Dense ideal Lattices from cyclic algebras

Abstract

In this thesis, we study the construction of algebraic lattices, tracing our steps back to the foundational concepts in the theory of the Geometry of Numbers introduced by Hermann Minkowski where lattices are built over number fields. Building upon this groundwork, we explore recent advancements, particularly the work of Hou Xiaolu [8], who extended this construction to quaternion algebras over number fields. We contribute to this theory by providing a generator matrix for her construction, which illuminates the geometric perspective and allows us to give a closed form volume formula. Additionally, we present a construction for the renowned E8 lattice. Finally, we lay down the foundations to broaden this construction to cyclic algebras over number fields, showcasing how quaternion algebras can be viewed as a special case, corresponding to a degree 2 case of cyclic algebras.