On the growth of \(p\)-class groups in \(p\)-class field towers
Let \(k\) be an algebraic number field, \(p\) a prime and \(G\) the Galois group of the maximal unramified \(p\)-extension of \(k\). Furthermore, let \(\{G^{(n)} \mid n=0,1, \dots\}\) be the derived series of \(G\) such that \(G^{(n+1)}\) is the Hilbert \(p\)-class field of the fixed field \(k^{(n)}\) of \(G^{(n)}\). The author investigates the question of whether the generator rank \(d_n\) of \(G^{(n)}\) tends to infinity with \(n\). He proves that this is the case if \(d_0\geq 2+2 (\varepsilon +1)^{1/2}\), where \(\varepsilon\) is the \(p\)-rank of the group of units of \(k\). He also shows that a conjecture of Fontaine and Mazur implies \(\lim_{n \to\infty} d_n= \infty\) if and only if \(G\) is infinite. For other results of the author we refer to the paper at hand and his related paper [Pac. J. Math. 176, 15-18 (1996; see Zbl 0879.11068 above)].
scientific article; zbMATH DE number 1013923
scientific article