SOME SPECIAL FAMILIES OF HYPERELLIPTIC CURVES

Abstract

Let [Formula: see text] denote the locus of hyperelliptic curves of genus g whose automorphism group contains a subgroup isomorphic to G. We study spaces [Formula: see text] for G≅ℤ n , ℤ 2 ⊕ℤ n , ℤ 2 ⊕A 4 , or SL 2 (3). We show that for G≅ℤ n , ℤ 2 ⊕ℤ n , the space [Formula: see text] is a rational variety and find generators of its function field. For G≅ℤ 2 ⊕A 4 , SL 2 (3) we find a necessary condition in terms of the coefficients, whether or not the curve belongs to [Formula: see text]. Further, we describe algebraically the loci of such curves for g≤12 and show that for all curves in these loci, the field of moduli is a field of definition.