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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.

MSC:
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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