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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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