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Computing ideal classes representatives in quaternion algebras


Authors: Ariel Pacetti and Nicolás Sirolli
Journal: Math. Comp. 83 (2014), 2479-2507
MSC (2010): Primary 11R52
DOI: https://doi.org/10.1090/S0025-5718-2014-02796-8
Published electronically: January 9, 2014
MathSciNet review: 3223343
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Abstract: Let $ K$ be a totally real number field and let $ B$ be a totally definite quaternion algebra over $ K$. Given a set of representatives for ideal classes for a maximal order in $ B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $ B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of discriminant $ 30$ in an algebra over the real quadratic field $ \mathbb{Q}[\sqrt {5}]$.


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Additional Information

Ariel Pacetti
Affiliation: Departamento de Matemática, Universidad de Buenos Aires - Pabellón I, Ciudad Universitaria (C1428EGA), Buenos Aires, Argentina
Email: apacetti@dm.uba.ar

Nicolás Sirolli
Affiliation: Departamento de Matemática, Universidad de Buenos Aires - Pabellón I, Ciudad Universitaria (C1428EGA), Buenos Aires, Argentina
Email: nsirolli@dm.uba.ar

DOI: https://doi.org/10.1090/S0025-5718-2014-02796-8
Received by editor(s): June 20, 2011
Received by editor(s) in revised form: January 6, 2012, November 30, 2012, and January 21, 2013
Published electronically: January 9, 2014
Additional Notes: The first author was partially supported by PIP 2010-2012 GI and UBACyT X867
The second author was partially supported by a CONICET Ph.D. Fellowship
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.