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Optimal Quotients of Jacobians With Toric Reduction and Component Groups
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- Journal:
- Canadian Journal of Mathematics / Volume 68 / Issue 6 / 01 December 2016
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1362-1381
- Print publication:
- 01 December 2016
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- Article
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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.
