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  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 ).
For a generalization of the concept of a Néron model to arbitrary schemes see .
|||A. Néron, "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux" Publ. Math. IHES , 21 (1964)|
|||B. Mazur, "Rational points of Abelian varieties with values in towers of number fields" Invent. Math. , 18 (1974) pp. 183–266|
|||M. Raynaud, "Modèles de Néron" C.R. Acad. Sci. Paris Sér. A , 262 (1966) pp. 345–347|
|||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|
|||A. Grothendieck (ed.) et al. (ed.) , Groupes de monodromie en géométrie algébrique. SGA 7 , Lect. notes in math. , 288 , Springer (1972)|
|[a1]||M. Artin, "Néron models" G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986) pp. 213–230|
Néron model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N%C3%A9ron_model&oldid=12071