In this talk, we shall study the growth of the etale wild kernels in various p-adic Lie extensions. The etale wild kernels (coined by Banaszak, Kolster, Nguyen Quand Do etc) are related to the special values of the Dedekind zeta function. In this talk, we reinterpret the etale wild kernel as an appropriate fine Selmer group in the sense of Coates-Sujatha. This viewpoint brings us to the problem of studying a control theorem of the said fine Selmer groups, which in turn allows us to minic the strategies developed by Greenberg. However, this improvisation is not a direct procedure, as one needs to estimate the growth of cohomology groups of open subgroups of p-adic Lie groups which is not accessible directly from the lie algebraic approach of Greenberg (one of the main issue is that the open subgroups have the same lie algebra and so the cohomology of the lie algebra cannot distinguish the cohomology groups of the subgroups). Among the tools used in estimatiing these cohomology groups, one notable ingredient is Tate's lemma which asserts the vanishing of the first $\Gamma$-cohomology groups of nonzero Tate twist of Qp/Zp.

Once we have such a control theorem, we apply them to obtain asymptotic growth formulas for the etale wild kernels in various said p-adic Lie extensions. The leading terms of the growth formulas are related to a certain Galois group, and an appropriate noncommutative variant of Greenberg's conjecture predicts that this said Galois group is not "too big". In particular, we shall see that Greenberg conjecture gives an asymptotic upper bound on the growth of the etale wild kernels. These upper bound are not necessarily always optimal. Indeed, building on calculations of Sharifi, we can give some examples which show that the etale wild kernels can grow much slower than the predicted estimate of Greenberg.

Finally, if time permits, we shall mention briefly on a joint work with Debanajana Kundu on the fine Selmer groups of elliptic curves which builds on a natural analogue/generalization of Tate's lemma in the elliptic situation.

Introductory reading:

Greenberg - Iwasawa Theory—Past and Present: http://www.fields.utoronto.ca/sites/default/files/uploads/Greenberg.pdf

Kolster, Movahhedi - Galois co-descent of étale wild kernels and capitulation: http://www.fields.utoronto.ca/sites/default/files/uploads/Kolster-Movahh...

Quang Do: Analogues supérieurs du noyau sauvage: http://www.fields.utoronto.ca/sites/default/files/uploads/NQD.pdf