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(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 $p^t \ge |G_P| = p^m$. +We use also the convention $u^{(0)}_{X/Y, P} = 1$. By Hasse--Arf theorem (cf. ???), the numbers $u_{X/Y, P}^{(t)}$ are integers. Define $n_{X/Y, P}$ by the equality $e_{X/Y, P} = p^{n_{X/Y, P}}$. +We abbreviate the last ramification jump to $u_{X/Y, P}$. \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 $P \in \pi^{-1}(Q)$.} % @@ -156,7 +157,7 @@ $u_{X/Y, Q}^{(t)} := u_{X/Y, P}^{(t)}$ etc. for arbitrary $P \in \pi^{-1}(Q)$.} % \[ H^1_{dR}(X) \cong J_{p^n}^{2 (g_Y - 1)} \oplus J_{p^n - p^{n-m} + 1}^2 \oplus \bigoplus_{\red{\substack{Q \in Y(k)\\ Q \neq Q_0}}} J_{p^n - p^n/e_{\red{Q}}}^2 - \oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t \ge 0} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}}, + \oplus \bigoplus_{\red{Q \in Y(k)}} \bigoplus_{t = 0}^{n_{X/Y, P}} J_{\red{p^n - p^{n+t}/e_Q}}^{u_Q^{(t+1)} - u_Q^{(t)}}, \] % where $e_Q := e_{X/Y, Q}$ and $u_Q^{(t)} := u_{X/Y, Q}^{(t)}$. @@ -182,7 +183,8 @@ and $\mc T^i M := T^i_{H'} M$ for any $k[H']$-module $M$.\\ 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: +Assume now that $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$. Let $P' \in X'(k)$ be the image of $P \in X(k)$. Then, if +$e_{X/Y, P} = p^n$, we have: % \begin{align*} i_{X/X', P}^{(t)} &= @@ -192,6 +194,8 @@ Moreover, if $X' \to Y$ is the $\ZZ/p^N$-subcover of $X \to Y$ for $N \le n$ and \end{cases}\\ i_{X'/Y, P'}^{(t)} &= i_{X/Y, P}^{(t)} \qquad \textrm{ for } t < N. \end{align*} +If $e_{X/Y, P} \le p^{n - N}$ then $i_{X/Y, P}^{(t)} = i_{X/X', P}^{(t)}$ +for all $t$. } % \begin{Lemma} \label{lem:G_invariants_\'{e}tale} @@ -318,16 +322,30 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \end{proof} % \begin{Lemma} \label{lem:u_equals_ul} + Assume that $Y' \to Y$ is a $\ZZ/p$-subcover of $X \to Y$. For any $Q \in Y(k)$: % \[ - p \cdot (u^{(n_Q)}_{X/Y, Q} - 1) = \sum_{Q'} \left( (u^{(n_{Q'})}_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right), + p \cdot (u_{X/Y, Q} - 1) = \sum_{Q'} \left( (u_{X/Y', Q'} - 1) + (p-1) \cdot (l^{(1)}_{Y'/Y, Q'} + 1) \right), \] % - where we sum over points $Q' \in Y'(k)$ lying above $Q$ and $n_Q := n_{X/Y, Q}$, $n_{Q'} := n_{X/Y', Q'}$. + where we sum over points $Q' \in Y'(k)$ lying above $Q$. \end{Lemma} \begin{proof} - ???? + % + Consider the following two cases. If $e_{Y'/Y, Q} = 1$ then + $l^{(1)}_{Y'/Y, Q} = - 1$ and $u_{X/Y, Q} = u_{X/Y', Q'}$ for all $p$ points $Q' \in Y'(k)$ in the preimage of $Q$. This easily implies the desired equality.\\ + If $e_{Y'/Y, Q} = 1$, then there exists a unique point $Q' \in Y'(k)$ + in the preimage of $Q$ through $Y' \to Y$. Moreover, $n_{X/Y, Q} = n_{X/Y', Q'}$. By using ????above formulas???: + % + \begin{align*} + p \cdot (u_{X/Y, Q} - 1) &= + p \cdot (i^{(0)}_{X/Y, Q} + \ldots + i^{(n_Q - 1)}_{X/Y, Q} - 1)\\ + &= (p-1) \cdot (i^{(0)}_{X/Y, Q} + 1) + \left( (i^{(0)}_{X/Y, Q} + p \cdot i^{(1)}) + p \cdot (i^{(2)}_{X/Y, Q} + i^{(3)}_{X/Y, Q} + \ldots) - 1 \right)\\ + &= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + (i^{(0)}_{X/Y', Q'} + i^{(1)}_{X/Y', Q'} + \ldots - 1)\\ + &= (p-1) \cdot (l^{(1)}_{X/Y, Q} + 1) + u_{X/Y', Q'} - 1. + \end{align*} + % \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:cyclic_de_rham}] @@ -377,10 +395,10 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. % \begin{align*} \dim_k \mc T^i \mc M &= - 2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'}^{(n_{Q'})} - 1)\\ + 2(g_{Y'} - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', 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) + &+ 2 + 2(\# R - 1) + \sum_{Q' \in Y'(k)} (u_{X/Y', Q'} - 1)\\ + &= p \cdot \left( 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q' \in Y(k)} (u_{X/Y, Q'} - 1) \right) \end{align*} % where @@ -392,7 +410,7 @@ 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_{Q \in Y(k)} (u_{X/Y, P}^{(n_Q)} - 1). + &= 2(g_Y - 1) + 2 + 2(\# R - 1) + \sum_{Q \in Y(k)} (u_{X/Y, P} - 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 @@ -415,7 +433,7 @@ shows that $m_{\sigma - 1}$ is well-defined and injective. \ker(\tr_{X/X''} : \mc M \to \mc M'') = \mc M^{(p^n - p^{n-1})} \] % - and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus: + and that $\tr_{X/X''}$ induces a $k$-linear isomorphism $T^{i + p^n - p^{n-1}} \mc M \to \mc T^i \mc M''$ for any $i \ge 1$. Thus, if $i \in [p^{n-1} - p^k, p^{n-1} - p^{k-1}]$: % \[ \dim_k T^{i + p^n - p^{n-1}} \mc M = \dim_k \mc T^i \mc M'' = ....