From: Boudewijn Moonen <Boudewijn.Moonen@ipb.uni-bonn.de>
Subject: Re: Riemann surface
Date: Thu, 29 Mar 2001 11:00:12 +0200
Newsgroups: sci.math
Summary: Automorphism group of Riemann surfaces of genus g

John Chamish wrote:
> 
> What is the automorphism group of a Riemann surface of genus g ?
> 
> Thanks


1) If g = 0, then the surface is the projectve line and so its
automorphism
group is the projective linear group.

2) If g = 1, then the surface is a torus. These tori are classified by
SL(2,Z)\H, where H := { z in C | im(z) > 0 } is the upper half plane.
Generically, then, the automorphisms are the translations. Additional
discrete symmetries arise for those tori corresponding to points of H
with nontrivial stabilizers. A more deetailed description can be found
in Chapter 4 of Hartshorne's "Algebraic Geometry"

3) If g >= 2, the automorphism group is finite; in fact there is the
famous Hurwitz bound 84(g - 1). (This bound is strict; there is a 
famous Riemann surface of genus 3 due to Klein where this bound is
achieved). The generic Riemann surface of genus g >= 3 has no
automorphisms at all.


Regards,


-- 
    Boudewijn Moonen
    Institut fuer Photogrammetrie der Universitaet Bonn
    Nussallee 15
    
    D-53115 Bonn
    
    GERMANY

    e-mail: Boudewijn.Moonen@ipb.uni-bonn.de
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