Computation of \(L(0,\chi)\) and of relative class numbers of CM-fields (Q2725475)

This well-documented paper continues the author's efforts to compute values of \(L\) functions in 1 or in 0. Functions under examination are the ones associated with a non-trivial Hecke character on the ray class group of a totally real field. These two values are connected by a functional equation, but while the value in 1 is traditionally prefered, the author provides an efficient algorithm to compute the value in 0. Complexity is in \(O((df)^{0.5+\varepsilon})\) where \(f\) is the norm of the conductor and \(d\) the discriminant of the base field. The strategy is to express \(L(0,\chi)\) over an integer basis. Denominators occurring are bounded by a result of \textit{J. Coates} and \textit{W. Sinnott} [Proc. Lond. Math. Soc. (3) 34, 365-384 (1977; Zbl 0354.12009)]; the integer coefficients are expressed as linear combinations of \(L(0,\chi^\ell)\) which in turn are computed via an approximate functional equation. This method differs notably from the one deduced from Shintani's method and which would imply a full parametrisation of a fundamental domain. Here the use of the approximate functional equation restricts the information required to the number of ideals of given norm \(\ll(df)^{0.5+\varepsilon}\). NEWLINENEWLINENEWLINESome examples are detailed and applications concerning the class number one problem for fields with complex multiplication are given. Other applications by various authors are developed in [\textit{Y. Lefeuvre}, Ann. Inst. Fourier 50, 67-103 (2000; Zbl 0952.11024), \textit{S. Louboutin} and \textit{Y.-H. Park}, Publ. Math. 57, 283-295 (2000; Zbl 0963.11065), \textit{Y.-H. Park}, Acta Arith. 101, 63-80 (2002; Zbl 0997.11091) and \textit{K.-Y. Chang} and \textit{S.-H. Kwon}, Acta Arith. 101, 53-61 (2002; Zbl 0998.11059)].