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

Abstract

Let F be a number field. 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 Greenberg’s conjecture (and its noncommutative analogue). Finally, we shall give some examples to illustrate our results.