On the modularity of elliptic curves over imaginary quadratic fields

Technical Summary: Modularity of Elliptic Curves over Imaginary Quadratic Fields

Problem Statement
The paper addresses the modularity of elliptic curves $E$ defined over imaginary quadratic fields $F$ and, more generally, over imaginary CM fields. While the modularity of elliptic curves over $\mathbb{Q}$ (Wiles, Taylor–Wiles, BCDT) and over real quadratic fields (FLHS15) is established, the case of imaginary quadratic fields has remained significantly more difficult. The primary obstruction is that the systems of Hecke eigenvalues conjecturally matching such elliptic curves contribute to the cohomology of locally symmetric spaces (such as Bianchi 3-manifolds) which are not Shimura varieties. Consequently, standard modularity lifting techniques, which rely heavily on the geometry of Shimura varieties, do not apply directly. Previous work (ACC+23, BCGP21) established potential modularity, and recent work (AKT23) established actual modularity for a positive proportion of curves under restrictive conditions on the residual Galois representations (specifically, requiring $p$ to be unramified or the representation to be ordinary).

Methodology
The authors combine two main strands of research:

1. Residual Modularity and Lifting: They utilize the residual modularity results of Allen–Khare–Thorne (AKT23) and apply a modularity lifting theorem in the potentially Barsotti–Tate case.
2. Local-Global Compatibility: The core technical innovation is a new local-global compatibility theorem for $p$-adic Galois representations associated with torsion in the cohomology of locally symmetric spaces.

To prove the lifting theorem, the authors establish a local-global compatibility result (Theorem 1.3) in the crystalline case. This result relates the global Galois representation $\rho_m$ (constructed by Scholze from the cohomology of locally symmetric spaces for $GL_n$) to the local crystalline deformation ring $R^{\square, \text{crys}}_{\bar{\rho}_m|_{G_{F_v}}}(\lambda_v)$.

Key technical ingredients include:

• P-Ordinary Hida Theory: The authors develop a $P$-ordinary version of Hida theory for the unitary group $\tilde{G}$ (quasi-split unitary group) and its Betti cohomology. This allows them to work with specific parts of the cohomology where the local behavior at $p$ is controlled.
• Degree Shifting and Boundary Cohomology: To relate the cohomology of the locally symmetric space to the boundary cohomology (where automorphy is better understood), they employ a "degree shifting" argument. A crucial new idea, suggested by Peter Scholze, involves increasing the level at auxiliary primes $\bar{v}'' | p$ to simplify the analysis of the boundary of the Borel–Serre compactification. This allows them to control the spectral sequence relating the cohomology groups without requiring $p > n^2$ or $p$ to be unramified in $F$.
• Characteristic 0 Lifts: By working with $P$-ordinary parts, they construct characteristic 0 lifts of the local representations that are crystalline with the correct Hodge–Tate weights, facilitating the factorization through the crystalline deformation ring.

Key Contributions and Results

1. Local-Global Compatibility (Theorem 1.3):
   The paper proves that for an imaginary CM field $F$ and a maximal ideal $\mathfrak{m}$ (non-Eisenstein, decomposed generic), the composition of the map from the local deformation ring to the global deformation ring and the map to the Hecke algebra factors through the crystalline deformation ring $R^{\square, \text{crys}}$. This holds under technical assumptions (e.g., $F$ contains an imaginary quadratic field, specific splitting conditions on $p$) but allows for arbitrary dimension $n$, arbitrary regular Hodge–Tate weights, and crucially, allows $p$ to be small and highly ramified in $F$. This generalizes previous results which were restricted to the ordinary or Fontaine–Laffaille cases.

2. Modularity over CM Fields (Theorem 1.2):
   Combining the new lifting theorem with AKT23, the authors prove that for an imaginary CM field $F$ (Galois over $\mathbb{Q}$, $\zeta_5 \notin F$), 100% of Weierstrass equations over $F$ (ordered by height) define a modular elliptic curve. This applies to curves where the image of the Galois representation on $E[3]$ or $E[5]$ satisfies a "decomposed generic" condition.

3. Modularity over Imaginary Quadratic Fields (Theorem 1.1):
   The authors establish the modularity of every elliptic curve $E/F$ where $F$ is an imaginary quadratic field such that the Mordell–Weil group of the modular curve $X_0(15)$ over $F$ is finite. This includes $F = \mathbb{Q}(\sqrt{-d})$ for $d = 1, 2, 3, 5$.

   • Strategy: They analyze the quadratic points on several modular curves of small level at 3 and 5 ($X(b_3, b_5)$, $X(s_3, b_5)$, $X(ns_3^\circ, b_5)$, $X(b_3, ns_5)$, $X(s_3, ns_5)$, $X(ns_3^\circ, ns_5)$).
   • Classification: They classify the elliptic curves for which both the 3-torsion and 5-torsion are "exceptional" (i.e., the Galois image is not decomposed generic).
   • Specific Cases:
     • For $X_0(15) = X(b_3, b_5)$ and $X(s_3, b_5)$, the curves have rank 0 over $\mathbb{Q}$. The authors verify modularity for the finite set of quadratic points.
     • For $X(ns_3^\circ, b_5)$, they identify two infinite families of imaginary quadratic points. One family consists of curves with rational $j$-invariants; the other consists of $\mathbb{Q}$-curves (isogenous to their conjugates). Both families are proven modular.
     • For $X(b_3, ns_5)$ (genus 2) and $X(s_3, ns_5), X(ns_3^\circ, ns_5)$ (genus 3), they use the relative symmetric power Chabauty method (Siksek, Box) to determine the quadratic points. They show that all such points either yield curves with rational $j$-invariants or are $\mathbb{Q}$-curves, thus establishing modularity.

Significance and Claims
The paper claims to establish the modularity of every elliptic curve over infinitely many imaginary quadratic fields, a significant step toward the general modularity conjecture for these fields. The authors explicitly state that their result applies to $F = \mathbb{Q}(\sqrt{-d})$ for $d=1, 2, 3, 5$ and, by Goldfeld's conjecture (verified by Smith for the relevant congruence conditions), to 100% of imaginary quadratic fields ordered by discriminant.

The significance of the work lies in:

• Breaking the "Ordinary" Barrier: The new local-global compatibility theorem allows for the treatment of crystalline representations with small, highly ramified primes, a regime where previous methods (relying on ordinary or Fontaine–Laffaille conditions) failed. This is essential for applying modularity lifting to the specific cases of $p=3$ and $p=5$ over imaginary quadratic fields.
• Completing the Quadratic Case: By combining the lifting theorem with a detailed analysis of quadratic points on modular curves, the authors remove the "decomposed generic" restriction for a large class of imaginary quadratic fields, proving modularity for all elliptic curves in those fields (assuming the rank condition on $X_0(15)$).
• Technical Advancement: The development of $P$-ordinary Hida theory and the refined degree-shifting argument for unitary Shimura varieties provide new tools for the study of Galois representations in the cohomology of locally symmetric spaces, potentially applicable to other modularity problems over CM fields.

The authors note that the modularity of elliptic curves over imaginary quadratic fields has historically been difficult due to the lack of direct Shimura variety connections. Their work bridges this gap by leveraging the potential modularity results of BCGP21 and ACC+23, refining the local-global compatibility to handle the specific arithmetic constraints of imaginary quadratic fields, and performing the necessary geometric analysis of modular curves to handle the exceptional cases where residual modularity fails.