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Mathematics of Computation

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Classification of integral lattices with
large class number

Authors: Rudolf Scharlau and Boris Hemkemeier
Journal: Math. Comp. 67 (1998), 737-749
MSC (1991): Primary 11E41; Secondary 11H55, 11--04
MathSciNet review: 1458224
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Abstract: A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of $\ell$-elementary lattices, for small prime level $\ell,$ are obtained. For instance, the class number of $12$-dimensional $7$-elementary even lattices of determinant $7^6$ is $395$; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.

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Additional Information

Rudolf Scharlau
Affiliation: Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany

Boris Hemkemeier
Affiliation: Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany

Keywords: Lattice, integral quadratic form, class number of genus, neighbour method, $p$-elementary lattice, extremal modular lattice
Received by editor(s): January 11, 1995
Received by editor(s) in revised form: October 7, 1996
Article copyright: © Copyright 1998 American Mathematical Society