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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.

11R70 \(K\)-theory of global fields
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
Full Text: DOI Numdam EuDML
[1] Assim, Jilali; Movahhedi, Abbas, Bounds for étale capitulation kernels \(, K\)-Theory, 33, 3, 199-213, (2004) · Zbl 1163.11347
[2] Coates, John, On \(K_2\) and some classical conjectures in algebraic number theory, Ann. of Math. (2), 95, 99-116, (1972) · Zbl 0245.12005
[3] Gras, Georges; Jaulent, Jean-François, Sur LES corps de nombres réguliers, Math. Z., 202, 3, 343-365, (1989) · Zbl 0704.11040
[4] Greenberg, Ralph, A note on \(K_2\) and the theory of \(\textbf{Z}_p\)-extensions, Amer. J. Math., 100, 6, 1235-1245, (1978) · Zbl 0408.12012
[5] Griffiths, R. A. W., A genus formula for étale Hilbert kernels in a cyclic p-power extension, (2005)
[6] Hutchinson, K., Tate kernels, étale K-theory and the Gross kernel, (2005)
[7] Iwasawa, Kenkichi, On \(\textbf{Z}_l\)-extensions of algebraic number fields, Ann. of Math. (2), 98, 246-326, (1973) · Zbl 0285.12008
[8] Kahn, Bruno, Descente galoisienne et \(K_2\) des corps de nombres \(, K\)-Theory, 7, 1, 55-100, (1993) · Zbl 0780.12007
[9] Kolster, M.; Movahhedi, A., Galois co-descent for étale wild kernels and capitulation, Ann. Inst. Fourier (Grenoble), 50, 1, 35-65, (2000) · Zbl 0951.11029
[10] Movahhedi, A., Sur les \(p\)-extensions des corps \(p\)-rationnels, (1988) · Zbl 0723.11054
[11] Movahhedi, A., Sur LES \(p\)-extensions des corps \(p\)-rationnels, Math. Nachr., 149, 163-176, (1990) · Zbl 0723.11054
[12] Movahhedi, A.; Nguyễ-Quang-Ḏỗ, T., Séminaire de Théorie des Nombres, Paris 1987-88, 81, Sur l’arithmétique des corps de nombres \(p\)-rationnels, 155-200, (1990), Birkhäuser Boston, Boston, MA · Zbl 0703.11059
[13] Soulé, C.\(, K\)-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math., 55, 3, 251-295, (1979) · Zbl 0437.12008
[14] Tate, John, Relations between \(K_2\) and Galois cohomology, Invent. Math., 36, 257-274, (1976) · Zbl 0359.12011
[15] Vauclair, D, Capitulation, Cup-produit et sous-modules finis dans les \(\textbf{Z}_p\)-extensions d’un corps de nombres, (2005)
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