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  "body": "\n## Definition\n\nLet $F$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\mathcal{O}_F$, and let $B$ be a [[quaternion algebra|ZWXW0T3EAC]] over $F$.\nAn *Eichler order* in $B$ is an $\\mathcal{O}_F$-[[order|1ZBX4TKVZ9]] $\\mathcal{O} \\subseteq B$ such that $\\mathcal{O} = \\mathcal{O}_1 \\cap \\mathcal{O}_2$ for two [[maximal orders|E5DH1CSW1N]] $\\mathcal{O}_1, \\mathcal{O}_2$ of $B$.\n\n## Remarks\n\nExample: $F = \\Q$, $B = M_2(\\Q)$, $N \\ge 1$ and $g = \\operatorname{diag}(1, N)$.\nThe order $g M_2(\\Z) g^{-1}$ consists of the matrices $\\begin{pmatrix} a \u0026 b/N \\\\ Nc \u0026 d \\end{pmatrix}$ with $a, b, c, d \\in \\Z$.\nIts intersection with $M_2(\\Z)$ is\n$$\\mathcal{O}_0(N) = \\set{\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} \\in M_2(\\Z) : N \\mid c},$$\nso $\\mathcal{O}_0(N)$ is an Eichler order; $N$ is called its level.\nNon-example: the order $\\Z + 2 M_2(\\Z)$ in $M_2(\\Q)$ is not an Eichler order.\nEvery maximal order of $M_2(\\Q)$ is the endomorphism ring of a lattice in $\\Q^2$.\nBy the elementary divisor theorem, two such lattices have a common adapted basis.\nSo every Eichler order of $M_2(\\Q)$ contains an [[idempotent|0N5NJZKKN3]] other than $0$ and $1$.\nAn 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 $e = 1$.\n",
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