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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: October 23, 2000
MathSciNet review: 1804516
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-00-02707-0
PII: S 0002-9947(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