Modularity of elliptic curves over imaginary quadratic fields

Theorem 1.1: for every imaginary quadratic $K$ and every $E/K$ there is an isobaric automorphic representation $\pi_E$ of $\mathrm{GL}_2(\mathbb A_K)$ with trivial central character and weight zero such that $r_\iota(\pi_E)\simeq H^1_{et}(E_{\bar K},\bar{\mathbb Q}_l)$ for every $l$ and $\iota$, with local-global compatibility at every finite place $v\nmid l$ and the expected parameter at the complex place; hence $L(E/K,s)=L(\pi_E,s-1/2)$. $\pi_E$ is cuspidal when $E$ has no CM. The method is a geometric prime switch: cyclic covers of a quadratic twist of $E$ give a chain of mod $p$, mod 3, mod $p$ congruences that carries automorphy from an auxiliary curve to $E$ through the Caraiani-Newton lifting theorem. It does not treat other CM fields or totally real fields.

Theorem 1.1: for every imaginary quadratic $K$ and every $E/K$ there is an isobaric automorphic representation $\pi_E$ of $\mathrm{GL}_2(\mathbb A_K)$ with trivial central character and weight zero such that $r_\iota(\pi_E)\simeq H^1_{et}(E_{\bar K},\bar{\mathbb Q}_l)$ for every $l$ and $\iota$, with local-global compatibility at every finite place $v\nmid l$ and the expected parameter at the complex place; hence $L(E/K,s)=L(\pi_E,s-1/2)$. $\pi_E$ is cuspidal when $E$ has no CM. The method is a geometric prime switch: cyclic covers of a quadratic twist of $E$ give a chain of mod $p$, mod 3, mod $p$ congruences that carries automorphy from an auxiliary curve to $E$ through the Caraiani-Newton lifting theorem. It does not treat other CM fields or totally real fields.