cw formula begin 3

article_de_rham_cyclic.tex

@@ -10,6 +10,7 @@
 \usepackage[T1]{fontenc}
 \usepackage{tikz, tikz-cd, stmaryrd, amsmath, amsthm, amssymb,
 hyperref, bbm, mathtools, mathrsfs}
+\usepackage[all]{xy}
 %\usepackage{upgreek}
 \newcommand{\upomega}{\boldsymbol{\omega}}
 \newcommand{\upeta}{\boldsymbol{\eta}}
@@ -189,7 +190,11 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$
 	\item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump.
 \end{itemize}
 %
-By Hasse--Arf theorem (cf. ???), if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers.
+By Hasse--Arf theorem (cf. 
+{\color{red}
+\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 $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
 $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
 Let
@@ -422,7 +427,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
 \end{proof}
 %
 \begin{Lemma} \label{lem:u_equals_ul}
-	Assume that $Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
+	Assume that $ {\color{red} \phi:} Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$.
 	Then:
 	%
 	\[
@@ -483,6 +488,15 @@ 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}
+\[
+	\xymatrix{
+		 & X \ar[rd] \ar[ld] \ar[dd]^{\pi}&  \\
+		Y'  \ar[rd]^{\phi} &   &  X'' \ar[ld]\\ 
+		 & Y &
+	}
+\]
+	}
 	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 $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$:
 	%