2017/12/05 by Luis Arenas-Carmona, Arenas-Carmona, Luis, Claudio Bravo +1
Computer Science · Mathematics · #11R52 #11S45 #16G30 #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Algorithm #Combinatorics #Commutative property #Computation #Discrete mathematics #FOS: Mathematics #Field (mathematics) #Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT) #Pure mathematics #Tree (set theory) #math.NT #msc:11R52 #msc:11S45 #msc:16G30
paper · pdf · doi:10.48550/arxiv.1712.01463
This article will be published under the title "Computing embedding numbers and branches of orders via extensions of the Bruhat-Tits tree"
openalex publication_date 2017/12/05 · arxiv created 2019/05/22 · arxiv updated 2019/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.