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arXiv · 1311.0387

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

Guillermo Mantilla-Soler·Marina Monsurrò
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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.

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Guillermo Mantilla-Soler, Marina Monsurrò. 2015-10-17. The shape of $\mathbb{Z}/\ell\mathbb{Z}$-number fields. https://doi.org/10.1007/s11139-015-9744-2

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