Maximal orders containing a given sub-order for local fields of even characteristic

Abstract

We extend, to the case of local fields of even characteristic, our previous computations for the set of maximal orders containing given quaternions as concrete sub-graphs of the Bruhat–Tits tree. Computing the relative position of such sub-graphs, the branches, is useful, for instance, to explicitly compute relative spinor images, thus solving the selectivity problem. They also play a role in the description of quotient graphs, which are useful to study matrix groups of arithmetical importance. As in earlier work, we concentrate on orders generated by two quaternions, but extending our method to larger generating sets is quite straightforward. In our previous work, the results were given mainly in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin–Scheier extensions, to take care of quaternions generating Galois field extensions. The latter extensions have no non-trivial elements of null trace. For this reason, we state our results in terms of a wider family of algebra presentations, in contrast to the aforementioned work, where we chose a pair of pure quaternions as generators of the order.