I am reading the paper by S. Norine and M. Baker, "Harmonic morphisms and hyperelliptic graphs" which describe various analogies between what they define as harmonic maps between graphs and holomorphic maps between Riemann surfaces. There is one analogy whose classical statement I didn't know and I cannot find any reference about.
In particular given a holomorphic map $\phi:X\to Y $ it induces a map between their jacobians $\phi^*:J(Y)\to J(X)$ . They then say that $\phi^*$ is injective if and only if $\phi:X\to Y$ has a non trivial unramified abelian subcover. I don't know even what is intended as a subcover. They states this fact without proof saying that it is classical, although I didn't find anything I the literature I know.