So I'm trying to understand some relation between meromorphic function and abelian differential (i.e. meromorphic differential) over a compact Riemann surface $M$. The basic question is whether the coefficient functions of abelian differential form a meromorphic function.
If $f:M\to\Bbb C_{\infty}$ is a meromorphic function then $f(z)dz$ in local coordinate $z$ on $M$ defines an abelian differential. If $\omega$ is an abelian differential and if I define $c_1$ to be an abelian differential with $dz$ for each local coordinate $z$, then $\omega/c_1$ is a meromorphic function so the coefficient function of $\omega$ defines a meromorphic function. So I think from the given meromorphic function, I can define an abelian differential whose local coefficient is the given one, and from the given abelian differential, I can get a meromorphic function which is a coefficient of the given one. Does it make sense?