  Let $p$ be an odd prime. We study Mazur's conjecture on the growth of the
Mordell--Weil ranks of an elliptic curve $E/\mathbb{Q}$ over
$\mathbb{Z}_p$-extensions of an imaginary quadratic field, where $p$ is a prime
of good reduction for $E$. In particular, we obtain criteria that may be
checked through explicit calculation, thus allowing for the verification of
Mazur's conjecture in specific examples.
