---
date: 2022-02-14 09:10
modification date: Monday 14th February 2022 09:10:58
title: good reduction
aliases: [good reduction, potentially good reduction, reduction]

---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #AG/AVs
- Refs:
	- #todo/add-references
- Links:
	- [smooth](Unsorted/smooth%20scheme.md)
	- [projective (schemes)](Unsorted/projective%20(schemes).md)
	- [geometrically integral](geometrically%20integral)
	- [geometrically irreducible](geometrically%20irreducible)
	- [model of a scheme](Unsorted/model%20of%20a%20scheme.md)

---


# good reduction

![](attachments/Pasted%20image%2020220727100646.png)

- In terms of equations: for $E\slice \QQ$ an [elliptic curve](MOCs/elliptic%20curve.md) $y^2=x^3+ax + b$ of [discriminants](Unsorted/discriminants.md) $\Delta(E) = -16(4a^3+27b^2)$, $E$ has **good reduction at $p$** for any prime $p\nmid \Delta(E)$.
- A variety $X\in\Var\slice k$ has **good reduction** if there exists a smooth proper $\OO_k$ scheme $\mcx$ whose generic fiber $\mcx_k$ is $k\dash$isomorphic to $X$.
	- Assume $k$ is a characteristic zero field with a complete discrete valuation and $X\slice k$ is smooth and proper.
- Alternatively: if $k\in \Field$ is complete wrt a diiscrete valuation $v: k\units\to \ZZ$, let $X\in\mathsf{smProj}\Var\slice k$ with $\dim X = d$.
  Then $X$ has **good reduction** if there exists a smooth model of $X$ over the valuation ring $\OO \subseteq k$, i.e. there exists homogeneous polynomials cutting out a $k\dash$variety in some $\PP^N\slice k$ which is isomorphic to $X$ whose coefficients are in the valuation ring of $v$ where reducing all of them modulo the maximal ideal $\mfm$ gives equations defining a smooth variety of dimension $d$ over the residue field $\kappa$.
	  - If $k$ is not complete (e.g. a number field), say $X$ has **good reduction at $v$** if $X_{k_v}\in\Var\slice{k_v}$ has good reduction.
	  - Nice varieties: smooth, projective, [geometrically integral](geometrically%20integral) or [geometrically irreducible](geometrically%20irreducible).

![](attachments/Pasted%20image%2020220502150144.png)

## Special cases: curves and abelian varieties

![](attachments/Pasted%20image%2020220502184321.png)
![](attachments/Pasted%20image%2020220502184333.png)

# Notes

- Consequence: if $E\slice{\QQpadic}$ has good reduction, there is a unique elliptic scheme $\mce\slice{\ZZpadic}$ with $\mce_{\QQpadic} = E$ and a structure map $\mce \to \spec \ZZpadic$.

- [Neron-Ogg-Shaferevich](Neron-Ogg-Shaferevich): for an [elliptic curve](MOCs/elliptic%20curve.md), $X$ has good reduction iff the $G_k$ representation on $H_\et^1(X; \ZZladic)$ is an [unramified Galois representation](unramified%20Galois%20representation), i.e the action of [inertia](Unsorted/ramification%20index.md) $I_k$ is trivial.
  
![](attachments/Pasted%20image%2020220727100751.png)


# Crystalline representations

For [elliptic curves](MOCs/elliptic%20curve.md), a first incarnation of [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md):

![](attachments/Pasted%20image%2020220502150233.png)