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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 \(p\).

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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