Riemann Surfaces Question (Complex Analysis)

Question

Let $X$ be a compact Riemann surface, and denote by $m_X$ the following field: $ m_X := \{ f:X \to \mathbb{P}_\mathbb{C} : f- \text{meromorphic} \} - \{\infty \} $

What is the natural injection of the field of rational functions $\mathbb{C}(z)$ into $m_X$ ?

p.s- $\mathbb{P}_\mathbb{C}$ denotes the Riemann sphere.

Thanks in advance !!!

Answer 1

Let me say that emphatically:

There is no canonical injection of $\mathbb C(z)$ into $\mathcal M(X)$

To give an embedding $\mathbb C(z) \hookrightarrow\mathcal M(X)$ exactly amounts to choosing a non-constant morphism $m:X\to \mathbb P^1(\mathbb C)$.
If such a choice is made, the deduced field embedding $\mathbb C(z) \hookrightarrow\mathcal M(X)$ will send $z\mapsto m$, where $m$ is now seen as a meromorphic function on $X$ .

Answer 2

There is no natural map. $m_X$ is a field and $\mathbb C \subset m_X$ is a field extension. Obviously every $f \in m_X$ that is transcendental over $\mathbb C$ defines such an injection.

The really hard part is to show that there exists a non-constant meromorphic function at all!