TITLE: Analogues of the Weierstrass p function for higher genus compact Riemann surfaces
QUESTION [9 upvotes]: There was a previous post on the correspondence between Riemann surfaces and algebraic geometry.  I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how you start with a compact Riemann surfaces and map them into projective space
Links between Riemann surfaces and algebraic geometry
In the case of a genus 1 surface X there's a very explicit construction.  Namely X can be realized as ℂ/L for a lattice L ≅ ℤxℤ.  From here the Weierstrass p function and its derivative can be constructed
http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
and these give you a map ℂ/L --> ℙ^2 via z |--> [p(z), p'(z), 1] which realizes X as a degree three curve in ℙ^2
QUESTION:
Say now X is a compact Riemann surface of genus g > 1.  As has been pointed out below I should restrict to say g = 1/2(d-1)(d-2) where d>3, because otherwise there is no hope to realize X as a nonsingular curve in ℙ^2.    
Is there   
1) a complex manifold Y that is a covering space of X such that X ≅ Y/G where G is the covering group of Y over X  
2) holomorphic functions f₁, f₂, f₃ from Y/G to ℂ∪∞  
such that z |--> [f₁,(z), f₂(z), f₃(z)] realizes X as a projective variety of dimension 1 in ℙ^2?
I'm told a good choice for Y would be the hyperbolic plane because then the 4g-gon representation of a genus g surface tiles the plane.

REPLY [15 votes]: For $X = \Delta/\Gamma$ a compact Riemann surface of genus $g>1$ a good analogue of the Weierstrass $p$-function is the Poincaré series $f_a(z) = \sum_{\gamma \in \Gamma} \gamma^*(q_a)$, where $q_a = dz^2/(z-a)$ is a quadratic differential with a simple pole at $a \in \Delta$.  This series gives a meromorphic quadratic differential on $X$ with a simple pole at any desired point and no other singularities.  If $b$ is close enough to $a$ then $f_b/f_a$ gives a meromorphic function on $X$ with a simple zero at $a$, providing local coordinates.  By compactness one can then choose finitely many $a_1,\ldots,a_n$ such that the corresponding $f_1,\ldots,f_n$ give an embedding of $X$ into $P^{n-1}$.