Nonrational Genus Zero Function Fields and the Bruhat–Tits Tree

Abstract

Let K be a function field with constant field k, and let ∞ be a fixed place of K. Let 𝒞 be the Dedekind domain consisting of all those elements of K which are integral outside ∞. 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 ∞ has degree one.) In this article, 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 ∞ 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.