Fields of moduli and fields of definition of odd signature curves

TL;DR

This paper investigates conditions under which algebraic curves, especially odd signature, q-gonal, and plane quartic curves, can be defined over their field of moduli, providing new criteria and explicit examples.

Contribution

It establishes that odd signature curves can be defined over their field of moduli and explores the definability of q-gonal and plane quartic curves, including counterexamples.

Findings

Odd signature curves can be defined over their field of moduli.

Non-normal q-gonal curves are definable over their field of moduli.

Certain plane quartics with specific automorphism groups can be defined over their field of moduli.

Abstract

Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ is of type $(0;c_1,...,c_k)$, where some $c_i$ appears an odd number of times. This result is applied to $q$-gonal curves and to plane quartics. For $q$-gonal curves, we prove that non-normal $q$-gonal curves can be defined over their field of moduli and we construct examples of normal $q$-gonal curves with field of moduli $\mathbb{R}$ that can not be defined over $\mathbb{R}$. For plane quartics, we prove that they can be defined over their field of moduli if the automorphism group is not isomorphic to either $C_2$ or $C_2\times C_2$.