On homomorphisms of Abelian schemes

In this paper we consider homomorphisms of abelian schemes $\pi_i\colon X_i \to S$ ($i=1,2$) over a connected smooth algebraic curve $S$ defined over the field of complex numbers. We prove that under certain natural conditions the canonical map
$$ \operatorname{Hom}_S(X_1,X_2)\to\operatorname{Hom}(R_1\pi_{1*}Z,R_1\pi_{2*}Z) $$
is an isomorphism.
Bibliography: 5 titles.