Gitter und Modulformen

TL;DR

This paper explores the construction and classification of dense, extremal modular lattices using theta series with harmonic coefficients, highlighting their symmetry, optimal density, and extremality in lattice theory.

Contribution

It introduces methods to classify and construct extremal modular lattices and proves their properties using theta series with harmonic coefficients.

Findings

Extremal lattices are the densest modular lattices in small dimensions.

Theta series with harmonic coefficients can classify and construct extremal lattices.

Some extremal lattices are shown to be strongly perfect and local maxima of density.

Abstract

A main goal in lattice theory is the construction of dense lattices. Most of the remarkable dense lattices in small dimensions have an additional symmetry, they are modular, i.e. similar to their dual lattice. Extremal lattices are densest modular lattices, whoses density is as high as the theory of modular forms allows it to be. The theory of theta series with harmonic coefficients allows to classify and to construct extremal lattices as well as to prove that some of them are strongly perfect and hence local maxima of the density function.