diff --git a/article_de_rham_cyclic.out b/article_de_rham_cyclic.out index b97ff67..f94b2b7 100644 --- a/article_de_rham_cyclic.out +++ b/article_de_rham_cyclic.out @@ -1,4 +1,5 @@ \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\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 2 \BOOKMARK [1][-]{section.3}{\376\377\0003\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}{}% 3 -\BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 4 +\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*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 5 diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 87aacdb..e394342 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 74a8ea0..de0364c 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -143,14 +143,14 @@ hyperref, bbm, mathtools, mathrsfs} 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???} \}. + 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} +\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: % \[ @@ -187,7 +187,7 @@ Note also that for $j \ge 1$: \end{itemize} -\begin{Lemma} +\begin{Lemma} \label{lem:G_invariants_etale} If the $G$-cover $X \to Y$ is \'{e}tale, then the natural map % \[ @@ -200,7 +200,7 @@ Note also that for $j \ge 1$: ???? \end{proof} % -\begin{Lemma} +\begin{Lemma} \label{lem:trace_surjective} If the $G$-cover $X \to Y$ is totally ramified, then the map % \[ @@ -213,7 +213,7 @@ Note also that for $j \ge 1$: ???? \end{proof} % -\begin{Lemma} +\begin{Lemma} \label{lem:TiM_isomorphism} For any $i \le p^n - 1$: % \[ @@ -224,14 +224,27 @@ Note also that for $j \ge 1$: \end{proof} % -\begin{proof}[Proof of Theorem ????] +\begin{Lemma} + Let $M$ be a $k[H]$-module. Let $T^i M$ be as above and + $\mc T^i M := T^i_{H'} M$ for $H' \le H$, $H' \cong \ZZ/p^{n-1}$. + If $\mc T^i M \cong \mc T^{i+1} M$ for some $i$ then: + % + \[ + T^{pi + p} M \cong T^{pi + p - 1} M \cong \ldots \cong T^{pi - p + 1} M. + \] +\end{Lemma} +\begin{proof} + ?? +\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 $M := H^1_{dR}(X)$. + Write also $\mc M := H^1_{dR}(X)$. By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules: % \[ - M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2 + \mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - 1 -m'} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2 \oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}} \] % @@ -245,34 +258,40 @@ Note also that for $j \ge 1$: \end{cases} \] - Therefore, for $???$ + Therefore, for $i \le p^n - p^{n-1}$ % \begin{align*} - \dim_k \mc T^i M = + \dim_k \mc T^i \mc 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 ??: + In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$. + On the other hand, by Lemma~\ref{lem:TiM_isomorphism}: % \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. + \dim_k \mc T^1 \mc M &= \dim_k T^1 \mc M + \ldots + \dim_k T^p \mc M\\ + &\ge \dim_k T^{p^n - p^{n-1}} \mc M + \ldots + \dim_k T^{p^n - p^{n-1}} \mc M + = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M. \end{align*} % - Since the left-hand side and right hand side are equal, we conclude by Lemma ??? + Since the left-hand side and right hand side are equal, we conclude by Lemma~\ref{lem:TiM_isomorphism} that % \[ - \dim_k T^1 M = \ldots = \dim_k T^{p^n - p^{n-1}} M = \frac{1}{p} \dim_k \mc T^1 M. + \dim_k T^1 \mc M = \ldots = \dim_k T^{p^n - p^{n-1}} \mc M = \frac{1}{p} \dim_k \mc T^1 \mc M. \] % - If the cover $X \to X''$ is \'{e}tale, then the cover $X \to Y$ must be also \'{e}tale. + We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by Lemma~\ref{lem:G_invariants_etale} we have + % + \[ + \dim_k T^1 \mc M = 2 g_{X''} + \] + + 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: + $X \to X''$ is not \'{e}tale. Then, by Lemma~\ref{lem:trace_surjective}, 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} = @@ -282,13 +301,13 @@ Note also that for $j \ge 1$: This implies that: % \[ - \ker(\tr_{X/X''} : M \to M'') = M^{(p^n - p^{n-1})} + \ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})} \] % and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} M \to \mc T^i M''$ for any $i \ge 1$. Thus: % \[ - \dim_k T^{i + p^n - p^{n-1}} M = \dim_k \mc T^i M'' = .... + \dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = .... \] % This ends the proof. @@ -296,7 +315,117 @@ Note also that for $j \ge 1$: \section{Hypoelementary covers} % -Assume now that $G = H \rtimes_{\chi} \ZZ/??n$. +Assume now that $G = H \rtimes_{\chi} \ZZ/c$. +% +\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_etale} + 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_etale}, $T^1 \mc M \cong H^1_{dR}(X'')$ + is also determined by ramification data (???). + + Assume now that $X \to Y$ is not etale. 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} \ No newline at end of file