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 $\mathbf{mu}_N\cong \mathbb{Z}/N\mathbb{Z}$ acting on the Jacobian $J_N:=\Jac(C_{f,N})$. For each $\ell$ distinct from the characteristic of the base field, the Tate module $T_\ell J_N$ is shown to be a free module over the ring $\mathbb{Z}_\ell[T]/(\sum_{i=0}^{N-1}T^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian.