Chapter

Algorithmic Number Theory

Volume 1838 of the series Lecture Notes in Computer Science pp 413-422

Fast Computation of Relative Class Numbers of CM-Fields

  • Stéphane LouboutinAffiliated withUniversité de Caen

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Abstract

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant \(d_{\textbf{L}}\). 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 f χ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ in \(O((d_{\textbf{L}}f_{x})^{0.5+\epsilon})\) 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 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. This paper being an abridged version of [Lou4], the reader will find there all the details glossed over here.

Keywords and phrases

CM-field relative class number Hecke L-function

1991 Mathematics Subject Classification

Primary 11R29 11R21 11Y35