X ---> X/C

article_de_rham_cyclic.out

@@ -2,5 +2,5 @@
 \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n\000\040\000a\000n\000d\000\040\000p\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i\000e\000s}{}% 2
 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 3
 \BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000P\000r\000o\000o\000f\000\040\000o\000f\000\040\000M\000a\000i\000n\000\040\000T\000h\000e\000o\000r\000e\000m}{}% 4
-\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000E\000x\000a\000m\000p\000l\000e\000s}{}% 5
+\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000A\000n\000\040\000e\000x\000a\000m\000p\000l\000e\000\040\040\023\000\040\000a\000\040\000s\000u\000p\000e\000r\000e\000l\000l\000i\000p\000t\000i\000c\000\040\000c\000u\000r\000v\000e\000\040\000w\000i\000t\000h\000\040\000a\000\040\000m\000e\000t\000a\000c\000y\000c\000l\000i\000c\000\040\000a\000c\000t\000i\000o\000n}{}% 5
 \BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 6

article_de_rham_cyclic.tex

@@ -834,9 +834,8 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	is determined by higher ramification data as well.
 \end{proof}
 %
-\section{Examples}
-%
-Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order~$p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$
+The method of proof of Main Theorem allows to obtain explicit formulas in the style of the result of Chevalley--Weil for
+particular group. 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}
@@ -864,9 +863,14 @@ $a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$
 	for some $A_W, B_W \in \ZZ$. ??
 \end{proof}
 %
-Let $p > 2$ be a prime and $p \nmid m$ an natural number. Fix a primitive root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$.
+
+\section{An example -- a superelliptic curve with a metacyclic action}
+%
+
+Let $p > 2$ be a prime and $p \nmid m$ an natural number. Let $k$ be an algebraically closed field of characteristic~ $p$.
+Fix a primitive root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$.
 Note that $\zeta^m \in \FF_p$.
-We compute now the equivariant structure of the de Rham cohomology for the superelliptic curve $X$ with the affine part given by:
+In this section we compute the equivariant structure of the de Rham cohomology for the superelliptic curve $X$ with the affine part given by:
 %
 \begin{equation*}
 	y^m = x^{p^n} - x.
@@ -887,24 +891,80 @@ This action is given by:
 	\rho(x, y) &= (\zeta^m \cdot x, \zeta \cdot y).
 \end{align*}
 %
+\begin{Proposition}
+	\[
+		H^1_{dR}(X) \cong ????.
+	\]
+\end{Proposition}
+%
 Note that $X/G \cong \PP^1$ and the quotient map is given by $(x, y) \mapsto (x^p - x)^{p-1}$. Indeed, ????. 
-We claim that the set of branch points is given by $B := \{ 0, \infty \} \cup B'$, where
+We claim that the set of branch points is given by $B := \{ Q_{\infty}, Q_0, Q_1, \ldots, Q_N \}$, where
+$N := \frac{p^{n-1} - 1}{p - 1}$, $Q_0 = 0$, $Q_{\infty} = \infty$ and $Q_1, \ldots, Q_N$ are
+the elements of the set
 %
 \[
-	B' := \{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}.
+	\{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}.
 \]
 %
-The set $B'$ has $\frac{p^{n-1} - 1}{p - 1}$ elements. We claim that:
+Write $C' := \langle \rho^{p-1} \rangle \cong \ZZ/m$ and note that $C'$ is in the center of $G$. We claim that:
 %
 \begin{itemize}
-	\item $G_{Q_0} = C$,
+	\item $G_{Q_0}$ is the conjugacy class of the subgroup $C$,
 	
-	\item $G_Q = \langle \rho^{p-1} \rangle \cong \ZZ/m$ for $Q \in B'$,
+	\item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$,
 	
-	\item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$.
+	
+	\item $G_{Q_i} = C'$ for $i = 1, \ldots, N$.
 \end{itemize}
 %
-Indeed, ????. 
+Indeed, ????. The ramification points of $\pi : X \to X/G$ are as follows:
+%
+\begin{itemize}
+	\item points $P_0^{(1)}, \ldots, P_0^{(p)}$ above $Q_0$
+	\item[] (their stabilizers are subgroups $C_1 = C$, $\ldots$, $C_p$
+	conjugated to $C$),
+	
+	\item point $P_{\infty}$ above $Q_{\infty}$ (its stabilizer is $G$),
+	
+	\item points $P_i^{(1)}, \ldots, P_i^{(p \cdot (p-1))}$ above $Q_i$ for $i = 1, \ldots, N$
+	\item[] (their stabilizers equal $C'$).
+\end{itemize}
+%
+The same points are in the ramification locus of the morphism $X \to X/C$ with the following
+ramification groups:
+%
+\begin{align*}
+	C_{P_0^{(1)}} &= C\\
+	C_{P_0^{(i)}} &= C' \qquad \textrm{ for } i > 1,\\
+	C_{P_{\infty}} &= C\\
+	C_{P_i^{(j)}} &= C' \qquad \textrm{ for } i = 1, \ldots, N, \, j = 1, \ldots, p \cdot (p-1).
+\end{align*}
+
+
+
+
+
+Note that $Y := X/H$ is given by the equation:
+%
+\[
+	y^m = z^{p^{n-1}} + \ldots + z^p + z.
+\]
+%
+Let $\psi : C \to k^{\times}$ be a primitive character. We claim that:
+%
+\begin{align*}
+	a^{dR}_{X, C}(\psi^i) &=
+	\begin{cases}
+		p \cdot N, & \textrm{ if } m \nmid i,\\
+		0, & \textrm{ otherwise. }
+	\end{cases}\\
+	a^{dR}_{Y, C}(\psi^i) &=
+	\begin{cases}
+		\frac{p^{n-1} - 1}{p - 1}, & \textrm{ if } m \nmid i,\\
+		0, & \textrm{ otherwise. }
+	\end{cases}
+\end{align*} 
+%
 
 
 %