Fixed points and homology of superelliptic Jacobians

TL;DR

This paper investigates the fixed points and homological properties of superelliptic Jacobians, revealing their structure under cyclic group actions and computing polarization degrees, advancing understanding of their algebraic and geometric features.

Contribution

It determines the fixed point group of the cyclic action on superelliptic Jacobians and analyzes the Tate module structure, providing new insights into their algebraic properties.

Findings

Fixed points of cyclic group action explicitly characterized.

Tate module is a free module over a specific ring.

Degree of polarization on the new part of the Jacobian computed.

Abstract

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.