## Lattices and modular forms. (Gitter und Modulformen.)(German)Zbl 1160.11334

Summary: 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, whose 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.

### MSC:

 11H31 Lattice packing and covering (number-theoretic aspects) 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11F11 Holomorphic modular forms of integral weight
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