Bounds for étale capitulation kernels. II (Q1017376)

Let \(p\) be an odd prime number and \(E/F\) be a cyclic extension of number fields of degree \(p^n\) with Galois group \(G\). The authors obtain lower bounds for the orders of the kernel and cokernel of the natural maps \[ f_i:K_{2i-2}^{\text{ét}}({\mathcal O}^S_F)\to K_{2i-2}^{\text{ét}}({\mathcal O}^S_E)^G \] where \(S\) is a finite set of primes of \(F\) containing the primes above \(p\). The lower bounds are given in terms of the maximal number of non \(p\)-adic primes of \(S\) satisfying a certain independence condition. These results generalise and extend the results of the second author and \textit{J. Assim} and \textit{A. Movahhedi} [\(K\)-theory 33, No. 3, 199--213 (2004; Zbl 1163.11347)], who dealt with the case \(E/F\) cyclic of prime order.