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On the group of components of a Néron model. (English) Zbl 0781.14029
Let $$K$$ be a complete discrete valuation field with residue characteristic $$p\geq 0$$. This article describes the structure of the group of components $$\Phi$$ of the Néron model of an abelian variety $$A/K$$. Let $$\ell\neq p$$ be any prime. A filtration of the $$\ell$$-part $$\Phi_ \ell$$ of the group $$\Phi$$ is defined using the monodromy filtration on the Tate module $$T_ \ell A$$. In particular, we show the existence of a subgroup $$\Theta_ \ell$$ of $$\Phi_ \ell$$ such that $$| \Theta_ \ell |$$ is bounded by an explicit constant depending only on the unipotent rank of $$A/K$$, and such that the group $$\Phi_ \ell/ \Theta_ \ell$$ can be generated by $$t_ K$$ elements, where $$t_ K$$ denotes the toric rank of $$A/K$$. When $$A/K$$ has purely additive reduction, our results on the structure of $$\Phi_ \ell$$ are then used to describe the prime-to-$$p$$ part of the torsion subgroup of $$A(K)$$.
In the second part of this paper, we present a description of the $$p$$- part of the group $$\Phi$$ when $$A/K$$ is a jacobian.

##### MSC:
 14K05 Algebraic theory of abelian varieties 14G15 Finite ground fields in algebraic geometry 14G05 Rational points
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