The Lopez-Ros Parameter

The so called Lopez-Ros parameter ρ is given by

This choice of the parameter ensures that the image of a cycle encircling the points 0 and 1 encircling the points is a closed curve on the surface. To see this, we would like to integrate ϕ1 and ϕ2 from 0 to 1. However, this is not possible for ϕ1, as the singularity at 1 is not integrable. To circumvent this difficulty, we replace ϕ1 by a 1-form ϕ1b with the same periods:

This is true (up to some careful sign checking) because the difference of ϕ1b and ϕ1 is the deivative of a function which is meromorphic on the torus:

Then the definition of ρ is checked by

So we obtain finally the parametrization