#### Vol. 15, No. 5, 2021

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Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

### Robert Harron

Vol. 15 (2021), No. 5, 1095–1125
##### 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 $\mathbf{Q}$-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).

##### Keywords
algebraic number theory, cubic fields, lattices, arithmetic statistics, equidistribution, geodesics, majorant space
##### Mathematical Subject Classification
Primary: 11R16
Secondary: 11E12, 11R45