210 lines
6.1 KiB
TeX
210 lines
6.1 KiB
TeX
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% !TeX spellcheck = en_GB
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}{\endmanualtheoreminner}
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\theoremstyle{definition}
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%\usepackage{refcheck}
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\numberwithin{equation}{section}
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\hyphenation{Woj-ciech}
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%opening
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\begin{document}
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\title[The de Rham...]{?? The de Rham cohomology of covers with cyclic $p$-Sylow subgroup}
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\author[A. Kontogeorgis and J. Garnek]{Aristides Kontogeorgis and J\k{e}drzej Garnek}
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\address{???}
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\email{jgarnek@amu.edu.pl}
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\subjclass[2020]{Primary 14G17, Secondary 14H30, 20C20}
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\keywords{de~Rham cohomology, algebraic curves, group actions,
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characteristic~$p$}
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\urladdr{http://jgarnek.faculty.wmi.amu.edu.pl/}
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\date{}
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\begin{abstract}
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????
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\end{abstract}
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\maketitle
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\bibliographystyle{plain}
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%
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\section{}
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%
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\section{Cyclic covers}
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%
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Let $u_{X/Y, P}^{(t)}$ (resp. $l_{X/Y, P}^{(t)}$) denote the $t$th upper (resp. lower)
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ramification jump of $X \to Y$ at $P$.
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%
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\begin{Theorem}
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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:
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%
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\[
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H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{P \neq P_0} J_{p^n - \frac{p^n}{e_{X/Y, P}}}^2
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\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{X/Y, P}^{(t+1)} - u_{X/Y, P}^{(t)}}.
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\]
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\end{Theorem}
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%
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Write $H := \ZZ/p^n = \langle \sigma \rangle$.
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For any $k[H]$-module $M$ denote:
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%
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\begin{align*}
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M^{(i)} &:= \ker ((\sigma - 1)^i : M \to M),\\
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T^i M &= T^i_H M := M^{(i)}/M^{(i-1)} \quad \textrm{ for } i = 1, \ldots, p^n.
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\end{align*}
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%
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Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????).
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In the inductive step we use also the group $\ZZ/p^{n-1}$. In this case
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we denote the irreducible $k[\ZZ/p^{n-1}]$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
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and $\mc T^i M := T^i_{\ZZ/p^{n-1}} M$ for any $k[\ZZ/p^{n-1}]$-module $M$.
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\begin{Lemma}
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If the $G$-cover $X \to Y$ is \'{e}tale, then the natural map
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%
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\[
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H^1_{dR}(Y) \to H^1_{dR}(X)^G
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\]
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%
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is an isomorphism.
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\end{Lemma}
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\begin{proof}
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????
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\end{proof}
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%
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\begin{Lemma}
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If the $G$-cover $X \to Y$ is totally ramified, then the map
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%
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\[
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\tr_{X/Y} : H^1_{dR}(X) \to H^1_{dR}(Y)
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\]
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%
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is an epimorphism.
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\end{Lemma}
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\begin{proof}
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????
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\end{proof}
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%
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\begin{proof}[Proof of Theorem ????]
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We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
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$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$.
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Write also $M := H^1_{dR}(X)$.
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By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
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%
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\[
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M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n-m ??} + 1}^2 \oplus \bigoplus_{P \neq P_0} \mc J_{p^n - \frac{p^{n-1}}{e_{X/Y', P}}}^2
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\oplus \bigoplus_P \bigoplus_{t = 0}^{n-1} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
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\]
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%
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Therefore, for $???$
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%
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\begin{align*}
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\dim_k \mc T^i M =
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\begin{cases}
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???,
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\end{cases}
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\end{align*}
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\end{proof}
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\bibliography{bibliografia}
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\end{document}
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