  Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, $p$ an odd prime of
good ordinary reduction such that $E[p]$ is an irreducible
$G_{\mathbb{Q}}$-module, and $K$ an imaginary quadratic field with all primes
dividing $Np$ split. We prove Iwasawa Main Conjectures for the
$\mathbb{Z}_p$-cyclotomic and $\mathbb{Z}_p$-anticyclotomic deformation of $E$
over $\mathbb{Q}$ and $K$ respectively, dispensing with any of the ramification
hypotheses on $E[p]$ in previous works. Using base change, the proofs are based
on Wan's divisibility towards a three-variable main conjecture for $E$ over a
quartic CM field containing $K$.
  As an application, we prove cases of the two-variable main conjecture for $E$
over $K$. The one-variable main conjectures imply the $p$-part of the
conjectural Birch and Swinnerton-Dyer formula if ${\rm ord}_{s=1}L(E,s)\leq 1$.
They are also an ingredient in the proof of Kolyvagin's conjecture and its
cyclotomic variant in our joint work with Grossi \cite{BCGS}.
