The shapes of Galois quartic fields

TL;DR

This paper classifies the geometric shapes of all Galois quartic fields, showing their distribution and invariance properties across different Galois groups and ramification types, with detailed asymptotic results.

Contribution

It provides a complete description of the shapes of Galois quartic fields, including distribution, invariance, and asymptotic counts, depending on Galois group and ramification.

Findings

Shapes are equidistributed in V_4 families as discriminant increases.

Shape is a complete invariant for some V_4 quartic fields.

Asymptotics for the number of fields with a given shape in C_4 case.

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.