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

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 $6$.