Equations for covering spaces of non-singular curves Suppose I have a Riemann surface $C$ given by an explicit equation or equations as a complex plane curve or space curve.  $C$ has a(n unramified) covering space $C'$ associated to each subgroup of finite index in $\pi_1(C)$.  General principles make $C'$ first a Riemann surface and then an algebraic curve.  But is there an explicit method for writing down an equation for $C'$ starting from the equation(s) for $C$ and, say, generators for the subgroup?  
 A: I can propose an algorithm for doing this for ``Belyi curves'', the curves whose coefficients are algebraic numbers. According to Belyi's theorem, such a curve of genus $g$ is defined by a combinatorial triangulation of an abstract genus $g$ surface. And this
triangulation can be explicitly computed (in principle). Then if you have a covering, it is defined by another triangulation which can be easily constructed
from the first one. And passing from a triangulation back to the equation is
a procedure which requires only solving algebraic equations.
We obtain that a finite covering of a Belyi curve is a Belyi curve again, and coefficients can be (in principle) computed. Of course this is a theoretical possibility, the explicit algorithm can be hard.
Belyi curves are dense in the space of all curves of given genus. When you slightly vary your curve, you may slightly vary the covering. This shows that in
principle there are formulas which express the coefficients of the covering curve
in terms of the coefficients of the original one in the general case.
