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.

I have added, and clarified some proofs. A version of this paper will appear soon in the Ramanujan Journal

Cited by in corpus (1)

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