A generalized Hermite constant and its computations for imaginary quadratic fields

TL;DR

This paper introduces a new projective Hermite constant for binary hermitian forms over imaginary quadratic fields, computes it for fields with small discriminants, and compares it with other generalizations of the classical Hermite constant.

Contribution

It defines the projective Hermite constant for imaginary quadratic fields and computes it for fields with small discriminants using geometric tools.

Findings

Computed the projective Hermite constants for fields with discriminant less than 70.

Identified hermitian forms attaining these constants.

Compared the projective Hermite constant with other generalizations.

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.