On the group of components of a Néron model. (English) Zbl 0781.14029

Lorenzini, Dino J.

J. Reine Angew. Math. 445, 109-160 (1993).

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.

Reviewer: D.J.Lorenzini (Athens GA)

MSC:

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

Keywords:

complete discrete valuation field; Néron model of an abelian variety; Tate module