I am working on the uniformization of algebraic curve problem.
Currently, my adviser gave me a question about building a Möbius transformation between algebraic curves, and then lifting it to the Riemann surface.
My question is how does one build a Möbius transformation between algebraic curves? How does this transformation make sense?
Say that I have curve $y^2 = x^2 - 1$, and my first idea is mapping $(x, y) \rightarrow (1/x, y)$, then I would have $y^2 = (1/x)^2 - 1$, but my adviser said it doesn't make sense.
So any one could gave me an idea? Thanks!