We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.
Field of moduli and field of definition for curves of genus 2
Fields of definition of dynamical systems on $\mathbb{P}^{1}$. Improvements on a result of Silverman
Plane model-fields of definition, fields of definition, the field of moduli of smooth plane curves
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