On the modularity of elliptic curves over imaginary quadratic fields

TL;DR

This paper proves the modularity of elliptic curves over infinitely many imaginary quadratic fields, extending known results by establishing new local-global compatibility theorems for Galois representations in complex settings.

Contribution

It introduces a new local-global compatibility theorem for p-adic Galois representations associated to torsion in cohomology, applicable to arbitrary dimension and ramification.

Findings

Proves modularity of elliptic curves over many imaginary quadratic fields.

Establishes a local-global compatibility theorem for Galois representations in the crystalline case.

Handles arbitrary dimension, large Hodge--Tate weights, and highly ramified primes.

Abstract

In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and assume that the modular curve $X_0(15)$, which is an elliptic curve of rank $0$ over $\mathbb{Q}$, also has rank $0$ over $F$. Then we prove that all elliptic curves over $F$ are modular. More generally, when $F/\mathbb{Q}$ is an imaginary CM field that does not contain a primitive fifth root of unity, we prove the modularity of elliptic curves $E/F$ under a technical assumption on the image of the representation of $\mathrm{Gal}(\overline{F}/F)$ on $E[3]$ or $E[5]$. The key new technical ingredient we use is a local-global compatibility theorem for the $p$-adic Galois representations associated to torsion in the cohomology of the relevant locally symmetric spaces. We establish this result in the crystalline case, under some technical assumptions, but allowing arbitrary dimension, arbitrarily large regular Hodge--Tate weights, and allowing $p$ to be small and highly ramified in the imaginary CM field $F$.