On the missing branches of the Bruhat-Tits tree

TL;DR

This paper extends the theory of branches in the Bruhat-Tits tree to orders spanning field extensions and generated by non-nilpotent pure quaternions, enabling more comprehensive computations of maximal orders.

Contribution

It develops a new framework for computing branches over field extensions and for orders generated by non-nilpotent pure quaternions, broadening previous methods.

Findings

Extended branch computation to orders spanning field extensions.

Derived explicit descriptions for branches generated by pure quaternions.

Enhanced visualization techniques for Bruhat-Tits tree analysis.

Abstract

Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree of the Bruhat-Tits tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the Bruhat-Tits tree in terms of balls in k. This is easier for orders spanning a split commutative sub-algebra, i.e., an algebra isomorphic to (k x k). In the present work, we develop a theory of branches over field extension that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions. In fact, the hypotheses on the generators are not essential.