Classification of Hermitian Forms with the Neighbour Method

The neighbour method of Kneser can be adapted to the hermitian case. Generalizing results of Hoffmann (1991), we show that it can be used to classify any genus in a hermitian space of dimension ≥ 2 by neighbour steps at suitable primes. The method was implemented for positive definite hermitian lattices (not necessarily free) over (d1/2). A table of class numbers of unimodular genera and the largest minima attained in those genera is given. We also describe a generalization of the LLL-algorithm to lattices in positive hermitian spaces over number fields.