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 R_k denote the ring of integers of k and let B be an R_k-order in L. Suppose that is an Eichler order of A of square-free level S. In this paper, we determine when there exists an embedding σ:L→A over k which gives an optimal embedding of B into in the sense that . 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].