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Mathematics of Computation

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Computation of relative class numbers
of CM-fields by using Hecke $L$-functions

Author: Stéphane Louboutin
Journal: Math. Comp. 69 (2000), 371-393
MSC (1991): Primary 11M20, 11R42; Secondary 11R29
Published electronically: May 21, 1999
MathSciNet review: 1648395
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Abstract: 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

Keywords: CM-field, relative class number, Hecke $L$-function, ray class field, dihedral field
Received by editor(s): April 16, 1997
Published electronically: May 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society