Hermite constant and Voronoi theory over a quaternion skew field

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 Voronoi's theorem on extreme forms to quaternionic Humbert forms.