Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-07T07:19:59.952Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime decomposition and the Iwasawa MU-invariant

Published online by Cambridge University Press:  26 April 2018

FARSHID HAJIR  and
CHRISTIAN MAIRE
FARSHID HAJIR
Affiliation:
Department of Mathematics & Statistics, University of Massachusetts, Amherst MA 01003, U.S.A. e-mail: [email protected]
CHRISTIAN MAIRE
Affiliation:
Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté et CNRS, 16 route de Gray, 25030 Besançon, France. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

For Γ = ℤp, Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large μ-invariants. In this work, we investigate other uniform pro-p groups which are realisable as Galois groups of towers of number fields with arbitrarily large μ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p−1. Both in Iwasawa's work, and in the present one, the size of the μ-invariant appears to be intimately related to the existence of primes that split completely in the tower.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 3 , May 2019 , pp. 599 - 617
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bolker, E.D. Inverse limits of solvable groups. Proc. Amer. Math. Soc. 14 (1963), 147–152.10.1090/S0002-9939-1963-0144963-8Google Scholar
[2] Boston, N. Some cases of the Fontaine–Mazur conjecture. J. Number Theory 42 (1992), no. 3, 285–291.10.1016/0022-314X(92)90093-5Google Scholar
[3] Boston, N. Some cases of the Fontaine–Mazur conjecture. II. J. Number Theory 75 (1999), no. 2, 161–169.10.1006/jnth.1998.2337Google Scholar
[4] Boston, N. Explicit deformation of Galois representations. Invent. Math. 103 (1991), 181–196.10.1007/BF01239511Google Scholar
[5] Coates, J., Schneider, P. and Sujatha, R. Modules over Iwasawa algebras. J. Inst. Math. Jussieu 2, issue 1, (2003), 73–108.10.1017/S1474748003000045Google Scholar
[6] Cohen, H. and Lenstra, H. W. Jr., Heuristics on Class groups on Number Fields. Number Theory, Noordwijkerhout 1983. Lecture Notes in Math. Vol. 1068 (Springer 1984), 33–62.Google Scholar
[7] Cuoco, A. A. Generalised Iwasawa invariants in a family. Compositio Math. 51 (1984), 89–103.Google Scholar
[8] Dixon, J. D., F Du Sautoy, M. P., Mann, A. and Segal, D. Analytic pro-p-groups. Camb. stud. adv. math. 61 (Cambridge University Press, 1999).10.1017/CBO9780511470882Google Scholar
[9] Ferrero, B. and Washington, L. C. The Iwasawa invariant μp vanishes for abelian number fields. Ann. Math. Second Series 109 (1979), no. 2, 377–395.10.2307/1971116Google Scholar
[10] Gildenhuysn, D., Herfort, W. N. and Ribes, L. Profinite Frobenius group. Arch. Math. 33 (1979/80), no. 6, 518–528.10.1007/BF01222795Google Scholar
[11] Gras, G. Class Field Theory, SMM (Springer, 2003).10.1007/978-3-662-11323-3Google Scholar
[12] Gras, G. Les θ-régulateurs locaux d'un nombre algébrique – Conjectures p-adiques. Canadian J. Math. 68 (2016), no. 3, 571–624.10.4153/CJM-2015-026-3Google Scholar
[13] Gras, G. Théorèmes de réflexion. J. Théor. Nombres Bordeaux 10 (1998), no. 2, 399–499.10.5802/jtnb.234Google Scholar
[14] Gras, G. and Jaulent, J.-F. Sur les corps de nombres réguliers. Math. Z. 202 (1989), 343–365.10.1007/BF01159964Google Scholar
[15] Greenberg, R. Galois representations with open image. Ann. Math. Québec 40 (2016), no. 1, 83–119.10.1007/s40316-015-0050-6Google Scholar
[16] Hajir, F. and Maire, C. On the invariant factors of class groups in towers of number fields. Canadian J. Math. 70 (2018), no 1, 142–172.10.4153/CJM-2017-032-9Google Scholar
[17] Harris, M. p-adic representations arising from descent on abelian varieties. Compositio Math. 39:2 (1979), 177–245. With correction: Compositio Math. 121:1 (2000), 105–108.Google Scholar
[18] Herfort, W. N. and Ribes, L. On automorphisms of free pro-p-groups I. Proc. Amer. Math. Soc. 108 (1990), no. 2, 287–295.Google Scholar
[19] Howson, S. Euler characteristics as invariants of Iwasawa modules. Proc. London Math. Soc. 85 (2002) no. 3, 634–658.10.1112/S0024611502013680Google Scholar
[20] Ihara, Y. How many primes decompose completely in an infinite unramified Galois extension of a global field?. J. Math. Soc. Japan 35 (1983), no. 4, 693–709.10.2969/jmsj/03540693Google Scholar
[21] Iwasawa, K. On the μ-invariants of ℤℓ-extensions. Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, pp. 1–11 (Kinokuniya, Tokyo, 1973).Google Scholar
[22] Jaulent, J.-F. and Quang Do, T. Nguyen Corps p-réguliers, corps p-rationnels et ramification restreinte. J. Théor. Nombres Bordeaux 5 (1993), 343–363.10.5802/jtnb.98Google Scholar
[23] Gonzálo-Sánchez, J. and Klopsch, B. Analytic pro-p groups of small dimensions. J. Group Theory 12 (2009), no. 5, 711–734.Google Scholar
[24] Klopsch, B. and Snopce, I. A characterisation of uniform pro-p groups. Q. J. Math. 65 (2014), no. 4, 1277–1291.10.1093/qmath/hau005Google Scholar
[25] Lazard, M. Groupes analytiques p-adiques. IHES, Publ. Math. 26 (1965), 389–603.Google Scholar
[26] Maire, C. Sur la dimension cohomologique des pro-p-extensions des corps de nombres. J. Théorie. ds Nombres Bordeaux 17 2 (2005), 575–606.10.5802/jtnb.509Google Scholar
[27] Movahhedi, A. and Quang Do, T. Nguyen Sur l'arithmétique des corps de nombres p-rationnels. Séminaire de Théorie des Nombres (Paris 1987–88), 155–200 Progr. Math. 81 (Birkhäuser Boston, Boston, MA, 1990).Google Scholar
[28] Movahhedi, A. Sur les p-extensions des corps p-rationnels. Math. Nachr. 149 (1990), 163–176.10.1002/mana.19901490113Google Scholar
[29] Quang Do, T. Nguyen Formations de classes et modules d'Iwasawa. Noordwijkerhout 1983, Lecture Notes in Math. Vol. 1068 (Springer 1984), 167–185.Google Scholar
[30] Perbet, G. Sur les invariants d'Iwasawa dans les extensions de Lie p-adiques (French) [On Iwasawa invariants in p-adic Lie extensions]. Algebra Number Theory 5 (2011), no. 6, 819–848.10.2140/ant.2011.5.819Google Scholar
[31] Pitoun, F. and Varescon, F. Computing the torsion of the p-ramified module of a number field. (English summary) Math. Comp. 84 (2015), no. 291, 371–383.10.1090/S0025-5718-2014-02838-XGoogle Scholar
[32] Ribes, L. and Zalesskii, P. Profinite Groups EMG 40 (Springer, 2010).10.1007/978-3-642-01642-4Google Scholar
[33] Serre, J.-P. Quelques applications du théorème de densité de Chebotarev. Publ. Math. IHES, vol. 54 (1981), 123–201.10.1007/BF02698692Google Scholar
[34] Venjakob, O. On the structure theory of the Iwasawa algebra of a p-adic Lie group. J. Eur. Math. Soc. 4: (2002), 271–311.10.1007/s100970100038Google Scholar
[35] Wingberg, K. Free pro-p-extensions of number fields. Preprint (2005).Google Scholar