Some Explicit Constructions of Integral Structures in Quaternion Algebras

Let B be an indefinite quaternion algebra over Q. Following the explicit characterization of the Eichler orders in B given by Hashimoto, we define explicit embeddings of these orders in the 2 × 2-ring of matrices over a p-adic ring. For a natural number N and a prime q not dividing the discriminant of B, we describe the two natural inclusions of an Eichler order of level Nq in an Eichler order of level N. Moreover, we provide bases for a chain of Eichler orders in B and prove results about the rank of their intersection.