Computing quaternion quotient graphs via representations of orders

Abstract

We give a method to describe the quotient of the local Bruhat–Tits tree TP for PGL2(K), where K is a global function field, by certain subgroups of PGL2(K) of arithmetical significance. In particular, we can compute the quotient of TP by an arithmetic subgroup PGL2(A), where A=AP is the ring of functions that are regular outside P , recursively for a place P of any degree, when K is a rational function field. We achieve this by proving that the infinite matrices whose coordinates are the numbers of neighbors of a vertex in TP corresponding to orders in a fixed isomorphism class commute for different places P, using tools from the theory of representations of orders. The latter result holds for every global function field K.