chevalley weil for de Rham

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 $e$.
+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 $e$.
-Then:
 %
-\begin{equation}
+\begin{Proposition} \label{prop:chevalley_weil}
-	H^0(X, \Omega_X) \cong \bigoplus_{M \in \Indec(k[G])} M^{\oplus a_M},
+	Keep the above notation and assume that $p \nmid \# G$. Then:
-\end{equation}
+	%
+	\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{Corollary}
-%
+	Keep the notation of Proposition~\ref{prop:chevalley_weil}. Then:
-\begin{align*}
+	\begin{equation} \label{eqn:cw_dR}
-	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),
+		H^1_{dR}(X) \cong k[G]^{\oplus 2g_X - 2} \oplus k^{\oplus 2}.
-\end{align*}
+	\end{equation}
-%
+	%
-and $N_{P, i}(M)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $M \otimes_{k[G_Q]} \theta_{X/Y, Q}$.
+	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 $X''$.
+	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$ and $X'' := X/\langle \sigma^{p^{n-1}} \rangle$ {\color{red}, see the diagram below.} 
-	{\color{red}
+	
 \[
 	\xymatrix{
 		 & X \ar[rd]^{\langle \sigma^{p^{n-1}} \rangle} \ar[ld]_{\langle \sigma^p \rangle =H'} \ar[dd]^{\pi}&  \\
@@ -500,7 +517,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 		 & Y &
 	}
 \]
-	}
+	{\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}.
+	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 $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
-	We prove this by induction on~$n$. If $n = 0$, then it follows by Chevalley--Weil theorem.
+	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$ 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