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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{Introduction}
%
The classical Chevalley--Weil formula (cf. ????) gives an explicit description
of the equivariant structure of the cohomology of a curve $X$ with a group action over a field of characteristic~$0$. Their formula depends on the so-called \emph{fundamental characters} of points $x \in X$ that are ramified in the cover $X \to X/G$. ????

It is hard to expect such a formula over fields of characteristic~$p$.
Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set of indecomposable $k[G]$-modules is infinite. If, moreover, $p > 2$ then the indecomposable $k[G]$-modules are considered impossible to classify (cf. ???). There are many results concerning equivariant structure of cohomologies for particular curves (cf. ????), groups (cf. ????). Also, one may expect that that (at least in the case of $p$-groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. ???). However, there is no hope of obtaining a result similar to the one of Chevalley and Weil.\\

This brings attention to groups with cyclic $p$-Sylow subgroup. For those, the set of indecomposable modules is finite (cf. ???). Even though for this ?????


The article~\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} shows that in this case,
the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the lower ramification groups and the fundamental characters of the ramification locus. The representation theory of those groups is


\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}
%
Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure
of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data. Indeed, see \cite{garnek_indecomposables} for an example of a family of $\ZZ/p \times \ZZ/p$-covers
with the same ramification data, but different $k[G]$-module structure of $H^0(X, \Omega_X)$ and
$H^1_{dR}(X)$.

\section{Cyclic covers}
%
For any $\ZZ/p^n$-cover $\pi : X \to Y$ and $P \in X(k)$ write $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) for the $t$th ramification jump at $P$.
We use also the convention $u^{(0)}_{X/Y, P} = 1$.
By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $n_{X/Y, P}$
by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$.
We abbreviate the last ramification jump to $u_{X/Y, P}$.
For any $Q \in Y(k)$ we denote also $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
%
\begin{Theorem} \label{thm:cyclic_de_rham}
	Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $m := \max \{ n_{X/Y, P} : P \in X(k) \}$. Pick arbitrary $Q_0 \in Y(k)$ with $n_{X/Y, Q_0} = m$. Then, as a $k[\ZZ/p^n]$-module
	$H^1_{dR}(X)$ is isomorphic to:
	%
	\begin{equation} \label{eqn:HdR_formula}
		J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{Q \in Y(k)\\ Q \neq Q_0}} J_{p^n - p^n/e_{Q}}^2
		\oplus \bigoplus_{Q \in Y(k)} \bigoplus_{t = 0}^{n_{X/Y, Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}},
	\end{equation}
	%
	where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$.
\end{Theorem}
%
\begin{Remark}
	Note that for $g_Y = 0$, ...
\end{Remark}

Write $H := \langle \sigma \rangle \cong \ZZ/p^n$.
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 $H' := \ZZ/p^{n-1}$. In this case
we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\

\noindent Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that:
%
\begin{align*}
	u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\
	l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}.
\end{align*}
%
Assume now that $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$. Let $P' \in X'(k)$ be the image of $P \in X(k)$. Then, if 
$e_{X/Y, P} = p^n$, we have:
%
\begin{align*}
	i_{X/X', P}^{(t)} &=
	\begin{cases}
		i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(N)} \cdot p^N, & t = 0\\
		p^N \cdot i_{X/Y, P}^{(N+t)}, & t = 1, \ldots, n-N-1.
	\end{cases}\\
	i_{X'/Y, P'}^{(t)} &= i_{X/Y, P}^{(t)} \qquad \textrm{ for } t < N.
\end{align*}
If $e_{X/Y, P} \le p^{n - N}$ then $i_{X/Y, P}^{(t)} = i_{X/X', P}^{(t)}$
for all $t$.

%
\begin{Lemma} \label{lem:G_invariants_\'{e}tale}
	If the $G$-cover $X \to Y$ is \'{e}tale, then
	%
	\[
		\dim_k H^1_{dR}(X)^G = 2g_Y.
	\]
	%
\end{Lemma}
\begin{proof}
	Let $\HH^i(Y, \mc F^{\bullet})$ be the $i$th hypercohomology of a complex $\mc F^{\bullet}$.  
	Write also $\mc H^i(G, -)$ for the $i$th derived functor of the functor
	%
	\[
		\mc F \mapsto \mc F^G.
	\]
	%
	Since $X \to Y$ is \'{e}tale, $\mc H^i(G, \pi_* \mc F) = 0$ for any $i > 0$ and any coherent sheaf $\mc F$ on $X$ by \cite[Proposition~2.1]{Garnek_equivariant}.
	Therefore the spectral sequence~\cite[(3.4)]{Garnek_equivariant} applied for the complex $\mc F^{\bullet} := \pi_* \Omega_{X/k}^{\bullet}$ yields $\RR^i \Gamma^G(\pi_* \Omega_{X/k}^{\bullet}) = \HH^1(Y, \pi_*^G \Omega_{X/k}^{\bullet}) = H^1_{dR}(Y)$, since $\pi_*^G \Omega_X^{\bullet} \cong \Omega_Y$ (cf. ???).
	On the other hand, the seven-term exact sequence applied for the spectral sequence~\cite[(3.5)]{Garnek_equivariant} yields:
	%
	\begin{align*}
		0 \to H^1(G, H^0_{dR}(X)^G) \to H^1_{dR}(Y) \to H^1_{dR}(X)^G \to H^2(G, H^0_{dR}(X)^G) \to K,
	\end{align*}
	%
	where:
	%
	\[
		K := \ker(H^2_{dR}(Y) \to H^2_{dR}(X)^G) = \ker(k \stackrel{\id}{\rightarrow} k) = 0.
	\]
	%
	Therefore, since $H^0_{dR}(X)^G \cong k$:
	%
	\begin{align*}
		\dim_k H^1_{dR}(X)^G = \dim_k H^1_{dR}(Y) - \dim_k H^1(G, k) + \dim_k H^2(G, k)\\
		= 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k) ????.
	\end{align*}
\end{proof}
%
\begin{Lemma} \label{lem:trace_surjective}
	Suppose that $G$ is a $p$-group.
	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}
%
By induction, it suffices to prove this in the case when $G = \ZZ/p$.
Consider the following commutative diagram:
%
\begin{center}
	% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKMgEYqtRizYduvbHgJFh5MfWatEIABIA9YgAoAGqQAEAHVMB5ALYwA5nQD6BgJRceIDHIGLSo6qskNHX0ATRNzaztHENcZDz55QWQAJmV-CXUtbWEHYCgAJU5DWPdPfgUUVL9xNTYdHLzCvRi3WXKkgGY0msCs4UNw0ysAY2MLJxK2xKIu6oDM+ubBkbGHFriy6ZQAFm75qVb4rwrkXbmMg84xGChbeCJQADMAJwgrJDIQHAgkZLiXt6-ajfJDbf6vd6IXZfH6IABs4MB8OBsIAHIjIQB2FFIACcGKQAFYcYhMQTEF0YUTyUoqRTyak6ZT9hpzDhnrkDAB6EKcQ4AyHQkGIYk9TJsjnAbm8-kQpBwknYsVsCWcnl8q6cIA
	\begin{tikzcd}
		0 \arrow[r] & {H^0(X, \Omega_X)} \arrow[r] \arrow[d, "\tr_{X/Y}"] & H^1_{dR}(X) \arrow[r] \arrow[d, "\tr_{X/Y}"] & {H^1(X, \mc O_X)} \arrow[r] \arrow[d, "\tr_{X/Y}"] & 0 \\
		0 \arrow[r] & {H^0(Y, \Omega_Y)} \arrow[r]                        & H^1_{dR}(Y) \arrow[r]                        & {H^1(Y, \mc O_Y)} \arrow[r]                        & 0
	\end{tikzcd}
\end{center}
%
where the rows are Hodge--de Rham exact sequences. Recall that by~\cite[Theorem~1]{Valentini_Madan_Automorphisms}, in this case $H^0(X, \Omega_X)$ contains
a copy of $k[G]^{\oplus g_Y}$ as a direct summand. Thus, since trace is injective on $k[G]^{\oplus g_Y}$, the dimension
of the image of
%
\begin{equation} \label{eqn:trace_H0_Omega}
	\tr_{X/Y} : H^0(X, \Omega_X) \to H^0(Y, \Omega_Y)
\end{equation}
%
is $g_Y$. Therefore the map~\eqref{eqn:trace_H0_Omega} is surjective. 
Similarly, by Serre's duality, also $H^1(X, \mc O_X)$ contains $k[G]^{\oplus g_Y}$ as a direct summand
and one shows similarly that the trace map
%
\begin{equation*} %\label{eqn:trace_H0_Omega}
	\tr_{X/Y} : H^1(X, \mc O_X) \to H^1(Y, \mc O_Y)
\end{equation*}
%
is surjective. Therefore, since the outer vertical maps in the diagram are surjective,
the trace map on the de Rham cohomology must be surjective as well.
%
\end{proof}
%
\begin{Lemma} \label{lem:TiM_isomorphism}
	For any $i \le p^n - 1$ we have the following $k$-linear monomorphism:
	%
	\[
		m_{\sigma - 1} : T^{i+1} M \hookrightarrow T^i M.
	\]
\end{Lemma}
\begin{proof}
%
We define $m_{\sigma - 1}$ as follows:
%
\[
	m_{\sigma - 1}(\ol x) := (\sigma - 1) \cdot x,
\]
%
where for $\ol x \in T^{i+1} M$ we picked any representative $x \in M^{(i+1)}$.
Indeed, if $x \in M^{(i+1)}$ then clearly $(\sigma - 1) \cdot x \in M^{(i)}$.
Moreover $(\sigma - 1) \cdot x \in M^{(i-1)}$ holds if and only if $x \in M^{(i)}$. This
shows that $m_{\sigma - 1}$ is well-defined and injective.
\end{proof}
%
\begin{Lemma} \label{lem:lemma_mcT_and_T}
	Let $M$ be a $k[H]$-module. Let $T^i M$ and $\mc T^i M$ be as above.
	If $\dim_k \mc T^i M = \dim_k \mc T^{i+1} M$ for some $i$ then:
	%
	\[
		\dim_k T^{pi + p} M = \dim_k T^{pi + p - 1} M = \ldots = \dim_k T^{pi - p + 1} M.
	\]
\end{Lemma}
\begin{proof}
	Note that $\mc T^i M = M^{(pi)}/M^{(pi - p)}$. This easily implies that:
	%
	\begin{align*}
		\dim_k \mc T^i M &= \dim_k T^{pi} M + \ldots + \dim_k T^{pi - p + 1} M\\
		&\ge \dim_k T^{pi+p} M + \ldots + \dim_k T^{pi+1} M
		= \dim_k \mc T^{i+1} M.
	\end{align*}
	%
	Since the left-hand side and right hand side are equal, we conclude by Lemma~\ref{lem:TiM_isomorphism}
\end{proof}
%
\begin{Lemma} \label{lem:u_equals_ul}
	Assume that $Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
	For any $Q \in Y(k)$:
	%
	\[
	p \cdot (u_{X/Y, Q} - 1) = \sum_{Q'} \left( (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right),
	\]
	%
	where we sum over points $Q' \in Y'(k)$ lying above $Q$.
\end{Lemma}
\begin{proof}
	%
	Consider the following two cases. If $e_{Y'/Y, Q} = 1$ then
	$l^{(1)}_{Y'/Y, Q} = - 1$ and $u_{X/Y, Q} = u_{X/Y', Q'}$ for all $p$ points $Q' \in Y'(k)$ in the preimage of $Q$. This easily implies the desired equality.\\
	If $e_{Y'/Y, Q} = 1$, then there exists a unique point $Q' \in Y'(k)$
	in the preimage of $Q$ through $Y' \to Y$. Moreover, $n_{X/Y, Q} = n_{X/Y', Q'}$. By using ????above formulas???:
	%
	\begin{align*}
		p \cdot (u_{X/Y, Q} - 1) &=
		p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(n_Q - 1)}_{X/Y, Q} - 1)\\
		&= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + \left( (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) + p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \right)\\
		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\
		&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + u_{X/Y', Q'} - 1.
	\end{align*}
	%
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}]
	We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$.
	Let also $\mc M := H^1_{dR}(X)$ and write $\mc M_0$ for the module~\eqref{eqn:HdR_formula}.
	We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$:
	%
	\[
	\mc M \cong \mc J_{p^{n-1}}^{2 p \cdot (g_Y - 1)} \oplus k^{\oplus 2}.
	\]
	%
	Therefore $\dim_k \mc T^2 \mc M = \ldots = \dim_k \mc T^{p^{n-1}} \mc M = 2 p (g_Y - 1)$,
	which by Lemma~\ref{lem:lemma_mcT_and_T} implies that
	%
	\[
		\dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1) = \dim_k T^p \mc M_0.
	\]
	%
	Thus, for $i = 2, \ldots, p$:
	%
	\[
		\dim_k T^i \mc M \ge 2(g_Y - 1) = \dim_k T^{p+1} \mc M.
	\]
	%
	On the other hand, by Lemma~\ref{lem:G_invariants_\'{e}tale} we have
	%
	$
	\dim_k T^1 \mc M = 2 g_Y = \dim_k T^1 \mc M_0
	$. Thus:
	%
	\begin{align*}
		\sum_{i = 2}^p \dim_k T^i \mc M = 2g_X - \dim_k T^1 \mc M - \sum_{i = p+1}^{p^n} \dim_k T^i \mc M = (p-1) \cdot 2(g_Y - 1).
	\end{align*}
	%
	Thus $\dim_k T^i \mc M = 2(g_Y - 1) = \dim_k T^i \mc M_0$ for every $i \ge 2$, which ends the proof in this case.
	
	Assume now that $X \to Y$ is not \'{e}tale. Therefore $X \to X''$ is also not \'{e}tale.
	By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
	%
	\[
		\mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{Q \in Y'(k)\\Q \neq Q_1}} \mc J_{p^{n-1} - p^{n-1}/e'_Q}^2
		\oplus \bigoplus_{Q \in Y'(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', Q}^{(t+1)} - u_{X/Y', Q}^{(t)}}
	\]
	%
	where $e'_Q := e_{X/Y', Q}$ and $Q_1 \in \pi^{-1}(Q_0)$.  Therefore, for $i \le p^{n-1} - p^{n-2}$, using the Riemann--Hurwitz formula (cf. \cite[Corollary~IV.2.4]{Hartshorne1977}) and Lemma~\ref{lem:u_equals_ul}:
	%
	\begin{align*}
		\dim_k \mc T^i \mc M &=
		2(g_{Y'} - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
		&= 2 p (g_Y - 1) + \sum_{Q' \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q'}^{(1)} + 1)\\
		 &+ 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
		&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right),\\
	\end{align*}
	%
	where
	%
	\[ B := \{ Q \in Y(k) : e_Q > 1 \} = \{ Q \in Y(k) : e'_Q > 1 \}. \]
	%
	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} for any $1 \le i \le p^n - p^{n-1}$:
	%
	\begin{align*}
		\dim_k T^i \mc M &= \frac{1}{p} \dim_k \mc T^1 \mc M\\
		&= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1)\\
		&= \dim_k T^i \mc M_0.
	\end{align*}
	%
	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
	in $\FF_p[x]$ we have the identity:
	%
	\[
		1 + x + \ldots + x^{p-1} = (x - 1)^{p-1}.
	\]
	%
	Therefore in the group ring $k[H]$ we have:
	%
	\[
		\tr_{X/X''} = \sum_{j = 0}^{p-1} (\sigma^{p^{n-1}})^j = (\sigma^{p^{n-1}} - 1)^{p-1} = 
		(\sigma - 1)^{p^n - p^{n-1}}.
	\]
	%
	This implies that:
	%
	\[
		\ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})}
	\]
	%
	and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{j + p^n - p^{n-1}} \mc M \to \mc T^j \mc M''$ for any $j \ge 1$.
	Recall the Iverson bracket:
	%
	\[
		\llbracket P \rrbracket = 
		\begin{cases}
			1, & \textrm{ if $P$ is true,}\\
			0, & \textrm{ if $P$ is false,}
		\end{cases}
	\]
	%
	and note that $\dim_k T^i J_l = \llbracket i \le l \rrbracket$ for $i > 0$. Therefore, if $i \in (p^n - p^{N+1}, p^n - p^N]$:
	%
	\begin{align*}
		\dim_k T^i \mc M &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\
		&+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\
		&+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_Q - 1} \llbracket t \ge n_Q + N - n \rrbracket \cdot (u_{Q}^{(t+1)} - u_{Q}^{(t)}).
	\end{align*}
	%
	Suppose now that
	$i = p^n - p^{n-1} + j$, where $j \in (p^{n-1} - p^{N+1}, p^{n-1} - p^N]$. Then, by induction assumption:
	%
	\begin{align*}
		\dim_k T^i \mc M &= \dim_k \mc T^j \mc M'' = 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < (n - 1) - (m - 1) \rrbracket\\
		&+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le (n-1) - (n_{X''/Y, Q}) \}\\
		 &+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_{X''/Y, Q}} \llbracket t \ge n_{X''/Y, Q} + N - (n - 1) \rrbracket \cdot (u_{X''/Y, Q}^{(t+1)} - u_{X''/Y, Q}^{(t)})\\
		 &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\
		 &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\
		 &+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_Q - 1} \llbracket t \ge n_Q + N - n \rrbracket \cdot (u_{Q}^{(t+1)} - u_{Q}^{(t)})\\
		 &= \dim_k T^i \mc M_0.
	\end{align*}
	%
	This ends the proof.
\end{proof}

\section{Hypoelementary covers}
%
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 $M^{\psi} := M \otimes_{k[C]} \psi$.
%
\begin{Proposition} \label{prop:main_thm_for_hypoelementary}
	Main Theorem holds for a group $G$ of the above form and $k = \ol k$.
\end{Proposition}
%
\begin{Lemma} 
	Let $M$ be a $k[G]$-module of finite dimension. The $k[G]$-structure of $M$
	is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$.
\end{Lemma}
\begin{proof}
	This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Recall that if $U$ is an indecomposable $k[G]$-module
	then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is a one-dimensional
	$k[C]$-module. Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$.
	It turns out that the map
	%
	\[
		U \mapsto (\dim_k U, \chi_U)
	\]
	%
	is a bijection between the set of indecomposable $k[G]$-modules and the set $\{ 1, \ldots, p^n - 1 \} \times \wh{C}$. Fix a character $\chi$ that generates $\wh{C}$.
	Write $U_{a, b}$ for the indecomposable $k[G]$-module with socle $\chi^a$
	and dimension $b$. Write
	%
	\[
		M \cong \bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}.
	\]
\end{proof}
%
\begin{Lemma} \label{lem:N+Nchi+...}
	Keep the above notation. Let $M$, $N$ be $k[C]$-modules. Assume that
	%
	\[
		M \cong N \oplus N^{\chi} \oplus \ldots \oplus N^{\chi^{p-1}}.
	\]
	%	
	Then $N$ is uniquely determined by $M$.
	%If $p-1 | j$, then $N_1 \cong N_2^{\chi^i}$ for some $i$.
\end{Lemma}
\begin{proof}
	Note that
	%
	\[
		M \cong N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}.
	\]
	%
	By tensoring this isomorphism by $\chi^i$ we obtain:
	%
	\begin{align*}
		M^{\chi^i} \cong (N^{\chi^i})^{\oplus 2} \oplus N^{\chi^{i+1}} \oplus N^{\chi^{i+2}} \oplus \ldots \oplus N^{\chi^{i + p-2}}
		\cong (N^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j}
	\end{align*}
	%
	for $i = 0, \ldots, p-2$. Therefore:
	%
	\begin{equation} \label{eqn:N+M=M}
		N^{\oplus p} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \oplus 
		\cong M^{\oplus (p-1)}.
	\end{equation}
	%
	Indeed, for the proof of~\eqref{eqn:N+M=M} note that
	%
	\begin{align*}
		N^{\oplus p} &\oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}}
		\cong N^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((N^{\chi^i})^{\oplus 2}
		\oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j} \right)\\
		&\cong \left( N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} \right)^{\oplus (p-1)}
		\cong M^{\oplus (p-1)}.
	\end{align*}
	%
	The isomorphism~\eqref{eqn:N+M=M} clearly proves the thesis.
\end{proof}
%
\begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary}
	For any $i \le p^n - 1$ the map~$m_{\sigma - 1}$ from Lemma~\ref{lem:TiM_isomorphism}
	yields a $k[C]$-equivariant monomorphism:
	%
	\[
	m_{\sigma - 1} : T^{i+1} M \hookrightarrow (T^i M)^{\chi^{-1}}.
	\]
\end{Lemma}
\begin{proof}
	By Lemma~\ref{lem:TiM_isomorphism} this map is injective. Thus it suffices to check that it is $k[C]$-equivariant.
	Note that we have the following identity in the ring~$k[C]$:
	%
	\[
		(\sigma - 1) \cdot \rho = \rho \cdot (\sigma^{\chi(\rho)^{-1}} - 1)
		= \rho \cdot (\sigma - 1) \cdot (1 + \sigma + \sigma^2 + \ldots + \sigma^{\chi(\rho)^{-1} - 1})
	\]
	%
	Note that $\sigma$ acts trivially on $T^i M$, so that for any $\ol x \in T^i M$:
	%
	\[
		(1 + \sigma + \sigma^2 + \ldots + \sigma^{\chi(\rho)^{-1} - 1}) \cdot \ol x = \chi(\rho)^{-1} \cdot \ol x.
	\]
	%
	This easily shows that
	%
	\[
		m_{\sigma - 1}(\rho \cdot \ol x) = \chi(\rho)^{-1} \cdot \rho \cdot m_{\sigma - 1}(\ol x),
	\]
	%
	which ends the 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 $i \ge 2$ by Lemma~\ref{lem:TiM_isomorphism_hypoelementary}:
	%
	\begin{equation} \label{eqn:TiM=T1M_chi_\'{e}tale}
		T^i \mc M \cong (T^2 \mc M)^{\chi^{-i+2}}
	\end{equation}
	%
	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:
	%
	\begin{align}
		\mc T^1 \mc M &\cong T^1 \mc M \oplus T^2 \mc M \oplus (T^2 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p - 2)}} \label{eqn:decomposition_of_mc_T1}\\
		\mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^n - p^{n-1}. \label{eqn:decomposition_of_mc_Ti}
	\end{align}
	%
	Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by ramification data,
	we have by Lemma~\ref{lem:N+Nchi+...} and by~\eqref{eqn:decomposition_of_mc_Ti} that $T^2 \mc M$ is determined by ramification data.
	Moreover, by induction hypothesis and by~\eqref{eqn:decomposition_of_mc_T1}, $T^1 \mc M$
	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:
	%
	\begin{equation} \label{eqn:TiM=T1M_chi}
		T^{i+1} \mc M \cong (T^1 \mc M)^{\chi^{-i}}
	\end{equation}
	%
	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 $i \le p^n - p^{n-1}$:
	%
	\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*}
	%
	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 a similar reasoning, $\tr_{X/X'}$ yields an isomorphism:
	%
	\[
		T^{i + p^n - p^{n-1}} \mc M \cong (\mc T^i \mc M'')^{\chi^{-1??}}.
	\]
	%
	Thus, by induction hypothesis for $\mc M''$, the $k[C]$-structure of $T^{i + p^n - p^{n-1}} \mc M$
	is determined by ramification data as well.
\end{proof}

\section{Proof of Main Theorem}
%
\begin{Lemma}
Suppose $M$ is a finitely generated $k[G]$-module. 	
\begin{enumerate}[leftmargin=*]
	\item The $k[G]$-module structure of $M$
	is uniquely determined by the restrictions $M|_H$ as $H$ ranges over all $p$-hypo-elementary subgroups of $G$.
	
	\item The $k[G]$-module structure of $M$ is uniquely determined by the $\ol k[G]$-module structure of $M \otimes_k \ol k$.
\end{enumerate}	
\end{Lemma}
\begin{proof}
	\begin{enumerate}[leftmargin=*]
		\item This follows easily from Conlon induction theorem (cf. \cite[Theorem~(80.51)]{Curtis_Reiner_Methods_II}), see e.g. \cite[Lemma~3.2]{Bleher_Chinburg_Kontogeorgis_Galois_structure}.
		
		\item This is \cite[Proposition~3.5. (iii)]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
	\end{enumerate}
\end{proof}

(Conlon induction ???) (algebraic closure ???)
\bibliography{bibliografia}
\end{document}