The shapes of Galois quartic fields
Abstract
We determine the shapes of all degree 4 number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the V4 case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of V4-quartic fields. For C4-quartic fields, each family is a one-dimensional space of tetragonal lattices and the shapes make up a discrete subset of points in these spaces. We prove asymptotics for the number of fields with a given shape in this case.