The shapes of pure cubic fields

Harron, Robert

doi:10.1090/proc/13309

The shapes of pure cubic fields
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Proc. Amer. Math. Soc. 145 (2017), 509-524 Request permission

Abstract:

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are rectangular and that they are equidistributed, in a regularized sense, when ordered by discriminant, in the one-dimensional space of all rectangular lattices. We do the same for Type II fields, which are however no longer rectangular. We obtain as a corollary of the determination of these shapes that the shape of a pure cubic field is a complete invariant determining the field within the family of all cubic fields.

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