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Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent
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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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The shapes of Galois quartic fields
All Galois quartic fields have shapes that are orthorhombic for V4 groups (with regularized equidistribution) and tetragonal for C4 groups (with explicit per-shape counting asymptotics).
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