diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index e935b4e..332bc41 100644 --- a/article_de_rham_cyclic.bbl +++ b/article_de_rham_cyclic.bbl @@ -5,11 +5,30 @@ F.~M. Bleher, T.~Chinburg, and A.~Kontogeorgis. \newblock Galois structure of the holomorphic differentials of curves. \newblock {\em J. Number Theory}, 216:1--68, 2020. +\bibitem{Curtis_Reiner_Methods_II} +C.~W. Curtis and I.~Reiner. +\newblock {\em Methods of representation theory. {V}ol. {II}}. +\newblock Pure and Applied Mathematics (New York). John Wiley \& Sons, Inc., + New York, 1987. +\newblock With applications to finite groups and orders, A Wiley-Interscience + Publication. + \bibitem{Garnek_equivariant} J.~Garnek. \newblock Equivariant splitting of the {H}odge-de {R}ham exact sequence. \newblock {\em Math. Z.}, 300(2):1917--1938, 2022. +\bibitem{garnek_indecomposables} +J.~Garnek. +\newblock Indecomposable direct summands of cohomologies of curves, 2024. +\newblock arXiv 2410.03319. + +\bibitem{Hartshorne1977} +R.~Hartshorne. +\newblock {\em {Algebraic geometry}}. +\newblock Springer-Verlag, New York-Heidelberg, 1977. +\newblock Graduate Texts in Mathematics, No. 52. + \bibitem{Serre1979} J.-P. Serre. \newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 54ef6f5..604f2dc 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 f65cd9c..1e339b2 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -113,7 +113,7 @@ 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} @@ -132,6 +132,11 @@ hyperref, bbm, mathtools, mathrsfs} % \section{Introduction} % +The classical Chevalley--Weil formula (cf. ????) gives an explicit description +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~$p$. +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 +with ??? + \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 $p$. @@ -140,29 +145,34 @@ hyperref, bbm, mathtools, mathrsfs} \end{mainthm} % Note that if $p > 2$ and the $p$-Sylow subgroup of $G$ is not cyclic, the structure -of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{??Garnek_indecomposables}. +of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{garnek_indecomposables}. \section{Cyclic covers} % -\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 $P$.} +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 $n_{X/Y, P}$ by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$. We abbreviate the last ramification jump to $u_{X/Y, P}$. -\red{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 $P \in \pi^{-1}(Q)$.} +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} - 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: + 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 + $H^1_{dR}(X)$ is isomorphic to: % - \[ - H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\red{\substack{Q \in Y(k)\\ Q \neq Q_0}}} J_{p^n - p^n/e_{\red{Q}}}^2 - \oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t = 0}^{n_{X/Y, P}} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}}, - \] + \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 Y(k)\\ Q \neq Q_0}} J_{p^n - p^n/e_{Q}}^2 + \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)}}, + \end{equation} % where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$. \end{Theorem} % +\begin{Remark} + Note that for $g_Y = 0$, ... +\end{Remark} + Write $H := \langle \sigma \rangle \cong \ZZ/p^n$. For any $k[H]$-module $M$ denote: % @@ -176,7 +186,7 @@ In the inductive step we use also the group $H' := \ZZ/p^{n-1}$. In this case we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\ -\noindent \red{Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that: +\noindent Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that: % \begin{align*} u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\ @@ -196,13 +206,13 @@ $e_{X/Y, P} = p^n$, we have: \end{align*} If $e_{X/Y, P} \le p^{n - N}$ then $i_{X/Y, P}^{(t)} = i_{X/X', P}^{(t)}$ for all $t$. -} + % \begin{Lemma} \label{lem:G_invariants_\'{e}tale} If the $G$-cover $X \to Y$ is \'{e}tale, then % \[ - \red{\dim_k H^1_{dR}(X)^G = 2g_Y.} + \dim_k H^1_{dR}(X)^G = 2g_Y. \] % \end{Lemma} @@ -351,7 +361,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$. - Write also $\mc M := H^1_{dR}(X)$. + Let also $\mc M := H^1_{dR}(X)$ and write $\mc M_0$ for the module~\eqref{eqn:HdR_formula}. We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: % \[ @@ -362,7 +372,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. which by Lemma~\ref{lem:lemma_mcT_and_T} implies that % \[ - \dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1). + \dim_k T^p \mc M = \ldots = \dim_k T^{p^n} \mc M = 2(g_Y - 1) = \dim_k T^p \mc M_0. \] % Thus, for $i = 2, \ldots, p$: @@ -374,14 +384,14 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. On the other hand, by Lemma~\ref{lem:G_invariants_\'{e}tale} we have % $ - \dim_k T^1 \mc M = 2 g_Y + \dim_k T^1 \mc M = 2 g_Y = \dim_k T^1 \mc M_0 $. Thus: % \begin{align*} \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). \end{align*} % - Thus $\dim_k T^i \mc M = 2(g_Y - 1)$ for every $i \ge 2$, which ends the proof in this case. + 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. Assume now that $X \to Y$ is not \'{e}tale. Therefore $X \to X''$ is also not \'{e}tale. By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules: @@ -391,26 +401,27 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \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)}} \] % - 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}: + 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}: % \begin{align*} \dim_k \mc T^i \mc M &= - 2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ + 2(g_{Y'} - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ &= 2 p (g_Y - 1) + \sum_{Q' \in Y'(k)} (p-1) \cdot (l_{Y'/Y, Q'}^{(1)} + 1)\\ - &+ 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ - &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right) + &+ 2 + 2(\# B - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ + &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# B - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right),\\ \end{align*} % where % - \[ R := \{ P \in X(k) : e_P > 1 \} = \{ P \in X(k) : e'_P > 1 \}. \] + \[ B := \{ Q \in Y(k) : e_Q > 1 \} = \{ Q \in Y(k) : e'_Q > 1 \}. \] % In particular, $\dim_k \mc T^1 \mc M = \ldots = \dim_k \mc T^{p^{n-1} - p^{n-2}} \mc M$. - Thus by Lemma~\ref{lem:lemma_mcT_and_T} + Thus by Lemma~\ref{lem:lemma_mcT_and_T} for any $1 \le i \le p^n - p^{n-1}$: % \begin{align*} - \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\\ - &= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1). + \dim_k T^i \mc M &= \frac{1}{p} \dim_k \mc T^1 \mc M\\ + &= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 1)\\ + &= \dim_k T^i \mc M_0. \end{align*} % 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 @@ -433,12 +444,38 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \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 $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus, if $i \in [p^{n-1} - p^k, p^{n-1} - p^{k-1}]$: + 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 $j \ge 1$. + Recall the Iverson bracket: % \[ - \dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = .... + \llbracket P \rrbracket = + \begin{cases} + 1, & \textrm{ if $P$ is true,}\\ + 0, & \textrm{ if $P$ is false,} + \end{cases} \] % + and note that $\dim_k T^i J_l = \llbracket i \le l \rrbracket$ for $i > 0$. Therefore, if $i \in (p^n - p^{N+1}, p^n - p^N]$: + % + \begin{align*} + \dim_k T^i \mc M &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\ + &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\ + &+ \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)}). + \end{align*} + % + Suppose now that + $i = p^n - p^{n-1} + j$, where $j \in (p^{n-1} - p^{N+1}, p^{n-1} - p^N]$. Then, by induction assumption: + % + \begin{align*} + \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\\ + &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le (n-1) - (n_{X''/Y, Q}) \}\\ + &+ \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)})\\ + &= 2 \cdot (g_Y - 1) + 2 \cdot \llbracket N < n - m \rrbracket\\ + &+ 2 \cdot \# \{ Q \in Y(k) \setminus \{Q_0\} : N \le n - n_Q \}\\ + &+ \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)})\\ + &= \dim_k T^i \mc M_0. + \end{align*} + % This ends the proof. \end{proof} @@ -448,7 +485,7 @@ Assume now that $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} 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 hypoelementary $G$ as above and $k = \ol k$. + Main Theorem holds for a group $G$ of the above form and $k = \ol k$. \end{Proposition} % \begin{Lemma} @@ -456,7 +493,22 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo is uniquely determined by the $k[C]$-structure of $T^1 M, \ldots, T^{p^n} M$. \end{Lemma} \begin{proof} - See \cite[????]{Bleher_Chinburg_Kontogeorgis_Galois_structure} for a proof. + 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 + then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is a one-dimensional + $k[C]$-module. Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$. + It turns out that the map + % + \[ + U \mapsto (\dim_k U, \chi_U) + \] + % + 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 $\wh{C}$. + Write $U_{a, b}$ for the indecomposable $k[G]$-module with socle $\chi^a$ + and dimension $b$. Write + % + \[ + M \cong \bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}. + \] \end{proof} % \begin{Lemma} \label{lem:N+Nchi+...} @@ -557,7 +609,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo % Thus, since by induction hypothesis $\mc T^i \mc M$ is determined by ramification data, 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. - Moreover, by induction hypothesis and \red{by~\eqref{eqn:decomposition_of_mc_T1}}, $T^1 \mc M$ + Moreover, by induction hypothesis and by~\eqref{eqn:decomposition_of_mc_T1}, $T^1 \mc M$ is also determined by ramification data. 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} @@ -577,7 +629,7 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo \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: + 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} \ No newline at end of file