example: y^m = x^p^n - x

article_de_rham_cyclic.tex

@@ -194,8 +194,9 @@ By Hasse--Arf theorem (cf.
 \cite[p. 76]{Serre1979}),
 }
  if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers.
-For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
+For any $Q \in Y(\ol k)$ we denote also by abuse of notation $e_{X/Y, Q} := e_{X/Y, P}$,
-$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
+$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$, $G_Q := G_P$, etc. for arbitrary $P \in \pi^{-1}(Q)$.
+Note that $G_Q$ is well-defined only up to conjugacy.
 Let
 %
 \[
@@ -259,6 +260,12 @@ Finally, we recall the classical Chevalley-Weil formula. For any $e \in \NN$, de
 	\end{align*}
 	%
 \end{Corollary}
+%
+\begin{Remark}
+	??? Note that if $H$ and $H'$ are conjugated subgroups of $G$ then $\dim_k W^H = \dim_k W^{H'}$. Thus the sum
+	in Corollary ??? is well-defined.
+\end{Remark}
+%
 \begin{proof}
 	Note that the category of $k[C]$-modules is semisimple. Hence, by the Hodge--de Rham exact sequence (??recall it earlier??) and Serre's duality (cf. ????):
 	%
@@ -857,17 +864,17 @@ $a_W'$ 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 root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$.
+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 $m \cdot (p-1)$.
 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:
 %
 \begin{equation*}
-	y^m = x^{p^m} - x.
+	y^m = x^{p^n} - x.
 \end{equation*}
 %
-Note that for $m = 2$ this curve was considered e.g. in \cite[Section~4]{Bleher_Wood_polydiffs_structure}.
+Note that for $m = n = 2$ this curve was considered e.g. in \cite[Section~4]{Bleher_Wood_polydiffs_structure}.
-It is a curve of genus $\frac 12 (p^2 - 1) (m-1)$ with an action of the group $G := H \rtimes_{\chi} C$,
+It is a curve of genus $\frac 12 (p^n - 1) (m-1)$ with an action of the group $G := H \rtimes_{\chi} C$,
-where $H := \langle \sigma \rangle \cong \ZZ/p$, $C := \langle \rho \rangle \cong \ZZ/(m \cdot p - m)$ and
+where $H := \langle \sigma \rangle \cong \ZZ/p$, $C := \langle \rho \rangle \cong \ZZ/(m \cdot (p - 1))$ and
 %
 \[
 	\chi : C \to H, \quad \rho \mapsto \sigma^{\zeta^m}.
@@ -880,47 +887,26 @@ This action is given by:
 	\rho(x, y) &= (\zeta^m \cdot x, \zeta \cdot y).
 \end{align*}
 %
-Recall representation theory of $D_{2(p - 1)}$ (cf. \cite[Example~8.2.3]{Steinberg_Representation_book}).
+Note that $X/G \cong \PP^1$ and the quotient map is given by $(x, y) \mapsto (x^p - x)^{p-1}$. Indeed, ????. 
-For $1 \le j \le p-2$ let $\chi_j$ be the character of the representation of $D_{2(p-1)}$
+We claim that the set of branch points is given by $B := \{ 0, \infty \} \cup B'$, where
-induced from
 %
 \[
-\ZZ/(p-1) = \langle \theta \rangle \to \FF_p^{\times}, \quad \theta \mapsto \zeta^j.
+	B' := \{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}.
 \]
 %
-One easily checks that $\chi_j$ is given by the matrices:
+The set $B'$ has $\frac{p^{n-1} - 1}{p - 1}$ elements. We claim that:
 %
-\begin{align*}
+\begin{itemize}
-	\theta \mapsto
+	\item $G_{Q_0} = C$,
-	\left(
+	
-	\begin{matrix}
+	\item $G_Q = \langle \rho^{p-1} \rangle \cong \ZZ/m$ for $Q \in B'$,
-		\zeta^j & 0\\
+	
-		0 & \zeta^{-j}
+	\item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$.
-	\end{matrix}
+\end{itemize}
-	\right),
-	\qquad 
-	s \mapsto
-	\left(
-	\begin{matrix}
-		0 & 1\\
-		1 & 0
-	\end{matrix}
-	\right).
-\end{align*}
 %
-Moreover, $\chi_j$ is irreducible and isomorphic to $\chi_{p - 1 - j}$.
+Indeed, ????. 
-Let also $\chi_0$ be the representation:
+
-%
+
-\[
-D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad
-s \mapsto -1.
-\]
-%
-We claim that as $k[C]$-modules: ??k or $\FF_p$??
-%
-\begin{equation}
-	H^1_{dR}(X) \cong ????.
-\end{equation}
 %