# 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.

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

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.