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 $V_4$ case, each family is a twodimensional 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 onedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 DOI:
 10.48550/arXiv.1908.03969
 arXiv:
 arXiv:1908.03969
 Bibcode:
 2019arXiv190803969H
 Keywords:

 Mathematics  Number Theory;
 11R16;
 11R45;
 11E12;
 11P21
 EPrint:
 37 pages. Comments welcome