this paper we shall use either quartic coverings of the projective line or quadratic coverings of elliptic curves with enough ramification points (instead of quadratic coverings of the projective line as in the class field tower constructions) to find for every genus g 3 curves with infinite geometric fundamental group defined over Q(i) or over F q (i) (where i is a forth root of unity) and we find even parametric families of such curves over every ground field containing i (cf. Theorem 5.22).