On the shapes of pure prime degree number fields

TL;DR

This paper investigates the geometric shapes of pure prime degree number fields, revealing their distribution on specific subspaces and establishing shape as a complete invariant, extending prior cubic field results.

Contribution

It characterizes the shape distribution of pure prime degree fields, linking their ramification behavior to subspace placement and extending shape analysis to Frobenius number fields.

Findings

Shapes lie on two specific subspaces depending on ramification.

Shapes are equidistributed on these subspaces when ordered by discriminant.

Shape uniquely determines the field within the pure prime degree family.

Abstract

For $p$ prime and $\ell = \frac{p-1}{2}$, we show that the shapes of pure prime degree number fields lie on one of two $\ell$-dimensional subspaces of the space of shapes, and which of the two subspaces is dictated by whether or not $p$ ramifies wildly. When the fields are ordered by absolute discriminant we show that the shapes are equidistributed, in a regularized sense, on these subspaces. We also show that the shape is a complete invariant within the family of pure prime degree fields. This extends the results of Harron, in [Har17], who studied shapes in the case of pure cubic number fields. Furthermore we translate the statements of pure prime degree number fields to statements about Frobenius number fields, $F_p = C_p\rtimes C_{p-1}$, with a fixed resolvent field. Specifically we show that this study is equivalent to the study of $F_p$-number fields with fixed resolvent field $\mathbb{Q}(\zeta_p)$.