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article_de_rham_cyclic.bbl
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@ -5,6 +5,18 @@ 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{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{Serre1979}
J.-P. Serre.
\newblock {\em {Local fields}}, volume~67 of {\em {Graduate Texts in
Mathematics}}.
\newblock Springer-Verlag, New York-Berlin, 1979.
\newblock Translated from the French by Marvin Jay Greenberg.
\bibitem{Valentini_Madan_Automorphisms}
R.~C. Valentini and M.~L. Madan.
\newblock Automorphisms and holomorphic differentials in characteristic~{$p$}.

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article_de_rham_cyclic.synctex.gz
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162
article_de_rham_cyclic.tex
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@ -27,6 +27,7 @@ hyperref, bbm, mathtools, mathrsfs}
\newcommand{\wt}{\widetilde}
\newcommand{\mf}{\mathfrak}
\newcommand{\ms}{\mathscr}
\newcommand{\red}[1]{{\color{red}#1}}
\renewcommand{\AA}{\mathbb{A}}
\newcommand{\II}{\mathbb{I}}
\newcommand{\HH}{\mathbb{H}}
@ -143,27 +144,22 @@ of $H^1_{dR}(X)$ isn't determined uniquely by the ramification data, see \cite{?
\section{Cyclic covers}
%
Let for any $\ZZ/p^n$-cover $X \to Y$
%
\begin{align*}
u_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_P^{(t)} \cong \ZZ/p^{n-t} \},\\
l_{X/Y, P}^{(t)} &:= \min \{ t \ge 0 : G_{P, t} \cong \ZZ/p^{n-t} \}.
\end{align*}
%
Note that if $G_P = \ZZ/p^n$, this coincides with the standard definition of
the $t$th upper (resp. lower) ramification jump of $X \to Y$ at $P$.
If $G_P = \ZZ/p^m$, then (??relation with usual jumps??). By Hasse--Arf theorem (cf. ???),
the numbers $u_{X/Y, P}^{(t)}$ are integers.
\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 $P$.}
We use also the convention $u^{(0)}_{X/Y, P} = 1$ and $u^{(t)}_{X/Y, P} := u^{(m)}_{X/Y, P}$, if $p^t \ge |G_P| = p^m$.
By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $n_{X/Y, P}$
by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$.
\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 $P \in \pi^{-1}(Q)$.}
%
\begin{Theorem} \label{thm:cyclic_de_rham}
Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k) \rangle = \ZZ/p^m$. Pick arbitrary $P_0 \in X(k)$ such that $G_{P_0} \cong \ZZ/p^m$. 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 $k[\ZZ/p^n]$-modules:
%
\[
H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\ P \neq P_0}} J_{p^n - p^n/e_P}^2
\oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-1} J_{p^n - p^t}^{u_{P}^{(t+1)} - u_{P}^{(t)}},
H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\red{\substack{Q \in Y(k)\\ Q \neq Q_0}}} J_{p^n - p^n/e_{\red{Q}}}^2
\oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t \ge 0} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}},
\]
%
where $e_P := e_{X/Y, P}$ and $u_P^{(t)} := u_{X/Y, P}^{(t)}$.
where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$.
\end{Theorem}
%
Write $H := \langle \sigma \rangle \cong \ZZ/p^n$.
@ -178,32 +174,62 @@ Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????)
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 $M$.\\
Note also that for $j \ge 1$:
%
\[
u_{X/Y, P}^{(j)} - u_{X/Y, P}^{(j-1)} = \frac{1}{p^{j-1}} (l_{X/Y, P}^{(j)} - l^{(j-1)}_{X/Y, P})
\]
%
(in particular, $u_{X/Y, P}^{(1)} = l_{X/Y, P}^{(1)}$). Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ then:
%
\begin{itemize}
\item $u_{X'/Y, P'}^{(t)} = u_{X/Y, P}^{(t)}$ for $t \le N$ (here $P'$ denotes the image of~$P$ on~$X'$),
\item $l_{X/X', P}^{(t)} = l_{X/Y, P}^{(t + N)}$ for $t \le n-N$.
\end{itemize}
\noindent \red{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)}\\
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}.
\end{align*}
%
Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ and $P'$ is the image of $P \in X(k)$ on $X'$ then:
%
\begin{align*}
i_{X/X', P}^{(t)} &=
\begin{cases}
i_{X/Y, P}^{(0)} + i_{X/Y, P}^{(1)} \cdot p + \ldots + i_{X/Y, P}^{(N)} \cdot p^N, & t = 0\\
p^N \cdot i_{X/Y, P}^{(N+t)}, & t = 1, \ldots, n-N-1.
\end{cases}\\
i_{X'/Y, P'}^{(t)} &= i_{X/Y, P}^{(t)} \qquad \textrm{ for } t < N.
\end{align*}
}
%
\begin{Lemma} \label{lem:G_invariants_\'{e}tale}
If the $G$-cover $X \to Y$ is \'{e}tale, then the natural map
If the $G$-cover $X \to Y$ is \'{e}tale, then
%
\[
H^1_{dR}(Y) \to H^1_{dR}(X)^G
\red{\dim_k H^1_{dR}(X)^G = 2g_Y.}
\]
%
is an isomorphism.
\end{Lemma}
\begin{proof}
????
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)\\
= 2g_Y - \dim_k H^1(G, k) + \dim_k H^2(G, k) ????.
\end{align*}
\end{proof}
%
\begin{Lemma} \label{lem:trace_surjective}
@ -291,35 +317,17 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
Since the left-hand side and right hand side are equal, we conclude by Lemma~\ref{lem:TiM_isomorphism}
\end{proof}
%
\begin{Lemma}
For any $P \in X(k)$:
\begin{Lemma} \label{lem:u_equals_ul}
For any $Q \in Y(k)$:
%
\[
p \cdot u^{(n)}_{X/Y, P} = u^{(n-1)}_{X/Y', P} + (p-1) \cdot l^{(1)}_{Y'/Y, Q},
p \cdot (u^{(n_Q)}_{X/Y, Q} - 1) = \sum_{Q'} \left( (u^{(n_{Q'})}_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right),
\]
%
where we sum over points $Q' \in Y'(k)$ lying above $Q$ and $n_Q := n_{X/Y, Q}$, $n_{Q'} := n_{X/Y', Q'}$.
\end{Lemma}
\begin{proof}
Note that:
%
\begin{align*}
u^{(n)}_{X/Y, P} - u^{(1)}_{X/Y, P} &= \sum_{j = 1}^{n-1} (u^{(j+1)}_{X/Y, P} - u^{(j)}_{X/Y, P}) = \sum_{j = 1}^{n-1} \frac{1}{p^j} (l^{(j+1)}_{X/Y, P} - l^{(j)}_{X/Y, P}).
\end{align*}
%
Similarly:
%
\[
u^{(n-1)}_{X/Y', P} - u^{(1)}_{X/Y', P} = \sum_{j = 1}^{n-2} \frac{1}{p^j} (l^{(j+1)}_{X/Y', P} - l^{(j)}_{X/Y', P}).
\]
We have:
%
\begin{align*}
p \cdot u^{(n)}_{X/Y, P} &= p \cdot (u^{(n)}_{X/Y, P} - u^{(1)}_{X/Y, P}) + u^{(1)}_{X/Y, Q}\\
&=p \cdot \sum_{j = 1}^{n-1} (u^{(j+1)}_{X/Y, P} - u^{(j)}_{X/Y, P}) + u^{(1)}_{X/Y, P}\\
&=p \cdot \sum_{j = 1}^{n-1} \frac{1}{p^j} (l^{(j+1)}_{X/Y, P} - l^{(j)}_{X/Y, P}) + u^{(1)}_{X/Y, P}\\
&=p \cdot \sum_{j = 1}^{n-1} \frac{1}{p^j} (l^{(j)}_{X/Y', P} - l^{(j-1)}_{X/Y', P}) + u^{(1)}_{X/Y, P}\\
\end{align*}
????
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}]
@ -361,25 +369,18 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
By induction hypothesis for $H'$ acting on $X$, we have the following isomorphism of $k[H']$-modules:
%
\[
\mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\P \neq P_0}} \mc J_{p^{n-1} - p^{n-1}/e'_P}^2
\oplus \bigoplus_{P \in X(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', P}^{(t+1)} - u_{X/Y', P}^{(t)}}
\mc M \cong \mc J_{p^{n-1}}^{2 (g_{Y'} - 1)} \oplus \mc J_{p^{n-1} - p^{n - m} + 1}^2 \oplus \bigoplus_{\substack{Q \in Y'(k)\\Q \neq Q_1}} \mc J_{p^{n-1} - p^{n-1}/e'_Q}^2
\oplus \bigoplus_{Q \in Y'(k)} \bigoplus_{t = 0}^{n-2} \mc J_{p^n - p^t}^{u_{X/Y', Q}^{(t+1)} - u_{X/Y', Q}^{(t)}}
\]
%
where $e'_P := e_{X/Y', P}$. Note that for any $P \in X(k)$:
%
\[
p \cdot u^{(n)}_{X/Y, P} = u^{(n-1)}_{X/Y', P} + (p-1) \cdot l^{(1)}_{Y'/Y, Q},
\]
%
where $Q$ denotes the image of~$P$ in~$Y'$. Indeed, ?????
Therefore, for $i \le p^{n-1} - p^{n-2}$, using the Riemann--Hurwitz formula (cf. ????):
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}:
%
\begin{align*}
\dim_k \mc T^i \mc M &=
2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 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_{P \in X(k)} (u_{X/Y', P}^{(n-1)} - 1)\\
&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1) \right)
2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'}^{(n_{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'}^{(n_{Q'})} - 1)\\
&= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'}^{(n_Q)} - 1) \right)
\end{align*}
%
where
@ -391,7 +392,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective.
%
\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_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1).
&= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P}^{(n_Q)} - 1).
\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
@ -529,19 +530,16 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo
\end{equation}
%
Observe that $\mc T^i \mc M$ has the filtration $\mc M^{(pi)} \supset \mc M^{(pi - 1)} \supset \ldots \supset \mc M^{(pi - p)}$ with subquotients $T^{pi} \mc M, \ldots, T^{pi - p} \mc M$.
Thus, since the category of $k[C]$-modules is semisimple, for $i \le p^n - p^{n-1}$:
Thus, since the category of $k[C]$-modules is semisimple:
%
\begin{align*}
\mc T^i \mc M &\cong
\begin{cases}
T^1 \mc M \oplus T^2 \mc M \oplus (T^2 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p - 2)}}, & i = 1\\
T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}}, & i > 1.
\end{cases}
\end{align*}
\begin{align}
\mc T^1 \mc M &\cong T^1 \mc M \oplus T^2 \mc M \oplus (T^2 \mc M)^{\chi^{-1}} \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p - 2)}} \label{eqn:decomposition_of_mc_T1}\\
\mc T^i \mc M &\cong T^2 \mc M \oplus \ldots \oplus (T^2 \mc M)^{\chi^{-(p-1)}} \quad \textrm{ for } 2 \le i \le p^n - p^{n-1}. \label{eqn:decomposition_of_mc_Ti}
\end{align}
%
Thus, since by induction hypothesis $\mc T^2 \mc M$ is determined by ramification data,
we have by Lemma~\ref{lem:N+Nchi+...} that $T^2 \mc M$ is determined by ramification data.
Moreover, by Lemma~\ref{lem:G_invariants_\'{e}tale} and induction hypothesis, $T^1 \mc M \cong H^1_{dR}(X'')$
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}}, $T^1 \mc M$
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}

9
ramification.tex
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@ -303,7 +303,10 @@ but maybe you mean
l_{X/Y, P}^{(t)} &:= \min \{ \nu \ge 0 : G_{P, \nu} \cong \ZZ/p^{n-t} \}-1.
\end{align}
}
{\color{green}
Adding one to usual jumps was unitentional. It doesn't change any thing in the formula for $H^1_{dR}(X)$ (we have differences there),
but let's return to the usual definition of ramification jumps.
}
\end{definition}
%
{\color{blue}
@ -335,7 +338,7 @@ We then have:
i_{j-1} =u_{X/Y,P}^{(j)}-u_{X /Y,P}^{(j-1)}= \frac{1}{p^{j-1}} (l_{X/ Y,P}^{j} - l_{X/ Y,P}^{j-1}) = \frac{1}{p^{j-1}} p^{j-1} i_{j-1}
\]
{\color{red} you have written it in the other way out, do you agree?}
{\color{green} Yes, it was the other way around!}
% Now the ramification jumps for a subgroup I thing are a little bit different from what you write.
@ -418,7 +421,7 @@ By induction hypothesis for $H'$ acting on $X$, we have the following isomorphis
\]
%
where $e'_P := e_{X/Y', P}$.
{\color{green} The formula above needs a correction -- I want to sum over branch locus in $Y(k)$! This matters if the cover is not completely ramified.}
%
\begin{align*}
\dim_k \mc T^i \mc M &=