Optimal quotients of Jacobians with toric reduction and component groups

TL;DR

This paper investigates the relationship between Jacobians with toric reduction and their elliptic curve quotients, providing examples and criteria for the surjectivity of induced maps on component groups.

Contribution

It offers the first examples where the induced map on component groups is not surjective and establishes criteria for surjectivity, addressing questions posed by Ribet and Takahashi.

Findings

Examples where the map on component groups is not surjective.

Criteria for when the induced map on component groups is surjective.

Discussion of surjectivity in the context of Jacobians of modular curves.

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 \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.