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).

中文翻译：

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固定二次求解器复三次场形状的等分布

我们展示了复杂三次场的形状位于由场的零迹形式定义的模块化表面的测地线上。我们还证明了 étale 中所有订单的一般性陈述$\mathbf{问}$-代数。将 Manjul Bhargava 和 Piper H 的方法应用于 Bhargava 和 Ariel Shnidman 的结果，我们证明了当复三次场的判别式趋于无穷大时，固定测地线上的形状关于双曲线测度变得均匀分布。我们还表明，复三次场的形状是一个完全不变量（在所有三次场的家族中）。