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# Optimal Quotients of Jacobians with Toric Reduction and Component Groups

Published:2016-08-19
Printed: Dec 2016
• Mihran Papikian,
Department of Mathematics, Pennsylvania State University, University Park, PA 16802
• Joseph Rabinoff,
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332
 Format: LaTeX MathJax PDF

## Abstract

Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J \to E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map $\Phi_J \to \Phi_E$ 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 $\Phi_J \to \Phi_E$ is surjective, and discuss when these criteria hold for the Jacobians of modular curves.
 Keywords: Jacobians with toric reduction, component groups, modular curves
 MSC Classifications: 11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G22 - Rigid analytic geometry 14G20 - Local ground fields

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