  Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic
field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology
classes for $E$ using $K$-CM points and conjectured they did not all vanish.
Conditional on this conjecture, he described the Selmer rank of $E$ using his
system of classes. We extend work of Wei Zhang to prove new cases of
Kolyvagin's conjecture by considering congruences of modular forms modulo large
powers of $p $. Additionally, we prove an analogous result, and give a
description of the Selmer rank, in a complementary "definite" case (using
certain modified $L$-values rather than CM points). Similar methods are also
used to improve known results on the Heegner point main conjecture of
Perrin-Riou. One consequence of our results is a new converse theorem, that
$p$-Selmer rank one implies analytic rank one, when the residual representation
has dihedral image.
