Perfect lattices over imaginary quadratic number fields

Braun, Oliver; Coulangeon, Renaud

doi:10.1090/S0025-5718-2014-02891-3

Perfect lattices over imaginary quadratic number fields
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by Oliver Braun and Renaud Coulangeon;

Math. Comp. 84 (2015), 1451-1467

DOI: https://doi.org/10.1090/S0025-5718-2014-02891-3

Published electronically: November 20, 2014

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Abstract:

We present an adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1. This includes a characterisation of extreme Hermitian forms which is analogous to the classic characterisation of extreme quadratic forms as well as a version of Voronoi’s famous algorithm which may be used to enumerate all perfect Hermitian forms for a given imaginary quadratic number field in dimensions 2 and 3. We also present an application of the algorithm which allows us to determine generators of the general linear group of an $\mathcal {O}_K$-lattice.

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