# Explicit universal covering map for higher genus algebraic curves

Suppose I have a projective plane curve $$C = V(F)$$ defined over $$\mathbb{C}$$, where $$F$$ is some homogeneous polynomial in three variables. For the sake of simplicity, let's assume that $$C$$ is irreducible, and that $$C$$ contains a rational point $$P$$.

If $$C$$ is smooth of degree 2 (so genus $$0$$), then we can obtain a universal conformal covering map $$\mathbb P^1(\mathbb C) \rightarrow C$$ by identifying each point in $$\mathbb P^1$$ with a line $$\ell$$ through $$P$$ in the usual way, and outputting the other intersection point of $$\ell$$ with $$C$$.

If $$C$$ is smooth of degree 3 (so genus $$1$$), then we can obtain a universal conformal covering map $$\mathbb C \rightarrow C$$ by applying a change of variables to convert $$C$$ to Weierstrass form $$F(x,y,z) = y^2 z - x^3 + g_2 xz^2 - g_3z^3,$$ and then using the Weierstrass $$\wp$$-function for the associated lattice: $$z \longmapsto [\wp(z) : \wp'(z) : 1],$$ where we send the poles of $$\wp$$ to the limiting points at infinity.

If $$C$$ is singular of geometric genus $$\le 1$$, then we can resolve the singularities via a normalization in order to obtain a smooth curve, followed by parameterizing as above.

Consider now the case where $$C$$ has genus $$g\ge 2$$. We then obtain a Riemann surface of genus $$g$$ punctured at the singular points. By the uniformization theorem, there is a conformal map $$\varphi_C:\mathbb H\rightarrow C$$, where $$\mathbb H$$ is the upper half-plane, corresponding to the universal covering of $$C$$. However, I am not aware of any explicit formula for this map, even for a specific instance of $$C$$.

Is there any known way to explicitly parameterize these higher genus curves by a conformal map?

• naive comment: These short sources may interest you. In the opposite direction, Serre, in A course in arithmetic, p.83, gives Eisenstein series for the coefficients of a plane elliptic curve in terms of its lattice, which curve is uniformized by the corresponding P-function. Partially analogous, Shafarevich, in Basic algebraic geometry, 1974 ed., p.393 ff., gives Poincare series for canonical embedding of a curve in terms of the discrete subgroup of Aut(D) uniformizing it. But how to make that group explicit, describe the embedded curve explicitly, or go from curve to uniformization? Commented Apr 26, 2022 at 17:45