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article_de_rham_cyclic.bbl
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@ -44,7 +44,7 @@ N.~Dummigan.
\newblock {\em Compositio Math.}, 119(2):111--132, 1999.
\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 {\em Math. Ann.}, 251(3):253--261, 1980.

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article_de_rham_cyclic.out
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\BOOKMARK [1][-]{section.1}{\376\377\0001\000.\000\040\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 1
\BOOKMARK [1][-]{section.2}{\376\377\0002\000.\000\040\000N\000o\000t\000a\000t\000i\000o\000n}{}% 2
\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

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article_de_rham_cyclic.tex
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@ -167,7 +167,7 @@ In the second step, we use similar methods to show the result for a group of the
$\ZZ/p^n \rtimes_{\chi} \ZZ/c$. Finally, we use Conlon induction theorem to deduce Main Theorem
for an arbitrary group with a cyclic $p$-Sylow subgroup.
%
\section{Notation}
\section{Notation and preliminaries}
%
Assume that $\pi : X \to Y$ is a $G$-cover of smooth projective curves over an field $k$
of characteristic $p$.
@ -190,7 +190,6 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$
\end{itemize}
%
By Hasse--Arf theorem (cf. ???), if the $p$-Sylow subgroup of $G$ is abelian, the numbers $u_{X/Y, P}^{(t)}$ are integers.
For any $Q \in Y(\ol k)$ we denote also by abuse of notation $G_Q := G_P$, $e_{X/Y, Q} := e_{X/Y, P}$,
$u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.
Let
@ -199,9 +198,45 @@ Let
B_{X/Y} := \{ Q \in Y(\ol k) : e_{X/Y, Q} > 1 \}
\]
%
be the branch locus of $\pi$. Recall that $\ZZ/p^n$ has $p^n$ indecomposable representations over a field of characteristic~$p$.
We denote them by $J_1, \ldots, J_{p^n}$. Observe that $J_i$ is given by the Jordan block of size $i$ and eigenvalue $1$.
be the branch locus of $\pi$. In the article we often use the Iverson bracket notation:
%
\[
\llbracket P \rrbracket =
\begin{cases}
1, & \textrm{ if $P$ is true,}\\
0, & \textrm{ if $P$ is false.}
\end{cases}
\]
%
We review now some facts from representation theory of finite groups.
Recall that $\ZZ/p^n$ has $p^n$ indecomposable representations over a field of characteristic~$p$.
We denote them by $J_1, \ldots, J_{p^n}$. Observe that $J_i$ is given by the Jordan block of size $i$ and eigenvalue $1$. Assume now that $G$ is a finite group with a normal cyclic $p$-Sylow subgroup $H = \langle \sigma \rangle \cong \ZZ/p^n$. Let $C := G/H$.
Recall that if $U$ is an indecomposable $k[G]$-module
then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is an indecomposable
$k[C]$-module. It turns out that the map
%
\begin{align*}
\Indec(k[G]) \to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
U \mapsto \left(U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}} \right)
\end{align*}
%
is a bijection (cf. \cite[p. 35--37, 42 -- 43]{Alperin_local_rep}). We write
$\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$.
Finally, we recall the classical Chevalley-Weil formula. Keep the above notation and assume that $p \nmid \# G$. For any $Q \in Y(k)$ let $\chi_Q : G_Q \to k^{\times}$ be the fundamental character of $G_Q$ acting on the tangent space of $Q$. Then:
%
\begin{equation}
H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} M^{\oplus a_M},
\end{equation}
%
where:
%
\begin{align*}
a_M := - \dim_k M + \sum_{Q \in Y(k)} \sum_{i = 0}^{e_{X/Y, Q} - 1} \left\langle \frac{-i}{e_{X/Y, Q}} \right\rangle \cdot N_{P, i}(M),
\end{align*}
%
and $N_{P, i}(M) := ???$.
\section{Cyclic covers}
%
\begin{Theorem} \label{thm:cyclic_de_rham}
@ -237,20 +272,9 @@ but it works for an arbitrary module).
Moreover, for $i > 0$:
%
\begin{equation} \label{eqn:dim_of_Ti_Jl}
\dim_k T^i J_l = \llbracket i \le l \rrbracket,
\dim_k T^i J_l = \llbracket i \le l \rrbracket.
\end{equation}
%
where we use the Iverson bracket notation:
%
\[
\llbracket P \rrbracket =
\begin{cases}
1, & \textrm{ if $P$ is true,}\\
0, & \textrm{ if $P$ is false.}
\end{cases}
\]
%
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 $M$.
@ -308,6 +332,7 @@ The equality $\dim_k H^1(G, k) = \dim_k H^2(G, k)$ does not hold for non-cyclic
\wedge(x_{1}, \ldots, x_s) \otimes \mathbb{F}_p[x_1, \ldots,x_s] & \text{ if } p>2
\end{cases}
\]
Therefore, for $s>1$ the degree one and two parts of the cohomological ring, which correspond to the first and second cohomology groups, have different dimensions.
\end{Remark}
}
%
@ -420,15 +445,15 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
Recall also that by \cite[???]{Serre1979} there exist integers $i_{X/Y, P}^{(0)}, i_{X/Y, P}^{(1)}, \ldots$ such that for every $t \ge 0$:
%
\begin{align*}
u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \ldots + i_{X/Y, P}^{(t-1)}\\
l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}.
u_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} + \cdots + i_{X/Y, P}^{(t-1)}\\
l_{X/Y, P}^{(t)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \cdots + i_{X/Y, P}^{(t-1)} \cdot p^{t-1}.
\end{align*}
%
Observe that:
%
\begin{align*}
i_{X/X', P}^{(0)} &= i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p,\\
i_{X/X', P}^{(t)} &= p \cdot (i_{X/Y, P}^{(t + 1)} + \ldots + i_{X/Y, P}^{(n-1)}) \quad \textrm{ for } t > 0.
i_{X/X', P}^{(t)} &= p \cdot (i_{X/Y, P}^{(t + 1)} + \cdots + i_{X/Y, P}^{(n-1)}) \quad \textrm{ for } t > 0.
\end{align*}
%
This implies that
@ -441,10 +466,10 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
%
\begin{align*}
p \cdot (u_{X/Y, Q} - 1) &=
p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(m_Q - 1)}_{X/Y, Q} - 1)\\
p \cdot (i^{(0)}_{X/Y, Q} + \cdots + i^{(m_Q - 1)}_{X/Y, Q} - 1)\\
&= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) \\
&+ p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \\
&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\
&+ p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \cdots) - 1 \\
&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \cdots - 1)\\
&= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (u_{X/Y', Q'} - 1).
\end{align*}
%
@ -584,32 +609,10 @@ 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}
This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience.
Assume that $G$ is a finite group with a normal cyclic $p$-Sylow subgroup $H = \langle \sigma \rangle \cong \ZZ/p^n$. Let $C := G/H$.
Recall that if $U$ is an indecomposable $k[G]$-module
then $U^{\sigma} := \ker(\sigma - 1)$ (the socle of $U$) is an indecomposable
$k[C]$-module. It turns out that the map
%
\begin{align*}
\Indec(k[G]) \to \Indec(k[C]) \times \{ 1, \ldots, p^n \}\\
U \mapsto (U^{\sigma}, \frac{\dim_k U}{\dim_k U^{\sigma}})
\end{align*}
%
is a bijection (cf. \cite[p. 35--37, 42 -- 43]{Alperin_local_rep}). We write
$\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k[C]) \times \{ 1, \ldots, p^n \}$.
Thus it comes from a character $\chi_U \in \wh{C} := \Hom(C, \CC)$.
It turns out that the map
This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Let $\psi : C \to k^{\times}$ be a primitive character. Write
%
\[
U \mapsto (\dim_k U, \chi_U)
\]
%
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
%
\[
M \cong \bigoplus_{a, b} M_{a, b}^{\oplus n(a, b)}.
M \cong \bigoplus_{a, b} \mc V(\psi^a, b)^{\oplus n(a, b)}.
\]
\end{proof}
%
@ -692,18 +695,18 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
\begin{proof}[Proof of Main Theorem]
As explained at the beginning of this section, it suffices to show this in the case when $G = H \rtimes_{\chi} C = \langle \sigma \rangle \rtimes_{\chi} \langle \rho \rangle \cong \ZZ/p^n \rtimes_{\chi} \ZZ/c$ and $k = \ol k$ by Lemma~\ref{lem:reductions}.
We prove this by induction on~$n$. If $n = 0$, then it follows by Chevalley--Weil theorem.
Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale. Then the results of \cite{Garnek_equivariant} imply that the Hodge--de Rham exact sequence splits and
Consider now two cases. Firstly, we assume that $X \to Y$ is \'{e}tale. Then by Lemma~\ref{lem:G_invariants_\'{e}tale} and \cite[Corollary~2.4]{Garnek_equivariant} we have $\dim_k H^1_{dR}(X)^H = 2g_Y = \dim_k H^0(X, \Omega_X)^H + \dim_k H^1(X, \mc O_X)^H$. Therefore the Hodge--de Rham exact sequence splits by \cite[Lemma~5.3]{Garnek_equivariant} and
%
\begin{equation}
H^1_{dR}(X) \cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X).
\end{equation}
\begin{align*}
H^1_{dR}(X) &\cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X)\\
&\cong H^0(X, \Omega_X) \oplus H^1(X, \mc O_X)^{\vee}
\end{align*}
%
By Lemma~\ref{lem:G_invariants_\'{e}tale} and \cite[Corollary~2.4]{Garnek_equivariant} we have $\dim_k H^1_{dR}(X)^H = 2g_Y = \dim_k H^0(X, \Omega_X)^H + \dim_k H^1(X, \mc O_X)^H$. Therefore the Hodge--de Rham exact sequence splits by \cite[Lemma~5.3]{Garnek_equivariant}.
Recall that by~\cite[Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data. This ends the proof in this case,
as $H^1(X, \mc O_X) \cong H^0(X, \Omega_X)^{\vee}$ by Serre's duality (cf. ???).\\
(the last isomorphism follows from Serre's duality, cf. ???).
Now it suffices to note that by~\cite[Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}
the $k[G]$-module structure of $H^0(X, \Omega_X)$ is determined by the higher ramification data. This ends the proof in this case.\\
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}
Assume now that $X \to Y$ is not \'{e}tale. Lemma~\ref{lem:TiM_isomorphism_hypoelementary} and proof of Theorem~\ref{thm:cyclic_de_rham}
yield an isomorphism of $k[C]$-modules:
%
\begin{equation} \label{eqn:TiM=T1M_chi}