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On Greenberg's generalized conjecture for families of number fields

Thong Nguyen Quang Do

TL;DR

The paper investigates Greenberg's generalized conjecture (GGC) for families of number fields, focusing on imaginary base fields and reducing GGC to pseudo-nullity conditions on carefully chosen special $\mathbb{Z}_p^2$-extensions. It develops an induction framework on the extension rank $d$, introduces and exploits Greenberg-type extensions, and derives a main criterion (via Theorem 10-5 and related results) linking pseudo-nullity of $X'(K^{(2)})$ to finiteness of Iwasawa invariants and the behavior of class-group data. A central thread is the control of obstruction kernels (via the Bertrandias-Payan module and related cohomology) and the propagation of vanishing conditions through towers, including the study of $(p,i)$-regular fields which provide robust families where parts of GGC hold. The results yield practical criteria to verify GGC in infinite families by reducing to finite-data conditions, and they illuminate how density results for undecomposed primes and $p$-rationality interact with GK-type finiteness issues in the non-semi-simple Iwasawa setting. Overall, the work advances the understanding of GGC by linking pseudo-nullity to explicit tower- and cohomology-controlled conditions and by constructing concrete families of imaginary fields where the conjecture can be tested or proved in parts.

Abstract

For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{Λ}$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the maximal abelian unramified pro-$p$-extension of $\tilde{k}$. Greenberg's generalized conjecture (GGC for short) asserts that the $\tildeΛ$-module $X(\tilde{k})$ is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special ${\mathbb Z}^2_p$-extension of $k$, and these conditions are partially or entirely fullfilled by certain families of number fields.

On Greenberg's generalized conjecture for families of number fields

TL;DR

The paper investigates Greenberg's generalized conjecture (GGC) for families of number fields, focusing on imaginary base fields and reducing GGC to pseudo-nullity conditions on carefully chosen special -extensions. It develops an induction framework on the extension rank , introduces and exploits Greenberg-type extensions, and derives a main criterion (via Theorem 10-5 and related results) linking pseudo-nullity of to finiteness of Iwasawa invariants and the behavior of class-group data. A central thread is the control of obstruction kernels (via the Bertrandias-Payan module and related cohomology) and the propagation of vanishing conditions through towers, including the study of -regular fields which provide robust families where parts of GGC hold. The results yield practical criteria to verify GGC in infinite families by reducing to finite-data conditions, and they illuminate how density results for undecomposed primes and -rationality interact with GK-type finiteness issues in the non-semi-simple Iwasawa setting. Overall, the work advances the understanding of GGC by linking pseudo-nullity to explicit tower- and cohomology-controlled conditions and by constructing concrete families of imaginary fields where the conjecture can be tested or proved in parts.

Abstract

For a number field and an odd prime , let be the compositum of all the -extensions of , the associated Iwasawa algebra, and the Galois group over of the maximal abelian unramified pro--extension of . Greenberg's generalized conjecture (GGC for short) asserts that the -module is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special -extension of , and these conditions are partially or entirely fullfilled by certain families of number fields.
Paper Structure (28 sections, 15 theorems, 63 equations)

This paper contains 28 sections, 15 theorems, 63 equations.

Table of Contents

  1. Introduction.
  2. Geenberg's generalized conjecture (GGC for short)
  3. The Kuz'min-Gross conjecture
  4. Induction on $d\geq 2$.
  5. Green $\mathbb{Z}_p^2$-extensions.
  6. Extension maps.
  7. Norm maps.
  8. Miscellaneous results on (co)capitulation
  9. (i) Computation of $(X'^{(2)})^{\Gamma}$.
  10. (ii) Computation of $(X'^{(2)})_{\Gamma}$.
  11. Undecomposed $p$-primes and density properties.
  12. Nullity of the obstructions kernels
  13. A survey of the Bertrandias-Payan module.
  14. (P1)
  15. (P2)
  16. ...and 13 more sections

Key Result

Lemma 1.1

Let $k$ be imaginary, and let $K^{(d)}/k$ be a $\mathbb{Z}^d_p$-extension ($d\geq 2$) which is normal over $\mathbb{Q}$. Then for any $p$-place $v$ of $K^{(d)}$, $\mathbb{Z}_p$-rank $\Gamma^{(d)}_v\geq d/s$, where $s$ is the number of $p$-places of $k$.

Theorems & Definitions (29)

  • Lemma 1.1
  • proof
  • Lemma 1.2: cp. tkat1, lemma 4.10
  • proof
  • Lemma 1.3: ba_ba_lo1, 2.10 and 2.11
  • Theorem 1.4
  • proof
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • ...and 19 more