On the structure of relative class groups. With an appendix of numerical examples
Let \(p\) be an odd prime and let \(\chi: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \overline{\mathbb{Q}}_ p^*\) be an odd \(p\)-adic Dirichlet character, which is not equal to the Teichmüller character. Let \(Cl(\chi)\) denote the \(\chi\)-part of the \(p\)-class group of the number field \(F=\mathbb{Q}^{\ker(\chi)}\) cut out by \(\chi\). It is a module over the \(p\)-adic ring \(R_ \chi= \mathbb{Z}_ p[ \text{im}(\chi)]\) and by a result of \textit{B. Mazur} and \textit{A. Wiles} [Invent. Math. 76, 179-330 (1986; Zbl 0545.12005)], then its cardinality is equal to the cardinality of \(R_ \chi/(B_{1,\chi^{-1}})\). Here \(B_{1,\chi^{- 1}}\) denotes a generalized Bernoulli number. The authors study the structure of \(Cl(\chi)\) as a module over the ring \(R_ \chi\). To this end they pick \(t\) primes \(l_ 1,\dots,l_ t\) that are \(\equiv 1\pmod p\) and split in \(F\). They show that if the Bernoulli number \(B_{1,\chi^{-1}\psi}\) has valuation \(t\) in the ring \(\mathbb{Z}_ p [\text{im}(\chi\psi)]\) for some character \(\psi\) of order \(p\) and conductor \(l_ 1\cdot l_ 2\cdot \dots\cdot l_ t\), then \(Cl(\chi)\) is generated over \(R_ \chi\) by the classes of the prime ideals over the \(l_ i\). Moreover, if the \(R_ \chi\)-rank of \(Cl(\chi)\) is equal to \(t\), then the collection of \(t\)-tuples of primes \(l_ 1,l_ 2,\dots,l_ t\) with this property has non-zero Chebotarev density. This result is related to \textit{V. A. Kolyvagin's} methods [Euler systems, The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 435-483 (1990; Zbl 0742.14017)]. Usually the module \(Cl(\chi)\) is cyclic as an \(R_ \chi\)-module. But there are examples of eigenspaces \(Cl(\chi)\) that are not cyclic as \(R_ \chi\)-modules. In all these examples the character \(\chi\) has order 2. The authors exhibit a different example: for one of the two 5-adic characters \(\chi\), the eigenspace \(Cl(\chi)\) of the quadratic field \(F= \mathbb{Q} (\sqrt {-541(37+6 \sqrt {37})}\) is not cyclic over \(R_ \chi=\mathbb{Z}_ 5\).
scientific article; zbMATH DE number 179546