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\(T\)-\(S\) capitulation. (French) Zbl 1254.11100
Let \(k\) be a number field, let \(S\) and \(T\) be disjoint finite sets of places of \(k\), such that \(T\) only contains non-archimedean places. Let \(k_{1,T}^S\) denote the maximal abelian extension of \(K\) that is at most moderately ramified at the places in \(T\) and such that the places in \(S\) split completely in \(K\). If we take \(T = \varnothing\) and let \(S = S_\infty\) denote the set of infinite (real) places, \(k_{1,T}^S\) is the usual Hilbert class field. This article generalizes the classical work on capitulation of ideal classes to such extensions \(k_{1,T}^S\); more exactly the author observes that \(\text{Gal}(k_{1,T}^S/k)\) is isomorphic to the ray class group \(\text{Cl}_{k,m}^S\) for a suitable modulus \(m\), and then asks what happens to the classes in this group when they are lifted to the ray class group \(\text{Cl}_{K,m(K)}^{S(K)}\), where \(m(K)\) and \(S(K)\) are certain extensions to \(K\) of \(m\) and \(S\), respectively. A generalization of Hilbert’s Theorem 94 to this situation was already given by J.-F. Jaulent [Sémin. Théor. Nombres, Univ. Bordeaux I 1987/1988, Exp. No. 17, 33 p. (1988; Zbl 0704.11046)]. A large part of this article deals with generalizing results originally due to K. Iwasawa [J. Math. Pures Appl. (9) 35, 189–192 (1956; Zbl 0071.26504)]; here the part of the ray class group in \(K\) fixed by the Galois group modulo the ray class group coming from below is mapped injectively into the cohomology group \(H^2(G,E_{K,m}^S)\), where \(E_{K,m}^S\) is the group of \(S\)-units congruent to \(1\) modulo \(m\), under the assumption that \(K/k\) is a finite normal moderately \(T\)-ramified extension; the author also provides an estimate for the size of the cokernel of this map. In the last section, the special case where \(K/k\) is cyclic is studied in detail.
G. Gras included the results of this article in his book [Class field theory. From theory to practice. Berlin: Springer (2003; Zbl 1019.11032)].

MSC:
11R37 Class field theory
11R29 Class numbers, class groups, discriminants
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