Differentials on a hyper elliptic curve that are preserved by involution can be pulled back from $\mathbb{P}^1$ from $\pi$, hence must be 0.

Question

This is the exercise 21.3.C from FOAG. I did the first two, but got stuck at the third one. I'm not entirely sure what it's asking.

Given a double cover $\pi: C\rightarrow \mathbb{P}^1$ branched over $2g+2$ distinct points. Show that $h^0(C,\Omega_{C/k}) = g$ as follows.

1. Show that $\frac{dx}{y}$ is a (regular) differential on $\mathrm{Spec} k[x,y]/(y^2 - f(x))$, i.e. an element of $\Omega_{(k[x,y]/(y^2 - f(x)))/k}$.
2. Show that for $0\leqslant i < g$, $x^i \frac{dx}{y}$ extends to a global differential $\omega_i$ on $C$, i.e. with no poles.
3. Use (1) to show that any differential $\omega\in H^0(C,\Omega_{C/k})$ preserved by the involution $i:y\mapsto -y$ must be pulled back from $\mathbb{P}^1$ by $\pi$, and hence must be zero. Show that every differential $\omega\in H^0(C,\Omega_{C/k})$ satisfies $i^*\omega = -\omega$.