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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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Computation of relative class numbers of CM-fields by using Hecke $L$-functions
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by Stéphane Louboutin PDF
Math. Comp. 69 (2000), 371-393 Request permission


We develop an efficient technique for computing values at $s=1$ of Hecke $L$-functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields $\mathbf { N}$ which are abelian extensions of some totally real subfield $\mathbf { L}$. We note that the smaller the degree of $\mathbf { L}$ the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing $\mathbf { L} =\mathbf { N}^+$ (the maximal totally real subfield of $\mathbf { N}$) we can choose $\mathbf { L}$ real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.
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Additional Information
  • Stéphane Louboutin
  • Affiliation: Université de Caen, Campus 2, Département de Mathématiques, 14032 Caen cedex, France
  • Email:
  • Received by editor(s): April 16, 1997
  • Published electronically: May 21, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 69 (2000), 371-393
  • MSC (1991): Primary 11M20, 11R42; Secondary 11R29
  • DOI:
  • MathSciNet review: 1648395