Néron model

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In algebraic geometry, a Néron model (or Néron minimal model, or minimal model) for an abelian variety A_K defined over a local field K is the "best possible" group scheme A_O defined over the ring of integers R of the local field K that becomes isomorphic to A_K after base change from R to K.

They were introduced by André Néron (1964).

[edit] Definition

Suppose that R is a Dedekind domain with field of fractions K, and suppose that A_K is an abelian variety over K. Then a Néron model is defined to be a universal smooth scheme A_R over R with a rational map to A_K. More precisely, this means that A_R is a smooth scheme over R with general fiber A_K, such that any rational map from a smooth scheme over R to A_R can be extended to a unique morphism.

Néron models exist and are unique (up to unique isomorphism) and are group schemes of finite type over R. The fiber of a Néron model over a closed point of Spec(R) is an algebraic group, but need not be an abelian variety: for example, it may be disconnected, or unipotent, or a torus.

[edit] References

• M. Artin, Néron models (in Arithmetic geometry, edited by G. Cornell, J. Silverman, ISBN 0387963111 Springer (1986) pp. 213–230).
• Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Neron Models ISBN 3540505873
• I.V. Dolgachev, "Néron model" SpringerLink Encyclopaedia of Mathematics (2001)
• André Néron Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Publications Mathématiques de l'IHÉS, 21 (1964), p. 5-128
• W. Stein, What are Néron models? (2003)