Fabio Caldarola^[a]

Invariants and coinvariants of class groups in \(\mathbb{Z}_p\)-extensions and Greenberg's Conjecture

Pages: 181-192
Received: 4 January 2016
Accepted in revised form: 23 February 2016
Mathematics Subject Classification (2010): 11R23, 11R29.
Keywords: Iwasawa modules, \(\mathbb{Z}_p\)-extensions, class groups, capitulation of ideals, lower central series.
Author address:
[a] : University of Calabria, Department of Mathematics and Computer Science, Cubo 31/B, ponte Pietro Bucci, Arcavacata di Rende, 87036, Italy

Abstract: Let \(K/k\) be a \(\mathbb{Z}_p\)-extension of a number field \(k\), \(k_n\) its \(n\)-th layer and \(A_n\) the \(p\)-class group of \(k_n\). In this paper we give two criteria, both based on the group of invariants \(B_n\) of \(A_n\), which imply the finiteness of the Iwasawa module \(X(K/k)\) and we discuss some of their consequences. The first criterion deals with stabilization and capitulation of the \(B_n\), while the second one uses the nilpotency of the Galois group \({\rm Gal}(L(K)/k)\), where \(L(K)\) is the maximal unramified abelian pro-\(p\)-extension of \(K\).

References

[1] A. Bandini, A note on \(p\)-ranks of class groups in \(\mathbb{Z}_p\)-extensions, JP J. Algebra Number Theory Appl. 9 (2007), 95-103. MR2407808
[2] A. Bandini, Greenberg's conjecture and capitulation in \(\mathbb{Z}_p^d\)-extensions, J. Number Theory 122 (2007), 121-134. MR2287114
[3] A. Bandini, Greenberg's conjecture for \(\mathbb{Z}_p^d\)-extensions, Acta Arith. 108 (2003), 357-368. MR1979904
[4] A. Bandini and F. Caldarola, Stabilization for Iwasawa modules in \(\mathbb{Z}_p\)-extensions, Rend. Semin. Mat. Univ. Padova, to appear.
[5] A. Bandini and F. Caldarola, Stabilization in non-abelian Iwasawa theory, Acta Arith. 169 (2015), 319-329. MR3371763
[6] T. Fukuda, Remarks on \(\mathbb{Z}_p\)-extensions of number fields, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 264-266. MR1303577
[7] R. Greenberg, Iwasawa theory - past and present, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo 2001, 335-385. MR1846466
[8] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. MR0401702
[9] A. Lannuzel and T. Nguyen Quang Do, Conjectures de Greenberg et extensions pro-\(p\)-libres d'un corps de nombres, Manuscripta Math. 102 (2000), 187-209. MR1771439
[10] J. V. Minardi, Iwasawa modules for \(\mathbb{Z}_p^d\)-extensions of algebraic number fields, Ph.D. Thesis, University of Washington, 1986. MR2635492
[11] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields, 2nd edition, Grundlehren Math. Wiss., 323, Springer-Verlag, Berlin 2008. MR2392026
[12] M. Ozaki, Non-abelian Iwasawa theory of \(\mathbb{Z}_p\)-extensions, J. Reine Angew. Math. 602 (2007), 59-94. MR2300452
[13] L. Ribes and P. Zalesskii, Profinite groups, 2nd edition, Ergeb. Math. Grenzgeb. (3), 40, Spinger-Verlag, Berlin 2010. MR2599132
[14] L. C. Washington, Introduction to cyclotomic fields, 2nd edition, Grad. Texts in Math., 83, Springer-Verlag, New York 1997. MR1421575