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

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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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     author = {L. V. Kuz'min},
     title = {Some duality theorems for cyclotomic $\Gamma$-extensions of algebraic number fields of $CM$~type},
     journal = {Izvestiya. Mathematics },
     pages = {441--498},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {1980},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1980_14_3_a0/}
}
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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/