# Power series for meromorphic differentials on compact Riemann surfaces

Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/\Gamma$ where I do know $\Gamma$ explicit. Is there a way to write down the power series of meromorphic functions, differentials, quadratic differentials, and so on explicitly if one does know these sections explicitly (for example in a hyperelliptic picture of the surface). I know that there is the subject of automorphic forms, but the literature I have seen about this concentrates on modular groups. Moreover it is typically written for number theorists. Is there literature for compact surfaces (for example $Y^2=Z^6-1$), and maybe readable for geometers.

Thank you.

• I would strongly recommend the book "Algebraic Curves and Riemann Surfaces" by Rick Miranda. – Kevin Jul 7 '12 at 16:18

## 1 Answer

If you have an equation such as $y^2=z^6-1$ you don't need the upper half plane, find $y$ as a power series of $z-a$ and express everything in terms of $z$. In your example $y=-\sum {1/2 \choose n}(-z^6)^{n}$, so you have the functions. The differentials are $f(z,y)dz/y$ so you are done there too and so on. To find an equation, you might need to find two functions first and I don't know how to do it for a random subgroup of $SL_2(\mathbb{R})$ (?). Reference: Griffiths and Harris, Principles of Algebraic Geometry.

• I'm sorry, mayebe my question was somehow unprecise. I do know these guys in terms of z. But I do hae to solve some ODE (involving these) whose solution will be trancendantal. So it might be better to work in other coordiantes than z. The first guess might be the coordinate on the universal covering. – Sebastian Mar 12 '10 at 15:33