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 shape of L, see Definition 1.2, 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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Mantilla-Soler, G., Monsurrò, M. The shape of \( \mathbb {Z}/\ell \mathbb {Z}\)-number fields. Ramanujan J 39, 451–463 (2016). https://doi.org/10.1007/s11139-015-9744-2
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DOI: https://doi.org/10.1007/s11139-015-9744-2