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Component groups of Néron models via rigid uniformization. (English) Zbl 0869.14020
Let \(R\) be a discrete valuation ring and \(K=\text{Quot} (R)\). The authors are interested in some aspects of the component group \(\Phi_E\) of the Néron model, if it exists, of an abelian variety \(E_K\). One can suppose \(R\) to be complete, because Néron models well behave with respect to completion, and then one can use rigid techniques as uniformizations of abelian varieties [M. Raynaud, Actes Congr. Internat. Math., Nice 1970, Part I, 473-477 (1971; Zbl 0223.14021) and S. Bosch and W. Lütkebohmert, Invent. Math. 78, 257-297 (1984; Zbl 0554.14015)]. The main results are about a filtration and a dual one of \(\Phi_E\). This has been first introduced by D. J. Lorenzini [J. Reine Angew. Math. 445, 109-160 (1993; Zbl 0781.14029)], only for the \(p\)-part of \(\Phi_E\) \((p=\) residue characteristic of \(R)\). The authors give a filtration for the whole group \(\Phi_E\), with a geometric interpretation of the different terms. Moreover, if \(j:R \hookrightarrow K\) is the natural inclusion, \(E\) can be seen as representing the sheaf \(j_*E_K\) on the small smooth site over \(R\); following this point of view, the authors extend their study to identity components or groups of components for Néron models in the category of sheaves over the small smooth site and even to complexes in the derived category over this site.

MSC:
14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
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