09-01
Enumerating perfect forms
Preprint series: 09-01, Preprints
- MSC:
- 11H55 Quadratic forms (reduction theory, extreme forms, etc.)
- 20G20 Linear algebraic groups over the reals, the complexes, the quaternions
- 90C99 None of the above but in this section
Abstract: A positive definite quadratic form is called perfect, if it is uniquely determined by its arithmetical minimum and the integral vectors attaining it. In this self-contained survey we explain how to enumerate perfect forms in $d$ variables up to arithmetical equivalence and scaling. We put an emphasis on practical issues concerning computer assisted enumerations. For the necessary theory of Voronoi we provide complete proofs based on Ryshkov polyhedra. This allows a very natural generalization to $T$-perfect forms, which are perfect with respect to a linear subspace $T$ in the space of quadratic forms. Important examples include Gaussian, Eisenstein and Hurwitz quaternionic perfect forms, for which we present new classification results in dimensions $8,10$ and 12.
Keywords: perfect forms, Voronoi reduction, finite group, Eisenstein lattice, Gaussian lattice, Hurwitz Quaternionic lattice, polyhedral vertex enumeration
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