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Component groups of purely toric quotients. (English) Zbl 1081.11040
Summary: Suppose $$\pi:J\to A$$ is an optimal quotient of abelian varieties over a $$p$$-adic field, optimal in the sense that $$\ker(\pi)$$ is connected. Assume that $$J$$ is equipped with a symmetric principal polarization $$\theta$$ (e.g., any Jacobian of a curve has such a polarization), that $$J$$ has semistable reduction, and that $$A$$ has purely toric reduction. In this paper, we express the group of connected components of the Néron model of $$A$$ in terms of the monodromy pairing on the character group of the torus associated to $$J$$. We apply our results in the case when $$A$$ is an optimal quotient of the modular Jacobian $$J_0(N)$$. For each prime $$p$$ that exactly divides $$N$$, we obtain an algorithm to compute the order of the component group of $$A$$ at $$p$$.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11G10 Abelian varieties of dimension $$> 1$$ 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14K15 Arithmetic ground fields for abelian varieties
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