I was currently studying the $Mittag-Leffler$ problems on Riemann surfaces i.e. to construct functions with specified Laurent tails at a finite number of points and such that the function is holomorphic everywhere else (i.e. it will not have any poles elsewhere). I also found that the fact that the first cohomology group of the divisors being finite dimensional helps us to reduce the problem to finitely many linear constraints.
Is there something more that we can infer on this problem i.e. can we say something even stronger?