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of $G$ and write $Y := X/H$. For any $k[C]$-mo \end{proof} % \begin{Lemma} \label{lem:N+Nchi+...} - Keep the above notation. Let $M$, $N$ be $k[C]$-modules. Assume that + Keep the above notation. Let $M$, $N$ be $k[C]$-modules. + \begin{enumerate}[(1)] + \item If $N \cong M \oplus M^{\chi} \oplus \ldots \oplus M^{\chi^{p-1}},$ + then $N$ is uniquely determined by $M$. + + \item If $N^{\oplus (p-1)} \cong M \oplus M^{\chi} \oplus \ldots \oplus M^{\chi^{p-2}}$, + then $M \cong N \cong N^{\chi}$. + \end{enumerate} % - \[ - M \cong N \oplus N^{\chi} \oplus \ldots \oplus N^{\chi^{p-1}}. - \] - % - 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 + (1) Note that % \[ - M \cong N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}. + N \cong M^{\oplus 2} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}}. \] % 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} + N^{\chi^i} \cong (M^{\chi^i})^{\oplus 2} \oplus M^{\chi^{i+1}} \oplus M^{\chi^{i+2}} \oplus \ldots \oplus M^{\chi^{i + p-2}} + \cong (M^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} M^{\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)}. + \begin{equation} \label{eqn:M+N=N} + M^{\oplus p} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} + \cong N^{\oplus (p-1)}. \end{equation} % - Indeed, for the proof of~\eqref{eqn:N+M=M} note that + Indeed, for the proof of~\eqref{eqn:M+N=N} 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)}. + M^{\oplus p} &\oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} + \cong M^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((M^{\chi^i})^{\oplus 2} + \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} M^{\chi^j} \right)\\ + &\cong \left( M^{\oplus 2} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \right)^{\oplus (p-1)} + \cong N^{\oplus (p-1)}. \end{align*} % - The isomorphism~\eqref{eqn:N+M=M} clearly proves the thesis. + The isomorphism~\eqref{eqn:M+N=N} clearly proves the thesis.\\ + (2) \end{proof} % \begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary} @@ -695,17 +699,15 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo 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. Note that the proof of Lemma~\ref{lem:G_invariants_\'{e}tale} implies - that there exists an exact sequence: + that there exists an exact sequence of $k[C]$-modules: % \[ - 0 \to k \to H^1_{dR}(Y) \to T^1 \mc M \to k \to 0. + 0 \to k \to H^1_{dR}(Y) \to T^1 \mc M \to k \to 0, \] % - Therefore, since the category of $k[C]$-modules is semisimple, $T^1 \mc M \cong H^1_{dR}(Y)$. By induction hypothesis we conclude that + where $k$ denotes the trivial $k[C]$-module. Therefore, since the category of $k[C]$-modules is semisimple, $T^1 \mc M \cong H^1_{dR}(Y)$. By induction hypothesis we conclude that $T^1 \mc M$ is determined by higher ramification data. - - - Recall that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma - 1)$ + Recall now 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}: % @@ -718,14 +720,19 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo % \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-1}. ???? \label{eqn:decomposition_of_mc_Ti} + \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-1}. \label{eqn:decomposition_of_mc_Ti} \end{align} % - Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by higher 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 higher ramification data. - Moreover, by induction hypothesis and by~\eqref{eqn:decomposition_of_mc_T1}, $T^1 \mc M$ - is also determined by higher ramification data. - + If $n \ge 2$, then by induction hypothesis, Lemma~\ref{lem:N+Nchi+...} and by~\eqref{eqn:decomposition_of_mc_Ti} the $k[C]$-module $T^2 \mc M$ is determined by higher ramification data. If $n = 1$, then the $k[C]$-module + % + \[ + \mc T^1 \mc M/T^1 \mc M = H^1_{dR}(X)/H^1_{dR}(Y) + \] + % + is of the form $N^{\oplus (p-1)}$ for a $k[C]$-module $N$ + by Lemma~????. Therefore by Lemma~????? we have $T^2 \mc M \cong N$. + This ends the proof, as $N$ is determined by the fundamental characters of the group action of $C$ on $X$.\\ + 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: %