Néron model
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In algebraic geometry, a Néron model (or Néron minimal model, or minimal model) for an Abelian variety AK defined over a local field K is the "best possible" group scheme AO defined over the ring of integers R of the local field K that becomes isomorphic to AK 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 AK is an Abelian variety over K. Then a Néron model is defined to be a universal smooth scheme AR over R with a rational map to AK. More precisely, this means that AR is a smooth scheme over R with general fiber AK, such that any rational map from a smooth scheme over R to AR 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
- Artin, Michael (1986), “Néron models”, in Cornell, G. & Silverman, Joseph H., Arithmetic geometry (Storrs, Conn., 1984), Berlin, New York: Springer-Verlag, pp. 213–230, MR861977
- Bosch, Siegfried; Lütkebohmert, Werner & Raynaud, Michel (1990), Néron models, vol. 21, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Berlin, New York: Springer-Verlag, MR1045822, ISBN 978-3-540-50587-7
- I.V. Dolgachev (2001), “Néron model”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Néron, André (1964), “Modèles minimaux des variétes abèliennes sur les corps locaux et globaux”, Publications Mathématiques de l'IHÉS 21: 5–128, MR0179172, ISSN 1618-1913, <http://www.numdam.org/item?id=PMIHES_1964__21__5_0>
- W. Stein, What are Néron models? (2003)
