Base change and Iwasawa Main Conjectures for ${\rm GL}_2$

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ 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 $\mathbb{Z}_p$-cyclotomic and $\mathbb{Z}_p$-anticyclotomic deformations of $E$ over $\mathbb{Q}$ 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 ${\rm ord}_{s=1}L(E,s)\leq 1$. They are also an ingredient in the proof of Kolyvagin's conjecture and its cyclotomic variant in our joint work with Grossi.