Optimal embeddings in quaternion algebras

Let A be a quaternion algebra over a number field k and assume that A satisfies the Eichler condition so that some infinite place of k is unramified in A. Let L be a quadratic extension of k which embeds in A. Let Rk denote the ring of integers of k and let B be an Rk-order in L. Suppose that E is an Eichler order of A of square-free level S. In this paper, we determine when there exists an embedding s : L¿A over k which gives an optimal embedding of B into E in the sense that s(L)nE = s(B). This generalises previous work of Eichler [M. Eichler, Zur Zahlentheorie der Quaternionenalgebren, J. Reine Angew. Math. 195 (1955) 127–155] and Chinburg and Friedman [T. Chinburg, E. Friedman, An embedding theorem for quaternion
algebras, J. London Math. Soc. 60 (1999) 33–44].