Eichler order

Remarks

Example: $F = \Q$, $B = M_2(\Q)$, $N \ge 1$ and $g = \operatorname{diag}(1, N)$. The order $g M_2(\Z) g^{-1}$ consists of the matrices $\begin{pmatrix} a & b/N \\ Nc & d \end{pmatrix}$ with $a, b, c, d \in \Z$. Its intersection with $M_2(\Z)$ is $$\mathcal{O}_0(N) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2(\Z) : N \mid c},$$ so $\mathcal{O}_0(N)$ is an Eichler order; $N$ is called its level. Non-example: the order $\Z + 2 M_2(\Z)$ in $M_2(\Q)$ is not an Eichler order. Every maximal order of $M_2(\Q)$ is the endomorphism ring of a lattice in $\Q^2$. By the elementary divisor theorem, two such lattices have a common adapted basis. So every Eichler order of $M_2(\Q)$ contains an idempotent other than $0$ and $1$. An idempotent $e$ of $\Z + 2M_2(\Z)$ is congruent to $0$ or $1$ modulo $2M_2(\Z)$; if $e \equiv 0$ then $e = e^k \in 2^k M_2(\Z)$ for all $k$, so $e = 0$, and symmetrically $1 - e \equiv 0$ forces $e = 1$.