Class number problems for dicyclic CM-fields (Q2714201)

A dicyclic group \(G\) of order \(4p\) (\(p\) an odd prime) is a semidirect product of a cyclic group \(H\) of order \(p\) and a cyclic group of order 4, where a generator \(b\) of the latter acts on \(H\) by inversion. Thus, the center of \(G\) is \(\{1,b^2\}\), and \(G\) modulo its center is dihedral of order \(2p\). NEWLINENEWLINENEWLINEThe article under review considers CM extensions \(N\) of the rationals with bicyclic Galois group \(G\); the central element \(b^2\) corresponds to complex conjugation. The main result is a lower bound of 4 for the minus class number (= relative class number) \( h^-_N = h_N/h_{N^+}\), and a complete list of cases where the minimal value \(4\) is attained: there are essentially only four such cases. This is in analogy with the problem of Gauss concerning class number one for imaginary quadratic \(N\). The proof consists in finding lower and upper bounds for class numbers through the analytic class number formula, hence in estimating values of zeta functions; many steps are quoted from earlier papers, mostly of Louboutin and collaborators. NEWLINENEWLINENEWLINEThe authors also prove the following statement which was observed earlier by Lefeuvre: if \(M\) denotes the unique quartic subfield of \(N\) (which is as above), then the quotient \(h^-_N/h^-_M\) is the square of an integer. The nice proof uses \(L\)-functions again. NEWLINENEWLINENEWLINEReviewer's remark: Most of the results (the \(l\)-valuations of the number in questions are even, except perhaps for \(l=2\) or \(l=p\)) might also be deduced by representation theory, in the style of Chap. II \S 2 of \textit{J. Tate}'s book [Les conjectures de Stark sur les functions \(L\) d'Artin en \(s=0\), Progress in Math. 47, Birkhäuser (1984; Zbl 0545.12009)].