Fixed points and homology of superelliptic Jacobians

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.

21 pages. Fixed some typos. Statement of Theorem 1.3 improved. Results unchanged

References in corpus (2)

• Superelliptic jacobians
• Endomorphism algebras of factors of certain hypergeometric Jacobians