Sylow subgroups of ideal class groups with moduli
Let K/k be a cyclic extension of number fields of prime degree \(\ell\). The authors generalize prior work on the Galois structure of the \(\ell\)- Sylow subgroup of the ideal class group of K, especially results of \textit{G. Gras} [Ann. Inst. Fourier 23, No.3, 1-48 (1973; Zbl 0276.12013); No.4, 1-44 (1974; Zbl 0262.12007)] and \textit{P. Morton} [J. Reine Angew. Math. 307/308, 373-398 (1979; Zbl 0395.12018)]. Here the authors consider the \(\ell\)-Sylow subgroup L of the ideal class group of K modulo \(f\infty\), where f is a nonzero integral ideal of k and \(\infty\) is a (possibly trivial) product of real places of k. Results on the structure of L are obtained under the assumption that the ideal f is prime to both \(\ell\) and the discriminant of K/k. For this purpose, a filtration of L of the type first considered by \textit{E. Inaba} [J. Fac. Sci. Univ. Tokyo, Sect. I 4, 61-115 (1940; Zbl 0024.01002)] is used. Let \(\sigma\) be a generator of the Galois group of K/k and define \(L_ i=\{x\in L:\;x\in \ker (\sigma -1)^ i\}.\) The authors calculate the orders of \(L_ 1\) and the successive quotients in the resulting filtration \(L_ 1\subseteq L_ 2\subseteq...\subseteq L\) of subgroups of L, and give an algorithm yielding the generators of the subgroups occurring there. Under an additional assumption on the triviality of norms of elements of L, the \(\ell^ r\)-rank of L for \(r\geq 1\) is expressed as a product of the orders of certain of the quotients.
scientific article; zbMATH DE number 4191813
scientific article