Math 2.0

Algebraic geometry Definition

Néron model

Builds on Discrete valuation ringAbelian varietySmooth morphismSeparated morphismMorphism of finite typeAbelian scheme

Definition

Let RR be a discrete valuation ring with fraction field KK. Let AKA_K be an abelian variety over KK. A Néron model of AKA_K is an RR-scheme NN with an isomorphism N×RK≅AKN \times_R K \cong A_K such that:

  1. NN is smooth, separated and of finite type over RR;
  2. for every smooth RR-scheme YY and every KK-morphism uK ⁣:Y×RK→AKu_K \colon Y \times_R K \to A_K, there is a unique RR-morphism u ⁣:Y→Nu \colon Y \to N that extends uKu_K.

Clause 2 is called the Néron mapping property.

Remarks

A Néron model exists for every abelian variety AKA_K, and it is unique up to unique isomorphism by clause 2. Clause 2 extends the group law of AKA_K, so NN is a group scheme over RR. Taking Y=Spec⁡RY = \operatorname{Spec} R in clause 2 shows that every point in AK(K)A_K(K) extends to a point in N(R)N(R), so N(R)=AK(K)N(R) = A_K(K).

Example: if AKA_K extends to an abelian scheme AA over RR, then AA is the Néron model of AKA_K. This is the case of good reduction.

Non-example: let EE be an elliptic curve over KK with bad reduction, and let WW be the closure in ℙR2\mathbb{P}^2_R of a Weierstrass equation for EE with coefficients in RR. The special fibre of WW is a singular cubic, so WW is not smooth over RR and fails clause 1.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Abelian scheme, concept, no evidence levelAbelian schemeconceptAbelian variety, concept, no evidence levelAbelian varietyconceptDiscrete valuation ring, concept, no evidence levelDiscrete valuation ri…conceptMorphism of finite type, concept, no evidence levelMorphism of finite ty…conceptSeparated morphism, concept, no evidence levelSeparated morphismconceptSmooth morphism, concept, no evidence levelSmooth morphismconceptNéron model, definition, no evidence levelNéron modeldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Abelian scheme concept
  2. Abelian variety concept
  3. Discrete valuation ring concept
  4. Morphism of finite type concept
  5. Separated morphism concept
  6. Smooth morphism concept
  7. Néron model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/N%C3%A9ron_model
About this page
TagW9DXVGAE3R
URIm2://W9DXVGAE3R
Kinddefinition
WritingW0 stub
MSC14K15, 11G10, 14L15
DefinesNéron model
UsesDiscrete valuation ring, Abelian variety, Smooth morphism, Separated morphism, Morphism of finite type, Abelian scheme
Changed2 November 2026

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