Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T16:17:43.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Continued Fractions Associated with $\text{S}{{\text{L}}_{3}}(\mathbf{Z})$ and Units in Complex Cubic Fields

Published online by Cambridge University Press:  20 November 2018

L. Ya. Vulakh
L. Ya. Vulakh*
Affiliation:
Department of Mathematics, The Cooper Union, 51 Astor Place, New York, NY 10003, USA, email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Continued fractions associated with $\text{G}{{\text{L}}_{3}}\left( \mathbf{Z} \right)$ are introduced and applied to find fundamental units in a two-parameter family of complex cubic fields.

Keywords

11R2711J7011J13fundamental unitscontinued fractionsdiophantine approximationsymmetric space
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 6 , 01 December 2002 , pp. 1305 - 1318
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Adams, B., Voronoi-algorithm expansion of two families with period length going to infinity. Math. Comp. 64 (1995), 1687–1704.Google Scholar
[2] Ash, A., Mumford, D., Rapoport, M. and Tai, Y., Smooth Compactification of Locally Symmetric Varieties. Math. Sci. Press, Brookline, Mass., 1975.Google Scholar
[3] Brentjes, A. T., Multi-dimensional continued fraction algorithms. Amsterdam, 1981.Google Scholar
[4] Cassels, J. W. S., An Introduction to the Geometry of Numbers. Springer-Verlag, 1959.Google Scholar
[5] Cohen, H., A Course in Computational Algebraic Number Theory. Graduate Texts in Math. 138, Springer-Verlag, 1996.Google Scholar
[6] Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups. Springer-Verlag, New York and Berlin, 1988.Google Scholar
[7] Davenport, H. and Mahler, K., Simultaneous Diophantine approximation. Duke Math. J. 13 (1946), 105–111.Google Scholar
[8] Davenport, H. and Rogers, C. A., Diophantine inequalities with an infinity of solutions. Philos. Trans. Roy. Soc. London 242 (1950), 311–344.Google Scholar
[9] Eberlein, P. B., Geometry of Nonpositively Curved Manifolds. The University of Chicago Press, 1996.Google Scholar
[10] Fröhlich, A. and Taylor, M., Algebraic Number Theory. Cambridge Stud. Adv. Math. 27, Cambridge Univ. Press, 1991.Google Scholar
[11] Grenier, D., Fundamental domains for the general linear group. Pacific J. Math. 132 (1988), 293–317.Google Scholar
[12] Kaneko, Kan, Voronoi-algorithm expansion of a family with period length going to infinity. SUT J. Math. 34 (1998), 49–62.Google Scholar
[13] Korkine, A. and Zolotarev, G., Sur les formes quadratiques. Math. Ann. 6 (1873), 366–389.Google Scholar
[14] Levesque, C. and Rhim, G., Two families of periodic Jacobi algorithms with period lengths going to infinity. J. Number Theory 37 (1991), 173–180.Google Scholar
[15] Margulis, G. A., Indefinite quadratic forms and unipotent flows on homogeneous spaces. C. R. Acad. Sci. Paris Ser. 1 Math. (10) 304 (1987), 251–257.Google Scholar
[16] Narkiewicz, Wladislaw, Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag, Berlin and New York, 1990.Google Scholar
[17] Ryshkov, S. S. and Baranovskii, E. P., Classical methods in the theory of lattice packings. Russian Math. Surveys 34 (1979), 1–68.Google Scholar
[18] Terras, A., Harmonic analysis on symmetric spaces and applications. II. Springer-Verlag, Berlin-New York, 1988.Google Scholar
[19] Voronoi, G., On a generalization of the Algorithm of Continued Fractions. Doctoral dissertation, Warsaw, 1896; Collected Works, Vol. 1. Izdat. Akad. Nauk Ukrain. SSR, Kiev 1952, 197–391 (Russian).Google Scholar
[20] Vulakh, L. Ya., The Markov spectrum for triangle groups. J. Number Theory 67 (1997), 11–28.Google Scholar
[21] Vulakh, L. Ya., Farey polytopes and continued fractions associated with discrete hyperbolic groups. Trans. Amer.Math. Soc. 351 (1999), 2295–2323.Google Scholar
[22] Vulakh, L. Ya., Diophantine approximation in Euclidean spaces. CRM Proceedings and Lecture Notes 19 (1999), 341–351.Google Scholar
[23] Vulakh, L. Ya., The Markov spectrum for Fuchsian groups. Trans. Amer.Math. Soc. 352 (2000), 4067–4094.Google Scholar
[24] Vulakh, L. Ya., Continued fractions associated with Bianchi groups and units in some quadric fields. Preprint.Google Scholar
[25] Vulakh, L. Ya., Units in some families of algebraic number fields. Preprint.Google Scholar
[26] Williams, H. C., The period length of Voronoi's algorithm for certain cubic orders. Publ. Math. Debrecen (3–4) 37 (1990), 245–265.Google Scholar