---
tag: W7PYDJZ8BC
kind: concept
name: Eichler order
msc: [11R52, 16H10]
specializes: [1ZBX4TKVZ9]
sources:
  - url: https://en.wikipedia.org/wiki/Eichler_order
same_as:
  wikidata: Q5348728
  wikipedia: Eichler order
---

## Summary

An Eichler order is an [[order|1ZBX4TKVZ9]] in a [[quaternion algebra|ZWXW0T3EAC]] that is the intersection of two [[maximal orders|E5DH1CSW1N]].
Every maximal order is an Eichler order, as the intersection of itself with itself.
In $M_2(\Q)$ the standard Eichler order of level $N$ is the ring of integer matrices whose lower left entry is divisible by $N$.
Its elements of determinant $1$ form the congruence subgroup $\Gamma_0(N)$.
Eichler orders in indefinite and definite quaternion algebras give Shimura curves and the Brandt matrices used to study modular forms.

## Explanations

### L1 human

In the rational $2 \times 2$ matrices, the integer matrices form a maximal order, and so does each of its conjugates.
Intersecting two of these gives a slightly smaller ring with a congruence condition.
For example, the integer matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $c$ divisible by $N$ form such an intersection.
This ring is behind the group $\Gamma_0(N)$, which is basic in the theory of modular forms.
Eichler orders carry the same idea to other quaternion algebras.