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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
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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 NĂ©ron 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 Jacobians with toric reduction, component groups, modular curves
MSC Classifications: 11G18, 14G22, 14G20 show english descriptions Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Rigid analytic geometry
Local ground fields
11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35]
14G22 - Rigid analytic geometry
14G20 - Local ground fields
 

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