Mazur’s Growth Number Conjecture in the Rank One Case

Abstract Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ 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.