text:
\section{Introduction}
\subsection{Mild difference modules}
Let $\scr{O}_t(*0)\coloneqq\C\conv{t}$ be the field of
convergent Laurent series in a variable $t$.
Let
$\phi_t\colon \scr{O}_t(*0)
\to
\scr{O}_t(*0)$
be an automorphism defined as
$\phi_t(f)(t)\coloneqq f(\tfrac{t}{1+t})$.
If we set $s=t^{-1}$,
we have $\phi_t(f)(s)=f(s+1)$.
By a difference module
(over
the difference field $(\scr{O}_t(*0),\phi_t)$),
we mean a
pair $(\scr{M},\psi)$
of a finite-dimensional
$\scr{O}_t(*0)$-vector space $\scr{M}$ and
an automorphism $\psi\colon \scr{M}\to \scr{M}$
of $\C$-vector spaces satisfying the relation
$\psi(fv)=\phi_t(f)\psi(v)$
for any $f\in \scr{O}_t(*0)$ and $v\in \scr{M}$.

There is a class of difference modules
called \textit{mild} \cite{Galois}*{\S 9}.
It is the class of
difference modules
whose formal fundamental solution is
the the matrices
of the following form:
$F(t)\bigoplus_{i=1}^j\exp(\frak{a}_i(t))t^{G_i}$.
Here, $F(t)$ is an
$(r\times r)$-matrix
whose coefficients
are formal power series
in $t^{-1/m}$ for positive integers $r,m$,
$\frak{a}_i(t)=\sum_{k=1}^m c_{i,k} t^{-k/m}$
($c_{i,k}\in\C$),
and $G_i$ is a constant $(r_i\times r_i)$-matrix
with $\sum_{i=1}^j r_i=r$.

The purpose of this paper
is to establish the Riemann-Hilbert correspondence for
mild difference modules as an analog of that for
germs of meromorphic connections
in one complex variable
by Deligne-Malgrange
\cites{Deligne,Malgrange}.
See also \cite{Sabbah}.

\subsection
{Stokes filtered
locally free sheaves
for mild difference modules}
To formulate the Riemann-Hilbert correspondence for
mild difference modules,
we introduce the notion of
Stokes filtered locally free sheaves
for
difference modules in \S \ref{SSt}.
We explain the notion briefly,
comparing with the
case of meromorphic connections.

In the case of germs of meromorphic connections,
we consider the notion of
a Stokes filtered local system on
$S^1=\{z\in\C\mid |z|=1\}$.
It is a pair
$(\cal{L},\cal{L}_\bullet)$
of a local system $\cal{L}$ of
finite-dimensional $\C$-vector spaces on $S^1$
and a filtration $\cal{L}_{\bullet}$
on $\cal{L}$ indexed by a
sheaf
$\cal{I}=\bigcup_{m\geq 0}
z^{-\frac{1}{m}}\C[z^{-\frac{1}{m}}]$
of ordered abelian groups.

The Deligne-Malgrange theorem claims that
there is an equivalence
(called the Riemann-Hilbert functor)  between
the category of germs of meromorphic
connections
and the category of
Stokes filtered local systems
on $S^1$.

When a Stokes filtered local system
$(\cal{L},\cal{L}_\bullet)$
corresponds to germs of a meromorphic connection
$\cal{M}$ by the Riemann-Hilbert functor,
the sheaf $\cal{L}$ is regarded as
the sheaf of flat sections of $\cal{M}$ on sectors
and
the filtration $\cal{L}_{\bullet}$
describes the growth rate of
the sections. The filtration $\cal{L}_\bullet$
is called
the Stokes filtration on $\cal{L}$
since the relation between the splittings
of $\cal{L}_\bullet$ on different domains
describes the classical Stokes phenomenon
of the solutions of the differential equation
associated to $\cal{M}$.

In the case of difference modules,
we consider a locally free sheaf $\scr{L}$
over a sheaf $\scr{A}_{\mathrm{per}}$
of rings over $S^1$.
Here, $\scr{A}_{\mathrm{per}}$
is a sheaf of rings over $S^1$
defined as follows:
\begin{align*}
    \scr{A}_{\mathrm{per}}(U)=
    \begin{cases}
    \C\conv{u^{-1}}&
    (U\subset (0,\pi),U\neq \emptyset)\\
    \C\conv{u}&(U\subset (-\pi,0),U\neq \emptyset)\\
    \C[u^{\pm 1}]
    &(U\cap\{e^{i\pi},e^0\}\neq\emptyset),
    \end{cases}
\end{align*}
where $U$ is assumed to be connected
and we set $(a,b)\coloneqq
\{e^{{\tt{i}}\theta}\in S^1\mid a<\theta<b\}$
for $a,b\in \mathbb{R}$ with $a<b$.
If we put $u=\exp(2\pi{\tt{i}} t^{-1})$,
we can regard $\scr{A}_{\mathrm{per}}$ as
a sheaf of rings of a certain
class of  $\phi_t$-invariant
(or, periodic with respect to
$s\mapsto s+1$) functions
(see \S \ref{Prel} for more details).

Then, we define
a filtration $\scr{L}_{\bullet}$
on $\scr{L}$
indexed by a sheaf of ordered
abelian groups.
We call it a Stokes filtration.
It will be turned out
that the filtration describes
the growth rate of the solutions of
the difference equation associated
to the difference module.
A new feature
of the filtration is the compatibility
of the action of $u$ with the filtration:
\begin{align*}
    u\scr{L}_{\leqslant \frak{a}}
    =\scr{L}_{\leqslant \frak{a}+2\pi{\tt{i}} t^{-1}},
\end{align*}
where $\frak{a}$ is an arbitrary index
(see \S \ref{SSt} for more details).

It is worth mentioning that the Stokes filtered
$\scr{A}_{\mathrm{per}}$-module
can be  non-graded
even if it
is rank one as a free
$\scr{A}_{\mathrm{per}}$-module.
This point will be explained in \S \ref{R1}.

\subsection{Main result}
Let ${\mathsf{Diffc}}^{\mathrm{mild}}$ be the category of
mild difference modules.
Let ${\mathsf{St(\scr{A}_\per)}}$ be the category of
the Stokes filtered locally free
$\scr{A}_{\mathrm{per}}$-modules.
Then, we can state the main result of the
present paper:
\begin{theorem}[Theorem \ref{Main}]\label{IMa}
There is a functor
${\mathrm{RH}}\colon {\mathsf{Diffc}}^{\mathrm{mild}} \to {\mathsf{St(\scr{A}_\per)}}$,
which is an equivalence of categories.
\end{theorem}
This result is analogous to
that of Deligne-Malgrange
\cites{Deligne,Malgrange}
(See
\cite{Sabbah}*{Theorem 5.8}).
The proof of this theorem is
similar to that can be found
in \cite{Sabbah}.
The main difference is
the definition of the functor
${\mathrm{RH}}$.
See Remark \ref{Periodic}
for more precise.

The author hopes that
this result contributes
to the intrinsic understanding of
linear difference modules.
In particular,
it would be interesting to
use the result
to
describe the Stokes structure of
the Mellin transformation
of a holonomic $\cal{D}$-module
concerning
recent progress
\cites{Bloch, Local, GS,
garcia2018mellin}
in the study
of the Mellin transformations
(see Remark \ref{Mellin}).

\subsection{Outline of the paper}
In \S \ref{Prel},
we prepare some notions used throughout the paper.
In \S \ref{SSt},
we introduce the notion of Stokes filtered
$\scr{A}_{\mathrm{per}}$-modules.
In \S \ref{RH&RS},
we formulate and prove the main theorem
assuming a theorem
proved in \S \ref{PRH}.
\subsection*{Acknowledgement}
The author would like to express his deep gratitude to
Claude Sabbah, who gave the author fruitful comments on
preliminary versions of this paper.
The author also would like to thank
Tatsuki Kuwagaki,
Takuro Mochizuki,
Fumihiko Sanda, and Takahiro Saito
for discussions and encouragement
in many occasions.
 The author is supported by JSPS KAKENHI Grant Number JP
 20K14280.
 \section{Preliminaries}\label{Prel}
In this section,
we prepare some notions used throughout the paper.
\subsection{An automorphism on a projective line }\label{Notations}
Let $\C$ be the set of complex numbers.
Set $\C^*\coloneqq \C\setminus \{0\}$.
Natural inclusion is denoted by
$\jmath\colon \C^*\to \C$.
When we distinguish a variable such as $t$,
we use the symbols $\C_t$ and $\C^*_t$.
Let
    $S^1=\{e^{{\tt{i}}\theta}\in\C\mid \theta\in \mathbb{R}\}$
be the unit circle,
where we set ${\tt{i}}=\sqrt{-1}$.
For two real numbers $a,b$ with $a<b$,
we set $(a,b)\coloneqq \{e^{{\tt{i}}\theta}\in S^1\mid a<\theta<b\}$.
\subsubsection{Real blowing up}\label{RBU}
We set
\begin{align*}
    \widetilde{\C}=\{(t,e^{{\tt{i}}\theta})\in \C\times S^1\mid
    t=|t|e^{{\tt{i}}\theta}\},
\end{align*}
which is called \textit{the real blowing up of $\C$ at the origin.}
When we distinguish a variable such as $t$,
we use the notation $\widetilde{\C}_t$.
There are maps
    $\varpi\colon \widetilde{\C}\longrightarrow \C$,

    $\widetilde{\jmath}\colon \C^*\hookrightarrow
    \widetilde{\C}$,
    and $\widetilde{\imath}\colon S^1 \hookrightarrow \widetilde{\C}$
defined by $\varpi(t,e^{{\tt{i}}\theta})=t$, $\widetilde{\jmath}(t)=(t,t/|t|)$,
and $\widetilde{\imath}(e^{{\tt{i}}\theta})=(0,e^{{\tt{i}}\theta})$, respectively.
We sometimes denote the boundary of $\widetilde{\C}_t$
by $S^1_t$ to distinguish a variable
such as $t$.

\subsubsection{Unit disc}\label{Unit}
Let $\Delta\coloneqq\{t\in\C\mid |t|<1\}$
be a unit open disc.
We set $\Delta^*\coloneqq \Delta\setminus \{0\}$ and
    $\widetilde{\Delta}\coloneqq \{(t,e^{{\tt{i}}\theta})\in \widetilde{\C}
    \mid t\in \Delta\}.$
Let $\varpi_\Delta\colon \widetilde{\Delta}\to\Delta$
be the projection.
When we distinguish a variable such as $t$,
we use the notations $\Delta_t$, $\Delta_t^*$ and $\widetilde{\Delta}_t$.
The natural inclusions
are denoted by
$\jmath_\Delta\colon \Delta^*\to \Delta$,
$\widetilde{\imath}_\Delta\colon S^1\to \widetilde{\Delta}$,
and
$\widetilde{\jmath}_\Delta\colon
\Delta^*\to \widetilde{\Delta}$.
Let $\varphi_t\colon \Delta_t\to \C_t$ be a holomorphic function
defined as
\begin{align*}
    \varphi_t(t)\coloneqq \frac{t}{1+t}.
\end{align*}
The map uniquely extends
to a continuous map
$\widetilde{\varphi}_t\colon \widetilde{\Delta}_t
\longrightarrow \widetilde{\C}_t$.

\subsubsection{Another coordinate}\label{Another}
Let $\C_s$ be a complex plane
with a coordinate $s$.
When we use
the two complex variables $s$ and $t$,
we implicitly assume the relation
\begin{align}\label{s=t}
    s=t^{-1}.
\end{align}
In other words,
we consider the complex projective line $\mathbb{P}^1$
covered by two open subsets $\C_s$ and $\C_t$
with the relation \eqref{s=t}.
Let $\varphi_s\colon \C_s\to \C_s$
be the map defined as
\begin{align*}
    \varphi_s(s)=s+1.
\end{align*}
Under the  relation \eqref{s=t},
the map $\varphi_s$
coincides with the map $\varphi_t$
on the domain $\Delta_t^*=\{s\in \C_s \mid |s|>1\}$.
Hence the maps $\varphi_s$ and $\varphi_t$ are glued to
an automorphism on $\mathbb{P}^1$.

\subsection{Sheaves of periodic functions}
For a point $x$ in a topological space
$X$ and a sheaf $\scr{F}$
on $X$,
let $\scr{F}_x$ denote the set of germs of $\scr{F}$ at $x$
(with some structure).
For a continuous map $f\colon X\to Y$ between topological spaces,
$f_*$ denotes the pushing forward of sheaves
and $f^{-1}$ denotes the pull back of sheaves.
If
$X$ is a complex manifold,
let $\scr{O}_X$ denote the sheaf of holomorphic functions on $X$.
If $X$ is a Riemann surface
and $D$ is a finite set of points,
then let $\scr{O}_X(*D)$ denote the sheaf
of meromorphic functions on $X$
whose poles are contained in $D $.

\subsubsection{Functions with fixed asymptotic behavior}
\label{Sheaves}
Using the notations in \S \ref{RBU}, set
\begin{align*}
    \widetilde{\scr{O}}
    \coloneqq
    \widetilde{\imath}^{-1}\widetilde{\jmath}_*\scr{O}_{\C^*},
\end{align*}
which is a sheaf on $S^1$.
There are subsheaves $\scr{A}^{\leqslant 0}$, and $\scr{A}^{<0}$
in $\widetilde{\scr{O}}$ characterized by
their asymptotic behavior as follows
(see \cite{Sabbah} for precise definitions):
\begin{itemize}

    \item $\scr{A}^{\leqslant 0}$ is the sheaf of holomorphic functions
    \textit{which are of moderate growth.}
    \item $\scr{A}^{<0}$ is the sheaf of holomorphic functions
    \textit{which are of rapid decay}.
\end{itemize}
To emphasize the coordinate function such as $t$,
 we use the notation $\widetilde{\scr{O}}_t$, $\scr{A}_t^{\leqslant 0}$, e.t.c.

\subsubsection{Periodic functions}\label{PER}
The map
$\widetilde{\varphi}_t\colon\widetilde{\Delta}_t
\to \widetilde{\C}_t$
defined in \S \ref{Unit} naturally
induces a morphism
   $ \widetilde{\varphi}_t^*\colon \widetilde{\jmath}_*\scr{O}_{\C^*}
    \longrightarrow
    \widetilde{\varphi}_{t*}\widetilde{\jmath}_{\Delta*}\scr{O}_{\Delta^*}$
defined as the composition with the map $\varphi$ restricted to
a suitable open subset.
We then set
\begin{align*}
    \widetilde{\phi}_t\coloneqq
    \widetilde{\imath}^{-1}(\widetilde{\varphi}_t^*)
    \colon \widetilde{\scr{O}}_t\to \widetilde{\scr{O}}_t,
\end{align*}
where we used the relation
$\widetilde{\varphi}_t\circ\widetilde{\imath}_\Delta=\widetilde{\imath}$.
Note that
the subsheaves
$\scr{A}^{\leqslant0}_t$
and $\scr{A}^{<0}_t$
are invariant under the automorphism $\widetilde{\phi}_t$.
We then set
$\nabla_{\widetilde{\phi}_t}\coloneqq \widetilde{\phi}_t-\mathrm{id}_{\widetilde{\scr{O}}}.$
The restrictions
of $\nabla_{\widetilde{\phi}_t}$
to  $\scr{A}^{\leqslant 0}_t$ and $\scr{A}^{<0}_t$
are also denoted by the same symbol.
Set
\begin{align}\label{u}
    u\coloneqq \exp(2\pi{\tt{i}} s)=\exp(2\pi{\tt{i}} t^{-1})
\end{align}
and $v=u^{-1}$.
Then, we have the equality $\nabla_{\widetilde{\phi}_t}(u)=0$.

\begin{definition}[Sheaves of periodic functions]
Let $\widetilde{\scr{O}}_{{\mathrm{per}}}$,
$\scr{A}^{< 0}_{\mathrm{per}}$,
and ${\scr{A}}_{\mathrm{per}}^{\leqslant 0}$
denote the kernel of
the operator $\nabla_{\widetilde{\phi}_t}$ on
$\widetilde{\scr{O}}$,
$\scr{A}^{< 0}$
and ${\scr{A}}^{\leqslant 0}$, respectively.
\end{definition}

\begin{lemma}[c.f. {\cite{Galois}*{p.117-118}}]
For a non-empty connected open subset
$U\subset S^1_t$,
we have the following descriptions of
$\widetilde{\scr{O}}_{\mathrm{per}}(U)$,
$\scr{A}^{\leqslant 0}_{\mathrm{per}}(U)$,
and
$\scr{A}^{<0}_{\mathrm{per}}(U)$$:$
\begin{itemize}
    \item If $U\subset (0,\pi)$, then
    \begin{align*}
        \widetilde{\scr{O}}_{\mathrm{per}}(U)=(\jmath_*\scr{O}_{\C_{v}^*})_0,
        \quad
        \scr{A}_{\mathrm{per}}^{\leqslant 0}(U)=\scr{O}_{v},
        \quad
        \scr{A}_{\mathrm{per}}^{<0}(U)=v\scr{O}_{v}.

    \end{align*}
    \item If $U\subset (-\pi,0)$, then
    \begin{align*}
        \widetilde{\scr{O}}_{\mathrm{per}}(U)=(\jmath_*\scr{O}_{\C_{u}^*})_0,
        \quad
        \scr{A}_{\mathrm{per}}^{\leqslant 0}(U)=\scr{O}_{u},
        \quad
        \scr{A}_{\mathrm{per}}^{<0}(U)=u\scr{O}_{u}.

    \end{align*}
    \item If $e^0\in U$ or $e^{\pi{\tt{i}}}\in U$, then
    \begin{align*}
        \widetilde{\scr{O}}_{\mathrm{per}}(U)=\scr{O}_{\C_{u}}(\C_u^*),
        \quad
        \scr{A}_{\mathrm{per}}^{\leqslant 0}(U)=\C,
        \quad
        \scr{A}_{\mathrm{per}}^{<0}(U)=0.

    \end{align*}
\end{itemize}
Here, the rings $(\jmath_*\scr{O}_{\C_u})_0$,
$(\jmath_*\scr{O}_{\C_v})_0$,
$\scr{O}_{u}\coloneqq\scr{O}_{\C_u,0}$,
and $\scr{O}_{v}\coloneqq\scr{O}_{\C_v,0}$
are naturally embedded into $\widetilde{\scr{O}}(U)$
in each case
via the relations \eqref{u} and $v=u^{-1}$.
\end{lemma}
\begin{proof}
For a section
$[f]\in \widetilde{\scr{O}}_{\mathrm{per}}(U)$,
there exists a representative
$f\in\widetilde{\scr{O}}(\cal{U})$
on an open subset
$\cal{U}\subset \widetilde{\C}$
with $\partial\widetilde{\C}\cap \cal{U}=U$.
We may assume that we have
$f(s)=f(s+1)$ for
$s\in \cal{V}\coloneqq
\cal{U}\cap\widetilde{\varphi}^{-1}
(\cal{U})$.
It then follows
that
$f$ can be regarded as a
holomorphic function on
$\exp(2\pi{\tt{i}}(\cal{V}\setminus U))\subset\C^*$.
Since we have
\begin{align*}
&|u|=\exp(2\pi|s|\Re({\tt{i}} e^{{\tt{i}}\arg (s)})),
\\
&\arg(u)=|s|\Im (2\pi{\tt{i}} e^{{\tt{i}}\arg (s)}) \mod 2\pi
\end{align*}
for $u=\exp(2\pi{\tt{i}} s)$,
if $U$ is an open interval in $(0,\pi)$,
then the family
$\{\infty\}\cup\exp(2\pi{\tt{i}}(\cal{U}\setminus U))$
of open subsets for
$\cal{U}\subset \widetilde{\C}$
with $\partial\widetilde{\C}\cap \cal{U}=U$
defines a fundamental system of neighborhoods around the
infinity of the $u$-plane,
which implies $\widetilde{\scr{O}}_{\mathrm{per}}(U)
=(\jmath_*\scr{O}_{\C_{v}^*})_0$.
Since $v=u^{-1}$ rapidly decays on $\cal{U}$,
we also have
$\scr{A}_{\mathrm{per}}^{\leqslant 0}(U)=\scr{O}_{v}$,
and $\scr{A}_{\mathrm{per}}^{<0}(U)=v\scr{O}_{v}$.
The other part of the lemma can be proved in
a similar way. We left them to the reader.
\end{proof}

\begin{definition}
We set
   $ \scr{A}_{\mathrm{per}}\coloneqq \sum_{n\in\mathbb{Z}}u^n\scr{A}^{\leqslant 0}_{\mathrm{per}}
   \subset\widetilde{\scr{O}}$.
\end{definition}
Let $\C[u^{\pm 1}]$ denote the ring of Laurent polynomials in $u$.
\begin{corollary}
For a connected open subset $\emptyset\neq U\subset S^1_t$,
we have the following:
\begin{align*}
    \scr{A}_{\mathrm{per}}(U)=\begin{cases}
    \scr{O}_v(*0) & (U\subset (0,\pi))\\
    \scr{O}_u(*0) & (U\subset (-\pi,0))\\
    \C[u^{\pm 1}] & (U\cap\{e^0,e^{\pi{\tt{i}}}\}\neq \emptyset)
    \end{cases}
\end{align*}
where we set $\scr{O}_{w}(*0)\coloneqq \scr{O}_{\C_w}(*\{0\})_0$
for $w=u,v$.  \qed
\end{corollary}

\subsection{Finite maps}\label{Ram}
Fix a positive integer $m$.
Let $\rho\colon \C_\tau\to \C_t$ be the $m$-th power map.
In other words, we consider another parameter $\tau$ with the relation
\begin{align}
    \tau^m=t.
\end{align}
There exists a real blow-up
$\widetilde{\rho}\colon \widetilde{\C}_\tau
    \longrightarrow \widetilde{\C}_t$
of $\rho$ defined as $\widetilde{\rho}(\tau,e^{{\tt{i}} \vartheta})\coloneqq
(\rho(\tau),e^{{\tt{i}} m\vartheta})$.
The restriction of $\widetilde{\rho}$
to the boundary is denoted by ${\rho}\colon S^1_\tau \to S^1_t$
or $\rho_m$ if we emphasize the dependence on $m$.
The multiplicative group of the $m$-th roots of unity
\begin{align*}
    \mu_m\coloneqq \{\zeta\in\C\mid \zeta^m=1\}.
\end{align*}
is regarded as the group of automorphisms on $S^1_\tau$ over $S^1_t$
in a natural way.
Let $\sigma_\zeta\colon S^1_\tau\to S^1_\tau$
be the automorphisms corresponding to $\zeta\in\mu_m$.

\section{Stokes filtered locally free sheaves}\label{SSt}
In this section,
we introduce the notion of a
Stokes filtered locally free sheaves over
the sheaf $\scr{A}_{\mathrm{per}}$ of rings,
which will be called a
 Stokes filtered
$\scr{A}_{\mathrm{per}}$-modules.
Concrete examples of Stokes filtered
$\scr{A}_{\mathrm{per}}$-modules will be given
in the next section.

\subsection{Sheaves of indexes}
In this subsection,
we prepare some sheaves of ordered abelian groups.
They will be used to
define filtrations on $\scr{A}_{\mathrm{per}}$-modules.

\subsubsection{A sheaf on $S^1_\tau$}
Fix $m\in \mathbb{Z}_{>0}$.
Recall the notations in \S \ref{Ram}
such as $\tau$
 and $S^1_\tau$

corresponding to $m$.
We recall that there is a sheaf
$\widetilde{\scr{O}}_{\tau}$ on ${S}^1_{\tau}$ defined in \S \ref{Sheaves}.

\begin{definition}\label{qtau}
Let $\scr{I}_{m,\tau}\subset \widetilde{\scr{O}}_\tau$
be a subsheaf of $\C$-vector spaces
locally generated by the sections
represented by
the functions of the form
\begin{align}\label{q(tau)}
    \frak{a}(\tau)=\sum_{\ell=1}^mc_\ell \tau^{-\ell}
\end{align}
where
$c_1,\dots,c_m\in \C$.
\end{definition}
\begin{remark}
    The motivation for introducing the sheaf
    $\scr{I}_{m,\tau}$
    comes from
    the definition
    of mild difference modules (Definition \ref{E&U}).
\end{remark}

\subsubsection{Sheaves on $S^1_t$}
Let $\rho\colon S^1_\tau\to S^1_t$ be the map defined in
\S \ref{Ram}.
Let
$\rho^*\colon \widetilde{\scr{O}}_t\to \rho_*\widetilde{\scr{O}}_\tau$
be the adjoint morphism.
We regard $\widetilde{\scr{O}}_t$ as a
subsheaf of $\rho_*\widetilde{\scr{O}}_\tau$
by this $\rho^*$.
\begin{definition}
For a positive integer $m$, we set
\begin{align*}
    \scr{I}_{m}
    \coloneqq \rho_{*}\scr{I}_{m,\tau}\cap \widetilde{\scr{O}}_t
    \subset\rho_*\widetilde{\scr{O}}_\tau.
\end{align*}
We then set
$\scr{I}\coloneqq \bigcup_{m=1}^\infty \scr{I}_m. $
\end{definition}

\subsubsection{Orders on local sections}
We shall define partial orders
on the space of sections of sheaves introduced above.
\begin{definition}
For an open subset
$U\subset S^1_t$
and sections $\frak{a},\frak{b}\in \scr{I}(U)$,
we define
\begin{align*}
    \frak{a}\leqslant_U\frak{b}&\Longleftrightarrow
    \exp(\frak{a}-\frak{b})\in \scr{A}^{\leqslant 0}(U), \text{ and }\\
    \frak{a}<_U\frak{b}&\Longleftrightarrow
    \exp(\frak{a}-\frak{b})\in \scr{A}^{< 0}(U).
\end{align*}
\end{definition}

\subsection{Stokes filtered $\scr{A}_{\mathrm{per}}$-modules}
\subsubsection{Pre-Stokes filtrations}\label{DefSt}
We shall define the notion of a pre-Stokes filtration on
a $\scr{A}_{{\mathrm{per}}}$-module as follows.
\begin{definition}\label{Pre-St}
Let $\scr{L}$ be a $\scr{A}_{\mathrm{per}}$-module.
\textit{A pre-Stokes filtration} on
$\scr{L}$
is a family
\begin{align*}
    \scr{L}_\bullet\coloneqq
    \{\scr{L}_{\leqslant \frak{a}}\subset \scr{L}_{|U}\mid
    U\subset S^1\colon\text{open subset},\ \frak{a}\in \scr{I}(U)\}
\end{align*}
of $\scr{A}_{{\mathrm{per}}|U}^{\leqslant 0}$-submodules in
$\scr{L}_{|U}$ for all open subsets $U$ with the following properties:
\begin{enumerate}
    \item If $\frak{a}_{|V}=\frak{b}$ for $V\subset U$, $\frak{a}\in \scr{I}(U)$, and
    $\frak{b}\in \scr{I}(V)$,
    then
    $\scr{L}_{\leqslant \frak{a}|V}=\scr{L}_{\leqslant \frak{b}}.$
    \item If $\frak{a}\leqslant_U\frak{b}$ for
          $\frak{a},\frak{b}\in \scr{I}(U)$,
          then
          $\scr{L}_{\leqslant \frak{a}}\subset
          \scr{L}_{\leqslant \frak{b}}.$
    \item For any $n\in \mathbb{Z}$ and $\frak{a}\in \scr{I}(U)$,
          we have

              $u^n\scr{L}_{\leqslant \frak{a}}
              =\scr{L}_{\leqslant \frak{a}+2n\pi{\tt{i}} s}.$

\end{enumerate}
\end{definition}
A pair $(\scr{L},\scr{L}_\bullet)$ of a $\scr{A}_{{\mathrm{per}}}$-module
and a pre-Stokes filtration on it is called
\textit{a pre-Stokes filtered $\scr{A}_{{\mathrm{per}}}$-module}.
A \textit{morphism} between two pre-Stokes filtered
$\scr{A}_{{\mathrm{per}}}$-modules
$(\scr{L},\scr{L}_\bullet)$,
and $(\scr{L}',\scr{L}_\bullet')$
is a morphism $\lambda\colon \scr{L}\to\scr{L}'$
of $\scr{A}_{{\mathrm{per}}}$-modules
such that
$\lambda_{|U}(\scr{L}_{\leqslant\frak{a}})\subset
\scr{L}'_{\leqslant \frak{a}}$
for any $\frak{a}\in \scr{I}{(U)}$.

\subsubsection{Grading}
For each $\frak{a}\in\scr{I}(U)$
and a
pre-Stokes filtered $\scr{A}_{{\mathrm{per}}}$-module
$\scr{L}=(\scr{L},\scr{L}_\bullet)$,
we set
\begin{align*}
    \scr{L}_{<\frak{a}}
    \coloneqq
    \sum_{\frak{b}<_U\frak{a}}
    \scr{L}_{\leqslant \frak{b}}
\end{align*}
and
${\mathrm{gr}}_\frak{a}\scr{L}\coloneqq
\scr{L}_{\leqslant \frak{a}}/\scr{L}_{<\frak{a}}$.
By condition (3) in the Definition,
we have the inclusion
$\scr{A}^{<0}_{|U}\cdot
\scr{L}_{\leqslant \frak{a}}\subset
\scr{L}_{<\frak{a}}.$
Hence it is natural to regard
${\mathrm{gr}}_\frak{a}\scr{L}$ as
a module over
$\scr{A}^{\leqslant 0}_{{\mathrm{per}}|U}/\scr{A}_{{\mathrm{per}}|U}^{<0}
=\C_U$.
Again by condition (3),
we have an isomorphism
\begin{align}\label{UN}
    u^n\colon {\mathrm{gr}}_{\frak{a}}\scr{L}\xrightarrow{\sim} {\mathrm{gr}}_{\frak{a}+2\pi{\tt{i}} s n}\scr{L}
\end{align}
for any $n\in \mathbb{Z}$.
Hence $\bigoplus_{\mathfrak{a}\in \scr{I}(U)}
    {\mathrm{gr}}_\frak{a}\scr{L}$
is naturally equipped with the structure of a sheaf
of $\C[u^{\pm 1}]$-modules over $U$.
Then the sheaf $\Sigma(\scr{L})$ of sets
defined as
\begin{align*}
    \Sigma(\scr{L})(U)\coloneqq
    \{\frak{a}\in \scr{I}(U)\mid {\mathrm{gr}}_{\frak{a}}\scr{L}\neq 0\},
\end{align*}
is equipped with the
the action of the additive group
$2\pi{\tt{i}} s\mathbb{Z}_{S^1}$
in a natural way.

\begin{lemma}\label{grLemma}
Let $(\scr{L},\scr{L}_\bullet)$ be
a pre-Stokes filtered
$\scr{A}_{{\mathrm{per}}}$-module.
There exists a
unique sheaf ${\mathrm{gr}}\scr{L}$ of
$\C[u^{\pm 1}]$-modules
such that for any open subset
$U\subset S^1$,
 we have
\begin{align*}
    {\mathrm{gr}}\scr{L}_{|U}=
    \bigoplus_{\mathfrak{a}\in \scr{I}(U)}
    {\mathrm{gr}}_\frak{a}\scr{L}.
\end{align*}
Moreover,
${\mathrm{gr}}\scr{L}$ is a local system of finitely generated
free $\C[u^{\pm 1}]$-modules
if and only if the following conditions
are satisfied
\begin{itemize}
    \item For each $\frak{a}\in \scr{I}(U)$,
          the sheaf ${\mathrm{gr}}_\frak{a}\scr{L}$
          is a local system of
          $\C$-vector spaces.
    \item The sheaf $\Sigma(\scr{L})$
          has locally finitely many
          $2\pi{\tt{i}} s \mathbb{Z}_{S^1}$-orbits.
\end{itemize}
\end{lemma}
\begin{proof}
For two connected
open subsets $V\subset U \subsetneq S^1_t$,
the restriction map
$\scr{I}(U)\to \scr{I}(V)$ is an isomorphism.
Then, by condition (1) in Definition \ref{Pre-St},
we have
\begin{align*}
    \bigoplus_{\frak{a}\in \scr{I}(U)}{\mathrm{gr}}_\frak{a}\scr{L}_{|V}
    =\bigoplus_{\frak{b}\in \scr{I}(V)}{\mathrm{gr}}_{\frak{b}}\scr{L}.
\end{align*}
The first assertion follows from this equality.
By \eqref{UN},
we have a natural isomorphism
\begin{align*}
    {\mathrm{gr}}_\frak{a}\scr{L}\otimes\C[u^{\pm 1}]
    \simeq \bigoplus_{n\in \mathbb{Z}} {\mathrm{gr}}_{\frak{a}+2\pi{\tt{i}} n }\scr{L}
\end{align*}
for each $\frak{a}\in \scr{I}{(U)}$.
The second assertion follows from this isomorphism.
\end{proof}
\subsubsection{Stokes filtration}
\begin{definition}
Let $\scr{L}$ be a $\scr{A}_{\mathrm{per}}$-module.
A pre-Stokes filtration $\scr{L}_\bullet$
on $\scr{L}$ is called
\textit{a Stokes filtration}
if the following conditions are satisfied:
\begin{enumerate}
    \item The graded sheaf ${\mathrm{gr}}\scr{L}$ is a local system of
    finitely generated
    $\C[u^{\pm 1}]$-modules.
    \item
    For each point $x\in S^1$, there exist a
    neighborhood $U$ of $x$ and
    an isomorphism
    \begin{align*}
        \eta\colon \scr{A}_{{\mathrm{per}}|U}\otimes_{\C[u^{\pm 1}]}
        {\mathrm{gr}}\scr{L}_{|U}\overset{\sim}{\longrightarrow}
        \scr{L}_{|U}
    \end{align*}
    of filtered $\scr{A}_{{\mathrm{per}}|U}$-modules
    such that ${\mathrm{gr}}(\eta)=\mathrm{id}_{{\mathrm{gr}}\scr{L}_{|U}}$.
\end{enumerate}
A pair $(\scr{L},\scr{L}_\bullet)$ of a $\scr{A}_{{\mathrm{per}}}$-module
and a Stokes filtration on it is called
\textit{a Stokes filtered $\scr{A}_{{\mathrm{per}}}$-module}.
The category of Stokes filtered $\scr{A}_{{\mathrm{per}}}$-module
is denoted by ${\mathsf{St(\scr{A}_\per)}}$,
which is defined to be a full subcategory of
the category of pre-Stokes filtered $\scr{A}_{{\mathrm{per}}}$-modules.
\end{definition}
\begin{remark}
    By Lemma \ref{grLemma},
    the sheaf $\scr{L}$ should be finitely generated and
    locally free over $\scr{A}_{\mathrm{per}}$ if it admits a Stokes filtration.
\end{remark}
\subsubsection{Graded Stokes filtered
$\scr{A}_{{\mathrm{per}}}$-modules
and some operations}
\begin{definition}
A Stokes filtered $\scr{A}_{\mathrm{per}}$-module $(\scr{G},\scr{G}_\bullet)$
is called \textit{graded}
if we have
an isomorphism
\begin{align*}
    \eta\colon \scr{G}\overset{\sim}{\longrightarrow}
    \scr{A}_{\mathrm{per}}\otimes{\mathrm{gr}}(\scr{G})
\end{align*}
of Stokes filtered $\scr{A}_{\mathrm{per}}$-modules such that
${\mathrm{gr}}(\eta)=\mathrm{id}$.
\end{definition}

For two (pre-)Stokes filtered $\scr{A}_{\mathrm{per}}$-modules
$(\scr{L}_1,\scr{L}_{1,\bullet})$
and
$(\scr{L}_2, \scr{L}_{2,\bullet})$
we can define the (pre-)Stokes filtrations
on the tensor product
$\scr{L}_1\otimes \scr{L}_2$
and the sheaf of internal-homs
${\mathscr{H}\! \! om}(\scr{L}_1,\scr{L}_2)$
as follows:
\begin{align*}
    (\scr{L}_1\otimes \scr{L}_2)_{\leqslant \frak{a}}
    &\coloneqq\sum_{\frak{b}\in\scr{I}(U)}
    \scr{L}_{1,\leqslant\frak{b}}
    \otimes_{\scr{A}_{\mathrm{per}}^{\leqslant 0}}
    \scr{L}_{2,\leqslant\frak{a}-\frak{b}},\\
    {\mathscr{H}\! \! om}(\scr{L}_1,\scr{L}_2)_{\leqslant\frak{a}}
    &\coloneqq
    \sum_{\frak{b}\in\scr{I}(U)}
    {\mathscr{H}\! \! om}_{\scr{A}_{\mathrm{per}}^{\leqslant 0}}(\scr{L}_{1,\leqslant \frak{b}},
    \scr{L}_{2,\leqslant \frak{a}+\frak{b}}).
\end{align*}
We also use the notation
$\scr{E}nd(\scr{L})\coloneqq {\mathscr{H}\! \! om}(\scr{L},\scr{L})$.

\subsection{Classification}
 For a Stokes $\scr{A}_{{\mathrm{per}}}$-module
 $\scr{L}=(\scr{L},\scr{L}_\bullet)$,
we set
\[{\scr{A}ut}^{<0}(\scr{L})\coloneqq
\mathrm{id}_{\scr{L}}+\scr{E}nd({\scr{L}})
_{<0}.\]
\begin{theorem}\label{ClSt}
Let $\scr{G}$ be a graded Stokes filtered
$\scr{A}_{\mathrm{per}}$-module.
Then, there is a natural
one-to-one correspondence
between the cohomology set
\begin{align*}
    H^1(S^1,
    {\scr{A}ut}^{<0}
    (\scr{G}))
\end{align*}
and the set of isomorphism classes of
pairs
$((\scr{L},\scr{L}_{\bullet}), \Xi_{{\mathrm{gr}}} )$ of
\begin{itemize}
\item
 a Stokes filtered
    $\scr{A}_{{\mathrm{per}}}$-module
    $\scr{L}=(\scr{L},\scr{L}_{\bullet})$,
    and
\item an isomorphism
    $\Xi_{\mathrm{gr}}\colon \scr{A}_{\mathrm{per}}\otimes_{\C[u^{\pm 1}]} {\mathrm{gr}}\scr{L}
    \xrightarrow{\sim}\scr{G}$ of Stokes filtered $\scr{A}_{\mathrm{per}}$-modules.
\end{itemize}
\end{theorem}
\begin{proof}
Since this is standard,
we only give the construction of the cohomology class.
Let $((\scr{L},\scr{L}_{\bullet}), \Xi_{{\mathrm{gr}}} )$
be the pair in the statement.
We can take a finite open covering
$S^1=\bigcup_{\alpha\in \Lambda} U_\alpha$
such that
\begin{itemize}
\item
$\Lambda=\mathbb{Z}/N\mathbb{Z}$ for a positive integer $N$,
\item
$U_\alpha$ is an open interval for each $\alpha\in \Lambda$, and
\item
$U_\alpha\cap U_\beta=\emptyset$ for
$\beta\notin \{\alpha-1,\alpha,\alpha+1\}$
\end{itemize}
and isomorphisms
\begin{align*}
    \eta_\alpha \colon \rho^{-1}
        \scr{A}_{{\mathrm{per}}|U_\alpha}
        \otimes_{\C[u^{\pm 1}]}
        {\mathrm{gr}}\scr{L}_{|U_\alpha}
        \overset{\sim}{\longrightarrow}
        \scr{L}_{|U_\alpha}
\end{align*}
of filtered $\rho^{-1}\scr{A}_{{\mathrm{per}}|U_\alpha}$-modules
    such that ${\mathrm{gr}}(\eta_\alpha)=
    \mathrm{id}_{{\mathrm{gr}}^{}\scr{L}_{|U_\alpha}}$.
Then, the tuple
\[\left(\Xi_{{\mathrm{gr}}|U_{\alpha,\alpha+1}}
\circ
\eta_{\alpha+1|U_{\alpha,\alpha+1}}^{-1}
\circ
\eta_{\alpha|U_{\alpha,\alpha+1}}
\circ
\Xi^{-1}_{{\mathrm{gr}}|U_{\alpha,\alpha+1}}
\right)_{\alpha\in \Lambda}\]
defines a class in
$H^1(S^1,
    {\scr{A}ut}^{<0}
    (\scr{G}))$,
    where we put $U_{\alpha,\alpha+1}=U_\alpha\cap U_{\alpha+1}$.
\end{proof}

\section{Riemann-Hilbert correspondence for
mild
difference modules}
\label{RH&RS}
 In this section,
 we formulate and prove a part of
Riemann-Hilbert correspondence for
mild difference modules
assuming a vanishing theorem (Theorem \ref{Vanishing}).

\subsection
{Definition of mild difference modules}
\label{SReg}
We fix some notations and terminologies of
difference modules and
recall the definition of
the mild difference
modules in the sense of
van der Put and Singer \cite{Galois}.
\subsubsection{Difference modules}
Recall that \textit{a difference ring}
is a pair $(R,\Phi)$
of a commutative ring $R$ with a unit $1$
and
a ring automorphism $\Phi$ on $R$.
If $R$ is a field, then $(R,\Phi)$
is called a difference field.
Let $\scr{O}_{t}(*0)
=\scr{O}_{\C_t}(*0)_0=\C\conv{t}$
be the field of convergent Laurent series in $t$.
Let
$\phi_t\colon \scr{O}_t(*0)\to \scr{O}_t(*0)$
be the automorphism defined by
$\phi_t(f)(t)\coloneqq f(t(1+t)^{-1})$.
Then the pair $(\scr{O}_t(*0),\phi_{t})$ is an
example  of a difference field.
The pair $(\widehat{\scr{O}}_t(*0),\widehat{\phi}_t)$
of the formal completion
$\widehat{\scr{O}}_t(*0)\coloneqq \C\pole{t}$ of $\scr{O}_{t}(*0)$
and the automorphism
$\widehat{\phi}_t$ defined in a similar way as $\phi_t$
is also an instance of a difference field.

A(n invertible) \textit{difference module}
over a difference ring $(R,\Phi)$
is a pair $(M,\Psi)$ of a
finitely generated $R$-module $M$
and an automorphism
$\Psi\colon M\to M$ of
abelian groups such that
$\Psi(rm)=\Phi(r)\Psi(m)$
for any $r\in R$ and $m\in M$.
We abbreviate $(M,\Psi)$
by omitting $\Psi$ if there is no
fear of confusion.
The category of difference modules
over $(\scr{O}_t(*0),\phi_t)$
is denoted by ${\mathsf{Diffc}}$, which is an abelian category.

For a difference module
$(\scr{M},\psi)$
over $(\scr{O}_t(*0),\phi_{t})$,
the automorphism $\psi$
naturally extends to an
automorphism $\widehat{\psi}$ on
$\widehat{\scr{M}}\coloneqq
\widehat{\scr{O}}_t(*0)\otimes_{\scr{O}_t{(*0)}}\scr{M}$.
The pair $(\widehat{\scr{M}},\widehat{\psi}{})$
is called the formal completion of
$(\scr{M},\psi{})$, which is a difference module over
$\widehat{\scr{O}}_t(*0)\coloneqq \C\pole{t}$.

\subsubsection{Regular singular
difference modules}
For a constant matrix
$G\in\mathrm{End}(\C^r)$, we let
$\scr{R}_{G}=(\scr{O}_{t}(*0)^{\oplus r},\psi_G)$
be a difference module defined by
$\psi_G=(1+t)^G\phi_t^{\oplus r}$, where
we set
$(1+t)^G=\exp(G\log (1+t))$
with $\log(1+ t)=-\sum_{n=1}^\infty n^{-1}(-t)^n$.
The formal completion of $\scr{R}_{G}$
is denoted by $\widehat{\scr{R}}_{G}$.

\begin{definition}\label{DReg}
A difference module $(\scr{M},\psi)$ over
$(\scr{O}_t(*0),\phi_{t})$
is called \textit{regular singular}
if its formal completion
$(\widehat{\scr{M}},\widehat{\psi})$
is isomorphic to $\widehat{\scr{R}}_G$
for a matrix $G\in\mathrm{End}(\C^r)$.
The trivial difference module $0$
is regarded as
a regular singular difference module.
\end{definition}
\subsubsection{Difference modules
over an extended field}
\label{Extended}
Fix a positive integer $m$.
We use the notations in \S \ref{Ram}.
There is a natural inclusion
$\rho^*\colon \scr{O}_t(*0)\to \scr{O}_\tau(*0)$.
Regard $t=\tau^m$ as an element of $\scr{O}_\tau(*0)$.
The pair of the field $\scr{O}_\tau(*0)$
and an automorphism
$\phi_\tau\colon \scr{O}_\tau(*0)\to \scr{O}_\tau(*0)$
defined as
$\phi_\tau(\tau)\coloneqq\tau(1+t)^{-1/m}$
forms a difference field.
The notion of formal completion is
defined analogously as in the case of
$(\scr{O}_t(*0),\phi_t)$.
For a difference module $\scr{M}$
over $(\scr{O}_t{(*0)},\phi_t)$,
the pull back
$\rho^*\scr{M}\coloneqq \scr{O}_{\tau}(*0)\otimes \scr{M}$
is a difference module over $(\scr{O}_{\tau}(*0),\phi_\tau)$
in a natural way.

\subsubsection{Mild
exponential factors}\label{ExF}
Fix a positive integer $m$ as in \S \ref{Extended}.
Let $\frak{a}=\frak{a}(\tau)$ be as \eqref{q(tau)}
in Definition \ref{qtau}.
We note that
$\exp(\phi_\tau(\frak{a})-\frak{a})$
is a well defined element in $\scr{O}_{\tau}$.
We define
a difference module
$\scr{E}^{\frak{a}}\coloneqq
(\scr{O}_\tau{(*0)},\psi_\frak{a})$
by
$\psi_\frak{a}(1)\coloneqq
\exp(\phi_\tau(\frak{a})-\frak{a})$.
The formal completion of
$\scr{E}^{\frak{a}}$ will be denoted by
$\widehat{\scr{E}}^{\frak{a}}{}$.
For two such difference modules
$\scr{E}^{\frak{a}}$ and $\scr{E}^{\frak{b}}$,
we have $\scr{E}^{\frak{a}}\simeq \scr{E}^{\frak{b}}$
if and only if $\frak{a}-\frak{b}\in 2\pi{\tt{i}} s\mathbb{Z}$.

\subsubsection{Formal decomposition and
mild difference modules}
We recall the definition
of mild difference modules:

\begin{definition}[{\cite[\S 7.1, p.71]{Galois}}]\label{E&U}
A difference module over
$(\scr{O}_\tau(*0),\phi_\tau)$
 is called
\begin{enumerate}
    \item \textit{mild elementary} if it is a direct sum
    of the modules of the form
    $\scr{E}^{\frak{a}}\otimes \rho^*\scr{R}_{G}$, and
    \item \textit{mild unramified}
    if it is formally isomorphic to
    a mild elementary module.
\end{enumerate}

A difference module $\scr{M}$
over $(\scr{O}_t(*0), \phi_t)$
is called \textit{mild} (resp. \textit{mild graded})
if there exists a positive integer
$m$ such that $\rho^*\scr{M}$
is mild unramified (resp. mild elementary).
The category of
mild difference modules
is denoted by ${\mathsf{Diffc}}^{\mathrm{mild}}$.
\end{definition}
\begin{remark}
The terms ``mild elementary", ``mild unramified",
and ``mild graded"
cannot be found in \cite{Galois}.
We introduced them to clarify the analogy to
the theory of differential modules.
 Mild difference modules form
    a special class of difference modules.
    The most general case
    is called \textit{wild} in \cite{Galois}.
\end{remark}
\begin{remark}\label{Mellin}
    Let
    $\mathbb{M}$ be a holonomic $\cal{D}$-module
    over $\mathbb{G}_m
    =\mathrm{Spec}(\C[x,x^{-1}])$.
    The algebraic Mellin transformation
    $\mathfrak{M}(\mathbb{M})$ of
    $\mathbb{M}$ is a difference module
    over the difference ring $(\C[s],\phi_s)$
    where $\phi_s(P(s))=P(s+1)$ for
    $P(s)\in \C[s]$.
    Then
    $\mathfrak{M}
    (\mathbb{M})
    \otimes_{\C[s]}\mathscr{O}_t(*0)$
    is mild if
    $\mathbb{M}$ is regular singular
    at $0$ and $\infty$
    by \cite[Theorem 1, Lemma 3]{garcia2018mellin}.
\end{remark}

\begin{lemma}\label{Graded}
Let $\scr{M}$ be a mild difference module over
    $(\scr{O}_t(*0),\phi_t)$.
Let $\scr{N}'$ be a mild elementary module over
$(\scr{O}_\tau(*0),\phi_\tau)$
such that
the formal completion $\widehat{\scr{N}}'$ is
isomorphic to
$\rho^*\widehat{\scr{M}}$.
Then, there exists a
mild graded difference module
$\scr{N}$ over $(\scr{O}_t(*0),\phi_t)$
such that
$\rho^*\scr{N}$ is isomorphic to $\scr{N}'$.
\end{lemma}
\begin{proof}
Since the formal completion of
$\scr{N}'$ is isomorphic to $\rho^*\scr{M}$,
it is naturally equipped with the action of $\mu_m$.
Since $\scr{N}'$ is elementary,
the action naturally lifts to the action on $\scr{N}'$.
The desired module $\scr{N}$
is the decent of
$\scr{N}'$ by this action.
\end{proof}
The module $\scr{N}$
in this lemma
will be called \textit{the graded module of $\scr{M}$}.

\subsection{Riemann-Hilbert functor}\label{RH-fun}
We shall define a functor from
the category
${\mathsf{Diffc}}^{\mathrm{mild}}$ of mild difference modules
over $(\scr{O}_{t}(*0),\phi_t)$
to the category ${\mathsf{St(\scr{A}_\per)}}$ of
Stokes filtered $\scr{A}_{{\mathrm{per}}}$-modules.

\subsubsection{Sheaves of difference rings}
The pair $(\widetilde{\scr{O}}_t,\widetilde{\phi}_{t})$
of the sheaf $\widetilde{\scr{O}}_t$ of rings on
$S^1_t$ and the automorphism
$\widetilde{\phi}_{t}$ on $\widetilde{\scr{O}}_t$
defined in \S \ref{PER}
is a sheaf of difference rings on $S^1_t$.
The pairs $(\scr{A}^{\leqslant 0},\widetilde{\phi}_t{})$
and $(\scr{A}^{< 0},\widetilde{\phi}_t{})$
are difference subrings in $(\widetilde{\scr{O}}_t,\widetilde{\phi}_{t})$.

Let
$\scr{O}_{t}(*0)_{S^1}$
be the constant sheaf on $S^1_t$ with fiber $\scr{O}_{t}(*0)$.
The automorphism induced from $\phi_t$ on
$\scr{O}_{t}(*0)_{S^1}$
is denoted by the same letter $\phi_t$.
We note that the sheaves $\widetilde{\scr{O}}_t$,
$\scr{A}^{\leqslant 0}_t$, and $\scr{A}^{<0}_t$ are
sheaves of difference algebras over
$(\scr{O}_{t}(*0)_{S^1},\phi_t)$
in  a natural way.
For a difference module $(\scr{M},\psi)$ over
$(\scr{O}_t(*0),\phi_t)$,
let $(\scr{M}_{S^1},\psi)$
denote the associated constant sheaf of
difference modules over
$(\scr{O}_{t}(*0)_{S^1},\phi_t)$.
For an open subset
$U\subset S^1$,
we also use the notation
\[\scr{M}_{|U}\coloneqq
(\scr{A}^{\leqslant 0}\otimes_{\scr{O}_t(*0)_{S^1}}
\scr{M}_{S^1})_{|U}
\]
for simplicity.

\subsubsection{Sheaves with
asymptotic behavior}
For
$\frak{a}\in \scr{I}(U)$ on an open subset $U\subset S^1_t$,
let $\scr{A}^{\leqslant \frak{a}}$ (resp.
$\scr{A}^{< \frak{a}}$) denote the subsheaf
$\exp(\frak{a})\scr{A}^{\leqslant 0}_{|U}$
(resp. $\exp(\frak{a})\scr{A}^{< 0}_{|U}$) in
$\widetilde{\scr{O}}_{|U}$.
For a local section
$\frak{a}\in \scr{I}(U)$,
the sheaves $\scr{A}^{\leqslant \frak{a}}$
and $\scr{A}^{<\frak{a}}$
are modules over $\scr{A}_{|U}^{\leqslant 0}$.

\subsubsection{The de Rham complexes
for difference modules}

For a sheaf of difference module $(\cal{M},\psi)$
over a sheaf of difference rings,
we set
\[\nabla_\psi\coloneqq \psi-\mathrm{id}_\cal{M}.\]

\begin{definition}

    For a difference module $\scr{M}$
    over $(\scr{O}_t,\phi_t)$, we set
\begin{align*}
    \widetilde{{\mathrm{DR}}}(\scr{M})
    &\coloneqq
    [\widetilde{\scr{O}}\otimes\scr{M}_{S^1}
    \xrightarrow{{\nabla}_{\widetilde{\psi}}}
    \widetilde{\scr{O}}\otimes \scr{M}_{S^1}],\\
    {\mathrm{DR}}_{\leqslant 0}(\scr{M})
    &\coloneqq
    [{\scr{A}}^{\leqslant 0}\otimes \scr{M}_{S^1}
    \xrightarrow{{\nabla}_{\widetilde{\psi}}}
    {\scr{A}}^{\leqslant 0}\otimes \scr{M}_{S^1}],
    \text{ and }\\
    {\mathrm{DR}}_{< 0}(\scr{M})
    &\coloneqq
    [{\scr{A}}^{< 0}\otimes \scr{M}_{S^1}
    \xrightarrow{{\nabla}_{\widetilde{\psi}}}
    {\scr{A}}^{< 0}\otimes \scr{M}_{S^1}],
\end{align*}
where the automorphisms on
$\widetilde{\scr{O}}\otimes
\scr{M}_{S^1}$,
${\scr{A}}^{\leqslant 0}\otimes \scr{M}_{S^1}$,
and ${\scr{A}}^{< 0}\otimes \scr{M}_{S^1}$
are denoted by $\widetilde{\psi}$,
and the  complexes are concentrated on degrees zero and one.
\end{definition}

The proof of the following vanishing theorem
will be given in the next section:
\begin{theorem}\label{Vanishing}
    Assume that $\scr{M}$ is a mild difference module
    over $(\scr{O}_t,\phi_t)$, then
    the complexes ${\mathrm{DR}}_{\leqslant 0}(\scr{M})$
    and ${\mathrm{DR}}_{< 0}(\scr{M})$
    have non-zero cohomology in degree zero
    at most.
    Moreover,
    the natural morphisms
    ${\mathrm{DR}}_{< 0}(\scr{M})\to
    {\mathrm{DR}}_{\leqslant 0}(\scr{M})\to
    \widetilde{{\mathrm{DR}}}(\scr{M})$
    induce injections

    $\scr{H}^0{\mathrm{DR}}_{< 0}(\scr{M})\to
    \scr{H}^0{\mathrm{DR}}_{\leqslant 0}(\scr{M})\to
    \scr{H}^0\widetilde{{\mathrm{DR}}}(\scr{M})$.

\end{theorem}
The following
corollary will also be proved
in the next section:
\begin{corollary}\label{Cor}
     Let $\scr{M}$ be a
        mild difference module
        and $\frak{a}\in \scr{I}(U)$.
        If we set
        \begin{align*}
            {\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})
            &=
            [\scr{A}^{\leqslant \frak{a}}\otimes
            \scr{M}_{|U}
            \xrightarrow{\nabla_{\widetilde{\psi}}}
            \scr{A}^{\leqslant \frak{a}}\otimes
            \scr{M}_{|U}
            ], \text{ and} \\
            {\mathrm{DR}}_{< \frak{a}}(\scr{M})
            &=
            [\scr{A}^{< \frak{a}}\otimes
            \scr{M}_{|U}
            \xrightarrow{\nabla_{\widetilde{\psi}}}
            \scr{A}^{< \frak{a}}\otimes
            \scr{M}_{|U}
            ],
            \end{align*}
            then these complexes have non-zero cohomology in degree zero at most.
       Moreover,
    the natural morphisms
    ${\mathrm{DR}}_{< \frak{a}}(\scr{M})\to
    {\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})\to
    \widetilde{{\mathrm{DR}}}(\scr{M})$
    induce injections

    $\scr{H}^0{\mathrm{DR}}_{< \frak{a}}(\scr{M})\to
    \scr{H}^0{\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})\to
    \scr{H}^0\widetilde{{\mathrm{DR}}}(\scr{M})$.
    The quotient
    \begin{align*}
    {\mathrm{gr}}_{\frak{a}}\scr{H}^0{\mathrm{DR}}(\scr{M})\coloneqq
     \scr{H}^0{\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})
     /\scr{H}^0{\mathrm{DR}}_{< \frak{a}}(\scr{M})
    \end{align*}
    is a local system of $\C$-vector space
    over $U$.
    \end{corollary}
\begin{remark}
    Let $U\subset S^1$ be an open subset
    and $\frak{a}\in \scr{I}(U)$.
    For a sheaf of difference module $\scr{N}$
    over
    $(\scr{A}^{\leqslant 0 }_{|U},
    \widetilde{\phi}_t)$,
    we can define the
    complexes
    ${\mathrm{DR}}_{\leqslant \frak{a}}(\scr{N})$
    and ${\mathrm{DR}}_{< \frak{a}}(\scr{N})$
    in a similar way.
    These complexes are natural in the sense
    that a morphism
    $\xi\colon \scr{N}\to \scr{N}'$
    of
    $(\scr{A}^{\leqslant 0 }_{|U},
    \widetilde{\phi}_t)$-modules induces
    morphisms of the complexes
    ${\mathrm{DR}}_{< \frak{a}}(\scr{N})\to
    {\mathrm{DR}}_{< \frak{a}}(\scr{N}')$ and
    ${\mathrm{DR}}_{\leqslant  \frak{a}}(\scr{N})\to
    {\mathrm{DR}}_{\leqslant \frak{a}}(\scr{N}')$
    in a natural way.
\end{remark}

Note that we have
$u^n\colon {\mathrm{DR}}_{\leqslant\frak{a}}(\scr{M})
\xrightarrow{\sim} {\mathrm{DR}}_{\leqslant\frak{a}+2\pi{\tt{i}} n s}(\scr{M})$
for $n\in \mathbb{Z}$.

\begin{lemma}
There exists a unique
sub sheaf ${\mathrm{Per}}(\scr{M})\subset \widetilde{{\mathrm{DR}}}{(\scr{M})}$
of $\scr{A}_{\mathrm{per}}$-modules
such that
    ${\mathrm{Per}}(\scr{M})_{|U}=
    \sum_{\frak{a}\in \scr{I}(U)}
    \scr{H}^0{\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})$
for any open subset $U\subsetneq S^1$.
\end{lemma}
\begin{proof}
If two open subsets $U$ and $V$ with
$\emptyset \neq V\subset  U\subsetneq S^1_t $
are connected,
the restriction map
$\scr{I}(U)\to \scr{I}{(V)}$
is an isomorphism.
Then the restriction map
$\sum_{\frak{a}\in \scr{I}(U)}
    {\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})
    \to \sum_{\frak{b}\in \scr{I}(V)}
    {\mathrm{DR}}_{\leqslant \frak{b}}(\scr{M})
$
is isomorphism.
The lemma follows from this fact.
\end{proof}
\begin{remark}\label{Periodic}
    We do not have
    ${\mathrm{Per}}(\scr{M})= \scr{H}^0\widetilde{{\mathrm{DR}}}(\scr{M})$
    in general.
    In other words, the filtration
    $\scr{H}^0{\mathrm{DR}}_{\leqslant \bullet}(\scr{M})$
    on $\scr{H}^0\widetilde{{\mathrm{DR}}}(\scr{M})$
    is not exhaustive in general.
    This is one of the differences between
    our setting and the case
    of meromorphic connections.
    We also use the notation
    ${\mathrm{Per}}$ for difference modules
    over
    $(\scr{A}_{|U}^{\leqslant 0},
    \widetilde{\phi}_t )$ in a similar way.
\end{remark}

We set
\begin{align*}
  \scr{H}^0{\mathrm{DR}}_{\bullet }(\scr{M})
  \coloneqq
  \{\scr{H}^0{\mathrm{DR}}_{\leqslant\frak{a}}(\scr{M})
  \mid U \subset S^1_t\colon \text{open subset, }
  \frak{a}\in \scr{I}(U)\}.
\end{align*}
It is easy to see that
$\scr{H}^0{\mathrm{DR}}_{\bullet}(\scr{M})$
is a pre-Stokes filtration on ${\mathrm{DR}}(\scr{M})$.
For a morphism $\xi\colon\scr{M}\to \scr{M}'$
of analytic difference modules,
we naturally obtain a morphism
${\mathrm{RH}}(\xi)\colon ({\mathrm{Per}}(\scr{M}),\scr{H}^0{\mathrm{DR}}_\bullet(\scr{M}))
\to ({\mathrm{Per}}(\scr{M}'),\scr{H}^0{\mathrm{DR}}_\bullet (\scr{M}'))$
of pre-Stokes filtered $\scr{A}_{{\mathrm{per}}}$-modules.
\begin{definition}
We define a functor
${\mathrm{RH}}\colon {\mathsf{Diffc}}^{\mathrm{mild}} \to{\mathsf{St}^{\rm pre}(\scr{A}_\per)}$
by
\begin{align*}
    {\mathrm{RH}}(\scr{M})\coloneqq
({\mathrm{Per}}(\scr{M}),\scr{H}^0{\mathrm{DR}}_\bullet(\scr{M}))
\end{align*}
for an object $\scr{M}\in {{\mathsf{Diffc}}}^{\mathrm{mild}}$
and ${\mathrm{RH}}(\xi)$ for a morphism $\xi$ in
${\mathsf{Diffc}}^{\mathrm{mild}}$.
\end{definition}
\subsubsection{An example}
Let $\scr{O}_{t}(*0)$ denote
the difference module
$(\scr{O}_{t}(*0),\phi_t)$
over itself.
\begin{theorem}\label{O}
${\mathrm{Per}}(\scr{O}_{t}(*0))=\scr{A}_{{\mathrm{per}}}$.
\end{theorem}
\begin{proof}
By definition,
we have
${\mathrm{DR}}_{\leqslant 2\pi{\tt{i}} n s}(\scr{O}_{t}(*0))
=u^n\scr{A}_{{\mathrm{per}}}^{\leqslant 0}$
for every $n\in \mathbb{Z}$.
It follows that
$\scr{A}_{{\mathrm{per}}}\subset {\mathrm{Per}}(\scr{O}_{t}(*0))$.
It then remains to prove the relation
\begin{align}\label{Key}
    \widetilde{\scr{O}}_{{\mathrm{per}}|U}\cap
    \scr{A}^{\leqslant \frak{a}}
    \subset \scr{A}_{{\mathrm{per}}|U}
\end{align}
for any $\frak{a}\in \scr{I}(U)$,
where $U\subset  S^1$
is an open interval.
To see the relation \eqref{Key},
we use the presentation
    $\frak{a}(t)=
    \sum_{\ell=1}^m c_\ell\tau^{-\ell}$
as in \eqref{q(tau)}.
Here, we fix a branch of
$\log t$ on  $U$
and put $\tau\coloneqq \exp(m^{-1}\log t)$
for a positive integer $m$.

The relation \eqref{Key} is shown as follows.
Take an integer $\ell_{\max}\coloneqq \max
\{\ell\mid c_\ell\neq 0\}$.
If $\ell_{\max}\neq m$,
then using the open subset
\begin{align*}
    U(\frak{a})\coloneqq
    \{e^{{\tt{i}}\theta}\in U\mid
    \Re( \exp(-{\tt{i}}\theta \ell_{\max}/m)c_{\ell_{\max}})>0\},
\end{align*}
where $\theta=\Im (\log t)$,
we have
\begin{align*}
    (\widetilde{\scr{O}}_{{\mathrm{per}}}\cap
    \scr{A}^{\leqslant \frak{a}})
    (V)=\begin{cases}
    \scr{A}_{{\mathrm{per}}}^{\leqslant 0}(V)
    &(V\subset U(\frak{a}))\\
    \scr{A}_{{\mathrm{per}}}^{< 0}(V)
    &(V\not\subset U(\frak{a})).
    \end{cases}
\end{align*}
This implies \eqref{Key} in this case.
The case
$\ell_{\max}=m$ and $c_m\in 2\pi{\tt{i}}\mathbb{Z}$
is reduced to the case above.
Assume that
$\ell_{\max}=m$ and
$c_m\notin 2\pi{\tt{i}}\mathbb{Z} $.
In this case,
for each point
$e^{{\tt{i}}\theta}\in U\setminus
\{e^0,e^{\pi{\tt{i}}}\}$,
there exists
an integer $n$
such that
\begin{align*}
    (\widetilde{\scr{O}}_{{\mathrm{per}}|U}\cap
    \scr{A}^{\leqslant \frak{a}})_{e^{{\tt{i}}\theta}}
    \subset
    u^n\scr{A}_{{\mathrm{per}},e^{{\tt{i}}\theta}}
    ^{\leqslant 0}
    \subset \scr{A}_{{\mathrm{per}}, e^{{\tt{i}}\theta}},
\end{align*}
which implies \eqref{Key} at $e^{{\tt{i}}\theta}$.
Since $c_m\notin 2\pi{\tt{i}} \mathbb{Z}$,
we have
$ (\widetilde{\scr{O}}_{{\mathrm{per}}|U}\cap
    \scr{A}^{\leqslant
    \frak{a}})_{\pm 1}=0$.
 \end{proof}

\subsubsection{Elementary examples}\label{EE}
Let $U\subset S^1$ be an open subset.
For each $\frak{a}\in \scr{I}(U)$ and
$G\in \mathrm{End}(\C^r)$
we consider a
$(\scr{A}^{\leqslant 0}_{|U},
\widetilde{\phi}_t)$-module
$\scr{E}^\frak{a}\otimes \scr{R}_{G|U}$,
where we have set
$\scr{E}^{\frak{a}}\coloneqq
(\scr{A}^{\leqslant 0}_{|U},
\exp(\widetilde{\phi}_t(\frak{a})-\frak{a})\widetilde{\phi}_t)$
similarly as in \S \ref{ExF}.
 Then we obtain the following:
\begin{corollary}\label{EM}
    ${\mathrm{Per}}(\scr{E}^{\frak{a}}
\otimes \scr{R}_{G|U}))
=\scr{A}_{\mathrm{per}}
\exp(-\frak{a})t^{-G}\C^r_{U}$.
\end{corollary}
\begin{proof}
The claim follows from
Theorem \ref{O}
and the following isomorphism
of $(\scr{A}^{\leqslant 0}_{|U},\widetilde{\phi}_t)$
-modules:
\begin{align*}
(\scr{A}
^{\leqslant \frak{b}+\frak{a}}
)^{\oplus r}
\xrightarrow{\sim}
\scr{A}^{\leqslant \frak{b}}
\otimes \scr{M},
\quad
\bm{f}\mapsto \exp(-\frak{a})t^{-G}\bm{f}
\end{align*}
where we put
$\bm{f}={}^t(f_1,\dots,f_r)$,
$\scr{M}=\scr{E}^{\frak{a}}
\otimes \scr{R}_{G|U}$, and
$\frak{b}\in\scr{I}(U)$.
\end{proof}
\begin{proof}[A proof of Corollary \ref{Cor}]
The fact
that Theorem \ref{Vanishing}
implies
Corollary \ref{Cor} easily follows from the
isomorphism
\begin{align*}
    {\mathrm{DR}}_{\leqslant \frak{a}}(\scr{M})
    \xrightarrow{\sim}
    {\mathrm{DR}}_{\leqslant 0} (\scr{E}^{\frak{a}}\otimes \scr{M}_{|U})
\end{align*}
given by the multiplication of
$\exp(-\frak{a})$.
\end{proof}

\subsection{Existence of local splittings}
\label{RH-reg}
In this subsection,
we
 prove the following:
\begin{theorem}\label{ThR}
If $\scr{M}\in {\mathsf{Diffc}}^{\mathrm{mild}}$,
then ${\mathrm{RH}}(\scr{M})$ is a
Stokes filtered $\scr{A}_{\mathrm{per}}$-module.
\end{theorem}

\subsubsection{Asymptotic expansions}
As a preliminary,
we recall the basic theory of asymptotic expansions.
Let $\cal{A}$ be the subsheaf of
$\widetilde{\scr{O}}$ whose sections have the
asymptotic expansion in $\widehat{\scr{O}}(*0)$.
By the Borel-Ritt theorem,
we have an exact sequence
\begin{align}\label{BR}
    0\longrightarrow \scr{A}^{<0}
    \longrightarrow \cal{A}
    \xrightarrow{{\mathrm{asy}}}\varpi^{-1}\widehat{\scr{O}}{(*0)}
    {\longrightarrow} 0
\end{align}
where the symbol `${\mathrm{asy}}$' denotes the asymptotic expansion.
We note that the relation $\cal{A}\subset \scr{A}^{\leqslant 0}$
holds
as subsheaves of $\widetilde{\scr{O}}$.

Take a positive integer $m$
and let $\rho\colon S^1_\tau\to S^1_t$
be a finite covering
as in \S \ref{Ram}.
Take a local section
$\frak{s}\colon U\to S^1_\tau$
of $\rho$ (i.e. fix a branch of $\tau=t^{1/m}$).
Then we have an
exact sequence
\begin{align}\label{pre-m}
    0\longrightarrow \scr{A}^{<0}_{t|U}
    \longrightarrow \cal{A}_{|U}^{(m)}
    \xrightarrow{{\mathrm{asy}}_{\frak{s}}}
    \widehat{\scr{O}}_\tau{(*0)}_{S^1|U}
    {\longrightarrow} 0
\end{align}
by the pull back of \eqref{BR},
where we put
$\cal{A}^{(m)}\coloneqq
\rho_*\cal{A}\cap \scr{A}^{\leqslant 0}\subset
\rho_*\widetilde{\scr{O}}_\tau $,
and
${\mathrm{asy}}_{\frak{s}}=\frak{s}^{-1}{\mathrm{asy}}$.

\subsubsection{Key fact}
Let $\scr{M}$
be a mild difference module.
By Lemma \ref{Graded},
we have a graded module $\scr{N}$
of $\scr{M}$.
Let $m$
be a positive integer and use the notations
in \S \ref{Ram} and \S \ref{Extended}.
Assume that there exists an
isomorphism
$\widehat{\Xi}\colon
\rho^*\widehat{\scr{M}}\xrightarrow{\sim}
\rho^*\widehat{\scr{N}}$
of difference modules
over
$(\widehat{\scr{O}}_{\tau}(*0),\widehat{\phi}_\tau)$.

\begin{theorem}
[{\cites{BF,Galois}}]\label{CReg}
Let $\scr{M},\scr{N},m,$ and $\widehat{\Xi}$
be as above.
Then, for any point $e^{{\tt{i}}\theta}\in S^1_t$,
there exist an open neighborhood $U$
of $e^{{\tt{i}}\theta}$,
an isomorphism
\begin{align*}
    \Xi_{U} &\colon
(\cal{A}^{(m)}{\otimes}
\scr{M}_{S^1})_{|U}
\xrightarrow{\sim}
(\cal{A}^{(m)}{\otimes}
\scr{N}_{S^1})_{|U},
\end{align*}
and a section $\frak{s}:U\to S^1_\tau$ such that
the formal completion
\begin{align*}
    \mathrm{id}_{\widehat{\scr{O}}_\tau(*0)}
    \otimes {\Xi}_{U}
    \colon
    \rho^*\widehat{\scr{M}}
    \longrightarrow
    \rho^*\widehat{\scr{N}}
\end{align*}
via ${\mathrm{asy}}_{\frak{s}}$
 coincides
with $\widehat{\Xi}$,
where we have used the
identifications
\begin{align*}
\widehat{\scr{O}}_\tau(*0)\otimes
    (\cal{A}^{(m)}{\otimes}
    \scr{M}_{S^1})_{|U}=\rho^*\widehat{\scr{M}},
    \text{ and }
\widehat{\scr{O}}_\tau(*0)\otimes
    (\cal{A}^{(m)}{\otimes}
    \scr{N}_{S^1})_{|U}=\rho^*\widehat{\scr{N}}.
\end{align*}
\end{theorem}
\begin{proof}
Take a basis
$e_1,\dots,e_r$ of $\scr{M}$ over
$\scr{O}_t(*0)$.
Take a basis $f_1,\dots,f_r$
of $\scr{N}$ over $\scr{O}_t(*0)$.
Let $\psi_{\scr{M}}$ and
$\psi_{\scr{N}}$ be the automorphisms on
$\scr{M}$ and $\scr{N}$, respectively.
Then we have
$\psi_{\scr{M}}
\bm{e}=
A(t) \bm{e}$
and
$\psi_{\scr{N}}
\bm{f}=
B(t) \bm{f}$
where we put
$\bm{e}={}^t(e_1\ \cdots\ e_r)$
and $\bm{f}={}^t(f_1\ \cdots\ f_r)$.
By assumption, there exists a invertible matrix
$\widehat{Y}(\tau)
\in {\mathrm{GL}}_r(\widehat{\scr{O}}_\tau(*0))$
which
satisfies the difference equation
$\widehat{\phi}_{\tau}(\widehat{Y})({\tau})
=B(\tau^m)\widehat{Y}(\tau )A(\tau^m)^{-1}$.
By \cite[Theorem 9.1, Theorem 11.1]{Galois},
which is essentially due to
Braaksma-Faber \cite[Theorem 4.1]{BF}
(or, more generally,
\cite[Theorem 11.7, Remarks 11.9, Theorem 11.10,
Remark 11.11]{Galois},
which uses the result of Immink \cites{Immink, Immink2})
for any point
$e^{{\tt{i}}\theta}\in S^1$
for sufficiently small
$U$ and a choice
$\frak{s}\colon U\to S^1_\tau$
of the branch of $\tau=t^{1/m}$,
there exists a
section
$Y(t)\in {\mathrm{GL}}_r(
\cal{A}^{(m)} (U))$
such that
${\mathrm{asy}}_{\frak{s}}(Y)=\widehat{Y}$ and
$\widetilde{\phi}_{t}({Y})
(t)=
B(t)
{Y}(t)
A(t)^{-1}.$
Set $\Xi_{U}\bm{e}=Y(t)\bm{f}$.
Then it follows that $\Xi_{U}$
is a morphism that satisfies the
conditions.
\end{proof}
\subsubsection{A proof of Theorem \ref{ThR}}
Let $\scr{M}$ be a
mild difference module.
Let $\scr{N}$ be the graded module of $\scr{M}$.

Then, by Theorem \ref{CReg}, for each point
$e\in S^1_t$,
there is an open neighborhood
$U$ of $e$
and
an isomorphism
\begin{align*}
    \Xi_U\colon \scr{M}_{|U}\xrightarrow{\sim}
    \scr{N}_{|U}
\end{align*}
of difference modules over
$(\scr{A}^{\leqslant 0}_{|U}, \widetilde{\phi}_t)$.
We obtain the theorem
by Corollary \ref{EM}.
\qed

\subsection{Statement of the main theorem}
We are now in the position
to state the main result of
this paper:

\begin{theorem}\label{Main}
The induced functor
\begin{align*}
    {\mathrm{RH}}\colon {\mathsf{Diffc}}^{\mathrm{mild}} \to {\mathsf{St(\scr{A}_\per)}},
    \quad
    \scr{M}\mapsto {\mathrm{RH}}(\scr{M})=
    ({\mathrm{Per}}(\scr{M}),\scr{H}^0 {\mathrm{DR}}_\bullet (\scr{M}))
\end{align*}
is an equivalence of categories.
\end{theorem}
We give a proof of the fact that
${\mathrm{RH}}$ is fully faithful in \S \ref{FF}.
Then we use the fact to prove that
${\mathrm{RH}}$ is essentially surjective
in \S \ref{ES}.
\subsection{Fully faithfulness}\label{FF}
We shall prove that the functor ${\mathrm{RH}}$
is fully faithful.
\subsubsection{}
Let $\scr{M}=(\scr{M},\psi)$ and
$\scr{M}'=(\scr{M},\psi')$ be  difference modules
over $(\scr{O}_t(*0),\phi_t)$.
The
module
${\mathscr{H}\! \! om}_{\scr{O}}(\mathscr{M},\scr{M}')$
of morphisms of $\mathscr{O}_t(*0)$-modules
from $\scr{M}$ to $\scr{M}'$
is equipped with the automorphism
$\psi_{\scr{M},\scr{M}'}(h)\coloneqq
\psi'\circ h\circ \psi^{-1}$
for $h\in {\mathscr{H}\! \! om}_{\scr{O}}(\mathscr{M},\scr{M}')$.
For simplicity of the notations,
we set
$(\scr{L},\scr{L}_{\bullet})\coloneqq
({\mathrm{Per}}(\scr{M}),\scr{H}^0{\mathrm{DR}}_\bullet(\scr{M}{}))$
and
$(\scr{L}',\scr{L}'_\bullet)
\coloneqq
({\mathrm{Per}}(\scr{M}'),\scr{H}^0{\mathrm{DR}}_\bullet(\scr{M}')).
$
\begin{lemma}\label{RH_0}
There is a natural morphism
\begin{align*}
    {\mathrm{RH}}_{\leqslant 0}\colon
    \scr{H}^0{\mathrm{DR}}_{\leqslant 0}({\mathscr{H}\! \! om}_{\scr{O}}
    (\scr{M},\scr{M}'))\longrightarrow
    {\mathscr{H}\! \! om}(\scr{L}, \scr{L}')_{\leqslant0}.
\end{align*}
\end{lemma}
\begin{proof}
A local section on an open subset $U$ of
$\scr{H}^0{\mathrm{DR}}_{\leqslant 0}({\mathscr{H}\! \! om}_{\scr{O}}(\scr{M},\scr{M}'))$
is regarded as a morphism
\begin{align*}
    \xi\colon \scr{M}_{|U}
    \to \scr{M}_{|U}
\end{align*}
which is compatible with the difference operators.
For each $\frak{a}\in\scr{I}(U)$,
it also sends
$\scr{A}^{\leqslant \frak{a}}
\otimes \scr{M}_{|U}$
into
$\scr{A}^{\leqslant \frak{a}}
\otimes\scr{M}'_{|U}$.
It follows that
$\xi$ sends $\scr{L}_{\leqslant \frak{a}}$
into $\scr{L}'_{\leqslant\frak{a}}$.
Hence it defines
a local section
${\mathrm{RH}}_{\leqslant 0}(\xi)
\in {\mathscr{H}\! \! om}(\scr{L},\scr{L'})_{\leqslant 0}$.
\end{proof}
\subsubsection{}\label{RH^tau}
For each point
$e\in S^1_t$, there is an open neighborhood
$U$ of $e$
such that
${\mathrm{RH}}_{\leqslant 0}$
is isomorphic to
the direct sum of the morphisms of the form
\begin{align*}
    &\scr{H}^0{\mathrm{DR}}_{\leqslant 0}
    ({\mathscr{H}\! \! om}
    (\scr{E}^{\frak{a}}\otimes\scr{R}_{G|U},
    \scr{E}^{\frak{b}}\otimes \scr{R}_{H|U}))\\
    &\longrightarrow
    {\mathscr{H}\! \! om}(\scr{A}_{\mathrm{per}}\exp(-\frak{a})
    t^{-G}\C^r_{U},
    \scr{A}_{\mathrm{per}}\exp(-\frak{b})
    t^{-H}\C^{r'}_{U}    )_{\leqslant 0}
\end{align*}
where $\frak{a},\frak{b}\in \scr{I}(U)$,
$G\in \mathrm{End}(\C^r)$, and
$H\in \mathrm{End}(\C^{r'})$ by Theorem \ref{CReg}.
It then follows that
the morphism ${\mathrm{RH}}_{\leqslant 0}$
is
an isomorphism.
\subsubsection{}
Let us set
$\scr{N}={\mathscr{H}\! \! om}_{\scr{O}}(\scr{M},\scr{M}')$
and we define ${\mathrm{DR}}(\scr{N})$ by the complex
$\scr{N}\xrightarrow{\psi-\mathrm{id}} \scr{N}$
in degree zero and one, where $\psi$ denotes
the automorphism on $\scr{N}$ naturally induced
from $\scr{M}$ and $\scr{M}'$.
Then we have $\scr{H}^0{\mathrm{DR}}(\scr{N})=
\mathrm{Hom}(\scr{M},\scr{M}')$.
By the projection formula,
we have
${\mathrm{DR}}(\scr{N})=\mathbb{R}\varpi_*{\mathrm{DR}}_{\leqslant 0}(\scr{N})$.
By Theorem \ref{Vanishing},
we obtain
$\mathbb{R}\varpi_*{\mathrm{DR}}_{\leqslant 0}(\scr{N})
=\mathbb{R}\varpi_*\scr{H}^0{\mathrm{DR}}_{\leqslant 0}(\scr{N})$
and hence
\begin{align*}
    \mathrm{Hom}(\scr{M},\scr{M}')=
    \varpi_*\scr{H}^0{\mathrm{DR}}_{\leqslant 0}
    ({\mathscr{H}\! \! om}_{\scr{O}}(\scr{M},\scr{M}')).
\end{align*}
Then, the morphism
${\mathrm{RH}}=\varpi_*{\mathrm{RH}}_{\leqslant 0}$
gives an isomorphism between
$\mathrm{Hom}(\scr{M},\scr{M}')$
and
$\varpi_*{\mathscr{H}\! \! om}(\scr{L},\scr{L}')_{\leqslant 0}
=\mathrm{Hom}_{{\mathsf{St(\scr{A}_\per)}}}(\scr{L},\scr{L}')$.
\subsection{Essential surjectivity}\label{ES}
To complete the proof of
Theorem \ref{Main},
we shall prove the essential surjectivity of
the functor ${\mathrm{RH}}$.
\subsubsection{Graded case}
We shall firstly show the
essential surjectivity of
${\mathrm{RH}}$ in the graded case:

\begin{lemma}\label{ESG}
Let $\scr{G}$ be a graded
Stokes filtered $\mathscr{A}_{\mathrm{per}}$-module.
    There exists
    a mild graded difference module
    $\scr{N}$ over $(\scr{O}_t(*0),\phi_t)$
    such that
    ${\mathrm{RH}}(\scr{N})\simeq \scr{G}$.
\end{lemma}
\begin{proof}
Take a positive integer
$m$ such that $\Sigma(\scr{G})\subset \scr{I}_m$.
Then
$\rho^{-1}\Sigma(\scr{G})\subset
\rho^{-1}\scr{I}_m=\scr{I}_{m,\tau}$
is a trivial sheaf of finitely many
$ (2\pi{\tt{i}} \mathbb{Z} s)$-orbits,
which admits a $\mu_m$-action.
Take a section $\frak{a}_i$ for each $i=1,\dots,\ell$
in the orbit, so that
\[\rho^{-1}_m\Sigma(\scr{G})
=\bigsqcup_{i=1}^\ell (\frak{a}_i+2\pi{\tt{i}} s \mathbb{Z}_{S^1}).\]
For each $i$,
let ${\mathrm{gr}}_{\frak{a}_{i}}(\rho^{-1}\scr{G})$ be
a local system of
$\C$-vector space
defined as follows:
\begin{align*}
    {\mathrm{gr}}_{\frak{a}_{i}}(\rho^{-1}\scr{G})(U)=
    {\mathrm{gr}}_{\frak{a}_{i,U}}\scr{G}(\rho(U))
\end{align*}
where $U$ is an open subset
such that $\rho_{|U}$ is a homeomorphism
and $\frak{a}_{i,U}\in \scr{I}_m(U)$
is a section such that
$\rho_{|U}^*\frak{a}_{i,U}=\frak{a}_{i|U}$.
Then, there is a matrix $G_i$
such that the monodromy of ${\mathrm{gr}}_{\frak{a}_i}(\scr{G})$
is given by $\exp(2\pi {\tt{i}} m G_i)$ for some basis at
a fiber.
Set
$\scr{N}'\coloneqq
\bigoplus_{i}\scr{E}^{\frak{a}_i}\otimes \rho^*\scr{R}_{G_i}$.
Then, by the $\mu_m$-invariant construction given above,
there is a
mild graded difference module
$\scr{N}$ such that $\rho^*\scr{N}\simeq \scr{N}'$.
We can easily check that ${\mathrm{RH}}(\scr{N})\simeq \scr{G}$.
\end{proof}

\subsubsection{Classification theorem}\label{MSthm}
For a difference module $\scr{N}$ over
$(\scr{O}_t(*0),\phi_t)$, we set
\begin{align*}
    {\scr{A}ut}^{<0}(\scr{N})
    =\mathrm{id}_{\scr{N}}+
    \scr{H}^0{\mathrm{DR}}_{<0}(\scr{E}nd(\scr{N})).
\end{align*}
The following Malgrange-Sibuya type classification theorem
for difference modules plays a key role
in the proof of essential surjectivity of ${\mathrm{RH}}$:
\begin{theorem}\label{ClMe}
Let $\scr{N}$ be a mild graded
difference module over
$(\scr{O}_{t}(*0),\phi_t)$.
Let
$m$ be a positive integer
such that $\rho^*_m\scr{N}$ is
mild elementary.
Then there is a natural one-to-one correspondence
between the set
\begin{align*}
    H^1(S^1 ,{\scr{A}ut}^{<0}(\scr{N}))
\end{align*}
and
the set of isomorphism classes of
pairs
$(\scr{M},\widehat{\Xi})$ of
\begin{itemize}
    \item a mild difference module
    $\scr{M}$ over $(\scr{O}_t(*0),\phi_t)$,
    and
    \item an isomorphism
    $\widehat{\Xi}\colon \rho^*\widehat{\scr{M}}
    \xrightarrow{\sim} \rho^*\widehat{\scr{N}}$
    of difference modules over
    $(\widehat{\scr{O}}_\tau(*0),\widehat{\phi}_\tau)$.
\end{itemize}
\end{theorem}
\begin{proof}
This theorem is essentially proved in
\cite[Theorem 11.12]{Galois}.
We shall recall the construction of
the pair
$(\scr{N},\widehat{\Xi})$ from the cohomology
class $[g]$
for the convenience of the reader.
Take a finite open covering
$S^1=\bigcup_{\alpha\in \Lambda} U_\alpha$
such that
\begin{itemize}
\item
$\Lambda=\mathbb{Z}/N\mathbb{Z}$ for a positive integer $N$
such that,
\item
$U_\alpha$ is an open interval for each
$\alpha\in \Lambda$, and
\item
$U_\alpha\cap U_\beta=\emptyset$ for
$\beta\notin \{\alpha-1,\alpha,\alpha+1\}$,
\end{itemize}
and a representative
$g=(g_\alpha)_\alpha\in
\prod_{\alpha\in \Lambda}
H^0(V_{\alpha},
{\scr{A}ut}^{<0}(\scr{N}))$,
where
$V_{\alpha}\coloneqq U_{\alpha}\cap U_{\alpha+1}$.
Fix a frame
$\scr{N}=
\bigoplus_{i=1}^r \scr{O}_t(*0)e_i$,
then, for each $\alpha$,
the section $g_\alpha$ is
identified with a section $(g_{\alpha,ij})\in
{\mathrm{GL}}_r(\cal{A}^{(m)}(V_{\alpha}))$
via $g_\alpha(e_i)=\sum_jg_{\alpha,ij}e_j$.
Then we obtain
the map
\[H^1(S^1, {\scr{A}ut}^{<0}(\scr{N}))
\to H^1(S^1,{\mathrm{GL}}_r(\cal{A}^{(m)})).\]
By the Malgrange-Sibuya theorem,
we have $H^1(S^1, {\mathrm{GL}}_r(\cal{A}^{(m)}))=1$.
It implies that
there exists a family
$(h_\alpha)_\alpha\in
\prod_{\alpha\in\Lambda}
H^0(U_\alpha, {\mathrm{GL}}_r(\cal{A}^{(m)}))$
such that
$g_\alpha=h_{\alpha}^{-1} h_{\alpha+1}$
on $V_{\alpha}$
(taking finer covering if necessary).
Let $\psi$ be
the difference operator on $\scr{N}$.
Using the frame,  we have
$\psi=A^0(t)\phi_t^{\oplus r} $
for some $A^0(t)\in {\mathrm{GL}}_r(\scr{O}_t(*0))$.
Note that we have the equality
$ \widetilde{\phi}_t(g_\alpha)
=A^{0}g_\alpha(A^0)^{-1}$.
We then set
\[\widetilde{A}_\alpha (t)
=\widetilde{\phi}_t(h_{\alpha})
A^0(t)h_\alpha^{-1}
\in {\mathrm{GL}}_r(\cal{A}^{(m)}(U_\alpha))\]
for each $\alpha\in \Lambda$.
Since we have
\begin{align*}
    \widetilde{A}_{\alpha}\widetilde{A}_{\alpha+1}^{-1}
    =\widetilde{\phi}_\tau(h_{\alpha})
A^0(t)h_\alpha^{-1}
h_{\alpha+1}
A^0(t)^{-1}\widetilde{\phi}_\tau(h_{\alpha+1}^{-1})
=\mathrm{id}
\end{align*}
on $V_{\alpha}$,
there exists a unique $A(t)\in {\mathrm{GL}}_r(\scr{O}_\tau(*0))$
such that $A(t)_{|U_\alpha}=\widetilde{A}_\alpha$.
Then $\psi\coloneqq A(t)\phi_t^{\oplus r}$ on
$\scr{M}\coloneqq
\scr{N}
=\bigoplus_{i=1}^r \scr{O}_\tau(*0)e_i$
defines a new difference module
$(\scr{N},\psi)$ with the trivial isomorphism
$\widehat{\Xi}$ between the formal completions.
\end{proof}

\subsubsection{End of the proof of Theorem \ref{Main}
$($Essential surjectivity$)$}
Let $(\scr{L},\scr{L}_{\bullet})$ be an object
in ${\mathsf{St(\scr{A}_\per)}}$.
There exists a positive integer
$m$ such that
$ \Sigma(\scr{L})\subset \scr{I}_m$.
By Theorem \ref{ClSt},
the pair $(\scr{L},\scr{L}_{\bullet})$
corresponds to a cohomology class
$[\scr{L},\scr{L}_{\bullet}]$
in
$H^1(S^1,
{\scr{A}ut}^{<0}(\scr{G}))$
with $\scr{G}=\scr{A}_{{\mathrm{per}}}\otimes {\mathrm{gr}}\scr{L}$.
There exists a
mild difference module
$\scr{N}$ over $(\scr{O}_t(*0),\phi_t)$
such that ${\mathrm{RH}}(\scr{N})=\scr{G}$ by Lemma \ref{ESG}.
By the fully-faithfulness of ${\mathrm{RH}}$,
we have an isomorphism
\begin{align*}
    H^1(S^1,{\scr{A}ut}^{<0}(\scr{N}))
    \simeq
    H^1(S^1,
    {\scr{A}ut}^{<0}(\scr{G})).
\end{align*}
Let $[\scr{M}]$ be the class in
$H^1(S^1, {\scr{A}ut}^{<0}(\scr{N}))$
which corresponds to
$[\scr{L},\scr{L}_{\bullet}]$
by the above isomorphism.
By Theorem \ref{ClMe},
there exists a difference module
$\scr{M}$ over $(\scr{O}_t(*0),\phi_t)$
which corresponds to the class $[\scr{M}]$.
Then, by the construction,
we have
${\mathrm{RH}}(\scr{M})\simeq
(\scr{L},\scr{L}_{\bullet})$,
which implies the essential surjectivity  of ${\mathrm{RH}}$.
\qed

\subsection{Rank one non-trivial examples}\label{R1}
\subsubsection{A regular singular difference module}
For a complex number $\alpha\in\C\setminus \mathbb{Z}$,
we consider a
difference module
$\scr{B}_{\alpha}\coloneqq (\scr{O}_t(*0),\psi_\alpha )$
defined as
$\psi_\alpha=(1+\alpha t)\phi_t$.
It is easy to see that
the formal completion
$\widehat{\scr{B}}_{\alpha}$
is isomorphic to
$\widehat{\scr{R}}_{\alpha}$,
where $\alpha$ is regarded as a $(1\times 1)$-matrix.
\begin{proposition}\label{REX}
We have the following description of
${\mathrm{Per}}({\scr{B}}_{\alpha}):$
\begin{align*}
    {\mathrm{Per}}({\scr{B}}_{\alpha})_{|U}=
    \begin{cases}
    \scr{A}_{{\mathrm{per}}|U} \Gamma(s)/\Gamma(s+\alpha)
    &(e^{\pi{\tt{i}}}\notin U)
    \\
    \scr{A}_{{\mathrm{per}}|U}
    (1-u)\Gamma(s)
    /
    (1-e^{2\pi {\tt{i}}\alpha }u)\Gamma(s+\alpha)
    &
    (e^{0}\notin U),
    \end{cases}
\end{align*}
where $U$ is an open subset in $S^1$,
the symbol $\Gamma(s)$ denotes the Gamma function,
and we set $s=t^{-1}$
and $u=\exp(2\pi{\tt{i}} s)$ as in $\S \ref{Another}$.
\end{proposition}
\begin{proof}
By the relation $\Gamma(s+1)=s\Gamma(s)$, the equality
\[\psi_\alpha (\Gamma(s)/\Gamma(s+\alpha))
=\Gamma(s)/\Gamma(s+\alpha)\]
holds,
which implies that
$\Gamma(s)/\Gamma(s+\alpha)\in
\widetilde{{\mathrm{DR}}}({\scr{B}_{\alpha}})(U_+)$
for $U_+=S^1\setminus \{e^{\pi{\tt{i}}}\}$.
By the Stirling formula,
we moreover obtain that
$\Gamma(s)/\Gamma(s+\alpha)\in
{\mathrm{DR}}_{\leqslant0}(\scr{B}_{\alpha})(U_+)$,
whose germ at each point $x\in U_+$ is not in
${\mathrm{DR}}_{<0}(\scr{B}_{\alpha})_x$.
By Theorem \ref{O},
we obtain the first half of the proposition.
The latter half can be proved in a similar way
by using the reflection formula:
$(1-u)\Gamma(s)=(-2\pi{\tt{i}})e^{\pi{\tt{i}} s}/\Gamma(1-s)$.
\end{proof}
\subsubsection{A mild difference module}
Let
$\scr{E}_\Gamma\coloneqq
(\scr{O}_t(*0),\psi_\Gamma)$
be a difference module with
$\psi_\Gamma=
\exp(\frak{l}(s+1)
-\frak{l}(s))t\phi_t$ where
$\frak{l}(s)=s\log s$.
It is not difficult to see
that the formal completion
$\widehat{\scr{E}}_\Gamma$
is isomorphic to
$\scr{E}^{\frak{a}_\Gamma}\otimes
\scr{R}_{-1/2}$
where $\frak{a}_\Gamma(s)=-s$.

\begin{proposition}
    We have the following description
    of ${\mathrm{Per}} (\scr{E}_\Gamma)$$:$
    \begin{align*}
        {\mathrm{Per}}(\scr{E}_\Gamma)_{|U}
        =\begin{cases}
            \mathscr{A}_{{\mathrm{per}}|U}s^{-s}\Gamma(s) &(e^{{\tt{i}}\pi}\notin U)\\
            \scr{A}_{{\mathrm{per}}|U}(1-u)s^{-s}\Gamma(s)&
            (e^0\notin U)
        \end{cases}
    \end{align*}
    where the notations except $s^{-s}$
    are defined as in Proposition \ref{REX}
    and we define
    $s^{-s}=\exp(-s\log s)$.
    Note that
    the choice of the branch of $log s$
    does not matter in the definition of the modules.
\end{proposition}
\begin{proof}
The proof is parallel to that for
Proposition \ref{REX} and left to the reader.
\end{proof}

\section{Proof of  Theorem \ref{Vanishing}}\label{PRH}
In this section,
we give a proof of Theorem \ref{Vanishing}
using the method in \cite{Immink}.
\subsection{Preliminary}
\subsubsection{Reduction}
By Theorem \ref{CReg},
we may assume that
for any $e^{{\tt{i}}\theta}\in S^1$,
there exists an open neighborhood
$U$ of $e^{{\tt{i}}\theta}$
such that the restriction of the
difference module
$\scr{M}$ in the statement of the
Theorem \ref{Vanishing}
to $U$
is
isomorphic to the direct sum of
the difference module of the form
$\scr{E}^{\frak{a}}\otimes \scr{R}_{G|U}$,
where notations are as in \S \ref{EE}.
We may also assume
that $G$
is of the form
$G=\gamma \mathrm{id}_{\C^r}+N$
where $N$ is a nilpotent Jordan matrix
and $\gamma \in \C$.
Hence it reduces to prove the following claim:
\begin{claim}\label{Claim}
    Assume that
    $ \scr{M}_{|U}=
\scr{E}^{\frak{a}}\otimes \scr{R}_{G|U}$
for $\frak{a}\in \scr{I}(U)$
and $G=\gamma\mathrm{id}_{\C^r}+N$ with
$\gamma\in\C$ and a nilpotent
matrix $N\in \mathrm{End}(\C^r)$.
Then, for any $\bm{f}\in
\scr{M}_{|U,e^{{\tt{i}}\theta}}$
there exists
$\Lambda_\theta(\bm{f}) \in \scr{M}_{|U,e^{{\tt{i}}\theta}}$
such that
$\nabla_{\widetilde{\psi}}
\Lambda_\theta(\bm{f})=\bm{f}$.
Moreover,
if $\bm{f}\in (\scr{A}^{< 0}\otimes \scr{M}_{S^1})
_{e^{{\tt{i}}\theta}}$,
then we have $\Lambda_\theta(\bm{f})\in
(\scr{A}^{< 0}\otimes \scr{M}_{S^1})_{e^{{\tt{i}}\theta}}$.
\end{claim}
The remaining part of the theorem
is easy and left to the
reader.
We shall prove this claim
in the case
$0<\theta<{\pi}$.
The proof in the case $\pi<\theta<2\pi$
is similar to that in
the case above.
The proof in the case $\theta\in\{0,\pi\}$
is essentially given in
\cite{Immink}*{\S 9.3}.

\subsubsection{Notations}
We shall fix some notations.
Let $c$ be a point in $\C$.
Let $\theta$ be real number
and $\varepsilon$ be a positive number.
Then we set
\begin{align*}
    S_c(\theta;\varepsilon)=\{s\in\C\mid
    s=c+Re^{{\tt{i}}\vartheta} \text{ for }
    |\theta-\vartheta|<\varepsilon
    \text{ and } R>0\}.
\end{align*}
For $\bm{f}\in
\scr{M}_{|U,e^{{\tt{i}}\theta}}$,
using the notation \S \ref{Another},
we take a representative
$f={}^t(f_1,\dots,f_r)$ of $\bm{f}$
defined on
$S_c(\theta;\varepsilon)$
for sufficiently small $\varepsilon>0$
and a suitable point $c\in\C$.
Without loss of generality,
we may assume that $|c|>1$ and that
there exist
 positive constants $C$ and
$N$ such that
\begin{align}\label{Moderate}
    |f(s)|\leq  C |s|^N
\end{align}
for $s\in S_c(\theta;\varepsilon)$,
where we put
$|f(s)|=\sqrt{\sum_{i=1}^r |f_i(s)|^2}$.
In the case where we have
$\bm{f}\in (\scr{A}^{< 0}\otimes \scr{M}_{S^1})
_{e^{{\tt{i}}\theta}}$,
for any
positive constant $N'$,
there exist a point $c'_{N'}\in S_c$
and a constant $C_{N'}$ such that
we have
\begin{align}\label{Rapid}
    |f(s)|\leq C_{N'}|s|^{-N'}
\end{align}
for $s\in S_{c_{N'}}(\theta,\varepsilon)$.

We fix the branch of
$\log s$ on
$S_c(\theta;\varepsilon)$
and set
$s^{-1/m}\coloneqq \exp(m^{-1}\log t)$.
Then,
$\frak{a}$
is expressed as
\begin{align*}
    \frak{a}(s)=\sum_{k=1}^m c_k s^{k/m}
    \quad (c_k\in\C).
\end{align*}
Taking the shift by $2\pi{\tt{i}} \mathbb{Z} s$ if necessarily,
we may assume that $c_m$ satisfies
the inequality
\begin{align}\label{mu}
    -2\pi+{R}(c_m,\theta,\varepsilon)
    +\mathrm{Im} (c_m)<0,
\end{align}
where we set
\begin{align*}
    R(c_m,\theta,\varepsilon)=
    \max\{0,-\mathrm{Re}(c_m)\}
    \max\{\tan^{-1}(\theta-\varepsilon),
    \tan^{-1}(\theta+\varepsilon)\}.
\end{align*}
We set
$A(s)=\exp(\frak{a}(s+1)-\frak{a}(s))
(1+s^{-1})^{G}$.
Then $\scr{M}_{|U}$
is identified with the module
$\left((\scr{A}_{|U}^{\leqslant 0}
)^{\oplus r},
A(s)\widetilde{\phi}_t^{\oplus r}
\right)$.
We also use the notation
$Y(s)=\exp(-\frak{a}(s))s^{G}$
with $s^G=\exp(G\log s)$,
which is also defined on
$S_c(\theta;\varepsilon) $.

\subsubsection{Preliminary estimate}
We shall recall an estimate by Immink:
\begin{lemma}[{\cite{Immink}*{Lemma 8.12}}]
\label{Immink-Estimate}
Let $ S\subset\mathbb{C}$
be a connected open subset
such that $|{\sf s}|\geq 1$ for ${\sf s}\in S$.
For $\mu\in\C$, $\nu\in \mathbb{R}$,
and a holomorphic function
${\Psi}\colon S\to \C$, we set
\begin{align*}
    {\Phi}(\zeta)\coloneqq \mu\zeta
    +\nu\log|\zeta|+\Psi(\zeta),
    \quad \quad (\zeta\in S).
\end{align*}
Assume that
we have the inequalities
\begin{align}
    &|\Psi(\zeta)|\leq C|\zeta|^{1-\epsilon}&
    (\zeta\in S)\\
    &|\Psi'(\zeta)|\leq C|\zeta|^{-\epsilon}
    &(\zeta\in S)
\end{align}
where $C$ and $\epsilon$ are positive numbers.
For ${\sf s}\in S $, and $\alpha \in\mathbb{R}$, set
\begin{align*}
    \ell(\mathsf{s};\alpha)\coloneqq
    \{\zeta\in \C\mid \zeta= {\sf s}+Te^{{\tt{i}}\alpha},  0<T\}.
\end{align*}
We also assume that
there exists a positive constant
$\delta$ the inequality
    $\mathrm{Re}(\mu e^{{\tt{i}}\alpha})\geq \delta$
    holds.
Further, let $P$
be a polynomial with positive coefficients.
Then there exists a positive number
$K$ which is fully determined by
the constants $\delta,\nu,C$
and $\epsilon$, and the polynomial $P$
such that
    \begin{align*}
        \int_\ell
        \left|\exp(\Phi({\sf s})-\Phi(\zeta))
        P(\log|{\sf s}/\zeta|)d\zeta
        \right|\leq K.
    \end{align*}
\end{lemma}

\subsection{Definition of the Splitting}

\subsubsection{Family of paths}
Take a point
$p\in S_c
(\theta;\varepsilon)$
and $q\in S_p
(\theta;\varepsilon)$ such that the point
$q+1$ is in $S_p(\theta;\varepsilon)$.
For $s\in S_q(\theta;\varepsilon)$,
set $s'=s+\tfrac{1}{2}$.
Then we define the family of
paths
\begin{align*}
    C(s)=
    p+\mathbb{R}_{>0}(s'-p)
    =\{p+T(s'-p)\mid  T> 0\}
\end{align*}
for each $s\in S_q(\theta;\varepsilon)$.
\subsubsection{Integral operator}
We consider the following integral
\begin{align}
\Lambda_\theta (f)(s)\coloneqq
-f(s)+
    Y(s)
    \int_{{{C}} (s)}
    \frac{Y(\zeta)^{-1}f(\zeta)}
    {1-e^{ 2\pi{\tt{i}} (s-\zeta) }}d\zeta
\end{align}
for the family of paths $C(s)$,
$s\in S_q(\theta;\varepsilon)$.
The proof of the following lemma
will be given
in the next subsection:
\begin{lemma}\label{Est}

  There exist
  positive constants $K$ and ${N}$ such that
  the inequality
   \begin{align}
      \left| Y(s)
       \int_{C(s)}
       \frac{Y(\zeta)^{-1}f(\zeta)}
       {1-e^{2\pi{\tt{i}}(s-\zeta)}}d\zeta\right|
       \leq K|s|^{{N}}
    \end{align}
  holds for $s\in S_{q}(\theta,\varepsilon)$.
  In particular, the integral is well defined.
  If moreover $f$ rapid decays,
  then for any positive $N'$
  there are positive constants $K',\varepsilon'$
  and a point $c_{N'}\in S_{c}(\theta,\varepsilon)$
  such that the inequality
  \begin{align}\label{rapid2}
      \left| Y(s)
       \int_{C(s)}
       \frac{Y(\zeta)^{-1}f(\zeta)}
       {1-e^{2\pi{\tt{i}}(s-\zeta)}}d\zeta\right|
       \leq K'|s|^{-N'}
    \end{align}
holds for $s\in S_{c_{N'}}(\theta,\varepsilon')$.

\end{lemma}
By this lemma, $\Lambda_\theta (f)(s)$
is a holomorphic function on
$S_q(\theta;\varepsilon)$.
We shall check that
this lemma implies
Claim \ref{Claim}
in the case $0<\theta<\pi$:
\begin{lemma}
We have
$\nabla_{\widetilde{\psi}}\Lambda_\theta(f)(s)=f(s)$
on $s\in S_q(\theta;\varepsilon)
\cap S_{q+1}(\theta;\varepsilon)$.
\end{lemma}
\begin{proof}
Note that $A(s)Y(s+1)=Y(s)$.
Then we have
\begin{align*}
    \nabla_{\widetilde{\psi}} \Lambda_\theta (f)
    =&-A(s)f(s+1)+A(s)Y(s+1)\int_{C(s+1)}
    \frac{Y(\zeta)^{-1}f(\zeta)}
    {1-e^{2\pi{\tt{i}}(s-\zeta)}}\\
    &+f(s)-Y(s)\int_{C(s)}
    \frac{Y(\zeta)^{-1}f(\zeta)}
    {1-e^{2\pi{\tt{i}}(s-\zeta)}}d\zeta\\
    =&-A(s)f(s+1)+f(s)+A(s)Y(s+1)\int_{C(s+1)-C(s)}\frac{Y(\zeta)^{-1}f(\zeta)}
    {1-e^{2\pi{\tt{i}}(s-\zeta)}}d\zeta\\
    =&-A(s)f(s+1)+f(s)+A(s)Y(s+1)Y(s+1)^{-1}f(s+1)\\
    =&
    f(s).
\end{align*}
This completes the proof.
\end{proof}
This lemma implies the first part of
Claim \ref{Claim}.
The last part follows from
\eqref{rapid2}.
\subsection{Estimates}
We shall give a proof of
the first part of Lemma \ref{Est}.
The latter part can
be shown in a similar way
and is left to the reader.
\subsubsection{}
By the definition
of $Y(s)$, the matrix-valued
function
$Y(s)Y(s')^{-1}$
is bounded on $S_c(\theta;\varepsilon)$,
and hence there is a positive
constant $K_1$ such that
\begin{align}\label{Y<}
    |Y(s)Y(\zeta)^{-1}|
    <
    K_1|Y(s')Y(\zeta)^{-1}|
    \quad s\in S_c(\theta;\varepsilon).
\end{align}
Since $\arg(s)$ is bounded on
$S_c(\theta;\varepsilon)$
and the matrix
$N$ in the sum $G=\gamma\mathrm{id}+N$
is a Jordan nilpotent matrix,
there is a
polynomial $P$ with positive coefficients
such that
\begin{align}\label{G<}
    |(s'/\zeta)^G|\leq
    |s'/\zeta|^{\mathrm{Re}(\gamma)}
    |P(\log |s'/\zeta|)|,
\end{align}
where $|(s'/\zeta)^G|$ denotes an operator norm.
We also have
a positive constant $K_2$,
which is independent of
$\zeta\in C(s)$
such that
\begin{align}\label{E}
    |1-e^{2\pi{\tt{i}} (s-\zeta)}|
    \leq
    K_2|e^{-2\pi{\tt{i}} (s'-\zeta)}|
\end{align}
for $s\in S_c(\theta;\varepsilon)$
and
$\zeta\in C(s)$.
\subsubsection{}
We shall divide
the path $C(s)$ into two terms
as follows:
\begin{align*}
    \ell_0(s)&=
    \{(p+T(s'-p)\mid  0<T\leq 1\},\\
    \ell_1(s)&=
    \{(p+T(s'-p)\mid   T\geq 1\}.
\end{align*}
We then set
\begin{align*}
    I_j(s)\coloneqq
    Y(s)
    \int_{{{\ell}_j} (s)}
    \frac{Y(\zeta)^{-1}f(\zeta)}
    {1-e^{ 2\pi{\tt{i}} (s-\zeta) }}d\zeta
\end{align*}
for $j=0,1$.
It is not difficult to see that
$I_0(s)$ is of moderate growth
(resp. rapidly decay)
when $f$ has the same property.
We shall give an estimate on $I_1(s)$.
Set $\bm{\psi}(s)\coloneqq \frak{a}(s)-c_m s$,
$\bm{\varphi}(\zeta)
\coloneqq \bm{\mu} \zeta+\bm{\nu}
\log \zeta+\bm{\psi}(\zeta)$
where we put $\bm{\mu}=c_m-2\pi{\tt{i}}$,
and $\bm{\nu}=-N+\mathrm{Re}(\gamma)$.
Then, combining
\eqref{Y<}, \eqref{G<},
and \eqref{E}, and
the assumption \eqref{Moderate},
there exist positive constants
$K_3$ and $N$ such that
\begin{align*}
    |I_1(s)|&\leq
     K_3|s'|^N
    \int_{\ell_1(s)} |
    \exp(\bm{\varphi}(s')-\bm{\varphi}
    (\zeta))P(\log|s'/\zeta|)
    d\zeta|
\end{align*}
Then we can apply
Lemma \ref{Immink-Estimate}
to the integral
by the inequality \eqref{mu}. \qed
\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


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Fortnite Holiday Decorations Locations: How to Destroy 10 Holiday Decorations
Check out where to find and how to destroy 10 Holiday Decorations in Fortnite Chapter 3
December 22nd, 2021 by Franklin Bellone Borges
Fortnite's Winterfest 2021 is here and players are already diving in and putting all they have into completing all of its holiday-themed challenges and quests, all thought up in true Fortnite fashion. With that said, one of the quests sends players to destroy 10 Holiday Decorations. But where can you find all 10 and how can you destroy them? To answer those questions we will now tell you where to find and how to destroy 10 Holiday Decorations in Fortnite Chapter 3.
Fortnite Holiday Decorations – Where to Find and How to Destroy
You can find a lot of decorations all over the island, but we recommend that you go to either Greasy Grove, located southwest of the center of the map, or Coney Crossroads, located at the upper part of the map, in the crossroads between the area, The Daily Bugle, and Shifty Shafts, as both are neighborhood/urban-like areas filled with houses and buildings, all filled with Holiday decorations. After spotting the Holyday decorations in the mentioned areas, just bring up your melee weapon and hit them hard, until they are destroyed. You can check out the exact location 0f both areas as they are shown in the game's interactive map below:
Now that you know how to destroy and where to find the holiday decorations in Fortnite, don't forget to check out where to find a chicken and how to fly with one, as well as where to find snowmen and how to run over them with your car on Fortune Chapter 3. Aside from the challenges available as part of Winterfest 2021, Fortnite's Chapter 3 also offers players the opportunity to get Spider-Man, as well as the ability to web swing like him. You can check out how to web swing in Fortnite here.
You can play Fortnite right now on PlayStation 5, PlayStation 4, Xbox One, Xbox Series S/X, Nintendo Switch, Android, and PC, exclusively via the Epic Games Store. Fortnite's Winterfest 2021 will run until January 6th, 2022, at 9 AM ET.
- This article was updated on December 22nd, 2021
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// ----------------------------------------------------------------------------
// Copyright 2013 Bloomberg Finance L.P.
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// Licensed under the Apache License, Version 2.0 (the "License");
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// limitations under the License.
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Paul Harris Fellowships
You may know that nearly one-third of Columbia Metro Rotarians are recognized as Paul Harris Fellows. What exactly does this mean?
The Paul Harris Fellowship is named for Paul Harris, who founded Rotary over 100 years ago. The Fellowship was established in his honor in 1957. A Paul Harris Fellow is an individual who contributes $1,000 or in whose name that amount is contributed. Every Paul Harris Fellow receives a pin, medallion and a certificate when he or she becomes a Fellow. This identifies the Paul Harris Fellow as an advocate of the Foundation's goals of world peace and international understanding.
The Rotary Foundation is a registered charity that supports the efforts of Rotary International to achieve world understanding and peace through international humanitarian, educational, and cultural exchange programs. It is supported solely by voluntary contributions from Rotarians and friends of the Foundation who share its vision of a better world.
Programs the Rotary Foundation supports include Polio Plus – a program designed to eradicate Polio from the world. It supports Humanitarians programs, including grants to fund club and district service projects – and matching grants for international projects. The Foundation also supports educational programs – like ambassadorial scholarships, Group Study Exchange, Rotary Grants for University Teachers, and Rotary World Peace Scholarships.


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NBC Olympics Selects Canon U.S.A. To Provide Field And Studio Equipment For Its Production Of 2016 Olympic Games In Rio
Canon to Enable Providing Viewers with Outstanding HDTV Picture Quality and Long-Zoom Image Detail
MELVILLE, N.Y., April 13, 2016 – NBC Olympics, a division of the NBC Sports Group, has selected Canon U.S.A., Inc., a leader in digital imaging solutions, to provide a wide variety of HDTV Field, Portable and Studio lenses for its production of the Games of the XXXI Olympiad, which take place in Rio de Janeiro, Brazil, from August 5 – August 21. The announcement was made today by David Mazza, CTO & SVP, NBC Sports & Olympics, and Yuichi Ishizuka, president and COO of Canon U.S.A., Inc.
Over 70 Canon broadcast lenses will be used during coverage including the DIGISUPER 95 TELE long-zoom field lens featuring a focal length of 1178mm (2356mm with 2x Extender); the DIGISUPER 27 studio lens providing crisp HD imagery in a studio environment; the HJ14ex4.3B wide-angle HD lens capturing a panoramic 96.3 degrees angle-of-view; and the HJ24ex7.5B HDTV field telephoto zoom lens, providing a remarkable 7.5-180mm zoom range in a compact, portable body.
"We are honored to once again be chosen as the lens provider for NBC," stated Yuichi Ishizuka, president and COO, Canon U.S.A., Inc.
"The lenses that Canon provides are an essential component to the visual system that captures the excitement and the emotion of the Games. Canon lenses are also light and robust, which our camera operators like," stated Christopher Adams, Vice President Venue Engineering, Olympics Production Engineering, NBC Sports Group.
About NBC Olympics
A division of the NBC Sports Group, NBC Olympics is responsible for producing, programming and promoting NBC Universal's Olympic coverage. It is renowned for its unsurpassed Olympic heritage, award-winning production, and ability to aggregate the largest audiences in U.S. television history. For more information on NBC Olympics' coverage of the Rio Olympics, please visit: http://nbcsportsgrouppressbox.com


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I love the hunt for beads: odd, large, ethnic, contrarian beads; antique or contemporary glass beads; rare and unusual colors; textures that lead to touching. Successful hunting inspires bold eclectic designs that result in my unique necklaces and bracelets.
Each Beadleful design starts with a centerpiece bead. I add whimsy, color excitement, texture, chunky beads, a fabulous clasp; then I design and string a collectable necklace.
I have been a bead jewelry artist since 1993 when, after 23 years in the corporate world, I retired to accompany my husband on his job assignment in Hong Kong. Southeast Asia became my handicraft fantasy world as I hunted for beads in the bazaars of Tibet and Nepal, in dusty shops in Beijing, on remote Indonesian islands reached by catamaran, in beautiful Bali, and in verdant Indochina—Laos, Cambodia, and Vietnam.
I also use vintage German pressed glass, contemporary American art glass and I admit I am in love with whimsical vintage plastic beads. For the 1998 Oscar television ceremony, Fay Wray, of King Kong fame, wore one of my necklaces featuring Venetian blown glass mixed with ethnic beads.
I offer my necklaces to discerning buyers who love the sparks my pieces create and the adornment they give!
Are you Priscilla Beadle who use to live in Salado Texas?
Thank you for your inquiry, but I am not that person.
I spent many years in Calif, but I am a Massachusetts native.
Hope you enjoyed my blog!
I read where you will be judging an event, in Rockland, Mass. Since photography will be part, of the event are you qualified to judge the photographs, in a satisfactory manner?
I don't blame you for asking that question after re-reading my statement. It could also be well posed by a sculptor, a printmaker, or a potter. I was chosen as a juror based on my broad appreciation and knowledge of the arts. Having had a liberal arts education and lived in Europe for five years, my exposure to two and three dimensional art is more sophisticated than many Americans. My husband and I have collected contemporary American artists' work since the late 80's and live every day surrounded by well composed, professionally executed, highly creative artwork. I hope your work rises to that level and will vie for the top prize among all the other excellent pieces I am looking forward to viewing.
Thank you for your reply. Take care. Mark.