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Néron models. (English) Zbl 0603.14028

Arithmetic geometry, Pap. Conf., Storrs/Conn. 1984, 213-230 (1986).
[For the entire collection see Zbl 0596.00007.]
Let R be a Dedekind domain with field of fractions K, and let \(A_ K\) be an abelian scheme over K. A Néron model \(A_ R\) for \(A_ K\) is a smooth group scheme over R whose general fibre is \(A_ K\) and such that the following condition is satisfied: if \(X_ R\) is a smooth R-scheme and \(f: X_ R\to A_ R\) is an arbitrary rational map, then f extends uniquely to a morphism \(X_ R\to A_ R\). The aim of this article is to prove the following fundamental result of Néron [see A. Néron, Publ. Math., Inst. Haut. Étud. Sci. 21 (1964; Zbl 0132.414)]: a Néron model \(A_ R\) always exists (and it is obviously unique) for every abelian variety \(A_ K\) over K. Some subsequent work of M. Raynaud on this problem is also discussed.
Reviewer: L.Bădescu

MSC:

14K15 Arithmetic ground fields for abelian varieties
14L15 Group schemes
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations