Kolyvagin's result on the vanishing of $\text{sha}(E\mathrm{/}K)[p^{\mathrm{\infty }}]$ and its consequences for anticyclotomic Iwasawa theory

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ 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$\sha(E/K)$arefiniteandtheirordersareprimeto$p$. 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 ${\mathbf{Z}}_p$-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.