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arXiv · 1907.07209

Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

Robert Harron
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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 \'etale 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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Robert Harron. 2019-07-16. Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent. https://doi.org/10.2140/ant.2021.15.1095

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