Let $\pi :C\rightarrow \mathbb{P}^{1}$ be a cyclic cover of degree $m$ of $\mathbb{P}^{1}$. So $C$ has an action of $\mathbb{Z}/m\mathbb{Z}$. Let $\xi$ be a primitive $m$-th root of unity. Consider the cohomology group $V_{\mathbb{Q}}=H^{1}(C,\mathbb{Q})$. In the book "Cylic covers, Calabi-Yau manifolds and complex multiplication" (by Jan Christian Rohde) page 80, we read: "In general there is no $\mathbb{Q}(\xi)$-structure on $H^{1}(C,\mathbb{Q})$ which turnes $H^{1}(C,\mathbb{Q})$ into a $\mathbb{Q}(\xi)$-vector space " and then he gives a direct sum decomposition with different $\mathbb{Q}(\xi^{r})$-structures. But in many books and articles on this subject it is just claimed that $V_{\mathbb{Q}}$ has an induced action of $\mathbb{Q}(\xi)$(Like: de Jong and Noot : "Jacobians with complex multiplication"). Which one is true? However, my second question is: even if there is an action of $\mathbb{Q}(\xi)$, I think we then must have two different $\mathbb{Q}$-vector space structures on $V_{\mathbb{Q}}$.
The first one is the usual one and one coming from $\mathbb{Q}\hookrightarrow \mathbb{Q}(\xi)$. Since if $dim_{\mathbb{Q}}V_{\mathbb{Q}}=g$, then by the induced action from $\mathbb{Q}\hookrightarrow \mathbb{Q}(\xi)$, it's dimension should be $dim_{\mathbb{Q}}V_{\mathbb{Q}}=m dimV_{\mathbb{Q}(\xi)}$ ($V_{\mathbb{Q}(\xi)}$ means $V_{\mathbb{Q}}$ as a $\mathbb{Q}(\xi)$-vector space ). But this two in general do not agree ($2g=g (C)$ and in general is not even divisible by $m$). So the $\mathbb{Q}$-structures should be different. Is this true? Can one give a good explanation of this?