On the division by an isogeny of the points of an elliptic curve

In the first part of this article we investigate the field of definition of the group $\nu^{-1}(E(K))$, where $\nu$ is an isogeny of degree $\rho$ of an elliptic curve $E$ over a local field $K$, with $[K:\mathbf Q_p]<\infty$. In the second part we show that local results have global consequences for various elliptic curves with complex multiplication. They are concerned with describing groups of rational points of Shafarevich–Tate groups and Mazur modules over $\Gamma$-extensions.
Bibliography: 16 titles.