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

Author:
A. W. Mason

Journal:
Trans. Amer. Math. Soc. **353** (2001), 749-767

MSC (2000):
Primary 20H25; Secondary 20E08, 14H05

DOI:
https://doi.org/10.1090/S0002-9947-00-02707-0

Published electronically:
October 23, 2000

MathSciNet review:
1804516

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Abstract | References | Similar Articles | Additional Information

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated *Bruhat-Tits building*. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is *infinite*. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.

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Additional Information

**A. W. Mason**

Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, United Kingdom

Email:
awm@maths.gla.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-00-02707-0

Keywords:
General linear group,
tree,
algebraic function field

Received by editor(s):
March 25, 1999

Published electronically:
October 23, 2000

Article copyright:
© Copyright 2000
American Mathematical Society