Computing quaternion quotient graphs via representation of orders

TL;DR

This paper develops a method to compute quaternion quotient graphs using the representation of orders, establishing a reciprocity law that enables recursive calculation of local quotient graphs over function fields.

Contribution

It introduces a novel reciprocity law linking quotient graphs at different places, facilitating recursive computation of local quotient graphs in quaternion algebras.

Findings

Established a reciprocity law between quotient graphs at different places.

Developed a recursive method to compute local quotient graphs.

Connected the structure of quotient graphs with conjugacy classes of maximal orders.

Abstract

We study the correspondence assigning the vertices of a certain quotient of the local Bruhat-Tits tree for the general linear group over a global function field, to conjugacy classes of maximal orders in some quaternion algebras. The interplay between quotient graphs and orders can be used to study representation of orders if the quotient graphs are known and conversely. We use this converse to find a reciprocity law between quotient graph at diferent places that suffices to compute, recursively, all local quotient graphs for a matrix algebra over a rational function field.