Fixed points and homology of superelliptic Jacobians

Abstract

Let $$\eta : C_{f,N}\rightarrow \mathbb {P}^1$$ 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 covering group $${{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}$$ acting on the Jacobian $$J_N:={{\mathrm{Jac}}}(C_{f,N})$$ . For each prime $$\ell $$ 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 study the subvarieties of $$J_N$$ and calculate the degree of the induced polarization on the new part $$J_N^\mathrm {new}$$ of the Jacobian.