Néron model

of an Abelian variety

A group scheme associated to an Abelian variety and having a certain minimality property. If $ R $ is a local Henselian discrete valuation ring with residue field $ k $ and field of fractions $ K $ and if $ A $ is an Abelian variety of dimension $ d $ over $ K $, then a Néron model of $ A $ is defined as a smooth commutative group scheme $ \mathfrak A $ over $ R $ whose generic fibre $ \mathfrak A _ {K} $ is isomorphic to $ A $, while the canonical homomorphism $ \mathfrak A ( R) \rightarrow \mathfrak A _ {K} ( K) $ 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 $ R $- isomorphism. A Néron model has the following minimality property: For any smooth $ R $- scheme $ \mathfrak X $ and any morphism $ \phi : \mathfrak X _ {K} \rightarrow \mathfrak A _ {K} $ of the generic fibres there exists a unique morphism $ \overline \phi \; : \mathfrak X \rightarrow \mathfrak A $ of $ R $- schemes induced by $ \phi $.

If $ S $ is a one-dimensional regular Noetherian scheme, $ \eta $ is a generic point of it, $ i : \eta \rightarrow S $ is its canonical imbedding, and $ A $ is an Abelian variety over $ k( \eta ) $, then a Néron model of $ A $ is defined as a smooth quasi-projective group scheme $ \mathfrak A $ over $ S $ that represents the sheaf $ i _ {*} A $ relative to the flat Grothendieck topology on $ S $( 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)

References

[a1]	M. Artin, "Néron models" G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986) pp. 213–230