Conjectures on the fundamental domain of the Hilbert modular group

Previous work has given some bounds on the fundamental domain of the Hilbert modular group in certain cases. In particular, the projection of the intersection of the fundamental domain with the manifold jz z0j D 1 into the plane determined by the imaginary parts of z and z0 is a region contained within a figure bounded by two lines and two hyperbolas. Some intense numerical computation gives a conjectured outline of the actual boundary of the above two-dimensional projection. We work with the case of Q. p 5/ and Q. p 2/ and observe that one of the bounding hyperbolas may be more accurately replaced by three or four currently unknown curves. Additional conjectures about the fundamental domain are listed.