X ---> X/C

article_de_rham_cyclic.bbl

@@ -1,150 +1,150 @@
- \def\cprime{$'$}
+ \def\cprime{$'$}
-\begin{thebibliography}{10}
+\begin{thebibliography}{10}
-
+
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+\bibitem{MR2035696}
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+Alejandro Adem and R.~James Milgram.
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+\newblock {\em Cohomology of finite groups}, volume 309 of {\em Grundlehren der
-  mathematischen Wissenschaften [Fundamental Principles of Mathematical
+  mathematischen Wissenschaften [Fundamental Principles of Mathematical
-  Sciences]}.
+  Sciences]}.
-\newblock Springer-Verlag, Berlin, second edition, 2004.
+\newblock Springer-Verlag, Berlin, second edition, 2004.
-
+
-\bibitem{Alperin_local_rep}
+\bibitem{Alperin_local_rep}
-J.~L. Alperin.
+J.~L. Alperin.
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+\newblock Cambridge University Press, Cambridge, 1986.
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+  representation theory of finite groups.
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+
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+\bibitem{Bleher_Camacho_Holomorphic_differentials}
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+F.~M. Bleher and N.~Camacho.
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-\newblock {\em J. Pure Appl. Algebra}, 227(10):Paper No. 107384, 27, 2023.
+\newblock {\em J. Pure Appl. Algebra}, 227(10):Paper No. 107384, 27, 2023.
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+
-\bibitem{Bleher_Chinburg_Kontogeorgis_Galois_structure}
+\bibitem{Bleher_Chinburg_Kontogeorgis_Galois_structure}
-F.~M. Bleher, T.~Chinburg, and A.~Kontogeorgis.
+F.~M. Bleher, T.~Chinburg, and A.~Kontogeorgis.
-\newblock Galois structure of the holomorphic differentials of curves.
+\newblock Galois structure of the holomorphic differentials of curves.
-\newblock {\em J. Number Theory}, 216:1--68, 2020.
+\newblock {\em J. Number Theory}, 216:1--68, 2020.
-
+
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+F.~M. Bleher and A.~Wood.
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+  {R}iemann-{R}och spaces.
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+\newblock {\em J. Algebra}, 631:756--803, 2023.
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+Z.~I. {Borevi\v{c}} and D.~K. Faddeev.
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-  finite groups.
+  finite groups.
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+C.~Chevalley, A.~Weil, and E.~Hecke.
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-\newblock {\em Methods of representation theory. {V}ol. {II}}.
+\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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+  Publication.
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+
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+\bibitem{Dummigan_99}
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+\newblock Complete {$p$}-descent for {J}acobians of {H}ermitian curves.
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+\newblock {\em Compositio Math.}, 119(2):111--132, 1999.
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+\bibitem{Ellingsrud_Lonsted_Equivariant_Lefschetz}
-G.~Ellingsrud and K.~L{\o}nsted.
+G.~Ellingsrud and K.~L{\o}nsted.
-\newblock An equivariant {L}efschetz formula for finite reductive groups.
+\newblock An equivariant {L}efschetz formula for finite reductive groups.
-\newblock {\em Math. Ann.}, 251(3):253--261, 1980.
+\newblock {\em Math. Ann.}, 251(3):253--261, 1980.
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+\bibitem{Garnek_equivariant}
-J.~Garnek.
+J.~Garnek.
-\newblock Equivariant splitting of the {H}odge-de {R}ham exact sequence.
+\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_p_gp_covers}
-J.~Garnek.
+J.~Garnek.
-\newblock {$p$}-group {G}alois covers of curves in characteristic {$p$}.
+\newblock {$p$}-group {G}alois covers of curves in characteristic {$p$}.
-\newblock {\em Trans. Amer. Math. Soc.}, 376(8):5857--5897, 2023.
+\newblock {\em Trans. Amer. Math. Soc.}, 376(8):5857--5897, 2023.
-
+
-\bibitem{Garnek_p_gp_covers_ii}
+\bibitem{Garnek_p_gp_covers_ii}
-J.~Garnek.
+J.~Garnek.
-\newblock $p$-group {G}alois covers of curves in characteristic $p$ {II}, 2023.
+\newblock $p$-group {G}alois covers of curves in characteristic $p$ {II}, 2023.
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+
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+\bibitem{garnek_indecomposables}
-J.~Garnek.
+J.~Garnek.
-\newblock Indecomposable direct summands of cohomologies of curves, 2024.
+\newblock Indecomposable direct summands of cohomologies of curves, 2024.
-\newblock arXiv 2410.03319.
+\newblock arXiv 2410.03319.
-
+
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+\bibitem{Gross_Rigid_local_systems_Gm}
-B.~H. Gross.
+B.~H. Gross.
-\newblock Rigid local systems on {$\Bbb G_m$} with finite monodromy.
+\newblock Rigid local systems on {$\Bbb G_m$} with finite monodromy.
-\newblock {\em Adv. Math.}, 224(6):2531--2543, 2010.
+\newblock {\em Adv. Math.}, 224(6):2531--2543, 2010.
-
+
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+\bibitem{Hartshorne1977}
-R.~Hartshorne.
+R.~Hartshorne.
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-\newblock Springer-Verlag, New York-Heidelberg, 1977.
+\newblock Springer-Verlag, New York-Heidelberg, 1977.
-\newblock Graduate Texts in Mathematics, No. 52.
+\newblock Graduate Texts in Mathematics, No. 52.
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+
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+\bibitem{Heller_Reiner_Reps_in_integers_I}
-A.~Heller and I.~Reiner.
+A.~Heller and I.~Reiner.
-\newblock Representations of cyclic groups in rings of integers. {I}.
+\newblock Representations of cyclic groups in rings of integers. {I}.
-\newblock {\em Ann. of Math. (2)}, 76:73--92, 1962.
+\newblock {\em Ann. of Math. (2)}, 76:73--92, 1962.
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-\newblock Indecomposable representations at characteristic {$p$}.
+\newblock Indecomposable representations at characteristic {$p$}.
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+\newblock {\em Duke Math. J.}, 21:377--381, 1954.
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-L.~Laurent and B.~K{\"{o}}ck.
+L.~Laurent and B.~K{\"{o}}ck.
-\newblock The canonical representation of the drinfeld curve.
+\newblock The canonical representation of the drinfeld curve.
-\newblock {\em Mathematische Nachrichten}, online first, 2024.
+\newblock {\em Mathematische Nachrichten}, online first, 2024.
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+\bibitem{Lusztig_Coxeter_orbits}
-G.~Lusztig.
+G.~Lusztig.
-\newblock Coxeter orbits and eigenspaces of {F}robenius.
+\newblock Coxeter orbits and eigenspaces of {F}robenius.
-\newblock {\em Invent. Math.}, 38(2):101--159, 1976/77.
+\newblock {\em Invent. Math.}, 38(2):101--159, 1976/77.
-
+
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+\bibitem{WardMarques_HoloDiffs}
-S.~Marques and K.~Ward.
+S.~Marques and K.~Ward.
-\newblock Holomorphic differentials of certain solvable covers of the
+\newblock Holomorphic differentials of certain solvable covers of the
-  projective line over a perfect field.
+  projective line over a perfect field.
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+\newblock {\em Math. Nachr.}, 291(13):2057--2083, 2018.
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-  Mathematics}}.
+  Mathematics}}.
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+\newblock Springer-Verlag, New York-Berlin, 1979.
-\newblock Translated from the French by Marvin Jay Greenberg.
+\newblock Translated from the French by Marvin Jay Greenberg.
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-B.~Steinberg.
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+\newblock {\em Representation theory of finite groups}.
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+R.~C. Valentini and M.~L. Madan.
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+\newblock Automorphisms and holomorphic differentials in characteristic~{$p$}.
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+\newblock {\em J. Number Theory}, 13(1):106--115, 1981.
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-  Cambridge Studies in Advanced Mathematics}.
+  Cambridge Studies in Advanced Mathematics}.
-\newblock Cambridge University Press, Cambridge, 1994.
+\newblock Cambridge University Press, Cambridge, 1994.
-
+
-\end{thebibliography}
+\end{thebibliography}

article_de_rham_cyclic.out

@@ -2,5 +2,5 @@
 \BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n\000\040\000a\000n\000d\000\040\000p\000r\000e\000l\000i\000m\000i\000n\000a\000r\000i\000e\000s}{}% 2
 \BOOKMARK [1][-]{section.3}{\376\377\0003\000.\000\040\000C\000y\000c\000l\000i\000c\000\040\000c\000o\000v\000e\000r\000s}{}% 3
 \BOOKMARK [1][-]{section.4}{\376\377\0004\000.\000\040\000P\000r\000o\000o\000f\000\040\000o\000f\000\040\000M\000a\000i\000n\000\040\000T\000h\000e\000o\000r\000e\000m}{}% 4
-\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000E\000x\000a\000m\000p\000l\000e\000s}{}% 5
+\BOOKMARK [1][-]{section.5}{\376\377\0005\000.\000\040\000A\000n\000\040\000e\000x\000a\000m\000p\000l\000e\000\040\040\023\000\040\000a\000\040\000s\000u\000p\000e\000r\000e\000l\000l\000i\000p\000t\000i\000c\000\040\000c\000u\000r\000v\000e\000\040\000w\000i\000t\000h\000\040\000a\000\040\000m\000e\000t\000a\000c\000y\000c\000l\000i\000c\000\040\000a\000c\000t\000i\000o\000n}{}% 5
 \BOOKMARK [1][-]{section*.1}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 6

article_de_rham_cyclic.tex

@@ -834,25 +834,24 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
 	is determined by higher ramification data as well.
 \end{proof}
 %
-\section{Examples}
+The method of proof of Main Theorem allows to obtain explicit formulas in the style of the result of Chevalley--Weil for
-%
+particular group. Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order~$p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$
-Assume that $G$ is a group with a normal $p$-Sylow subgroup $H$ of order~$p$. Let $C := G/H$. Then $G = H \rtimes_{\chi} C$
 for a homomorphism $\chi : C \to \FF_p^{\times}$. 
 %
 \begin{Proposition}
-Keep the above notation. {\color{red} Assume that $k$ is algebraically closed.} If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then:
+	Keep the above notation. {\color{red} Assume that $k$ is algebraically closed.} If $G$ acts on a curve $X$ and the cover $X \to X/H$ is not \'{e}tale, then:
-%
+	%
-\[
+	\[
 	H^1_{dR}(X) \cong \bigoplus_{W \in \Indec(C)} \mc V(W, p)^{\oplus a^{dR}_{Y, C}(W)} \oplus \mc V(W, p-1)^{\oplus b_W},
-\]	
+	\]	
-%
+	%
-where for any $W \in \Indec(k[C])$ the number $a_W$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $X$,
+	where for any $W \in \Indec(k[C])$ the number $a_W$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $X$,
-$a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ and
+	$a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$ and
-%
+	%
-\begin{align*}
+	\begin{align*}
-	b_W &:= a^{dR}_{X, C}(W) - \frac 1p  \sum_{i = 0}^{p-2} a^{dR}_{X, C}(W \otimes \chi^i) - a^{dR}_{Y, C}(W \otimes \chi).
+		b_W &:= a^{dR}_{X, C}(W) - \frac 1p  \sum_{i = 0}^{p-2} a^{dR}_{X, C}(W \otimes \chi^i) - a^{dR}_{Y, C}(W \otimes \chi).
-\end{align*}
+	\end{align*}
-%
+	%
 \end{Proposition}
 \begin{proof}
 	Theorem~\ref{thm:cyclic_de_rham} easily implies that
@@ -864,9 +863,14 @@ $a_W'$ is as in the equality~\eqref{eqn:cw} for the action of $C$ on $Y := X/H$
 	for some $A_W, B_W \in \ZZ$. ??
 \end{proof}
 %
-Let $p > 2$ be a prime and $p \nmid m$ an natural number. Fix a primitive root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$.
+
+\section{An example -- a superelliptic curve with a metacyclic action}
+%
+
+Let $p > 2$ be a prime and $p \nmid m$ an natural number. Let $k$ be an algebraically closed field of characteristic~ $p$.
+Fix a primitive root of unity $\zeta \in \ol{\FF}_p^{\times}$ of order $m \cdot (p-1)$.
 Note that $\zeta^m \in \FF_p$.
-We compute now the equivariant structure of the de Rham cohomology for the superelliptic curve $X$ with the affine part given by:
+In this section we compute the equivariant structure of the de Rham cohomology for the superelliptic curve $X$ with the affine part given by:
 %
 \begin{equation*}
 	y^m = x^{p^n} - x.
@@ -887,24 +891,80 @@ This action is given by:
 	\rho(x, y) &= (\zeta^m \cdot x, \zeta \cdot y).
 \end{align*}
 %
+\begin{Proposition}
+	\[
+		H^1_{dR}(X) \cong ????.
+	\]
+\end{Proposition}
+%
 Note that $X/G \cong \PP^1$ and the quotient map is given by $(x, y) \mapsto (x^p - x)^{p-1}$. Indeed, ????. 
-We claim that the set of branch points is given by $B := \{ 0, \infty \} \cup B'$, where
+We claim that the set of branch points is given by $B := \{ Q_{\infty}, Q_0, Q_1, \ldots, Q_N \}$, where
+$N := \frac{p^{n-1} - 1}{p - 1}$, $Q_0 = 0$, $Q_{\infty} = \infty$ and $Q_1, \ldots, Q_N$ are
+the elements of the set
 %
 \[
-	B' := \{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}.
+	\{ (\alpha^p - \alpha)^{p-1} : \alpha \in \FF_{p^n} \setminus \FF_p \}.
 \]
 %
-The set $B'$ has $\frac{p^{n-1} - 1}{p - 1}$ elements. We claim that:
+Write $C' := \langle \rho^{p-1} \rangle \cong \ZZ/m$ and note that $C'$ is in the center of $G$. We claim that:
 %
 \begin{itemize}
-	\item $G_{Q_0} = C$,
+	\item $G_{Q_0}$ is the conjugacy class of the subgroup $C$,
 	
-	\item $G_Q = \langle \rho^{p-1} \rangle \cong \ZZ/m$ for $Q \in B'$,
+	\item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$,
 	
-	\item $G_{Q_{\infty}} = G$ and the lower ramification jump at $Q_{\infty}$ equals $m$.
+	
+	\item $G_{Q_i} = C'$ for $i = 1, \ldots, N$.
 \end{itemize}
 %
-Indeed, ????. 
+Indeed, ????. The ramification points of $\pi : X \to X/G$ are as follows:
+%
+\begin{itemize}
+	\item points $P_0^{(1)}, \ldots, P_0^{(p)}$ above $Q_0$
+	\item[] (their stabilizers are subgroups $C_1 = C$, $\ldots$, $C_p$
+	conjugated to $C$),
+	
+	\item point $P_{\infty}$ above $Q_{\infty}$ (its stabilizer is $G$),
+	
+	\item points $P_i^{(1)}, \ldots, P_i^{(p \cdot (p-1))}$ above $Q_i$ for $i = 1, \ldots, N$
+	\item[] (their stabilizers equal $C'$).
+\end{itemize}
+%
+The same points are in the ramification locus of the morphism $X \to X/C$ with the following
+ramification groups:
+%
+\begin{align*}
+	C_{P_0^{(1)}} &= C\\
+	C_{P_0^{(i)}} &= C' \qquad \textrm{ for } i > 1,\\
+	C_{P_{\infty}} &= C\\
+	C_{P_i^{(j)}} &= C' \qquad \textrm{ for } i = 1, \ldots, N, \, j = 1, \ldots, p \cdot (p-1).
+\end{align*}
+
+
+
+
+
+Note that $Y := X/H$ is given by the equation:
+%
+\[
+	y^m = z^{p^{n-1}} + \ldots + z^p + z.
+\]
+%
+Let $\psi : C \to k^{\times}$ be a primitive character. We claim that:
+%
+\begin{align*}
+	a^{dR}_{X, C}(\psi^i) &=
+	\begin{cases}
+		p \cdot N, & \textrm{ if } m \nmid i,\\
+		0, & \textrm{ otherwise. }
+	\end{cases}\\
+	a^{dR}_{Y, C}(\psi^i) &=
+	\begin{cases}
+		\frac{p^{n-1} - 1}{p - 1}, & \textrm{ if } m \nmid i,\\
+		0, & \textrm{ otherwise. }
+	\end{cases}
+\end{align*} 
+%
 
 
 %