Nonvanishing of holomorphic differentials in genus one ($\eta \ne 0⟹(\eta )=0$)

Every nonzero holomorphic differential form on a compact Riemann surface of genus one is nowhere zero. Its canonical divisor is an effective divisor of degree $2g-2=0$, and hence has every coefficient zero. In particular, such a differential trivializes the canonical bundle. This assertion concerns the differential itself, not the individual factors in a representation $fdh$.