Higher analogues of Stickelberger's theorem.
Let \(p\) be an odd prime number, \(F\) a totally real number field, \(E/F\) an abelian CM extension. For any odd integer \(n \geq 1,\) the \(p\)-part of the Coates-Sinnott conjecture (resp. its étale version) predicts that a certain higher Stickelberger ideal \(S_n (E/F, p) \subset {{\mathbb Z}}_p [G(E/F)]\) annihilates \(K_{2n}({\mathcal O}_E)_p\) (resp. \(K_{2n}^{\text{ét}} ({\mathcal O}_E)\simeq H^{2}_{\text{ét}} ({\mathcal O}_E[1/p], {\mathbb Z}_p (n+1))).\) In this note, the author shows the étale version assuming two restrictive hypotheses on \((p, n)\): (i) \(p\) does not divide \(n | G(E/F)|\) (in particular, we are in the semi-simple case), (ii) the étale \(K\)-group coincides with the étale wild kernel. The key point is the construction in [\textit{G. Banaszak}, Ann. Math. 135, 325--360 (1992; Zbl 0756.11037)], of a natural map \(\wedge\) which, roughly speaking, ``splits'' Quillen's localization map \(K_{2n} (E)_p \to {\bigoplus_v} K_{2n-1}(k_v)_p\) ``up to a higher Stickelberger element'' (thm. IV-1, op. cit.). But note that: -- The proof does not avoid some ``hand waving'', since in Banaszak, op. cit., the map \(\wedge\) is constructed only for \(F = \mathbb Q,\) with a simple remark concerning the possibility to extend the construction to any totally real field \(F,\) and in propos. 3.2 of the present paper, the semi-simplicity hypothesis is justified only by a cryptic allusion to \textit{A. Wiles}' proof of some cases of the Brumer conjecture [Ann. Math. (2) 131, 555--565 (1990; Zbl 0719.11082)]. -- Because the natural setting for étale \(K\)-groups is Iwasawa theory, and also because of the allusion to Wiles' proof, perhaps what the author has in mind is the Main Conjecture over \(F\) (or Wiles' theorem) [Ann. Math. (2) 131, 493--540 (1990)], which applies perfectly well in the semisimple case. But then the reviewer cannot see the connection with the map \(\wedge.\) -- Actually, the Equivariant Main Conjecture proved by Ritter-Weiss and Burns-Greither (assuming the vanishing of the \(\mu_p\)-invariant) allows to show the étale version of the Coates-Sinnott conjecture for abelian \(E/F\) [see \textit{D. Burns} and \textit{C. Greither}, Doc. Math., J. DMV Extra Vol., 157--185 (2003; Zbl 1142.11371), or the reviewer, J. Théor. Nombres Bordeaux 17, 667--692 (2005; Zbl 1091.11041). The latter paper even contains a global version, up to 2-primary parts).
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