The Ramanujan Journal

, Volume 39, Issue 3, pp 451–463 | Cite as

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



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.


Number fields Shape Lattices 

Mathematics Subject Classification

11R04 11R20 11R80 11R33 11H99 11E12 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de los AndesBogotáColombia
  2. 2.Università Eropea di RomaRomeItaly

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