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Computation of L(0, χ) and of relative class numbers of CM-fields

Published online by Cambridge University Press:  22 January 2016

Stéphane Louboutin
Stéphane Louboutin*
Affiliation:
Institut de Mathématiques de Luminy, UPR 9016, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France, [email protected]
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Abstract

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Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant dL. Then, L(0, χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Let fχ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ) in O((dLfx)0.5+∊) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at s = 0 of such abelian Hecke L-functions over totally real number fields L. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of L(0, χ) over totally real number fields of degree 2 and 6.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 161 , March 2001 , pp. 171 - 191
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

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