}
 induced from $F_{X/S}^* : \mcO_{X_S^{(1)}} \migi F_{X/S *}(\mcO_X)$ is an isomorphism (cf. e.g., ~\cite{Bou}, Definition 1.2,  for the definition of the ordinariness of semistable curves over an algebraically closed field).
   \ede
%-----------------------------------------------------------------------[end definition]-------------------
\vspace{3mm}


It is well-known that the dual of $\mbR^1 f^{(1)}_* (F^*_{X/S})$ coincides with 
\begin{align}
f_*^{(1)}(C'_{X/S}) : ( f^{(1)}_* (F_{X/S*}(\omega_{X/S}))=) \  f_* (\omega_{X/S}) \migi f^{(1)}_*(\omega_{X_S^{(1)}/S})
\end{align}
  via Grothendieck-Serre duality.
%the Cartier operator 
%\begin{equation} \label{E01}
%C^{(\mcO_X, d')} : f_* (\omega_{X/S}) \migi f^{(1)}_*(\omega_{X_S^{(1)}/S})
%\end{equation}
% associated with $d'$ (cf. ~\cite{Og}, Proposition 1.2.4,  for the definition of the Cartier operator associated with an integrable vector bundle).
In particular, 
{\it  $\mfX$ is  ordinary if and only if $f_*^{(1)}(C'_{X/S})$ is an isomorphism}.
Recall   that the locus of $\overline{\mfM}_{g,r}$ classifying ordinary pointed stable curves is open and dense.


%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{P01}\leavevmode\\
 \ \ \ 
$\mfX$ is ordinary if and only if $f_*^{(1)}(C_{X/S}) : f_* (\Omega_{X^\mr{log}/S^\mr{log}}) \migi f^{(1)}_*(\Omega_{X_S^{(1)\mr{log}}/S^\mr{log}})$ is an isomorphism.
\epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
It suffices to consider the case where $r >0$.
Consider the morphism of short exact sequences 
\begin{align} \label{E1111}
\xymatrix{
0 \ar[r] & F_*( \omega_{X/S} )\ar[r]^{\hspace{-3mm}\mr{incl}} \ar[d]^{C'_{X/S}} &  F_*(\Omega_{X^\mr{log}/S^\mr{log}}) \ar[r] \ar[d]^{C_{X/S}} &  \bigoplus_{i=1}^r F_*(\sigma^{(1)}_{i*} (\mcO_S)) \ar[r] \ar[d]^{\bigoplus \mr{id}} & 0 \\
0 \ar[r] & \omega_{X_S^{(1)}/S} \ar[r]^{\hspace{-3mm}\mr{incl}} & \Omega_{X_S^{(1)\mr{log}}/S^\mr{log}} \ar[r] & \bigoplus_{i=1}^r \sigma_{i*} (\mcO_S) \ar[r]  & 0.
}
\end{align}
By taking the higher direct  images via $f^{(1)}$, (\ref{E1111}) gives rise to  a morphism of complexes
\begin{align} \label{DD025}
\xymatrix{
0 \ar[r] \ar[d]^{\mr{id}} &  f_* (\omega_{X/S}) \ar[r] \ar[d]^{f_{*}^{(1)}(C'_{X/S})} &  f_* (\Omega_{X^\mr{log}/S^\mr{log}})    \ar[r] \ar[d]^{f_{*}^{(1)}(C_{X/S})} & \mcO_S^{\oplus r} \ar[r] \ar[d]^{\mr{id}} & \mbR^1 f_* (\omega_{X/S}) \ar[r] \ar[d]^{\mbR^1 f_*^{(1)} (C'_{X/S})}& 0 \ar[d]^{\mr{id}} \\
0 \ar[r] & f^{(1)}_*(\omega_{X_S^{(1)}/S}) \ar[r] & f^{(1)}_*(\Omega_{X_S^{(1)\mr{log}}/S^\mr{log}}) \ar[r] &  \mcO_S^{\oplus r} \ar[r] & \mbR^1  f^{(1)}_*(\omega_{X_S^{(1)}/S})  \ar[r] & 0.
}
\end{align}
But, the morphism 
\begin{align}
\mbR^1 f_*^{(1)} (C') :  \mbR^1 f_* (\omega_{X/S}) \migi \mbR^1f_*^{(1)} (\omega_{X_S^{(1)}/S})
\end{align}
(i.e., the second vertical arrow from the right in (\ref{DD025})) is an isomorphism since it coincides, by Grothendieck-Serre duality, with the dual of the identity morphism of $\mcO_S$.
Hence, by the five lemma, $f_*^{(1)}(C_{X/S})$ is an isomorphism if and only if $f_*^{(1)}(C'_{X/S})$ is an isomorphism.
This completes the proof of Proposition \ref{P01}.
\end{proof}
%\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------

\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S16}
We shall give the definition of an \'{e}tale covering between pointed proper smooth curves, appeared in the statement of the main results.

%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{Y50199}\leavevmode\\
% \ \ \  
\vspace{-5mm}
\begin{itemize}
\item[(i)]
 An  {\bf \'{e}tale covering of $\mfX$} is a triple 
 \begin{align}
 (\mfY, w, v),
 \end{align}
 where
 \begin{itemize}
 \item[$\bullet$]
$\mfY$ denotes  a pointed stable curve $\mfY  := (Y/T, \{ \sigma'_{i} \}_{i=1}^{r'})$ over a $k$-scheme $T$ (of type $(g', r')$ for some pair of nonnegative integers $(g', r')$ with $2g'-2+r' >0$);
\item[$\bullet$]
 $w$ and $v$ denote  morphisms of $k$-schemes $w : Y \migi X$, $v : T \migi S$,
\end{itemize}
 satisfying the following conditions:
 \begin{itemize}
 \item[(i-1)]
 the square diagram
 \begin{align} \label{C01}
\xymatrix{
Y \ar[r]^w \ar[d] & X  \ar[d]
\\
T  \ar[r]_v & S
}
\end{align}
is commutative, where the two vertical arrows denote the  structure morphisms of  pointed stable curves;
\item[(i-2)]
the  morphism $w_v  :  Y \migi X \times_S T$ over $T$ induced by (\ref{C01}) is finite and \'{e}tale  satisfying  that $(\mr{pr} \circ w_v)^{-1}(D_{\mfX}) = D_{\mfY}$, where $\mr{pr}$ denotes the projection $X \times_S T \migi X$.
%, where $D_{\mfX}$ and $D_\mfY$ denote the divisors on $X \times_S T$ and $Y$ defined as the union of the marked points $\{ \sigma_i \times  \mr{id}_T\}_{i=1}^r$ and $\{ \sigma'_i \}_{i=1}^{r'}$ respectively.
 %  a finite \'{e}tale morphism
 \end{itemize}
 (Hence,  the natural  morphism $w^* (\Omega_{X^\mr{log}/S^\mr{log}}) \migi \Omega_{Y^\mr{log}/T^\mr{log}}$ is an isomorphism.)
% If there is no fear of confusion, then we shall simply denote such an \'{e}tale covering  by $\mfY$.
 % (or $\mfY/\mfX$).
  An {\bf \'{e}tale covering of $\mfX$ over $S$} is an \'{e}tale covering $(\mfY, w, v)$ of $\mfX$ such that $\mfY$ is a pointed stable curve over $S$ and $v = \mr{id}_S$.
  We shall abbreviate  $(\mfY, w, \mr{id}_S)$ to $(\mfY, w)$ for simplicity.
  \item[(ii)]
  We shall say that an \'{e}tale covering $(\mfY, w, v)$ of $\mfX$ (over $S$) is {\bf Galois} (resp., {\bf abelian}) if $w_v$ is a Galois covering  (resp., an abelian  covering).
% If $\mfw   : \mfY \migi \mfX$ is a Galois covering, then 
 In this situation, we shall write 
  \begin{align}
  \mr{Gal} (\mfY/\mfX) := \mr{Gal} (Y/X \times_S T)
  \end{align}
   and refer to it as the {\bf Galois group} of $(\mfY, w,v)$ over $\mfX$.
\item[(iii)]
Let $(\mfY, w_Y)$ and $(\mfZ, w_Z)$ be an \'{e}tale coverings of $\mfX$ over $S$ (where we denote by $Y$ and $Z$ the underlying semistable curves respectively).
We shall say that $(\mfY, w_Y)$ is {\bf isomorphic to $(\mfZ, w_Z)$} if
there exists an isomorphism  $h : Y \isom Z$ over $X$ such that $h^{-1}(D_{\mfZ}) = D_\mfY$.
(Thus, the isomorphism class of an \'{e}tale covering $(Y, w, \mr{id}_S)$ depends only on the \'{e}tale morphism $w : Y \migi X$, i.e., not on the marked points.)
  \end{itemize}
   \ede
%-----------------------------------------------------------------------[end definition]-------------------
\vspace{3mm}

%Next, let us consider the pull-back of an integrable vector bundle via an \'{e}tale covering of pointed stable curves.
Finally, we shall consider  a relation between 
the sheaf of horizontal sections of an integrable vector bundle and its pull-back via an \'{e}tale covering.


%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{P0d13d}\leavevmode\\
 \ \ \ 
%the pull-back of an integrable vector bundle  via an \'{e}tale covering and its direct image via the relative Frobenius morphism.
Let $\mfY := (Y/S, \{ \sigma'_{i} \}_{i=1}^{r'})$ be an \'{e}tale covering of $\mfX$ and denote by $w : Y \migi X$ the structure morphism of $Y$ over $X$.
Let us consider  the commutative square diagram of $k$-schemes
\begin{align} \label{DD0111}
\xymatrix{
Y \ar[r]^w \ar[d]_{F_{Y/S}} & X \ar[d]^{F_{X/S}} \\
Y^{(1)}_S \ar[r]_{w^{(1)}} & X^{(1)}_S
}
\end{align}
Then, for each $\mcO_{X}$-module $\mcF$, the morphism
\begin{align} \label{E5002}
w^{(1)* } (F_{X/S*}(\mcF)) \migi F_{Y/S*} (w^* (\mcF))
\end{align}
 of $\mcO_{Y^{(1)}_S}$-module induced, via adjunction, from this square diagram is 
  an  isomorphism and  functorial in $\mcF$.
Moreover, if $(\mcF, \nabla_\mcF)$ is an integrable vector bundle on $\mfX$, then the isomorphism (\ref{E5002}) restricts to an isomorphism
\begin{align} \label{EE01}
w^{(1)* } (F_{X/S*}(\mr{Ker} (\nabla_\mcF))) \isom F_{Y/S*} (\mr{Ker} (\nabla_{w^* (\mcF)}))
\end{align}
of $\mcO_{Y^{(1)}_S}$-modules.
%Suppose further  that $T = S$ and 
%let $\mfw := (w, v) : \mfY \migi \mfX$ be an \'{e}tale covering over $S$ (i.e.,  $v = \mr{id}_S$) and $(\mcF, \nabla_\mcF)$ an integrable  vector bundle on $\mfX$.
%$w^{(1)*}(F_{X/S*}(\mcF)) \isom F_{Y}(w^* (\mcF))$
\epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
%The assertion  that  (\label{E5002}) is an isomorphism follows directly from the fact that
The former assertion may be immediately verified.
Indeed, 
the   \'{e}taleness of $w$ implies that  the diagram (\ref{DD0111})
is  cartesian, and hence,  (\ref{E5002}) is an isomorphism.
%we have a canonical  isomorphism 
%, which is functorial with respect to $\mcF$.

Next, let us consider the latter assertion.
The isomorphism $w^* (\Omega_{X^\mr{log}/S^\mr{log}}) \isom \Omega_{Y^\mr{log}/S^\mr{log}}$ (due to the \'{e}taleness of $w$) and the  isomorphism  (\ref{E5002})  of the case where $\mcF$ is replaced with $\Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcF$ yield
a composite isomorphism
\begin{align} \label{E5001}
w^{(1)* } (F_{X/S*}(\Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcF)) \isom & \ F_{Y/S*} (w^* (\Omega_{X^\mr{log}/S^\mr{log}}\otimes \mcF)) \\ 
\isom  &  \ F_{Y/S*} (\Omega_{Y^\mr{log}/S^\mr{log}} \otimes w^*(\mcF)). \notag
\end{align}
One verifies immediately that the square diagram
\begin{align} \label{E5004}
\begin{CD}
w^{(1)* } (F_{X/S*}(\mcF)) @> w^{(1)*}(F_{X/S*}(\nabla_\mcF)) >> w^{(1)* } (F_{X/S*}(\Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcF))    \\
@V (\ref{E5002}) V \wr V @V \wr V (\ref{E5001}) V
\\
F_{Y/S*} (w^* (\mcF)) @> F_{Y/S*} (\nabla_{w^*(\mcF)}) >> F_{Y/S*} (\Omega_{Y^\mr{log}/S^\mr{log}}\otimes w^*(\mcF))
\end{CD}
\end{align}
is commutative.
In particular, 
by taking the kernels  of  the upper and  lower  horizontal arrows, we obtain  the isomorphism (\ref{EE01}), as desired.
\end{proof}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%--[ begin  section1]---%%%%%%
\vspace{10mm}
\section{$\mr{GL}_n$-do'pers and ordinariness} \vspace{3mm}



In this section, we   shall 
recall  the definition of a $\mr{GL}_n$-oper, as well as a $\mr{GL}_n$-do'per (cf. Definition \ref{y0108} and Definition \ref{y01019}), and then, introduce 
 the notion  of  ($\circledast$-)ordinarinesses of  $\mr{GL}_n$-do'pers  (cf. Definition \ref{y05019}).
Moreover, we shall discuss  several properties concerning the ordinariness.
%discuss the definition of a $\mr{GL}_n$-do'per (cf. Definition \ref{y0108} and Definition \ref{y01019}) and 
%   ($\circledast$-)ordinarinesses of  $\mr{GL}_n$-do'pers (cf. Definition \ref{y05019}).
%Also, we consider basic properties on the ordinariness and give the proof of Theorem A.
At the end of this section, the proof of  Theorem A, being  one of our main results, will  be given.
In the following, {\it we fix a positive integer $n$ with $n <p$}.
%In particular, we are interested in some  conditions equivalent to the condition of being ordinary.
%Let $g$, $r$, and  $k$ be  as in Introduction, and $n$  a positive integer with $n < p$.
%Also, let $S$ be a $k$-scheme and 
%$\mfX_{/S} := (f : X \stackrel{}{\migi} S,  \{ \sigma_i : S \migi X\}_{i=1}^r)$
% a pointed stable  curve over $S$ of type $(g,r)$.


\vspace{5mm}
%-----------------------------------------------------------------------[end lemma]-------------------

%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S21}

%Let $n$ be a positive integer and $\hslash \in k$.
%Suppose further that $n \geq 2$.
%Let $k$ be a field of arbitrary characteristic.
%, and $u : U \migi X$ an \'{e}tale morphism of $k$-schemes.
% Also, let $U$ be an open subscheme of $X$, equipped with a log structure $U^{\mr{log}}$ obtained by restricting the log structure of $X^\mr{log}$.
We first recall (cf. ~\cite{Wak5}, Definition 4.2.1 and ~\cite{Wak6}, Definition 4.2.1) the definition  of a $\mr{GL}_n$-oper as follows.
 Let    $S^\mr{log}$ be a log scheme over $k$ and $f^\mr{log} : X^\mr{log} \migi S^\mr{log}$ be a log-curve over $S^\mr{log}$.



%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{y0108}\leavevmode\\
%\vspace{-5mm}
%\begin{itemize}
% \item[(i)]
% Let $n$ be a positive integer and $\hslash \in \Gamma (S, \mcO_S)$. 
 \vspace{-5mm}
 \begin{itemize}
 \item[(i)]
 A {\bf $\mr{GL}_n$-oper} on  $X^\mr{log} /S^\mr{log}$ is   a collection of  data 
 \begin{equation} 
 \label{GL1}
 \mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j=0}^n),\end{equation}
where
\begin{itemize}
\item[$\bullet$]
$\mcF$ is a vector bundle  on $X$ of rank $n$;
\item[$\bullet$]
$\nabla_\mcF$ is an $S$-log connection
 $\mcF \migi \Omega_{X^\mr{log}/S^\mr{log}} \otimes\mcF$ on $\mcF$;
 %, i.e., an $f^{-1}(\mcO_S)$-linear morphism satisfying the Leibniz rule :
% \begin{equation}
% \nabla_\mcF (a \cdot v) = da \otimes v + a \cdot  \nabla_\mcF (v) 
% \end{equation}
%  (for any local sections $a \in \mcO_X$ and $v \in \mcF$);
\item[$\bullet$]
$\{ \mcF^j \}_{j=0}^n$ is   a decreasing filtration: 
\begin{equation}
0 = \mcF^n \subseteq \mcF^{n-1} \subseteq \dotsm \subseteq  \mcF^0= \mcF
\end{equation}
 on $\mcF$ by vector bundles on $X$,
\end{itemize}
 satisfying the following conditions:
\begin{itemize}
\item[(i-1)]
the subquotients $\mcF^j / \mcF^{j+1}$ ($ 0\leq j\leq n-1$)  are line bundles;
%locally free $\mcO_X$-modules of rank $m$.
%invertible sheaves.
\item[(i-2)]
$\nabla_\mcF (\mcF^j) \subseteq  \Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcF^{j-1}$ ($1 \leq j \leq n-1$);
\item[(i-3)]
the {\it $\mcO_X$-linear} morphisms
\begin{equation} \label{GL2}
\mfk \mfs^j_{\mcF^\heartsuit} :  \mcF^j/\mcF^{j+1} \stackrel{}{\migi}   \Omega_{X^\mr{log}/S^\mr{log}} \otimes (\mcF^{j-1}/\mcF^j)\end{equation}
($1 \leq j \leq n-1$)
defined by assigning $\overline{a} \mapsto \overline{\nabla_\mcF (a)}$ for any local section $a \in \mcF^j$ (where $\overline{(-)}$'s denote the images in the respective quotients), which are  well-defined by virtue of the condition (i-2),  are isomorphisms.
 %where the first arrow denotes the morphism induced by  $\nabla_\mcF$ (which is well-defined by virtue of  the condition (2)),
%is an isomorphism.
%(We shall often refer the morphism $\mr{KS}_i$ to the {\it $i$-th Kodaira Spencer map})
% and the morphism $\mr{gr}^i \mcF \migi \mr{gr}^{i-1}\mcF \otimes \omega_{X/S}$ induced by $\nabla$ 
\end{itemize}
For simplicity, we shall refer to a $\mr{GL}_n$-oper on $X^{\mfX \text{-} \mr{log}}/S^{\mfX \text{-} \mr{log}}$ (where $\mfX$ denotes a pointed stable curve whose underlying curve coincides with $X$) as a {\bf $\mr{GL}_n$-oper on $\mfX$}.
\item[(ii)]
Let $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j =0}^n)$ and $\mcG^\heartsuit := (\mcG, \nabla_\mcG, \{\mcG^j \}_{j=0}^n)$ be $\mr{GL}_n$-opers on $X^\mr{log} /S^\mr{log}$.
An {\bf isomorphism of $\mr{GL}_n$-opers} from $\mcF^\heartsuit$ to $\mcG^\heartsuit$ is an isomorphism  of  integrable vector bundles from $(\mcF, \nabla_\mcF)$  to $(\mcG, \nabla_\mcG)$ that preserves the respective filtrations $\{ \mcF^j \}_{j =0}^n$ and $\{ \mcG^j \}_{j =0}^n$.
\end{itemize}

%\item[(ii)]
%\end{itemize}  
  \ede
%-----------------------------------------------------------------------[end definition]-------------------
%\vspace{3mm}
%-----------------------------------------------------------------------[begin definition]------------------
%-----------------------------------------------------------------------[begin definition]------------------
%\vspace{3mm}
\bde \label{y01019}\leavevmode\\
 \ \ \ We shall say that a $\mr{GL}_n$-oper $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j\}_{j=0}^n)$ is {\bf dormant} if the $p$-curvature of $\nabla_\mcF$ vanishes identically on $X$  (cf. e.g., ~\cite{Wak5}, Definition 3.2.1,  for the definition of $p$-curvature).
For convenience, we shall abbreviate a ``dormant $\mr{GL}_n$-oper" to a ``{\bf $\mr{GL}_n$-do'per}".
  \ede
%-----------------------------------------------------------------------[end definition]-------------------


\vspace{3mm}
%----------------------------------------------------------------------[begin subsection]-------------
%\subsection{} \label{S22}

Let us  consider  the  pull-back of a $\mr{GL}_n$-oper.
%  via a finite \'{e}tale morphism.
Let $T^\mr{log}$ be a log scheme over $k$ and $Y^\mr{log}$ a log-curve over $T^\mr{log}$.
Suppose that we are give a $\mr{GL}_n$-oper $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \sigma_i \}_{i=1}^r)$ on $X^\mr{log}/S^\mr{log}$ and a 
pair $(w^\mr{log}, v^\mr{log})$ consisting of morphisms
$w^\mr{log} : Y^\mr{log} \migi X^\mr{log}$, $v^\mr{log} : T^\mr{log}  \migi S^\mr{log}$ which make
the square diagram
% which fit into the following 
%commutative  diagram
\begin{align}
\xymatrix{
Y^\mr{log} \ar[r]^{w^\mr{log}} \ar[d] & X^\mr{log} \ar[d]
\\
T^\mr{log} \ar[r]^{v^\mr{log}} & S^\mr{log}
}
\end{align}
commute and such that  the induced morphism $Y^\mr{log} \migi X^\mr{log} \times_{S^\mr{log}} T^\mr{log}$ is log \'{e}tale.
Write $w^* (\mcF)^j := w^*(\mcF^j)$ ($j =0, \cdots, n$).
Then, since $w^\mr{log}$ induces an isomorphism
$w^* (\Omega_{X^\mr{log}/S^\mr{log}}) \isom \Omega_{Y^\mr{log}/T^\mr{log}}$, 
 the  collection of data
\begin{equation} \label{E02}
w^*(\mcF)^\heartsuit := (w^*(\mcF), \nabla_{w^*(\mcF)}, \{ w^*(\mcF^j) \}_{j=0}^n)
\end{equation}
forms a $\mr{GL}_n$-opers on $Y^\mr{log}/T^\mr{log}$.
If, moreover,  $\mcF^\heartsuit$ is dormant, then $w^*(\mcF)^\heartsuit$ is dormant.

%-----------------------------------------------------------------------[begin definition]------------------
%\vspace{3mm}
\bde \label{yy9}\leavevmode\\
 \ \ \ 
 We shall refer to the $\mr{GL}_n$-oper  $w^*(\mcF)^\heartsuit$ on $Y^\mr{log}/T^\mr{log}$ defined above as the {\bf pull-back of $\mcF^\heartsuit$ via $(w^\mr{log}, v^\mr{log})$}.
   \ede
%-----------------------------------------------------------------------[end definition]-------------------





\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S23}

Now, we consider  certain ordinarinesses of  $\mr{GL}_n$-do'pers on pointed stable curves.
Let  $S$ be a scheme over $k$, 
$\mfX := (f : X \migi S, \{ \sigma_i \}_{i=1}^r)$  a pointed stable curve over $S$,  and 
$\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j=0}^n)$
 a $\mr{GL}_n$-do'per on $\mfX$.
% and denote by 
%$\mcE nd^{0} (\mcF)$ ($\subseteq \mcE nd (\mcF)$) the sheaf of $\mcO_X$-linear endomorphisms of $\mcF$ with vanishing trace.
Consider an $\mcO_{X}$-module  
\begin{equation}
\mcE nd^\circledast (\mcF) \  (\subseteq \mcE nd (\mcF))
\end{equation}
defined to be  the sheaf of $\mcO_X$-linear endomorphisms of $\mcF$ with vanishing trace.
In the following, 
\begin{align}
\text{{\it the symbol $\Box$ denotes either the presence or absence of $\circledast$.}} \notag 
\end{align}
The filtration $\{ \mcF^j \}_{j=0}^n$ carries a decreasing filtration $\{ \mcE nd^\Box(\mcF)^j \}_{j \in \mbZ}$ on $\mcE nd^\Box(\mcF)$
% (resp., $\{ \mcE nd^\circledast(\mcF)^j \}_{j \in \mbZ}$) 
 given by 
\begin{align}
\mcE nd^\Box(\mcF)^j &:= \{ f \in \mcE nd^\Box (\mcF) \ | \ f (\mcF^l) \subseteq \mcF^{l+j} \ \text{for any $l$} \}.
% (\text{resp.,} \ \mcE nd^\circledast (\mcF)^j &:= \{ f \in \mcE nd^\circledast (\mcF) \ | \ f (\mcF^l) \subseteq \mcF^{l+j} \} ) \notag
\end{align}
Let  
\begin{align}
\nabla_\mcF^{\mr{ad} \Box} &: \mcE nd^\Box (\mcF) \migi \Omega_{X^\mr{log}/S^\mr{log}}\otimes \mcE nd^\Box (\mcF) 
\end{align}
denotes the $S$-log connection on $\mcE nd^\Box (\mcF)$
% (resp., $\mcE nd^\circledast (\mcF)$)
   induced naturally by the $S$-log connection $\nabla_\mcF$.
By the definition of a $\mr{GL}_n$-oper,  the restriction of   $\nabla_\mcF^{\mr{ad}\Box}$ to $ \mcE nd^\Box (\mcF)^j$  (for each  $j \in \mbZ$) determines 
%one may obtain, by restricting  $\nabla_\mcF^\mr{ad}$, 
%(resp., $\nabla_\mcF^{\mr{ad}\circledast}$), 
an $f^{-1}(\mcO_S)$-linear morphism
%$\nabla_\mcF$ induces naturally an $S$-connection  on  
\begin{align} \label{E03}
\nabla_\mcF^{\mr{ad} \Box, j} &:  \mcE nd^\Box (\mcF)^j \migi  \Omega_{X^\mr{log}/S^\mr{log}}\otimes \mcE nd^\Box (\mcF)^{j-1}. 
%(\text{resp.,} \ \nabla_\mcF^{\mr{ad}\circledast, j} &:  \mcE nd^\circledast (\mcF)^j \migi  \Omega_{X^\mr{log}/S^\mr{log}}\otimes \mcE nd^\circledast (\mcF)^{j-1}) \notag
\end{align}
%for each $j \in \mbZ$.
%According to ~\cite{Wak5},  t
Since the composite 
\begin{align} \label{DD090}
f_*(\mcE nd^\Box (\mcF)^j) & \stackrel{f_*(\nabla_\mcF^{\mr{ad}\Box, j})}{\migi} f_*(\Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcE nd^\Box (\mcF)^{j-1})  \\
&  \hspace{5mm} \migiincl  f_*(\Omega_{X^\mr{log}/S^\mr{log}} \otimes \mcE nd^\Box (\mcF))  \notag \\
& \hspace{5mm} \migi  f_*(\mr{Coker}(\nabla_\mcF^{\mr{ad} \Box})) \notag
\end{align}
becomes the  zero map,  
the composite of the second and third  arrows in (\ref{DD090}) gives rise to 
%we have 
an $\mcO_S$-linear   morphism
\begin{align} \label{E06}
\Theta^{\Box, j}_{\mcF^\heartsuit} : \mr{Coker} (f_*(\nabla_\mcF^{\mr{ad}\Box, j})) \migi f_* (\mr{Coker} (\nabla_\mcF^{\mr{ad}\Box})).
\end{align}



%Thus, we obtain an $\mcO_S$-linear composite
%\begin{equation}
%\eta^{\Box j}_{\mcF^\heartsuit} : \mr{Coker}(f_*(\nabla_\mcF^{\mr{ad}\Box, j})) \stackrel{{'e}_\sharp [\nabla_\mcF^{\mr{ad}\Box, j}]}{\migi} \mbR^1f_*(\mcK^\bullet [\nabla_\mcF^{\mr{ad}\Box, j}])  \stackrel{}{\migi} \mbR^1f_* (\mcK^\bullet [\nabla_\mcF^{\mr{ad}\Box}]), 
%\end{equation}
%where the second arrow denotes the $\mcO_S$-linear  morphism induced, via the functor $\mbR^1f_*(-)$,   by a natural  morphism of complexes
%$\mcK^\bullet [\nabla_\mcF^{\mr{ad}\Box, j}] \migi \mcK^\bullet [\nabla_\mcF^{\mr{ad}\Box}]$.
%Here, 
%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{y05019}\leavevmode\\
 \ \ \  We shall say that a  $\mr{GL}_n$-do'per $\mcF^\heartsuit$ is {\bf ordinary} (resp., {\bf $\circledast$-ordinary}) if 
 $\Theta^{ 1}_{\mcF^\heartsuit}$ (resp., $\Theta^{\circledast, 1}_{\mcF^\heartsuit}$) is injective.
  \ede
%-----------------------------------------------------------------------[end definition]-------------------



\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S24}

Let $\mfX$ and  $\mcF^\heartsuit$ be  as above and $(\mcL, \nabla_\mcL)$ an integrable line bundle on $\mfX$.
%, i.e., a pair consisting of a line bundle on $X$ and an $S$-connection on $\mcL$.
%It is immediately verified that 
Then, the collection of data
\begin{equation}
\mcF^\heartsuit_{(\mcL, \nabla_\mcL)} := (\mcF \otimes \mcL, \nabla_{\mcF}\otimes \nabla_\mcL, \{ \mcF^j \otimes \mcL \}_{j =0}^n)
\end{equation}
(cf. (\ref{DD091}) for the definition of  $\nabla_\mcF \otimes \nabla_\mcL$)
forms a $\mr{GL}_n$-oper on $\mfX$.
Moreover,   one verifies that the $p$-curvature of
  $\nabla_\mcL$ 
  %has vanishing $p$-curvature
   vanishes identically on $X$ if and only if  $\mcF^\heartsuit_{(\mcL, \nabla_\mcL)}$ is dormant.

Then, we shall define an equivalence relation in the set of $\mr{GL}_n$-do'pers on $\mfX$, as follows.

%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{y0108t}\leavevmode\\
 \ \ \  Let $\mcF^\heartsuit$ and $\mcG^\heartsuit$ be $\mr{GL}_n$-do'pers on $\mfX$.
We shall say that $\mcF^\heartsuit$ is {\bf equivalent to $\mcG^\heartsuit$} if
$\mcF^\heartsuit$ is isomorphic to $\mcG^\heartsuit_{(\mcL, \nabla_\mcL)}$ for some  integrable line bundle $(\mcL, \nabla_\mcL)$ on $\mfX$ (with vanishing $p$-curvature).
%\item[(ii)]
%\end{itemize}  
  \ede
%-----------------------------------------------------------------------[end definition]-------------------
\vspace{3mm}


%-----------------------------------------------------------------------[begin remark]------------------
\begin{rema} \label{y01974f} \leavevmode\\
 \ \ \ 
 Let 
 %$\mcF^\heartsuit$ and
  $(\mcL, \nabla_\mcL)$ be an integrable line bundle on $\mfX$.
 % in the beginning of this subsection.
% Note that there exists a canonical isomorphism between the $\mcO_X$-modules 
$\mcE nd^\Box (\mcF)$ is canonically identified with $\mcE nd^\Box (\mcF \otimes \mcL)$ and 
% One may identify, in a natural manner,  $\mcE nd^\Box (\mcF)$ with $\mcE nd^\Box (\mcF \otimes \mcL)$.
%Also, we have the equal
the $S$-log connection $\nabla^{\mr{ad}\Box}_\mcF$ coincides with $\nabla_{\mcF \otimes \mcL}^{\mr{ad}\Box}$  via this identification.
It follows that $\mcF^\heartsuit$ is $\Box$-ordinary if and only if $\mcF^\heartsuit_{(\mcL, \nabla_\mcL)}$ is $\Box$-ordinary.
In particular, {\it the $\Box$-ordinariness of  $\mr{GL}_n$-do'pers depends only on their  equivalence classes}.
 \end{rema}
%-----------------------------------------------------------------------[end remark]-------------------





\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S25}


%-----------------------------------------------------------------------[begin remark]------------------
%\begin{rema} \label{y01974} \leavevmode\\
% \ \ \
 We shall consider the case where $n =1$.
%\begin{itemize}
%\item[(i)]
Any  integrable line bundle $(\mcL, \nabla_\mcL)$ may be equipped  with a trivial  $1$-step filtration $\{ \mcL^j \}_{j =0}^1$
%Consider the $1$-step filtration on $\mcO_X$ 
 given by $\mcL^0 := \mcL$ and $\mcL^1 := 0$.
 This filtration allows us to consider  $(\mcL, \nabla_\mcL)$ as a  $\mr{GL}_1$-oper $\mcL^\heartsuit := (\mcL, \nabla_\mcL, \{ \mcL^j \}_{j=0}^1)$ on $\mfX$.
 By this way, {\it we may identify  any integrable line bundle  (resp., any integrable line bundle with vanishing $p$-curvature) with a $\mr{GL}_1$-oper (resp., a $\mr{GL}_1$-do'per)}.
 In particular,  one obtains the {\bf trivial $\mr{GL}_1$-do'per} 
 \begin{align} \label{E05}
 \mcO_X^\heartsuit := (\mcO_X, d, \{ \mcO^j_X \}_{j=0}^1)
 \end{align}
on $\mfX$.
It is clear   that any two $\mr{GL}_1$-do'pers are equivalent.
In other words, {\it any $\mr{GL}_1$-do'per  is equivalent to the trivial  $\mr{GL}_1$-do'per  $\mcO_X^\heartsuit$}.

Next, consider the $\Box$-ordinariness.
According to the above discussion and the discussion in Remark \ref{y01974f},
 an arbitrary  $\mr{GL}_1$-do'per  is $\Box$-ordinary if and only if  $\mcO_X^\heartsuit$ is $\Box$-ordinary.
 Thus, it suffices to  consider   the case of  the trivial $\mr{GL}_1$-do'per. 
%It follows from the above discussion that
%to verify the $1^\Box$-ordinariness of a $\mr{GL}_1$-do'per on $\mfX_{/S}$, 
% is $1^\Box$-ordinary if and only if 
% it suffices to verify the $1^\Box$-ordinariness of 
% the $\mr{GL}_1$-do'per $\mcO_X^\heartsuit$.

%In the case where $\Box$ denotes  the presence   of $\circledast$, 
On the one hand, if $\Box = \circledast$, then  $\mcO_X^\heartsuit$ is necessarily $\circledast$-ordinary  since $\mcE nd^\circledast (\mcO_X)$ $=$ $0$. 
On the other hand, suppose that  $\Box$ denotes the  absence  of $\circledast$.
We have the equality $\nabla_{\mcO_X}^{\mr{ad}} = d$, and  $\nabla_{\mcO_X}^{\mr{ad}, 1}$ coincides with the zero map $0 \migi \Omega_{X^\mr{log}/S^\mr{log}}$.
%,  it follows from ?
 Thus, $\Theta^1_{\mcO_X^\heartsuit}$ coincides with 
 $f_*^{(1)} (C_{X/S})$
  %: f_* (\Omega_{X^\mr{log}/S^\mr{log}}) \migi f_*^{(1)} (\Omega_{X_S^{(1) \mr{log}}/S^\mr{log}})$
   (cf. (\ref{E1002})).
 % The composite ${'' e}_\flat [\nabla_{\mcO_X}^{\mr{ad}\circledast}] \circ \eta^{\circledast 1}_{\mcO_X^\heartsuit}$ may be identified with   the Cartier operator
%$C^{(\mcO_X, d_{X^\mr{log}/S^\mr{log}})} : f_* (\Omega_{X^\mr{log}/S^\mr{log}}) \migi f^{(1)}_* (\Omega_{X^{(1)\mr{log}}/S^\mr{log}})$  associated with $(\mcO_X, d)$.
% (cf., e.g.,  ? for the definition of the Cartier operator associated with an  integrable vector bundle).
It follows from Proposition \ref{P01} that  {\it $\mcO_X^\heartsuit$ is ordinary if and only if
$\mfX$ is ordinary in the sense of Definition \ref{Y5019}}.
This observation implies  the following proposition  for $n =1$,
%Thus, we have  proved  the following proposition,
 by which {\it one may consider the ordinariness of $\mr{GL}_n$-do'pers  may be thought of as a generalization of the classical  ordinariness of  algebraic curves in positive characteristic}.
%\end{itemize}

 %\end{rema}
%-----------------------------------------------------------------------[end remark]-------------------

%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{g00078}\leavevmode\\
 \ \ \ 
%Suppose that $r =0$ and  $X$ is smooth over $S$, and 
%let $\mcF^\heartsuit$ be a $\mr{GL}_n$-do'per on $\mfX_{/S}$.
Let $\mcF^\heartsuit$ be a $\mr{GL}_n$-do'per on $\mfX$.
Then, $\mcF^\heartsuit$ is ordinary if and only if $\mfX$ is ordinary  and $\mcF^\heartsuit$ is $\circledast$-ordinary.
If, moreover, $n =1$, then $\mcF^\heartsuit$ is necessarily $\circledast$-ordinary.
 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
Since we have verified the latter assertion, it suffices to prove the former assertion.
Let us write $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j=0}^n)$.
The morphism
$\mcO_X \oplus \mcE nd^\circledast (\mcF) \migi \mcE nd (\mcF)$ given by assigning
$(a, v) \mapsto  a \cdot  \mr{id}_\mcF +v$ is  (since  $n$ is  prime to $p$) an isomorphism
%Since $n$ is supposed to be prime to $p$, there exists canonically  an isomorphism
%$\mcE nd (\mcF) \isom \mcE nd^\circledast (\mcF) \oplus \mcO_X$
 that is compatible with  $d  \oplus \nabla_\mcF^{\mr{ad}\circledast}$ and $\nabla_\mcF^{\mr{ad}}$.
This isomorphism restricts to   isomorphisms 
\begin{align}
\mcO_X \oplus \mcE nd^\circledast (\mcF)^0 \isom \mcE nd (\mcF)^0 \ \ \text{and} \ \ \{ 0 \}\oplus \mcE nd^\circledast (\mcF)^1 \isom \mcE nd (\mcF)^1
\end{align}
%$\mcO_X \oplus \mcE nd^\circledast (\mcF)^0 \isom \mcE nd (\mcF)^0$ and  moreover, an isomorphism $0 \oplus \mcE nd^\circledast (\mcF)^1 \isom \mcE nd (\mcF)^1$
 that are compatible with
$\nabla_{\mcO_X}^{\mr{ad}, 1} \oplus \nabla_\mcF^{\mr{ad}\circledast, 1}$ and $\nabla_\mcF^{\mr{ad}, 1}$.
Hence, we  have  a commutative square diagram
\begin{align}
\begin{CD}
f_* (\Omega_{X^\mr{log}/S^\mr{log}}) \oplus \mr{Coker} (f_*(\nabla_\mcF^{\mr{ad} \circledast, 1}))@> \sim >> \mr{Coker} (f_*(\nabla_\mcF^{\mr{ad}, 1}))
\\
@V   \Theta^1_{\mcO_X^\heartsuit} \oplus \Theta_{\mcF^\heartsuit}^{\circledast, 1}  VV @VV  \Theta^1_{\mcF^\heartsuit} V
\\
f^{(1)}_* (\Omega_{X_S^{(1)\mr{log}}/S^\mr{log}}) \oplus f_*(\mr{Coker} (\nabla_\mcF^{\mr{ad} \circledast})) @> \sim >> f_*(\mr{Coker} (\nabla_\mcF^{\mr{ad}})).
\end{CD}
\end{align}
This diagram shows that
%$\Theta_{\mcO_X^\heartsuit} \oplus \Theta_{\mcF^\heartsuit}^{\circledast, 1} = \Theta^1_{\mcF^\heartsuit}$ via this isomorphism, 
%and hence, 
 $\Theta_{\mcF^\heartsuit}^1$ is injective if and only if 
  $\Theta^1_{\mcO_X^\heartsuit} $ is injective (equivalently,  $\mfX$ is ordinary by  the discussion preceding Proposition \ref{g00078}) and  $\Theta_{\mcF^\heartsuit}^{\circledast, 1}$ is injective. 
This  completes  the proof of Proposition \ref{g00078}.
\end{proof}
%\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------


\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S26}

According to Proposition \ref{g00078}, the study of the ordinariness of $\mr{GL}_n$-do'pers reduces to 
the study of the $\circledast$-ordinariness. 
In this subsection, we shall explain that  these conditions may be closely related to  a  local property of  the moduli of $\mr{GL}_n$-do'pers.
% an aspect 
%To begin with, we shall prove Proposition \ref{ghu} described below.
%Suppose  that  $r =0$ and  $X$ is smooth over $S$.
%Thus, to understand the ordinariness of $\mr{GL}_n$-do'per, we have to know, in more detail, the $\circledast$-ordinariness.
Toward  explaining it, 
%In the following Proposition \ref{ghu},
  we first recall  (from ~\cite{Wak5})  several  conditions each of which is  equivalent to the $\circledast$-ordinariness (cf.  Proposition \ref{ghu} below).

For  a $\mr{GL}_n$-do'per
 $\mcF^{\heartsuit} := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j=0}^n)$  on $\mfX$,  we shall denote by
\begin{equation}
\gamma_{\mcF^\heartsuit} : F_{X/S*}(\mr{Ker}(\nabla_\mcF)) \migi F_{X/S*} (\mcF/\mcF^1)
\end{equation}
the  $\mcO_{X^{(1)}_S}$-linear composite of the  inclusion $F_{X/S*}(\mr{Ker} (\nabla_\mcF)) \migiincl F_{X/S*}(\mcF)$ and the quotient  $F_{X/S*}(\mcF) \migisurj F_{X/S*}(\mcF /\mcF^1)$.
% induced by the quotient $\mcF \migisurj \mcF/\mcF^1$.



%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{ghu}\leavevmode\\
 \ \ \ 
%Suppose that $n \geq 2$, and 
Let $\mcF^\heartsuit$ be as above,  and 
%$\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j =0}^n)$ be a $\mr{GL}_n$-do'per on $\mfX_{/S}$. S
suppose that $S$ is affine.
Then, the following three conditions (i)-(iii) are mutually equivalent to each other:
\begin{itemize}
\item[(i)]
$\mcF^\heartsuit$ is $\circledast$-ordinary.
\item[(ii)]
The morphism $\Theta^{\circledast, 0}_{\mcF^\heartsuit}$ is injective.
\item[(iii)]
%For any $S$-scheme $T$, 
The dormant $\mfs \mfl_n$-oper  associated with  $\mcF^\heartsuit$
%  the base-change of $\mcF^\heartsuit$ via $T \migi S$
   has no nontrivial first order deformation.
(Here, we refer to  ~\cite{Wak5}, Definition 2.2.1 (i) and Definition 3.6.1, for the definition of a dormant $\mfg$-oper on a pointed stable curve for a semisimple Lie algebra $\mfg$ and to ~\cite{Wak5}, \S\,4.3 and Corollary 4.13.3,  for the discussion concerning the dormant $\mfs \mfl_n$-oper associated with  a dormant  $\mr{GL}_n$-oper.)

\end{itemize}
Suppose further that $r = 0$,  $X$ is  smooth over $S$, and is ordinary.
Then, these conditions are also equivalent to the following condition:
\begin{itemize}
\item[(iv)]
%If $s$ denotes the composite
%\begin{align}
%\mr{Ker}(\nabla_\mcF) \migiincl F_* (\mcF) \migisurj F_* (\mcF/ \mcF^1)
%\end{align}
%of natural morphisms, then
The equality 
\begin{align}
\mr{Hom}_{\mcO_{X^{(1)}_S}}(F_{X/S*}(\mr{Ker}(\nabla_\mcF)), \mr{Coker}(\gamma_{\mcF^\heartsuit}))) =0
\end{align} 
is satisfied.
\end{itemize}
 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
The equivalence (i) $\Leftrightarrow$ (ii) follows from ~\cite{Wak5}, Lemma 5.2.1 (i).
The equivalence (ii) $\Leftrightarrow$ (iii) follows from ~\cite{Wak5}, Theorem D and Corollary 5.12.1 (i).
Moreover,  the equivalence (iii) $\Leftrightarrow$ (iv) follows from ~\cite{Wak5} Proposition 8.3.3 and  Proposition 8.4.1 (and the discussion in ~\cite{Wak5}, \S\,8.1).
\end{proof}
\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------




%-----------------------------------------------------------------------[begin remark]------------------
%\begin{rema} \label{y11974f} \leavevmode\\
% \ \ \ 
By the equivalence (i) $\Leftrightarrow$ (iii) in Proposition \ref{ghu},   the $\circledast$-ordinariness of $\mr{GL}_n$-do'pers may be linked to  the study in the paper ~\cite{Wak5} concerning the geometric structure of the moduli stack of dormant $\mfs \mfl_n$-opers.
 In fact, denote by 
 \begin{align} \label{DD077}
 \mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}
 \end{align}
  be the sheaf  associated with the  \'{e}tale  presheaf on $\overline{\mfM}_{g,r}$
  % (:= the moduli stack classifying pointed proper smooth curves of type $(g,r)$ over $k$)
 %  (with respect to  the \'{e}tale topology) 
    which, to any $k$-scheme $T$ equipped with an \'{e}tale morphism $T \migi \overline{\mfM}_{g,r}$
%  \'{e}tale $\overline{\mfM}_{g,r}$-scheme $T$
   classifying  pointed stable curve $\mfY$ over $T$, assigns the set of equivalence  classes of  $\mr{GL}_n$-do'pers on $\mfY$.
 Also, denote by 
 \begin{align} \label{N01}
 \mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_n, g,r}
 \end{align}
 the \'{e}tale sheaf on $\overline{\mfM}_{g,r}$ which, to any
 $T$ (as well as $\mfY$) as above, 
  assigns the set of isomorphism   classes of dormant $\mfs \mfl_n$-opers on $\mfY$.
%(Notice that  the moduli stack indicated by  $\mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_n, g,r}$ as in  (\ref{N01}) differs from one  introduced in ~\cite{Wak5}.
%Indeed, our moduli stack $\mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_n, g,r}$ classifies only dormant $\mfs \mfl_n$-opers whose underlying curves are smooth.)
By ~\cite{Wak5}, Theorem C, Theorem  D,  and the isomorphism (11),  these sheaves may be represented by finite (relative) $\overline{\mfM}_{g,r}$-schemes
 and 
 the assignment from each $\mr{GL}_n$-do'per to its associated  dormant  $\mfs \mfl_n$-oper defines 
 an isomorphism
 \begin{align} \label{gg099}
\Lambda : \mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}  \isom \mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_n, g,r}
 \end{align}
of (relative) $\overline{\mfM}_{g,r}$-schemes. 
Let $\mcF^\heartsuit$ be a $\mr{GL}_n$-do'per $\mcF^\heartsuit$ on $\mfX$ and 
$s : S \migi \mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r} $  the $S$-rational point of $\mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r} $ classifying $\mcF^\heartsuit$.
Then, the equivalence (i) $\Leftrightarrow$ (iii) in Proposition \ref{ghu} implies  that
{\it $\mcF^\heartsuit$ is $\circledast$-ordinary if and only if 
the image of the composite $\Lambda \circ s$ lies in the unramified locus of $ \mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_n, g,r}$ relative to  $\overline{\mfM}_{g,r}$}.

By applying results in ~\cite{Wak5}, we have the following proposition.
 

 
%  \end{rema}
%-----------------------------------------------------------------------[end remark]-------------------



 %-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{ghu3}\leavevmode\\
 \vspace{-5mm}
\begin{itemize}
\item[(i)]
Denote by 
\begin{align}
{^\circledast \mfO} \mfp_{\mr{GL}_n, g, r}^{^\text{Zzz...}}
\end{align}
the locus  of  $\mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}$ classifying $\circledast$-ordinary $\mr{GL}_n$-do'pers.
Then,  it forms   an open substack  which 
%There exits an open substack $\mcU$ of $\mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}$
%satisfying the following properties
%\begin{itemize}
%\item[(1)]
  is dense in any irreducible component of $\mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}$
that dominates $\overline{\mfM}_{g,r}$, and ${^\circledast \mfO} \mfp_{\mr{GL}_n, g, r}^{^\text{Zzz...}}/\overline{\mfM}_{g,r}$ is finite and \'{e}tale.
% via the natural forgetting morphism $\mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r} \migi \overline{\mfM}_{g,r}$.
In particular, any $\mr{GL}_n$-do'per on the pointed proper smooth  curve classified by the generic point of $\mfM_{g,r}$ ($\subseteq \overline{\mfM}_{g,r}$) is ordinary (and hence, $\circledast$-ordinary).
%\item[(2)]
%\end{itemize}


\item[(ii)]
  Any $\mr{GL}_n$-do'per on a  totally degenerate pointed  stable curve (e.g.,  the projective line together with $3$ marked points)
   is necessarily ordinary, and hence, $\circledast$-ordinary  (cf. ~\cite{Wak5}, Definition 6.3.1, for the definition of a totally degenerate pointed  stable curve).
\end{itemize}
\epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
Assertions (i) and (ii) follows from ~\cite{Wak5}, Theorem F,  and the fact that any pointed totally degenerate stable curve is ordinarily in the sense of Definition \ref{Y5019}.
%Also, assertion (ii) follows from ~\cite{Wak5}, .
\end{proof}
%\vspace{3mm}
%------------------------------------------------
 
 %-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bco \label{C001}\leavevmode\\
 \ \ \ Let $S^\mr{log}$ be a log scheme over $k$ such that $S = \mr{Spec} (R)$ for some local ring $R$ over $k$.
 Also, let $\mfX$ be a pointed stable curve over $S^\mr{log}$ whose  spacial fiber is totally degenerate.
Then, any $\mr{GL}_n$-do'per on $\mfX$ is ordinary, and hence, $\circledast$-ordinary.
 \eco
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
The assertion follows directly from Proposition \ref{ghu3} (i) and (ii).
%THe assertion follows from Proposition \ref{ghu} (ii) and the open nature (in $\mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}$) of the $n^\circledast$-ordinariness resulted in Proposition \ref{ghu} (i). 
\end{proof}
\vspace{3mm}
%------------------------------------------------
 
%In particular, we have the following



\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S27}

%In this subsection, w
We shall analyze the behavior, according to the clutching morphism of the underlying pointed stable curves, of the $\circledast$-ordinariness.
% of $\mr{GL}_n$-do'pers.

Let $S^\mr{log}$ be a log scheme over $k$ and $\mfX := (X/S, \{ \sigma_i \}_{i=1}^r)$ a pointed stable curve  over $S^\mr{log}$ of type $(g,r)$ (where $2g-2+r >0$).
Suppose that  $\mfX$ may be obtained by gluing together $m$ ($\geq 1$) pointed stable curves $ \mfX_{j} := (X_j/S, \{ \sigma_{j, i} \}_{i=1}^{r_j})$ ($j =1, \cdots, m$)  along some marked points in $\bigcup_{j=1}^m \{ \sigma_{j,i} \}_{i=1}^{r_j} $.
(More precisely, $\mfX$ is obtained by gluing together  $\{ \mfX_{j }\}_{j =1}^m$ by means of a certain {\it clutching data} for a pointed stable curve of type $(g,r)$. We refer to  ~\cite{Wak5}, Definition 6.1.1, for the definition of a clutching data.)
In particular, for each $j$, we have a natural morphism $\mfC \mfl \mfu \mft_j : X_j \migi X$  of $S$-schemes.
Let us  consider the log structure on $X_j$ pulled-back from the log structure of $X^{\mfX_{}\text{-}\mr{log}}$ via $\mfC \mfl \mfu \mft_j$.
We shall  denote  the resulting log scheme by $ X^{\mfX_{}\text{-}\mr{log}}_j$.
The structure morphism $X_j \migi S$ of  $X_j/S$ extends to a morphism
$X_j^{\mfX_{}\text{-}\mr{log}} \migi S^{\mfX_{}\text{-}\mr{log}}$ of log schemes.
Moreover, the natural morphism $ X^{\mfX_{}\text{-}\mr{log}}_j \migi X_j$ of log schemes  (where we consider $X_j$ as being equipped with the trivial log structure) 
extends to a commutative square diagram
\begin{align} \label{DD099}
\xymatrix{
X_j^{\mfX \text{-} \mr{log}} \ar[r]^{w_j^\mr{log}} \ar[d] & X_j^{\mfX_j \text{-} \mr{log}}\ar[d]
\\
S^{\mfX \text{-} \mr{log}} \ar[r]^{v_j^\mr{log}} &  S^{\mfX_j \text{-} \mr{log}},
}
\end{align}
where 
the underlying morphisms $w_j$ and $v_j$ of the horizontal arrows coincide with the identity morphisms of $X_j$ and $S$ respectively.
%the endomorphisms of $X_j$ and $S$ underlying  the upper and lower horizontal arrows respectively coincide with the identity morphisms.
One verifies that
the morphism
% the diagram yields
%write $ X^{\mfX_{/S}\text{-}\mr{log}}_j$ for the log scheme obtained as $X$ equipped with log structure pull-backed the log structure of $X^{\mfX_{/S}\text{-}\mr{log}}$ via the closed immersion $\mfC \mfl \mfu \mft_j$.
%The natural morphism $ X^{\mfX_{/S}\text{-}\mr{log}}_j \migi  X_j^{\mfX_{j/S}\text{-}\mr{log}} $ of log schemes yields 
%a {\it log \'{e}tale} morphism
\begin{equation}
w_{v_j} : X^{\mfX_{}\text{-}\mr{log}}_j \migi  X_j^{\mfX_{j}\text{-}\mr{log}} \times_{S^{\mfX_{j}\text{-}\mr{log}}} S^{\mfX_{}\text{-}\mr{log}} 
\end{equation}
over $S^{\mfX_{}\text{-}\mr{log}}$ induced by the diagram (\ref{DD099}) is log \'{e}tale.
%In particular, we have a canonical isomorphism $\Omega_{X^{\mfX\text{-}\mr{log}}/S^{\mfX\text{-}\mr{log}}} \isom  \Omega_{X_j^{\mfX_j\text{-}\mr{log}}/S^{\mfX_j\text{-}\mr{log}}}$.
%(Note that the  endomorphism $X_j$ underlying  $\mfe_j$ coincides with the identity morphism.)
%It follows that  $\mfe_j$ induces an isomorphism
%The morphism $w_j$ induces an  isomorphism
%\begin{equation}
%w_j^*(\Omega_{X^{\mfX\text{-}\mr{log}}/S^{\mfX\text{-}\mr{log}}}) \isom \Omega_{X_j^{\mfX\text{-}\mr{log}}/S^{\mfX\text{-}\mr{log}}}
%\end{equation}
%of $\mcO_{X_j}$-modules.
%$\Omega_{X^{\mfX_{/S}\text{-}\mr{log}}_j/S^{\mfX_{/S}\text{-}\mr{log}}} \isom $

Now, suppose that we are given a  $\mr{GL}_n$-do'per  $\mcF^\heartsuit$
on $\mfX$.
By restricting via  $\mfC \mfl \mfu \mft_j$, we obtain, from $\mcF^\heartsuit$,  a $\mr{GL}_n$-do'per  $\mcF |^\heartsuit_{X_j^{\mfX \text{-} \mr{log}}}$ on $X_j^{\mfX \text{-} \mr{log}}/S^{\mfX \text{-} \mr{log}}$.
Moreover, since $w_j^\mr{log}$ induces a canonical isomorphism $\Omega_{X^{\mfX\text{-}\mr{log}}/S^{\mfX\text{-}\mr{log}}} \isom  \Omega_{X_j^{\mfX_j\text{-}\mr{log}}/S^{\mfX_j\text{-}\mr{log}}}$,
%By restricting to $\mfX_j$, we obtain a $\mr{GL}_n$-do'per $\mcF^\heartsuit$ on $\mfX$
%Hence, 
%if we are given a $\mr{GL}_n$-do'per $\mcF^\heartsuit$ on $\mfX$, then, 
%by pulling-back the data in $\mcF^\heartsuit$ via $\mfC \mfl \mfu \mft$,
 $\mcF |^\heartsuit_{X_j^{\mfX \text{-} \mr{log}}}$ may be naturally identified with  a $\mr{GL}_n$-do'per 
 \begin{align}
 \mcF |^\heartsuit_{\mfX_{j}}
 \end{align}
  on $\mfX_{j/S}$.
 %  (cf. the discussion following Definition \ref{y01019}).
The following proposition (which will be used in the proof of Theorem B) is verified.

 %-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{ghu2}\leavevmode\\
% \vspace{-5mm}
 \ \ \ Let  $\mfX$, $\mfX_j$ ($j =1, \cdots, m$), and  $\mcF^\heartsuit$ be as above.
 % a $\mr{GL}_n$-do'per on $\mfX_{/S}$,
Then, $\mcF^\heartsuit$ is $\circledast$-ordinary  if and only if 
for any $j \in \{ 1, \cdots, m \}$ 
the $\mr{GL}_n$-do'per $\mcF |^\heartsuit_{\mfX_{j}}$ 
% obtained by restricting $\mcF^\heartsuit$ via the  log \'{e}tale morphisms
%$\mfX_{a/S} \migiincl \mfX_{/S}$
 is $\circledast$-ordinary.
 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
The assertion follows from the equivalence (i) $\Leftrightarrow$ (iii) in Proposition \ref{ghu} and ~\cite{Wak5}, Theorem 6.2.2.
\end{proof}
\vspace{3mm}
%------------------------------------------------
 

\vspace{3mm}
%----------------------------------------------------------------------[begin subsection]-------------
\subsection{The proof of Theorem A} \label{S28}

%\leavevmode\\
%\vspace{-4mm}
%\vspace{5mm}
%----------------------------------------------------------------------[begin subsection]-------------
%\subsection{The proof of Theorem A} \label{y2gH901fg}
%By applying Proposition \ref{ghu},
%we prove Theorem A asserted in Introduction.
%\begin{proof}[The proof of Theorem A]
Now, we shall prove  one of our main results, Theorem A.
Let $\mfX$, $(\mfY, w)$,  and $\mcF^\heartsuit$ be as in \S\,0.2.
It follows from  the assumption  and Proposition \ref{g00078} that $\mfY$ is ordinary and $w^*(\mcF)^\heartsuit$ is $\circledast$-ordinary.
For completing the proof, it suffices to prove that $\mfX$ is ordinary and $\mcF^\heartsuit$ is $\circledast$-ordinary (by Proposition \ref{g00078} again).

First, we shall show that $\mfX$ is ordinary.
Consider the square diagram
\begin{align} \label{DD097}
\begin{CD}
\Gamma (X, \Omega_{X^\mr{log}/k})  @> \Gamma (X_k^{(1)}, C_{X/k}) >> \Gamma (X_k^{(1)}, \Omega_{X^{(1)\mr{log}}_k/k})
\\
@V  VV @VV V
\\
\Gamma (Y, \Omega_{Y^\mr{log}/k})  @>> \Gamma (Y_k^{(1)}, C_{Y/k}) >  \Gamma (Y_k^{(1)}, \Omega_{Y^{(1)\mr{log}}_k/k}), 
\end{CD}
\end{align}
where the left-hand and right-hand   vertical arrows are the morphisms induced from $w$ and $w^{(1)}$ respectively.
By Proposition \ref{P01}, the ordinariness of $\mfY$ implies the injectivity of the lower horizontal arrow in (\ref{DD097}).
Hence, 
%the lower horizontal arrow is, by the ordinariness of  $\mfY_{/k}$, an isomorphism.
since the two vertical arrows in (\ref{DD097})  are injective,  the upper horizontal arrow is injective.
By Proposition \ref{P01} again,  $\mfX$ turns out be  ordinary.

Next, we shall show that $\mcF^\heartsuit$ is $\circledast$-ordinary. 
%Hence, it suffices to prove that $\mcF^\heartsuit$ is $n^\circledast$-ordinary (cf. Proposition \ref{g00078}).
Recall that the isomorphism
\begin{align} \label{E5007}
w^{(1)*}(F_{X/k*}(\mr{Ker}(\nabla_\mcF))) \isom F_{Y/k*}(\mr{Ker} (\nabla_{w^*(\mcF)}))
\end{align}
obtained by Proposition \ref{P0d13d}.
%is an isomorphism (cf. (\ref{EE01})).
Also,  the functorial isomorphism (\ref{E5002}) induces  an isomorphism 
\begin{align} \label{E5006}
w^{(1)*}(F_{X/k*}(\mcF/\mcF^1)) \isom F_{Y/k*} (w^*(\mcF)/w^*(\mcF)^1).
\end{align}
The above two isomorphisms make the square diagram
\begin{align}
\begin{CD}
w^{(1)*}(F_{X/k*}(\mr{Ker}(\nabla_\mcF))) @> w^{(1)*}(\gamma_{\mcF^\heartsuit}) >> w^{(1)*}(F_{X/k*}(\mcF/\mcF^1)) 
\\
@V (\ref{E5007}) V \wr V @V \wr V (\ref{E5006}) V
\\
F_{Y/k*}(\mr{Ker} (\nabla_{w^*(\mcF)})) @> \gamma_{w^*(\mcF)^\heartsuit} >> F_{Y/k*} (w^*(\mcF)/w^*(\mcF)^1)
\end{CD}
\end{align}
commute.
Hence, we have an isomorphism
\begin{align} \label{EE02}
w^{(1)*}(\mr{Coker}(\gamma_{\mcF^\heartsuit})) \isom \mr{Coker}(\gamma_{w^*(\mcF)^\heartsuit}).
\end{align}
%In particular, we have  canonical isomorphisms of $\mcO_{Y^{(1)}}$-modules
%\begin{equation}
%w^{(1)*}(\mr{Ker}(\nabla_\mcF)) \isom F_{Y/k*}(\mr{Ker}(\nabla_{w^*(\mcF)}))
%\end{equation}
%and
%\begin{equation}
%w^{(1)*}(\mr{Coker}(\gamma_{\mcF^\heartsuit})) \isom \mr{Coker}(\gamma_{w^*(\mcF^\heartsuit)}).
%\end{equation}
By means of (\ref{E5007}) and (\ref{EE02}), we obtain a composite injection
\begin{align} \label{EE03}
& \ \ \   \ \ \mr{Hom}_{\mcO_{X^{(1)}}}(F_{X/k*}(\mr{Ker}(\nabla_\mcF)), \mr{Coker}(\gamma_{\mcF^\heartsuit})) \\
 & \migiincl \mr{Hom}_{\mcO_X}(w^{(1)*}(F_{X/k*}(\mr{Ker}(\nabla_\mcF))), w^{(1)*}(\mr{Coker}(\gamma_{\mcF^\heartsuit})))\notag \\
& \isom \mr{Hom}_{\mcO_X} (F_{Y/k*}(\mr{Ker}(\nabla_{w^*(\mcF)})), \mr{Coker}(\gamma_{w^*(\mcF)^\heartsuit})).\notag
\end{align}
But, by the equivalence (i) $\Leftrightarrow$  (iv) in Proposition \ref{ghu}, the  hypothesis that  $w^*(\mcF)^\heartsuit$ is  $\circledast$-ordinary implies the equality
\begin{equation}
\mr{Hom}_{\mcO_X} (F_{Y/k*}(\mr{Ker}(\nabla_{w^*(\mcF)})), \mr{Coker}(\gamma_{w^*(\mcF)^\heartsuit})) = 0.
\end{equation}
Consequently, the composite injection  (\ref{EE03}) shows the equality
\begin{align}
 \mr{Hom}_{\mcO_{X^{(1)}}}(F_{X/k*}(\mr{Ker}(\nabla_\mcF)), \mr{Coker}(\gamma_{\mcF^\heartsuit})) = 0,
 \end{align}
  equivalently, $\mcF^\heartsuit$ is $\circledast$-ordinary by Proposition \ref{ghu} again.
This
%It follows from Corollary ? that $\mcF^\heartsuit$ is $n^\circledast$-ordinary, and hence,
 completes the proof of Theorem A.
%the equality ? implies that $\mr{Hom}(\mr{Ker}(\nabla_\mcF), \mr{Coker}(s)) =0$.
%Thus, it follows from ? that $\mcF^\heartsuit$ is ordinary.
%\end{proof}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%--[ begin  section1]---%%%%%%
\vspace{10mm}
\section{Cyclic log \'{e}tale  coverings of a pointed projective line} \vspace{3mm}


In this section, 
we consider $\mr{GL}_n$-do'pers on the projective line with some  marked points.
The goal of this section is to prove (cf. Proposition \ref{PP01}) that the pull-back of any $\mr{GL}_n$-do'per  on the  projective line with three marked points  via a cyclic  log \'{e}tale covering    is necessarily  $\circledast$-ordinary.
This assertion will be the key ingredient of the proof of Theorem B.
\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S31}

For each value  $s  \in k \sqcup \{ \infty \}$, denote by $[s]$ the $k$-rational point of the projective line 
\begin{align}
\mbP := \mr{Proj} (k[x,y])
\end{align}
 over $k$ determined by $s$.
Let $l$ be a positive  integer 
%which is greater then $1$ and 
  prime to  $p$,  and  denote by  $\mu_l := \{   \zeta_1, \cdots, \zeta_l \}$ the group of $l$-power roots of unity.
%, and 
%assume that all the $l$-power roots of unity $\mu_l := \{   \zeta_1, \cdots, \zeta_l \}$.
% (equipped with an ordering) in an algebraically closed field of $k$ are contained in $k$.
%write $\zeta_1, \cdots, \zeta_l$ for the set of 
 The collection of data
\begin{equation}
\mfP^{(l+2)\mr{pt}}_{} := (\mbP/k,  \{ [0], [\zeta_1], \cdots, [\zeta_l], [\infty] \})
\end{equation}
forms a pointed stable curve of type $(0, l+2)$.
In particular, 
\begin{equation}
\mfP^{3\mr{pt}}_{} := (\mbP/k, \{ [0], [1], [\infty]\})
\end{equation}
 (i.e, the $\mfP^{(l+2)\mr{pt}}$ of the case  $l =1$) is a {\it unique} (up to isomorphism) pointed stable curve of type $(0,3)$.


\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S32}



The Frobenius twist $\mbP^{(1)}_k$ of  $\mbP/k$  is evidently isomorphic to  $\mbP$.
The discussion in this subsection will be applied  to the discussion in \S\,3.3, where we deal with $\mbP^{(1)}_k$, via a natural identification $\mbP^{(1)}_k \cong \mbP$.
% $\mbP$ be the projective line  over $k$, which 

Let $n$ be a positive integer.
Recall the Birkhoff-Grothendieck's theorem,  which asserts  that for any vector bundle $\mcV$ on $\mbP$ of rank $n$ is isomorphic to a direct sum of $n$ line bundles:
\begin{equation}
\mcV \cong \bigoplus_{j= 1}^n \mcO_{\mbP} (w_j),
\end{equation}
where $w_{j_1} \leq w_{j_2}$ if $j_1 < j_2$.
The nondecreasing sequence  of integers $(w_j)_{j=1}^n$  depends only on the isomorphism class of the vector bundle $\mcV$.
% is uniquely determined.

%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \leavevmode\\
 \ \ \ 
 In the above situation, we shall say that $\mcV$ is {\bf of type $(w_j)_{j=1}^n$}.
Also, we shall say that $\mcV$ is {\bf of homogeneous type} if it is of type $(w_j)_{j=1}^n$ with $w_1 = w_n$ (equivalently, $w_1 = w_2 = \cdots = w_n$).
 \ede
\vspace{3mm}
%-----------------------------------------------------------------------[end definition]-------------------



%One verifies easily the following lemma.

By induction on $n$, one may verify immediately the following lemma (cf. ~\cite{Wak5}), Lemma 7.7.1.


%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\ble\label{311789} \leavevmode\\
 \ \ \ 
Suppose that we are given,
for $\square = 1,2$,  
a rank $n$ vector bundle  $\mcV_\square$ on $\mbP$ of type $(w_{\square, j})_{j =1}^n$, where $\{ w_{\square, j}\}_{1 \leq j \leq n}$  denotes 
a nondecreasing sequence of integers, 
 and given 
% $\mcV_2$ rank $r$ vector bundle on $\mbP^{(1)}$ of type $(m^2_i)_{i=1}^r$.
% Suppose that  there exists 
  an  injection $\mcV_1 \migiincl \mcV_2$ of $\mcO_{\mbP}$-modules.
 Then, for any $j \in \{ 1, \cdots, n\}$,  the inequality $w_{1, j} \leq w_{2, j}$ is satisfied. 
 \ele
%-----------------------------------------------------------------------[begin proof]-------------------
%\begin{proof}
%\end{proof}
\vspace{3mm}
%----------------------------------------------------------
The above lemma deduces immediately the following lemma (from an argument similar to the argument in the proof of ~\cite{Wak5}, Lemma 7.7.2), which  will be used in the proof of Proposition \ref{PP01}.


%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\ble \label{6234} \leavevmode\\
 \ \ \ 
 Let $s$ be an integer and  $\{ \mcV_l\}_{l\in \mbZ_{\geq 0}}$  a set of rank $n$ vector bundles (indexed by the set of nonnegative integers $\mbZ_{\geq 0}$) on $\mbP$ satisfying the following two conditions:
\begin{itemize}
\item[$\bullet$]
each $\mcV_l$ is of degree $s + l$ and of type $(w_{l, j})_{j=1}^n$, where $\{ w_{l, j} \}_{j=1}^n$ denotes   a nondecreasing sequence  of integers (hence $\sum_{j=1}^n w_{l, j} = s+l$); 
\item[$\bullet$]
the inequality   $w_{l, n} - w_{l, 1} \leq 1$ (for each $l \geq 0$) is satisfied.
\end{itemize}
 % $\{ w_{l, j} \}_{l \in \mbZ_{\geq 0}, 1 \leq j \leq n}$ ,  and  such that and satisfies  .
Also, suppose that we are given  a sequence of $\mcO_{\mbP}$-linear injections 
 \begin{equation} \mcV_0 \migiincl \mcV_1 \migiincl \mcV_2 \migiincl \mcV_3 \migiincl \cdots. \end{equation}
 Then, there exists $l_0 \in \mbZ_{\geq 0}$ such that $\mcV_{l_0}$ is of homogeneous type. 
 %of type $(w_{l_0, j})_{j=1}^n$ satisfying the equality  $w_{l_0, n} = w_{l_0, 1}$.
 \ele
%-----------------------------------------------------------------------[begin proof]-------------------
\vspace{3mm}
\bpr \label{500987}\leavevmode\\
 \ \ \ Let $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j=0}^n)$ be a $\mr{GL}_n$-do'per on $\mfP^{(l+2)\mr{pt}}$ and $\{ w_l \}_{l=1}^n$ a nondecreasing sequence of integers.
%$\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^i \}_{i=1}^n)$
% be a dormant $(\mr{GL}_n, 1, \mbU)$-oper on $\mfP_{/k}$.
Suppose that the rank $n$ vector bundle $F_{\mbP/k*}(\mr{Ker}(\nabla_\mcF))$  on $\mbP^{(1)}_k$  is of type $(w_l)_{l=1}^n$, i.e., 
 %$F_{\mbP/k*}(\mr{Ker}(\nabla_\mcF))$
  decomposes into a direct sum of  $n$ line bundles: 
\begin{equation} F_{\mbP^1/k*}(\mr{Ker}(\nabla_\mcF)) \isom  \bigoplus_{l = 1}^n \mcO_{\mbP^{(1)}_k} (w_l).
\label{decom}
\end{equation}
%(satisfying that  $w_{j_1} \leq w_{j_2}$ if $j_1 < j_2$).
%where $m_i \leq m_j$ if $i \leq j$.
%Let $m_i$  ($i = 1, \cdots, n$) be as in (\ref{decom}).
Then, we have the inequality
%\begin{equation}
%\mr{max}\{ m_i \}\leq 2 
$w_n -w_1 \leq l +1$, equivalently, $| w_i -w_j | \leq l +1$ for any $i$, $j \in \{1, \cdots, n \}$.
%\label{ineq1}
%\end{equation}
 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
%After possibly reordering the multiset $\{ m_i \}_{i=1}^n$, we may assume that
%$m_i \leq m_j$ if $1 \leq i \leq j \leq n$.
%SUppose
According to the decomposition (\ref{decom}), the rank $n$ vector bundle 
\begin{equation}
\mcA_{\mr{Ker}(\nabla_\mcF)} := F_{\mbP/k}^*(F_{\mbP/k*}(\mr{Ker}(\nabla_\mcF)))
\end{equation}
 on $\mbP$ decomposes into the direct sum
\begin{equation} \label{61298}
\mcA_{\mr{Ker}(\nabla_\mcF)} \isom  \bigoplus_{l = 1}^n \mcO_{\mbP} (p \cdot w_l).
\end{equation}
Recall from ~\cite{Wak5}, \S\,7.2, (777),  that $\mcA_{\mr{Ker}(\nabla_\mcF)}$ admits a canonical decreasing  filtration 
$\{ \mcA^j_{\mr{Ker}(\nabla_\mcF)} \}_{j=0}^n$ with  $\mr{gr}_\mcF^j := \mcA^j_{\mr{Ker}(\nabla_\mcF)} /\mcA^{j+1}_{\mr{Ker}(\nabla_\mcF)}$ ($j =0, \cdots, n-1$).
It follows from ~\cite{Wak5}, Proposition 7.3.3 and  the latter assertion of ~\cite{Wak5}, Corollary 7.3.4
of the case where $(g,r) = (0,3)$
(cf. ~\cite{Wak5}, the proof of Proposition 7.6.3)
% (and the condition that $(g,r) = (0,3)$)
  that
\begin{equation} \label{429996}
\mr{deg}(\mr{gr}_\mcF^{n-1}) < \mr{deg}(\mr{gr}_\mcF^{n-2}) < \cdots < \mr{deg}(\mr{gr}_\mcF^{0}).
\end{equation}
Now, let us write 
\begin{align}
\xi' & : \mcO_{\mbP} (p \cdot w_n) \migiincl  \bigoplus_{l = 1}^n \mcO_{\mbP} (p \cdot w_l) \stackrel{ (\ref{61298})^{-1}}{\migi}\mcA_{\mr{Ker}(\nabla_\mcF)} \\
  \big(\text{resp.},  \ \xi'' & : \mcA_{\mr{Ker}(\nabla_\mcF)} \stackrel{ (\ref{61298})}{\migi}  \bigoplus_{l = 1}^n \mcO_{\mbP} (p \cdot w_l) \migisurj  \mcO_{\mbP} (p \cdot w_1)  \  \big), \notag 
\end{align}
 where the first (resp., second) arrow denotes 
  the inclusion into the $n$-th factor (resp., the projection onto the $1$-st factor).
  % upon
% identifying $\mcA_{\mr{Ker}(\nabla_\mcF)}$ with $\bigoplus_{l = 1}^n \mcO_{\mbP^{1}} (p \cdot w_l)$ by means of the decomposition (\ref{61298}).
Also,  write
\begin{align}
  \ & n' := \mr{max} \{ j \ | \  \mr{Im}(\xi') \subseteq \mcA_{\mr{Ker}(\nabla_\mcF)}^j \} \\
 (\text{resp.},  \  & n'' := \mr{max} \{ j \ | \  \xi'( \mcA_{\mr{Ker}(\nabla_\mcF)}^j ) \neq 0\} \ ). \notag
 \end{align} 
Then, $\xi'$ (resp., $\xi''$) induces a {\it nonzero} morphism
\begin{align}
& \overline{\xi}' : \mcO_{\mbP} (p \cdot w_n) \migi \mcA_{\mr{Ker}(\nabla_\mcF)}^{n'}/\mcA_{\mr{Ker}(\nabla_\mcF)}^{n'+1} \ (= \mr{gr}_\mcF^{n'})  \\ 
(\text{resp.,} \  & \overline{\xi}'' :  (\mr{gr}_\mcF^{n''} = ) \ \mcA_{\mr{Ker}(\nabla_\mcF)}^{n''}/\mcA_{\mr{Ker}(\nabla_\mcF)}^{n''+1} \migi \mcO_{\mbP} (p \cdot w_1))  \notag
\end{align}
between line bundles (on the smooth curve $\mbP^1$). In particular $\overline{\xi}'$ and $ \overline{\xi}''$ are injective, and hence, we have
\begin{equation} \label{722465}
p \cdot w_n = \mr{deg} (\mcO_{\mbP} (p \cdot w_n)) \leq \mr{deg}( \mr{gr}_\mcF^{n'}) \leq  \mr{deg}( \mr{gr}_\mcF^{0})
\end{equation}
and
\begin{equation} \label{733465}
p \cdot w_1 =  \mr{deg} (\mcO_{\mbP} (p \cdot w_1)) \geq \mr{deg}( \mr{gr}_\mcF^{n''}) \geq \mr{deg}( \mr{gr}_\mcF^{n-1}),
\end{equation}
where the last inequalities in both  (\ref{722465}) and (\ref{733465})   follow from (\ref{429996}).
But, it follows from the latter assertion of ~\cite{Wak5}, Corollary 7.3.4,  that
\begin{align} \label{1}
& \ \ \   \ \mr{deg}( \mr{gr}_\mcF^{0}) - \mr{deg}( \mr{gr}_\mcF^{n-1})  \\
& = - (n-1) \cdot (2 \cdot 0 - 2 + l +2) + \sum_{i=1}^{l+2} (w_{i,n}-w_{i,1}) \notag \\
& < -(n-1) \cdot l + (l+2) \cdot p.\notag
\end{align}
By combining  (\ref{722465}), (\ref{733465}), and (\ref{1}), we obtain
\begin{equation}
w_n -w_1 \leq \frac{1}{p} \cdot (\mr{deg}( \mr{gr}_\mcF^{0}) - \mr{deg}( \mr{gr}_\mcF^{n-1})) < l+2 - \frac{(n-1)\cdot l}{p}.
\end{equation}
This implies (since $w_n -w_1 \in \mbZ$) that $w_n -w_1 \leq l+1$, as desired. 
\end{proof}
%\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------





\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S33}



Consider the endomorphism $\pi : \mbP \migi \mbP$ of $\mbP$ corresponding to the $k$-algebra  endomorphism of $k[x,y]$ given by assigning $x \mapsto x^l$ and $y \mapsto y^l$.
This morphism 
is unramified over $\mbP \setminus \{ [0],[\infty]\}$ and satisfies that
$\pi^{-1}([0]) = \{[0]\}$, $\pi^{-1}([\infty]) = \{[\infty]\}$, and $\pi^{-1}([1]) = \{[\zeta_1], \cdots, [\zeta_l]\}$.
Also, it extends to a log \'{e}tale Galois covering 
\begin{equation} \label{DE01}
\pi^\mr{log} : \mbP^{\mfP^{(l+2)\mr{pt}}\text{-} \mr{log}} \migi \mbP^{\mfP^{3\mr{pt}}\text{-} \mr{log}}
%\pi : \mfP^{(l+2)\mr{pt}}_{/k} \migi \mfP^{3\mr{pt}}_{/k}
\end{equation}
%(cf. ? for the notational convention concerning $\pi$)
over $k$  with Galois group $\mu_l$.




%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \label{PP01} \leavevmode\\
 \ \ \
 % Let $\mcE^\spadesuit := (\mcE_\mbB, \nabla_\mcE)$ be
 Let $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j =0}^n)$  be a $\mr{GL}_n$-do'per  on $\mfP^{3\mr{pt}}$ (which is  $\circledast$-ordinary due to Proposition \ref{ghu} (i) or  (ii)).
 Then, the pull-back $\pi^* (\mcF)^\heartsuit$ via $\pi^\mr{log}$ (i.e., the pull-bak of $\mcF^\heartsuit$ via $(\pi^\mr{log}, \mr{id}_k)$ in the sense of Definition \ref{yy9}) is a $\circledast$-ordinary $\mr{GL}_n$-do'per on $\mfP^{(l+2)\mr{pt}}$.
 % dormant $\mfs \mfl_n$-oper on $ \mfP^{(l+2)\mr{pt}}_{/k}$.

 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
By Remark \ref{y01974f} and  Lemma \ref{001} proved below, one may assume, after possibly  twisting by some integrable line bundle with vanishing $p$-curvature,  that
$F_{\mbP/k*}(\mr{Ker} (\nabla_\mcF))$ is of homogenous type.
 Since $\pi^{(1)*} (F_{\mbP/k*}(\mr{Ker} (\nabla_\mcF))) \cong F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)}))$ by Lemma \ref{h01} proved below,
$F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)}))$ is  of homogenous type,  and hence, 
$\mcE nd (F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)})))$ is isomorphic to a direct sum of finite copies of $\mcO_{\mbP^{(1)}_k}$.
This implies the equality 
\begin{align}
H^1 (\mbP^{(1)}_k, \mcE nd (F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)})))) =0.
\end{align}
Here, recall from well-known generalities of deformation theory that  the space of first order deformations of $F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)}))$  is canonically isomorphic to
$H^1 (\mbP^{(1)}_k, \mcE nd (F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)}))))$.
 Thus, $F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF)}))$  has no nontrivial  first order deformation.
 The assertion follows from  this fact together with ~\cite{Wak5}, Corollary 7.5.2.
\end{proof}
\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------

To compete the proof of Proposition \ref{PP01}, it remains to prove Lemmas \ref{h01} and  \ref{001}.
We first prove Lemma \ref{h01},  which   asserts that  there exists an  isomorphism of the form  (\ref{EE01})
% for our covering $\pi^\mr{log}$  
where $w$ is replaced with  our covering $\pi^\mr{log}$ 
(which is not \'{e}tale but log \'{e}tale). 
%-----------------------------------------------------------------------[begin lemma]------------------
\vspace{3mm}
\ble\label{h01} \leavevmode\\
 \ \ \ Let $(\mcV, \nabla_\mcV)$ be an integrable vector bundle  on $\mfP^{3\mr{pt}}$ with vanishing $p$-curvature.
Then, the natural morphism
\begin{align} \label{ghhk}
\pi^{(1)*} (F_{\mbP/k*}(\mr{Ker} (\nabla_\mcV))) \migi  F_{\mbP/k*} (\mr{Ker} (\nabla_{\pi^*(\mcV)}))
\end{align}
of $\mcO_{\mbP}$-modules is an isomorphism.
 \ele
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
Since (\ref{ghhk}) is an isomorphism on the open subset $\mbP^{(1)}\setminus \{ [0], [\infty] \}$ (cf.  (\ref{EE01})),
it suffices to prove that (\ref{ghhk})  is an isomorphism on the formal neighborhoods of $[0]$ and $[\infty]$.

By similarity of  the argument,  we shall only consider the case of $[0]$.
Let us fix 
%$[s] \in \{ [0], [\infty] \}$ and
 a local parameter  $t$ of $\mbP$ at $[0]$.
 Then, the formal neighborhood of $[0]$ in $\mbP$ (resp., $\mbP^{(1)}_k$) is isomorphic to $\mr{Spf}(k[[t]])$ (resp., $\mr{Spf}(k[[t^p]])$) and 
 the local parameter $t$ gives 
% we have
 % , which 
%%defining the closed subscheme $[s]$ of $\mbP^{1(1)}$.
% The parameter $t$
%  gives
   an identification $\Omega_{\mbP^{}/k} |_{\mr{Spf}(k[[t]])} \isom \widetilde{k[[t]]}$  (where $\widetilde{(-)}$ denotes the coherent sheaf associated with $(-)$).
 % allow us to   identify  the sections of  $\Omega_{\mbP^1/k}$ over $\mr{Spf}(k[[t]])$ with $k[[t]]$
It follows from ~\cite{O2}, Corollary 2.10 (or, ~\cite{Wak5}, the discussion following Lemma 7.3.2) that  the $t^p$-adic completion of $(F_{\mbP/k*}(\mcV), F_{\mbP/k*}(\nabla_\mcV))$ %(considered as )
%may be obtained  from 
%corresponds to
is isomorphic to
 the direct sum $\widetilde{k[[t]]}^{\oplus n}$ together with a  $\widetilde{k[[t^p]]}$-linear morphism $\bigoplus_{j =1}^n \nabla_{m_j}$, where,
 for each $m \in \mbZ$,  $\nabla_m$ denotes   the map $\widetilde{k[[t]]} \migi \widetilde{k[[t]]}$ determined by $a \mapsto t \cdot \frac{\partial a}{\partial t} - m \cdot a$ for any $a \in k[[t]]$.
 Hence, to complete the proof, one may assume, without loss of generality,  that $(\mcV, \nabla_\mcV) |_{\mr{Spf}(k[[t]])} \cong (\widetilde{k[[t]]}, \nabla_m)$ (where $m \in \mbZ$).
  %Now, since if $s \in \mu_l$, then th formal completion of $w^{(1)}$ is an isomorphism.
% Hence, it suffices to consider the case where 
% By  similarly of an argument, we only consider the case where $s =0$.
We shall identify, in a natural fashion,  the formal neighborhood of $[0]$ in the domain $\mbP$ of $\pi$ (resp., the domain $\mbP^{(1)}$ of $\pi^{(1)}$) with $\mr{Spf} (k[[t^{\frac{1}{l}}]])$ (resp., $\mr{Spf} (k[[t^{\frac{p}{l}}]])$).
%We have $\pi^{-1} ([0]) = \{ [0]\}$, and
The ring homomorphism   
$k[[t]] \migi  k[[t^\frac{1}{l}]]$ (resp., $k[[t^p]] \migi  k[[t^\frac{p}{l}]]$) obtained as the formal completion of  $\pi$ (resp., $\pi^{(1)}$) at $[0]$ may be given by assigning $t \mapsto (t^{\frac{1}{l}})^l$ (resp., $t^p \mapsto (t^{\frac{p}{l}})^l$).
  Since  $F_{\mbP/k*}(\mr{Ker}(\nabla_\mcF)) |_{\mr{Spf}(k[[t^p]])}$ is isomorphic to $\widetilde{k[[t^p]]} \cdot t^m$, we have
 % the $k[[t^\frac{p}{m}]]$
  \begin{align} \label{E602}
  \pi^{(1)*}(F_{\mbP/k*}(\mr{Ker}(\nabla_\mcF))) |_{\mr{Spf}(k[[t^\frac{p}{l}]])}
  \isom \widetilde{k[[t^{\frac{p}{l}}]]} \cdot t^m.
  \end{align}
   %  is isomorphic to $\widetilde{k[[t^{\frac{p}{l}}]]} \cdot t^m$
  %$k[[t^\frac{p}{m}]] \otimes_{k[[t^p]]} k[[t^p]] \cdot t^l \cong k[[t^\frac{p}{m}]] t^l$.
  One the other hand, 
  %since $w$ is log \'{e}tale, one may 
  % the $k[[t^\frac{p}{l}]] $-linear map 
  one verifies that  $\nabla_{\pi^*(\mcV)} |_{\mr{Spf} (k[[t^\frac{1}{l}]])}$  is isomorphic to 
  the map given by assigning $a \mapsto t^{\frac{1}{l}} \cdot \frac{\partial a}{\partial (t^{\frac{1}{l}})} - lm \cdot a$ ($a \in k[[t^\frac{1}{l}]]$).
  Hence, we have 
  \begin{align} \label{E601}
  F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^* (\mcV)})) |_{\mr{Spf} (k[[t^\frac{p}{l}]])} \cong \widetilde{k[[t^{\frac{p}{l}}]]} \cdot (t^{\frac{1}{l}})^{lm}.
  \end{align}
  By (\ref{E602}) and (\ref{E601}), the morphism (\ref{ghhk}) is an isomorphism on the formal neighborhood of $[0]$.
  This completes the proof of Lemma \ref{h01}.
\end{proof}
\vspace{3mm}
%-----------------------------------------------------------------------[end lemma]-------------------

By means of Lemma \ref{h01}, we prove the following Lemma \ref{001}.

 %-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\ble \label{001} \leavevmode\\
  \ \ \ For any  $\mr{GL}_n$-do'per $\mcF^\heartsuit := (\mcF, \nabla_\mcF, \{ \mcF^j \}_{j =0}^n)$    on $\mfP^{3\mr{pt}}$,
 there exists a $\mr{GL}_n$-do'per $\mcG^\heartsuit : = (\mcG, \nabla_\mcG, \{ \mcG^j \}_{j =0}^n)$ on $\mfP^{3\mr{pt}}$ equivalent to $\mcF^\heartsuit$ such that
the vector bundle  $F_{\mbP/k*}(\mr{Ker} (\nabla_\mcG))$ on $\mbP^{(1)}_k$ is of homogeneous type.
 \ele
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
We shall   apply the discussion in the proof of  ~\cite{Wak5}, Proposition 7.7.4, to $\mcF^\heartsuit$.
 Here,  note  that there necessarily exists a $(n, 1)$-determinant data $\mbU$ for $X^\mr{log}$ over $S^\mr{log}$ (cf. ~\cite{Wak5}, Definition 4.9, for the definition of  a $(n, 1)$-determinant data) for which $\mcF^\heartsuit$ is isomorphic to some dormant $(\mr{GL}_n, 1, \mbU)$-oper. 
By applying the discussion in {\it loc.\,cit.}, we obtain 
 a collection  of data
 \begin{equation}
\{ \mcF^\heartsuit_m \}_{m \in \mbZ_{\geq 0}}, \hspace{5mm} \{ i_m \}_{m \in \mbZ_{\geq 0}},
 \end{equation}
 consisting of 
 \begin{itemize}
 \item[$\bullet$]
 $\mr{GL}_n$-do'pers $\mcF^\heartsuit_m := (\mcF_m, \nabla_{\mcF_m}, \{ \mcF_m^j \}_{j=0}^n)$ ($m \in \mbZ_{\geq 0}$) on $\mfP^{3\mr{pt}}$ which  are  equivalent to $\mcF^\heartsuit$ and satisfy the equalities 
 \begin{align}
 \mr{deg} (F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_{m +1}}))) = \mr{deg} (F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_{m}}))) + 1,
 \end{align}
  and
 \item[$\bullet$]
a sequence of $\mcO_{\mbP_k^{(1)}}$-linear injections
%each $i_m$ is an  $\mcO_{\mbP_k^{1(1)}}$-linear injections
 %$i_m : \mr{Ker} (\nabla_{\mcF_m}) \migi \mr{Ker} (\nabla_{\mcF_{m+1}})$
 \begin{equation}
F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_0})) \migiincl F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_1})) \migiincl F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_3})) \migiincl  \cdots.
%\mr{Ker}(\nabla_{\mcF_0}) \migiincl
\end{equation}
 \end{itemize}
Suppose that there exists $m_1 \geq 0$ for which  $F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_{m_1}}))$ is of type $(w_{m_1, j})_{j=1}^n$ with $w_{m_1, n} -w_{m_1, 1} \geq 2$.
Let us take a positive integer $l$ with $l \geq 2$  and take $\pi^\mr{log}$ as at the beginning of \S\,\ref{S33} for this $l$.
Then, $F_{\mbP/k*}(\mr{Ker} (\nabla_{\pi^*(\mcF_{m_1})}))$ (which is isomorphic to $\pi^{(1)*}(F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_{m_1}})))$   by Lemma \ref{h01})
% the vector bundle $\pi^{(1)*}(\mr{Ker}(\nabla_{\mcF_{m_1}}))$, which is isomorphic to $\mr{Ker}(\pi^*(\nabla_{\mcF_{m_1}}))$
% by Lemma \ref{h01},
  is of type $(l \cdot w_{m_1, j})_{j=1}^n$.
 But, it follows from Proposition \ref{500987} that
 \begin{equation}
 (l \cdot 2 \leq ) \  l \cdot w_{m_1, n} - l \cdot w_{m_1, 1} \leq l +1.
 \end{equation}
 This is a contradiction, and implies that
  $F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_m}))$ (for any $m$) is of type $(w_j)_{j =1}^n$ with $w_n -w_1 \leq 1$. 
Thus, 
%the assertion follows from 
by  Lemma  \ref{6234},
 there exists $m_2 \geq 0$ such that the $\mr{GL}_n$-do'per $\mcF^\heartsuit_{m_2}$
 satisfies the required  conditions,
%   for which $F_{\mbP/k*}(\mr{Ker}(\nabla_{\mcF_{m_2}}))$ is of homogeneous type, 
   as desired.
 \end{proof}
%\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------


 





%\vspace{5mm}
%-----------------------------------------------------------------------[end lemma]-------------------



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%--[ begin  section1]---%%%%%%
\vspace{10mm}
\section{$\mr{GL}_n$-do'pers on a totally degenerate curve of genus $1$} \vspace{3mm}

%This section is devoted to prove Theorem B.
The goal of this section is to prove Theorem B.
To this end, we apply  a usual  degeneration  technique for underlying curves of $\mr{GL}_n$-do'pers.
By means of the result of the previous section, we first  prove  Theorem B for  the case of elliptic curves  and abelian coverings given by multiplication by  integers (cf. Proposition \ref{ghjk00098}).
Then, we complete the proof of Theorem B by applying this tentative result and considering a stable curve obtained by gluing together some elliptic curves.




\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S42}

%Before beginning the proof of Theorem B,
% Let us consider the degeneration of the endomorphism of an elliptic curve given by multiplication by a positive integer $l$ prime to $p$.
In this subsection, we consider a certain stable curve obtained by gluing together some copies of the projective line, and  consider the log structure of the projective line  restricted  from the natural log structure of this stable curve. 

For a  positive integer $m$,  we shall denote by 
\begin{equation}
\mfC^{l}_{m} := (C_m/k , \{ \sigma^{l}_{m, i} \}_{i=1}^{ m l})
\end{equation}
the pointed stable curve over $k$ of type $(1, m l)$ determined uniquely  by the following two conditions:
\begin{itemize}
\item[(i)]
$C_m$ is a (proper) semistable curve over $k$ of the form 
\begin{align}
C_m := \bigcup_{j =1}^{m} P_{j},
\end{align}
 where each $P_{j}$ is (a copy of) the projective line $\mbP$ with $\{ \sigma^{l}_{m, i} \}_{(j-1)l<  i \leq  jl}\subseteq  P_j(k)$;
\item[(ii)]
there exist mutually distinct  $k$-rational points  of  $C_m$:
\begin{align}
\delta_{1} \cdots, \delta_{m}
\end{align}
  such that 
\begin{equation}
P_{i} \cap P_{j} = \begin{cases}
\delta_{i} & \text{if $j = i+1$ mod $m$} \\
\emptyset & \text{if otherwise.}
 \end{cases}
\end{equation}
(Hence, the set $\{ \delta_{1} \cdots, \delta_{m} \}$ coincides with the set of of nodes of $C_m$).
%one may write  the set of node of $C^a$ as $\delta^a_1 \cdots, \delta^a_{r\cdot a}$
\end{itemize}
In particular, for each $j = 1, \dots, m$,  we obtain (after ordering  the set of marked points) a pointed stable curve
\begin{equation}
\mfP^{(l+2) \mr{pt}}_{m, j} := (P_{j}, \{ \sigma^{l}_{m, i} \}_{(j-1)l < i\leq j l} \cup \{ \delta_{j-1}, \delta_{j} \})
\end{equation}
(where $\delta_{0} := \delta_{m}$) over $k$   of type $(0,l+2)$.
Denote by 
\begin{equation}
P_{j}^{\mfC_{m}^l\text{-}\mr{log}}
\end{equation}
 the log scheme  over $\mr{Spec}(k)^{\mfC_{m}^l\text{-}\mr{log}}$ which is defined as the $k$-scheme  $P_{j}$ equipped  with the log structure  pulled-back from  the log structure of $C_m^{\mfC_{m}^l\text{-}\mr{log}}$
  via the natural closed immersion $P_{j} \migi C_m$.
Then, we have the commutative square  diagram
\begin{align}
\xymatrix{
P_{j}^{\mfC_{m}^l\text{-}\mr{log}} \ar[r] \ar[d] & P_{j}^{\mfP^{(l+2)\mr{pt}}_{m, j}\text{-}\mr{log}} \ar[d] \\
\mr{Spec}(k)^{\mfC_{m}^l\text{-}\mr{log}} \ar[r] & \mr{Spec}(k)^{\mfP^{(l+2)\mr{pt}}_{m, j}\text{-}\mr{log}}.
}
\end{align}
The following lemma is immediately verified.

 %-----------------------------------------------------------------------[begin lemma]------------------
\vspace{3mm}
\ble \label{L002} \leavevmode\\
 \ \ \
The resulting morphism
%   yields a log \'{e}tale morphism
\begin{equation} \label{DE02}
\Phi_{m, j}^l :  P_{j}^{\mfC_{m}^l\text{-}\mr{log}}  \migi 
%\xi^m_{j} : P_i^{\mfC^m_{/s^\mr{log}}\text{-}\mr{log}} \migi 
P_{j}^{\mfP^{(l+2)\mr{pt}}_{m, j}\text{-}\mr{log}}    \times_{k^{\mfP^{(l+2)\mr{pt}}_{m, j}\text{-}\mr{log}}}
%\times_{ \mr{Spec}(k)^{\mfP^{(l+2)\mr{pt}}_{m, j}\text{-}\mr{log}} }
 k^{\mfC_{m}^l\text{-}\mr{log}}
\end{equation}
 over $\mr{Spec}(k)^{\mfC_{m}^l\text{-}\mr{log}}$ is log \'{e}tale and 
 its  underlying morphism of $k$-schemes coincides with the identity morphism of $P_{j}$ ($= \mbP$).
\ele
%-----------------------------------------------------------------------[begin proof]-------------------






\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S43}



 Let us consider the degeneration of the endomorphism of an elliptic curve given by multiplication by a positive integer $l$ prime to $p$.

Let $m$ be a positive integer,  and let $v : \mr{Spec} (k) \migi \overline{\mfM}_{1, m}$ be a $k$-rational point of the moduli stack $\overline{\mfM}_{1, m}$ classifying   the pointed stable curve $\mfC_{m}^1$.
The divisor at infinity of $\overline{\mfM}_{1, m}$ around $v$ may be locally given  as $\prod_{i=1}^m t_i =0$ for some local functions $t_i$ ($i = 1, \cdots, m$).
Then, the choice of sections $t_i$'s  determines
% the formal neighborhood of $v$ in  $\overline{\mfM}_{1, m}$ is given as
a morphism
$v_{R_1}  : \mr{Spec} (R_1) \migi \overline{\mfM}_{1, m}$, where $R_1:= k[[ t_1, \cdots, t_m]]$.
Denote by
\begin{align}
\mfC^1_{m, R_1} := (C_{m, R_1}/R_1, \{ \sigma_{m, R_1, i}^1 \}_{i =1}^{m})
\end{align}
the pointed  stable curve over $R_1$ of type $(1, m)$ classified by $v_{R_1}$. 
(Hence, the spacial fiber is isomorphic to $\mfC_{m}^1$ and its  underlying semistable curve of the generic fiber is smooth.)
Denote by  $\mr{Spec} (R_1)^\mr{log}$  the log scheme defined as $\mr{Spec} (R_1)$ equipped with
the log structure associated with the homomorphism
%\begin{align}
$\mbN^{\oplus m} \migi R_1$ given by assigning $(a_i)_{i=1}^m \mapsto \prod_{i=1}^m t_i^{a_i}$.
Then, there exists a canonical isomorphism 
\begin{align}
\eta_1^\mr{log} : \mr{Spec}(R_1)^{\mfC^1_{m, R_1} \text{-}\mr{log}} \isom \mr{Spec} (R_1)^\mr{log}
\end{align}
%\end{align}
(whose underlying morphism of schemes coincides with the identity).

Let us  write  $R_l :=  k [[ t^{\frac{1}{l}}_1, \cdots, t^{\frac{1}{l}}_m]]$  (resp., $R_\infty := \varinjlim_{p \nmid n} k [[ t^{\frac{1}{n}}_1, \cdots, t^{\frac{1}{n}}_m]]$)
and equip  $\mr{Spec} (R_l)$ (resp., $\mr{Spec} (R_\infty)$) with a log structure associated with the homomorphism $(\frac{1}{l}\mbN)^{\oplus m} \migi R_l$  (resp., $\varinjlim_{p \nmid n} (\frac{1}{n}\mbN)^{\oplus m} \migi R_\infty$) given by $(a_i)_{i=1}^m \mapsto \prod_{i=1}^m t_i^{a_i}$.
Write $\mr{Spec}(R_l)^\mr{log}$ (resp., $\mr{Spec}(R_\infty)^\mr{log}$) for  the resulting log scheme and write
$\mr{Spec} (k)^\mr{log}$ for  the log scheme defined as $\mr{Spec} (k)$ equipped with
the log structure pulled-back from the log structure of $\mr{Spec}(R_\infty)^\mr{log}$
via the closed immersion $\mr{Spec} (k) \migi \mr{Spec}(R_\infty)$.
Then,  the sequence of homomorphisms  $R_1 \migiincl R_l \migiincl R_\infty \migisurj k$ yield a sequence of morphisms of log schemes
\begin{align} \label{EE013}
\mr{Spec} (k)^\mr{log} & \migi \mr{Spec}(R_\infty)^\mr{log} \migi \mr{Spec}(R_l)^\mr{log} \\
& \migi \mr{Spec}(R_1)^\mr{log} \ (\stackrel{(\eta_1^\mr{log})^{-1}}{\isom} \mr{Spec}(R_1)^{\mfC_{m, R_1}^1 \text{-} \mr{log}}) \notag 
\end{align}
  (by which $\mr{Spec}(R_\infty)^\mr{log}$ may be thought of as a universal Kummer \'{e}tale covering of $\mr{Spec}(R)^{\mfC^1_{m, R}\text{-}\mr{log}}$).

Let us fix an algebraically closed field $K$ together with an inclusion $R_\infty \migiincl K$, which induces a morphism $\mr{Spec}(K) \migi \mr{Spec} (R_\infty)^\mr{log}$.
Then, let
\begin{align}
\mfC_{m, K}^1 := (C_{m, K}, \{ \sigma^1_{m, K, i} \}_{i=1}^m)
\end{align}
be the base-change of $\mfC^1_{m, R_1}$ via this morphism. 
Write 
%Write  $C_{m, K} :=C_{m, R_1} \times_{R_1} K$  and 
\begin{align}
C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}^\Box} := C_{m, R_1}^{\mfC^1_{m,R_1}\text{-}\mr{log}^\Box}  \times_{R_1^{\mfC^1_{m,R_1}\text{-}\mr{log}} } R_\infty^\mr{log}.
%    \  C_{r, \widetilde{k}}^{\mfC^1_{r}\text{-}\mr{log}} := C_{r}^{\mfC^1_{r}\text{-}\mr{log}}\times_{k^{\mfC^1_r\text{-}\mr{log}}, \breve{v}} \mr{Spec} (k)^\mr{log}.
\end{align}
%Let us consider the natural  morphism 
where $\Box$ denotes either the present or absence of the prime ``$ \ '  \ $".
(Hence, we have $C_{m, K} \cong  C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}'} \times_{R_\infty^\mr{log}} K$).
The natural morphisms   $C_{m, K} \migi C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}'}$ and $C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}} \migi C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}'}$ give rise to a diagram  of logarithmic fundamental groups 
\begin{align} \label{DDD01}
\xymatrix{
&  \pi^{\mr{ket}}_1 (C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}})  \ar[d] & 
\\
\pi^{\mr{et}}_1 (C_{m, K})  \ar[r] &  \pi^{\mr{ket}}_1 (C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}'}) \ \Big( \ar[r]^{\hspace{-8mm}\sim} &  \pi^{\mr{ket}}_1 (C_m^{\mfC_m^1 \text{-} \mr{log}'} \times_{k^{\mfC_m^1 \text{-}\mr{log}}}k^\mr{log}) \Big)
}
\end{align}
where all the arrows are determined up to   choices of base point, i.e., up to  composition with inner automorphism.
 The vertical arrow in (\ref{DDD01}) is surjective and
the horizontal arrow   becomes an isomorphism after taking maximal pro-$l$ quotient $(-)^{(l)}$.
Here, $[l]_{C_{m, K}}$ denotes the endomorphism of $C_{m, K}$ determined by multiplication by $l$, which is an abelian \'{e}tale covering of degree $l^2$.
There exist uniquely (up to isomorphism) a log-curve $C_{m \cdot l, R_\infty}^\mr{log}$  over $R_\infty^\mr{log}$ and   a Kummer \'{e}tale covering 
\begin{align}
[l]^\mr{log}_{C_{m, R_\infty}} : C_{m\cdot l, R_\infty}^\mr{log} \migi C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}}
\end{align}
% of $C_{m, \widetilde{R}}^{\mfC^1_{m,R}\text{-}\mr{log}}$  
 over $\mr{Spec} (R_\infty)^\mr{log}$  corresponding to the abelian covering  $[l]_{C_{m, K}}$
 % of $C_{m, K}$
   via the resulting surjection
 \begin{align}
  \pi^{\mr{ket}}_1 (C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}})  \migisurj  \pi^{\mr{ket}}_1 (C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}'})^{(l)} \isom  \pi^{\mr{et}}_1 (C_{m, K})^{(l)}.
 \end{align} 
% $ \pi^{\mr{et}}_1 (C_{m, K})^{(l)}  \isom \pi^{\mr{ket}}_1 (C_{m, R_\infty}^{\mfC^1_{m,R_1}\text{-}\mr{log}})^{(l)}$.
(Namely, $C_{m \cdot l, R_\infty} \times_{R_\infty} K \cong C_{m,K}$ and  the fiber  of $[l]_{C_{m, R_\infty}}$ over $K$ coincides with $[l]_{C_{m, K}}$.)
One may find   a collection of data described as follows:
\begin{itemize}
\item[$\bullet$]
  a pointed stable curve  
\begin{align}
\mfC^{l}_{m \cdot  l, R_l} := (C_{m \cdot l, R_l}/R_l, \{ \sigma_{m \cdot l,  R_l,  i}^l \}_{i=1}^{m \cdot l^2})
\end{align}
 of type $(1, m \cdot l^2)$  over $R_l$ (which is uniquely determined up to change of  ordering in the marked points  $\{ \sigma_{m \cdot l,  R_l,  i}^l \}_{i=1}^{m \cdot l^2}$);
\item[$\bullet$]
an isomorphism  
\begin{align}
\eta_l^\mr{log} : \mr{Spec}(R_l)^\mr{log} \isom \mr{Spec}(R_l)^{\mfC^{l}_{m \cdot  l, R_l} \text{-} \mr{log}}
\end{align}
 of log schemes whose underlying morphism of schemes coincides with the identity;
\item[$\bullet$]
a commutative diagram
%a cartesian rectangular diagram
\begin{align}
\xymatrix{
C_{m\cdot l, R_\infty}^\mr{log} \ar[r] \ar[dd]  
%\ar@{}[rd]|{\Box}
 & C_{ml, R_l}^{\mfC^{l}_{m \cdot l, R_l} \text{-} \mr{log}}  \ar[r] \ar[d]  &  C_{m, R_1}^{\mfC^{1}_{m, R_1} \text{-} \mr{log}} \ar[d]
\\
 & \mr{Spec} (R_l)^{\mfC^{l}_{m \cdot l, R_l} \text{-} \mr{log}} \ar[d]_{\wr}^{\eta_l^\mr{log}} \ar[r] &  \mr{Spec}(R_1)^{\mfC^{1}_{m, R_1} \text{-} \mr{log}} \ar[d]_{\wr}^{\eta_1^\mr{log}}
\\
\mr{Spec}(R_\infty)^\mr{log} \ar[r] & \mr{Spec}(R_l)^\mr{log} \ar[r]^{}& \mr{Spec}(R_1)^\mr{log},
}
\end{align}
where both the left-hand rectangle and the right-hand lower square  are cartesian, the two  lower horizontal arrows are obtained in (\ref{EE013}), and  all the upper vertical arrows except  $\eta_1^\mr{log}$ and $\eta_l^\mr{log}$ are the structure morphisms of pointed stable curves.
\end{itemize}
The fact that the left-hand rectangle is cartesian gives an  isomorphism 
\begin{align} \label{EE012}
C_{m\cdot l, R_\infty}^\mr{log} \isom  C_{m \cdot l, R_l}^{\mfC^{l}_{m \cdot l, R_l} \text{-} \mr{log}} \times_{\mr{Spec}(R_l)^\mr{log}} \mr{Spec}(R_\infty)^\mr{log}
\end{align}
 over $\mr{Spec}(R_\infty)^\mr{log}$.
Also, one verifies that the special fiber of $\mfC_{m\cdot l, R_l}^l$ is isomorphic to
$\mfC^{l}_{m \cdot l} := (C_{m \cdot l} / k, \{ \sigma_{m \cdot l,  i}^l \}_{i=1}^{m \cdot l^2})$.
Denote by 
\begin{align}
[l]_{C_m}^\mr{log} :   C_{m \cdot l, k}^{\mfC^l_{m \cdot l} \text{-} \mr{log}} \times_{k^{\mfC^l_{m \cdot l} \text{-} \mr{log}} } k^\mr{log}  \migi  C_{m}^{\mfC^1_{m} \text{-} \mr{log}} \times_{k^{\mfC^1_{m} \text{-} \mr{log}} } k^\mr{log}
%\mfC^{l}_{r l} := (C_{rl, \widetilde{R}}/\widetilde{R}, \{ \sigma_{rl, i}^l \}_{i=1}^{r l^2}), \ \ \ [l]_{\widetilde{R}} : C_{rl, \widetilde{R}} \migi C^{\mfC^1_{r, R}\text{-}\mr{log}}_{r, \widetilde{R}},
\end{align}
%(which may be thought of as an abelian \'{e}tale covering of $C_{1, R} \times_R \overline{K}$), 
the base-change of $[l]^\mr{log}_{C_{m, R_\infty}}$ over $k^\mr{log}$ under  the isomorphism (\ref{EE012}).
The image  of each $P_j \subseteq C_{m \cdot l}$ ($j = 1, \cdots, m \cdot l$) via $[l]_{C_m}$  is contained in
$P_{s_j} \subseteq C_{m}$ for some $s_j \in \{ 1, \cdots, m \}$.
Thus, we obtain its restriction of the following form
\begin{align}
[l]_{C_m}^\mr{log} |_{P_j} : P_j^{\mfC^l_{m \cdot l} \text{-}\mr{log}} \times_{k^{\mfC^l_{m \cdot l} \text{-} \mr{log}} } k^\mr{log}  \migi P_{s_j}^{\mfC^1_m \text{-}\mr{log}} \times_{k^{\mfC^1_{m} \text{-} \mr{log}} } k^\mr{log}.  
\end{align}
%its restriction to the strict closed subschemes.
By a straightforward argument, one verifies  the following lemma.


 %-----------------------------------------------------------------------[begin lemma]------------------
\vspace{3mm}
\ble \label{L001} \leavevmode\\
 \ \ \
The square diagram
\begin{align}
\begin{CD}
P_j^{\mfC^l_{m\cdot l} \text{-}\mr{log}} \times_{k^{\mfC^l_{m \cdot l} \text{-} \mr{log}} } k^\mr{log} @> [l]_{C_m}^\mr{log} |_{P_j} >> P_{s_j}^{\mfC^1_m \text{-}\mr{log}} \times_{k^{\mfC^1_{m} \text{-} \mr{log}} } k^\mr{log}  
\\
@V \Phi^l_{m \cdot l, j} \times \mr{id}_{k^\mr{log}} VV @VV \Phi^1_{m, s_j} \times \mr{id}_{k^\mr{log}} V
\\
 P_j^{\mfP_{m, j}^{(l+2) \mr{pt}}\text{-}\mr{log}} \times_k k^\mr{log} @> \pi^\mr{log} >>  P_{s_j}^{\mfP_{m, j}^{3 \mr{pt}}\text{-}\mr{log}} \times_k k^\mr{log}
%P_j^{\mfC^l_m \text{-}\mr{log}}@>>> P_j^{\mfP_{m, j}^{(l+2) \mr{pt}}\text{-}\mr{log}} \times_{k^{\mfC^l_m \text{-}\mr{log}}} k^\mr{log}
%\\
%@VVV @VVV
%\\
%C_{rl, k}^{\mfC^l_{rl} \text{-} \mr{log}} \times_{k^{\mfC^l_{rl} \text{-} \mr{log}} } k^\mr{log}  @>>> C_{r, k}^{\mfC^l_{r} \text{-} \mr{log}} \times_{k^{\mfC^l_{r} \text{-} \mr{log}} } k^\mr{log}
\end{CD}
\end{align}
(cf.  (\ref{DE01}) for the definition of $\pi^\mr{log}$)
is commutative and cartesian.
\ele
%-----------------------------------------------------------------------[begin proof]-------------------




 
 By applying  the above  lemma, one may  prove the following proposition.
 
 %-----------------------------------------------------------------------[begin lemma]------------------
\vspace{3mm}
\bpr \label{ghjk00098} \leavevmode\\
 \ \ \
%Let $m$ be a positive integer and 
Write  $\mfC^l_{m \cdot l, K}$ for  the base-change of $\mfC_{m \cdot l, R_l}^l$ via the morphism 
$\mr{Spec} (K) \migi \mr{Spec} (R_\infty)$.
In particular, we obtain an abelian covering 
%Also, write 
$(\mfC^l_{m \cdot l, K}, [l]_{C_{m, K}})$ 
%$[\mfl]_{C_{m, K}} : \mfC^l_{m \cdot l, K} \migi \mfC^1_{m, K}$ 
%for the abelian  covering
 of $\mfC^1_{m, K}$ over $K$  defined as the base-change of $[l]_{C_{m, R_\infty}}$.
%That is to say, it is a unique (up to isomorphism) abelian  covering  whose underlying morphism of schemes coincides with $[l]_{C_{m, K}}$.
%(Hence, the  morphism of schemes underlying $[l]_{C_{m, K}}$ coincides with  the endomorphism of $C_{m, K}$ given by multiplication by $l$.)
  Then, the pull-back
 $[l]_{C_{m, K}}^* (\mcF)^\heartsuit$ of any $\mr{GL}_n$-do'per $\mcF^\heartsuit$ on $\mfC^1_{m, K}$  is $\circledast$-ordinary.
%  via the endomorphism $[l]$ of 
%    is ordinary $\mr{GL}_n$-oper on $\mfC$.
 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
By Proposition \ref{ghu3} (i),  $\mfO \mfp_{\mr{GL}_n, 1,m}^{^\text{Zzz...}} \times_{\overline{\mfM}_{1, m}, v_{R_1}} \mr{Spec}(R_1)$ is isomorphic to a disjoint union of finite  copies of $\mr{Spec} (R_1)$.
Hence, 
there exists a morphism $\widetilde{v}_{R_1} : \mr{Spec} (R_1) \migi \mfO \mfp_{\mr{GL}_n, 1,m}^{^\text{Zzz...}}$ whose composite with $\mr{Spec} (K) \migi \mr{Spec} (R_1)$ coincides with the classifying morphism of $\mcF^\heartsuit$.
If $\mcF^\heartsuit_{R_1}$ denotes the $\mr{GL}_n$-do'per on $\mfC_{m, R_1}^1$,
 then its pull-back via  $C_{m, R_\infty}^{\mfC^1_{m, R_1}\text{-} \mr{log}} \migi C_{m, R_1}^{\mfC^1_{m, R_1}\text{-} \mr{log}}$ (resp., the composite $C_{m \cdot l, R_\infty}^\mr{log} \stackrel{[l]^\mr{log}_{C_{m, R_\infty}}}{\migi} C_{m, R_\infty}^{\mfC^1_{m, R_1}\text{-} \mr{log}} \migi C_{m, R_1}^{\mfC^1_{m, R_1}\text{-} \mr{log}}$) forms  a $\mr{GL}_n$-do'per 
$\mcF^\heartsuit_{m, R_\infty^\mr{log}}$ on the log-curve $C_{m, R_\infty}^{\mfC^1_{m, R_1}\text{-} \mr{log}}/\mr{Spec}(R_\infty)^\mr{log}$ (resp.,  a $\mr{GL}_n$-do'per $\mcF^\heartsuit_{m \cdot l, R_\infty^\mr{log}}$ on the log-curve $C_{m \cdot l, R_\infty}^\mr{log}/\mr{Spec}(R_\infty)^\mr{log}$).
Denote by 
$\mcF^\heartsuit_{m, k^\mr{log}}$
 (resp., $\mcF^\heartsuit_{m \cdot l, k^\mr{log}}$) 
 the $\mr{GL}_n$-do'per on  
 $C_m^{\mfC^1_m \text{-} \mr{log}} \times_{k^{\mfC^1_m \text{-} \mr{log}}} k^\mr{log} /k^\mr{log}$  (resp., $C_{m \cdot l}^{\mfC^l_{m \cdot l} \text{-} \mr{log}} \times_{k^{\mfC^l_{m\cdot l} \text{-} \mr{log}}} k^\mr{log} /k^\mr{log}$)
defined as the restriction of 
 $\mcF^\heartsuit_{m, R_\infty^\mr{log}}$ (resp., $\mcF^\heartsuit_{m \cdot l, R_\infty^\mr{log}}$) to the special fiber.
Since the generic fiber of $\mcF^\heartsuit_{m \cdot l, R_\infty^\mr{log}}$ is isomorphic  to $[l]_{C_{m, K}}^* (\mcF)^\heartsuit$, it suffices (by  Corollary \ref{C001}) to verify  the $\circledast$-ordinariness of  $\mcF^\heartsuit_{m \cdot l, k^\mr{log}}$.
To this end, we first observe from Lemma \ref{L002}  that 
for each $j \in \{ 1, \cdots, m \cdot l \}$,  
%the $\mr{GL}_n$-do'per
the restriction  $\mcF_{m, k^\mr{log}} |_{P_{s_j}}^\heartsuit$ 
of $\mcF^\heartsuit_{m, k^\mr{log}}$ to 
  $P_{s_j}^{\mfC^1_m \text{-} \mr{log}} \times_{k^{\mfC^1_m \text{-} \mr{log}}} k^\mr{log} /k^\mr{log}$ may be identified with a $\mr{GL}_n$-do'per  
on $P_{s_j}^{\mfP_{m, j}^{3 \mr{pt}}\text{-}\mr{log}} \times k^\mr{log}/k^\mr{log}$ via $\Phi_{m, s_j}^1 \times  \mr{id}_{k^\mr{log}}$ (cf. (\ref{DE02})).
By Proposition \ref{PP01}, the pull-back $\pi^* (\mcF_{m, k^\mr{log}} |_{P_{s_j}})^\heartsuit$ via $\pi^\mr{log}$ of this $\mr{GL}_n$-do'per 
is $\circledast$-ordinary.
By Lemma \ref{L002} again, this  pull-back may be identified with 
a $\mr{GL}_n$-do'per on 
$P_j^{\mfC^l_{m\cdot l} \text{-}\mr{log}} \times_{k^{\mfC^l_{m \cdot l} \text{-} \mr{log}} } k^\mr{log}/k^\mr{log}$, which is isomorphic to  the restriction of $\mcF^\heartsuit_{m \cdot l, k^\mr{log}}$.
Thus, it follows from Proposition \ref{ghu2} that $\mcF^\heartsuit_{m \cdot l, k^\mr{log}}$ is $\circledast$-ordinary.
This completes the proof of Proposition \ref{ghjk00098}.
 \end{proof}
%\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------




\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{The proof of Theorem B} \label{S44}

This subsection is devoted to  prove Theorem B.
Let $G$ be a finite abelian group with $p \nmid \sharp (G)$. 



%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{D001}\leavevmode\\
 \ \ \ 
Let $S$ be a scheme over $k$ and $\mfX := (X/S, \{ \sigma_i \}_i)$ a pointed proper smooth  curve over $S$.
We shall say that  $\mfX$ is {\bf $\circledast_{n, G}$-ordinary}
 if 
 for any   $\circledast$-ordinary  $\mr{GL}_n$-do'per  $\mcF^\heartsuit$ on $\mfX$
 and any abelian covering $(\mfY, w, v)$ of $\mfX$ with Galois group $\mr{Gal} (\mfY/\mfX) \cong G$ such that $v$ is \'{e}tale, the pull-back $w^*(\mcF)^\heartsuit$ is $\circledast$-ordinary.
 
 
% it satisfies the following conditions:
% let us take a quadruple 
%\begin{align}
%(T, \mfY, \mfw, \mcF^\heartsuit),
%\end{align}
% where
% \begin{itemize}
% \item[(i)]
% $T$ is a scheme over $k$;
% \item[(ii)]
%$\mfY := (Y/T, \{ \sigma'_i \}_{i})$ is a pointed proper smooth  curve over $T$;
% \item[(iii)]
% $\mfw$   is  an abelian  Galois covering $\mfY \migi \mfX$ of $\mfX$  with Galois group isomorphic to $G$;
% \item[(iv)]
% $\mcF^\heartsuit$ is a $\circledast$-ordinary  $\mr{GL}_n$-do'per on $\mfX$.
% \end{itemize}
% Then, the pull-back $\mfw^* (\mcF)^\heartsuit$ is $\circledast$-ordinary. 
 \ede
\vspace{3mm}
%-----------------------------------------------------------------------[end definition]-------------------




%, the $\mr{GL}_n$-d'oper $w'^*(\mcF^\heartsuit)$ is ordinary.
Denote by 
\begin{equation}
{^\circledast \mfM}_{g, r, n, G}
\end{equation}
the substack of $\mfM_{g,r}$ classifying $\circledast_{n, G}$-ordinary pointed proper smooth curves.
The proof of Theorem B reduces to proving the assertion that 
${^\circledast \mfM}_{g, r, n , G}$ is a dense open substack of $\mfM_{g,r}$.
Thus, it suffices to prove the following Lemma \ref{L022} and Lemma \ref{L023}.


%-----------------------------------------------------------------------[begin theorem]------------------
\vspace{3mm}
\ble \label{L022} \leavevmode\\
 \ \ \ 
 ${^\circledast \mfM}_{g, r, n, G}$ is an open substack of $\mfM_{g,r}$
\ele
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
Denote by  $\mfN_{g, r, n, G}$
the \'{e}tale sheaf on $\mfM_{g,r}$ associated with the presheaf which, to any \'{e}tale scheme $S$ over  $\mfM_{g,r}$ classifying  a pointed stable curve $\mfX$, assigns the set of isomorphism classes of \'{e}tale coverings $(\mfY, w, \mr{id}_S)$ of $\mfX$ over $S$ with Galois group $\mr{Gal} (\mfY/\mfX) \cong G$.
Then, $\mfN_{g, r, n, G}$ may be represented by a relative finite scheme over $\mfM_{g,r}$.
% the moduli stack (over the category of $k$-schemes) classifying, for each $k$-scheme $S$, the groupoid of  triples 
%\begin{equation}
%\mbX := (\mfX, \mfY, \mfw)
%\end{equation}
%where  $\mfX$ and $\mfY$ are pointed proper smooth curves  over $S$ of type $(g,r)$ and $( (g-1) \cdot \sharp (G)+1, r \cdot \sharp (G))$ respectively  (hence $S^{\mfX \text{-} \mr{log}} = S^{\mfY \text{-} \mr{log}} = S$), and $\mfw$ is an abelian  \'{e}tale covering  $\mfY \migi \mfX$ with Galois group $G$.
The assignment  $(\mfY, w, \mr{id}_S)  \mapsto w^* (\mcF)^\heartsuit$
%$\mbX \mapsto \mfw^*(\mcF)^\heartsuit$
%\begin{equation}
%(\mfw', \mcG^\heartsuit) \mapsto w'^* (\mcG^\heartsuit)
%\end{equation}
 defines a morphism
\begin{equation}
\xi : \mfN_{g, r, n, G} \times_{\overline{\mfM}_{g, r}} \mfO \mfp_{\mr{GL}_n, g, r}^\ZZZ \migi \mfO \mfp^\ZZZ_{\mr{GL}_n, (g-1) \cdot \sharp (G) +1, r  \cdot \sharp (G)}.
\end{equation}
Recall from  Proposition \ref{ghu} (i) that  ${^\circledast \mfO} \mfp_{\mr{GL}_n, (g-1) \cdot \sharp (G) +1, r  \cdot \sharp (G)}^{^\text{Zzz...}}$ is an open substack of  $\mfO \mfp_{\mr{GL}_n, (g-1) \cdot \sharp (G) +1, r  \cdot \sharp (G)}^{^\text{Zzz...}}$, and hence,
% (cf. Proposition \ref{ghu} (i) for the definition of $\mfO \mfp_{\mr{GL}_n, l(g-1), lr}^{^\text{Zzz...}, n^\circledast}$), equivalently, 
 \begin{align} \label{E0011}
 (\mfN_{g, r,n, G} \times_{\overline{\mfM}_{g, r}} \mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r})  \setminus \xi^{-1}({^\circledast \mfO} \mfp_{\mr{GL}_n, (g-1) \cdot \sharp (G) +1, r  \cdot \sharp (G)}^{^\text{Zzz...}})
 \end{align}
is a closed substack of $\mfN_{g, r,n,G} \times_{\overline{\mfM}_{g, r}} \mfO \mfp_{\mr{GL}_n, g, r}^\ZZZ$.
On the other hand, 
the natural projection 
\begin{align} \label{E0012}
 \mfN_{g, r, n, G} \times_{\overline{\mfM}_{g, r}} \mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r} \migi \mfM_{g,r}.
\end{align}
is proper since
both 
$\mfN_{g, r, n,G}/\mfM_{g,r}$  and  $ \mfO \mfp^{^\text{Zzz...}}_{\mr{GL}_n, g,r}/\overline{\mfM}_{g,r}$  are  proper (cf. ~\cite{Wak5}, Theorem C).
Hence, 
  ${^\circledast \mfM}_{g, r, n, G}$, which 
 coincides  with the complement of the image of (\ref{E0011}) via (\ref{E0012}), is open in $\mfM_{g,r}$, as desired.
%$\overline{\mfM}_{g, r}^{(n,l)^\circledast\text{-}\mr{ord}}$ turns out to  
 % form an open substack  of $\overline{\mfM}_{g, r}$.
% consisting of pointed hyperbolic curve $X$ over $k$ of type $(g,r)$ and a 
%This implies the proof of Proposition \ref{ghui0}.
\end{proof}
%-----------------------------------------------------------------------[end theorem]-------------------



%-----------------------------------------------------------------------[begin lemma]------------------
\vspace{3mm}
\ble \label{L023} \leavevmode\\
 \ \ \ 
Any geometric generic point  $\mr{Spec}(L) \migi \mfM_{g,r}$ (where $L$ denotes  an algebraically closed field over $k$) of $\mfM_{g,r}$ lies in  ${^\circledast \mfM}_{g, r, n, G}$.
 \ele
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
Denote by $\mfC_L := (C_L/L, \{ \sigma_{L, i} \}_{i=1}^r)$ the pointed proper smooth curve classified by the  point $\mr{Spec}(L) \migi \mfM_{g,r}$.
Let $\mcF^\heartsuit$ be a $\mr{GL}_n$-do'per on $\mfC_L$ and 
$(\mfC'_L, w_L)$ (where $\mfC'_L : = (C'_L/L, \{ \sigma'_{L, i} \}_{i})$) 
an abelian   covering of $\mfC_L$ over $L$  with Galois group $\mr{Gal} (\mfC'_L/ \mfC_L) \cong G$. 
%$\mfw_L := (w_L, \mr{id}_L)  : \mfC'_L \migi \mfC_L$ (where $\mfC'_L : = (C'_L/L, \{ \sigma'_{L, i} \}_{i})$) be an abelian   covering with Galois group $G$, and let 
What we have to prove is that
%To complete the proof of this lemma, we show that 
$w_L^*(\mcF)^\heartsuit$ is $\circledast$-ordinary.
(Here,  since  there is no nontrivial \'{e}tale covering of the projective line over $L$, we only consider  the case where $g \geq 1$.)
%First, the case of $g=0$ is clear since there is no nontrivial \'{e}tale covering of the projective line over $L$.


First, we shall consider the case where $g =1$.
We may assume, without loss of generality, that  $L =K$ and $r =m$ as in \S\,\ref{S43} (hence, $\mfC_K$ is isomorphic to $\mfC_{m, K}^1$).
One may find  an \'{e}tale covering  $w'_K :  C_{m, K} \migi C'$  such that $w_K \circ w'_K = [l]_{C_{m, K}}$ 
%whose composite with $w_K$ coincides with $[l]_{C_K}$
 for some positive integer $l$ prime to $p$.
By Theorem A, the proof reduces to proving  that  $[l]_{C_{m, K}}^*(\mcF)^\heartsuit$   is $\circledast$-ordinary.
Hence, the assertion follows   follows from Proposition \ref{ghjk00098}.
(Moreover, by combining with Theorem A, it holds that  
%a general pointed proper smooth curve  of genus $1$
$\mfC_K$ is $\circledast_{n, G'}$-ordinary for any quotient group $G'$ of $G$. Thus, it follows from Lemma \ref{L022} that any general pointed proper smooth curve is $\circledast_{n, G'}$-ordinary.)

Next, we consider the case  $g \geq 2$.
%To this end, suppose  that the  claim $(**)$ of  the case where $g =1$ has proven (cf. Lemma \ref{ghjk} below).
Let $\mfC_k := (C_k/k, \{ \sigma_{k, i} \}_{i =1}^r)$
%\{ \sigma_{0, i} \}_{i=1}^r)$
 be a pointed stable  curve of type $(g, r)$  over $k$
 obtained (from the argument in \S\,\ref{S27}) by gluing together $g$ pointed elliptic curves $\mfC_{k, j} := (C_{k, j}/k, \{ \sigma_{k, j, i} \}_i)$  
 %of type $(1, r_{j})$ 
% (for some $r_j \geq 0$ satisfying the equality $\sum_{j =1}^g r_j = r$) 
  (by means of a certain clutching data) in such a way that the dual graph of $\mfC_k$ is tree.
By the above discussion, one may find such a curve $\mfC_k$ satisfying furthermore that 
% that  Let us suppose further that
 each $\mfC_{k, j}$ is $\circledast_{n, G'}$-ordinary for  any quotient group $G'$ of $G$.
After possibly replacing $L$ with its extension field, 
there exist an inclusion $R \migiincl L$ (over $k$), where $R := k[[t_1, \cdots, t_m ]]$ (for some $m$), and   a pointed stable curve
\begin{align}
\mfC_{R} := (C_R/R, \{ \sigma_{R, i} \}_{i=1}^r)
\end{align}
over $R$  of type $(g,r)$  satisfying the following properties:
\begin{itemize}
\item[$\bullet$]
If $s$ denotes the closed point of $\mr{Spec}(R)$, then the fiber of $\mfC_{R}$
%$C \migi \mr{Spec}(R)$
 over  $s$
is isomorphic to  $\mfC_k$;
\item[$\bullet$]
If $\eta$ denotes the  geometric generic point  $\mr{Spec}(L) \migi \mr{Spec}(R)$ corresponding to the inclusion $R \migiincl L$, then fiber  of
 %$C \migi \mr{Spec}(R)$
 $\mfC_{R}$
  over  $\eta$ is isomorphic to $\mfC_L$.
  %a smooth curve whose classifying point dominates $\mfM_{g}$.
\end{itemize}
One verifies from the definition of $\mfC_R$ that  there exists a Galois  covering $w_R : C'_R \migi C_R$ over $R$ whose base-change via $\eta$ coincides with $w_L : C'_L \migi C_L$.
(Indeed, the  natural map $\pi_1^{\text{\'{e}t}} (C_L) \migi \pi_1^{\text{\'{e}t}} (C_R)$ between \'{e}tale fundamental groups  is bijective after taking   maximal abelian pro-prime-to-$p$ quotient.)
One may find a set of marked points $\{ \sigma'_{R, i} \}_i$ of $C'_R$ for which $\mfC'_R :=(C'_R/R, \{ \sigma'_{R, i} \}_i)$ forms a pointed stable curve over $R$ and
 $(\mfC'_L, w_L)$ extends to an \'{e}tale covering 
 $(\mfC'_R, w_R)$ of $\mfC_R$.
 % $\mfw_R := (w_R, \mr{id}_R) \ \mfC'_R \migi \mfC_R$.
The  special fiber $\mfC'_k := ( C'_R \times_{R, s} k, \{ \sigma'_{R, i} \times \mr{id}_k \}_i)$ of $\mfC'_R$
may be obtained by gluing together some pointed proper smooth curves $\mfC'_{k, j}$ ($j = 1, 2, \cdots $).
For each $j$, the fiber 
$(\mfC'_k, w_k)$ of $(\mfC'_R, w_R)$ over $s$
%$w_k : \mfC'_k \migi \mfC_k$ of $w_R$ over  $s$
   restricts to 
an abelian \'{e}tale  covering $(\mfC'_{k, j}, w_{k, j})$ of $\mfC_{k, s_j}$
%$\mfw_{k, j} ; \mfC'_{k, j} \migi \mfC_{k, s_j}$ 
with Galois group $G_j$ (where $s_j \in \{ 1, \cdots, g\}$ and  $G_j$ denotes some quotient group of $G$).
% $C'_k := C'_R \times_{R, s} k$.
%For each $j$, there exists a set of marked points $\{ \sigma'_{}$
Here, observe that since $\mfO \mfp_{\mr{GL}_n, g,r}^{^\text{Zzz...}}$ is finite and generically \'{e}tale  over $\overline{\mfM}_{g,r}$, there exists 
 a $\mr{GL}_n$-do'per $\mcF_R^\heartsuit$  on $\mfC_R$ which restricts to $\mcF^\heartsuit$.
 By the condition on each $\mfC_{k, j}$ assumed earlier,
 the restrictions 
 $\mcF |_{\mfC_{k, s_j}}^\heartsuit$ of $\mcF^\heartsuit$ to $\mfC_{k, s_j}$, as well as its  pull-back via $w_{k, j}$,  is  $\circledast$-ordinary.
 %  $w^* (\mcF_k)^\heartsuit |_{\mfC_{k, l_j}}$ and moreover, $\mfw_{k, l}^* (\mcF )$ is $n^\circledast$-ordinarly.
% By  the assertion for  case of $g =1$ proved earlier,
% the pull-back $\mfw_{k, l}^* (\mcF)^\heartsuit$ is $n^\circledast$-ordinary.
Hence, it follows from Proposition \ref{ghu2} that $w_k^* (\mcF)^\heartsuit$ is $\circledast$-ordinary.
%FInally, by 
 %, and hence, $w^* (\mcF)^\heartsuit$ is $n^\circledast$-ordinary.
Finally, by Corollary \ref{C001}, $w_L^* (\mcF)^\heartsuit$ is $\circledast$-ordinary, and hence, we complete the proof of Lemma \ref{L023}.
\end{proof}
\vspace{3mm}
%-----------------------------------------------------------------------[end lemma]-------------------
In particular, we have completed the proof of Theorem B.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%--[ begin  section1]---%%%%%%
\vspace{10mm}
\section{Ordinariness  for a semisimple Lie algebra $\mfg$} \vspace{3mm}



%\vspace{5mm}

As shown  in Proposition \ref{ghu} (or Proposition \ref{ghu3}, (i)), the ordinariness, as well as the $\circledast$-ordinariness, of $\mr{GL}_n$-do'pers is closely related to  the  unramifiedness of the moduli stack $\mfO \mfp_{\mfs \mfl_n, g,r}^{^\text{Zzz...}}$ at their classifying morphisms.
In a similar vein, 
one may define the  ordinariness of dormant $\mfg$-opers for a certain class of  semisimple Lie algebras $\mfg$  and consider the sorts of problems discussed   before for dormant $\mfg$-opers.
%result  similar to the main result of the present paper.


\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S51}


For example, let $\mfg$ be a semisimple Lie algebra over $k$ whose rank
% we must assume that
  is, in a certain sense (cf. the conditions $\text{(Char)}_0$ and $\text{(Char)}^\mbW_p$ described in ~\cite{Wak5}, \S\,2.1),  sufficiently small relative to the characteristic $p$ of $k$. 
%In particular, the case $\mfg = \mfs \mfl_{p-1}$ should be omitted.)
Once we  fix a certain collection of data concerning such a $\mfg$ (e.g., a choice of an $\mfs \mfl_2$-triple $\mfs \mfl_2 \migiincl \mfg$ and a pinning that are compatible  in a certain sense (cf. ~\cite{Wak5}, \S\,2.1)), it makes sense to speak about   the notion of a dormant $\mfg$-oper on a pointed stable curve.
In the following, when we mention a semisimple $\mfg$, we always assume that such a collection of data have been chosen.
Roughly speaking, a dormant $\mfg$-oper on a pointed stable  curve $\mfX := (X/S, \{ \sigma_i \}_i)$ is a pair  $\mcE^\spadesuit := (\mcE, \nabla_\mcE)$ consisting  of  an $\mr{Aut}^0 (\mfg)$-torsor $\mcE$ over $X$ (where $\mr{Aut}^0 (\mfg)$ denotes the identity component of the group of Lie algebra automorphisms of $\mfg$)  and  an (integrable) $S$-log  connection $\nabla_\mcE$ on $\mcE$ satisfying certain conditions, including  vanishing $p$-curvature.
We refer to ~\cite{Wak5}, Definition 2.2.1, and Definition 3.6.1, for the precise definition of a dormant $\mfg$-oper.
As in the previous  discussion, we shall abbreviate a dormant  $\mfg$-oper to a {\bf $\mfg$-do'per}.
If, moreover,  
$(\mfY, w, v)$ is 
%$\mfw : \mfY \migi \mfX$ is 
an \'{e}tale covering of $\mfX$, then by pulling-back the data in $\mcE^\spadesuit$ via $w$, one may obtain a $\mfg$-do'per $w^* (\mcE)^\spadesuit$ on $\mfY$, which we refer to as the {\bf pull-back of $\mcE^\spadesuit$}.



Denote by $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$ the moduli stack classifying pairs $(\mfX, \mcE^\spadesuit)$ consisting of a pointed stable curve   of type $(g,r)$ over $k$  and a $\mfg$-do'per $\mcE^\spadesuit$ on it.
By forgetting the data of a $\mfg$-do'per, we have a morphism $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r} \migi \overline{\mfM}_{g,r}$.
Denote by  
\begin{align}
{^\circledcirc \mfO} \mfp^{^\text{Zzz...}}_{\mfg, g,r}
\end{align}
the  unramified (equivalently, \'{e}tale, due to  ~\cite{Wak5}, Corollary 5.12.2) locus of $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$ over $\overline{\mfM}_{g,r}$; it forms a possibly empty open substack of $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$.




%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{D0001} \leavevmode\\
 \ \ \ 
Let $\mfX$ be a pointed stable curve  over a $k$-scheme $S$ of type $(g,r)$.
We shall say that a $\mfg$-do'per $\mcE^\spadesuit$ on $\mfX_{}$ is {\bf $\circledast_\mfg$-ordinary} if its  classifying map $S \migi \mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}}$   factors through the open immersion ${^\circledcirc \mfO} \mfp^{^\text{Zzz...}}_{\mfg, g,r} \migiincl \mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$.
 \ede
\vspace{3mm}
%-----------------------------------------------------------------------[end definition]-------------------

The notion of $\circledast_{\mfs \mfl_n}$-ordinariness  is, via the isomorphism (\ref{gg099}), equivalent to the $\circledast$-ordinariness of $\mr{GL}_n$-do'pers discussed so far. 
More precisely, 
let $\mcF^\heartsuit$ be a $\mr{GL}_n$-do'per and denote by $\mcE^\spadesuit$ the $\mfs \mfl_n$-do'per associated with $\mcF^\heartsuit$.
Then,  it follows from the equivalence (i) $\Leftrightarrow$ (iii) in Proposition \ref{ghu} that 
$\mcF^\heartsuit$ is $\circledast$-ordinary  if and only if $\mcE^\spadesuit$ is $\circledast_{\mfs \mfl_n}$-ordinary. 
%-----------------------------------------------------------------------[begin remark]------------------
The result described in Theorem B generalizes to some $\mfg$.
In fact, we shall consider simple Lie algebras $\mfg$ introduced  in the following definition. 
% To obtain a result similar to Theorem ? for $\mfg$ (not only for $\mfg = \mfs \mfl_n$),
% we consider a class of Lie algebras $\mfg$ as defined in Definition ? below, for  which one may relate $\mfs \mfl_n$-opers with  $\mfg$-opers.
 
 %-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{D00012} \leavevmode\\
 \ \ \ 
We shall say that $\mfg$ is {\bf admissibly of classical type $A$} if 
 $\mfg$ is isomorphic to  either $\mfs \mfl_n$ (for $n < p$), $\mfs \mfo_{2n+1}$ (for $n < \frac{p-1}{2}$), or $\mfs \mfp_{2n}$ (for $n <  \frac{p}{2}$).
\ede
\vspace{3mm}
%-----------------------------------------------------------------------[end definition]-------------------

If $\mfg$ is admissibly of classical type $A$, 
%If $\mfg$ is taken to be either $\mfs \mfl_n$ (for $n < p$), $\mfs \mfo_{2n+1}$ (for $n < \frac{p-1}{2}$), or $\mfs \mfp_{2n}$ (for $n <  \frac{p}{2}$), 
%then   $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$ is finite and generically \'{e}tale over $\mfM_{g,r}$ (cf. ?).
%In particular, for such a $\mfg$, 
then  the locus ${^\circledcirc \mfO} \mfp^{^\text{Zzz...}}_{\mfg, g,r}$ is dense and open in each irreducible component of $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$ that dominates $\overline{\mfM}_{g,r}$ (cf. ~\cite{Wak5}, Theorem F), as in the case of $\mfs \mfl_n$ (cf. Proposition \ref{ghu3} (i)).
Moreover, note that  one may find   an embedding
 $\mfg \migi \mfs \mfl_m$ (for some $m\leq p-1$) that preserves the respective pinnings and $\mfs \mfl_2$-triples (cf. ~\cite{Wak5}, \S\,7.9).
This embedding yields 
a closed immersion
 \begin{align}
 \iota_{\mfg \migi \mfs \mfl_m} : \mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r} \migi \mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_m, g,r}
 \end{align}
  over $\overline{\mfM}_{g,r}$.
 
 

%-----------------------------------------------------------------------[begin theorem]------------------
\vspace{3mm}
\bpr \label{PPP2} \leavevmode\\
 \ \ \ 
Let $\mfX$ be a pointed stable  curve  over a $k$-scheme  of type $(g,r)$ and $\mcE^\spadesuit$ a $\mfg$-do'per on $\mfX$.
Denote by 
$\mcE_{\mfs \mfl_m}^\spadesuit$ the $\mfs \mfl_m$-do'per on $\mfX$ classified by
the composite  $\iota_{\mfg \migi \mfs \mfl_m} \circ s : S \migi \mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_m, g,r}$, where  $s$ denotes  
%$\iota_{\mfg \migi \mfs \mfl_m}$ and
 the classifying morphism $s : S \migi  \mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r}$ of $\mcE^\spadesuit$.
Suppose that $\mcE_{\mfs \mfl_m}^\spadesuit$ is $\circledast_{\mfs \mfl_m}$-ordinary.
Then, $\mcE^\spadesuit$ is $\circledast_\mfg$-ordinary.
% This closed immersion associates,  to each $\mfg$-do'per $\mcE^\spadesuit$ on a pointed stable curve $\mfX$, an $\mfs \mfl_m$-do'per $\mcE_{\mfs \mfl_m}^\spadesuit$ on $\mfX$.
\epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
The assertion follows from the observation that $\mfO \mfp^{^\text{Zzz...}}_{\mfg, g,r} $ is unramified (relative to $\overline{\mfM}_{g,r}$) at the point $s$
since $\iota_{\mfg \migi \mfs \mfl_m}$ is a closed immersion and $\mfO \mfp^{^\text{Zzz...}}_{\mfs \mfl_m, g,r}$ is unramified at the point  $s \circ \iota_{\mfg \migi \mfs \mfl_m}$.
% at an $S$-rational  point $S$ if   is uniramified.
%  Hence, the assertion follows from the definition of ($\mfg$- and $\mfs \mfl_m$-)ordinariness of do'pers.
\end{proof}
%-----------------------------------------------------------------------[end theorem]-------------------




 
 By means of the above proposition, we have the following generalization of Theorem B.
 
  


 % \end{rema}
%-----------------------------------------------------------------------[end remark]-------------------


%-----------------------------------------------------------------------[begin theorem]------------------
\vspace{3mm}
\bt \leavevmode\\
 \ \ \ 
Let $\mfg$ be a semisimple Lie algebra over $k$ which is admissibly  of classical type $A$, 
% If $\mfg$ is taken to be either $\mfs \mfl_n$ (for $n < p$), $\mfs \mfo_{2n+1}$ (for $n < \frac{p-1}{2}$), or $\mfs \mfp_{2n}$ (for $n <  \frac{p}{2}$).
  $\mfX$  a pointed proper smooth curve of type $(g,r)$ over $k$,  
  $(\mfY, w, \mr{id}_k)$ 
%   $w : \mfY \migi \mfX$
    an abelian covering of $\mfX$ over $k$, and $\mcE^\spadesuit$  a  $\mfg$-do'per on $\mfX$.
  Suppose the following conditions:
  \begin{itemize}
  \item[(i)]
   $\mcE^\spadesuit$ is $\circledast_{\mfg}$-ordinary.
  \item[(ii)]
  $(\mfY, w, \mr{id}_k)$ is abelian with $p \nmid \sharp (\mr{Gal}(\mfY/\mfX))$;
  \item[(iii)]
  $\mfX$ is general in the moduli stack $\mfM_{g,r}$;
 % \item[(iii)]
  \end{itemize}
   %  with , and 
 %Suppose that  $\mfX$ is general in $\mfM_{g,r}$
 % (in the moduli space classifying proper smooth curve over $k$ of genus $g$) and
 %and $\mcE^\spadesuit$ is ordinary.
 Then, 
 the pull-back $w^* (\mcE)^\spadesuit$
 is $\circledast_{\mfg}$-ordinary.
 % $\mfg$-do'per  $\mfw^* (\mcE)^\spadesuit$ pulled-back 
% the dormant $\mfg$-oper   $w^*(\mcE^\spadesuit)$ on $Y$ obtained from $\mcE^\spadesuit$  by pulling-back via $w$ is ordinary.
 \et
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
%Let $\mcE^\spadesuit$ be as in the statement, and, for simplicity, let us  write $\mcE'^\spadesuit := \mfw^* (\mcE)^\spadesuit$.
%Then, b
By  Theorem B (and the discussion following Definition \ref{D0001}), 
the pull-back  (via $w$) of the $\mfs \mfl_m$-do'per  $\mcE_{\mfs \mfl_m}^\spadesuit$ associated (via $\iota_{\mfg \migi \mfs \mfl_m}$) with $\mcE^\spadesuit$ 
%, which is isomorphic to 
%$\mfw^* (\mcE')^\heartsuit$,
 is $\circledast_{\mfs \mfl_m}$-ordinary.
But, since this $\mfs \mfl_m$-do'per on $\mfY$ is isomorphic to the $\mfs \mfl_m$-do'per associated (via $\iota_{\mfg \migi \mfs \mfl_m}$) with the $\mfg$-do'per $w^* (\mcE)^\spadesuit$, Proposition \ref{PPP2} implies that $w^* (\mcE)^\spadesuit$ is $\circledast_\mfg$-ordinary.
\end{proof}
%-----------------------------------------------------------------------[end theorem]-------------------

\vspace{5mm}
 %----------------------------------------------------------------------[begin subsection]-------------
\subsection{} \label{S52}

We shall conclude the paper  with  the study of  a relationship between $\circledast_{\mfs \mfl_2}$-ordinariness and $\circledast_\mfg$-ordinariness.
 %-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \label{D000d1} \leavevmode\\
 \ \ \ 
We shall say that a pointed stable curve $\mfX$ over a $k$-scheme $S$ is {\bf $\circledast_\mfg$-ordinary} if 
for any $S$-scheme $T$, any $\mfg$-do'per on the base-change of $\mfX$ over $T$ is $\circledast_\mfg$-ordinary.
\ede
\vspace{3mm}
%-----------------------------------------------------------------------[end definition]-------------------





Denote by  ${^\circledast \overline{\mfM}}_{\mfg, g,r}$
%$\mfM_{g,r}^{\text{$\mfg$-ord}}$
  the substack of $\overline{\mfM}_{g,r}$ classifying $\circledast_\mfg$-ordinary curves.
One verifies that ${^\circledast \overline{\mfM}}_{\mfg, g,r}$ coincides with  the complement of the image, via the forgetting map $\mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}} \migi \overline{\mfM}_{g,r}$ (which is proper),   of the complement of ${^\circledast \mfO} \mfp_{\mfg, g,r}^{^\text{Zzz...}}$ in $\mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}}$.
In particular, 
%(since $\mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}} \migi \overline{\mfM}_{g,r}$ is finite) $\overline{\mfM}_{g,r}^{\text{$\mfg$-ord}}$
 ${^\circledast \overline{\mfM}}_{\mfg, g,r}$  is a possibly empty open substack of $\overline{\mfM}_{g,r}$.
% If, moreover, $r =0$, then it is nonempty (cf. Theorem C (ii)).
Then, one may prove the following proposition.

%-----------------------------------------------------------------------[begin theorem]------------------
\vspace{3mm}
\bt \leavevmode\\
 \ \ \ 
${^\circledast \overline{\mfM}}_{\mfg, g,r}$  is an open substack of  ${^\circledast \overline{\mfM}}_{\mfs \mfl_2, g,r}$.
If, moreover,  $r=0$, then  ${^\circledast \overline{\mfM}}_{\mfg, g,0}$ (and hence, ${^\circledast \overline{\mfM}}_{\mfs \mfl_2, g,r}$) is a dense open substack of $\overline{\mfM}_{g,0}$.
\et
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
The former assertion follows from  the fact that the image of $\mfO \mfp_{\mfs \mfl_2, g,r}^{^\text{Zzz...}}  \setminus {^\circledast \mfO} \mfp_{\mfs \mfl_2, g,r}^{^\text{Zzz...}} $ via  the closed immersion $\mfO \mfp_{\mfs \mfl_2, g,r}^{^\text{Zzz...}} \migi \mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}}$ discussed in  ~\cite{Wak5}, (134) and  Corollary 2.7.6 (i),  is contained in $\mfO \mfp_{\mfg, g,r}^{^\text{Zzz...}}  \setminus {^\circledast \mfO} \mfp_{\mfg, g,r}^{^\text{Zzz...}}$.
The latter assertion follows from ~\cite{Wak5}, Theorem C (ii) (and the former assertion).
\end{proof}
%-----------------------------------------------------------------------[end theorem]-------------------



\vspace{10mm}

%Now we shall prove Theorem ? for arbitrary $g$.

%\begin{proof}[proof of Theorem A]


%\end{proof}

 
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\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}







%-----------------------------------------------------------------------[begin definition]------------------
\vspace{3mm}
\bde \leavevmode\\
 \ \ \ 
 \ede
\vspace{3mm}
%-----------------------------------------------------------------------[end definition]-------------------



%-----------------------------------------------------------------------[begin corollary]------------------
\vspace{3mm}
\bco \leavevmode\\
 \ \ \ \eco
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
\end{proof}
%-----------------------------------------------------------------------[end corollary]-------------------





%-----------------------------------------------------------------------[begin proposition]------------------
\vspace{3mm}
\bpr \leavevmode\\
 \ \ \ 
 \epr
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
\end{proof}
\vspace{3mm}
%-----------------------------------------------------------------------[end proposition]-------------------

%-----------------------------------------------------------------------[begin lemma]------------------
\vspace{3mm}
\ble \leavevmode\\
 \ \ \ 
 \ele
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}

\end{proof}
\vspace{3mm}
%-----------------------------------------------------------------------[end lemma]-------------------

%-----------------------------------------------------------------------[begin theorem]------------------
\vspace{3mm}
\bt \leavevmode\\
 \ \ \ 
\et
%-----------------------------------------------------------------------[begin proof]-------------------
\begin{proof}
\end{proof}
%-----------------------------------------------------------------------[end theorem]-------------------

%-----------------------------------------------------------------------[begin conjecture]------------------
\vspace{3mm}
\begin{conj} \leavevmode\\
\end{conj}
\vspace{3mm}
%-----------------------------------------------------------------------[end conjecture]-------------------


%-----------------------------------------------------------------------[begin remark]------------------
\vspace{3mm}
\begin{rem}{(, ~\cite[P.9]{JP})}
\end{rem}
\vspace{3mm}
%-----------------------------------------------------------------------[end remark]-------------------




1602.07061.extracted_bib
{ACGH D. Abramovich, Q. Chen, D. Gillam, Y. Huang, M. Olsson, M. Satriano, S. Sun. Logarithmic geometry and moduli. Handbook of Moduli, Vol.\,I, Adv. Lect. Math. {24, Int. Press, Somerville, MA, (2013), pp.\,1-61.
1602.07061.extracted_bib
{BD1 A. Beilinson, V. Drinfeld, ``Quantization of Hitchin's integrable system and Hecke eigensheaves." Available at: %http://math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf. http://math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf
1602.07061.extracted_bib
{BD2 A. Beilinsion, V. Drinfeld, Opers. arXiv: math. AG/0501398v1, (2005).
1602.07061.extracted_bib
{Bou I. I. Bouw, The $p$-rank of curves and covers of curves. {Courbes semi-stables et groupe fondamental en g\'{eom\'{etrie alg\'{ebrique, Progr. Math. {187 (2000), pp. 267-277.
1602.07061.extracted_bib
{DM P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. {Publ. Math. I.H.E.S. {36 (1969), pp. 75-110.
1602.07061.extracted_bib
{Hoshi1 Y. Hoshi, The exactness of the log homotopy sequence. {Hiroshima Math. J. {39 (2009), pp. 61-121.
1602.07061.extracted_bib
{JRXY K. Joshi, S. Ramanan, E. Z. Xia, and J. K. Yu, On vector bundles destabilized by Frobenius pull-back. Composito Math. {142 (2006), pp. 616-630.
1602.07061.extracted_bib
{JP K. Joshi, C. Pauly, Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves. arXiv: math. AG/0912.3602 (2009).
1602.07061.extracted_bib
{KaFu F. Kato, Log smooth deformation and moduli of log smooth curves. Internat. J. Math. {11, (2000), pp. 215-232.
1602.07061.extracted_bib
{KaKa K. Kato, Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory, John Hopkins Univ. Press, Baltimore, (1989), pp. 191-224.
1602.07061.extracted_bib
{Kal N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Etudes Sci. Publ. Math. {39 (1970), pp. 175-232.
1602.07061.extracted_bib
{Kn2 F. F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M_{g,r$. Math. Scand. {52 (1983), pp. 161-199.
1602.07061.extracted_bib
{Mzk1 S. Mochizuki, A theory of ordinary $p$-adic curves. Publ. RIMS {32 (1996), pp. 957-1151.
1602.07061.extracted_bib
{Mzk2 S. Mochizuki, {Foundations of $p$-adic Teichm\"{uller theory. American Mathematical Society, (1999).
1602.07061.extracted_bib
{Nak S. Nakajima, On generalized Hasse-Witt invariants of an algebraic curve. Galois groups and their representations (Nagoya, 1981) (Y. Ihara, ed.), Adv. Stud. Pure Math. {2, North Holland/Kinokuniya, (1983), pp. 69-88.
1602.07061.extracted_bib
{Og A. Ogus, $F$-Crystals, Griffiths Transversality, and the Hodge Decomposition. Ast\'{erisque {221, Soc. Math. de France, (1994).
1602.07061.extracted_bib
{Og3 A. Ogus, Higgs cohomology, $p$-curvature, and the Cartier isomorphism. Compositio. Math. {140 (2004), pp. 145-164.
1602.07061.extracted_bib
{O2 B. Osserman, Logarithmic connections with vanishing $p$-curvature. J. Pure and Applied Algebra {213 (2009), pp. 1651-1664.
1602.07061.extracted_bib
{Ray1 M. Raynaud, Rev\^{etements des courbes en caract\'{eristique $p >0$ et ordinarit\'{e. Compositio Math. {123 (2000), pp. 73-88.
1602.07061.extracted_bib
{Wak Y. Wakabayashi, An explicit formula for the generic number of dormant indigenous bundles. Publ. Res. Inst. Math. Sci. {50 (2014), pp. 383-409.
1602.07061.extracted_bib
{Wak2 Y. Wakabayashi, Spin networks, Ehrhart quasi-polynomials, and combinatorics of dormant indigenous bundles. RIMS Preprint {1786 (2013).
1602.07061.extracted_bib
{Wak3 Y. Wakabayashi, The symplectic nature of the space of dormant indigenous bundles on algebraic curves. RIMS Preprint {1801 (2014).
1602.07061.extracted_bib
{Wak5 Y. Wakabayashi, A theory of dormant opers on pointed stable curves --- a proof of Joshi's conjecture ---. arXiv: math. AG/1411.1208v2, (2014).
1602.07061.extracted_bib
{Wak6 Y. Wakabayashi, A theory of opers on pointed stable curves in positive characteristic I --- generalities ---. manuscript.
1602.07061.extracted_bib
{Wak7 Y. Wakabayashi, A theory of opers on pointed stable curves in positive characteristic II ---deformation and factorization of opers ---. manuscript. %arXiv: math. AG/1411.1208v1, (2014).
1602.07061.extracted_bib
{Wak8 Y. Wakabayashi, A theory of opers on pointed stable curves in positive characteristic III --- a proof of Joshi's conjecture ---. manuscript. %
1602.07061.extracted_bib
{JP %Joshi, K., Pauly, C. %Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves. %math. AG/0912.3602, (2009).