  Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good
reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we
study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic
field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory
of characters following the method of Greenberg--Vatsal. As a result of our
study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point
main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and
Kolyvagin and the $p$-part of the Birch--Swinnerton-Dyer formula in analytic
rank $1$ for Eisenstein primes $p$.
