Modularity of elliptic curves over number fields

Let $K$ be a number field with adele ring $\mathbb A_K$, and let $E/K$ be an elliptic curve. For a finite place $v$ of $K$ where $E$ has good reduction, let $k_v$ be the residue field, $q_v=\#k_v$, and

$$a_v(E)=q_v+1-\#\tilde E(k_v),$$

where $\tilde E$ is the reduction of $E$ at $v$.

Call $E$ modular if there is a cuspidal automorphic representation $\pi=\bigotimes_v'\pi_v$ of $\mathrm{GL}_2(\mathbb A_K)$ with the following properties.

1. $\pi$ has trivial central character, and $\pi_\infty$ has the same infinitesimal character as the trivial representation of $\mathrm{GL}_2(K\otimes_{\mathbb Q}\mathbb R)$. Equivalently, $\pi$ is cohomological with trivial coefficients: it contributes to the cohomology of the locally symmetric spaces $Y_U=\mathrm{GL}_2(K)\backslash\mathrm{GL}_2(\mathbb A_K)/U\,K_\infty$ attached to $\mathrm{GL}_2$ over $K$.
2. For all but finitely many finite places $v$ where $E$ has good reduction, $\pi_v$ is unramified and its Satake parameters $\{\alpha_v,\beta_v\}$ satisfy

$$q_v^{1/2}(\alpha_v+\beta_v)=a_v(E),\qquad \alpha_v\beta_v=1 .$$

Condition 2 says that $L(s,E)$ and $L(s-\tfrac12,\pi)$ have the same Euler factors at almost all places. By strong multiplicity one, $\pi$ is unique if it exists.

Conjecture. For every number field $K$ and every elliptic curve $E/K$ with $\operatorname{End}_K(E)=\mathbb Z$, $E$ is modular.

The hypothesis $\operatorname{End}_K(E)=\mathbb Z$ excludes only curves with complex multiplication by an imaginary quadratic field $M\subseteq K$. For those curves the $\ell$-adic Galois representation is a sum of two characters, so no cuspidal $\pi$ exists. Deuring proved that their $L$-functions are products of two Hecke $L$-functions, which is the correct form of modularity in that case. Curves with complex multiplication not defined over $K$ are modular in the sense above by automorphic induction. So the open content is the case of curves without complex multiplication over $\bar K$.

Thorne states a refined form, his Conjecture 2.6, which also asks that $\pi_v$ match the Weil–Deligne representation of $E$ at $v$ under the local Langlands correspondence, at every finite place $v$ [1].

Background

Over $\mathbb Q$ this is the Taniyama–Shimura–Weil conjecture. Wiles and Taylor–Wiles proved it for semistable curves [2, 3], which implied Fermat's Last Theorem. Breuil, Conrad, Diamond and Taylor completed the proof for all curves over $\mathbb Q$ [4].

Over a totally real field, $\pi$ corresponds to a Hilbert modular form of parallel weight $2$, and $\ell$-adic Galois representations attached to such forms are classical. Over other fields the spaces $Y_U$ are not algebraic varieties, and their cohomology has torsion. Over CM fields, Galois representations attached to these torsion classes now exist, and the ten-author paper proved modularity lifting theorems in this setting [5].

Modularity is the input for many results on elliptic curves over number fields, including cases of the Birch–Swinnerton-Dyer conjecture and Fermat-type equations. Thorne's survey describes the state of the subject [1].