Computing ideal classes representatives in quaternion algebras
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- by Ariel Pacetti and Nicolás Sirolli PDF
- Math. Comp. 83 (2014), 2479-2507 Request permission
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 $\mathbb {Q}[\sqrt {5}]$.References
- Juliusz Brzeziński, A characterization of Gorenstein orders in quaternion algebras, Math. Scand. 50 (1982), no. 1, 19–24. MR 664504, DOI 10.7146/math.scand.a-11940
- J. Brzeziński, On orders in quaternion algebras, Comm. Algebra 11 (1983), no. 5, 501–522. MR 693798, DOI 10.1080/00927878308822861
- J. Brzezinski, On automorphisms of quaternion orders, J. Reine Angew. Math. 403 (1990), 166–186. MR 1030414, DOI 10.1515/crll.1990.403.166
- Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
- Caterina Consani and Jasper Scholten, Arithmetic on a quintic threefold, Internat. J. Math. 12 (2001), no. 8, 943–972. MR 1863287, DOI 10.1142/S0129167X01001118
- Lassina Dembélé and Steve Donnelly, Computing Hilbert modular forms over fields with nontrivial class group, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 371–386. MR 2467859, DOI 10.1007/978-3-540-79456-1_{2}5
- H. M. Edgar, R. A. Mollin, and B. L. Peterson, Class groups, totally positive units, and squares, Proc. Amer. Math. Soc. 98 (1986), no. 1, 33–37. MR 848870, DOI 10.1090/S0002-9939-1986-0848870-X
- Benedict H. Gross and Mark W. Lucianovic, On cubic rings and quaternion rings, J. Number Theory 129 (2009), no. 6, 1468–1478. MR 2521487, DOI 10.1016/j.jnt.2008.06.003
- Irving Kaplansky, Submodules of quaternion algebras, Proc. London Math. Soc. (3) 19 (1969), 219–232. MR 240142, DOI 10.1112/plms/s3-19.2.219
- Markus Kirschmer and John Voight, Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput. 39 (2010), no. 5, 1714–1747. MR 2592031, DOI 10.1137/080734467
- Stefan Lemurell, Quaternion orders and ternary quadratic forms, 2011, http://arxiv.org/abs/1103.4922.
- Arnold Pizer, On the arithmetic of quaternion algebras. II, J. Math. Soc. Japan 28 (1976), no. 4, 676–688. MR 432600, DOI 10.2969/jmsj/02840676
- Arnold Pizer, An algorithm for computing modular forms on $\Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340–390. MR 579066, DOI 10.1016/0021-8693(80)90151-9
- Ariel Pacetti and Fernando Rodriguez Villegas, http://www.ma.utexas.edu/users/villegas/cnt/cnt.html
- Ariel Pacetti and Fernando Rodriguez Villegas, Computing weight 2 modular forms of level $p^2$, Math. Comp. 74 (2005), no. 251, 1545–1557. With an appendix by B. Gross. MR 2137017, DOI 10.1090/S0025-5718-04-01709-0
- Ariel Pacetti and Gonzalo Tornaría, Shimura correspondence for level $p^2$ and the central values of $L$-series, J. Number Theory 124 (2007), no. 2, 396–414. MR 2321370, DOI 10.1016/j.jnt.2006.10.011
- W. A. Stein et al, Sage Mathematics Software (Version 4.7.2), The Sage Development Team, 2011, http://www.sagemath.org.
- Jude Socrates and David Whitehouse, Unramified Hilbert modular forms, with examples relating to elliptic curves, Pacific J. Math. 219 (2005), no. 2, 333–364. MR 2175121, DOI 10.2140/pjm.2005.219.333
- Marie-France Vignéras, Simplification pour les ordres des corps de quaternions totalement définis, J. Reine Angew. Math. 286(287) (1976), 257–277. MR 429841, DOI 10.1515/crll.1976.286-287.257
- Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
- John Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, 2010, http://arxiv.org/abs/1004.0994
Additional Information
- Ariel Pacetti
- Affiliation: Departamento de Matemática, Universidad de Buenos Aires - Pabellón I, Ciudad Universitaria (C1428EGA), Buenos Aires, Argentina
- MR Author ID: 759256
- Email: apacetti@dm.uba.ar
- Nicolás Sirolli
- Affiliation: Departamento de Matemática, Universidad de Buenos Aires - Pabellón I, Ciudad Universitaria (C1428EGA), Buenos Aires, Argentina
- MR Author ID: 1067127
- ORCID: 0000-0002-0603-4784
- Email: nsirolli@dm.uba.ar
- 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 - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 2479-2507
- MSC (2010): Primary 11R52
- DOI: https://doi.org/10.1090/S0025-5718-2014-02796-8
- MathSciNet review: 3223343