Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent
The notion of the shape, of a number field \(K\), was introduced by \textit{D. Terr} [The distribution of shapes of cubic orders. University of California (Ph.D. thesis) (1997)]; wich is a certain lattice of rank \( [K : \mathbb Q] -1\) attached to \(K\) considered up to rotation, reflection, and scaling. In particular, he proved that the shapes of cubic fields are equidistributed in the space of shapes of rank \(2\) lattices as the discriminant of the cubic field goes to infinity. Other authors, to other number fields, generalized this result. The author investigate, first, the distribution of more specific families of number fields. Then he 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. Applying a method of \textit{M. Bhargava} and the author [Compos. Math. 152, No. 6, 1111--1120 (2016; Zbl 1347.11074)] to results of Bhargava and Ariel Shnidman, he proves 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. He also show that the shape of a complex cubic field is a complete invariant within the family of all cubic fields.
scientific article