Hyperelliptic curves with reduced automorphism group A 5

Abstract

We study genus g hyperelliptic curves with reduced automorphism group A 5 and give equations y 2 = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x 2 or x 5. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y 2 = F(x) or y 2 = x F(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x 2 or x 5. While similar equations have been given in (Bujalance et al. in Mm. Soc. Math. Fr. No. 86, 2001) over $${\mathbb R}$$, this is the first time that these equations are given over the field of moduli of the curve.