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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.

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions

Citations:

Zbl 1318.11141
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Full Text: arXiv Euclid

References:

[1] A. Brumer: On the units of algebraic number fields , Mathematika 14 (1967), 121-124. · Zbl 0171.01105
[2] B. Ferrero and L.C. Washington: The Iwasawa invariant \(\mu_{p}\) vanishes for abelian number fields , Ann. of Math. (2) 109 (1979), 377-395. · Zbl 0443.12001
[3] R. Greenberg: On a certain \(l\)-adic representation , Invent. Math. 21 (1973), 117-124. · Zbl 0268.12004
[4] R. Greenberg: On \(p\)-adic \(L\)-functions and cyclotomic fields . II, Nagoya Math. J. 67 (1977), 139-158. · Zbl 0373.12007
[5] C. Greither: Class groups of abelian fields, and the main conjecture , Ann. Inst. Fourier (Grenoble) 42 (1992), 449-499. · Zbl 0729.11053
[6] T. Itoh, Y. Mizusawa and M. Ozaki: On the \(\mathbb{Z}_{p}\)-ranks of tamely ramified Iwasawa modules , Int. J. Number Theory 9 (2013), 1491-1503. · Zbl 1318.11141
[7] \begingroup K. Iwasawa: On \(\mathbf{Z}_{l}\)-extensions of algebraic number fields , Ann. of Math. (2) 98 (1973), 246-326. \endgroup · Zbl 0285.12008
[8] K. Iwasawa: Riemann-Hurwitz formula and \(p\)-adic Galois representations for number fields , TĂ´hoku Math. J. (2) 33 (1981), 263-288. · Zbl 0468.12004
[9] \begingroup C. Khare and J.-P. Wintenberger: Ramification in Iwasawa modules , 2010). \endgroup arXiv:
[10] Y. Mizusawa and M. Ozaki: On tame pro-\(p\) Galois groups over basic \(\mathbb{Z}_{p}\)-extensions , Math. Z. 273 (2013), 1161-1173. · Zbl 1291.11131
[11] J. Neukirch, A. Schmidt and K. Wingberg: Cohomology of Number Fields, second edition, Grundlehren der Mathematischen Wissenschaften 323 , Springer, Berlin, 2008. · Zbl 1136.11001
[12] L. Salle: On maximal tamely ramified pro-2-extensions over the cyclotomic \(\mathbb{Z}_{2}\)-extension of an imaginary quadratic field , Osaka J. Math. 47 (2010), 921-942. · Zbl 1263.11097
[13] T. Tsuji: Semi-local units modulo cyclotomic units , J. Number Theory 78 (1999), 1-26. · Zbl 0948.11042
[14] T. Tsuji: On the Iwasawa \(\lambda\)-invariants of real abelian fields , Trans. Amer. Math. Soc. 355 (2003), 3699-3714. · Zbl 1038.11072
[15] L.C. Washington: Introduction to Cyclotomic Fields, second edition, Graduate Texts in Mathematics 83 , Springer, New York, 1997. · Zbl 0966.11047
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