Exponents of the ideal class groups of CM number fields (Q1566390)

Let \(K\) be a number field, let \(d_K\) be its discriminant and let \(e_K\) be the exponent of the ideal class group of \(K\). Restricting to the case when \(K\) is an imaginary quadratic field, it was proved independently by \textit{D. W. Boyd} and \textit{H. Kisilevsky} [Proc. Am. Math. Soc. 31, 433--436 (1972; Zbl 0252.12002)] and \textit{P. Weinberger} [Acta. Arith. 22, 117--124 (1973; Zbl 0217.04202)] that, assuming ERH, \(e_K \gg \log(| d_K| )/\log\log(| d_K| )\). Hence, \(e_K\) goes to infinity as \(| d_K| \) goes to infinity. In the paper under review this result is extended (under GRH) to the more general case of CM number fields of fixed degree: the authors prove that \(e_K \gg_n \log(| d_K| )/\log\log(| d_K| )\), where the constant implied depends on \(n=[K:\mathbb Q]\) only. It is to be remarked that, by the time of this review, another paper on the subject has appeared [\textit{F. Amoroso} and \textit{R. Dvornicich}, Monatsh. Math. 138, 85--94 (2003; Zbl 1040.11077)]. Again under GRH, it is proven that \(e_K\) goes to infinity with \(| d_K| \) for arbitrary CM fields; the lower bound has the form \(e_K\gg (\log(| d_K| ))^{{1\over 2}-\epsilon}\), where the implied contant does not depend on the degree of \(K\). Moreover, specializing the more general results of that paper to CM fields of fixed degree, one reobtains the estimate \(e_K \gg_n \log(| d_K| )/\log\log(| d_K| )\) as a special case.