Exposition of Riemann Surfaces

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2 total messages Started by David V Feldman Fri, 25 Dec 1992 04:13

#142

Author: David V Feldman
Date: Fri, 25 Dec 1992 04:13

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I am wondering if anyone has ever given an exposition of the
basic theory of Riemann surfaces that systematically did everything
on the universal covering space.  The idea would be that having
global coordinates would allow all objects such as differentials
of various sorts to be viewed as functions with appropriate
transformation properties, and theorems like Riemann-Roch, Serre
Duality, Abel-Jacobi might be accessible, in some guise, to readers
with only a basic knowledge of complex variables, but no significant
background in topology, sheaves, cohomology and the like.  I am
especially thinking here of compact Riemann surfaces of genus greater
than 2.

If no such exposition exists, in part, or in whole, I am interested
in opinions on what parts of the theory would be awkward from this
point of view.

David Feldman

Re: Exposition of Riemann Surfaces

#155

Author: William F. Hammond
Date: Sat, 26 Dec 1992 22:16

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In article <1he1p8INNlh0@mozz.unh.edu> dvf@kepler.unh.edu
(David V Feldman) writes:

> I am wondering if anyone has ever given an exposition of the
> basic theory of Riemann surfaces that systematically did everything
> on the universal covering space.  The idea would be that having
> global coordinates would allow all objects such as differentials
> of various sorts to be viewed as functions with appropriate
> transformation properties,  . . .

This is the theory of "automorphic functions" in dimension 1.
For a first reference I would suggest:  C. L. Siegel, Topics in
Complex Function Theory, volume II (Automorphic Functions and Abelian
Integrals), English translation by A. Shenitzer & M. Tretkoff,
Interscience Tracts in Pure & Appl. Math., No. 25, Wiley-Interscience,
1971.

----------------------------------------------------------------------
William F. Hammond                   Dept. of Mathematics & Statistics
518-442-4625                         SUNYA, Albany, NY 12222 (U.S.A.)
hammond@csc.albany.edu               FAX: 518-442-4731
----------------------------------------------------------------------

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