Given a germ of an analytic function on a (compact, for simplicity) Riemann surface, how can one see (locally) whether this is a "germ of meromorphic function"? I.e. if I do analytic continuation along various paths, how can I be sure sure that I will never see an essential singularity?

Another formulation of this question is, how can one determine whether a convergent taylor series determines a meromorphic function on the universal covering space of the Riemann surface?

The fact that there will be no essential singularity certainly implies something, e.g. when our Riemann surface is CP^1, then for a taylor series to be meromorphic, it must be rational. But how do one check this locally, in a nbhd of a point?

Thanks

P.S. I don't really know how to tag this question. Suggest a tag in comment please if possible.