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 $mathbf{mu}_Ncong mathbb{Z}/Nmathbb{Z}$ 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.