Cyclic covers and Ihara's Question

TL;DR

This paper investigates the field generated by the $ ell$-power torsion of Jacobians of superelliptic curves over number fields, extending previous results by decomposing Galois representations and relating torsion fields to branch set data.

Contribution

It provides a detailed description of the torsion fields for superelliptic Jacobians using cyclic cover decompositions and extends earlier work by Ihara and Anderson.

Findings

Torsion fields are described in terms of branch set and reduction type.

Decomposition of Galois representations into block triangular form.

Extension of earlier results on torsion rationality over specific Galois extensions.

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 $\mathbf{P}^1$.