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Néron model

In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety AK defined over the field of fractions K of a Dedekind domain R is the "push-forward" of AK from Spec(K) to Spec(R), in other words the "best possible" group scheme AR defined over R corresponding to AK.

They were introduced by André Néron (1961, 1964) for abelian varieties over the quotient field of a Dedekind domain R with perfect residue fields, and Raynaud (1966) extended this construction to semiabelian varieties over all Dedekind domains.

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Suppose that R is a Dedekind domain with field of fractions K, and suppose that AK is a smooth separated scheme over K (such as an abelian variety). Then a Néron model of AK is defined to be a smooth separated scheme AR over R with fiber AK that is universal in the following sense.

If X is a smooth separated scheme over R then any K-morphism from XK to AK can be extended to a unique R-morphism from X to AR (Néron mapping property).

In particular, the canonical map A R ( R ) A K ( K ) {\displaystyle A_{R}(R)\to A_{K}(K)} is an isomorphism. If a Néron model exists then it is unique up to unique isomorphism.

In terms of sheaves, any scheme A over Spec(K) represents a sheaf on the category of schemes smooth over Spec(K) with the smooth Grothendieck topology, and this has a pushforward by the injection map from Spec(K) to Spec(R), which is a sheaf over Spec(R). If this pushforward is representable by a scheme, then this scheme is the Néron model of A.

In general the scheme AK need not have any Néron model. For abelian varieties AK Néron models exist and are unique (up to unique isomorphism) and are commutative quasi-projective group schemes over R. The fiber of a Néron model over a closed point of Spec(R) is a smooth commutative algebraic group, but need not be an abelian variety: for example, it may be disconnected or a torus. Néron models exist as well for certain commutative groups other than abelian varieties such as tori, but these are only locally of finite type. Néron models do not exist for the additive group.

  • The formation of Néron models commutes with products.
  • The formation of Néron models commutes with étale base change.
  • An Abelian scheme AR is the Néron model of its generic fibre.

The Néron model of an elliptic curve AK over K can be constructed as follows. First form the minimal model over R in the sense of algebraic (or arithmetic) surfaces. This is a regular proper surface over R but is not in general smooth over R or a group scheme over R. Its subscheme of smooth points over R is the Néron model, which is a smooth group scheme over R but not necessarily proper over R. The fibers in general may have several irreducible components, and to form the Néron model one discards all multiple components, all points where two components intersect, and all singular points of the components.

Tate's algorithm calculates the special fiber of the Néron model of an elliptic curve, or more precisely the fibers of the minimal surface containing the Néron model.

Néron model
Neron model Language Watch Edit In algebraic geometry the Neron model or Neron minimal model or minimal model for an abelian variety AK defined over the field of fractions K of a Dedekind domain R is the push forward of AK from Spec K to Spec R in other words the best possible group scheme AR defined over R corresponding to AK They were introduced by Andre Neron 1961 1964 for abelian varieties over the quotient field of a Dedekind domain R with perfect residue fields and Raynaud 1966 extended this construction to semiabelian varieties over all Dedekind domains Contents 1 Definition 2 Properties 3 The Neron model of an elliptic curve 4 See also 5 ReferencesDefinition EditSuppose that R is a Dedekind domain with field of fractions K and suppose that AK is a smooth separated scheme over K such as an abelian variety Then a Neron model of AK is defined to be a smooth separated scheme AR over R with fiber AK that is universal in the following sense If X is a smooth separated scheme over R then any K morphism from XK to AK can be extended to a unique R morphism from X to AR Neron mapping property In particular the canonical map A R R A K K displaystyle A R R to A K K is an isomorphism If a Neron model exists then it is unique up to unique isomorphism In terms of sheaves any scheme A over Spec K represents a sheaf on the category of schemes smooth over Spec K with the smooth Grothendieck topology and this has a pushforward by the injection map from Spec K to Spec R which is a sheaf over Spec R If this pushforward is representable by a scheme then this scheme is the Neron model of A In general the scheme AK need not have any Neron model For abelian varieties AK Neron models exist and are unique up to unique isomorphism and are commutative quasi projective group schemes over R The fiber of a Neron model over a closed point of Spec R is a smooth commutative algebraic group but need not be an abelian variety for example it may be disconnected or a torus Neron models exist as well for certain commutative groups other than abelian varieties such as tori but these are only locally of finite type Neron models do not exist for the additive group Properties EditThe formation of Neron models commutes with products The formation of Neron models commutes with etale base change An Abelian scheme AR is the Neron model of its generic fibre The Neron model of an elliptic curve EditThe Neron model of an elliptic curve AK over K can be constructed as follows First form the minimal model over R in the sense of algebraic or arithmetic surfaces This is a regular proper surface over R but is not in general smooth over R or a group scheme over R Its subscheme of smooth points over R is the Neron model which is a smooth group scheme over R but not necessarily proper over R The fibers in general may have several irreducible components and to form the Neron model one discards all multiple components all points where two components intersect and all singular points of the components Tate s algorithm calculates the special fiber of the Neron model of an elliptic curve or more precisely the fibers of the minimal surface containing the Neron model See also EditMinimal model programReferences EditArtin Michael 1986 Neron models in Cornell G Silverman Joseph H eds Arithmetic geometry Storrs Conn 1984 Berlin New York Springer Verlag pp 213 230 MR 0861977 Bosch Siegfried Lutkebohmert Werner Raynaud Michel 1990 Neron models Ergebnisse der Mathematik und ihrer Grenzgebiete 3 21 Berlin New York Springer Verlag doi 10 1007 978 3 642 51438 8 ISBN 978 3 540 50587 7 MR 1045822 I V Dolgachev 2001 1994 Neron model Encyclopedia of Mathematics EMS Press Neron Andre 1961 Modeles p minimaux des varietes abeliennes Seminaire Bourbaki 7 MR 1611194 Zbl 0132 41402 Neron Andre 1964 Modeles minimaux des varietes abeliennes sur les corps locaux et globaux Publications Mathematiques de l IHES 21 5 128 doi 10 1007 BF02684271 MR 0179172 Raynaud Michel 1966 Modeles de Neron Comptes Rendus de l Academie des Sciences Serie A B 262 A345 A347 MR 0194421 W Stein What are Neron models 2003 Retrieved from https en wikipedia org w index php title Neron model amp oldid 1052129479, wikipedia, wiki, book,

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