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A generalized Hermite constant for imaginary quadratic fields

Authors: Wai Kiu Chan, María Inés Icaza and Emilio A. Lauret
Journal: Math. Comp. 84 (2015), 1883-1900
MSC (2010): Primary 11H50, 11H55
Published electronically: January 22, 2015
MathSciNet review: 3335896
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Abstract: We introduce the projective Hermite constant for positive definite binary hermitian forms associated with an imaginary quadratic number field $ K$. It is a lower bound for the classical Hermite constant, and these two constants coincide when $ K$ has class number one. Using the geometric tools developed by Mendoza and Vogtmann for their study of the homology of the Bianchi groups, we compute the projective Hermite constants for those $ K$ whose absolute discriminants are less than 70, and determine the hermitian forms that attain the projective Hermite constants in these cases. A comparison of the projective hermitian constant with some other generalizations of the classical Hermite constant is also given.

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Wai Kiu Chan
Affiliation: Department of Mathematics and Computer science, Wesleyan University, Middletown, Connecticut 06459–0128

María Inés Icaza
Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

Emilio A. Lauret
Affiliation: FaMAF-CIEM, Universidad Nacional de Córdoba, Ciudad Universitaria 5000, Córdoba, Argentina

Keywords: Minima of hermitian forms, extreme hermitian forms, Hermite constant
Received by editor(s): January 10, 2011
Received by editor(s) in revised form: July 5, 3013, and October 8, 2013
Published electronically: January 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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