# A basis for the holomorphic differentials of a hyper-elliptic Riemann surface

Bumped in to this problem while trying to understand hyper-elliptic Riemann surfaces, and being a bit new to the subject of Riemann surfaces, was not that much confident on a couple of things.

Consider the hyper-elliptic Riemann surface $$\mathcal{R}$$ defined as the affine curve

$$y^2=z(z-1)(z-a)(z-b)(z-c)(z-d)=P(z)$$,

where $$1 and compactified at $$z=\infty$$ with two points and with local coordinate $$\zeta=\frac{1}{z}$$.

1. Find the genus.

2. Find a basis for the holomorphic differentials of $$\mathcal{R}$$.

3. Consider the divisor of points $$D=(z_1,y_1)+(z_2,y_2)$$, where $$y_j$$ are one of the solutions of the equation $$y_j^2=P(z_j)$$. Find $$i(D)$$ (the dimension of the space of holomorphic differentials vanishing at those two points) assuming that $$z_1\neq z_2$$.

4. Find $$i(D)$$ assuming that $$z_1=z_2$$ and $$y_1=-y_2$$.

Obviously, (1) is straightforward. Any help/suggestions on the other three parts will be very helpful. I believe I have to use Riemann-Roch for (3) and (4) but not sure how.

• Perhaps this will help. Commented Nov 5, 2018 at 1:26

A suitable basis is $$\frac{z^k}{y}dz$$, $$k=0,\dots,g-1$$. You need to verify that these differentials are holomorphic. (3) and (4) can be solved using this basis.