explicit formula for sylow subgroup Zp

article_de_rham_cyclic.tex

@@ -187,7 +187,10 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$
 	\item $u^{(0)}_{X/Y, P} := 1$ for any ramified point $P \in X(\ol k)$
 	(note that this is not a standard convention),
 	
-	\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump.
+	\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump,
+	
+	\item $\theta_{X/Y, P} : G_P \to \Aut_k(\mf m_P/\mf m_P^2) \cong k^{\times}$
+	is the fundamental character of~$P$.
 \end{itemize}
 %
 By Hasse--Arf theorem (cf. 
@@ -221,14 +224,15 @@ then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is an indecomposable
 $k[C]$-module. It turns out that the map
 %
 \begin{align*}
-	\Indec(k[G]) \to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
+	\Indec(k[G]) &\to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
-	U \mapsto \left(U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}} \right)
+	U &\mapsto \left(U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}} \right)
 \end{align*}
 %
 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 $Q \in Y(k)$ let $\chi_Q : G_Q \to k^{\times}$ be the fundamental character of $G_Q$ acting on the tangent space of $Q$. Then:
+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:
 %
 \begin{equation}
 	H^0(X, \Omega_X) \cong \bigoplus_{M \in \Indec(k[G])} M^{\oplus a_M},
@@ -237,10 +241,10 @@ Finally, we recall the classical Chevalley-Weil formula. Keep the above notation
 where:
 %
 \begin{align*}
-	a_M := - \dim_k M + \sum_{Q \in Y(k)} \sum_{i = 0}^{e_{X/Y, Q} - 1} \left\langle \frac{-i}{e_{X/Y, Q}} \right\rangle \cdot N_{P, i}(M),
+	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 $N_{P, i}(M) := ???$.
+and $N_{P, i}(M)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $M \otimes_{k[G_Q]} \theta_{X/Y, Q}$.
 
 \section{Cyclic covers}
 %
@@ -788,7 +792,15 @@ Keep the above notation. If $G$ acts on a curve $X$ and the cover $X \to X/H$ is
 	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},
 \]	
 %
-where $b_M := \ldots$, $c_M := \ldots$.
+where
+%
+\begin{align*}
+	b_M &:= (1  - \frac 1p) \cdot \dim_k M + \sum_{Q \in (X/G)(k)} \sum_{i < e_{X/Y, Q}??} 
+\end{align*}
+%
+
+
+ $b_M := \ldots$, $c_M := \ldots$.
 \end{Proposition}
 \begin{proof}
 	???