Optimal Quotients of Jacobians with Toric Reduction and Component Groups
Printed: Dec 2016
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
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.
Jacobians with toric reduction, component groups, modular curves
11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35]
14G22 - Rigid analytic geometry
14G20 - Local ground fields