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On the codescent of étale wild kernels in $p$-adic Lie extensions

Meng Fai Lim

Abstract

Let $F$ be a number field and $p$ an odd prime. We estimate the kernels and cokernels of the codescent maps of the étale wild kernels over various $p$-adic Lie extensions. For this, we propose a novel approach of viewing the étale wild kernel as an appropriate fine Selmer group in the sense of Coates-Sujatha. This viewpoint reduces the problem to a control theorem of the said fine Selmer groups, which in turn allows us to employ the strategies developed by Mazur and Greenberg. As applications of our estimates on the kernels and cokernels of the codescent maps, we establish asymptotic growth formulas for the étale wild kernels in the various said $p$-adic Lie extensions. We then relate these growth formulas to the Greenberg's conjecture (and its noncommutative analogue). Finally, we shall give some examples to illustrate our results.

On the codescent of étale wild kernels in $p$-adic Lie extensions

Abstract

Let be a number field and an odd prime. We estimate the kernels and cokernels of the codescent maps of the étale wild kernels over various -adic Lie extensions. For this, we propose a novel approach of viewing the étale wild kernel as an appropriate fine Selmer group in the sense of Coates-Sujatha. This viewpoint reduces the problem to a control theorem of the said fine Selmer groups, which in turn allows us to employ the strategies developed by Mazur and Greenberg. As applications of our estimates on the kernels and cokernels of the codescent maps, we establish asymptotic growth formulas for the étale wild kernels in the various said -adic Lie extensions. We then relate these growth formulas to the Greenberg's conjecture (and its noncommutative analogue). Finally, we shall give some examples to illustrate our results.

Paper Structure

This paper contains 23 sections, 30 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Uniform pro-$p$ groups
  3. Iwasawa invariants
  4. Some estimates on cohomology groups
  5. Arithmetic preliminaries
  6. Tate twist
  7. Iwasawa cohomology groups
  8. Étale wild kernel
  9. Iwasawa $\mu$-conjecture
  10. Greenberg's conjecture
  11. Codescent in $p$-adic extensions
  12. $\mathbb{Z}_{p}$-extension
  13. $\mathbb{Z}_{p}^d$-extension
  14. Multi-False-Tate extensions
  15. $GL_2$-extension
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $G$ be a uniform group and $N$ a closed normal subgroup of $G$ such that $R:=G/N$ is uniform. Then $N$ is also uniform. Furthermore, writing $N_n= N^{p^n}$, $G_n=G^{p^n}$, and $R_n=R^{p^n}$, we have $N_n = G_n\cap N$ and $G_n/N_n\cong R_n$.

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1: Tate's Lemma
  • proof
  • Lemma 3.2
  • proof
  • ...and 53 more