##  [Néron Model](/neron-model-0) 

 Definition

A smooth, separated model of a smooth commutative algebraic group (notably an abelian variety) over a discrete valuation ring that satisfies the Néron mapping property: every morphism from a smooth scheme over the base to the generic fibre extends uniquely to the model.

 

 

 

 

 

 





## Principle

Principle

Characterize integral and reduction behaviour of a group variety by a universal extension property for morphisms from smooth test schemes; smoothness and separatedness preserve good reduction data without imposing properness, so the model reflects arithmetic and component-group structure of the special fibre.

 

 

 

 

 





## Demonstration

Demonstration

For an elliptic curve over a p-adic field, its Néron model over the valuation ring yields the connected component of the special fibre, a finite component group measuring multiplicative or additive reduction, and a map identifying integral points with sections of the model.

 

 

 

 

## Misapplication

Misapplication

Confusing a Néron model with a minimal regular or proper model and expecting properness (Néron models are generally not proper), or attempting to construct a Néron model for a noncommutative or singular variety without verifying hypotheses.

 

 

 

 

 





## Consequence

Consequence

The Néron model organizes reduction data, provides a natural home for integral points and local height decompositions, and yields invariants such as the component group and reduction type used in the study of local-global problems and heights.

 

 

 

 

## Reversal

Reversal

Instead of the Néron mapping property one could insist on properness (minimal regular model) or on semistable reduction; reversing these requirements produces models better for intersection theory but worse for universal extension of morphisms from smooth schemes.

 

 

 

 

 





## Boundary

Boundary

Defined over discrete valuation rings (or Dedekind bases in families) for smooth commutative groups and abelian varieties; it excludes arbitrary schemes, noncommutative groups, and bases without a suitable valuation structure, and it presupposes the existence of a smooth separated model.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often conflated with minimal regular models, stable models or integral models that are proper; the Néron model differs by privileging the mapping property and smoothness over properness, which leads to different arithmetic invariants and applications.

 

 

 

 

 





## Synthesis

Synthesis

A Néron model is the canonical smooth integral incarnation of a smooth commutative group over a valuation ring: by enforcing a universal mapping property it isolates the reduction and component-group data essential for integral points, local heights and arithmetic comparison.