Namespaces
Variants
Actions

Difference between revisions of "Néron model"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (links)
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
n0662901.png
 +
$#A+1 = 29 n = 0
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/N066/N.0606290 N\Aeeron model
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''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 $  R $
 +
is a local Henselian [[discrete valuation ring]] with residue field $  k $
 +
and [[field of fractions]] $  K $
 +
and if $  A $
 +
is an Abelian variety of dimension $  d $
 +
over $  K $,  
 +
then a Néron model of $  A $
 +
is defined as a smooth commutative group scheme $  \mathfrak A $
 +
over $  R $
 +
whose generic fibre $  \mathfrak A _ {K} $
 +
is isomorphic to $  A $,  
 +
while the canonical homomorphism $  \mathfrak A ( R) \rightarrow \mathfrak A _ {K} ( K) $
 +
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 $  R $-
 +
isomorphism. A Néron model has the following minimality property: For any smooth $  R $-
 +
scheme $  \mathfrak X $
 +
and any morphism $  \phi : \mathfrak X _ {K} \rightarrow \mathfrak A _ {K} $
 +
of the generic fibres there exists a unique morphism $  \overline \phi \; : \mathfrak X \rightarrow \mathfrak A $
 +
of $  R $-
 +
schemes induced by $  \phi $.
  
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 $  S $
 +
is a one-dimensional regular Noetherian scheme, $  \eta $
 +
is a generic point of it, $  i : \eta \rightarrow S $
 +
is its canonical imbedding, and $  A $
 +
is an Abelian variety over $  k( \eta ) $,  
 +
then a Néron model of $  A $
 +
is defined as a smooth quasi-projective group scheme $  \mathfrak A $
 +
over $  S $
 +
that represents the sheaf $  i _ {*} A $
 +
relative to the flat Grothendieck topology on $  S $(
 +
see [[#References|[4]]]).
  
 
For a generalization of the concept of a Néron model to arbitrary schemes see [[#References|[3]]].
 
For a generalization of the concept of a Néron model to arbitrary schemes see [[#References|[3]]].
Line 9: Line 49:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Néron,  "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux"  ''Publ. Math. IHES'' , '''21'''  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Mazur,  "Rational points of Abelian varieties with values in towers of number fields"  ''Invent. Math.'' , '''18'''  (1974)  pp. 183–266</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Raynaud,  "Modèles de Néron"  ''C.R. Acad. Sci. Paris Sér. A'' , '''262'''  (1966)  pp. 345–347</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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 &amp; Masson  (1968)  pp. 12–30</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Grothendieck (ed.)  et al. (ed.) , ''Groupes de monodromie en géométrie algébrique. SGA 7'' , ''Lect. notes in math.'' , '''288''' , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Néron,  "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux"  ''Publ. Math. IHES'' , '''21'''  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Mazur,  "Rational points of Abelian varieties with values in towers of number fields"  ''Invent. Math.'' , '''18'''  (1974)  pp. 183–266</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Raynaud,  "Modèles de Néron"  ''C.R. Acad. Sci. Paris Sér. A'' , '''262'''  (1966)  pp. 345–347</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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 &amp; Masson  (1968)  pp. 12–30</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Grothendieck (ed.)  et al. (ed.) , ''Groupes de monodromie en géométrie algébrique. SGA 7'' , ''Lect. notes in math.'' , '''288''' , Springer  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Artin,  "Néron models"  G. Cornell (ed.)  J. Silverman (ed.) , ''Arithmetic geometry'' , Springer  (1986)  pp. 213–230</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Artin,  "Néron models"  G. Cornell (ed.)  J. Silverman (ed.) , ''Arithmetic geometry'' , Springer  (1986)  pp. 213–230</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


of an Abelian variety

A group scheme associated to an Abelian variety and having a certain minimality property. If $ R $ is a local Henselian discrete valuation ring with residue field $ k $ and field of fractions $ K $ and if $ A $ is an Abelian variety of dimension $ d $ over $ K $, then a Néron model of $ A $ is defined as a smooth commutative group scheme $ \mathfrak A $ over $ R $ whose generic fibre $ \mathfrak A _ {K} $ is isomorphic to $ A $, while the canonical homomorphism $ \mathfrak A ( R) \rightarrow \mathfrak A _ {K} ( K) $ 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 $ R $- isomorphism. A Néron model has the following minimality property: For any smooth $ R $- scheme $ \mathfrak X $ and any morphism $ \phi : \mathfrak X _ {K} \rightarrow \mathfrak A _ {K} $ of the generic fibres there exists a unique morphism $ \overline \phi \; : \mathfrak X \rightarrow \mathfrak A $ of $ R $- schemes induced by $ \phi $.

If $ S $ is a one-dimensional regular Noetherian scheme, $ \eta $ is a generic point of it, $ i : \eta \rightarrow S $ is its canonical imbedding, and $ A $ is an Abelian variety over $ k( \eta ) $, then a Néron model of $ A $ is defined as a smooth quasi-projective group scheme $ \mathfrak A $ over $ S $ that represents the sheaf $ i _ {*} A $ relative to the flat Grothendieck topology on $ S $( 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=48032
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article