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# Mathematical Research Letters

## Volume 14 (2007)

### Number 2

### Fields of Moduli of Hyperelliptic Curves

Pages: 249 – 262

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n2.a8

#### Author

#### Abstract

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.