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Generalised Iwasawa invariants and the growth of class numbers

  • Sören Kleine ORCID logo EMAIL logo
Published/Copyright: September 1, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 1

Abstract

We study the generalised Iwasawa invariants of ℤpd-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion ℤp⁢[[T1,…,Td]]-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of ℤpd-extensions of K, i.e., that the generalised Iwasawa invariants of a ℤpd-extension 𝕂 of K bound the invariants of all ℤpd-extensions of K in an open neighbourhood of 𝕂. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain ℤp2-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

Keywords: Generalised Iwasawa invariants; pseudo-null Iwasawa modules; growth of class numbers; Greenberg’s generalised conjecture
MSC 2010: 11R23; 13C05

Communicated by Jan Bruinier

Acknowledgements

I would like to thank C. Greither for carefully reading a preliminary version of this article; I am grateful for his comments on the proof of the rank inequality in Section 3. Moreover, I am very grateful to the anonymous referee for suggesting several improvements of the proofs of Theorems 3.2 and 5.4; the exposition of the manuscript has profited considerably from the referee’s comments.

References

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Received: 2019-05-07
Revised: 2020-08-04
Published Online: 2020-09-01
Published in Print: 2021-01-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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  2. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
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  4. Completely positive maps of order zero on pro-𝐶∗-algebras
  5. Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series
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https://doi.org/10.1515/forum-2019-0119
Keywords for this article
Generalised Iwasawa invariants; pseudo-null Iwasawa modules; growth of class numbers; Greenberg’s generalised conjecture

Articles in the same Issue

  1. Frontmatter
  2. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
  3. On three-variable expanders over finite valuation rings
  4. Completely positive maps of order zero on pro-𝐶∗-algebras
  5. Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series
  6. Special value formula for the twisted triple product L-function and an application to the restricted L2-norm problem
  7. Generalised Iwasawa invariants and the growth of class numbers
  8. Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth
  9. The geometric sieve for quadrics
  10. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
  11. Commutative algebraic monoid structures on affine surfaces
  12. Representations induced from the Zelevinsky segment and discrete series in the half-integral case
  13. A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)
  14. From subcategories to the entire module categories
  15. Coherent state transform for Landau levels on quasi-tori
  16. On rational homotopy and minimal models
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