On the failure of pseudo-nullity of Iwasawa modules
Authors:
Yoshitaka Hachimori and Romyar T. Sharifi
Journal:
J. Algebraic Geom. 14 (2005), 567-591
DOI:
https://doi.org/10.1090/S1056-3911-05-00396-6
Published electronically:
March 24, 2005
MathSciNet review:
2129011
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References |
Additional Information
Abstract: Consider the family of CM-fields which are pro-$p$ $p$-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic $\mathbf {Z}_p$-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-$p$ extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given $p$-adic Lie groups. The proof uses Kida’s formula for the growth of $\lambda$-invariants in cyclotomic ${\mathbf Z}_p$-extensions of CM-fields. In fact, we give a new proof of Kida’s formula which includes a slight weakening of the usual $\mu = 0$ assumption. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.
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[BH]bh P. Balister and S. Howson, Note on Nakayama’s Lemma for compact $\Lambda$-modules, Asian Math. J. 1 (1997), 224-229.
[Bha]bha A. Bhave, Ph.D. thesis, TIFR, Bombay, in preparation.
[CH]CH J. Coates and S. Howson, Euler characteristics and elliptic curves II, J. Math. Soc. Japan 53 (2001), 175–235.
[CSS]css J. Coates, P. Schneider, and R. Sujatha, Modules over Iwasawa algebras, J. Inst. Math. Jussieu 2 (2003), 73–108.
[CS]cs J. Coates and R. Sujatha, Fine Selmer groups of elliptic curves over $p$-adic Lie extensions, preprint .
[DdMS]DdMS J. Dixon, M. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, 1999.
[FM]fm J.-M. Fontaine and B. Mazur, Geometric Galois Representations, Elliptic curves, modular forms, and Fermat’s Last Theorem, International Press, Boston, 1995, 41–78.
[FW]FeWa B. Ferrero and L. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), 377–395.
[Gr]greenberg R. Greenberg, Iwasawa theory—past and present, Class Field Theory—Its Centenary and Prospect, Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, 335–385.
[HM]HM Y. Hachimori and K. Matsuno, An analogue of Kida’s formula for the Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), 581–601.
[HV]HV Y. Hachimori and O. Venjakob, Completely faithful Selmer groups over Kummer extensions, Doc. Math. Extra Volume: Kazuya Kato’s Fiftieth Birthday (2003), 443–478.
[Hr1]Ha1 M. Harris, $p$-adic representations arising from descent on abelian varieties, Compositio Math. 39 (1979), 177–245.
[Hr2]Ha M. Harris, Correction to: $p$-adic representations arising from descent on abelian varieties, Compositio Math. 121 (2000), 105–108.
[Ho1]Ho S. Howson, Iwasawa theory of elliptic curves for $p$-adic Lie extensions, Ph.D. thesis, University of Cambridge, 1998.
[Ho2]Ho2S. Howson, Euler characteristics as invariants of Iwasawa modules, Proc. London Math. Soc. (3) 85 (2002), 634–658.
[IKY]iky Y. Ihara, M. Kaneko, and A. Yukinari, On some properties of the universal power series for Jacobi sums, Galois representations and arithmetic algebraic geometry, Adv. Stud. Pure Math., vol. 12, Math. Soc. Japan, Tokyo, 1987, 65–86.
[Iw1]Iw K. Iwasawa, Riemann-Hurwitz formula and $p$-adic Galois representations for number fields, Tohoku Math. J. (2) 33 (1981), 263–288.
[Iw2]iwasawa K. Iwasawa, On cohomology groups of units for ${\mathbf Z}_p$-extensions, Amer. J. Math. 105 (1983), 189–200.
[Ja]jannsen U. Jannsen, Iwasawa modules up to isomorphism, Algebraic number theory – in honor of K. Iwasawa, Adv. Stud. Pure Math., vol. 17, Math. Soc. Japan, Tokyo, 1989, 171–207.
[Ki]Ki Y. Kida, $l$-extensions of CM-fields and cyclotomic invariants, J. Number Theory 12 (1980), 519–528.
[Ku]kuzmin L. Kuz’min, Some duality theorems for cyclotomic $\Gamma$-extensions of algebraic number fields of CM type, Math. USSR-Izv. 14 (1980), 441–498.
[La]lang S. Lang, Cyclotomic fields I and II, Combined 2nd ed., Springer-Verlag, New York, 1990.
[NSW]nsw J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Springer-Verlag, Berlin, 2000.
[Oh]ohta M. Ohta, On cohomology groups attached to towers of algebraic curves, J. Math. Soc. Japan 45 (1993), 131–183.
[Ra]Ra R. Ramakrishna, Deforming an even representation, Invent. Math. 132 (1998), 563–580.
[Sh1]massey R. Sharifi, Massey products and ideal class groups, preprint, arXiv:math.NT/ 0308165.
[Sh2]paireis R. Sharifi, Iwasawa theory and the Eisenstein ideal, Preprint, arXiv:math.NT/ 0501236.
[Ve1]venj-str O. Venjakob, On the structure theory of the Iwasawa algebra of a $p$-adic Lie group, J. Eur. Math. Soc. 4 (2002), 271–311.
[Ve2]venjakob O. Venjakob, A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory, J. Reine Angew. Math. 559 (2003), 153–191.
[Ve3]Ve3 O. Venjakob, On the Iwasawa theory of $p$-adic Lie extensions, Compositio Math. 138 (2003), 1–54.
Additional Information
Yoshitaka Hachimori
Affiliation:
CICMA, Department of Mathematics and Statistics, Concordia University, Montréal, Québec H3G 1M8, Canada
Email:
yhachi@mathstat.concordia.ca
Romyar T. Sharifi
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
Email:
sharifi@math.mcmaster.ca
Received by editor(s):
June 27, 2004
Published electronically:
March 24, 2005
Additional Notes:
The first author was partially supported by Gakushuin University and the 21st Century COE Program at the Graduate School of Mathematical Sciences of the University of Tokyo. The second author was supported by the Max Planck Institute for Mathematics.
