Fields of moduli of hyperelliptic curves

Details Fields of moduli of hyperelliptic curves Fields of moduli of hyperelliptic curves

2004-07-06

arxiv.org/abs/math/0407088v1

Mathematics

Number Theory

14 pages

Scientific paper

Let F be an algebraically closed field with char(F) not equal to 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let \iota be the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/<\iota> is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/<\iota> is cyclic.

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