302 lines
8.8 KiB
TeX
302 lines
8.8 KiB
TeX
% !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{Introduction}
|
|
%
|
|
\begin{mainthm}
|
|
Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
|
|
Let $X$ be a curve with an action of~$G$ over a field $k$ of characteristic $p$.
|
|
The $k[G]$-module structure of $H^1_{dR}(X)$ is uniquely determined by the lower ramification groups and the fundamental characters of closed
|
|
points $x$ of $X$ that are ramified in the cover $X \to X/G$.
|
|
\end{mainthm}
|
|
|
|
\section{Cyclic covers}
|
|
%
|
|
Let for any $\ZZ/p^n$-cover $X \to Y$
|
|
%
|
|
\begin{align*}
|
|
u_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_P^{(t)} \cong \ZZ/p^{n-t???} \},\\
|
|
l_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_{P, t} \cong \ZZ/p^{n-t???} \}.
|
|
\end{align*}
|
|
%
|
|
Note that if $G_P = \ZZ/p^n$, this coincides with the standard definition of
|
|
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 $M$.
|
|
|
|
Note also that for $j \ge 1$:
|
|
%
|
|
\[
|
|
l_{X/Y, P}^{(j)} - l_{X/Y, P}^{(j-1)} = \frac{1}{p^{j-1}} (u_{X/Y, P}^{(j)} - u^{(j-1)}_{X/Y, P})
|
|
\]
|
|
%
|
|
(in particular, $u_{X/Y, P}^{(1)} = l_{X/Y, P}^{(1)}$). Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ then:
|
|
%
|
|
\begin{itemize}
|
|
\item $u_{X'/Y, P}^{(t)} = u_{X'/Y, P}^{(t)}$ for $t \le N$,
|
|
|
|
\item $l_{X/X', P}^{(t)} = l_{X/X', P}^{(t + N)}$ for $t \le n-N$.
|
|
\end{itemize}
|
|
|
|
|
|
\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{Lemma}
|
|
For any $i \le p^n - 1$:
|
|
%
|
|
\[
|
|
(\sigma - 1) : T^{i+1} M \hookrightarrow T^i M.
|
|
\]
|
|
\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:
|
|
%
|
|
\[
|
|
M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2
|
|
\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
|
|
\]
|
|
%
|
|
where
|
|
%
|
|
\[
|
|
m' :=
|
|
\begin{cases}
|
|
n-1, & \textrm{ if } m = n,\\
|
|
m, & \textrm{ otherwise.}
|
|
\end{cases}
|
|
\]
|
|
|
|
Therefore, for $???$
|
|
%
|
|
\begin{align*}
|
|
\dim_k \mc T^i M =
|
|
\begin{cases}
|
|
???,
|
|
\end{cases}
|
|
\end{align*}
|
|
%
|
|
In particular, $\dim_k \mc T^1 M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} M$.
|
|
On the other hand, by Lemma ??:
|
|
%
|
|
\begin{align*}
|
|
\dim_k \mc T^1 M &= \dim_k T^1 M + \ldots + \dim_k T^p M\\
|
|
&\ge \dim_k T^{p^n - p^{n-1}} M + \ldots + \dim_k T^{p^n - p^{n-1}} M
|
|
= \dim_k \mc T^{p^{n-1} - p^{n-2}} M.
|
|
\end{align*}
|
|
%
|
|
Since the left-hand side and right hand side are equal, we conclude by Lemma ???
|
|
that
|
|
%
|
|
\[
|
|
\dim_k T^1 M = \ldots = \dim_k T^{p^n - p^{n-1}} M = \frac{1}{p} \dim_k \mc T^1 M.
|
|
\]
|
|
%
|
|
If the cover $X \to X''$ is \'{e}tale, then the cover $X \to Y$ must be also \'{e}tale.
|
|
Thus the proof follows in this case by~\cite{Nakajima??Inventiones}. Suppose now that
|
|
$X \to X''$ is not \'{e}tale. Then, by Lemma ???, the map $\tr_{X/X''} : H^1_{dR}(X) \to H^1_{dR}(X'')$ is surjective. Moreover, note that in the group ring $k[H]$ we have:
|
|
%
|
|
\[
|
|
\tr_{X/X''} = \sum_{j = 0}^{p-1} (\sigma^{p^{n-1}})^j = (\sigma^{p^{n-1}} - 1)^{p-1} =
|
|
(\sigma - 1)^{p^n - p^{n-1}}.
|
|
\]
|
|
%
|
|
This implies that:
|
|
%
|
|
\[
|
|
\ker(\tr_{X/X''} : M \to M'') = M^{(p^n - p^{n-1})}
|
|
\]
|
|
%
|
|
and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} M \to \mc T^i M''$ for any $i \ge 1$. Thus:
|
|
%
|
|
\[
|
|
\dim_k T^{i + p^n - p^{n-1}} M = \dim_k \mc T^i M'' = ....
|
|
\]
|
|
%
|
|
This ends the proof.
|
|
\end{proof}
|
|
|
|
\section{Hypoelementary covers}
|
|
%
|
|
Assume now that $G = H \rtimes_{\chi} \ZZ/??n$.
|
|
|
|
\bibliography{bibliografia}
|
|
\end{document} |