---
tag: K0EQTBRPE3
kind: concept
name: Néron model
msc: [14K15, 11G10, 14L15]
sources:
  - url: https://en.wikipedia.org/wiki/N%C3%A9ron_model
same_as:
  wikidata: Q7071448
  wikipedia: Néron model
---

## Summary

Let $R$ be a [[discrete valuation ring|MGM3CNJW9V]] with fraction field $K$.
The Néron model of an [[abelian variety|R7W03TXRS9]] over $K$ is its best smooth model over $R$.
It is a smooth [[group scheme|TJ04VQP4NB]] of finite type over $R$, characterized by a universal property for maps from smooth $R$-schemes.
André Néron proved that it always exists.
Its special fibre describes the reduction of the abelian variety, even when the reduction is bad.

## Explanations

### L1 human

An abelian variety given by equations with rational coefficients can be reduced modulo a prime, but the reduction may become singular.
One would like a version over the $p$-adic integers that is as good as possible.
The Néron model is that version: it is smooth, and every rational point extends to a point of the model.
When the variety has good reduction, the Néron model is just the abelian scheme given by good equations.