  Let $E$ be an elliptic curve defined over ${\bf Q}$ and $K$ an imaginary
quadratic field satisfying the Heegner hypothesis. A classical result of
Kolyvagin states that, under suitable assumptions, if the basic Heegner point
$y_K \in E(K)$ is not divisible by an odd prime $p$, then the groups $E(K)/{\bf
Z} y_K$ and $\sha(E/K)$ are finite and their orders are prime to $p$. In this
article we develop the following themes: firstly, we discuss improvements of
Kolyvagin's result, following Cha (2005) and Lawson and Wuthrich (2016).
Secondly, we prove an abstract Iwasawa-theoretical result which allows us to
deduce, under several additional assumptions, that similar vanishing holds for
all layers in the anticyclotomic ${\bf Z}_p$-extension of $K$. Analogous
results hold for CM points on simple quotients of Jacobians of Shimura curves
over totally real fields; this will be discussed in a separate article.
