% !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} \label{thm:cyclic_de_rham} 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: % \[ H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{P \neq P_0} J_{p^n - p^n/e_P}^2 \oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{P}^{(t+1)} - u_{P}^{(t)}}, \] % where $e_P := e_{X/Y, P}$ and $u_P^{(t)} := u_{X/Y, P}^{(t)}$. \end{Theorem} % Write $H := \langle \sigma \rangle \cong \ZZ/p^n$. 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 $H' := \ZZ/p^{n-1}$. In this case we denote the irreducible $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-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} \label{lem:G_invariants_\'{e}tale} 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} \label{lem:trace_surjective} 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} \label{lem:TiM_isomorphism} For any $i \le p^n - 1$: % \[ (\sigma - 1) : T^{i+1} M \hookrightarrow T^i M. \] \end{Lemma} \begin{proof} \end{proof} % \begin{Lemma} \label{lem:lemma_mcT_and_T} Let $M$ be a $k[H]$-module. Let $T^i M$ and $\mc T^i M$ be as above. If $\dim_k \mc T^i M = \dim_k \mc T^{i+1} M$ for some $i$ then: % \[ \dim_k T^{pi + p} M = \dim_k T^{pi + p - 1} M = \ldots = \dim_k T^{pi - p + 1} M. \] \end{Lemma} \begin{proof} By Lemma~\ref{lem:TiM_isomorphism}: % \begin{align*} \dim_k \mc T^i M &= \dim_k T^{pi} M + \ldots + \dim_k T^{pi - p + 1} 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~\ref{lem:TiM_isomorphism} \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$. Write also $\mc M := H^1_{dR}(X)$. We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by Lemma~\ref{lem:G_invariants_\'{e}tale} we have % \[ \dim_k T^1 \mc M = 2 g_Y \] % Moreover, by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: % \[ \mc M \cong \mc J_{p^{n-1}}^{2 p \cdot (g_Y - 1)} \oplus k^{\oplus 2}. \] % Therefore $\dim_k \mc T^2 \mc M = \ldots = \dim_k \mc T^{p^{n-1}} \mc M = 2 p (g_Y - 1)$, which implies that % \[ \dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1). \] % by Lemma~\ref{lem:lemma_mcT_and_T}. Thus, for $i = 2, \ldots, p$: % \[ \dim_k T^i \mc M \ge 2(g_Y - 1) = \dim_k T^{p+1} \mc M \] % On the other hand: % \begin{align*} \sum_{i = 2}^p \dim_k T^i \mc M = 2g_X - \dim_k T^1 \mc M - \sum_{i = p+1}^{p^n} \dim_k T^i \mc M = (p-1) \cdot 2(g_Y - 1). \end{align*} % Thus $\dim_k T^i \mc M = 2(g_Y - 1)$ for every $i \ge 2$, which ends the proof in this case. Assume now that $X \to Y$ is not \'{e}tale. Therefore $X \to X''$ is also not \'{e}tale. By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules: % \[ \mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2 \oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}} \] % where $e'_P := e_{X/Y', P}$ and % \[ m' := \begin{cases} n-1, & \textrm{ if } m = n,\\ m, & \textrm{ otherwise.} \end{cases} \] % Therefore, for $i \le p^n - p^{n-1}$ % \begin{align*} \dim_k \mc T^i \mc M = \begin{cases} ???, \end{cases} \end{align*} % In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$. Thus by Lemma~\ref{lem:lemma_mcT_and_T} % \[ \dim_k T^1 \mc M = \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M. \] % By Lemma~\ref{lem:trace_surjective} since $X \to X''$ is not \'{e}tale, the map $\tr_{X/X''} : H^1_{dR}(X) \to H^1_{dR}(X'')$ is surjective. Recall that in $\FF_p[x]$ we have the identity: % \[ 1 + x + \ldots + x^{p-1} = (x - 1)^{p-1}. \] % Therefore 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: % \[ \ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})} \] % and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus: % \[ \dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = .... \] % This ends the proof. \end{proof} \section{Hypoelementary covers} % Assume now that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$. % \begin{Proposition} \label{prop:main_thm_for_hypoelementary} Main Theorem holds for a hypoelementary $G$ as above and $k = \ol k$. \end{Proposition} % \begin{Lemma} Let $M$ be a $k[G]$-module of finite dimension. The $k[G]$-structure of $M$ is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$. \end{Lemma} \begin{proof} ??? \end{proof} % \begin{Lemma} \label{lem:N+Nchi+...} Let $N_1$, $N_2$ be $k[G]$-modules. Assume that for some $j$ % \[ N_1 \oplus N_1^{\chi} \oplus \ldots \oplus N_1^{\chi^j} \cong N_2 \oplus N_2^{\chi} \oplus \ldots \oplus N_2^{\chi^j}. \] % If $\GCD(j, p-1) = 1$, then $N_1 \cong N_2$. If $p-1 | j$, then $N_1 \cong N_2^{\chi^i}$ for some $i$. \end{Lemma} \begin{proof} \end{proof} % \begin{Lemma} \label{lem:chevalley_weil_for_Z/p} If $X$ has a $G$-action and $Y := X/H$, then as $k[C]$-modules: % \[ H^1_{dR}(X) \cong H^1_{dR}(Y) \oplus N^{p-1} \] % for a $k[C]$-module $N$ such that $N^{\chi} \cong N$. \end{Lemma} \begin{proof} ??Chevalley--Weil?? ??is it really needed ?? \end{proof} % \begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary} For any $i \le p^n - 1$: % \[ (\sigma - 1) : T^{i+1} M \hookrightarrow (T^i M)^{\chi^{-1}}. \] \end{Lemma} \begin{proof} \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:main_thm_for_hypoelementary}] We prove this by induction on $n$. If $n = 0$, then it follows by Chevalley--Weil theorem. Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale. Recall that by proof of Theorem~\ref{thm:cyclic_de_rham}, the map $(\sigma - 1)$ is an isomorphism of $k$-vector spaces between $T^{i+1} \mc M$ and $T^i \mc M$ for $i = 2, \ldots, p^n$. This yields an isomorphism of $k[C]$-modules for $i \ge 2$: % \begin{equation} \label{eqn:TiM=T1M_chi_\'{e}tale} T^i \mc M \cong (T^2 \mc M)^{\chi^{-i+2}} \end{equation} % Observe that $\mc T^i \mc M$ has the filtration $\mc M^{(pi)} \supset \mc M^{(pi - 1)} \supset \ldots \supset \mc M^{(pi - p)}$ with subquotients $T^{pi} \mc M, \ldots, T^{pi - p} \mc M$. Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$: % \begin{align*} \mc T^i \mc M &\cong \begin{cases} T^1 \mc M \oplus T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-p + 1}}, & i = 1\\ T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-p}}, & i > 1. \end{cases} \end{align*} % Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by ramification data, we have by Lemma~\ref{lem:N+Nchi+...} that $T^2 \mc M$ is determined by ramification data. Moreover, by Lemma~\ref{lem:G_invariants_\'{e}tale}, $T^1 \mc M \cong H^1_{dR}(X'')$ is also determined by ramification data (???). Assume now that $X \to Y$ is not \'{e}tale. Analogously as in the previous case, Lemma~\ref{lem:TiM_isomorphism_hypoelementary} and proof of Theorem~\ref{thm:cyclic_de_rham} yield an isomorphism of $k[C]$-modules: % \begin{equation} \label{eqn:TiM=T1M_chi} T^{i+1} \mc M \cong (T^1 \mc M)^{\chi^{-i}} \end{equation} % for $i \le p^n - p^{n-1}$. Observe that $\mc T^i M$ has the filtration $\mc M^{(pi)} \supset \mc M^{(pi - 1)} \supset \ldots \supset \mc M^{(pi - p)}$ with subquotients $T^{pi} \mc M, \ldots, T^{pi - p + 1} \mc M$. Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$: % \begin{align*} \mc T^i \mc M &\cong T^{pi - p + 1} \mc M \oplus \ldots \oplus T^{pi} \mc M\\ &\cong T^1 \mc M \oplus (T^1 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^1 \mc M)^{\chi^{-p}}. \end{align*} % By induction assumption, the $k[C]$-module structure of $\mc T^i \mc M$ is uniquely determined by the ramification data. Thus, by Lemma~\ref{lem:N+Nchi+...} for $N := T^1 \mc M$ and by~\eqref{eqn:TiM=T1M_chi} the $k[C]$-structure of the modules $T^i \mc M$ is uniquely determined by the ramification data for $i \le p^n - p^{n-1}$. By similar reasoning, $\tr_{X/X'}$ yields an isomorphism: % \[ T^{i + p^n - p^{n-1}} \mc M \cong (\mc T^i \mc M'')^{\chi^{-1??}}. \] % Thus, by induction hypothesis for $\mc M''$, the $k[C]$-structure of $T^{i + p^n - p^{n-1}} \mc M$ is determined by ramification data as well. \end{proof} \section{Proof of Main Theorem} % (Conlon induction ???) (algebraic closure ???) \bibliography{bibliografia} \end{document}