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

TL;DR

This work establishes Iwasawa Main Conjectures for GL$_2$-type objects by combining base change to quartic CM fields with Wan’s three-variable MC and the BSTW two-variable zeta element. It proves cyclotomic MC for $E/\mathbb{Q}$ and anticyclotomic MC for $E/K$, under mild irreducibility conditions on $E[p]$ and without ramification hypotheses that earlier results required. The authors extend these results to two-variable MC over $K$ under a vexing-primes condition and, with further hypotheses, obtain integral equalities linking Selmer groups to two-variable $p$-adic $L$-functions, including the $p$-adic Waldspurger formula in the BDP framework. Applications include $p$-part BSD for $E/\mathbb{Q}$ when $\mathrm{ord}_{s=1}L(E,s)\le 1$ and progress toward Kolyvagin-type conjectures, demonstrating the broad impact of these MCs on BSD, Kolyvagin systems, and related conjectures.

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.

Theorems & Definitions (40)

• Conjecture 1.1.1: Mazur's Main Conjecture
• Theorem 1.1.2
• Remark 1.1.3
• Conjecture 1.2.1: Heegner point Main Conjecture
• Theorem 1.2.2
• Remark 1.2.3
• Theorem 1.2.4
• Corollary 1.3.1
• proof
• Remark 1.3.2