From 00c319f4c075e8d74db349324c76b65a715bbccb Mon Sep 17 00:00:00 2001 From: jgarnek Date: Thu, 17 Oct 2024 15:02:25 +0200 Subject: [PATCH] sketch of pf for Zpn --- article_de_rham_cyclic.out | 5 ++- article_de_rham_cyclic.synctex.gz | Bin 18057 -> 25714 bytes article_de_rham_cyclic.tex | 68 ++++++++++++++++++++++++++++-- 3 files changed, 67 insertions(+), 6 deletions(-) diff --git a/article_de_rham_cyclic.out b/article_de_rham_cyclic.out index 32e2153..b97ff67 100644 --- a/article_de_rham_cyclic.out +++ b/article_de_rham_cyclic.out @@ -1,3 +1,4 @@ -\BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040}{}% 1 +\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*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 3 +\BOOKMARK 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zQcwtDQL0V;bI8;2_0ete{LDnhuz(>H#hPIc?j}$G2z)lUqKI_ty(C06%urPNe}*xs zWk86U(nj@NC<)j#fK(ftR_u6AGc{ZwH8dez@ro$P$`%#JyXg80OvF)C0Q{U z17D356q&bmw`B++)td8>zNDR!LaL-vNO14q!f#XA!(IooC!~r#=}S!Ie3q=DwC{;2 z**a;LokC~^J1`O$W(=hh*Z`uztrnJt#I0;^rCeVbdH&@GwHm{pBDKdq;0tB-=Kq-X z_=o?$@%YE(_aDpSAOH3K@#*r*`_I=GZ~poI`&SldJZiB({^9ZJ;r83p_b(T3{(n1q J6oi{B0RX+nMCt$l diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 8a1c2b0..74a8ea0 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -112,7 +112,7 @@ hyperref, bbm, mathtools, mathrsfs} %opening \begin{document} -\title[The de Rham...]{?? The de Rham cohomology of covers with cyclic $p$-Sylow subgroup} +\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} @@ -129,8 +129,15 @@ hyperref, bbm, mathtools, mathrsfs} \maketitle \bibliographystyle{plain} % -\section{} +\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 $X \to Y$ @@ -206,7 +213,17 @@ Note also that for $j \ge 1$: ???? \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: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$. @@ -224,7 +241,7 @@ Note also that for $j \ge 1$: m' := \begin{cases} n-1, & \textrm{ if } m = n,\\ - n, & \textrm{ otherwise.} + m, & \textrm{ otherwise.} \end{cases} \] @@ -236,7 +253,50 @@ Note also that for $j \ge 1$: ???, \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} \ No newline at end of file