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| In [[algebraic geometry]], the '''Néron model''' (or '''Néron minimal model''', or '''minimal model''')
| | I'm Luther but I never really liked that title. Some time in the past I chose to live in Arizona but I need to move for my family. Playing croquet is some thing I will by no means give up. Interviewing is how I make a residing and it's some thing I really enjoy.<br><br>Here is my web page :: [http://enjoythisworld.com/UserProfile/tabid/464/userId/33651/Default.aspx extended auto warranties] |
| for an [[abelian variety]] ''A<sub>K</sub>'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''A<sub>K</sub>'' from Spec(''K'') to Spec(''R''), in other words the "best possible" group scheme ''A<sub>R</sub>'' defined over ''R'' corresponding to ''A<sub>K</sub>''.
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| They were introduced by {{harvs|txt|authorlink=André Néron|first=André |last=Néron|year1=1961|year2=1964}} for abelian varieties over the quotient field of a Dedekind domain ''R'' with perfect residue fields, and {{harvtxt|Raynaud|1966}} extended this construction to semiabelian varieties over all Dedekind domains.
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| ==Definition==
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| Suppose that ''R'' is a [[Dedekind domain]] with field of fractions ''K'', and suppose that ''A<sub>K</sub>'' is an abelian variety over ''K'', or more generally a smooth separated scheme over ''K''. Then a '''Néron model''' of ''A<sub>K</sub>'' is defined to be a universal [[Separated_morphism|separated]] [[smooth morphism|smooth]] scheme ''A<sub>R</sub>'' over ''R'' with the following Néron mapping property.
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| This means that ''A<sub>R</sub>'' is a separated smooth scheme over ''R'' with general fiber ''A<sub>K</sub>'', such that if ''X'' is a smooth scheme over ''R'' then any ''K''-morphism from ''X''<sub>''K''</sub>
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| to ''A<sub>K</sub>'' can be extended to a unique morphism from ''X'' to ''A<sub>R</sub>'' '''(Néron mapping property)'''. In particular, the canonical map <math>A_R(R)\to A_K(K)</math> is an isomorphism. | |
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| In terms of sheaves, any scheme ''A'' over Spec(''K'') represents a sheaf for the flat 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''.
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| For abelian varieties Néron models exist and are unique (up to unique isomorphism) and are commutative quasi-projective [[group scheme]]s 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.
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| == Properties ==
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| * The formation of Néron models commutes with products.
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| * The formation of Néron models commutes with étale base change.
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| * An [[Abelian scheme]] ''A''<sub>''R''</sub> is the Néron model of its generic fibre.
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| ==The Néron model of an elliptic curve==
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| The Néron model of an ellitpic curve ''A''<sub>''K''</sub> 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.
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| [[Tate's algorithm]] describes the fibers of the Néron model of an elliptic curve, or more precisely the fibers of the minimal surface containing the Néron model.
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| ==References==
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| * {{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | editor1-last=Cornell | editor1-first=G. | editor2-last=Silverman | editor2-first=Joseph H. | editor2-link=Joseph H. Silverman | title=Arithmetic geometry (Storrs, Conn., 1984) | publisher=[[Springer-Verlag]] | location=Berlin, New York | id={{MathSciNet | id = 861977}} | year=1986 | chapter=Néron models | pages=213–230}}
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| * {{Citation | last1=Bosch | first1=Siegfried | last2=Lütkebohmert | first2=Werner | last3=Raynaud | first3=Michel | author3-link= Michel Raynaud | title=Néron models | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] (3)| isbn=978-3-540-50587-7 | id={{MathSciNet | id = 1045822}} | year=1990 | volume=21}}
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| *{{springer|id=N/n066290|authorlink=I.V. Dolgachev|author=I.V. Dolgachev|title=Néron model }}
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| *{{citation|last=Néron|first= André |title=Modèles p-minimaux des variétés abéliennes. |series=Séminaire Bourbaki|volume= 7 |year=1961|issue= 227|mr= 1611194 | zbl= 0132.41402
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| |url= http://www.numdam.org/item?id=SB_1961-1962__7__65_0}}
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| * {{Citation | last1=Néron | first1=André | author1-link= André Néron | title=Modèles minimaux des variétes abèliennes sur les corps locaux et globaux | url=http://www.numdam.org/item?id=PMIHES_1964__21__5_0 | id={{MathSciNet | id = 0179172}} | year=1964 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=21 | pages=5–128 | doi=10.1007/BF02684271}}
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| *{{citation|mr=0194421|last=Raynaud|first= Michel|title=Modèles de Néron|journal=C. R. Acad. Sci. Paris Sér. A-B|volume= 262 |year=1966 |pages=A345–A347}}
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| *W. Stein, [http://wstein.org/edu/Fall2003/252/lectures/10-20-03/10-20-03-Neron_models.pdf What are Néron models?] (2003)
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| {{DEFAULTSORT:Neron model}}
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| [[Category:Algebraic geometry]]
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| [[Category:Number theory]]
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I'm Luther but I never really liked that title. Some time in the past I chose to live in Arizona but I need to move for my family. Playing croquet is some thing I will by no means give up. Interviewing is how I make a residing and it's some thing I really enjoy.
Here is my web page :: extended auto warranties