conlon; intro - beginning
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@ -5,11 +5,30 @@ F.~M. Bleher, T.~Chinburg, and A.~Kontogeorgis.
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\newblock Galois structure of the holomorphic differentials of curves.
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\newblock {\em J. Number Theory}, 216:1--68, 2020.
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\bibitem{Curtis_Reiner_Methods_II}
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C.~W. Curtis and I.~Reiner.
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\newblock {\em Methods of representation theory. {V}ol. {II}}.
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\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc.,
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New York, 1987.
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\newblock With applications to finite groups and orders, A Wiley-Interscience
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Publication.
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\bibitem{Garnek_equivariant}
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J.~Garnek.
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\newblock Equivariant splitting of the {H}odge-de {R}ham exact sequence.
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\newblock {\em Math. Z.}, 300(2):1917--1938, 2022.
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\bibitem{garnek_indecomposables}
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J.~Garnek.
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\newblock Indecomposable direct summands of cohomologies of curves, 2024.
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\newblock arXiv 2410.03319.
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\bibitem{Hartshorne1977}
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R.~Hartshorne.
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\newblock {\em {Algebraic geometry}}.
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\newblock Springer-Verlag, New York-Heidelberg, 1977.
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\newblock Graduate Texts in Mathematics, No. 52.
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\bibitem{Serre1979}
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J.-P. Serre.
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\newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in
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@ -113,7 +113,7 @@ hyperref, bbm, mathtools, mathrsfs}
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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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\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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@ -132,6 +132,11 @@ hyperref, bbm, mathtools, mathrsfs}
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%
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\section{Introduction}
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%
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The classical Chevalley--Weil formula (cf. ????) gives an explicit description
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of the equivariant structure of the cohomology of a curve with a group action over a field of characteristic~$0$. It is hard to expect such a formula over fields of characteristic~$p$.
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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. ???). This brings attention to groups
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with ???
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\begin{mainthm}
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Suppose that $G$ is a group with a $p$-cyclic Sylow subgroup.
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Let $X$ be a curve with an action of~$G$ over a field $k$ of characteristic $p$.
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@ -140,29 +145,34 @@ hyperref, bbm, mathtools, mathrsfs}
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\end{mainthm}
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%
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Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure
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of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{??Garnek_indecomposables}.
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of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{garnek_indecomposables}.
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\section{Cyclic covers}
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%
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\red{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 $P$.}
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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 $P$.
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We use also the convention $u^{(0)}_{X/Y, P} = 1$.
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By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $n_{X/Y, P}$
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by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$.
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We abbreviate the last ramification jump to $u_{X/Y, P}$.
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\red{For any $Q \in Y(k)$ we denote also $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
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$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.}
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For any $Q \in Y(k)$ we denote also $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
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$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
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%
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\begin{Theorem} \label{thm:cyclic_de_rham}
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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 $k[\ZZ/p^n]$-modules:
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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
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$H^1_{dR}(X)$ is isomorphic to:
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%
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\[
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H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\red{\substack{Q \in Y(k)\\ Q \neq Q_0}}} J_{p^n - p^n/e_{\red{Q}}}^2
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\oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t = 0}^{n_{X/Y, P}} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}},
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\]
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\begin{equation} \label{eqn:HdR_formula}
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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
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\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)}},
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\end{equation}
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%
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where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$.
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\end{Theorem}
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%
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\begin{Remark}
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Note that for $g_Y = 0$, ...
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\end{Remark}
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Write $H := \langle \sigma \rangle \cong \ZZ/p^n$.
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For any $k[H]$-module $M$ denote:
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%
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@ -176,7 +186,7 @@ In the inductive step we use also the group $H' := \ZZ/p^{n-1}$. In this case
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we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$
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and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\
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\noindent \red{Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that:
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\noindent Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that:
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%
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\begin{align*}
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u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\
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@ -196,13 +206,13 @@ $e_{X/Y, P} = p^n$, we have:
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\end{align*}
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If $e_{X/Y, P} \le p^{n - N}$ then $i_{X/Y, P}^{(t)} = i_{X/X', P}^{(t)}$
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for all $t$.
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}
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%
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\begin{Lemma} \label{lem:G_invariants_\'{e}tale}
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If the $G$-cover $X \to Y$ is \'{e}tale, then
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%
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\[
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\red{\dim_k H^1_{dR}(X)^G = 2g_Y.}
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\dim_k H^1_{dR}(X)^G = 2g_Y.
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\]
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%
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\end{Lemma}
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@ -351,7 +361,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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\begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}]
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We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$,
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$H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$.
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Write also $\mc M := H^1_{dR}(X)$.
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Let also $\mc M := H^1_{dR}(X)$ and write $\mc M_0$ for the module~\eqref{eqn:HdR_formula}.
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We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$:
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%
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\[
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@ -362,7 +372,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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which by Lemma~\ref{lem:lemma_mcT_and_T} implies that
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%
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\[
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\dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1).
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\dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1) = \dim_k T^p \mc M_0.
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\]
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%
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Thus, for $i = 2, \ldots, p$:
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@ -374,14 +384,14 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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On the other hand, by Lemma~\ref{lem:G_invariants_\'{e}tale} we have
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%
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$
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\dim_k T^1 \mc M = 2 g_Y
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\dim_k T^1 \mc M = 2 g_Y = \dim_k T^1 \mc M_0
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$. Thus:
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%
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\begin{align*}
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\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).
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\end{align*}
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%
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Thus $\dim_k T^i \mc M = 2(g_Y - 1)$ for every $i \ge 2$, which ends the proof in this case.
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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.
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Assume now that $X \to Y$ is not \'{e}tale. Therefore $X \to X''$ is also not \'{e}tale.
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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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@ -391,26 +401,27 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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\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)}}
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\]
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%
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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. ????) and Lemma~\ref{lem:u_equals_ul}:
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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}:
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%
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\begin{align*}
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\dim_k \mc T^i \mc M &=
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2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
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2(g_{Y'} - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
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&= 2 p (g_Y - 1) + \sum_{Q' \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q'}^{(1)} + 1)\\
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&+ 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
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&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right)
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&+ 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\
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&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right),\\
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\end{align*}
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%
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where
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%
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\[ R := \{ P \in X(k) : e_P > 1 \} = \{ P \in X(k) : e'_P > 1 \}. \]
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\[ B := \{ Q \in Y(k) : e_Q > 1 \} = \{ Q \in Y(k) : e'_Q > 1 \}. \]
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%
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In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$.
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Thus by Lemma~\ref{lem:lemma_mcT_and_T}
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Thus by Lemma~\ref{lem:lemma_mcT_and_T} for any $1 \le i \le p^n - p^{n-1}$:
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%
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\begin{align*}
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\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\\
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&= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1).
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\dim_k T^i \mc M &= \frac{1}{p} \dim_k \mc T^1 \mc M\\
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&= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1)\\
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&= \dim_k T^i \mc M_0.
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\end{align*}
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%
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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
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@ -433,12 +444,38 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
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\ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})}
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\]
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%
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and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus, if $i \in [p^{n-1} - p^k, p^{n-1} - p^{k-1}]$:
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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 $j \ge 1$.
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Recall the Iverson bracket:
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%
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\[
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\dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = ....
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\llbracket P \rrbracket =
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\begin{cases}
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1, & \textrm{ if $P$ is true,}\\
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0, & \textrm{ if $P$ is false,}
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\end{cases}
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\]
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%
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and note that $\dim_k T^i J_l = \llbracket i \le l \rrbracket$ for $i > 0$. Therefore, if $i \in (p^n - p^{N+1}, p^n - p^N]$:
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%
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\begin{align*}
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\dim_k T^i \mc M &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\
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&+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\
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&+ \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)}).
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\end{align*}
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%
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Suppose now that
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$i = p^n - p^{n-1} + j$, where $j \in (p^{n-1} - p^{N+1}, p^{n-1} - p^N]$. Then, by induction assumption:
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%
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\begin{align*}
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\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\\
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&+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le (n-1) - (n_{X''/Y, Q}) \}\\
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&+ \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)})\\
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&= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\
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&+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\
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&+ \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)})\\
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&= \dim_k T^i \mc M_0.
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\end{align*}
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%
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This ends the proof.
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\end{proof}
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@ -448,7 +485,7 @@ Assume now that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi}
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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 $M^{\psi} := M \otimes_{k[C]} \psi$.
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%
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\begin{Proposition} \label{prop:main_thm_for_hypoelementary}
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Main Theorem holds for a hypoelementary $G$ as above and $k = \ol k$.
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Main Theorem holds for a group $G$ of the above form and $k = \ol k$.
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\end{Proposition}
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%
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\begin{Lemma}
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@ -456,7 +493,22 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
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is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$.
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\end{Lemma}
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\begin{proof}
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See \cite[????]{Bleher_Chinburg_Kontogeorgis_Galois_structure} for a proof.
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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
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then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is a one-dimensional
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$k[C]$-module. Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$.
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It turns out that the map
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%
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\[
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U \mapsto (\dim_k U, \chi_U)
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\]
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%
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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 $\wh{C}$.
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Write $U_{a, b}$ for the indecomposable $k[G]$-module with socle $\chi^a$
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and dimension $b$. Write
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%
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\[
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M \cong \bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}.
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\]
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\end{proof}
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%
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\begin{Lemma} \label{lem:N+Nchi+...}
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@ -557,7 +609,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
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%
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Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by ramification data,
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we have by Lemma~\ref{lem:N+Nchi+...} and by~\eqref{eqn:decomposition_of_mc_Ti} that $T^2 \mc M$ is determined by ramification data.
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Moreover, by induction hypothesis and \red{by~\eqref{eqn:decomposition_of_mc_T1}}, $T^1 \mc M$
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Moreover, by induction hypothesis and by~\eqref{eqn:decomposition_of_mc_T1}, $T^1 \mc M$
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is also determined by ramification data.
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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}
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@ -577,7 +629,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
|
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\end{align*}
|
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%
|
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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 $i \le p^n - p^{n-1}$.
|
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By similar reasoning, $\tr_{X/X'}$ yields an isomorphism:
|
||||
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??}}.
|
||||
@ -589,6 +641,23 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
|
||||
|
||||
\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}
|
||||
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Reference in New Issue
Block a user