  Let J be a Jacobian variety with toric reduction over a local field K. Let J
-> E be an optimal quotient defined over K, where E is an elliptic curve. We
give examples in which the functorially induced map \Phi_J -> \Phi_E on
component groups of the N\'eron models is not surjective. This answers a
question of Ribet and Takahashi. We also give various criteria under which
\Phi_J -> \Phi_E is surjective, and discuss when these criteria hold for the
Jacobians of modular curves.
