View source for Néron model

''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" />.

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]]]).

For a generalization of the concept of a Néron model to arbitrary schemes see [[#References|[3]]].

====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>



====Comments====


====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>