Higher analogs of wild kernel. (Analogues supérieurs du noyau sauvage.) (French) Zbl 0783.11042

Nguyen Quang Do, Thong

Sémin. Théor. Nombres Bordx., Sér. II 4, No. 2, 263-271 (1992).

Let \(F\) be a number field, \(p\) an odd prime, \(S\) the set of the \(p\)-adic primes of \(F\) and \(o_ F^ S\) the ring of \(S\)-integers of \(F\). It is well-known that for \(i\geq 2\) the \(p\)-adic Chern characters \[ ch_{i,2}: K_{2i-2} (o_ F^ S)\otimes \mathbb{Z}_ p\to H^ 2(o_ F^ S,\mathbb{Z}_ p(i)), \qquad ch_{i,1}: K_{2i-1} (o_ F^ S)\otimes \mathbb{Z}_ p\to H^ 1(o_ F^ S,\mathbb{Z}_ p(i)) \] are surjective and isomorphisms for \(i=2\). Moreover, \(ch_{i,1}\) is split surjective. The author constructs a canonical partial splitting of \(ch_{i,2}\) on the subgroup \[ \text{ Ш}_ S^ 2(\mathbb{Z}_ p (i)):= \ker(H^ 2 (o_ F^ S, \mathbb{Z}_ p(i))\to \oplus_{v\in S} H^ 2(F_ v, \mathbb{Z}_ p(i))). \] This group should be viewed as the \(p\)-primary part of a higher wild kernel, this being true for \(i=2\). The author also shows that \(\text{ Ш}_ S^ 2(\mathbb{Z}_ p(i))\) is annihilated by the higher Stickelberger ideal \(S_{i-1}(F)\) introduced by Coates and Sinnott, in case \(F\) is abelian over \(\mathbb{Q}\).

Reviewer: M.Kolster (Hamilton / Ontario)

MSC:

11R70	\(K\)-theory of global fields
11R23	Iwasawa theory
11R34	Galois cohomology

Keywords:

\(p\)-adic Chern characters; higher wild kernel; higher Stickelberger ideal

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