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Classification of Hermitian forms with the neighbour method. (English) Zbl 0936.68129
Summary: The neighbour method of Kneser can be adapted to the Hermitian case. Generalizing results of D. W. Hoffmann [Manuscr. Math. 71, No. 4, 399–429 (1991; Zbl 0729.11020)], we show that it can be used to classify any genus in a Hermitian space of dimension \(\geq 2\) by neighbour steps at suitable primes. The method was implemented for positive definite Hermitian lattices (not necessarily free) over \(\mathbb Q(\sqrt d)\). 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.

11E41 Class numbers of quadratic and Hermitian forms
11H50 Minima of forms
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11Y16 Number-theoretic algorithms; complexity
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