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zo?V{SVDN7*l|@|b7(V-TS%VL{cs(zrl<6CGqV(?Oct7LW^@*)gc-RTXHsbmkbF(Vz_;_B)IKi}aK_SGBP*fY)*YFhGJ{{!no(3}7O diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 0dbf6e0..b67c239 100644 --- a/article_de_rham_cyclic.tex +++ b/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 $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: %