On the shapes of pure prime-degree number fields
Erik Holmes1
140 St. George Street, Room 6290, Toronto, ON M5S 2E4, Canada
Journal de théorie des nombres de Bordeaux, Volume 37 (2025) no. 1, pp. 1-48

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 [15], 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 with a fixed resolvent field. Specifically we show that this study is equivalent to the study of $F_p$-number fields, $F_p = C_p\rtimes C_{p-1},$ with fixed resolvent field $\mathbb{Q}(\zeta _p)$.

Pour un nombre premier $p$ et $\ell = \frac{p-1}{2}$, nous montrons que la shape (forme) d’un corps de nombres pur de degré premier appartient à l’un de deux sous-espaces de dimension $\ell $ de l’espace des formes, selon que $p$ est sauvagement ramifié ou non. Lorsque ces corps sont ordonnés par leur discriminant absolu, nous démontrons que les formes sont équidistribuées, au sens régularisé, sur ces sous-espaces. Nous montrons également que la forme constitue un invariant complet au sein de la famille des corps purs de degré premier. Ce travail généralise les résultats de Harron dans [15], qui étudie les formes dans le cas des corps cubiques purs. En outre, nous traduisons ces résultats en énoncés concernant les corps de Frobenius avec un corps résolvant fixé. Plus précisément, nous montrons que cette étude est équivalente à celle des corps de nombres de groupe de Galois $F_p = C_p \rtimes C_{p-1}$ et de corps résolvant fixé $\mathbb{Q}(\zeta _p)$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1311
Classification: 11R21, 11R45, 11P21, 11E12
Keywords: Number fields, lattices, equidistribution, carefree tuples

Erik Holmes  1

1 40 St. George Street, Room 6290, Toronto, ON M5S 2E4, Canada
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Erik Holmes. On the shapes of pure prime-degree number fields. Journal de théorie des nombres de Bordeaux, Volume 37 (2025) no. 1, pp. 1-48. doi: 10.5802/jtnb.1311
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