Abstract

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number fieldLof discriminantdL. Then,L(0,χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values ats= 0 of such abelian HeckeL-functions over totally real number fieldsL. Letfχdenote the norm of the finite part of the conductor ofχ. Then, roughly speaking, we can computeL(0,χ) inO((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 ats= 0 of such abelian HeckeL-functions over totally real number fieldsL. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations ofL(0,χ) over totally real number fields of degree 2 and 6.

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