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z(p@kPLfcmaq3s8aAT;7u=fizPmh;+cKKGSIxbItgA1Kl-V!%W6Mmu{RS1lpnHRxzq zB^$IIfPul1y4Qa5q&7#e;9i(ruvuzISsoGiVO#st8zkatmkLt&01;EI9@5n;X6 zx&n4!uhvz3Kq_4gRU!wzRZtw;49>eN%EFZe!EhzY@ ziY7ba>uRWqaDgRgs0y9MW(7r^MD|^nn(@+xawH9IgKzOTxU_N=4xTy}+r=VdyyJ}S z-TdWnGC1K(?>=pRxJgkDWLCpIKTRoI^I^AcEW(r;HPIv^iCmg-COEWKR!A>1wepO*A2)Rp3H9W5oz&Qsu_Ohnk`B{I~-*+ZWDmjsvzgF!kfLKoj@fKWupSAAkM%@4w<0_W8HZ z-~HDwA3p!`-Jd@E`1yZ6{OLCzzyJK#4=>;S`{%#?{Kvn&e)pfp&;R!Mr(b^k>BkSh zeE0L`58r=Y{`IFH{`}#)^Y1_W_~8$qKYjm~Km73DKm7TpAAa-w-(TN-L#y!J|NYx9 zzkdIhKmPf{^Sj^v?eBm66&HRvz59k{JxKrOmmfa=^z*NO|I72c{|5>wz7nX52mmh7 BaOD61 diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index fd78367..e7d600c 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -187,7 +187,10 @@ Throughout the paper we will use the following notation for any $P \in X(\ol k)$ \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. + \item $u_{X/Y, P} := u_{X/Y, P}^{(m_{X/Y, P})}$ is the last ramification jump, + + \item $\theta_{X/Y, P} : G_P \to \Aut_k(\mf m_P/\mf m_P^2) \cong k^{\times}$ + is the fundamental character of~$P$. \end{itemize} % By Hasse--Arf theorem (cf. @@ -221,14 +224,15 @@ 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) + \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: +Finally, we recall the classical Chevalley-Weil formula. Keep the above notation and assume that $p \nmid \# G$. For any $e \in \NN$, denote by $\chi_e$ the primitive character of a cyclic group of order $e$. +Then: % \begin{equation} H^0(X, \Omega_X) \cong \bigoplus_{M \in \Indec(k[G])} M^{\oplus a_M}, @@ -237,10 +241,10 @@ Finally, we recall the classical Chevalley-Weil formula. Keep the above notation 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), + a_M := - \dim_k M + \sum_{Q \in Y(k)} \sum_{i = 0}^{e_{X/Y, Q} - 1} \frac{e_{X/Y, Q} - i}{e_{X/Y, Q}} \cdot N_{P, i}(M), \end{align*} % -and $N_{P, i}(M) := ???$. +and $N_{P, i}(M)$ is the multiplicity of the character $\chi_{e_Q}^i$ in the $k[G_Q]$-module $M \otimes_{k[G_Q]} \theta_{X/Y, Q}$. \section{Cyclic covers} % @@ -788,7 +792,15 @@ Keep the above notation. If $G$ acts on a curve $X$ and the cover $X \to X/H$ is H^1_{dR}(X) \cong \bigoplus_{M \in \Indec(C)} \mc V(M, p-1)^{\oplus b_M} \oplus \mc V(M, p)^{\oplus c_M}, \] % -where $b_M := \ldots$, $c_M := \ldots$. +where +% +\begin{align*} + b_M &:= (1 - \frac 1p) \cdot \dim_k M + \sum_{Q \in (X/G)(k)} \sum_{i < e_{X/Y, Q}??} +\end{align*} +% + + + $b_M := \ldots$, $c_M := \ldots$. \end{Proposition} \begin{proof} ???