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ze|3mKYAvT_6$+^oa*!HS4ib(&Nr##SiDW$d@by7z?8>(!Wvr5>(^&fNq=_XnpkwTC z{Eh{&0FVtZBMN>ehz|Ew;1Gu+Z0QVVEcCC0**-#{KZB`9LU$as^B>l~__tqw`~3?} za^AfD&(H5ZeSZD+(}&MrKK%6V^W(=~f4_Wu|L)_b_a8pL{*Mnoe0uxo@2~&#%g5h; z`tz5MpWeQI_xfKyef;f*k58}v{P$nqeR}`)%e&97|NY%hpZ@#ZfBj1<=3J_rDK(9(ba diff --git a/article_de_rham_cyclic.tex b/article_de_rham_cyclic.tex index 72ae431..fd78367 100644 --- a/article_de_rham_cyclic.tex +++ b/article_de_rham_cyclic.tex @@ -231,7 +231,7 @@ $\mc V(M, i)$ for the $k[G]$-module corresponding to a pair $(M, i) \in \Indec(k 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: % \begin{equation} - H^0(X, \Omega_X) \cong \bigoplus_{W \in \Indec(k[G])} M^{\oplus a_M}, + H^0(X, \Omega_X) \cong \bigoplus_{M \in \Indec(k[G])} M^{\oplus a_M}, \end{equation} % where: @@ -630,8 +630,33 @@ Let $X$ be a curve with an action of $G$ and write $Y := X/H$. For any $k[C]$-mo This is basically \cite[proof of Theorem~1.1]{Bleher_Chinburg_Kontogeorgis_Galois_structure}. We sketch the proof for reader's convenience. Let $\psi : C \to k^{\times}$ be a primitive character. Write % \[ - M \cong \bigoplus_{a, b} \mc V(\psi^a, b)^{\oplus n(a, b)}. + M \cong \bigoplus_{i = 1}^{p^n} \bigoplus_{W \in \Indec(C)} \mc V(W, i)^{\oplus n(W, i)}. \] + % + Note that as $k[C]$-modules: + % + \[ + T^j \mc V(W, i) \cong + \begin{cases} + W^{\chi^{-j + 1}}, & \textrm{ if } j \le i,\\ + 0, & \textrm{ if } j > i. + \end{cases} + \] + % + Hence: + % + \[ + T^j M \cong \bigoplus_{i = j}^{p^n} \bigoplus_{W \in \Indec(C)} (W^{\chi^{-j + 1}})^{\oplus n(W, i)} + \] + % + and the $k[C]$-module structure of $T^j M$ determines uniquely + the numbers: + % + \[ + \sum_{i = j}^{p^n} n(W, i) + \] + % + for every $W \in \Indec(k[C])$. This easily implies that the numbers $n(W, 1)$, $\ldots$, $n(W, p^n)$ are uniquely determined by the $k[C]$-structure of $T^1 M$, $\ldots$, $T^{p^n} M$. The proof follows. \end{proof} % \begin{Lemma} \label{lem:N+Nchi+...}