Namespaces
Variants
Actions

Difference between revisions of "Néron model"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Neron model to Néron model over redirect: accented title)
m (links)
Line 1: Line 1:
 
''of an Abelian variety''
 
''of an Abelian variety''
  
A [[Group scheme|group scheme]] associated to an [[Abelian variety|Abelian variety]] and having a certain minimality property. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662901.png" /> is a local Henselian discrete valuation ring with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662902.png" /> and field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662903.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662904.png" /> is an Abelian variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662905.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662906.png" />, then a Néron model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662907.png" /> is defined as a smooth commutative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662908.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662909.png" /> whose generic fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629010.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629011.png" />, while the canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629012.png" /> is an isomorphism. This concept was introduced by A. Néron [[#References|[1]]] in the case of a perfect field. In the local case a Néron model exists and is uniquely determined up to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629013.png" />-isomorphism. A Néron model has the following minimality property: For any smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629014.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629015.png" /> and any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629016.png" /> of the generic fibres there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629018.png" />-schemes induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629019.png" />.
+
A [[Group scheme|group scheme]] associated to an [[Abelian variety|Abelian variety]] and having a certain minimality property. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662901.png" /> is a local Henselian [[discrete valuation ring]] with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662902.png" /> and [[field of fractions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662903.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662904.png" /> is an Abelian variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662905.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662906.png" />, then a Néron model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662907.png" /> is defined as a smooth commutative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662908.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n0662909.png" /> whose generic fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629010.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629011.png" />, while the canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629012.png" /> is an isomorphism. This concept was introduced by A. Néron [[#References|[1]]] in the case of a perfect field. In the local case a Néron model exists and is uniquely determined up to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629013.png" />-isomorphism. A Néron model has the following minimality property: For any smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629014.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629015.png" /> and any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629016.png" /> of the generic fibres there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629018.png" />-schemes induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629019.png" />.
  
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629020.png" /> is a one-dimensional regular Noetherian scheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629021.png" /> is a generic point of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629022.png" /> is its canonical imbedding, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629023.png" /> is an Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629024.png" />, then a Néron model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629025.png" /> is defined as a smooth quasi-projective group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629026.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629027.png" /> that represents the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629028.png" /> relative to the flat Grothendieck topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629029.png" /> (see [[#References|[4]]]).
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629020.png" /> is a one-dimensional regular Noetherian scheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629021.png" /> is a generic point of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629022.png" /> is its canonical imbedding, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629023.png" /> is an Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629024.png" />, then a Néron model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629025.png" /> is defined as a smooth quasi-projective group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629026.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629027.png" /> that represents the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629028.png" /> relative to the flat Grothendieck topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066290/n06629029.png" /> (see [[#References|[4]]]).

Revision as of 20:53, 28 November 2014

of an Abelian variety

A group scheme associated to an Abelian variety and having a certain minimality property. If is a local Henselian discrete valuation ring with residue field and field of fractions and if is an Abelian variety of dimension over , then a Néron model of is defined as a smooth commutative group scheme over whose generic fibre is isomorphic to , while the canonical homomorphism 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 -isomorphism. A Néron model has the following minimality property: For any smooth -scheme and any morphism of the generic fibres there exists a unique morphism of -schemes induced by .

If is a one-dimensional regular Noetherian scheme, is a generic point of it, is its canonical imbedding, and is an Abelian variety over , then a Néron model of is defined as a smooth quasi-projective group scheme over that represents the sheaf relative to the flat Grothendieck topology on (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)


Comments

References

[a1] M. Artin, "Néron models" G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986) pp. 213–230
How to Cite This Entry:
Néron model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=N%C3%A9ron_model&oldid=35048
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article