From 3ef78ffca35f3a1d1f53015cc3622e26d6204480 Mon Sep 17 00:00:00 2001 From: jgarnek Date: Fri, 25 Oct 2024 11:18:30 +0200 Subject: [PATCH] irreducible -> indecomposable; ccorrect def of X''; Hasse Arf --- article_de_rham_cyclic.synctex.gz | Bin 63467 -> 65378 bytes article_de_rham_cyclic.tex | 24 +++++++++++++++--------- 2 files changed, 15 insertions(+), 9 deletions(-) diff --git a/article_de_rham_cyclic.synctex.gz b/article_de_rham_cyclic.synctex.gz index ce17ce83f2101c1d16eed770249617ff3c645f11..2e2ffa131e97521362363a8089f5c5b981a6bb7d 100644 GIT binary patch literal 65378 zcmV){Kz+X-iwFP!000003hcdEk7c*A9{8?b;S1zLCLkSf9_r1o(Li^@aHG53a6>vU zN~Kb%Zm9;>a9#3+|9gKQ*W$1l+>JZpa12j=l|vXFHc|pmuJ5{ z`}O6szkGcD`DAal?Eme%cmMwB$EQF1pKqVOe);Cb5C7ri|Bm|o0MZ|Reaip7dHRC? z{_^4DZznVR)y%&8)qH&Vmx^7f~{c$6xBtL3HtP)cEt`K%E{({o?tXfjB>o z_{R^w4a8bKj{5Dpv1qMtqyG8rn~xuU_ym#v<>#L&t$7S>{beltE7bM%%V+Q3e*Ep- z^Zwh&+1*ezz~24x^4W(UUOoTqpPv3YlKUITpMQM&_E!%7`pd{5{Gl!T?%hv61N4`l 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By Hasse--Arf theorem (cf. ???), +the numbers $u_{X/Y, P}^{(t)}$ are integers. % \begin{Theorem} \label{thm:cyclic_de_rham} - Suppose that $\pi : X \to Y$ is a $\ZZ/p^n$-cover. Let $\langle G_P : P \in X(k) \rangle = \ZZ/p^m = G_{P_0}$ for $P_0 \in X(k)$. Then, as $k[\ZZ/p^n]$-modules: + 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: % \[ H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\substack{P \in X(k)\\ P \neq P_0}} J_{p^n - p^n/e_P}^2 @@ -171,7 +176,7 @@ For any $k[H]$-module $M$ denote: % Recall that $\dim_k T^i M$ determines the structure of $M$ completely (cf. ????). In the inductive step we use also the group $H' := \ZZ/p^{n-1}$. In this case -we denote the irreducible $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$ +we denote the indecomposable $k[H']$-modules by $\mc J_1, \ldots, \mc J_{p^{n-1}}$ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\ Note also that for $j \ge 1$: % @@ -260,8 +265,8 @@ We define $m_{\sigma - 1}$ as follows: m_{\sigma - 1}(\ol x) := (\sigma - 1) \cdot x, \] % -where for $\ol x \in T^i M$ we picked any representative $x \in M^{(i)}$. -Indeed, if $x \in M^{(i+1)} := \ker((\sigma - 1)^{i+1})$ then clearly $(\sigma - 1) x \in M^{(i)}$. +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 $(\sigma - 1) \cdot x \in M^{(i)}$. 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. \end{proof} @@ -275,7 +280,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \] \end{Lemma} \begin{proof} - By Lemma~\ref{lem:TiM_isomorphism}: + Note that $\mc T^i M = M^{(pi)}/M^{(pi - p)}$. This easily implies that: % \begin{align*} \dim_k \mc T^i M &= \dim_k T^{pi} M + \ldots + \dim_k T^{pi - p + 1} M\\ @@ -288,7 +293,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] We use the following notation: $H' := \langle \sigma^p \rangle \cong \ZZ/p^{n-1}$, - $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/H''$. + $H'' := H/\langle \sigma^{p^{n-1}} \rangle \cong \ZZ/p^{n-1}$, $Y' := X/H'$, $X'' := X/\langle \sigma^{p^{n-1}} \rangle$. Note that $H''$ naturally acts on $X''$. Write also $\mc M := H^1_{dR}(X)$. We consider now two cases. If the cover $X \to Y$ is \'{e}tale, then by induction assumption, since $2(g_{Y'} - 1) = p \cdot 2 \cdot (g_Y - 1)$: % @@ -329,12 +334,13 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \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)}} \] % - where $e'_P := e_{X/Y', P}$. Note that + where $e'_P := e_{X/Y', P}$. Note that for any $P \in X(k)$: % \[ - ??? p \cdot u^{(n)} = u^{(n-1)} + (p-1) \cdot l^{(1)} ???. + p \cdot u^{(n)}_{X/Y, P} = u^{(n-1)}_{X/Y', P} + (p-1) \cdot l^{(1)}_{Y'/Y, Q}, \] % + where $Q$ denotes the image of~$P$ in~$Y'$. Therefore, for $i \le p^{n-1} - p^{n-2}$, using the Riemann--Hurwitz formula (cf. ????): % \begin{align*} @@ -342,7 +348,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. 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) + &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{P \in X(k)} (u_{X/Y, P}^{(n)} - 1) \right) \end{align*} % where