On Humbert-Minkowski’s constant for a number field

Baeza, R.; Icaza, M.

doi:10.1090/S0002-9939-97-03940-3

On Humbert-Minkowski’s constant for a number field
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by R. Baeza and M. I. Icaza PDF

Proc. Amer. Math. Soc. 125 (1997), 3195-3202 Request permission

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

Baeza, R.: The volume of the space of Humbert reduced forms. Preprint. Universidad de Chile, 1994.
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, DOI 10.1007/978-1-4757-2016-7
Eberhard Freitag, Hilbert modular forms, Springer-Verlag, Berlin, 1990. MR 1050763, DOI 10.1007/978-3-662-02638-0
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.
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Band 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0152507
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, DOI 10.1007/978-3-662-08287-4