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On the modularity of elliptic curves over imaginary quadratic fields

Ana Caraiani, James Newton

TL;DR

The article proves modularity for elliptic curves over infinitely many imaginary quadratic fields and extends results to imaginary CM fields not containing a primitive fifth root of unity, under residual-image hypotheses. It develops a comprehensive framework for the cohomology of locally symmetric spaces, notably introducing $P$-ordinary Hida theory for Betti cohomology and performing precise degree-shifting analyses via parabolic induction and boundary cohomology. Central innovations include crystalline local-global compatibility for torsion Galois representations, determinant-valued deformation rings, and a refined LG-compatible framework that links automorphic and Galois data through Kisin-type deformation rings. This work enables modularity lifting in CM settings and provides tools with potential BSD-type and Sato–Tate Ramanujan-type applications, leveraging boundary phenomena and level/weight-variance control. The results contribute a robust pathway from torsion cohomology to actual modularity statements across a wide array of CM fields.

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 $F$.

On the modularity of elliptic curves over imaginary quadratic fields

TL;DR

The article proves modularity for elliptic curves over infinitely many imaginary quadratic fields and extends results to imaginary CM fields not containing a primitive fifth root of unity, under residual-image hypotheses. It develops a comprehensive framework for the cohomology of locally symmetric spaces, notably introducing -ordinary Hida theory for Betti cohomology and performing precise degree-shifting analyses via parabolic induction and boundary cohomology. Central innovations include crystalline local-global compatibility for torsion Galois representations, determinant-valued deformation rings, and a refined LG-compatible framework that links automorphic and Galois data through Kisin-type deformation rings. This work enables modularity lifting in CM settings and provides tools with potential BSD-type and Sato–Tate Ramanujan-type applications, leveraging boundary phenomena and level/weight-variance control. The results contribute a robust pathway from torsion cohomology to actual modularity statements across a wide array of CM fields.

Abstract

In this paper, we establish the modularity of every elliptic curve , where runs over infinitely many imaginary quadratic fields, including for . More precisely, let be imaginary quadratic and assume that the modular curve , which is an elliptic curve of rank over , also has rank over . Then we prove that all elliptic curves over are modular. More generally, when is an imaginary CM field that does not contain a primitive fifth root of unity, we prove the modularity of elliptic curves under a technical assumption on the image of the representation of on or . The key new technical ingredient we use is a local-global compatibility theorem for the -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 to be small and highly ramified in the imaginary CM field .
Paper Structure (47 sections, 117 theorems, 327 equations)

This paper contains 47 sections, 117 theorems, 327 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation
  4. The cohomology of locally symmetric spaces
  5. Preliminaries
  6. Locally symmetric spaces
  7. Hecke operators and coefficient systems
  8. The general linear group and the quasi-split unitary group
  9. Explicit Hecke operators
  10. Automorphic Galois representations and middle degree cohomology
  11. $P$-ordinary Hida theory
  12. $P$-ordinary Hida theory at finite level
  13. The $P$-ordinary part of a smooth representation
  14. Independence of level
  15. A variant at parahoric level
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $F$ be an imaginary quadratic field such that the Mordell--Weil group $X_0(15)(F)$ is finite. Then every elliptic curve $E/F$ is modular.

Theorems & Definitions (244)

  • Theorem 1.1: Corollary \ref{['cor:modularity IQF improved']}
  • Theorem 1.2: Corollary \ref{['cor:100 percent']}
  • Remark 1.2.1
  • Theorem 1.3
  • Remark 1.3.1
  • Remark 1.3.2
  • Proposition 2.1.3
  • proof
  • Lemma 2.1.4
  • proof
  • ...and 234 more