On the codescent of \'etale wild kernels in $p$-adic Lie extensions

TL;DR

This paper investigates the behavior of étale wild kernels in p-adic Lie extensions of number fields, providing new estimates and growth formulas by connecting them to fine Selmer groups and control theorems.

Contribution

It introduces a novel approach of viewing étale wild kernels as fine Selmer groups, enabling the derivation of growth formulas and relations to Greenberg's conjecture in p-adic Lie extensions.

Findings

Established asymptotic growth formulas for étale wild kernels.

Connected growth behavior to Greenberg's conjecture and its noncommutative analogue.

Provided examples illustrating the theoretical results.

Abstract

Let $F$ be a number field and $p$ an odd prime. We estimate the kernels and cokernels of the codescent maps of the \'etale wild kernels over various $p$-adic Lie extensions. For this, we propose a novel approach of viewing the \'etale 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 \'etale 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.