Maximal class numbers of CM number fields
The problem of finding bounds for the class number \(h\) of a number field goes back at least to the time of Gauß. Under the assumptions of the generalized Riemann hypothesis (GRH) and Artin's conjecture on the entirety of Artin \(L\)-functions, \textit{W. Duke} [Compos. Math. 136, No. 1, 103--115 (2003; Zbl 1013.11072)] has shown that for families of number fields in which the degree, signature and Galois group are fixed one can deduce bounds of the form \[ h\leq C|d|^{1/2}\frac{(\log\log |d|)^{n-1}}{(\log |d|)^b}, \] where \(d\) is the discriminant, \(n\) the degree over \(\mathbb Q\) and \(C\), \(b\geq 0\) depend on the family under consideration. A natural question to ask is whether such bounds are actually sharp or not. In the paper under review it is shown that for CM number fields with a fixed maximal real subfield \(F\), of degree at least \(2\), the answer to the above question is negative (again under the assumption of GRH). Actually, the authors obtain a stronger bound in the considered situation, namely \[ h\ll_F|d|^{1/2}(\log\log|d|)^{1+o(1)}, \] and show that this bound is the best possible.
scientific article; zbMATH DE number 5690729