I'm confused about this definition in chapter one of Griffith's Introduction to Algebraic Curves:
Suppose C is a compact Riemann Surface, with $f\in K(C), p\in C.$ Select a local coordinate $z$ in a neighborhood of the point $p$ such that $z(p)=0.$ Then in a neighborhood of $p$ $$f=z^{\nu}h(z),$$ where $h(z)$ is a holomorphic function, $h(0)\neq 0$, and $\nu\in\mathbb{Z}$.
I was wondering what it means for $z(p)$ to be equal to zero, since $z$ is a local coordinate. What does this mean? Does this mean that $z$ is a function? How do we use a local coordinate as a function in this way?
Yes, the local coordinate $z$ is really a homeomorphism $z : U \to \mathbb{C}$, going from a small piece of the Riemann surface $C$ to some open subset of $\mathbb{C}$. By possibly translating $z(U) \subset \mathbb{C}$, we can assume that $z(p) = 0 \in \mathbb{C}$, i.e. z maps $p$ to the origin. That way, the chart $(U, z)$ is "centered" around $p$.
The atlas of such charts $\{(U, z)\}$ comes with the initial data of a Riemann surface.