Construction of Arakelov-modular Lattices over Totally Definite Quaternion Algebras

TL;DR

This paper explores the construction of Arakelov-modular lattices using totally definite quaternion algebras over totally real number fields, providing new existence results especially over the rationals.

Contribution

It generalizes the definition of Arakelov-modular lattices to number fields and proves the existence of such lattices over  for prime levels.

Findings

Existence of Arakelov-modular lattices over  for any prime .

Extension of the definition of Arakelov-modular lattices to totally real number fields.

Construction method for lattices from totally definite quaternion algebras.

Abstract

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.