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

摘要

Let C be a smooth projective curve over a field k. For each closed point Q of C let C = C (C,Q,k) be the coordinate ring of the affine curve obtained by removing Q from C. Serre has proved that GL(2)(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)(C) attached to the terminal vertices of T are in one-one correspondence with the elements of Cl(C), the ideal class group of C. This extends an earlier result of Nagao for the simplest case C = k[t]. Serre's proof is based on applying the theory of groups acting on trees to the quotient graph (X) over bar = GL(2)(C)X, where X is the associated Bruhat-Tits building. To determine (X) over bar he makes extensive use of the theory of vector bundles (of rank 2) over C. In this paper we determine (X) over bar 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 S, say. In this paper we prove that S is bounded, even when Cl(C) is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of GL(2)(C), involving unipotent and elementary matrices. [References: 9]