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

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

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

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

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

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

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

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

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.