Hermite constant and Voronoï theory over a quaternion skew field (Q854529)

Let \(D\) be a quaternion skew field over a global field. The paper gives an intrinsic definition of the Hermite constant \(\gamma _n(D)\) of \(D\) independent of the choice of a splitting field. An upper bound for \(\gamma _n(D)\) is obtained. In the case where the ground field is a number field, the authors define quaternionic Humbert forms and characterize \(\gamma _n(D)\) as the maximum of the Hermite invariants over all Humbert forms in \(D^n\). The local maxima of these Hermite invariants are characterized à la Voronoi as the Humbert forms that are perfect and eutactic.