The shape of $\mathbb{Z}/\ell\mathbb{Z}$-number fields

Abstract

Let $\ell$ be a prime and let $L/\mathbb{Q}$ be a Galois number field with Galois group isomorphic to $\mathbb{Z}/\ell\mathbb{Z}$. We show that the {\it shape} of $L$ is either $\frac{1}{2}\mathbb{A}_{\ell-1}$ or a fixed sub lattice depending only on $\ell$; such a dichotomy in the value of the shape only depends on the type of ramification of $L$. This work is motivated by a result of Bhargava and Shnidman, and a previous work of the first named author, on the shape of $\mathbb{Z}/3\mathbb{Z}$ number fields.