On tamely ramified Iwasawa modules for the cyclotomic \(\mathbb {Z}_{p}\)-extension of abelian fields. (English) Zbl 1300.11111

This paper generalizes results of a previous paper of T. Itoh et al. [Int. J. Number Theory 9, No. 6, 1491–1503 (2013; Zbl 1318.11141)], using comparable techniques. The setting is as follows: \(k\) is an absolutely abelian field, \(S\) is a finite set of rational (!) primes that does not contain the fixed odd prime \(p\), and \(X_S(k_\infty)\) is the Galois group of the maximal \(S\)-ramified abelian \(p\)-extension over \(k_\infty\) (the cyclotomic \(\mathbb Z_p\)-extension over \(k\)). The author calculates the rank of \(X_S(k_\infty)\) over \(\mathbb Z_p\). If \(S\) is empty, this is just the usual \(\lambda\) invariant, and in fact the results are in terms of \(\lambda\); in other words, he computes the rank of \(Q_S:=X_S(k_\infty)/X(k_\infty)\). A little more is done: the author actually gives the ranks of the \(\chi\)-parts of the latter quotient, with \(\chi\) running over the characters of Gal\((k/\mathbb Q)\). The principal difference with respect to the three-author paper quoted above is that the latter just treated \(k=\mathbb Q\).
The quotient module \(Q_S=X_S(k_\infty)/X(k_\infty)\) has a standard description via class field theory, which we only give in very general terms: it is given as the quotient of \(S\)-semilocal units modulo (the image of) global units. (We neglect that in addition one has to complete the modules \(p\)-adically, and that everything happens in a projective limit over the Iwasawa tower.). Now something very interesting happens. As his Theorem 1.1 (generalization of Theorem A of Itoh-Mizusawa-Ozaki) the author proves: If \(S=\{q\}\) is a singleton, then \(Q_S\) has rank zero (in fact \(Q_S\) is finite). In a way this is a tame analog of Leopoldt’s conjecture: the image of the global units in the \(q\)-semilocal units is “as large as it can be”. Equally vaguely one can say: if \(S\) has more than one element, then there are not enough global units to make the quotient \(Q_S\) finite. The paper under review gives precise formulas for the ranks of the \(\chi\)-parts of \(Q_S\) in section 6, which we do not reproduce here. At the end, one finds some nice examples.
Reviewer’s remark in conclusion: It would be very interesting to explore whether Theorem 1.1 can also be proved for (some?) nonabelian number fields.


11R23 Iwasawa theory
11R18 Cyclotomic extensions


Zbl 1318.11141
Full Text: arXiv Euclid


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