Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent
type: Item
instance of: scholarly article
title: Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent
zbMATH Open document ID: 1481.11107
author: Robert Harron
published in: Algebra \& Number Theory
publication date: 2021-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1907.07209
review text: The notion of the shape, of a number field \(K\), was introduced by \textit{D. Terr} [The distribution of shapes of cubic orders. University of California (Ph.D. thesis) (1997)]; wich is a certain lattice of rank \( [K : \mathbb Q] -1\) attached to \(K\) considered up to rotation, reflection, and scaling. In particular, he proved that the shapes of cubic fields are equidistributed in the space of shapes of rank \(2\) lattices as the discriminant of the cubic field goes to infinity. Other authors, to other number fields, generalized this result. The author investigate, first, the distribution of more specific families of number fields. Then he show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field's trace-zero form. Applying a method of \textit{M. Bhargava} and the author [Compos. Math. 152, No. 6, 1111--1120 (2016; Zbl 1347.11074)] to results of Bhargava and Ariel Shnidman, he proves that the shapes lying on a fixed geodesic become equidistributed with respect to the hyperbolic measure as the discriminant of the complex cubic field goes to infinity. He also show that the shape of a complex cubic field is a complete invariant within the family of all cubic fields.
reviewed by: Abdelmalek Azizi
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 7381103
zbMATH Keywords: algebraic number theory
zbMATH Keywords: cubic fields
zbMATH Keywords: lattices
zbMATH Keywords: arithmetic statistics
zbMATH Keywords: equidistribution
zbMATH Keywords: geodesics
zbMATH Keywords: majorant space
MaRDI profile type: Publication
OpenAlex ID: W3196012927
arXiv ID: 1907.07209
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DOI: 10.2140/ANT.2021.15.1095
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description: scientific article