Cyclic covers and Ihara’s question

Abstract

Let $\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\ell$-power degree, we describe the field generated by the $\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\ell$ and the branch set is defined over a pro-$\ell$ extension of $k(\mu_{\ell^\infty})$ unramified away from $\ell$, then the $\ell$-power torsion of the Jacobian is rational over the maximal such extension. By decomposing the covering into a chain of successive cyclic $\ell$-coverings, the mod $\ell$ Galois representation attached to the Jacobian is decomposed into a block triangular form. The blocks on the diagonal of this form are further decomposed in terms of the Tate twists of certain subgroups $W_s$ of the quotients of the Jacobians of successive coverings. The result is a natural extension of earlier work by Anderson and Ihara, who demonstrated that a stricter condition on the branch locus guarantees the $\ell$-power torsion of the Jacobian is rational over the fixed field of the kernel of the canonical pro-$\ell$ outer Galois representation attached to an open subset of $\mathbf{P}^1$.