Eichler order

Summary

An Eichler order is an order in a quaternion algebra that is the intersection of two maximal orders. 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.