## Reflexion theorems. (Théorèmes de réflexion.)(French)Zbl 0949.11058

The author gives here a wide generalization of the so-called Spiegelungssatz of Leopoldt involving $$S-T$$-ray class groups (for arbitrary finite sets of places), Kummer radicals and torsion submodules of the Galois groups associated to classical abelian $$p$$-extensions, tame and higher kernels of $$K$$-theory for number fields and so on… This long paper of one hundred pages, which includes a general approach of the mirror equalities and inequalities, a technical description of the main situations and a careful discussion of the intricate case $$p = 2$$, actually appears to be the reference on this subject.
First emblematic Spiegelungssätze are the old result of A. Scholz [J. Reine Angew. Math. 166, 201–203 (1932; Zbl 0004.05104)] on the 3-rank of ideal classes of quadratic fields and the classical paper of H. W. Leopoldt on cyclotomic fields [J. Reine Angew. Math. 199, 165–174 (1958; Zbl 0082.25402)]. Further extensions were given by S. N. Kuroda [J. Number Theory 2, 287–297 (1970; Zbl 0222.12013)] for generalized class groups, B. Oriat [Astérisque 61, 169–175 (1979; Zbl 0403.12014)], B. Oriat and P. Satgé [J. Reine Angew. Math. 307–308, 134–159 (1979; Zbl 0395.12015)] in a non semi-simple situation, the reviewer [Prog. Math. 75, 183–220 (1988; Zbl 0679.12007)] in cyclotomic towers, and others.
In the paper under review the main result is a nice theorem of reflexion (Th. 5.18) which, in the simplest case where $$S \cup T$$ contains both the $$p$$-adic places and the infinite ones, gives the following striking identity on the $$p$$-ranks of the $$\chi$$- components of the generalized class groups : $rg_{\chi^*} ({\mathcal C \ell}_T^S) - rg_{\chi} ({\mathcal C \ell}_{S^*}^{T^*}) = \rho_{\chi}(T, S).$ Here $$\chi \mapsto \chi^*$$ is the classical mirror involution between the characters and $$\rho_{\chi}(T, S)$$ is a quite elementary algebraic expression which only depends on the Galois properties of the sets of places $$S$$ and $$T$$. Most classical results then follow by specializing $$S$$ and $$T$$.

### MSC:

 11R37 Class field theory 11R70 $$K$$-theory of global fields
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