# 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)