Let $C$ be a smooth, projective curve of genus >1 over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, $Aut(J(C))$ is isomorphic to $Aut(C)$ if $C$ is hyperelliptic and to $Aut(C) \times \mathbb{Z}/2$ otherwise.

Is there something similar for the endomorphisms of $J(C)$. My guess is that one has to relate them to correspondences on the curve, but I have been unable to find a reference.