The existence of meromorphic functions on Riemann surfaces In Miranda's book on algebraic curves and Riemann surfaces, Miranda writes:

It is a basic and highly nontrivial
  result that a compact Riemann surface
  has nonconstant meromorphic functions
  on it [...] The theory involved in
  producing meromorphic functions for an
  unknown compact Riemann surface is
  rather technical analysis and
  functional analysis. After one has
  access to meromorphic functions,
  however, the theory is completely
  algebraic, or at least can be made to
  be so.

I've seen this claim a number of other places as well. It seems unnatural to use real analysis to prove a theorem about Riemann surfaces, which are geometric/algebraic objects. Is there genuinely no purely geometric/algebraic way to realize an abstract Riemann surface as a branched cover of the Riemann sphere?
 A: This deep fact is essentially the same as the uniformization theorem. The problem is
how to construct at least one holomorphic or meromorphic form with prescribed singularity.
All known proofs use some Analysis, and none of them is simple. Once you have Uniformization,
it is easy to construct holomorphic forms.
A good modern proof (in full generality) is contained in the book of J. Hubbard, Teichmuller Theory.
The complexity of the proof depends on how exactly you define a Riemann surface.
A Riemann surface is a 1-dimensional complex manifold.
A 1-dimensional complex manifold is separable (has a countable base). This fact is a part of the
modern statement of the Uniformization theorem.
However, if you include this separability to the DEFINITION of a Riemann surface, the proof substantially simplifies. Especially simple proof (assuming separabiity)
can be found in Goluzin's book Geometric theory
of functions of a complex variable.
All Riemann surfaces arising in real life are easily seen to be separable (give me an example if I am wrong), so there is no real harm if we include this in the definition:-)
But with or without this separability condition, all proofs of existence of holomorphic forms
or of the uniformization use Analysis. Complex or real, I don't see a sharp distinction between
these two. 
A: (Deleted incorrect suggestion).
I think one can use uniformization and the construction of automorphic functions on the universal cover to produce meromorphic functions. A google search for these terms found these notes. 
