2009/09/04 by A. W. Mason, Andreas Schweizer · 5 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Combinatorics #Degree (music) #Discrete mathematics #Field (mathematics) #Function field #Genus #Geometric and Algebraic Topology #Mathematics #Pure mathematics #Quotient #Zero (linguistics) #math.GR #math.NT #msc:11F06 #msc:11R58 #msc:20E08 #msc:20H10
paper · pdf · doi:10.1080/00927870902828926
published in Communications in Algebra 37(12), 4241-4258 (Taylor & Francis) · 25 pages, contains standard LaTeX graphics
arxiv created 2009/09/04 · openalex publication_date 2009/11/24 · arxiv updated 2016/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.