diff --git a/article_de_rham_cyclic.bbl b/article_de_rham_cyclic.bbl index e7e97c7..e935b4e 100644 --- a/article_de_rham_cyclic.bbl +++ b/article_de_rham_cyclic.bbl @@ -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~{$p$}. diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index 495728a..0b9cb48 100644 Binary files a/article_de_rham_cyclic.synctex.gz and b/article_de_rham_cyclic.synctex.gz differ diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 5e32b2f..cf11131 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -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} diff --git a/ramification.tex b/ramification.tex index 0d80184..af474c4 100644 --- a/ramification.tex +++ b/ramification.tex @@ -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 &=