On some branches of the Bruhat-Tits tree

TL;DR

This paper presents an algorithm to compute and describe the shape of specific subtrees in the Bruhat-Tits tree related to orders in PSL_2(k), with applications to class field theory and representation fields.

Contribution

It introduces an explicit algorithm for the largest subtree containing a given suborder in the Bruhat-Tits tree and characterizes its shape using invariants, with applications to class field computations.

Findings

Algorithm for subtree computation based on generators of H

Classification of subtree shapes via two invariants when finite

Full invariant table for orders generated by orthogonal pure quaternions

Abstract

We give an algorithm to explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL_2(k), whose vertices correspond to orders containing a given suborder H, in terms of a set of generators for H. The shape of this subtree is described, when it is finite, by a set of two invariants. We use our method to provide a full table for the invariants of an order generated by a pair of orthogonal pure quaternions. 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 O of orders of maximal rank in a fixed central simple algebra, containing an isomorphic copy of H. Some further applications are described here.