Fields of Moduli of Hyperelliptic Curves

Let $X$ be a hyperelliptic curve defined over a field $K$ of characteristic not equal to $2$. Let $\iota$ be the hyperelliptic involution of $X$. We show that $X$ can be defined over its field of moduli if $\Aut(X)/\langle \iota\rangle$ is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when $\Aut(X)/\langle \iota\rangle$ is cyclic.

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Published 1 January 2007