Abstract. In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.

Abstract. We develop an efficient technique for computing values at s =1 of Hecke L-functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields N which are abelian extensions of some totally real subfield L. We note that the smaller the degree of L the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing L = N + (the maximal totally real subfield of N) we can choose L real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants. 1.

Abstract. It was well known that it is easy to compute relative class numbers of abelian CM-fields by using generalized Bernoulli numbers (see Theorem 4.17 in Introduction to cyclotomic fields by L. C. Washington, Grad. Texts in Math., vol. 83, Springer-Verlag, 1982). Here, we provide a technique for computing the relative class number of any CM-field. 1. Statement of the results Proposition 1. Let n ≥ 1 be an integer and α>1be real. Set Pn(x) = ∑n−1

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