Even Unimodular Lattices from Quaternion Algebras

Abstract

We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the $E_8$ lattice (resp. $E_8^2$ and ${\Lambda }_{16}$) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of $8$.

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• Division Algebras and Wireless Communication
• Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer