On the field of moduli of superelliptic curves

TL;DR

This paper investigates conditions under which superelliptic curves of genus at least 2 can be defined over their field of moduli, highlighting the role of the reduced automorphism group and providing specific criteria and examples.

Contribution

It establishes that superelliptic curves with non-cyclic reduced automorphism groups are definable over their field of moduli, and offers sufficient conditions in cyclic cases, along with genus-specific examples.

Findings

Curves with non-trivial, non-cyclic reduced automorphism groups are definable over their field of moduli.

Provides sufficient conditions for definability in cyclic automorphism group cases.

Identifies families of superelliptic curves of genus ≤ 10 that may not be definable over their field of moduli.

Abstract

A superelliptic curve $\X$ of genus $g\geq 2$ is not necessarily defined over its field of moduli but it can be defined over a quadratic extension of it. While a lot of work has been done by many authors to determine which hyperelliptic curves are defined over their field of moduli, less is known for superelliptic curves. In this paper we observe that if the reduced group of a genus $g\geq 2$ superelliptic curve $\X$ is different from the trivial or cyclic group, then $\X$ can be defined over its field of moduli; in the cyclic situation we provide a sufficient condition for this to happen. We also determine those families of superelliptic curves of genus at most $10$ which might not be definable over their field of moduli.