Holomorphic differentials on $\mathbb{P^1}$?

Question

I'm studying "The Arithmetic of Elliptic curves by J. H. Silverman". The author has shown in example 4.5 (page 32), that we have not any holomorphic differentials on $\mathbb{P^1}$. I don't understand the term "holomorphic" for differentials. I've seen some definitions for holomorphic functions, but what does it mean for a differential? For instance, in the above example, I cannot understand how $div(dt)=-2(\infty)$? where "$t$" is a coordinate function on $\mathbb{P^1}$.
Thanks in advance.

Answer 1

Any $\omega\in\Omega_C$ is holomorphic if $\operatorname{ord}_P(g)\geq0$ for all $P\in C$, where $\omega=g\,dt$ and $t$ is a uniformizer at $P$. Note that $g\in\bar{K}(C)$.

Explanation:

On page 31 in Proposition 4.3(a) it is stated that

For every $\omega\in\Omega_C$ there exists a unique function $g\in\bar{K}(C)$, depending on $\omega$ and $t$, satisfying $$\omega=g\,dt.$$ We denote $g$ by $\omega/dt$.

Then on the same page in Proposition 4.3(c) it is stated that

The quantity $$\operatorname{ord}_P(\omega/dt)$$ depends only on $\omega$ and $P$, independent of the choice of uniformizer $t$. We call this value the order of $\omega$ at $P$ and denote it by $\operatorname{ord}_P(\omega)$.

On page 32 there is the definition

The differential $\omega\in\Omega_C$ is regular (or holomorphic) if $$\operatorname{ord}_P(\omega)\geq0\quad\text{for all }P\in C.$$