Locally Divergent Orbits on Hilbert Modular Spaces

TL;DR

This paper investigates the structure of locally divergent orbits under torus actions on Hilbert modular spaces, proving density and closure properties that confirm or contradict existing conjectures depending on the rank and field type.

Contribution

It characterizes the closures of locally divergent orbits on Hilbert modular spaces of rank 2, confirming Margulis's orbit rigidity conjecture in most cases and providing counterexamples in specific scenarios.

Findings

For r > 2, non-closed locally divergent orbits are dense.

For r = 2, closures are finite unions of torus orbits.

Results contradict Margulis's conjecture in certain cases.

Abstract

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.