The Generalization of Nagao's Theorem to Other Subrings of the Rational Function Field

Abstract

Let ν be a (discrete) valuation of the rational function field k(t), where k is a field, and let 𝒞_ν be the intersection of the valuation rings of all the valuations of k(t), other than that of ν. It is well-known that either ν = ν_π, the valuation determined by some irreducible polynomial π in k[t], or ν = ν_∞, the “valuation at infinity”. In this paper we prove that GL ₂(𝒞_ν), where ν = ν_π, is the fundamental group of a certain tree of groups. The tree has finitely many vertices and its terminal vertices correspond with the elements of the ideal class group of 𝒞_ν. This extends a previous result of Nagao for the special case ν = ν_∞. In this case 𝒞_ν = k[t] and Nagao proves that GL ₂(k[t]) is an amalgamated product of a pair of groups. As a consequence we show that, when the degree of π is at least 4, GL ₂(𝒞_ν) has a free, non-cyclic quotient whose kernel contains (for example) all the unipotent matrices. This represents a two-dimensional anomaly.

Key Words:

• Rational function field
• Subring
• General linear group
• Bruhat-Tits tree

Mathematics Subject Classification 2000:

• Primary 20H25
• Secondary 20E08

Acknowledgments