Some duality theorems for cyclotomic $\Gamma$-extensions of algebraic number fields of $CM$ type

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.
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