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

TL;DR

This paper extends Kolyvagin's classical results on the finiteness of certain groups related to elliptic curves over imaginary quadratic fields, and proves vanishing results in anticyclotomic Iwasawa theory under additional assumptions.

Contribution

It improves upon Kolyvagin's results by incorporating recent developments and establishes a general Iwasawa-theoretic framework for vanishing results across all layers of the anticyclotomic ${f Z}_p$-extension.

Findings

Enhanced conditions for the finiteness of Sha groups.

Proved vanishing of Sha in all layers of the anticyclotomic extension.

Applicable to CM points on Shimura curve quotients.

Abstract

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.