A generalized Hermite constant for imaginary quadratic fields

Chan, Wai Kiu; Icaza, María; Lauret, Emilio

doi:10.1090/S0025-5718-2015-02903-2

A generalized Hermite constant for imaginary quadratic fields
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by Wai Kiu Chan, María Inés Icaza and Emilio A. Lauret PDF

Math. Comp. 84 (2015), 1883-1900 Request permission

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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