Théorèmes de réflexion.
Résumé
The author gives a wide generalization of the so-called Spiegelungssatz of Leopoldt and extends most classical and unclassical results on Kummer duality. This long paper of one hundred pages, which includes a general approach to the mirror equalities and inequalities in a semisimple context, a technical description of the main situations and a careful discussion of the special case p = 2, actually appears to be the reference on this subject. The first emblematic Spiegelungssätze are the old theorem of A. Scholz [J. Reine Angew. Math. 166 (1932), 201–203; Zbl 004.05104] on the 3-rank of ideal classes of quadratic fields and the classical result of H.-W. Leopoldt on cyclotomic fields [J. Reine Angew. Math. 199 (1958), 165– 174; MR0096633 (20 #3116)]. Further extensions were given by S.-N. Kuroda [J. Number Theory 2 (1970), 282–297; MR0311624 (47 #186)] for generalized class groups, B. Oriat [in Journées Arithmétiques de Luminy (Luminy, 1978), 169–175, Ast ́erisque, 61, Soc. Math. France, Paris, 1979; see MR0556662 (80j:10003)], Oriat and P. Satgé [J. Reine Angew. Math. 307/308 (1979), 134–159; MR0534216 (80f:12007)] in a non-semisimple situation, the reviewer [in Séminaire de Théorie des Nombres, Paris 1986–87, 183–220, Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988; MR0990512 (90g:11146)] in cyclotomic towers, and many others. In the paper under review the main result is a nice theorem of reflexion for generalized class groups C(S,T) (Theorem 5.18) which, in the simplest case where S ∪ T contains both the p-adic places and the infinite ones, gives the following striking identity on the p-ranks of the χ-components of the generalized class groups: rgχ∗ (C(S,T) ) − rgχ(C( T∗, S∗) ) = ρχ(T , S ). Here χ → χ∗ is the classical mirror involution between the p-adic characters of the Galois group and ρχ(T , S ) is a quite elementary algebraic expression which only depends on the Galois properties of the finite sets of places S and T (T∗ and S∗ depending easily of T, S and infinite places) . By specializing S and T , this formula and those obtained without the restriction on the places above p ∞, from which most classical results easily follow, first give rise to various generalizations of many isolated results (especially in the intricate case p = 2) and also to interesting rank formulas for various arithmetic invariants including tame and higher kernels of K-theory for number fields (from a review by Jean-François Jaulent) . NOTE: These questions are developped in our book "Class Field Theory" (Chap. II, § 5), Springer Monographs in Mathematics, Springer second corrected printing 2005............. The whole paper is available (in PDF format) on:---- http://www.numdam.org/item?id=JTNB_1998__10_2_399_0 ........................................................................................................................................................................................................................ FOR A COMPLETE VIEW OF MY PUBLICATIONS, PLEASE LOOK AT MY HOME PAGE: http://monsite.orange.fr/maths.g.mn.gras/