Fixed points and homology of superelliptic Jacobians

Let $\eta :C_{f,N}\to {\mathbb{P}}^1$ be a cyclic cover of ${\mathbb{P}}^1$ of degree $N$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic group ${\mathbf{m}\mathbf{u}}_N\cong \mathbb{Z}\mathrm{/}N\mathbb{Z}$ acting on the Jacobian $J_N:=\text{Jac}(C_{f,N})$. For each $\mathrm{\ell }$ distinct from the characteristic of the base field, the Tate module $T_{\mathrm{\ell }}{J}_N$ is shown to be a free module over the ring ${\mathbb{Z}}_{\mathrm{\ell }}[T]\mathrm{/}({\sum }_{i=0}^{N-1}{T}^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian.