commit 2a54cefac7c8583ca904c98cb1149cb2d7bfb445 Author: jgarnek Date: Thu Oct 17 13:22:42 2024 +0200 first commit diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl new file mode 100644 index 0000000..9b431fa --- /dev/null +++ b/article_de_rham_cyclic.bbl @@ -0,0 +1,3 @@ +\begin{thebibliography}{} + +\end{thebibliography} diff --git a/article_de_rham_cyclic.out b/article_de_rham_cyclic.out new file mode 100644 index 0000000..32e2153 --- /dev/null +++ b/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 diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz new file mode 100644 index 0000000..b4404f3 Binary files /dev/null and b/article_de_rham_cyclic.synctex.gz differ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex new file mode 100644 index 0000000..2e1b6bf --- /dev/null +++ b/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} \ No newline at end of file