Nagoya Mathematical Journal

Computation of {$L(0,\chi)$} and of relative class numbers of {CM}-fields

Stéphane Louboutin

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Abstract

Let $\chi$ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field ${\bf L}$ of discriminant $d_{\bf L}$. Then, $L(0,\chi )$ 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 ${\bf L}$. Let $f_\chi$ denote the norm of the finite part of the conductor of $\chi$. Then, roughly speaking, we can compute $L(0,\chi )$ in $O((d_{\bf L}f_\chi )^{0.5 +\epsilon})$ 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 ${\bf L}$. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of $L(0,\chi )$ over totally real number fields of degree $2$ and $6$.

Article information

Source
Nagoya Math. J., Volume 161 (2001), 171-191.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631557

Mathematical Reviews number (MathSciNet)
MR1820217

Zentralblatt MATH identifier
0985.11054

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R21: Other number fields 11Y40: Algebraic number theory computations

Citation

Louboutin, Stéphane. Computation of {$L(0,\chi)$} and of relative class numbers of {CM}-fields. Nagoya Math. J. 161 (2001), 171--191. https://projecteuclid.org/euclid.nmj/1114631557


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