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Logarithmic approach of the étale wild kernels of number fields. (Approche logarithmique des noyaux étales sauvages des corps de nombres.) (French) Zbl 1163.11075

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.

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
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References:

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