On some branches of the Bruhat-Tits tree

Abstract

We explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL2(k), whose vertices correspond to maximal orders containing a fixed order generated by a pair of orthogonal pure quaternions. In other words, we compute the set of maximal integral valued lattices in a ternary quadratic space, whose discriminant is a unit, containing a pair of orthogonal vectors, extending thus previous computations by Schulze-Pilot. The maximal order setting makes these computations simpler. The method presented here can be applied to arbitrary sub-orders or sublattices. The shape of this subtree is described, when it is finite, by a set of two invariants. In a previous work, the first author showed that determining the shape of these local subtrees allows the computation of representation fields, a class field determining the set of isomorphism classes, in a genus of Eichler orders, containing an isomorphic copy of a given order. Some further applications are described here.