Let C be the complex curve $y^{2} = x^{8} - 1$ and $\sigma(x,y) = (\zeta x,y)$ be automorphism of order $r$ = 8, where $\zeta$ is a primitive $r$-th root of unity. How to calculate basis of $H^{0}(K_{C})$?
I know that the answer is $\frac{dx}{y},\frac{xdx}{y},\frac{x^{2}dx}{y}$. I've tried to prove it by showing that every global section is a linear combinations of these, but failed.