Nonrational genus zero function fields and the Bruhat-Tits tree

TL;DR

This paper provides a detailed description of the quotient graph G extbackslash T for a specific genus zero function field, revealing new quaternionic vertex stabilizer subgroups in the Bruhat-Tits tree structure.

Contribution

It extends the understanding of G extbackslash T for genus zero function fields with degree two places, introducing quaternionic stabilizers not seen in previous cases.

Findings

Describes G extbackslash T for genus zero, degree two place case

Identifies quaternionic type vertex stabilizer subgroups

Expands known examples of Bruhat-Tits tree structures

Abstract

Let K be a function field with constant field k and let "infinity" be a fixed place of K. Let C be the Dedekind domain consisting of all those elements of K which are integral outside "infinity". The group G=GL_2(C) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat-Tits tree, T. Classical Bass-Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G\T. The shape of this quotient graph (for any G) is described in a fundamental result of Serre. However there are very few known examples for which a detailed description of G\T is known. (One such is the rational case, C=k[t], i.e. when K has genus zero and "infinity" has degree one.) In this paper we give a precise description of G\T for the case where the genus of K is zero, K has no places of degree one and "infinity" has degree two. Among the known examples a new feature here is the appearance of vertex stabilizer subgroups (of G) which are of quaternionic type.