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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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\author [A. Kontogeorgis and J. Garnek] { Aristides Kontogeorgis and J\k { e} drzej Garnek}
\address { ???}
\email { jgarnek@amu.edu.pl}
\subjclass [2020] { Primary 14G17, Secondary 14H30, 20C20}
\keywords { de~Rham cohomology, algebraic curves, group actions,
characteristic~$ p $ }
\urladdr { http://jgarnek.faculty.wmi.amu.edu.pl/}
\date { }
\begin { abstract}
????
\end { abstract}
\maketitle
\bibliographystyle { plain}
%
\section { Introduction}
%
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The classical Chevalley--Weil formula
(cf. \cite { Chevalley_ Weil_ Uber_ verhalten} ,
{ \color { red}
\cite { Ellingsrud_ Lonsted_ Equivariant_ Lefschetz} )
}
gives an explicit description
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of the equivariant structure of the cohomology of a curve $ X $ with a group action over a field of characteristic~$ 0 $ . Their formula depends on the so-called \emph { fundamental characters} of points $ x \in X $ that are ramified in the cover $ X \to X / G $ . ????
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. \cite { Prest} ). There are many results concerning equivariant structure of cohomologies for particular groups
(see e.g.~\cite { Valentini_ Madan_ Automorphisms} for the case of cyclic groups, \cite { WardMarques_ HoloDiffs} for abelian groups, \cite { Bleher_ Chinburg_ Kontogeorgis_ Galois_ structure} for groups with a cyclic Sylow subgroup, or \cite { Bleher_ Camacho_ Holomorphic_ differentials} for the Klein group) or curves (cf. \cite { Lusztig_ Coxeter_ orbits} , \cite { Dummigan_ 99} , \cite { Gross_ Rigid_ local_ systems_ Gm} , \cite { laurent_ kock_ drinfeld} ). Also, one may expect that that (at least in the case of $ p $ -groups) determining cohomologies comes down to Harbater--Katz--Gabber covers (cf. \cite { Garnek_ p_ gp_ covers} , \cite { Garnek_ p_ gp_ covers_ ii} ). However, there is no hope of obtaining a result similar to the one of Chevalley and Weil.\\
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This brings attention to groups with cyclic $ p $ -Sylow subgroup. For those, the set of
{ \color { red} equivalence classes of }
indecomposable modules is finite (cf. \cite { Higman} , \cite { Borevic_ Faddeev} , \cite { Heller_ Reiner_ Reps_ in_ integers_ I} ). While their representation theory still
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seems a bit too complicated to derive a general formula for the cohomologies,
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the article~\cite { Bleher_ Chinburg_ Kontogeorgis_ Galois_ structure} proved that
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the $ k [ G ] $ -module structure of $ H ^ 0 ( X, \Omega _ X ) $ is determined by the higher ramification data (i.e. higher ramification groups and the fundamental characters of the ramification locus). The main result of this article is a similar statement for the de Rham cohomology.
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%
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\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 $ .
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The $ k [ G ] $ -module structure of $ H ^ 1 _ { dR } ( X ) $ is uniquely determined by the
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higher ramification data of the cover $ X \to X / G $ and the genus of $ X $ .
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\end { mainthm}
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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 higher ramification data. Indeed, see \cite { garnek_ indecomposables} for a construction of $ G $ -covers with the same higher ramification data, but varying $ k [ G ] $ -module structure of $ H ^ 0 ( X, \Omega _ X ) $ and~$ H ^ 1 _ { dR } ( X ) $ .\\
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We elaborate now on the proof of Main Theorem. The first step is to prove Main Theorem for $ G = \ZZ / p ^ n $ . In this case we give an explicit formula for the structure of $ H ^ 1 _ { dR } ( X ) $ , depending only on the ramification indices, ramification jumps and genus of the quotient
curve (see Theorem~\ref { thm:cyclic_ de_ rham} ). This formula is proven inductively, by applying induction twice: once for the curve $ X $ with an action of $ \ZZ / p ^ { n - 1 } $ and once for the curve $ X'' : = X / ( \ZZ / p ) $ .
In the second step, we use similar methods to show the result for a group of the form
$ \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.
%
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\section { Notation and preliminaries}
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%
Assume that $ \pi : X \to Y $ is a $ G $ -cover of smooth projective curves over an field $ k $
of characteristic $ p $ .
Throughout the paper we will use the following notation for any $ P \in X ( \ol k ) $ :
\begin { itemize}
\item $ e _ { X / Y, P } $ is the ramification index at $ P $ ,
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\item $ m _ { X / Y, P } : = \ord _ p ( e _ { X / Y, P } ) $ is the maximal power of~$ p $
dividing the ramification index,
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\item $ m _ { X / Y } : = \max \{ m _ { X / Y, P } : P \in X ( k ) \} $ ,
\item $ u _ { X / Y, P } ^ { ( t ) } $ (resp. $ l _ { X / Y, P } ^ { ( t ) } $ ) is the $ t $ th upper (resp. lower) ramification jump
at $ P $ for $ t \ge 1 $ ,
\item $ u ^ { ( 0 ) } _ { X / Y, P } : = 1 $ for any ramified point $ P \in X ( \ol k ) $
(note that this is not a standard convention),
\item $ u _ { X / Y, P } : = u _ { X / Y, P } ^ { ( m _ { X / Y, P } ) } $ is the last ramification jump.
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\end { itemize}
%
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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 } $ ,
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$ u _ { X / Y, Q } ^ { ( t ) } : = u _ { X / Y, P } ^ { ( t ) } $ etc. for arbitrary $ P \in \pi ^ { - 1 } ( Q ) $ .
Let
%
\[
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B_ { X/Y} := \{ Q \in Y(\ol k) : e_ { X/Y, Q} > 1 \}
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\]
%
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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}
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%
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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 ) : = ??? $ .
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\section { Cyclic covers}
%
\begin { Theorem} \label { thm:cyclic_ de_ rham}
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Let $ k $ be an algebraically closed field of characteristic~$ p $ .
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Suppose that $ \pi : X \to Y $ is a $ \ZZ / p ^ n $ -cover. Pick arbitrary $ Q _ 0 \in Y ( k ) $
with $ m _ { X / Y, Q _ 0 } = m _ { X / Y } $ . 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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\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 B\\ Q \neq Q_ 0} } J_ { p^ n - p^ n/e_ { Q} } ^ 2
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\oplus \bigoplus _ { Q \in B} \bigoplus _ { t = 0} ^ { m_ { 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 $ B : = B _ { X / Y } $ , $ e _ Q : = e _ { X / Y, Q } $ and $ u _ Q ^ { ( t ) } : = u _ { X / Y, Q } ^ { ( t ) } $ , $ m : = m _ { X / Y, Q } $ , $ m _ Q : = m _ { X / Y, Q } $ .
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\end { Theorem}
%
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\begin { Remark}
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Note that for $ g _ Y = 0 $ the first exponent is negative. However, since $ m _ { X / Y } = n $ (as $ \PP ^ 1 $ doesn't have any \' { e} tale covers), the first two summands in~\eqref { eqn:HdR_ formula} cancel out. Thus also in this case the module~\eqref { eqn:HdR_ formula} is well-defined.
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\end { Remark}
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Write $ H : = \langle \sigma \rangle \cong \ZZ / p ^ n $ .
For any $ k [ H ] $ -module $ M $ denote:
%
\begin { align*}
M^ { (i)} & := \ker ((\sigma - 1)^ i : M \to M),\\
T^ i M & = T^ i_ H M := M^ { (i)} /M^ { (i-1)} \quad \textrm { for } i = 1, \ldots , p^ n.
\end { align*}
%
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Recall that $ \dim _ k T ^ i M $ ,
{ \color { red} for $ i = 1 , \ldots ,p ^ n $ }
determines the structure of $ M $ completely (see \cite [p. 108] { Valentini_ Madan_ Automorphisms} -- they give the argument for $ M : = H ^ 0 ( X, \Omega _ X ) $ ,
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but it works for an arbitrary module).
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Moreover, for $ i > 0 $ :
%
\begin { equation} \label { eqn:dim_ of_ Ti_ Jl}
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\dim _ k T^ i J_ l = \llbracket i \le l \rrbracket .
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\end { equation}
%
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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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%
\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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%
\[
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\dim _ k H^ 1_ { dR} (X)^ G = 2g_ Y - \dim _ k H^ 1(G, k) + \dim _ k H^ 2(G, k).
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\]
%
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In particular, if $ G \cong \ZZ / p ^ n $ then $ \dim _ k H ^ 1 _ { dR } ( X ) ^ G = 2 g _ Y $ .
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\end { Lemma}
\begin { proof}
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Let $ \HH ^ i ( Y, \mc F ^ { \bullet } ) $ be the $ i $ th hypercohomology of a complex $ \mc F ^ { \bullet } $ .
Write also $ \mc H ^ i ( G, - ) $ for the $ i $ th derived functor of the functor
%
\[
\mc F \mapsto \mc F^ G.
\]
%
Since $ X \to Y $ is \' { e} tale, $ \mc H ^ i ( G, \pi _ * \mc F ) = 0 $ for any $ i > 0 $ and any coherent sheaf $ \mc F $ on $ X $ by \cite [Proposition~2.1] { Garnek_ equivariant} .
Therefore the spectral sequence~\cite [(3.4)] { Garnek_ equivariant} applied for the complex $ \mc F ^ { \bullet } : = \pi _ * \Omega _ { X / k } ^ { \bullet } $ yields $ \RR ^ i \Gamma ^ G ( \pi _ * \Omega _ { X / k } ^ { \bullet } ) = \HH ^ 1 ( Y, \pi _ * ^ G \Omega _ { X / k } ^ { \bullet } ) = H ^ 1 _ { dR } ( Y ) $ , since $ \pi _ * ^ G \Omega _ X ^ { \bullet } \cong \Omega _ Y $ (cf. ???).
On the other hand, the seven-term exact sequence applied for the spectral sequence~\cite [(3.5)] { Garnek_ equivariant} yields:
%
\begin { align*}
0 \to H^ 1(G, H^ 0_ { dR} (X)^ G) \to H^ 1_ { dR} (Y) \to H^ 1_ { dR} (X)^ G \to H^ 2(G, H^ 0_ { dR} (X)^ G) \to K,
\end { align*}
%
where:
%
\[
K := \ker (H^ 2_ { dR} (Y) \to H^ 2_ { dR} (X)^ G) = \ker (k \stackrel { \id } { \rightarrow } k) = 0.
\]
%
Therefore, since $ H ^ 0 _ { dR } ( X ) ^ G \cong k $ :
%
\begin { align*}
\dim _ k H^ 1_ { dR} (X)^ G = \dim _ k H^ 1_ { dR} (Y) - \dim _ k H^ 1(G, k) + \dim _ k H^ 2(G, k)\\
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= 2g_ Y - \dim _ k H^ 1(G, k) + \dim _ k H^ 2(G, k).
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\end { align*}
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%
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Finally, note that if $ G $ is cyclic then $ \dim _ k H ^ 1 ( G, k ) = \dim _ k H ^ 2 ( G, k ) $ by
{ \color { red}
\cite [th. 6.2.2] { Weibel} .
}
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\end { proof}
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{ \color { red}
\begin { Remark}
The equality $ \dim _ k H ^ 1 ( G, k ) = \dim _ k H ^ 2 ( G, k ) $ does not hold for non-cyclic groups. For example it is known \cite [cor. II.4.3,th. II.4.4] { MR2035696} that the cohomological ring for the elementary abelian group $ \mathbb { F } _ p ^ s $ is given by
\[
H^ * (G, \mathbb { F} _ p)=
\begin { cases}
\mathbb { F} _ 2[x_ 1, \ldots ,x_ s] & \text { if } p=2 \\
\wedge (x_ { 1} , \ldots , x_ s) \otimes \mathbb { F} _ p[x_ 1, \ldots ,x_ s] & \text { if } p>2
\end { cases}
\]
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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.
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\end { Remark}
}
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%
\begin { Lemma} \label { lem:trace_ surjective}
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Suppose that $ G $ is a $ p $ -group.
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If the $ G $ -cover $ X \to Y $ is totally ramified, then the map
%
\[
\tr _ { X/Y} : H^ 1_ { dR} (X) \to H^ 1_ { dR} (Y)
\]
%
is an epimorphism.
\end { Lemma}
\begin { proof}
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%
By induction, it suffices to prove this in the case when $ G = \ZZ / p $ .
Consider the following commutative diagram:
%
\begin { center}
% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKMgEYqtRizYduvbHgJFh5MfWatEIABIA9YgAoAGqQAEAHVMB5ALYwA5nQD6BgJRceIDHIGLSo6qskNHX0ATRNzaztHENcZDz55QWQAJmV-CXUtbWEHYCgAJU5DWPdPfgUUVL9xNTYdHLzCvRi3WXKkgGY0msCs4UNw0ysAY2MLJxK2xKIu6oDM+ubBkbGHFriy6ZQAFm75qVb4rwrkXbmMg84xGChbeCJQADMAJwgrJDIQHAgkZLiXt6-ajfJDbf6vd6IXZfH6IABs4MB8OBsIAHIjIQB2FFIACcGKQAFYcYhMQTEF0YUTyUoqRTyak6ZT9hpzDhnrkDAB6EKcQ4AyHQkGIYk9TJsjnAbm8-kQpBwknYsVsCWcnl8q6cIA
\begin { tikzcd}
0 \arrow [r] & { H^ 0(X, \Omega _ X)} \arrow [r] \arrow [d, "\tr_{X/Y}"] & H^ 1_ { dR} (X) \arrow [r] \arrow [d, "\tr_{X/Y}"] & { H^ 1(X, \mc O_ X)} \arrow [r] \arrow [d, "\tr_{X/Y}"] & 0 \\
0 \arrow [r] & { H^ 0(Y, \Omega _ Y)} \arrow [r] & H^ 1_ { dR} (Y) \arrow [r] & { H^ 1(Y, \mc O_ Y)} \arrow [r] & 0
\end { tikzcd}
\end { center}
%
where the rows are Hodge--de Rham exact sequences. Recall that by~\cite [Theorem~1] { Valentini_ Madan_ Automorphisms} , in this case $ H ^ 0 ( X, \Omega _ X ) $ contains
a copy of $ k [ G ] ^ { \oplus g _ Y } $ as a direct summand. Thus, since trace is injective on $ k [ G ] ^ { \oplus g _ Y } $ , the dimension
of the image of
%
\begin { equation} \label { eqn:trace_ H0_ Omega}
\tr _ { X/Y} : H^ 0(X, \Omega _ X) \to H^ 0(Y, \Omega _ Y)
\end { equation}
%
is $ g _ Y $ . Therefore the map~\eqref { eqn:trace_ H0_ Omega} is surjective.
Similarly, by Serre's duality, also $ H ^ 1 ( X, \mc O _ X ) $ contains $ k [ G ] ^ { \oplus g _ Y } $ as a direct summand
and one shows similarly that the trace map
%
\begin { equation*} %\label{eqn:trace_H0_Omega}
\tr _ { X/Y} : H^ 1(X, \mc O_ X) \to H^ 1(Y, \mc O_ Y)
\end { equation*}
%
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is surjective. Therefore, since the outer vertical maps in the diagram are surjective,
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the trace map on the de Rham cohomology must be surjective as well.
%
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\end { proof}
%
\begin { Lemma} \label { lem:TiM_ isomorphism}
For any $ i \le p ^ n - 1 $ we have the following $ k $ -linear monomorphism:
%
\[
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m_ { \sigma - 1} : T^ { i+1} M \hookrightarrow T^ i M.
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\]
\end { Lemma}
\begin { proof}
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%
We define $ m _ { \sigma - 1 } $ as follows:
%
\[
m_ { \sigma - 1} (\ol x) := (\sigma - 1) \cdot x,
\]
%
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where for $ \ol x \in T ^ { i + 1 } M $ we picked any representative $ x \in M ^ { ( i + 1 ) } $ .
Indeed, if $ x \in M ^ { ( i + 1 ) } $ then clearly $ ( \sigma - 1 ) \cdot x \in M ^ { ( i ) } $ .
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Moreover $ ( \sigma - 1 ) \cdot x \in M ^ { ( i - 1 ) } $ holds if and only if $ x \in M ^ { ( i ) } $ . This
shows that $ m _ { \sigma - 1 } $ is well-defined and injective.
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\end { proof}
%
\begin { Lemma} \label { lem:lemma_ mcT_ and_ T}
Let $ M $ be a $ k [ H ] $ -module. Let $ T ^ i M $ and $ \mc T ^ i M $ be as above.
If $ \dim _ k \mc T ^ i M = \dim _ k \mc T ^ { i + 1 } M $ for some $ i $ then:
%
\[
\dim _ k T^ { pi + p} M = \dim _ k T^ { pi + p - 1} M = \ldots = \dim _ k T^ { pi - p + 1} M.
\]
\end { Lemma}
\begin { proof}
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Note that $ \mc T ^ i M = M ^ { ( pi ) } / M ^ { ( pi - p ) } $ . This easily implies that:
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%
\begin { align*}
\dim _ k \mc T^ i M & = \dim _ k T^ { pi} M + \ldots + \dim _ k T^ { pi - p + 1} M\\
& \ge \dim _ k T^ { pi+p} M + \ldots + \dim _ k T^ { pi+1} M
= \dim _ k \mc T^ { i+1} M.
\end { align*}
%
Since the left-hand side and right hand side are equal, we conclude by Lemma~\ref { lem:TiM_ isomorphism}
\end { proof}
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%
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\begin { Lemma} \label { lem:u_ equals_ ul}
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Assume that $ Y' \to Y $ is a $ \ZZ / p $ -subcover of $ X \to Y $ .
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Then:
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%
\[
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p \cdot \sum _ { Q \in B_ { X/Y} } (u_ { X/Y, Q} - 1) = \sum _ { Q' \in B_ { X/Y'} } (u_ { X/Y', Q'} - 1)
+ (p-1) \cdot \sum _ { Q \in B_ { Y'/Y} } (l^ { (1)} _ { Y'/Y, Q} + 1).
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\]
%
\end { Lemma}
\begin { proof}
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%
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Pick a point $ Q \in B _ { X / Y } $ . If $ Q \not \in B _ { Y' / Y } $ then
$ u _ { X / Y, Q } = u _ { X / Y', Q' } $ for all $ p $ points $ Q' \in Y' ( k ) $ in the preimage of $ Q $ and:
%
\begin { equation} \label { eqn:Q_ not_ in_ B'}
p \cdot (u_ { X/Y, Q} - 1) = \sum _ { Q'} (u_ { X/Y', Q'} - 1).
\end { equation}
%
Assume now that $ Q \in B _ { Y' / Y } $ . Then there exists a unique point $ Q' \in Y' ( k ) $
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in the preimage of $ Q $ through $ Y' \to Y $ . Moreover, $ m _ { X / Y, Q } = n $ , $ m _ { X / Y', Q' } = n - 1 $ .
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Recall also that by
{ \color { red}
\cite [Example p.76] { Serre1979}
}
there exist integers $ i _ { X / Y, P } ^ { ( 0 ) } , i _ { X / Y, P } ^ { ( 1 ) } , \ldots $ such that for every $ t \ge 0 $ :
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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)} + \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} .
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\end { align*}
%
Observe that:
%
\begin { align*}
i_ { X/X', P} ^ { (0)} & = i_ { X/Y, P} ^ { (0)} + i_ { X/Y, P} ^ { (1)} \cdot p,\\
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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.
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\end { align*}
%
This implies that
%
\begin { equation} \label { eqn:Q_ in_ B'}
p \cdot (u_ { X/Y, Q} - 1) = (u_ { X/Y', Q'} - 1) + (p-1) \cdot (l^ { (1)} _ { X/Y, Q} + 1).
\end { equation}
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%
Indeed, using the above formulas:
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%
\begin { align*}
p \cdot (u_ { X/Y, Q} - 1) & =
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p \cdot (i^ { (0)} _ { X/Y, Q} + \cdots + i^ { (m_ Q - 1)} _ { X/Y, Q} - 1)\\
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& = (p-1) \cdot (i^ { (0)} _ { X/Y, Q} + 1) + (i^ { (0)} _ { X/Y, Q} + p \cdot i^ { (1)} ) \\
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& + 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)\\
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& = (p-1) \cdot (l^ { (1)} _ { X/Y, Q} + 1) + (u_ { X/Y', Q'} - 1).
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\end { align*}
%
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The proof follows by summing~\eqref { eqn:Q_ not_ in_ B'} and~\eqref { eqn:Q_ in_ B'} over all $ Q \in B _ { X / Y } $ .
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\end { proof}
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\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 } $ ,
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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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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 ) $ :
%
\[
\mc M \cong \mc J_ { p^ { n-1} } ^ { 2 p \cdot (g_ Y - 1)} \oplus k^ { \oplus 2} .
\]
%
Therefore $ \dim _ k \mc T ^ 2 \mc M = \ldots = \dim _ k \mc T ^ { p ^ { n - 1 } } \mc M = 2 p ( g _ Y - 1 ) $ ,
which by Lemma~\ref { lem:lemma_ mcT_ and_ T} implies that
%
\[
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\dim _ k T^ p \mc M = \ldots = \dim _ k T^ { p^ n} \mc M = 2(g_ Y - 1) = \dim _ k T^ p \mc M_ 0.
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\]
%
Thus, for $ i = 2 , \ldots , p $ :
%
\[
\dim _ k T^ i \mc M \ge 2(g_ Y - 1) = \dim _ k T^ { p+1} \mc M.
\]
%
On the other hand, by Lemma~\ref { lem:G_ invariants_ \' { e} tale} we have
%
$
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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:
%
\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*}
%
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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.
By induction hypothesis for $ H' $ acting on $ X $ , we have the following isomorphism of $ k [ H' ] $ -modules:
%
\[
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\mc M \cong \mc J_ { p^ { n-1} } ^ { 2 (g_ { Y'} - 1)} \oplus \mc J_ { p^ { n-1} - p^ { n - m} + 1} ^ 2 \oplus \bigoplus _ { \substack { Q \in Y'(k)\\ Q \neq Q_ 1} } \mc J_ { p^ { n-1} - p^ { n-1} /e'_ Q} ^ 2
\oplus \bigoplus _ { Q \in Y'(k)} \bigoplus _ { t = 0} ^ { n-2} \mc J_ { p^ n - p^ t} ^ { u_ { X/Y', Q} ^ { (t+1)} - u_ { X/Y', Q} ^ { (t)} }
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\]
%
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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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%
\begin { align*}
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\dim _ k \mc T^ i \mc M & =
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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(\# 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*}
%
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} 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^ i \mc M & = \frac { 1} { p} \dim _ k \mc T^ 1 \mc M\\
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& = 2(g_ Y - 1) + 2 + 2(\# B - 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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%
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
in $ \FF _ p [ x ] $ we have the identity:
%
\[
1 + x + \ldots + x^ { p-1} = (x - 1)^ { p-1} .
\]
%
Therefore in the group ring $ k [ H ] $ we have:
%
\[
\tr _ { X/X''} = \sum _ { j = 0} ^ { p-1} (\sigma ^ { p^ { n-1} } )^ j = (\sigma ^ { p^ { n-1} } - 1)^ { p-1} =
(\sigma - 1)^ { p^ n - p^ { n-1} } .
\]
%
This implies that:
%
\[
\ker (\tr _ { X/X''} : \mc M \to \mc M'') = \mc M^ { (p^ n - p^ { n-1} )}
\]
%
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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 $ . ?? Therefore, if $ i \in ( p ^ n - p ^ { N + 1 } , p ^ n - p ^ N ] $ :
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%
\begin { align*}
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\dim _ k T^ i \mc M_ 0 & = 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 - m_ Q \} \\
& + \sum _ { Q \in Y(k)} \sum _ { t = 0} ^ { m_ Q - 1} \llbracket t \ge m_ Q + N - n \rrbracket \cdot (u_ { Q} ^ { (t+1)} - u_ { Q} ^ { (t)} ).
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\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 \\
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& + 2 \cdot \# \{ Q \in Y(k) \setminus \{ Q_ 0\} : N \le (n-1) - (m_ { X''/Y, Q} ) \} \\
& + \sum _ { Q \in Y(k)} \sum _ { t = 0} ^ { m_ { X''/Y, Q} } \llbracket t \ge m_ { 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 - m_ Q \} \\
& + \sum _ { Q \in Y(k)} \sum _ { t = 0} ^ { m_ Q - 1} \llbracket t \ge m_ Q + N - n \rrbracket \cdot (u_ { Q} ^ { (t+1)} - u_ { Q} ^ { (t)} )\\
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& = \dim _ k T^ i \mc M_ 0.
\end { align*}
%
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This ends the proof.
\end { proof}
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\section { Proof of Main Theorem}
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%
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\begin { Lemma} \label { lem:reductions}
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}
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%
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By Lemma~\ref { lem:reductions} we may assume that $ G = H \rtimes _ { \chi } C = \langle \sigma \rangle \rtimes _ { \chi } \langle \rho \rangle \cong \ZZ / p ^ n \rtimes _ { \chi } \ZZ / c $ and that $ k $ is algebraically closed.
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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%
\begin { Lemma}
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Let $ k $ and $ G $ be as above. Assume that $ M $ is a $ k [ G ] $ -module of finite dimension. The $ k [ G ] $ -structure of $ M $
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is uniquely determined by the $ k [ C ] $ -structure of $ T ^ 1 M, \ldots , T ^ { p ^ n } M $ .
\end { Lemma}
\begin { 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. Let $ \psi : C \to k ^ { \times } $ be a primitive character. Write
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%
\[
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M \cong \bigoplus _ { a, b} \mc V(\psi ^ a, b)^ { \oplus n(a, b)} .
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\]
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\end { proof}
%
\begin { Lemma} \label { lem:N+Nchi+...}
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Keep the above notation. Let $ M $ , $ N $ be $ k [ C ] $ -modules.
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If
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%
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\[
N \cong M \oplus M^ { \chi } \oplus \ldots \oplus M^ { \chi ^ { p-1} } ,
\]
then $ N $ is uniquely determined by $ M $ .
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\end { Lemma}
\begin { proof}
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Note that
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%
\[
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N \cong M^ { \oplus 2} \oplus M^ { \chi } \oplus M^ { \chi ^ 2} \oplus \ldots \oplus M^ { \chi ^ { p-2} } .
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\]
%
By tensoring this isomorphism by $ \chi ^ i $ we obtain:
%
\begin { align*}
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N^ { \chi ^ i} \cong (M^ { \chi ^ i} )^ { \oplus 2} \oplus M^ { \chi ^ { i+1} } \oplus M^ { \chi ^ { i+2} } \oplus \ldots \oplus M^ { \chi ^ { i + p-2} }
\cong (M^ { \chi ^ i} )^ { \oplus 2} \oplus \bigoplus _ { \substack { j = 0\\ j \neq i} } ^ { p-2} M^ { \chi ^ j}
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\end { align*}
%
for $ i = 0 , \ldots , p - 2 $ . Therefore:
%
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\begin { equation} \label { eqn:M+N=N}
M^ { \oplus p} \oplus N^ { \chi } \oplus N^ { \chi ^ 2} \oplus \ldots \oplus N^ { \chi ^ { p-2} }
\cong N^ { \oplus (p-1)} .
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\end { equation}
%
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Indeed, for the proof of~\eqref { eqn:M+N=N} note that
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%
\begin { align*}
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M^ { \oplus p} & \oplus N^ { \chi } \oplus N^ { \chi ^ 2} \oplus \ldots \oplus N^ { \chi ^ { p-2} }
\cong M^ { \oplus p} \oplus \bigoplus _ { i = 1} ^ { p-2} \left ((M^ { \chi ^ i} )^ { \oplus 2}
\oplus \bigoplus _ { \substack { j = 0\\ j \neq i} } ^ { p-2} M^ { \chi ^ j} \right )\\
& \cong \left ( M^ { \oplus 2} \oplus M^ { \chi } \oplus M^ { \chi ^ 2} \oplus \ldots \oplus M^ { \chi ^ { p-2} } \right )^ { \oplus (p-1)}
\cong N^ { \oplus (p-1)} .
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\end { align*}
%
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The isomorphism~\eqref { eqn:M+N=N} clearly proves the thesis.
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\end { proof}
%
\begin { Lemma} \label { lem:TiM_ isomorphism_ hypoelementary}
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For any $ i \le p ^ n - 1 $ the map~$ m _ { \sigma - 1 } $ from Lemma~\ref { lem:TiM_ isomorphism}
yields a $ k [ C ] $ -equivariant monomorphism:
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%
\[
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m_ { \sigma - 1} : T^ { i+1} M \hookrightarrow (T^ i M)^ { \chi ^ { -1} } .
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\]
\end { Lemma}
\begin { proof}
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By Lemma~\ref { lem:TiM_ isomorphism} this map is injective. Thus it suffices to check that it is $ k [ C ] $ -equivariant.
Note that we have the following identity in the ring~$ k [ C ] $ :
%
\[
(\sigma - 1) \cdot \rho = \rho \cdot (\sigma ^ { \chi (\rho )^ { -1} } - 1)
= \rho \cdot (\sigma - 1) \cdot (1 + \sigma + \sigma ^ 2 + \ldots + \sigma ^ { \chi (\rho )^ { -1} - 1} )
\]
%
Note that $ \sigma $ acts trivially on $ T ^ i M $ , so that for any $ \ol x \in T ^ i M $ :
%
\[
(1 + \sigma + \sigma ^ 2 + \ldots + \sigma ^ { \chi (\rho )^ { -1} - 1} ) \cdot \ol x = \chi (\rho )^ { -1} \cdot \ol x.
\]
%
This easily shows that
%
\[
m_ { \sigma - 1} (\rho \cdot \ol x) = \chi (\rho )^ { -1} \cdot \rho \cdot m_ { \sigma - 1} (\ol x),
\]
%
which ends the proof.
%
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\end { proof}
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\begin { proof} [Proof of Main Theorem]
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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} .
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We prove this by induction on~$ n $ . If $ n = 0 $ , then it follows by Chevalley--Weil theorem.
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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 = 2 g _ 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
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%
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\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*}
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%
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(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.\\
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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}
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yield an isomorphism of $ k [ C ] $ -modules:
%
\begin { equation} \label { eqn:TiM=T1M_ chi}
T^ { i+1} \mc M \cong (T^ 1 \mc M)^ { \chi ^ { -i} }
\end { equation}
%
for $ i \le p ^ n - p ^ { n - 1 } $ . Observe that $ \mc T ^ i M $ has the filtration $ \mc M ^ { ( pi ) } \supset \mc M ^ { ( pi - 1 ) } \supset \ldots \supset \mc M ^ { ( pi - p ) } $ with subquotients $ T ^ { pi } \mc M, \ldots , T ^ { pi - p + 1 } \mc M $ .
Thus, since the category of $ k [ C ] $ -modules is semisimple, for $ i \le p ^ n - p ^ { n - 1 } $ :
%
\begin { align*}
\mc T^ i \mc M & \cong T^ { pi - p + 1} \mc M \oplus \ldots \oplus T^ { pi} \mc M\\
& \cong T^ 1 \mc M \oplus (T^ 1 \mc M)^ { \chi ^ { -1} } \oplus \ldots \oplus
(T^ 1 \mc M)^ { \chi ^ { -p} } .
\end { align*}
%
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By induction assumption, the $ k [ C ] $ -module structure of $ \mc T ^ i \mc M $ is uniquely determined by the higher 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 higher ramification data for $ i \le p ^ n - p ^ { n - 1 } $ .
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By a similar reasoning, $ \tr _ { X / X' } $ yields an isomorphism:
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%
\[
T^ { i + p^ n - p^ { n-1} } \mc M \cong (\mc T^ i \mc M'')^ { \chi ^ { -1??} } .
\]
%
Thus, by induction hypothesis for $ \mc M'' $ , the $ k [ C ] $ -structure of $ T ^ { i + p ^ n - p ^ { n - 1 } } \mc M $
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is determined by higher ramification data as well.
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\end { proof}
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%
\section { Examples}
%
\noindent Let $ p > 2 $ . Consider the Mumford curve
%
\[
X : (x^ p - x) \cdot (y^ p - y) = 1.
\]
%
It is a curve of genus $ ( p - 1 ) ^ 2 $ and an action of the group $ ( \ZZ / p \times \ZZ / p ) \rtimes D _ { 2 ( p - 1 ) } $ given by:
%
\begin { align*}
\sigma _ 0(x, y) & = (x+1, y),\\
\sigma _ 1(x, y) & = (x, y+1),\\
s(x, y) & = (y, x),\\
\theta (x, y) & = (\zeta \cdot x, \zeta ^ { -1} \cdot y) \quad \textrm { for } \FF _ p^ { \times } = \langle \zeta \rangle .
\end { align*}
%
Recall representation theory of $ D _ { 2 ( p - 1 ) } $ (cf. \cite [Example~8.2.3] { Steinberg_ Representation_ book} ).
For $ 1 \le j \le p - 2 $ let $ \chi _ j $ be the character of the representation of $ D _ { 2 ( p - 1 ) } $
induced from
%
\[
\ZZ /(p-1) = \langle \theta \rangle \to \FF _ p^ { \times } , \quad \theta \mapsto \zeta ^ j.
\]
%
One easily checks that $ \chi _ j $ is given by the matrices:
%
\begin { align*}
\theta \mapsto
\left (
\begin { matrix}
\zeta ^ j & 0\\
0 & \zeta ^ { -j}
\end { matrix}
\right ),
\qquad
s \mapsto
\left (
\begin { matrix}
0 & 1\\
1 & 0
\end { matrix}
\right ).
\end { align*}
%
Moreover, $ \chi _ j $ is irreducible and isomorphic to $ \chi _ { p - 1 - j } $ .
Let also $ \chi _ 0 $ be the representation:
%
\[
D_ { 2(p-1)} \to \FF _ p^ { \times } , \qquad \theta \mapsto 1, \qquad
s \mapsto -1.
\]
%
We claim that as $ k [ C ] $ -modules: ??k or $ \FF _ p $ ??
%
\begin { equation}
H^ 1_ { dR} (X) \cong V_ 0^ { \oplus (p-1)} \oplus \bigoplus _ { j = 1} ^ { \frac { p-1} { 2} } V_ j^ { \oplus 2(p-1)} .
\end { equation}
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{ \color { gray}
Basis of holomorphic differentials:
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%
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\[
\omega _ { a, b} = \frac { x^ a \cdot y^ b \, dx} { (x^ p - x)} \qquad 0 \le a, b \le p-2.
\]
}
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\bibliography { bibliografia,AKGeneral}
%
% \bibliography{AKGeneral}
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\end { document}