Base change and Iwasawa Main Conjectures for GL₂

Let E/Q be an elliptic curve of conductor N, p an odd prime of good ordinary reduction such that Ep is an irreducible Gₐ-module, and K an imaginary quadratic field with all primes dividing Np split. We prove Iwasawa Main Conjectures for the Zₚ-cyclotomic and Zₚ-anticyclotomic deformation of E over Q and K respectively, dispensing with any of the ramification hypotheses on Ep 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 ordₒ=₁L (E, s) 1. They are also an ingredient in the proof of Kolyvagin's conjecture and its cyclotomic variant in our joint work with Grossi BCGS.