Locally Divergent Orbits on Hilbert Modular Spaces

We describe the closures of locally divergent orbitsunder the action of tori on Hilbert modular spaces of rank r = 2. In particular, we prove that if D is a maximal R-split torus acting on a real Hilbert modular space then every locally divergent non-closed orbit is dense for r > 2 and its closure is a finite union of tori orbits for r = 2. Our results confirm an orbit rigidity conjecture of Margulis in all cases except for (i) r = 2 and, (ii) r > 2 and the Hilbert modular space corresponds to a CM-field; in the cases (i) and (ii) our results contradict the conjecture. As an application, we describe the set of values at integral points of collections of non-proportional, split, binary, quadratic forms over number fields.