Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Humbert-Minkowski's constant
for a number field


Authors: R. Baeza and M. I. Icaza
Journal: Proc. Amer. Math. Soc. 125 (1997), 3195-3202
MSC (1991): Primary 11E12, 11H50; Secondary 11R29, 15A48
MathSciNet review: 1403112
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We use Humbert's reduction theory to introduce an obstruction for the unimodularity of minimal vectors of positive definite quadratic forms over totally real number fields. Using this obstruction we obtain an inequality relating the values of a generalized Hermite constant for such fields, which over the field of rational numbers leads to a well-known result of Mordell.


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

  • [B] Baeza, R.: The volume of the space of Humbert reduced forms. Preprint. Universidad de Chile, 1994.
  • [C-S] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369 (89a:11067)
  • [Fr] Eberhard Freitag, Hilbert modular forms, Springer-Verlag, Berlin, 1990. MR 1050763 (91c:11025)
  • [I] Icaza, M.I.: Hermite constant and extreme forms for algebraic number fields. To appear in Journal of London Math. Soc. (2) 55, 11-22, 1997. CMP 97:05
  • [H] Pierre Humbert, Théorie de la réduction des formes quadratiques définies positives dans un corps algébrique K fini, Comment. Math. Helv. 12 (1940), 263–306 (French). MR 0003002 (2,148a)
  • [OM] O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Bd. 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0152507 (27 #2485)
  • [Si] Carl Ludwig Siegel, Lectures on the geometry of numbers, Springer-Verlag, Berlin, 1989. Notes by B. Friedman; Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter; With a preface by Chandrasekharan. MR 1020761 (91d:11070)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11E12, 11H50, 11R29, 15A48

Retrieve articles in all journals with MSC (1991): 11E12, 11H50, 11R29, 15A48


Additional Information

R. Baeza
Affiliation: Department of Mathematics, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago-Chile
Email: rbaeza@abello.dic.uchile.cle

M. I. Icaza
Affiliation: Department of Mathematics, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago-Chile
Email: icazap@abello.dic.uchile.cle

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03940-3
PII: S 0002-9939(97)03940-3
Keywords: Quadratic forms, Hermite constant, reduction theory.
Received by editor(s): January 18, 1996
Received by editor(s) in revised form: June 13, 1996
Additional Notes: The first author was partially supported by Fondecyt 1950392 and European Union # CI1*-CT93-0353
The second author was supported by Fondecyt 3940002 and European Union # CI1*-CT93-0353
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society