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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]
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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 

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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