Optimal quotients of Jacobians with toric reduction and component groups

Abstract

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 Φ_J -> Φ_E 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 Φ_J -> Φ_E is surjective, and discuss when these criteria hold for the Jacobians of modular curves.

arXiv comment19 pages. Reorganization and many edits; new proofs of Theorems 4.2 and 4.6