Galois co-descent for étale wild kernels and capitulation (Q1965851)

For a number field \(K\) and a prime \(p\), let us denote by \(S\) a finite set of primes containing the primes above \(p\) and the archimedian primes. For \(i\geq 2\), the higher étale wild kernels \[ WK_{2i-2}^{\text{ét}} (F):= \operatorname {Ker} (H_{\text{ét}}^2 ({\mathcal O}_F^S, \mathbb{Z}_p(i))\to \bigoplus_{v\in S} H^2(F_v, \mathbb{Z}_p(i))) \] are natural generalizations of the \(p\)-part of the classical wild kernel (inside \(K_2\)). They play a similar role in étale cohomology, étale \(K\)-theory and Iwasawa theory as the \(p\)-primary part \(A_F'\) of the \((p)\)-class group of \(F\). In this paper, given a cyclic extension \(L/F\) of degree \(p\) and \(G= \operatorname {Gal} (L/F)\), the authors study Galois co-descent for \(WK_{2i-2}^{\text{ét}} (L)\). The analogous problem for the étale \(K\)-groups \(K_{2i-2}^{\text{ét}} ({\mathcal O}_L^S)\) had been settled previously by \textit{J. Assim} [Manuscr. Math. 86, 499-518 (1995; Zbl 0835.11043)] in terms of primitive ramification. But the difficulty here lies in the cokernel \(K_{2i-2}^{\text{ét}} ({\mathcal O}_L^S)/ WK_{2i-2}^{\text{ét}} (L)\), and it is overcome by interpreting this quotient in terms of a Brauer group. The authors show that the transfer map \(WK_{2i-2}^{\text{ét}} (L)_G\to WK_{2i-2}^{\text{ét}} (F)\) is onto except in a very special situation, and they give a formula for the order of the cokernel in the style of genus theory. As an application, they determine all Galois \(p\)-extensions \(F/\mathbb{Q}\) such that \(WK_2^{\text{ét}} (F)= 0\). A related problem concerns the étale capitulation kernels \(\operatorname {Cap}_{i-1} (F_\infty)\) \((i\geq 2)\), where \(F_\infty= \bigcup_n F_n\) is the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). These capitulation kernels are defined as \[ \varprojlim_n \operatorname {Ker} (K_{2i-2}^{\text{ét}} ({\mathcal O}_n^S)\to K_{2i-2}^{\text{ét}} ({\mathcal O}_\infty^S)) \] (with obvious notations) and are higher analogs of \[ \operatorname {Cap}_0 (F_\infty)= \varprojlim_n \operatorname {Ker} (A_n')\to A_\infty'). \] For a totally real field \(F\), it is shown that \(\operatorname {Cap}_{i-1} (F_\infty) \simeq WK_{2i-2}^{\text{ét}} (F_n)\) for large \(n\) and all odd \(i\geq 3\), if and only if Greenberg's conjecture holds for \(F(\mu_p)^+\). Therefore the codescent results for the étale wild kernels imply similar results for the capitulation kernels.