On some branches of the Bruhat-Tits tree

2014/07/27 by Luis Arenas-Carmona, Arenas-Carmona, Luis, Ignacio Saavedra +1 · 2 citations
Mathematics · #11S45-11R52 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11S45-11R52

paper · pdf · doi:10.48550/arxiv.1407.7177

arxiv created 2014/07/27 · arxiv updated 2014/07/29

Abstract

We give an algorithm to explicitly compute the largest subtree, in the local Bruhat-Tits tree for PSL₂(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.

Cited by

• Computing quaternionic representations via twisted forms of Bruhat-Tits trees
• Two dimensional integral representations via branches of the Bruhat-Tits tree

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