Construction of Arakelov-modular Lattices over Totally Definite Quaternion Algebras

We study ideal lattices constructed from totally definite quaternion algebras over totally real number fields, and generalize the definition of Arakelov-modular lattices over number fields. In particular, we prove for the case where the totally real number field is $\mathbb{Q}$, that for $\ell$ a prime integer, there always exists a totally definite quaternion over $\mathbb{Q}$ from which an Arakelov-modular lattice of level $\ell$ can be constructed.