Reflexion theorems (Q1292626)

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\dots{} 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 \textit{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 \textit{H. W. Leopoldt} on cyclotomic fields [J. Reine Angew. Math. 199, 165--174 (1958; Zbl 0082.25402)]. Further extensions were given by \textit{S. N. Kuroda} [J. Number Theory 2, 287--297 (1970; Zbl 0222.12013)] for generalized class groups, \textit{B. Oriat} [Astérisque 61, 169--175 (1979; Zbl 0403.12014)], \textit{B. Oriat} and \textit{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 \(T\).