example - k[C] structure; main thm section instead of hypoel
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@ -103,6 +103,12 @@ J.-P. Serre.
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\newblock Springer-Verlag, New York-Berlin, 1979.
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\newblock Translated from the French by Marvin Jay Greenberg.
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\bibitem{Steinberg_Representation_book}
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B.~Steinberg.
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\newblock {\em Representation theory of finite groups}.
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\newblock Universitext. Springer, New York, 2012.
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\newblock An introductory approach.
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\bibitem{Valentini_Madan_Automorphisms}
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R.~C. Valentini and M.~L. Madan.
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\newblock Automorphisms and holomorphic differentials in characteristic~{$p$}.
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@ -1,6 +1,6 @@
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\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000E\000x\000a\000m\000p\000l\000e\000s}{}% 5
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\BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 6
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@ -167,49 +167,52 @@ of characteristic $p$.
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Throughout the paper we will use the following notation for any $P \in X(\ol k)$:
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\begin{itemize}
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\item $e_{X/Y, P}$ is the ramification index at $P$,
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\item $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) is the $t$th upper (resp. lower) ramification jump
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at $P$,
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\item $m_{X/Y, P} := \ord_p(e_{X/Y, P})$ is the maximal power of~$p$
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dividing the ramification index,
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\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump,
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\item $m_{X/Y} := \max \{ m_{X/Y, P} : P \in X(k) \}$.
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\item $m_{X/Y} := \max \{ m_{X/Y, P} : P \in X(k) \}$,
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\item $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) is the $t$th upper (resp. lower) ramification jump
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at $P$ for $t \ge 1$,
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\item $u^{(0)}_{X/Y, P} := 1$ for any ramified point $P \in X(\ol k)$
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(note that this is not a standard convention),
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\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump.
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\end{itemize}
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%
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For any $Q \in Y(k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
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By Hasse--Arf theorem (cf. ???), if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers.
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For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
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$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
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Let
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%
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\[
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B_{X/Y} := \{ Q \in Y(k) : e_{X/Y, Q} > 1 \}
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B_{X/Y} := \{ Q \in Y(\ol k) : e_{X/Y, Q} > 1 \}
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\]
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%
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be the branch locus of $\pi$.
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be the branch locus of $\pi$. Recall that $\ZZ/p^n$ has $p^n$ indecomposable representations over a field of characteristic~$p$.
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We denote them by $J_1, \ldots, J_{p^n}$. Observe that $J_i$ is given by the Jordan block of size $i$ and eigenvalue $1$.
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%
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\section{Cyclic covers}
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%
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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 $P$.
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We use also the convention $u^{(0)}_{X/Y, P} = 1$.
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By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $m_{X/Y, P}$
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by the equality $e_{X/Y, P} = p^{m_{X/Y, P}}$.
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We abbreviate the last ramification jump to $u_{X/Y, P}$.
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For any $Q \in Y(k)$ we denote also $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
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$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
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%
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\begin{Theorem} \label{thm:cyclic_de_rham}
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Let $k$ be an algebraically closed field of characteristic~$p$.
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Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $m := \max \{ m_{X/Y, P} : P \in X(k) \}$. Pick arbitrary $Q_0 \in Y(k)$ with $m_{X/Y, Q_0} = m$. Then, as a $k[\ZZ/p^n]$-module
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Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Pick arbitrary $Q_0 \in Y(k)$
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with $m_{X/Y, Q_0} = m_{X/Y}$. Then, as a $k[\ZZ/p^n]$-module
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$H^1_{dR}(X)$ is isomorphic to:
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%
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\begin{equation} \label{eqn:HdR_formula}
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J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{Q \in B\\ Q \neq Q_0}} J_{p^n - p^n/e_{Q}}^2
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\oplus \bigoplus_{Q \in B} \bigoplus_{t = 0}^{m_{X/Y, Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}},
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\oplus \bigoplus_{Q \in B} \bigoplus_{t = 0}^{m_{Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}},
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\end{equation}
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%
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where $B := B_{X/Y}$, $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$.
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where $B := B_{X/Y}$, $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$, $m := m_{X/Y, Q}$, $m_Q := m_{X/Y, Q}$.
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\end{Theorem}
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%
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\begin{Remark}
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Note that for $g_Y = 0$, ...
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Note that for $g_Y = 0$ the first exponent is negative. However, since $m_{X/Y} = n$ (as $\PP^1$ doesn't have any \'{e}tale covers), the first two summands in~\eqref{eqn:HdR_formula} cancel out. Thus also in this case the module~\eqref{eqn:HdR_formula} is well-defined.
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\end{Remark}
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Write $H := \langle \sigma \rangle \cong \ZZ/p^n$.
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@ -247,9 +250,10 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.
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If the $G$-cover $X \to Y$ is \'{e}tale, then
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%
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\[
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\dim_k H^1_{dR}(X)^G = 2g_Y.
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\dim_k H^1_{dR}(X)^G = 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k).
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\]
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%
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In particular, if $G \cong \ZZ/p^n$ then $\dim_k H^1_{dR}(X)^G = 2g_Y$.
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\end{Lemma}
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\begin{proof}
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Let $\HH^i(Y, \mc F^{\bullet})$ be the $i$th hypercohomology of a complex $\mc F^{\bullet}$.
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@ -277,8 +281,10 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.
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%
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\begin{align*}
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\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)\\
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= 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k) ????.
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= 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k).
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\end{align*}
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%
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Finally, note that if $G$ is cyclic then $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ by ????.
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\end{proof}
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%
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\begin{Lemma} \label{lem:trace_surjective}
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@ -406,13 +412,14 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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\begin{equation} \label{eqn:Q_in_B'}
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p \cdot (u_{X/Y, Q} - 1) = (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1).
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\end{equation}
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Indeed, by using the above formulas:
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%
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Indeed, using the above formulas:
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%
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\begin{align*}
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p \cdot (u_{X/Y, Q} - 1) &=
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p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(m_Q - 1)}_{X/Y, Q} - 1)\\
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&= (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)\\
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&= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) \\
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&+ p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \\
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&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\
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&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (u_{X/Y', Q'} - 1).
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\end{align*}
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@ -526,17 +533,30 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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This ends the proof.
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\end{proof}
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\section{Hypoelementary covers}
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\section{Proof of Main Theorem}
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%
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Assume now that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$.
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\begin{Lemma} \label{lem:reductions}
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Suppose $M$ is a finitely generated $k[G]$-module.
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\begin{enumerate}[leftmargin=*]
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\item The $k[G]$-module structure of $M$
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is uniquely determined by the restrictions $M|_H$ as $H$ ranges over all $p$-hypo-elementary subgroups of $G$.
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\item The $k[G]$-module structure of $M$ is uniquely determined by the $\ol k[G]$-module structure of $M \otimes_k \ol k$.
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\end{enumerate}
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\end{Lemma}
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\begin{proof}
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\begin{enumerate}[leftmargin=*]
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\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}.
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\item This is \cite[Proposition~3.5. (iii)]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
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\end{enumerate}
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\end{proof}
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%
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By Lemma~\ref{lem:reductions} we may assume that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and that $k$ is algebraically closed.
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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 $M^{\psi} := M \otimes_{k[C]} \psi$.
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%
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\begin{Proposition} \label{prop:main_thm_for_hypoelementary}
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Main Theorem holds for a group $G$ of the above form and $k = \ol k$.
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\end{Proposition}
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%
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\begin{Lemma}
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Let $M$ be a $k[G]$-module of finite dimension. The $k[G]$-structure of $M$
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Let $k$ and $G$ be as above. Assume that $M$ is a $k[G]$-module of finite dimension. The $k[G]$-structure of $M$
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is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$.
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\end{Lemma}
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\begin{proof}
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@ -635,7 +655,8 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
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%
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\end{proof}
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\begin{proof}[Proof of Proposition~\ref{prop:main_thm_for_hypoelementary}]
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\begin{proof}[Proof of Main Theorem]
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As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and $k \ol k$ by Lemma~\ref{lem:reductions}.
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We prove this by induction on $n$. If $n = 0$, then it follows by Chevalley--Weil theorem.
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Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale.
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Recall that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma - 1)$
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@ -685,26 +706,76 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
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Thus, by induction hypothesis for $\mc M''$, the $k[C]$-structure of $T^{i + p^n - p^{n-1}} \mc M$
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is determined by higher ramification data as well.
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\end{proof}
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\section{Proof of Main Theorem}
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%
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\begin{Lemma}
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Suppose $M$ is a finitely generated $k[G]$-module.
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\begin{enumerate}[leftmargin=*]
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\item The $k[G]$-module structure of $M$
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is uniquely determined by the restrictions $M|_H$ as $H$ ranges over all $p$-hypo-elementary subgroups of $G$.
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\item The $k[G]$-module structure of $M$ is uniquely determined by the $\ol k[G]$-module structure of $M \otimes_k \ol k$.
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\end{enumerate}
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\end{Lemma}
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\begin{proof}
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\begin{enumerate}[leftmargin=*]
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\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}.
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\item This is \cite[Proposition~3.5. (iii)]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
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\end{enumerate}
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\end{proof}
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\section{Examples}
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%
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\noindent Let $p > 2$. Consider the Mumford curve
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%
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\[
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X : (x^p - x) \cdot (y^p - y) = 1.
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\]
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%
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It is a curve of genus $(p-1)^2$ and an action of the group $(\ZZ/p \times \ZZ/p) \rtimes D_{2(p-1)}$ given by:
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%
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\begin{align*}
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\sigma_0(x, y) &= (x+1, y),\\
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\sigma_1(x, y) &= (x, y+1),\\
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s(x, y) &= (y, x),\\
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\theta(x, y) &= (\zeta \cdot x, \zeta^{-1} \cdot y) \quad \textrm{ for } \FF_p^{\times} = \langle \zeta \rangle.
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\end{align*}
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%
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Recall representation theory of $D_{2(p - 1)}$ (cf. \cite[Example~8.2.3]{Steinberg_Representation_book}).
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For $1 \le j \le p-2$ let $\chi_j$ be the character of the representation of $D_{2(p-1)}$
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induced from
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%
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\[
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\ZZ/(p-1) = \langle \theta \rangle \to \FF_p^{\times}, \quad \theta \mapsto \zeta^j.
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\]
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%
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One easily checks that $\chi_j$ is given by the matrices:
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%
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\begin{align*}
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\theta \mapsto
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\left(
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\begin{matrix}
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\zeta^j & 0\\
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0 & \zeta^{-j}
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\end{matrix}
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\right),
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\qquad
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s \mapsto
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\left(
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\begin{matrix}
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0 & 1\\
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1 & 0
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\end{matrix}
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\right).
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\end{align*}
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%
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Moreover, $\chi_j$ is irreducible and isomorphic to $\chi_{p - 1 - j}$.
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Let also $\chi_0$ be the representation:
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%
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\[
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D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad
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s \mapsto -1.
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\]
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%
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We claim that as $k[C]$-modules: ??k or $\FF_p$??
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%
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\begin{equation}
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H^1_{dR}(X) \cong V_0^{\oplus (p-1)} \oplus \bigoplus_{j = 1}^{\frac{p-1}{2}} V_j^{\oplus 2(p-1)}.
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\end{equation}
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{\color{gray}
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Basis of holomorphic differentials:
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%
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\[
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\omega_{a, b} = \frac{x^a \cdot y^b \, dx}{(x^p - x)} \qquad 0 \le a, b \le p-2.
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\]
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}
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(Conlon induction ???) (algebraic closure ???)
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\bibliography{bibliografia}
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\end{document}
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