The shapes of Galois quartic fields

H, Piper; Harron, Robert

doi:10.1090/tran/8137

The shapes of Galois quartic fields
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by Piper H and Robert Harron PDF

Trans. Amer. Math. Soc. 373 (2020), 7109-7152 Request permission

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 $V_4$ 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 $V_4$-quartic fields. For $C_4$-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.

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