Optimal Quotients of Jacobians With Toric Reduction and Component Groups

Abstract

AbstractLet 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 on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which is surjective and discuss when these criteria hold for the Jacobians of modular curves.