Jérôme Chaubert
This work concerns the study of Euclidean minima of maximal orders in central simple algebras. In the first part, we define the concept of ideal lattice in the non-commutative case. Let A be a semi-simple algebra over Q. An ideal lattice over A is a triple (I, α, τ) where I is an ideal of A, α is a unit in AR = A ⊗Q R fixed by τ and τ is a positive involution on AR, in other words, trAR/R(xαxτ) > 0 for all x ∈ IR. We study ideal lattices, especially their Hermite invariant, to link the concepts of Euclidean minimum of a maximal order and ideal lattices of the form (Λ, α, τ). Namely, we show that we can bound the Euclidean minimum of Λ by the Hermite invariant of the ideal lattices of Λ. This upper bound allows us to calculate the Euclidean minima of some infinite families of maximal orders, when A is a quaternion skew field over Q. In the second part, we look for other ways to find upper and lower bounds of the euclidean minimum of a maximal order in a quaternion skew field. This research leads us to an exhaustive list of euclidean quaternion skew fields over Q, and their euclidean minima. The quadratic case has also been studied but remains partially unsolved. At the end, we give bounds for Euclidean minima of some maximal orders of quaternion skew fields over cyclotomic fields.
EPFL2007
