The classical Riemann-Roch theorem for Riemann surfaces makes a connection between the dimension of the space of meromorphic functions and the dimension of the space of meromorphic forms. Thus it is useful for the existence of meromorphic functions on a Riemann surface with prescribed singularities (divisor).

By taking differential, a meromorphic function is equivalent to a meromorphic form with vanishing periods, up to a constant, like that $1/z$ is mapped to $z^{-2} dz$. In this sense, Riemann-Roch theorem is a relation between the dimension of meromorphic forms with all periods vanishing and that of arbitrary meromorphic forms.

However, the mapping $f \to df$ is not surjective to all meromorphic forms with vanishing periods, but to only the Abelian differentials of the second kind.

**My question is:** Is there a generalized Riemann-Roch theorem, about the relation between the dimension of the space of Abelian differentials of the *third* kind with all periods vanishing and prescribed singularities, and the dimension of a related space of meromorphic forms?

I have a *concrete problem* in mind: On an elliptic curve, construct a meromorphic form with three singular points and both the two periods vanishing. Around the three singular points, the local behaviours are $(1/z + O(1))dz$, $(-1/z + O(1))dz$ and $(1/z^2 + O(1))dz$. I want to know the existence and uniqueness of the meromorphic form. This question seems not to be a direct consequence of the generalized Riemann-Roch theorem conjured above, but I think they must be related. If you can answer or give a hint to the concrete problem, I also appreciate.