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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 7109-7152
  • MSC (2010): Primary 11R16, 11R45, 11E12, 11P21
  • DOI: https://doi.org/10.1090/tran/8137
  • MathSciNet review: 4155202