conlon; intro - beginning

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

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 $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
+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 $P \in \pi^{-1}(Q)$.}
+$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ 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:
 	%
-	\[
+	\begin{equation} \label{eqn:HdR_formula}
-	H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\red{\substack{Q \in Y(k)\\ Q \neq Q_0}}} J_{p^n - p^n/e_{\red{Q}}}^2
+		J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{Q \in Y(k)\\ Q \neq Q_0}} J_{p^n - p^n/e_{Q}}^2
-	\oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t = 0}^{n_{X/Y, P}} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}},
+		\oplus \bigoplus_{Q \in Y(k)} \bigoplus_{t = 0}^{n_{X/Y, Q}} J_{p^n - p^{n+t}/e_Q}^{u_Q^{(t+1)} - u_Q^{(t)}},
-	\]
+	\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)\\
+		 &+ 2 + 2(\# B - 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)
+		&= 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\\
+		\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).
+		&= 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}