Optimal Quotients of Jacobians With Toric Reduction and Component Groups

Let I 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 Neron 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.