Base Change and Iwasawa Main Conjectures for GL₂

Let E be an elliptic curve defined over Q of conductor N, p an odd prime of good ordinary reduction such that E[p] is an irreducible Galois 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 deformations of E over Q and K, respectively, dispensing with any of the ramification hypotheses on E[p] in previous works. The strategy employs base change and the two-variable zeta element associated to E over K, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for E over a quartic CM field containing K and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for E over K. The aforementioned one-variable main conjectures imply the p-part of the conjectural Birch and Swinnerton-Dyer formula for E if ords=1 L(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 [1].