Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

TL;DR

This paper provides an explicit description of the Galois descent obstruction for hyperelliptic curves with cyclic automorphism groups, offering criteria for descent and methods to find models over minimal fields.

Contribution

It introduces an explicit arithmetic criterion for hyperelliptic descent and describes the Galois obstruction using arithmetic dihedral invariants.

Findings

Obstruction vanishes when invariants are trivial, enabling descent over the field of moduli.

If obstruction does not vanish, a hyperelliptic model exists over a quadratic extension.

Provides explicit models and criteria for hyperelliptic descent with cyclic automorphism groups.

Abstract

This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic model over a degree 2 extension of this field.