Locally Divergent Orbits on Hilbert Modular Spaces

We describe the closures of locally divergent orbits under the action of tori on Hilbert modular spaces of rank r≥2. In particular, we prove that if D is a maximal $\mathbb{R}$-split torus acting on a real Hilbert modular space, then every locally divergent nonclosed 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 G. A. 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 nonproportional, split, binary, quadratic forms over number fields.