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Néron model

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of an Abelian variety

A group scheme associated to an Abelian variety and having a certain minimality property. If is a local Henselian discrete valuation ring with residue field and field of fractions and if is an Abelian variety of dimension over , then a Néron model of is defined as a smooth commutative group scheme over whose generic fibre is isomorphic to , while the canonical homomorphism is an isomorphism. This concept was introduced by A. Néron [1] in the case of a perfect field. In the local case a Néron model exists and is uniquely determined up to an -isomorphism. A Néron model has the following minimality property: For any smooth -scheme and any morphism of the generic fibres there exists a unique morphism of -schemes induced by .

If is a one-dimensional regular Noetherian scheme, is a generic point of it, is its canonical imbedding, and is an Abelian variety over , then a Néron model of is defined as a smooth quasi-projective group scheme over that represents the sheaf relative to the flat Grothendieck topology on (see [4]).

For a generalization of the concept of a Néron model to arbitrary schemes see [3].

References

[1] A. Néron, "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux" Publ. Math. IHES , 21 (1964)
[2] B. Mazur, "Rational points of Abelian varieties with values in towers of number fields" Invent. Math. , 18 (1974) pp. 183–266
[3] M. Raynaud, "Modèles de Néron" C.R. Acad. Sci. Paris Sér. A , 262 (1966) pp. 345–347
[4] M. Raynaud, "Caractéristique d'Euler–Poincaré d'un faisceau et cohomologie des variétés abéliennes (d'après Ogg–Shafarévitch et Grothendieck)" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 12–30
[5] A. Grothendieck (ed.) et al. (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7 , Lect. notes in math. , 288 , Springer (1972)


Comments

References

[a1] M. Artin, "Néron models" G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986) pp. 213–230
How to Cite This Entry:
Néron model. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=N%C3%A9ron_model&oldid=35048
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article