Lattice packings through division algebras
In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensions.
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