Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

An algorithm to compute relative cubic fields


Author: Anna Morra
Journal: Math. Comp. 82 (2013), 2343-2361
MSC (2010): Primary 11R16, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-2013-02686-5
Published electronically: March 14, 2013
MathSciNet review: 3073205
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be an imaginary quadratic number field with class number $ 1$. We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions $ L/K$ up to a bound $ X$ on the norm of the relative discriminant ideal. The main tools are Taniguchi's [18] generalization of Davenport-Heilbronn parametrisation of cubic extensions, and reduction theory for binary cubic forms over imaginary quadratic fields. Finally, we give numerical data for $ K=\mathbb{Q}(i)$, and we compare our results with ray class field algorithm results, and with asymptotic heuristics, based on a generalization of Roberts' conjecture [19].


References [Enhancements On Off] (What's this?)

  • 1. K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213-1237. MR 1415795 (97m:11159)
  • 2. K. Belabas, Paramétrisation de structures algébriques et densités de discriminants [d'après Bhargava], Astérisque (2005), no. 299, pp. 267-299, Séminaire Bourbaki. Vol. 2003/2004. MR 2167210 (2006k:11057)
  • 3. K. Belabas, L'algorithmique de la théorie algébrique des nombres, dans Théorie algorithmique des nombres et équations diophantiennes (N. Berline, A. Plagne, C. Sabbah eds.) Ed. de l'École Polytechnique, 85-153, (2005). MR 2224342 (2007a:11167)
  • 4. H. Cohen, Advanced Topics in Computational Number Theory, Graduate Texts in Math. 193, Springer-Verlag, 2000. MR 1728313 (2000k:11144)
  • 5. J. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Mathematica, 51, no. 3 (1984), 275-324. MR 743014 (85j:11063)
  • 6. J. Cremona, Reduction of binary cubic and quartic forms, London Mathematical Society ISSN 1461-1570, 1999. MR 1693411 (2000f:11040)
  • 7. J. Cremona, Reduction of binary forms over imaginary quadratic fields, slides of the talk given in Bordeaux (2007), can be found at http://www.warwick.ac.uk/staff/J.E.Cremona/papers/jec_bordeaux.pdf.
  • 8. B. Datskovsky and D. J. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116-138. MR 936994 (90b:11112)
  • 9. H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields (ii), Proc. Roy. Soc. Lond. A 322 (1971), pp. 405-420. MR 0491593 (58:10816)
  • 10. J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Harmonic analysis and number theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. MR 1483315 (98g:11058)
  • 11. G. Julia, Etude sur les formes binaires non quadratiques à indéterminées réelles ou complexes, Mémoires de l'Académie des Sciences de l'Institut de France 55 (1917), 1-296. Also in Julia's Oeuvres, vol. 5.
  • 12. K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math J. 11, Issue 3, (1964), 257-262. MR 0166188 (29:3465)
  • 13. G. Malle, On the distribution of Galois groups, J. Number Theory 92 (2002), 315-329. MR 1884706 (2002k:12010)
  • 14. G. Malle, The totally real primitive number fields of discriminant at most $ 10^9$, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, (2006), 114-123. MR 2282919 (2007j:11179)
  • 15. A. Morra, Comptage asymptotique et algorithmique d'extensions cubiques relatives, Thèse (in english), Université Bordeaux 1, 2009. Available online at http://tel.archives-ouvertes.fr/docs/00/52/53/20/PDF/these.pdf
  • 16. PARI/GP, version 2.5.0, Bordeaux, 2011, http://pari.math.u-bordeaux.fr/.
  • 17. R. G. Swan, Generators and relations for certain special linear groups, Advances in Mathematics 6, (1971) 1-77. MR 0284516 (44:1741)
  • 18. T. Taniguchi, Distribution of discriminants of cubic algebras, preprint 2006, arXiv:math.NT/0606109v1.
  • 19. T. Taniguchi and F. Thorne, Secondary terms in counting functions for cubic fields, preprint 2011, arXiv:math.NT/1102.2914v1.
  • 20. E. Whitley, Modular symbols and elliptic curves over imaginary quadratic number fields, PhD Thesis, Exeter (1990).
  • 21. T. Womack, Explicit descent on elliptic curves, PhD Thesis, Nottingham (2003).

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11R16, 11Y40

Retrieve articles in all journals with MSC (2010): 11R16, 11Y40


Additional Information

Anna Morra
Affiliation: Université Rennes 1, IRMAR, 263 avenue du Général Leclerc, CS74205, 35042 Rennes Cedex, France

DOI: https://doi.org/10.1090/S0025-5718-2013-02686-5
Received by editor(s): March 21, 2011
Received by editor(s) in revised form: August 26, 2011, and February 5, 2012
Published electronically: March 14, 2013
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society