An algorithm for the principal ideal problem in indefinite quaternion   algebras
  Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm. 
Aurel Page (IMB, INRIA Bordeaux - Sud-Ouest)
math.NT