Extending proofs to non-free lattices

Develop methods that remove reliance on the free-lattice assumption and establish analogous results for non-free classical positive definite quadratic lattices over totally real number fields, thereby extending the paper’s arguments to the full lattice setting.

Background

The paper’s techniques largely operate in the setting of free lattices (quadratic forms). Although the authors also work in the language of lattices, key proofs depend on freeness and the classical assumption, limiting generality.

They explicitly note that it is not clear how to adapt these arguments to non-free lattices, and that existing techniques in the literature likewise do not straightforwardly extend, marking a methodological open problem.