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A Note on Asymptotic Class Number Upper Bounds in p-adic Lie Extensions

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Abstract

Let p be an odd prime and F∞ a p-adic Lie extension of a number field F with Galois group G. Suppose that G is a compact pro-pp-adic Lie group with no torsion and that it contains a closed normal subgroup H such that G/H ≅ ℤp. Under various assumptions, we establish asymptotic upper bounds for the growth of p-exponents of the class groups in the said p-adic Lie extension. Our results generalize a previous result of Lei, where he established such an estimate under the assumption that H ≅ ℤp.

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References

  1. Ardakov, K., Brown, K. A.: Primeness, semiprimeness and localisation in Iwasawa algebras. Trans. Amer. Math. Soc., 359(4), 1499–1515 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Coates, J., Fukaya, T., Kato, K., et al.: The GL 2 main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Ètudes Sci., 101, 163–208 (2005)

    Article  MATH  Google Scholar 

  3. Coates, J., Schneider, P., Sujatha, R.: Modules over Iwasawa algebra. J. Inst. Math. Jussieu, 2(1), 73–108 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cuoco, A., Monsky, P.: Class numbers in \(\mathbb{Z}_p^d\)-extensions. Math. Ann., 255(2), 235–258 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dixon, J., Du Sautoy, M. P. F., Mann, A., et al.: Analytic Pro-p Groups, 2nd edn, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1999

    Book  MATH  Google Scholar 

  6. Ferrero, B., Washington, L. C.: The Iwasawa invariant µ p vanishes for abelian number fields. Ann. of Math., 109, 377–395 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gillard, R.: Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abeliennes. J. Reine Angew. Math., 358, 76–91 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Goodearl, K. R., Warfield, R. B.: An introduction to non-commutative Noetherian rings, London Math. Soc. Stud. Texts 61, Cambridge University Press, Cambridge, 2004

    Book  MATH  Google Scholar 

  9. Greenberg, R.: Iwasawa theory-past and present, in: Class field Theory-its centenary and prospect. Adv. Stud. Pure Math., 30, 335–385 (2001)

    Article  MATH  Google Scholar 

  10. Hachimori, Y., Sharifi, R.: On the failure of pseudo-nullity of Iwasawa modules. J. Algebraic Geom., 14(3), 567–591 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Howson, S.: Euler characteristics as invariants of Iwasawa modules. Proc. London Math. Soc. (3), 85(3), 634–658 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Howson, S.: Structure of central torsion Iwasawa modules. Bull. Soc. Math. France, 130(4), 507–535 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Iwasawa, K.: On Γ-extensions of algebraic number fields. Bull. Amer. Math. Soc., 65, 183–226 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lam, T. Y.: Lectures on Modules and Rings, Grad. Texts in Math. 189, Springer, 1999

    Book  Google Scholar 

  15. Lei, A.: Estimating class numbers over metabelian extensions. Acta Arith., 180(4), 347–364 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liang, D., Lim, M. F.: On the Iwasawa asymptotic class number formula for \(\mathbb{Z}_p^r\rtimes\mathbb{Z}_p\)-extensions, Acta Arith., published online, DOI: 10.4064/aa180412-12-7

  17. Lim, M. F.: Notes on the fine Selmer groups. Asian J. Math., 21(2), 337–362 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lim, M. F.: Comparing the π-primary submodules of the dual Selmer groups. Asian J. Math., 21(6), 1153–1182 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lim, M. F., Sharifi, R.: Neková⌣ duality over p-adic Lie extensions of global fields. Doc. Math., 18, 621–678 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Monsky, P.: Fine estimate for the growth of e n in \(\mathbb{Z}_p^d\)-extensions, in Algebraic Number Theory-in honor of K. Iwasawa, ed. J. Coates, R. Greenberg, B. Mazur and I. Satake, Adv. Stud. Pure Math. 17, 1989, pp. 309–330

    Google Scholar 

  21. Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, 2nd edn., Grundlehren Math. Wiss. 323, Springer-Verlag, Berlin, 2008

    Google Scholar 

  22. Neumann, A.: Completed group algebras without zero divisors. Arch. Math. (Basel), 51(6), 496–499 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ochi, Y., Venjakob, O.: On ranks of Iwasawa modules over p-adic Lie extensions. Math. Proc. Cambridge Philos. Soc., 135, 25–43 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Perbet, G.: Sur les invariants d’Iwasawa dans les extensions de Lie p-adiques. Algebra Number Theory, 5, 819–848 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sharifi, R.: On Galois groups of unramified pro-p extensions. Math. Ann., 342(2), 297–308 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Venjakob, O.: On the structure theory of the Iwasawa algebra of a p-adic Lie group. J. Eur. Math. Soc. (JEMS), 4(3), 271–311 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Venjakob, O.: A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math., 559, 153–191 (2003)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author like to thank Antonio Lei for many insightful discussion on his paper [15] and the subject on the asymptotic class number formulas in general. The author would also like to thank Dingli Liang for his interest and discussion on the subject of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P. R. China

    Meng Fai Lim

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  1. Meng Fai Lim
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Correspondence to Meng Fai Lim.

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Supported by National Natural Science Foundation of China (Grant Nos. 11550110172 and 11771164)

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Lim, M.F. A Note on Asymptotic Class Number Upper Bounds in p-adic Lie Extensions. Acta. Math. Sin.-English Ser. 35, 1481–1490 (2019). https://doi.org/10.1007/s10114-019-8410-9

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  • DOI: https://doi.org/10.1007/s10114-019-8410-9

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