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\begin{document}

\title[The de Rham...]{?? The de Rham cohomology of covers with cyclic $p$-Sylow subgroup}
\author[A. Kontogeorgis and J. Garnek]{Aristides Kontogeorgis and J\k{e}drzej Garnek}
\address{???}
\email{jgarnek@amu.edu.pl}
\subjclass[2020]{Primary 14G17, Secondary 14H30, 20C20} 
\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}
%
\section{}
%
\section{Cyclic covers}
%
Let $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) denote the $t$th upper (resp. lower)
ramification jump of $X \to Y$ at $P$.
%
\begin{Theorem}
	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:
	%
	\[
	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{P \neq P_0} J_{p^n - \frac{p^n}{e_{X/Y, P}}}^2
	\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{X/Y, P}^{(t+1)} - u_{X/Y, P}^{(t)}}.
	\]
\end{Theorem}
%
Write $H := \ZZ/p^n = \langle \sigma \rangle$.
For any $k[H]$-module $M$ denote:
%
\begin{align*}
	M^{(i)} &:= \ker ((\sigma - 1)^i : M \to M),\\
	T^i M &= T^i_H M := M^{(i)}/M^{(i-1)} \quad \textrm{ for } i = 1, \ldots, p^n.
\end{align*}
%
Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????).
In the inductive step we use also the group $\ZZ/p^{n-1}$. In this case
we denote the irreducible $k[\ZZ/p^{n-1}]$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
and $\mc T^i M := T^i_{\ZZ/p^{n-1}} M$ for any $k[\ZZ/p^{n-1}]$-module $M$.

\begin{Lemma}
	If the $G$-cover $X \to Y$ is \'{e}tale, then the natural map
	%
	\[
		H^1_{dR}(Y) \to H^1_{dR}(X)^G
	\]
	%
	is an isomorphism.
\end{Lemma}
\begin{proof}
	????
\end{proof}
%
\begin{Lemma}
	If the $G$-cover $X \to Y$ is totally ramified, then the map
	%
	\[
	\tr_{X/Y} : H^1_{dR}(X) \to H^1_{dR}(Y)
	\]
	%
	is an epimorphism.
\end{Lemma}
\begin{proof}
	????
\end{proof}
%

\begin{proof}[Proof of Theorem ????]
	We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$.
	Write also $M := H^1_{dR}(X)$.
	By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
	%
	\[
		M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n-m ??} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2
		\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
	\]
	%
	Therefore, for $???$
	%
	\begin{align*}
		\dim_k \mc T^i M =
		\begin{cases}
			???, 
		\end{cases}
	\end{align*}
\end{proof}

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\end{document}