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

TL;DR

This paper proves Iwasawa Main Conjectures for elliptic curves over certain number fields using base change and zeta elements, removing previous ramification restrictions and advancing understanding of BSD and Kolyvagin's conjecture.

Contribution

It establishes Iwasawa Main Conjectures for elliptic curves over both cyclotomic and anticyclotomic extensions without ramification restrictions, employing a novel base change approach.

Findings

Proves Iwasawa Main Conjectures for elliptic curves over $Q$ and $K$.

Deduces cases of the two-variable main conjecture for $E$ over $K$.

Supports the $p$-part of the Birch and Swinnerton-Dyer conjecture for rank ≤ 1.

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.