Geodesic
Some duality theorems for cyclotomic $\Gamma$-extensions of algebraic number fields of $CM$ type
Izvestiya. Mathematics, Tome 14 (1980) no. 3, pp. 441-498 
L. V. Kuz'min. Some duality theorems for cyclotomic $\Gamma$-extensions of algebraic number fields of $CM$ type. Izvestiya. Mathematics, Tome 14 (1980) no. 3, pp. 441-498. http://geodesic.mathdoc.fr/item/IM2_1980_14_3_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

For an odd prime $l$ and a cyclotomic $\Gamma$ – $l$-extension $k_\infty/k$ of a field $k$ of $CM$ type, a compact periodic $\Gamma$-module $A_l(k)$, analogous to the Tate module of a function field, is defined. The analog of the Weil scalar product is constructed on the module $A_l(k)$. The properties of this scalar product are examined, and certain other duality relations are determined on $A_l(k)$. It is proved that, in a finite $l$-extension $k'/k$ of $CM$ type, the $\mathbf Z_l$-ranks of $A_l(k)$ and $A_l(k')$ are connected by a relation similar to the Hurwitz formula for the genus of a curve. Bibliography: 7 titles.

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[2] Kuzmin L. V., “Modul Teita polei algebraicheskikh chisel”, Izv. AN SSSR. Ser. matem., 36 (1972), 267–327 | MR

[3] Iwasawa K., “On $Z_l$-extensions of algebraic number fields”, Ann. Math., 98:2 (1973), 246–326 | DOI | MR | Zbl

[4] Greenberg R., “On the Iwasawa invariants of totally real number fields”, Amer. J. Math., 98:1 (1976), 263–284 | DOI | MR | Zbl

[5] Maklein S., Gomologiya, Mir, M., 1966

[6] Kartan A., Eilenberg S., Gomologicheskaya algebra, IL, M., 1960

[7] Bryumer A., “O novykh rabotakh Ivasavy i Leopoldta”, Matematika, 13:5 (1969), 95–102