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On the computation of overorders by:3 (Source: Crossref)

    The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper, we consider the related problems of computing all minimal overorders as well as all overorders of a given order. We use techniques from algorithmic representation theory and the theory of minimal integral ring extensions to obtain efficient and practical algorithms, whose implementation is publicly available.

    AMSC: 11Y40, 11R04


    • 1. H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963) 8–28. CrossrefGoogle Scholar
    • 2. A. J. Berrick and M. E. Keating, An Introduction to Rings and Modules with K-Theory in View, Cambridge Studies in Advanced Mathematics, Vol. 65 (Cambridge University Press, Cambridge, 2000). CrossrefGoogle Scholar
    • 3. G. Bisson, Computing endomorphism rings of elliptic curves under the GRH, J. Math. Cryptol. 5(2) (2011) 101–113. Google Scholar
    • 4. G. Bisson, Endomorphism rings in cryptography, theses, Technische Universiteit Eindhoven and Institut National Polytechnique de Lorraine (2011). Google Scholar
    • 5. G. Bisson, Computing endomorphism rings of abelian varieties of dimension two, Math. Comp. 84(294) (2015) 1977–1989. Crossref, Web of ScienceGoogle Scholar
    • 6. N. Bourbaki, Elements of mathematics, Commutative Algebra (Hermann, Paris, 1972). Google Scholar
    • 7. J. Brzeziński, On orders in quaternion algebras, Comm. Algebra 11(5) (1983) 501–522. Crossref, Web of ScienceGoogle Scholar
    • 8. L. M. Butler, Subgroup Lattices and Symmetric Functions, Memoirs of the American Mathematical Society, Vol. 112, No. 539 (American Mathematical Society, Providence, RI, 1994). Google Scholar
    • 9. H. Cohen, A course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, Vol. 138 (Springer, Berlin, 1993). CrossrefGoogle Scholar
    • 10. C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. 1 (John Wiley & Sons, New York, 1990). Google Scholar
    • 11. D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, Vol. 150, (Springer, New York, 1995). With a view toward algebraic geometry. CrossrefGoogle Scholar
    • 12. D. Ferrand and J.-P. Olivier, Homomorphismes minimaux d’anneaux, J. Algebra 16 (1970) 461–471. Crossref, Web of ScienceGoogle Scholar
    • 13. C. Fieker, W. Hart, T. Hofmann and F. Johansson, Nemo/Hecke: Computer algebra and number theory packages for the Julia programming language, in ISSAC’17—Proc. 2017 ACM Int. Symp. Symbolic and Algebraic Computation (ACM, New York, 2017), pp. 157–164. CrossrefGoogle Scholar
    • 14. C. Fieker, T. Hofmann and C. Sircana, On the construction of class fields, in ANTS XIII—Proc. Thirteenth Algorithmic Number Theory Symp. Open Book Series, Vol. 2 (Mathematical Science Publication, Berkeley, CA, 2019), pp. 239–255. CrossrefGoogle Scholar
    • 15. C. Friedrichs, Berechnung von Maximalordnungen über Dedekindringen, Ph.D. thesis, Technische Universität Berlin (2000). Google Scholar
    • 16. P. Furtwängler, Über die Führer von Zahlringen, Anzeiger Wien 56 (1919) 75. Google Scholar
    • 17. C. Greither, On the two generator problem for the ideals of a one-dimensional ring, J. Pure Appl. Algebra 24(3) (1982) 265–276. Crossref, Web of ScienceGoogle Scholar
    • 18. D. F. Holt, B. Eick and E. A. O’Brien, Handbook of Computational Group Theory, Discrete Mathematics and its Applications (Chapman & Hall/CRC, Boca Raton, FL, 2005). CrossrefGoogle Scholar
    • 19. G. Ivanyos and L. Rónyai, Finding maximal orders in semisimple algebras over Q, Comput. Complexity 3(3) (1993) 245–261. CrossrefGoogle Scholar
    • 20. H. W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. (N.S.) 26(2) (1992) 211–244. Crossref, Web of ScienceGoogle Scholar
    • 21. G. Lettl and C. Prabpayak, Conductor ideals of orders in algebraic number fields, Arch. Math. (Basel) 103(2) (2014) 133–138. Crossref, Web of ScienceGoogle Scholar
    • 22. G. Lettl and C. Prabpayak, Orders in cubic number fields, J. Number Theory 166 (2016) 415–423. Crossref, Web of ScienceGoogle Scholar
    • 23. S. Marseglia, Computing square-free polarized abelian varieties over finite fields, preprints (2018); arXiv:1805.10223. Google Scholar
    • 24. S. Marseglia, Computing the ideal class monoid of an order, preprint (2018); arXiv:1805.09671. Google Scholar
    • 25. R. A. Parker, The computer calculation of modular characters (the meat-axe), in Computational Group Theory, Durham, 1982 (Academic Press, London, 1984), pp. 267–274. Google Scholar
    • 26. M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Encyclopedia of Mathematics and its Applications, Vol. 30 (Cambridge University Press, Cambridge, 1989). CrossrefGoogle Scholar
    • 27. I. Reiner, Maximal Orders, London Mathematical Society Monographs. New Series, Vol. 28 (Oxford University Press, Oxford, 2003). CrossrefGoogle Scholar
    • 28. A. Reinhart, A note on conductor ideals, Comm. Algebra 44(10) (2016) 4243–4251. Crossref, Web of ScienceGoogle Scholar
    • 29. H. Zassenhaus, Über die konstruktive Behandlung mathematischer Probleme, Lectures at the Rhine–Westphalia Academy of Sciences, No. 307 (Westdeutscher Verlag, Opladen, 1982), pp. 7–52. CrossrefGoogle Scholar
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