On the field of moduli of superelliptic curves

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.