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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Classification of integral lattices with large class number
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by Rudolf Scharlau and Boris Hemkemeier PDF
Math. Comp. 67 (1998), 737-749 Request permission

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
  • Email: Rudolf.Scharlau@mathematik.uni-dortmund.de
  • Boris Hemkemeier
  • Affiliation: Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany
  • Email: Boris.Hemkemeier@mathematik.uni-dortmund.de
  • Received by editor(s): January 11, 1995
  • Received by editor(s) in revised form: October 7, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 737-749
  • MSC (1991): Primary 11E41; Secondary 11H55, 11--04
  • DOI: https://doi.org/10.1090/S0025-5718-98-00938-7
  • MathSciNet review: 1458224