Meromorphic differential on a Riemann surface ($\omega =f(z)dz$)

A meromorphic differential is specified in each holomorphic chart by a meromorphic coefficient $f(z)$, with the coefficients transforming by the coordinate derivative so that the one-form agrees on overlaps. The derivative of a meromorphic function gives such a differential, and the ratio of two nonzero differentials is a meromorphic function. Orders of zeros and poles are independent of chart because coordinate derivatives are nonvanishing holomorphic factors. On a compact connected surface the sum of orders of a nonzero differential is $2g-2$, allowing its divisor to determine the genus.