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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The shapes of Galois quartic fields


Authors: Piper H and Robert Harron
Journal: Trans. Amer. Math. Soc. 373 (2020), 7109-7152
MSC (2010): Primary 11R16, 11R45, 11E12, 11P21
DOI: https://doi.org/10.1090/tran/8137
Published electronically: July 29, 2020
MathSciNet review: 4155202
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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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Additional Information

Piper H
Affiliation: Department of Mathematics, Keller Hall, University of Hawai‘i at Mānoa, Honolulu, Hawaii 96822
Email: piper@math.hawaii.edu

Robert Harron
Affiliation: Department of Keller Hall, University of Hawai‘i at Mānoa, Honolulu, Hawaii 96822
MR Author ID: 987029
Email: robert.harron@gmail.com

Keywords: Quartic fields, equidistribution, lattices, carefree tuples
Received by editor(s): August 11, 2019
Received by editor(s) in revised form: January 12, 2020
Published electronically: July 29, 2020
Additional Notes: The second author is partially supported by a Simons Collaboration Grant.
Communicated by: Henri Darmon
Article copyright: © Copyright 2020 American Mathematical Society