\keywords{de~Rham cohomology, algebraic curves, group actions,
characteristic~$p$}
\urladdr{http://jgarnek.faculty.wmi.amu.edu.pl/}
\date{}
\begin{abstract}
????
\end{abstract}
\maketitle
\bibliographystyle{plain}
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\section{Introduction}
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\begin{mainthm}
Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
Let $X$ be a curve with an action of~$G$ over a field $k$ of characteristic $p$.
The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the lower ramification groups and the fundamental characters of closed
points $x$ of $X$ that are ramified in the cover $X \to X/G$.
\end{mainthm}
\section{Cyclic covers}
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Let for any $\ZZ/p^n$-cover $X \to Y$
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\begin{align*}
u_{X/Y, P}^{(t)}&:= \min\{ t \ge 0 : G_P^{(t)}\cong\ZZ/p^{n-t}\},\\
l_{X/Y, P}^{(t)}&:= \min\{ t \ge 0 : G_{P, t}\cong\ZZ/p^{n-t}\}.
\end{align*}
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Note that if $G_P =\ZZ/p^n$, this coincides with the standard definition of
the $t$th upper (resp. lower) ramification jump of $X \to Y$ at $P$.
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\begin{Theorem}\label{thm:cyclic_de_rham}
Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k)\rangle=\ZZ/p^m = G_{P_0}$ for $P_0\in X(k)$. Then, as $k[\ZZ/p^n]$-modules:
In particular, $\dim_k \mc T^1\mc M =\ldots=\dim_k \mc T^{p^{n-1}- p^{n-2}}\mc M$.
Thus by Lemma~\ref{lem:lemma_mcT_and_T}
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\[
\dim_k T^1 \mc M = \ldots = \dim_k T^{p^n - p^{n-1}}\mc M = \frac{1}{p}\dim_k \mc T^1 \mc M = ????.
\]
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By Lemma~\ref{lem:trace_surjective} since $X \to X''$ is not \'{e}tale, the map $\tr_{X/X''} : H^1_{dR}(X)\to H^1_{dR}(X'')$ is surjective. Recall that
Assume now that $G = H \rtimes_{\chi} C =\langle\sigma\rangle\rtimes_{\chi}\langle\rho\rangle\cong\ZZ/p^n \rtimes_{\chi}\ZZ/c$.
Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-module $M$ and any character $\psi$ of $H$ we write $M^{\psi} := M \otimes_{k[G]}\psi$.
(\sigma - 1) : T^{i+1} M \hookrightarrow (T^i M)^{\chi^{-1}}.
\]
\end{Lemma}
\begin{proof}
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:main_thm_for_hypoelementary}]
We prove this by induction on $n$. If $n =0$, then it follows by Chevalley--Weil theorem.
Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale.
Recall that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma-1)$
is an isomorphism of $k$-vector spaces between $T^{i+1}\mc M$ and $T^i \mc M$ for
$i =2, \ldots, p^n$. This yields an isomorphism of $k[C]$-modules for $i \ge2$ by Lemma~\ref{lem:TiM_isomorphism_hypoelementary}:
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\begin{equation}\label{eqn:TiM=T1M_chi_\'{e}tale}
T^i \mc M \cong (T^2 \mc M)^{\chi^{-i+2}}
\end{equation}
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Observe that $\mc T^i \mc M$ has the filtration $\mc M^{(pi)}\supset\mc M^{(pi -1)}\supset\ldots\supset\mc M^{(pi - p)}$ with subquotients $T^{pi}\mc M, \ldots, T^{pi - p}\mc M$.
Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$:
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\begin{align*}
\mc T^i \mc M &\cong
\begin{cases}
T^1 \mc M \oplus T^2 \mc M \oplus\ldots\oplus (T^2 \mc M)^{\chi^{-p + 1}}, & i = 1\\
T^2 \mc M \oplus\ldots\oplus (T^2 \mc M)^{\chi^{-p}}, & i > 1.
\end{cases}
\end{align*}
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Thus, since by induction hypothesis $\mc T^2\mc M$ is determined by ramification data,
we have by Lemma~\ref{lem:N+Nchi+...} that $T^2\mc M$ is determined by ramification data.
Moreover, by Lemma~\ref{lem:G_invariants_\'{e}tale} and induction hypothesis, $T^1\mc M \cong H^1_{dR}(X'')$
is also determined by ramification data (???).
Assume now that $X \to Y$ is not \'{e}tale. Analogously as in the previous case, Lemma~\ref{lem:TiM_isomorphism_hypoelementary} and proof of Theorem~\ref{thm:cyclic_de_rham}
yield an isomorphism of $k[C]$-modules:
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\begin{equation}\label{eqn:TiM=T1M_chi}
T^{i+1}\mc M \cong (T^1 \mc M)^{\chi^{-i}}
\end{equation}
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for $i \le p^n - p^{n-1}$. Observe that $\mc T^i M$ has the filtration $\mc M^{(pi)}\supset\mc M^{(pi -1)}\supset\ldots\supset\mc M^{(pi - p)}$ with subquotients $T^{pi}\mc M, \ldots, T^{pi - p +1}\mc M$.
Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$:
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\begin{align*}
\mc T^i \mc M &\cong T^{pi - p + 1}\mc M \oplus\ldots\oplus T^{pi}\mc M\\
&\cong T^1 \mc M \oplus (T^1 \mc M)^{\chi^{-1}}\oplus\ldots\oplus
(T^1 \mc M)^{\chi^{-p}}.
\end{align*}
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By induction assumption, the $k[C]$-module structure of $\mc T^i \mc M$ is uniquely determined by the ramification data. Thus, by Lemma~\ref{lem:N+Nchi+...} for $N := T^1\mc M$ and by~\eqref{eqn:TiM=T1M_chi} the $k[C]$-structure of the modules $T^i \mc M$ is uniquely determined by the ramification data for $i \le p^n - p^{n-1}$.
By similar reasoning, $\tr_{X/X'}$ yields an isomorphism:
