Fields of moduli and definition of hyperelliptic curves of odd genus

摘要

Mestre has shown that if a hyperelliptic curve C of even genus is defined over a subfield then C can be hyperelliptically defined over the same field k. In this paper, for all genera g > 1, mod 4, hence odd, we construct an explicit hyperelliptic curve defined over which can not be hyperelliptically defined over .