  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.
