Past exam of the algebraic curves course of the University of Oxford / 2023 / 3 / c

The generator $\eta =fdh$ is a nonzero holomorphic differential form. Its canonical divisor $(\eta )$ is an effective divisor, since there are no poles. In genus one, the degree of a divisor formula in part (a) givesAn effective divisor of degree zero has every coefficient zero. ThusThis proves nonvanishing of holomorphic differentials in genus one. It concerns the differential itself: a zero or pole of the particular coefficient $f$ may be cancelled by the local behavior of $dh$.