first commit

article_de_rham_cyclic.bbl

@@ -0,0 +1,3 @@
+\begin{thebibliography}{}
+
+\end{thebibliography}

article_de_rham_cyclic.out

@@ -0,0 +1,3 @@
+\BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040}{}% 1
+\BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 2
+\BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 3

article_de_rham_cyclic.tex

@@ -0,0 +1,210 @@
+% !TeX spellcheck = en_GB
+\RequirePackage[l2tabu, orthodox]{nag}
+\documentclass[a4paper,12pt]{amsart}
+%\usepackage[margin=32mm,bottom=40mm]{geometry}
+%\renewcommand{\baselinestretch}{1.1}
+\usepackage{microtype}
+\usepackage[charter]{mathdesign}
+\let\circledS\undefined
+%
+\usepackage[T1]{fontenc}
+\usepackage{tikz, tikz-cd, stmaryrd, amsmath, amsthm, amssymb,
+hyperref, bbm, mathtools, mathrsfs}
+%\usepackage{upgreek}
+\newcommand{\upomega}{\boldsymbol{\omega}}
+\newcommand{\upeta}{\boldsymbol{\eta}}
+\newcommand{\dd}{\boldsymbol{d}}
+\usepackage[shortlabels]{enumitem}
+\usetikzlibrary{arrows}
+\usetikzlibrary{positioning}
+\usepackage[utf8x]{inputenc}
+% \usepackage[MeX]{polski}
+\newcommand{\bb}{\textbf}
+\newcommand{\uu}{\underline}
+\newcommand{\ol}{\overline}
+\newcommand{\mc}{\mathcal}
+\newcommand{\wh}{\widehat}
+\newcommand{\wt}{\widetilde}
+\newcommand{\mf}{\mathfrak}
+\newcommand{\ms}{\mathscr}
+\renewcommand{\AA}{\mathbb{A}}
+\newcommand{\II}{\mathbb{I}}
+\newcommand{\HH}{\mathbb{H}}
+\newcommand{\ZZ}{\mathbb{Z}}
+\newcommand{\CC}{\mathbb{C}}
+\newcommand{\RR}{\mathbb{R}}
+\newcommand{\PP}{\mathbb{P}}
+\newcommand{\QQ}{\mathbb{Q}}
+\newcommand{\LL}{\mathbb{L}}
+\newcommand{\NN}{\mathbb{N}}
+\newcommand{\FF}{\mathbb{F}}
+\newcommand{\VV}{\mathbb{V}}
+\newcommand{\ddeg}{\textbf{deg}\,}
+\DeclareMathOperator{\SSh}{-Sh}
+\DeclareMathOperator{\Ind}{Ind}
+\DeclareMathOperator{\pr}{pr}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Sh}{Sh}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\sgn}{sgn}
+\DeclareMathOperator{\Divv}{Div}
+\DeclareMathOperator{\Coind}{Coind}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\Tot}{Tot}
+\DeclareMathOperator{\Span}{Span}
+\DeclareMathOperator{\res}{res}
+\DeclareMathOperator{\Gl}{Gl}
+\DeclareMathOperator{\Sl}{Sl}
+\DeclareMathOperator{\GCD}{GCD}
+\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\Proj}{Proj}
+\DeclareMathOperator{\Ext}{Ext}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\cha}{char}
+\DeclareMathOperator{\Cl}{Cl}
+\DeclareMathOperator{\Jac}{Jac}
+\DeclareMathOperator{\Lie}{Lie}
+\DeclareMathOperator{\GSp}{GSp}
+\DeclareMathOperator{\Sp}{Sp}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\qlog}{qlog}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\divv}{div}
+\DeclareMathOperator{\mmod}{-mod}
+\DeclareMathOperator{\ev}{ev}
+\DeclareMathOperator{\Indec}{Indec}
+\DeclareMathOperator{\pole}{pole}
+\theoremstyle{plain}
+\newtheorem{Theorem}{Theorem}[section]
+\newtheorem*{mainthm}{Main Theorem}
+\newtheorem{Remark}[Theorem]{Remark}
+\newtheorem{Lemma}[Theorem]{Lemma}
+\newtheorem{Corollary}[Theorem]{Corollary}
+\newtheorem{Conjecture}[Theorem]{Conjecture}
+\newtheorem{Proposition}[Theorem]{Proposition}
+\newtheorem{Setup}[Theorem]{Setup}
+\newtheorem{Example}[Theorem]{Example}
+\newtheorem{manualtheoreminner}{Theorem}
+\newenvironment{manualtheorem}[1]{%
+	\renewcommand\themanualtheoreminner{#1}%
+	\manualtheoreminner
+}{\endmanualtheoreminner}
+\newtheorem{Question}[Theorem]{Question}
+
+\theoremstyle{definition}
+\newtheorem{Definition}[Theorem]{Definition}
+
+%\theoremstyle{remark}
+
+
+
+\renewcommand{\thetable}{\arabic{section}.\arabic{Theorem}}
+
+%\usepackage{refcheck}
+\numberwithin{equation}{section}
+\hyphenation{Woj-ciech}
+%opening
+\begin{document}
+
+\title[The de Rham...]{?? The de Rham cohomology of covers with cyclic $p$-Sylow subgroup}
+\author[A. Kontogeorgis and J. Garnek]{Aristides Kontogeorgis and J\k{e}drzej Garnek}
+\address{???}
+\email{jgarnek@amu.edu.pl}
+\subjclass[2020]{Primary 14G17, Secondary 14H30, 20C20} 
+\keywords{de~Rham cohomology, algebraic curves, group actions,
+	characteristic~$p$}
+\urladdr{http://jgarnek.faculty.wmi.amu.edu.pl/}
+\date{}  
+
+\begin{abstract}
+	????
+\end{abstract}
+
+\maketitle
+\bibliographystyle{plain}
+%
+\section{}
+%
+\section{Cyclic covers}
+%
+Let $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) denote the $t$th upper (resp. lower)
+ramification jump of $X \to Y$ at $P$.
+%
+\begin{Theorem}
+	Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k) \rangle = \ZZ/p^m = G_{P_0}$ for $P_0 \in X(k)$. Then, as $k[\ZZ/p^n]$-modules:
+	%
+	\[
+	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{P \neq P_0} J_{p^n - \frac{p^n}{e_{X/Y, P}}}^2
+	\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{X/Y, P}^{(t+1)} - u_{X/Y, P}^{(t)}}.
+	\]
+\end{Theorem}
+%
+Write $H := \ZZ/p^n = \langle \sigma \rangle$.
+For any $k[H]$-module $M$ denote:
+%
+\begin{align*}
+	M^{(i)} &:= \ker ((\sigma - 1)^i : M \to M),\\
+	T^i M &= T^i_H M := M^{(i)}/M^{(i-1)} \quad \textrm{ for } i = 1, \ldots, p^n.
+\end{align*}
+%
+Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????).
+In the inductive step we use also the group $\ZZ/p^{n-1}$. In this case
+we denote the irreducible $k[\ZZ/p^{n-1}]$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
+and $\mc T^i M := T^i_{\ZZ/p^{n-1}} M$ for any $k[\ZZ/p^{n-1}]$-module $M$.
+
+\begin{Lemma}
+	If the $G$-cover $X \to Y$ is \'{e}tale, then the natural map
+	%
+	\[
+		H^1_{dR}(Y) \to H^1_{dR}(X)^G
+	\]
+	%
+	is an isomorphism.
+\end{Lemma}
+\begin{proof}
+	????
+\end{proof}
+%
+\begin{Lemma}
+	If the $G$-cover $X \to Y$ is totally ramified, then the map
+	%
+	\[
+	\tr_{X/Y} : H^1_{dR}(X) \to H^1_{dR}(Y)
+	\]
+	%
+	is an epimorphism.
+\end{Lemma}
+\begin{proof}
+	????
+\end{proof}
+%
+
+\begin{proof}[Proof of Theorem ????]
+	We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
+	$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$.
+	Write also $M := H^1_{dR}(X)$.
+	By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
+	%
+	\[
+		M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n-m ??} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2
+		\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
+	\]
+	%
+	Therefore, for $???$
+	%
+	\begin{align*}
+		\dim_k \mc T^i M =
+		\begin{cases}
+			???, 
+		\end{cases}
+	\end{align*}
+\end{proof}
+
+\bibliography{bibliografia}
+\end{document}