Curves which cannot be defined over an extension of degree at most two over the field of moduli

TL;DR

This paper disproves a conjecture by providing examples of hyperelliptic curves that cannot be defined over any extension of degree two of their fields of moduli, challenging assumptions about the definability of algebraic curves.

Contribution

It presents explicit counterexamples of hyperelliptic curves that cannot be defined over degree two extensions of their fields of moduli, refuting the conjecture.

Findings

Counterexamples of hyperelliptic curves not definable over degree two extensions

Disproof of the conjecture on defining algebraic curves over small extensions

Implications for the understanding of fields of moduli and definition

Abstract

It has been conjectured that every algebraic curve may be defined either over its field of moduli or over an extension of degree two of it. In this paper we provide a negative answer to it by giving examples of hyperelliptic curves which cannot be defined over an extension of degree at least two over their fields of moduli.