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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)
##### Keywords:
capitulation; Tate kernel; $$K$$-group; étale cohomology
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##### References:
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