Fast Computation of Relative Class Numbers of CM-Fields

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.