Fields of moduli and definition of hyperelliptic curves of odd genus (Q707957)

A hyperelliptic curve is said to be hyperelliptically defined over a field \(k \subset \mathbb{C}\) if it is birationally equivalent to a curve of the form \(y^2=q(x)\), where \(q(x)\) is a polynomial with simple roots and coefficients in \(k\). In [Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 313-334 (1991 ; Zbl 0752.14027)] it was shown that `being defined over \(k\)' and `being hyperelliptically defined over \(k\)' are equivalent when the genus is even. However, in this article, the author constructs an explicit hyperelliptic curve defined over \(\mathbb{Q}\) which cannot be hyperelliptically defined over \(\mathbb{Q}\) for all \(g>1\) with \(g \equiv 1 \pmod{4}\). The proof goes as follows. In [Arch. Math. 86, No. 5, 398--408 (2006; Zbl 1095.14028)], the authors showed that a certain hyperelliptic curve \(C\) of genus \(g \equiv 1 \pmod{4}\) could not be hyperelliptically defined over \(\mathbb{Q}\). However, in the present article, it is proved that the field of moduli of \(C\) is \(\mathbb{Q}\). Since this curve has automorphism group \((\mathbb{Z}/2\mathbb{Z})^2\), which is not cyclic, [Math. Res. Lett. 14, No. 2, 249--262 (2007; Zbl 1126.14036)] shows that the field of moduli is a field of definition.