Serre’s generalization of Nagao’s theorem: An elementary approach

Abstract:

Let $C$ be a smooth projective curve over a field $k$. For each closed point $Q$ of $C$ let $\mathcal {C} = \mathcal {C}(C, Q, k)$ be the coordinate ring of the affine curve obtained by removing $Q$ from $C$. Serre has proved that $GL_2(\mathcal {C})$ is isomorphic to the fundamental group, $\pi _1(G, T)$, of a graph of groups $(G, T)$, where $T$ is a tree with at most one non-terminal vertex. Moreover the subgroups of $GL_2(\mathcal {C})$ attached to the terminal vertices of $T$ are in one-one correspondence with the elements of $\operatorname {Cl}(\mathcal {C})$, the ideal class group of $\mathcal {C}$. This extends an earlier result of Nagao for the simplest case $\mathcal {C} = k[t]$. Serre’s proof is based on applying the theory of groups acting on trees to the quotient graph $\overline {X} = GL_2(\mathcal {C}) \backslash X$, where $X$ is the associated Bruhat-Tits building. To determine $\overline {X}$ he makes extensive use of the theory of vector bundles (of rank 2) over $C$. In this paper we determine $\overline {X}$ using a more elementary approach which involves substantially less algebraic geometry. The subgroups attached to the edges of $T$ are determined (in part) by a set of positive integers $\mathcal {S}$, say. In this paper we prove that $\mathcal {S}$ is bounded, even when Cl$(\mathcal {C})$ is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of $GL_2(\mathcal {C})$, involving unipotent and elementary matrices.