Jaulent, Jean-François; Michel, Alexis Logarithmic approach of the étale wild kernels of number fields. (Approche logarithmique des noyaux étales sauvages des corps de nombres.) (French) Zbl 1163.11075 J. Number Theory 120, No. 1, 72-91 (2006). For a number field \(F\) and a prime number \(\ell,\) the \(\ell\)-étale wild kernels \(W K_{2i}^{\text{ét}} (F)\) are defined as localization kernels \[ \text{Ker}\, (H^2_{\text{ét}} ({\mathcal O}_F[1/\ell], {\mathbb Z}_\ell (i)+1)) \to \displaystyle\bigoplus_{v| p}\;H^2_{\text{ét}}(F_v, {\mathbb Z}_\ell (i+1)). \] The name comes from Tate’s result (a particular case of the Quillen-Lichtenbaum conjecture) which states that \(WK_2^{\text{ét}} (F)\) is isomorphic to the \(\ell\)-part of the usual wild kernel for \(K_2.\) If the prime \(\ell\) is such that \(\text{Gal}(F(\mu_{\ell^\infty})/F)\) is pro-cyclic, the \(WK_{2i}^{\text{ét}} (F)'s\) are canonically isomorphic to the co-invariants of the Tate twists \(X(i)\) of a standard Iwasawa module \(X\) [P. Schneider, Math. Z. 168, 181–205 (1979; Zbl 0421.12024)]. In this paper, the authors single out the zero twist \(WK_0 (F),\) which is isomorphic by class field theory to the so-called group of “logarithmic classes” defined and studied by Jaulent and his students. Provided \(F\) contains a primitive \(\ell^n\)-th root of unity, the étale wild kernels mod \(\ell^n\) are obviously all isomorphic up to adequate twists, and the “logarithmic approach” announced in the title consists in deriving a certain number of properties of these kernels mod \(\ell^n,\) building on results obtained previously for the logarithmic classes. This is presented as a unifying approach (in favor of the log classes), but actually, one can only compare the kernels mod \(\ell^n,\) not the whole kernels themselves (which would contradict known results on the special values of \(L\)-functions). Hence the applications are mainly limited to, say, vanishing theorems, but whose “log” proofs are often essentially the same as the direct cohomological proofs (compare thm. 4 and the subsequent “Spiegelung” corollaries with [e.g. M. Kolster, Seminar on number theory, Paris, France, 1991–92. Boston, MA: Birkhäuser. Prog. Math. 116, 37–62 (1994; Zbl 1043.19500)]. Sometimes certain statements look different because they are expressed in class-field theoretic terms, but they are not stronger than the direct cohomological results (cf. e.g. [M. Kolster and A. Movahhedi, Ann. Sci. Inst. Fourier 50, 35–65 (2000; Zbl 0951.11029)]). When looking for a “unifying heuristic”, one could equally choose any fixed étale kernel, e.g. the \(\ell\)-part of the classical wild kernel for \(K_2,\) since they are all isomorphic mod \(\ell^n.\) Therefore the only argument in favor of the “log” classes would be their algorithmic applications, but here the reviewer is not competent. Reviewer: Thong Nguyen Quang Do (Besançon) Cited in 5 Documents MSC: 11R23 Iwasawa theory 19D50 Computations of higher \(K\)-theory of rings 11R70 \(K\)-theory of global fields 19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) 11R65 Class groups and Picard groups of orders Keywords:logarithmic classes; étale wild kernels Citations:Zbl 0421.12024; Zbl 1043.19500; Zbl 0951.11029 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Banaszak, G., Algebraic \(K\)-theory of number fields and ring of integers and the Stickelberger ideal, Ann. of Math., 135, 325-360 (1992) · Zbl 0756.11037 [2] Banaszak, G., Generalization of the Moore sequence and the wild kernel for higher \(K\)-groups, Compos. Math., 86, 281-305 (1993) · Zbl 0778.11066 [3] Banaszak, G., Euler systems for higher \(K\)-theory of number fields, J. Number Theory, 56, 213-252 (1996) · Zbl 0851.19003 [4] Borel, A., Stable real cohomology of arithmetic groups, Ann. Sci. 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