Golden lattices

TL;DR

This paper introduces the concept of golden lattices, a special class of dense, extremal modular lattices derived from extremal even unimodular lattices, utilizing the properties of the golden ratio.

Contribution

It defines golden lattices as even unimodular []lattices with extremal Hilbert theta series and provides a construction method from known extremal lattices.

Findings

Constructed new families of dense modular lattices.

Established a link between golden ratio and lattice extremality.

Demonstrated the density and extremality of the resulting lattices.

Abstract

Let $\vartheta := \frac{-1+\sqrt{5}}{2}$ be the golden ratio. A golden lattice is an even unimodular $\Z[\vartheta ]$-lattice of which the Hilbert theta series is an extremal Hilbert modular form. We construct golden lattices from extremal even unimodular lattices and obtain families of dense modular lattices.