Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

Harron, Robert

doi:10.2140/ant.2021.15.1095

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Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

Robert Harron

Vol. 15 (2021), No. 5, 1095–1125

DOI: 10.2140/ant.2021.15.1095

Abstract

We 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. We also prove a general such statement for all orders in étale $Q$ -algebras. Applying a method of Manjul Bhargava and Piper H to results of Bhargava and Ariel Shnidman, we prove 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. We also show that the shape of a complex cubic field is a complete invariant (within the family of all cubic fields).

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Keywords

algebraic number theory, cubic fields, lattices, arithmetic statistics, equidistribution, geodesics, majorant space

Mathematical Subject Classification

Primary: 11R16

Secondary: 11E12, 11R45

Milestones

Received: 19 August 2019

Revised: 8 June 2020

Accepted: 21 July 2020

Published: 30 June 2021

Authors

Robert Harron
Department of Mathematics University of Hawai‘i at Mānoa Honolulu, HI United States

Vol. 15, No. 5, 2021