A commutative order in a quaternion algebra is called selective if it embeds into some, but not all, of the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one indefinite quaternion algebra. Here we prove that the order generated by a cubic root of unity is selective for any definite quaternion algebra over the rationals with type number 3 or larger. The proof extends to a few other closely related orders.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-4-5,
author = {Luis Arenas-Carmona},
title = {Roots of unity in definite quaternion orders},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {381-393},
zbl = {06480405},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-4-5}
}
Luis Arenas-Carmona. Roots of unity in definite quaternion orders. Acta Arithmetica, Tome 168 (2015) pp. 381-393. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-4-5/