Mazur's Growth Number Conjecture in the Rank One Case

Let $p\ge 5$ be a prime number. Let $\mathsf{E}\mathrm{/}\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under mild hypotheses, we show that if the $p$-adic height of the Heegner point of $\mathsf{E}$ over $K$ is non-zero, then Mazur’s conjecture on the growth of Selmer coranks in the ${\mathbb{Z}}_p^2$-extension of $K$ holds.