Under the Iwasawa main conjecture and explicit numerical conditions, Mazur's growth number conjecture holds for non-anticyclotomic Z_p-extensions and propagates to p-congruent modular forms.
Mazur's Growth Number Conjecture in the Rank One Case
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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.
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Mazur's growth number conjecture and congruences
Under the Iwasawa main conjecture and explicit numerical conditions, Mazur's growth number conjecture holds for non-anticyclotomic Z_p-extensions and propagates to p-congruent modular forms.