Resumen de Fields of Moduli of Hyperelliptic Curves

Bonnie Huggins

• 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.