diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 7ca63f7..16ace2c 100644 Binary files a/article_de_rham_cyclic.synctex.gz and b/article_de_rham_cyclic.synctex.gz differ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index e7d600c..f7598a7 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -231,20 +231,37 @@ $k[C]$-module. It turns out that the map is a bijection (cf. \cite[p. 35--37, 42 -- 43]{Alperin_local_rep}). We write $\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$. -Finally, we recall the classical Chevalley-Weil formula. Keep the above notation and assume that $p \nmid \# G$. For any $e \in \NN$, denote by $\chi_e$ the primitive character of a cyclic group of order $e$. -Then: +Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, denote by $\chi_e$ the primitive character of a cyclic group of order $e$. % -\begin{equation} - H^0(X, \Omega_X) \cong \bigoplus_{M \in \Indec(k[G])} M^{\oplus a_M}, -\end{equation} +\begin{Proposition} \label{prop:chevalley_weil} + Keep the above notation and assume that $p \nmid \# G$. Then: + % + \begin{equation} \label{eqn:cw} + H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} W^{\oplus a_W}, + \end{equation} + % + where: + % + \begin{align*} + a_W := (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} \sum_{i = 1}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{Q, i}(W) + \llbracket W \cong k \rrbracket, + \end{align*} + % + and $N_{Q, i}(W)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $W \otimes_{k[G_Q]} \theta_{X/Y, Q}$. +\end{Proposition} % -where: -% -\begin{align*} - a_M := - \dim_k M + \sum_{Q \in Y(k)} \sum_{i = 0}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{P, i}(M), -\end{align*} -% -and $N_{P, i}(M)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $M \otimes_{k[G_Q]} \theta_{X/Y, Q}$. +\begin{Corollary} + Keep the notation of Proposition~\ref{prop:chevalley_weil}. Then: + \begin{equation} \label{eqn:cw_dR} + H^1_{dR}(X) \cong k[G]^{\oplus 2g_X - 2} \oplus k^{\oplus 2}. + \end{equation} + % + where: + % + \begin{align*} + a_W^{dR} := 2 (g_Y - 1) \cdot \dim_k W + \sum_{Q \in Y(k)} (e_{X/Y, Q} - 1) \cdot \dim_k W + 2 \cdot \llbracket W \cong k \rrbracket, + \end{align*} +\end{Corollary} + \section{Cyclic covers} % @@ -263,9 +280,9 @@ and $N_{P, i}(M)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[ \end{Theorem} % \begin{Remark} - 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. + Note that this formula is well-defined for $g_Y = 0$, even though the first exponent is negative. Indeed, 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. \end{Remark} - +% Write $H := \langle \sigma \rangle \cong \ZZ/p^n$. For any $k[H]$-module $M$ denote: % @@ -491,8 +508,8 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \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 $X''$. - {\color{red} + $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$ and $X'' := X/\langle \sigma^{p^{n-1}} \rangle$ {\color{red}, see the diagram below.} + \[ \xymatrix{ & X \ar[rd]^{\langle \sigma^{p^{n-1}} \rangle} \ar[ld]_{\langle \sigma^p \rangle =H'} \ar[dd]^{\pi}& \\ @@ -500,7 +517,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. & Y & } \] - } + {\color{red} Note that $H''$ naturally acts on $X''$ and $X''/H'' \cong Y$.} 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 $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: % @@ -740,8 +757,9 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo \end{proof} \begin{proof}[Proof of Main Theorem] - 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}. - We prove this by induction on~$n$. If $n = 0$, then it follows by Chevalley--Weil theorem. + 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}. {\color{red} Let $Y := X/H$. Similarly as in the proof of Theorem~\ref{thm:cyclic_de_rham}, we write $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, + $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$ and $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Observe that the ramification datum of the covers $X'' \to Y$ and $X \to Y'$ depends only on the ramification data of $X \to Y$. + We prove the result 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. Then by Lemma~\ref{lem:G_invariants_\'{e}tale} and \cite[Corollary~2.4]{Garnek_equivariant} we have $\dim_k H^1_{dR}(X)^H = 2g_Y = \dim_k H^0(X, \Omega_X)^H + \dim_k H^1(X, \mc O_X)^H$. Therefore the Hodge--de Rham exact sequence splits by \cite[Lemma~5.3]{Garnek_equivariant} and % \begin{align*} @@ -782,28 +800,32 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo % \section{Examples} % -Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order $p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$ +Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order~$p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$ for a homomorphism $\chi : C \to \FF_p^{\times}$. % \begin{Proposition} -Keep the above notation. If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then: +Keep the above notation. {\color{red} Assume that $k$ is algebraically closed.} If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then: % \[ - H^1_{dR}(X) \cong \bigoplus_{M \in \Indec(C)} \mc V(M, p-1)^{\oplus b_M} \oplus \mc V(M, p)^{\oplus c_M}, + H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(C)} \mc V(W, p)^{\oplus a_W'} \oplus \mc V(W, p-1)^{\oplus b_W}, \] % -where +where for any $W \in \Indec(k[C])$ the number $a_W$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $X$, +$a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ and % \begin{align*} - b_M &:= (1 - \frac 1p) \cdot \dim_k M + \sum_{Q \in (X/G)(k)} \sum_{i < e_{X/Y, Q}??} + b_W &:= \frac 1p \left( (p-1) \cdot a_W - \sum_{i = 1}^{p-2} a_{W \otimes \chi^i} \right) - a'_{W \otimes \chi}. \end{align*} % - - - $b_M := \ldots$, $c_M := \ldots$. \end{Proposition} \begin{proof} - ??? + Theorem~\ref{thm:cyclic_de_rham} easily implies that + % + \[ + H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(C)} \mc V(W, p)^{\oplus A_W} \oplus \mc V(W, p-1)^{\oplus B_W} + \] + % + for some $A_W, B_W \in \ZZ$. ?? \end{proof} \noindent Let $p > 2$. Consider the Mumford curve