---
date: 2022-01-15 21:49
modification date: Monday 24th January 2022 18:52:00
title: "order"
aliases: [order, maximal order]
---

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

---

- Tags: 
	- #NT/algebraic 
- Refs:
	- #todo/add-references
- Links:
	- [ring of integers](Unsorted/ring%20of%20integers.md)
	- [elliptic curves](MOCs/elliptic%20curve.md)
	- [[CM]]

---

# order

Idea: orders are like lattices!

# Order for general rings

An **order** $\OO$ is a [Noetherian](Unsorted/Noetherian.md) integral domain of [dimension](dimension.md) one with nonzero [conductor](Unsorted/conductor.md).
Equivalently, 
$$\intcl \OO\in \mods{\OO}^\fg.$$
![](attachments/Pasted%20image%2020220224124457.png)

# Order for k-algebras

Setup: take $R$ a [Noetherian](Unsorted/Noetherian.md) integral domain with $K\da \ff(R)$, and let $\Lambda\in\Alg\slice K^\fd$.
Then an $R\dash$**order** in $\Lambda$ is a ring  $X$ with the structure of an $R\dash$lattice, so $X\leq \Lambda \in \mods{R}^\fg$ such that $\spanof_K X =\Lambda$, i.e. $X$ is an $R\dash$submodule of $\Lambda$ which spans it as a $K\dash$vector space.

Equivalently, if $K\in \Alg\slice \QQ^\fd$ with $\dim_\QQ K = n$, an **order** in $K$ is a subring $\OO \leq K$ that is a free $\ZZ\dash$module of rank $n$, so $K = \OO \tensor_\ZZ \QQ$.

> Must be a lattice and a unital ring, so must contain 1.

# Examples

- Any [Dedekind domain](Unsorted/Dedekind%20domain.md)  which is not a field is an order.
- $X\da \Endo(E)$ for $E\in \Ell\slice \QQ$ an [elliptic curve](elliptic%20curve.md) is a $\ZZ\dash$order in some $\Lambda \in \Alg\slice \QQ$ with $\dim_\QQ X \in \ts{1,2,4}$.
	- In dimensions 1 and 2, $\Lambda = \QQ$ or $\QQ(\sqrt d)$ is an imaginary [quadratic field](quadratic%20field).
	In dimension 4, $L$ is a [quaternion algebra](quaternion%20algebra).
- $\ZZ$ is the only order in $\QQ$.
- $2\ZZ\leq \QQ$ is not an order: it is a lattice but not a ring.
- $X\da \ts{{a\over 2^n} \st a,n \in \ZZ} \leq \QQ$ is not an order: it is a ring but not a lattice since it is infinitely generated as a $\ZZ\dash$module.
- For [number fields](number%20fields) $K\in \Numberfield$:
	- $\OO_K \subseteq K$ is the unique **maximal order**.
	Note that generally, maximal orders for arbitrary $K\in \Alg\slice \QQ$ need not be unique.
	- Every other order $\OO$ must be a subset of $\OO_K$ and $[\OO_K: \OO]$ must be finite since $\OO$ contains a $\QQ\dash$basis of $K$
	- For $K$ imaginary quadratic, the orders $\OO$ are all of the form $\ZZ + n\OO_K$ for $n\in \ZZ_{\geq 0}$.
		The integer $n$ here is the [conductor](conductor.md) of the order, and equals the index $[\OO_K: \OO] < \infty$.
	- For $K = \QQ(a_1,\cdots, a_n)$, the ring $\OO\da \ZZ[a_1,\cdots, a_n]$ is an order.
- The [integral closure](Unsorted/integrally%20closed.md) of an order is always [Dedekind](Unsorted/Dedekind%20domain.md).
- Not every ring whose integral closure is Dedekind is an order: Nagata constructs Noetherian domains of dimension one whose conductor is *zero*.
	- In common ANT situation (see [AKLB setup](AKLB%20setup.md)), $B\slice A$ is finitely generated so every intermediate ring is finitely generated, including those whose integral closure is $B$ -- so if $A[\alpha], B$ have the same fraction field, so $L = K(\alpha)$, then $A[\alpha]$ is an order in $B$.
- ![](attachments/Pasted%20image%2020220224124659.png)

# Orders in k-algebras

![](attachments/Pasted%20image%2020220123190527.png)

![](attachments/Pasted%20image%2020220126093016.png)

# Orders for elliptic curves

The ring of endomorphisms of an elliptic curve can be of one of three forms:
	- $\ZZ$
	- $\mfo \subseteq K$ an order in an imaginary quadratic field
	- $\mfo \subseteq A \in \Alg\slice{\QQ}$ a quaternion algebra

# Conductors

![](attachments/Pasted%20image%2020220730134641.png)
Relation to discriminants:
![](attachments/Pasted%20image%2020220730134812.png)