% !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} \newcommand{\red}[1]{{\color{red}#1}} \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} % The classical Chevalley--Weil formula (cf. ????) gives an explicit description of the equivariant structure of the cohomology of a curve $X$ with a group action over a field of characteristic~$0$. Their formula depends on the so-called \emph{fundamental characters} of points $x \in X$ that are ramified in the cover $X \to X/G$. ???? It is hard to expect such a formula over fields of characteristic~$p$. Indeed, if $G$ is a finite group with a non-cyclic $p$-Sylow subgroup, the set of indecomposable $k[G]$-modules is infinite. If, moreover, $p > 2$ then the indecomposable $k[G]$-modules are considered impossible to classify (cf. \cite{Prest}). There are many results concerning equivariant structure of cohomologies for particular groups (see e.g.~\cite{Valentini_Madan_Automorphisms} for the case of cyclic groups, \cite{WardMarques_HoloDiffs} for abelian groups, \cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} for groups with a cyclic Sylow subgroup, or \cite{Bleher_Camacho_Holomorphic_differentials} for the Klein group) or curves (cf. \cite{Lusztig_Coxeter_orbits}, \cite{Dummigan_99}, \cite{Gross_Rigid_local_systems_Gm}, \cite{laurent_kock_drinfeld}). Also, one may expect that that (at least in the case of $p$-groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. \cite{Garnek_p_gp_covers}, \cite{Garnek_p_gp_covers_ii}). However, there is no hope of obtaining a result similar to the one of Chevalley and Weil.\\ This brings attention to groups with cyclic $p$-Sylow subgroup. For those, the set of indecomposable modules is finite (cf. \cite{Higman}, \cite{Borevic_Faddeev}, \cite{Heller_Reiner_Reps_in_integers_I}). While their representation theory seems too complicated to derive a formula for the cohomologies, the article~\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} proved that the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data (i.e. higher ramification groups and the fundamental characters of the ramification locus). The main result of this article is a similar statement for the de Rham cohomology. % \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 higher ramification data of the cover $X \to X/G$. \end{mainthm} % Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure of $H^1_{dR}(X)$ isn't determined uniquely by the higher ramification data. Indeed, see \cite{garnek_indecomposables} for a construction of $G$-covers with the same higher ramification data, but different $k[G]$-module structure of $H^0(X, \Omega_X)$ and $H^1_{dR}(X)$. \section{Cyclic covers} % For any $\ZZ/p^n$-cover $\pi : X \to Y$ and $P \in X(k)$ write $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) for the $t$th ramification jump at $P$. We use also the convention $u^{(0)}_{X/Y, P} = 1$. By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $n_{X/Y, P}$ by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$. We abbreviate the last ramification jump to $u_{X/Y, P}$. For any $Q \in Y(k)$ we denote also $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$, $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$. % \begin{Theorem} \label{thm:cyclic_de_rham} Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $m := \max \{ n_{X/Y, P} : P \in X(k) \}$. Pick arbitrary $Q_0 \in Y(k)$ with $n_{X/Y, Q_0} = m$. Then, as a $k[\ZZ/p^n]$-module $H^1_{dR}(X)$ is isomorphic to: % \begin{equation} \label{eqn:HdR_formula} J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{Q \in Y(k)\\ Q \neq Q_0}} J_{p^n - p^n/e_{Q}}^2 \oplus \bigoplus_{Q \in Y(k)} \bigoplus_{t = 0}^{n_{X/Y, Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}}, \end{equation} % where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$. \end{Theorem} % \begin{Remark} Note that for $g_Y = 0$, ... \end{Remark} 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 indecomposable $k[H']$-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 $M$.\\ \noindent Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that: % \begin{align*} u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\ l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}. \end{align*} % Assume now that $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$. Let $P' \in X'(k)$ be the image of $P \in X(k)$. Then, if $e_{X/Y, P} = p^n$, we have: % \begin{align*} i_{X/X', P}^{(t)} &= \begin{cases} i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(N)} \cdot p^N, & t = 0\\ p^N \cdot i_{X/Y, P}^{(N+t)}, & t = 1, \ldots, n-N-1. \end{cases}\\ i_{X'/Y, P'}^{(t)} &= i_{X/Y, P}^{(t)} \qquad \textrm{ for } t < N. \end{align*} If $e_{X/Y, P} \le p^{n - N}$ then $i_{X/Y, P}^{(t)} = i_{X/X', P}^{(t)}$ for all $t$. % \begin{Lemma} \label{lem:G_invariants_\'{e}tale} If the $G$-cover $X \to Y$ is \'{e}tale, then % \[ \dim_k H^1_{dR}(X)^G = 2g_Y. \] % \end{Lemma} \begin{proof} Let $\HH^i(Y, \mc F^{\bullet})$ be the $i$th hypercohomology of a complex $\mc F^{\bullet}$. Write also $\mc H^i(G, -)$ for the $i$th derived functor of the functor % \[ \mc F \mapsto \mc F^G. \] % Since $X \to Y$ is \'{e}tale, $\mc H^i(G, \pi_* \mc F) = 0$ for any $i > 0$ and any coherent sheaf $\mc F$ on $X$ by \cite[Proposition~2.1]{Garnek_equivariant}. Therefore the spectral sequence~\cite[(3.4)]{Garnek_equivariant} applied for the complex $\mc F^{\bullet} := \pi_* \Omega_{X/k}^{\bullet}$ yields $\RR^i \Gamma^G(\pi_* \Omega_{X/k}^{\bullet}) = \HH^1(Y, \pi_*^G \Omega_{X/k}^{\bullet}) = H^1_{dR}(Y)$, since $\pi_*^G \Omega_X^{\bullet} \cong \Omega_Y$ (cf. ???). On the other hand, the seven-term exact sequence applied for the spectral sequence~\cite[(3.5)]{Garnek_equivariant} yields: % \begin{align*} 0 \to H^1(G, H^0_{dR}(X)^G) \to H^1_{dR}(Y) \to H^1_{dR}(X)^G \to H^2(G, H^0_{dR}(X)^G) \to K, \end{align*} % where: % \[ K := \ker(H^2_{dR}(Y) \to H^2_{dR}(X)^G) = \ker(k \stackrel{\id}{\rightarrow} k) = 0. \] % Therefore, since $H^0_{dR}(X)^G \cong k$: % \begin{align*} \dim_k H^1_{dR}(X)^G = \dim_k H^1_{dR}(Y) - \dim_k H^1(G, k) + \dim_k H^2(G, k)\\ = 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k) ????. \end{align*} \end{proof} % \begin{Lemma} \label{lem:trace_surjective} Suppose that $G$ is a $p$-group. 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} % By induction, it suffices to prove this in the case when $G = \ZZ/p$. Consider the following commutative diagram: % \begin{center} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKMgEYqtRizYduvbHgJFh5MfWatEIABIA9YgAoAGqQAEAHVMB5ALYwA5nQD6BgJRceIDHIGLSo6qskNHX0ATRNzaztHENcZDz55QWQAJmV-CXUtbWEHYCgAJU5DWPdPfgUUVL9xNTYdHLzCvRi3WXKkgGY0msCs4UNw0ysAY2MLJxK2xKIu6oDM+ubBkbGHFriy6ZQAFm75qVb4rwrkXbmMg84xGChbeCJQADMAJwgrJDIQHAgkZLiXt6-ajfJDbf6vd6IXZfH6IABs4MB8OBsIAHIjIQB2FFIACcGKQAFYcYhMQTEF0YUTyUoqRTyak6ZT9hpzDhnrkDAB6EKcQ4AyHQkGIYk9TJsjnAbm8-kQpBwknYsVsCWcnl8q6cIA \begin{tikzcd} 0 \arrow[r] & {H^0(X, \Omega_X)} \arrow[r] \arrow[d, "\tr_{X/Y}"] & H^1_{dR}(X) \arrow[r] \arrow[d, "\tr_{X/Y}"] & {H^1(X, \mc O_X)} \arrow[r] \arrow[d, "\tr_{X/Y}"] & 0 \\ 0 \arrow[r] & {H^0(Y, \Omega_Y)} \arrow[r] & H^1_{dR}(Y) \arrow[r] & {H^1(Y, \mc O_Y)} \arrow[r] & 0 \end{tikzcd} \end{center} % where the rows are Hodge--de Rham exact sequences. Recall that by~\cite[Theorem~1]{Valentini_Madan_Automorphisms}, in this case $H^0(X, \Omega_X)$ contains a copy of $k[G]^{\oplus g_Y}$ as a direct summand. Thus, since trace is injective on $k[G]^{\oplus g_Y}$, the dimension of the image of % \begin{equation} \label{eqn:trace_H0_Omega} \tr_{X/Y} : H^0(X, \Omega_X) \to H^0(Y, \Omega_Y) \end{equation} % is $g_Y$. Therefore the map~\eqref{eqn:trace_H0_Omega} is surjective. Similarly, by Serre's duality, also $H^1(X, \mc O_X)$ contains $k[G]^{\oplus g_Y}$ as a direct summand and one shows similarly that the trace map % \begin{equation*} %\label{eqn:trace_H0_Omega} \tr_{X/Y} : H^1(X, \mc O_X) \to H^1(Y, \mc O_Y) \end{equation*} % is surjective. Therefore, since the outer vertical maps in the diagram are surjective, the trace map on the de Rham cohomology must be surjective as well. % \end{proof} % \begin{Lemma} \label{lem:TiM_isomorphism} For any $i \le p^n - 1$ we have the following $k$-linear monomorphism: % \[ m_{\sigma - 1} : T^{i+1} M \hookrightarrow T^i M. \] \end{Lemma} \begin{proof} % We define $m_{\sigma - 1}$ as follows: % \[ m_{\sigma - 1}(\ol x) := (\sigma - 1) \cdot x, \] % where for $\ol x \in T^{i+1} M$ we picked any representative $x \in M^{(i+1)}$. Indeed, if $x \in M^{(i+1)}$ then clearly $(\sigma - 1) \cdot x \in M^{(i)}$. Moreover $(\sigma - 1) \cdot x \in M^{(i-1)}$ holds if and only if $x \in M^{(i)}$. This shows that $m_{\sigma - 1}$ is well-defined and injective. \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} Note that $\mc T^i M = M^{(pi)}/M^{(pi - p)}$. This easily implies that: % \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^{pi+p} M + \ldots + \dim_k T^{pi+1} M = \dim_k \mc T^{i+1} 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{Lemma} \label{lem:u_equals_ul} Assume that $Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$. For any $Q \in Y(k)$: % \[ p \cdot (u_{X/Y, Q} - 1) = \sum_{Q'} \left( (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right), \] % where we sum over points $Q' \in Y'(k)$ lying above $Q$. \end{Lemma} \begin{proof} % Consider the following two cases. If $e_{Y'/Y, Q} = 1$ then $l^{(1)}_{Y'/Y, Q} = - 1$ and $u_{X/Y, Q} = u_{X/Y', Q'}$ for all $p$ points $Q' \in Y'(k)$ in the preimage of $Q$. This easily implies the desired equality.\\ If $e_{Y'/Y, Q} = 1$, then there exists a unique point $Q' \in Y'(k)$ in the preimage of $Q$ through $Y' \to Y$. Moreover, $n_{X/Y, Q} = n_{X/Y', Q'}$. By using ????above formulas???: % \begin{align*} p \cdot (u_{X/Y, Q} - 1) &= p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(n_Q - 1)}_{X/Y, Q} - 1)\\ &= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + \left( (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) + p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \right)\\ &= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\ &= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + u_{X/Y', Q'} - 1. \end{align*} % \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$. 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 $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: % \[ \mc M \cong \mc J_{p^{n-1}}^{2 p \cdot (g_Y - 1)} \oplus k^{\oplus 2}. \] % Therefore $\dim_k \mc T^2 \mc M = \ldots = \dim_k \mc T^{p^{n-1}} \mc M = 2 p (g_Y - 1)$, which by Lemma~\ref{lem:lemma_mcT_and_T} implies that % \[ \dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1) = \dim_k T^p \mc M_0. \] % 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, by Lemma~\ref{lem:G_invariants_\'{e}tale} we have % $ \dim_k T^1 \mc M = 2 g_Y = \dim_k T^1 \mc M_0 $. Thus: % \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 $\dim_k T^i \mc M = 2(g_Y - 1) = \dim_k T^i \mc M_0$ 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: % \[ \mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{Q \in Y'(k)\\Q \neq Q_1}} \mc J_{p^{n-1} - p^{n-1}/e'_Q}^2 \oplus \bigoplus_{Q \in Y'(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', Q}^{(t+1)} - u_{X/Y', Q}^{(t)}} \] % where $e'_Q := e_{X/Y', Q}$ and $Q_1 \in \pi^{-1}(Q_0)$. Therefore, for $i \le p^{n-1} - p^{n-2}$, using the Riemann--Hurwitz formula (cf. \cite[Corollary~IV.2.4]{Hartshorne1977}) and Lemma~\ref{lem:u_equals_ul}: % \begin{align*} \dim_k \mc T^i \mc M &= 2(g_{Y'} - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ &= 2 p (g_Y - 1) + \sum_{Q' \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q'}^{(1)} + 1)\\ &+ 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right),\\ \end{align*} % where % \[ B := \{ Q \in Y(k) : e_Q > 1 \} = \{ Q \in Y(k) : e'_Q > 1 \}. \] % 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} for any $1 \le i \le p^n - p^{n-1}$: % \begin{align*} \dim_k T^i \mc M &= \frac{1}{p} \dim_k \mc T^1 \mc M\\ &= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1)\\ &= \dim_k T^i \mc M_0. \end{align*} % 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 $T^{j + p^n - p^{n-1}} \mc M \to \mc T^j \mc M''$ for any $j \ge 1$. Recall the Iverson bracket: % \[ \llbracket P \rrbracket = \begin{cases} 1, & \textrm{ if $P$ is true,}\\ 0, & \textrm{ if $P$ is false,} \end{cases} \] % and note that $\dim_k T^i J_l = \llbracket i \le l \rrbracket$ for $i > 0$. Therefore, if $i \in (p^n - p^{N+1}, p^n - p^N]$: % \begin{align*} \dim_k T^i \mc M_0 &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\ &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\ &+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_Q - 1} \llbracket t \ge n_Q + N - n \rrbracket \cdot (u_{Q}^{(t+1)} - u_{Q}^{(t)}). \end{align*} % Suppose now that $i = p^n - p^{n-1} + j$, where $j \in (p^{n-1} - p^{N+1}, p^{n-1} - p^N]$. Then, by induction assumption: % \begin{align*} \dim_k T^i \mc M &= \dim_k \mc T^j \mc M'' = 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < (n - 1) - (m - 1) \rrbracket\\ &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le (n-1) - (n_{X''/Y, Q}) \}\\ &+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_{X''/Y, Q}} \llbracket t \ge n_{X''/Y, Q} + N - (n - 1) \rrbracket \cdot (u_{X''/Y, Q}^{(t+1)} - u_{X''/Y, Q}^{(t)})\\ &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\ &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\ &+ \sum_{Q \in Y(k)} \sum_{t = 0}^{n_Q - 1} \llbracket t \ge n_Q + N - n \rrbracket \cdot (u_{Q}^{(t+1)} - u_{Q}^{(t)})\\ &= \dim_k T^i \mc M_0. \end{align*} % This ends the proof. \end{proof} \section{Hypoelementary covers} % Assume now that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$. Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-module $M$ and any character $\psi$ of $H$ we write $M^{\psi} := M \otimes_{k[C]} \psi$. % \begin{Proposition} \label{prop:main_thm_for_hypoelementary} Main Theorem holds for a group $G$ of the above form 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} This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Recall that if $U$ is an indecomposable $k[G]$-module then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is a one-dimensional $k[C]$-module. Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$. It turns out that the map % \[ U \mapsto (\dim_k U, \chi_U) \] % is a bijection between the set of indecomposable $k[G]$-modules and the set $\{ 1, \ldots, p^n - 1 \} \times \wh{C}$. Fix a character $\chi$ that generates $\wh{C}$. Write $U_{a, b}$ for the indecomposable $k[G]$-module with socle $\chi^a$ and dimension $b$. Write % \[ M \cong \bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}. \] \end{proof} % \begin{Lemma} \label{lem:N+Nchi+...} Keep the above notation. Let $M$, $N$ be $k[C]$-modules. Assume that % \[ M \cong N \oplus N^{\chi} \oplus \ldots \oplus N^{\chi^{p-1}}. \] % Then $N$ is uniquely determined by $M$. %If $p-1 | j$, then $N_1 \cong N_2^{\chi^i}$ for some $i$. \end{Lemma} \begin{proof} Note that % \[ M \cong N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}}. \] % By tensoring this isomorphism by $\chi^i$ we obtain: % \begin{align*} M^{\chi^i} \cong (N^{\chi^i})^{\oplus 2} \oplus N^{\chi^{i+1}} \oplus N^{\chi^{i+2}} \oplus \ldots \oplus N^{\chi^{i + p-2}} \cong (N^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j} \end{align*} % for $i = 0, \ldots, p-2$. Therefore: % \begin{equation} \label{eqn:N+M=M} N^{\oplus p} \oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \oplus \cong M^{\oplus (p-1)}. \end{equation} % Indeed, for the proof of~\eqref{eqn:N+M=M} note that % \begin{align*} N^{\oplus p} &\oplus M^{\chi} \oplus M^{\chi^2} \oplus \ldots \oplus M^{\chi^{p-2}} \cong N^{\oplus p} \oplus \bigoplus_{i = 1}^{p-2} \left((N^{\chi^i})^{\oplus 2} \oplus \bigoplus_{\substack{j = 0\\j \neq i}}^{p-2} N^{\chi^j} \right)\\ &\cong \left( N^{\oplus 2} \oplus N^{\chi} \oplus N^{\chi^2} \oplus \ldots \oplus N^{\chi^{p-2}} \right)^{\oplus (p-1)} \cong M^{\oplus (p-1)}. \end{align*} % The isomorphism~\eqref{eqn:N+M=M} clearly proves the thesis. \end{proof} % \begin{Lemma} \label{lem:TiM_isomorphism_hypoelementary} For any $i \le p^n - 1$ the map~$m_{\sigma - 1}$ from Lemma~\ref{lem:TiM_isomorphism} yields a $k[C]$-equivariant monomorphism: % \[ m_{\sigma - 1} : T^{i+1} M \hookrightarrow (T^i M)^{\chi^{-1}}. \] \end{Lemma} \begin{proof} By Lemma~\ref{lem:TiM_isomorphism} this map is injective. Thus it suffices to check that it is $k[C]$-equivariant. Note that we have the following identity in the ring~$k[C]$: % \[ (\sigma - 1) \cdot \rho = \rho \cdot (\sigma^{\chi(\rho)^{-1}} - 1) = \rho \cdot (\sigma - 1) \cdot (1 + \sigma + \sigma^2 + \ldots + \sigma^{\chi(\rho)^{-1} - 1}) \] % Note that $\sigma$ acts trivially on $T^i M$, so that for any $\ol x \in T^i M$: % \[ (1 + \sigma + \sigma^2 + \ldots + \sigma^{\chi(\rho)^{-1} - 1}) \cdot \ol x = \chi(\rho)^{-1} \cdot \ol x. \] % This easily shows that % \[ m_{\sigma - 1}(\rho \cdot \ol x) = \chi(\rho)^{-1} \cdot \rho \cdot m_{\sigma - 1}(\ol x), \] % which ends the 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 $i \ge 2$ by Lemma~\ref{lem:TiM_isomorphism_hypoelementary}: % \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: % \begin{align} \mc T^1 \mc M &\cong T^1 \mc M \oplus T^2 \mc M \oplus (T^2 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p - 2)}} \label{eqn:decomposition_of_mc_T1}\\ \mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^n - p^{n-1}. \label{eqn:decomposition_of_mc_Ti} \end{align} % Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by higher ramification data, we have by Lemma~\ref{lem:N+Nchi+...} and by~\eqref{eqn:decomposition_of_mc_Ti} that $T^2 \mc M$ is determined by higher ramification data. Moreover, by induction hypothesis and by~\eqref{eqn:decomposition_of_mc_T1}, $T^1 \mc M$ is also determined by higher 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 higher 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 higher ramification data for $i \le p^n - p^{n-1}$. By a 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 higher ramification data as well. \end{proof} \section{Proof of Main Theorem} % \begin{Lemma} Suppose $M$ is a finitely generated $k[G]$-module. \begin{enumerate}[leftmargin=*] \item The $k[G]$-module structure of $M$ is uniquely determined by the restrictions $M|_H$ as $H$ ranges over all $p$-hypo-elementary subgroups of $G$. \item The $k[G]$-module structure of $M$ is uniquely determined by the $\ol k[G]$-module structure of $M \otimes_k \ol k$. \end{enumerate} \end{Lemma} \begin{proof} \begin{enumerate}[leftmargin=*] \item This follows easily from Conlon induction theorem (cf. \cite[Theorem~(80.51)]{Curtis_Reiner_Methods_II}), see e.g. \cite[Lemma~3.2]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. \item This is \cite[Proposition~3.5. (iii)]{Bleher_Chinburg_Kontogeorgis_Galois_structure} \end{enumerate} \end{proof} (Conlon induction ???) (algebraic closure ???) \bibliography{bibliografia} \end{document}