Hermite constant and Voronoï theory over a quaternion skew field

Abstract

The Hermite constant $\gamma_n(D)$ of a quaternion skew field $D$ over a global field is defined and studied. We obtain an upper bound of $\gamma_{n}(D)$. In the case that the base field is a number field, we introduce the notion of quaternionic Humbert forms over $D$. Then $\gamma_{n}(D)$ is characterized as a critical value of the Hermite invariants for $n$-ary quaternionic Humbert forms. We extend Voronoï's theorem on extreme forms to quaternionic Humbert forms.