Title. Larsen's conjecture for elliptic curves over Q with analytic rank at most one

Abstract.
Larsen conjectured that for an abelian variety $A$ over a finitely generated infinite field $k$, the rank of $A((k^{\mathrm{sep}})^G)$ is infinite for every topologically finitely generated subgroup $G$ of $G_k$. In this talk, I will explain a proof of this conjecture for elliptic curves $E/\mathbb{Q}$ of analytic rank at most one. The argument uses the Heegner point method.