Cohomology of metacyclic groups and class numbers of subfields of cyclotomic extensions (Q1322518)

Let \(p\) be an odd prime, \(\xi\) a primitive \(p\)-th root of unity, \(h_ p= h_ p^ - h_ p^ +\) the class number of the \(p\)-th cyclotomic field \(\mathbb{Q}(\zeta)\), where \(h^ +_ p\) means the class number of the maximal real subfield \(\mathbb{Q}(\zeta+ \zeta^{-1})\) of \(\mathbb{Q}(\zeta)\). The well- known Kummer's criterion states that \(p\mid h_ p^ -\) if and only if \(p\mid B_{2k}\) for some \(1\leq k\leq (p-3)/2\). (\(B_ n\) means the Bernoulli number.) The \(p\)-Sylow subgroup \(U\) of the class group of \(\mathbb{Q}(\zeta)\) (called sometimes ``the irregular class group'') can be expressed as a direct sum \(U= \bigoplus_{i=0}^{p-2} U^{\omega^ i}\), where \(\omega\) is the Teichmüller character. The ``minus part'' \(U^ -\) of \(U\) is defined as this sum over the odd \(i\). Then \(| U^ -|\) equals the \(p\)-part of \(h_ p^ -\). The core of this paper consists of the following theorem: Theorem 1.1. Let \(i\) be odd, \(3\leq i\leq p-2\). Then \(U^{\omega^ i}\neq 0\) if and only if \(p\) divides \(B_{p-i}\). The direction ``\(\Leftarrow\)'' is due to \textit{K. Ribet} [Invent. Math. 34, 151-162 (1976; Zbl 0338.12003)] and the converse direction ``\(\Rightarrow\)'' is often quoted as the Herbrand theorem [\textit{J. Herbrand}, J. Math. Pures Appl., IX. Sér. 11, 417-441 (1932; Zbl 0006.00802)]. In fact it had been shown by \textit{F. Pollaczek} [Math. Z. 21, 1-38 (1924), \S6]. Theorem 1.1 can be generalized for subfields of \(\mathbb{Q}(\zeta)\) in the natural way. The author gives a cohomological interpretation of these statements using the spectral sequence of \textit{M. Atiyah} [Inst. Haut. Étud. Sci., Publ. Math. 9, 247-288 (1961; Zbl 0107.023)]. The Bernoulli number \(B_{2k}/ 2k\) enters as the quotient of the orders of the coefficients in two adjacent dimensions for the used cohomology theory. The \(p\)-divisibility condition amounts to the non-vanishing of a certain characteristic class. The paper contains 6 sections: 1. Variants of Kummer's criterion, 2. Representation theory, 3. Algebraic and topological \(K\)-theory, 4. Completion theorems, 5. Arithmetic interpretation, 6. Appendix: Symplectic groups.