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Bounds for étale capitulation kernels. II. (Bornes pour les noyaux de capitulations. II.) (English) Zbl 1171.11059
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 J. Assim and A. Movahhedi [\(K\)-theory 33, No. 3, 199–213 (2004; Zbl 1163.11347)], who dealt with the case \(E/F\) cyclic of prime order.

MSC:
11R70 \(K\)-theory of global fields
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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References:
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