diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index e9745ac..334e3eb 100644 --- a/article_de_rham_cyclic.bbl +++ b/article_de_rham_cyclic.bbl @@ -103,6 +103,12 @@ J.-P. Serre. \newblock Springer-Verlag, New York-Berlin, 1979. \newblock Translated from the French by Marvin Jay Greenberg. +\bibitem{Steinberg_Representation_book} +B.~Steinberg. +\newblock {\em Representation theory of finite groups}. +\newblock Universitext. Springer, New York, 2012. +\newblock An introductory approach. + \bibitem{Valentini_Madan_Automorphisms} R.~C. Valentini and M.~L. Madan. \newblock Automorphisms and holomorphic differentials in characteristic~{$p$}. diff --git a/article_de_rham_cyclic.out b/article_de_rham_cyclic.out index ce00cc0..ec8b0c6 100644 --- a/article_de_rham_cyclic.out +++ b/article_de_rham_cyclic.out @@ -1,6 +1,6 @@ \BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 1 \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n}{}% 2 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 3 -\BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000H\000y\000p\000o\000e\000l\000e\000m\000e\000n\000t\000a\000r\000y\000\040\000c\000o\000v\000e\000r\000s}{}% 4 -\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000P\000r\000o\000o\000f\000\040\000o\000f\000\040\000M\000a\000i\000n\000\040\000T\000h\000e\000o\000r\000e\000m}{}% 5 +\BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000P\000r\000o\000o\000f\000\040\000o\000f\000\040\000M\000a\000i\000n\000\040\000T\000h\000e\000o\000r\000e\000m}{}% 4 +\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000E\000x\000a\000m\000p\000l\000e\000s}{}% 5 \BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 6 diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 3df364c..e06555e 100644 Binary files a/article_de_rham_cyclic.synctex.gz and b/article_de_rham_cyclic.synctex.gz differ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 13f5185..d17e436 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -167,49 +167,52 @@ of characteristic $p$. Throughout the paper we will use the following notation for any $P \in X(\ol k)$: \begin{itemize} \item $e_{X/Y, P}$ is the ramification index at $P$, - \item $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) is the $t$th upper (resp. lower) ramification jump - at $P$, + \item $m_{X/Y, P} := \ord_p(e_{X/Y, P})$ is the maximal power of~$p$ dividing the ramification index, - \item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump, - \item $m_{X/Y} := \max \{ m_{X/Y, P} : P \in X(k) \}$. + + \item $m_{X/Y} := \max \{ m_{X/Y, P} : P \in X(k) \}$, + + \item $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) is the $t$th upper (resp. lower) ramification jump + at $P$ for $t \ge 1$, + + \item $u^{(0)}_{X/Y, P} := 1$ for any ramified point $P \in X(\ol k)$ + (note that this is not a standard convention), + + \item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump. \end{itemize} % -For any $Q \in Y(k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$, +By Hasse--Arf theorem (cf. ???), if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers. + +For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$, $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$. Let % \[ - B_{X/Y} := \{ Q \in Y(k) : e_{X/Y, Q} > 1 \} + B_{X/Y} := \{ Q \in Y(\ol k) : e_{X/Y, Q} > 1 \} \] % -be the branch locus of $\pi$. +be the branch locus of $\pi$. Recall that $\ZZ/p^n$ has $p^n$ indecomposable representations over a field of characteristic~$p$. +We denote them by $J_1, \ldots, J_{p^n}$. Observe that $J_i$ is given by the Jordan block of size $i$ and eigenvalue $1$. % \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 $m_{X/Y, P}$ -by the equality $e_{X/Y, P} = p^{m_{X/Y, P}}$. -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} Let $k$ be an algebraically closed field of characteristic~$p$. - Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $m := \max \{ m_{X/Y, P} : P \in X(k) \}$. Pick arbitrary $Q_0 \in Y(k)$ with $m_{X/Y, Q_0} = m$. Then, as a $k[\ZZ/p^n]$-module + Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Pick arbitrary $Q_0 \in Y(k)$ + with $m_{X/Y, Q_0} = m_{X/Y}$. 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 B\\ Q \neq Q_0}} J_{p^n - p^n/e_{Q}}^2 - \oplus \bigoplus_{Q \in B} \bigoplus_{t = 0}^{m_{X/Y, Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}}, + \oplus \bigoplus_{Q \in B} \bigoplus_{t = 0}^{m_{Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}}, \end{equation} % - where $B := B_{X/Y}$, $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$. + where $B := B_{X/Y}$, $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$, $m := m_{X/Y, Q}$, $m_Q := m_{X/Y, Q}$. \end{Theorem} % \begin{Remark} - Note that for $g_Y = 0$, ... + Note that for $g_Y = 0$ the first exponent is negative. However, since $m_{X/Y} = n$ (as $\PP^1$ doesn't have any \'{e}tale covers), the first two summands in~\eqref{eqn:HdR_formula} cancel out. Thus also in this case the module~\eqref{eqn:HdR_formula} is well-defined. \end{Remark} Write $H := \langle \sigma \rangle \cong \ZZ/p^n$. @@ -247,9 +250,10 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$. If the $G$-cover $X \to Y$ is \'{e}tale, then % \[ - \dim_k H^1_{dR}(X)^G = 2g_Y. + \dim_k H^1_{dR}(X)^G = 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k). \] % + In particular, if $G \cong \ZZ/p^n$ 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 $\mc F^{\bullet}$. @@ -277,8 +281,10 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$. % \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) ????. + = 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k). \end{align*} + % + Finally, note that if $G$ is cyclic then $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ by ????. \end{proof} % \begin{Lemma} \label{lem:trace_surjective} @@ -406,13 +412,14 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \begin{equation} \label{eqn:Q_in_B'} p \cdot (u_{X/Y, Q} - 1) = (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1). \end{equation} - - Indeed, by using the above formulas: + % + Indeed, using the above formulas: % \begin{align*} p \cdot (u_{X/Y, Q} - 1) &= p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(m_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 (i^{(0)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) \\ + &+ p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \\ &= (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*} @@ -526,17 +533,30 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. This ends the proof. \end{proof} -\section{Hypoelementary covers} +\section{Proof of Main Theorem} % -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{Lemma} \label{lem:reductions} + 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} +% +By Lemma~\ref{lem:reductions} we may assume that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and that $k$ is algebraically closed. 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 $M$ + Let $k$ and $G$ be as above. Assume that $M$ is 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} @@ -635,7 +655,8 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo % \end{proof} -\begin{proof}[Proof of Proposition~\ref{prop:main_thm_for_hypoelementary}] +\begin{proof}[Proof of Main Theorem] + As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and $k \ol k$ by Lemma~\ref{lem:reductions}. 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 $(\sigma - 1)$ @@ -685,26 +706,76 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo 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} +\section{Examples} +% +\noindent Let $p > 2$. Consider the Mumford curve +% +\[ + X : (x^p - x) \cdot (y^p - y) = 1. +\] +% +It is a curve of genus $(p-1)^2$ and an action of the group $(\ZZ/p \times \ZZ/p) \rtimes D_{2(p-1)}$ given by: +% +\begin{align*} + \sigma_0(x, y) &= (x+1, y),\\ + \sigma_1(x, y) &= (x, y+1),\\ + s(x, y) &= (y, x),\\ + \theta(x, y) &= (\zeta \cdot x, \zeta^{-1} \cdot y) \quad \textrm{ for } \FF_p^{\times} = \langle \zeta \rangle. +\end{align*} +% +Recall representation theory of $D_{2(p - 1)}$ (cf. \cite[Example~8.2.3]{Steinberg_Representation_book}). +For $1 \le j \le p-2$ let $\chi_j$ be the character of the representation of $D_{2(p-1)}$ +induced from +% +\[ + \ZZ/(p-1) = \langle \theta \rangle \to \FF_p^{\times}, \quad \theta \mapsto \zeta^j. +\] +% +One easily checks that $\chi_j$ is given by the matrices: +% +\begin{align*} + \theta \mapsto + \left( + \begin{matrix} + \zeta^j & 0\\ + 0 & \zeta^{-j} + \end{matrix} + \right), + \qquad + s \mapsto + \left( + \begin{matrix} + 0 & 1\\ + 1 & 0 + \end{matrix} + \right). +\end{align*} +% +Moreover, $\chi_j$ is irreducible and isomorphic to $\chi_{p - 1 - j}$. +Let also $\chi_0$ be the representation: +% +\[ + D_{2(p-1)} \to \FF_p^{\times}, \qquad \theta \mapsto 1, \qquad + s \mapsto -1. +\] +% +We claim that as $k[C]$-modules: ??k or $\FF_p$?? +% +\begin{equation} + H^1_{dR}(X) \cong V_0^{\oplus (p-1)} \oplus \bigoplus_{j = 1}^{\frac{p-1}{2}} V_j^{\oplus 2(p-1)}. +\end{equation} + + + +{\color{gray} +Basis of holomorphic differentials: +% +\[ + \omega_{a, b} = \frac{x^a \cdot y^b \, dx}{(x^p - x)} \qquad 0 \le a, b \le p-2. +\] +} + -(Conlon induction ???) (algebraic closure ???) \bibliography{bibliografia} \end{document} \ No newline at end of file