Component Groups of Purely Toric Quotients

Suppose $\pi:J\rightarrow A$ 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~$p$.

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Published 1 January 2001