Let X be a compact Riemann surface of genus $g$ corresponding to the equation $w^3 = P(z)$, P is a polynomial without multiple roots and $degP = 8$ It's known that there are $g$ holomorphic differentials on this surface. How can I construct them explicitly?
If my curve was defined as hyperelliptic one, $w^2=P(z)$, then I would use $\frac{z^k \cdot dz}{w}$ for $k = 0, \cdots ,g-1$. $\frac{z^k dz}{w} = \frac{2z^k dw}{P'(z)}$, we would check that the form is holomorphic at $z \to \infty$ and $\omega \to 0$. Mention here that $deg P = 2g+2$. (I don't understand to the full how we do this). Then $\frac{z^k dz}{w}$ is well defined when $z$ is a coordinate. and $\frac{2z^k dw}{P'(z)}$ is well defined when $w$ is a cordinate.
By implicit function theorem $w$ is a coordinate everywhere where $P'(z) \neq 0$ so the second representation works and we get over the case of $w=0$.
$z$ is a coordinate everywhere where $2w \neq 0$. so the first representation works and when $z \to \infty$ then we have a double pole from $dz$ and $g+1$-ble pole from $\frac{1}{w}$
Our curve equation means that $dw = \frac{P'(z)}{3w^2} dz$ so if we try the same way it is done for hyperelliptic curve, then we should seek the answer as $\frac{z^k dz}{w^2}= \frac{3z^k dw}{P'(z)}$.
How can we check the holomorphoucity of forms more rigorously and how can I construct the forms in the case $w^3 = P(z)$