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Yes, this is a very active area -- one of the major themes of current research in number theory.

Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than modularity. An elliptic curve $E$ over a number field $K$ is said to be potentially modular if there is a finite extension $L / K$ such that $E$ becomes modular over $L$. This notion of potential modularity has been much studied by Richard Taylor and his coauthors, and turns out to be almost as good for most purposes as knowing modularity over $K$.

It's now known, for instance, that any elliptic curve over a totally real number field $K$ becomes modular over some totally real extension $L / K$; a bit of googling turns up www2.math.kyushu-u.ac.jp/~virdol/basechange2.pdf http://www2.math.kyushu-u.ac.jp/~virdol/basechange2.pdf (which shows that one can choose $L$ in a rather specific way, using work of Taylor and Skinner-Wiles to do the heavy lifting).

I'm not an expert in the area, but my impression from talking to genuine experts is that current methods are very much limited to the case where the elliptic curve is defined over a field which is either totally real or CM -- outside these situations modularity is much less well understood.

(EDIT: I should add that there are some totally real fields for which one can show modularity, rather than just potential modularity; Jarvis and Manoharmayum have shown, for instance, that every semistable elliptic curve over $\mathbb{Q}(\sqrt{2})$ is modular.)

Yes, this is a very active area -- one of the major themes of current research in number theory.

Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than modularity. An elliptic curve $E$ over a number field $K$ is said to be potentially modular if there is a finite extension $L / K$ such that $E$ becomes modular over $L$. This notion of potential modularity has been much studied by Richard Taylor and his coauthors, and turns out to be almost as good for most purposes as knowing modularity over $K$.

It's now known, for instance, that any elliptic curve over a totally real number field $K$ becomes modular over some totally real extension $L / K$; a bit of googling turns up www2.math.kyushu-u.ac.jp/~virdol/basechange2.pdf (which shows that one can choose $L$ in a rather specific way, using work of Taylor and Skinner-Wiles to do the heavy lifting).

I'm not an expert in the area, but my impression from talking to genuine experts is that current methods are very much limited to the case where the elliptic curve is defined over a field which is either totally real or CM -- outside these situations modularity is much less well understood.

(EDIT: I should add that there are some totally real fields for which one can show modularity, rather than just potential modularity; Jarvis and Manoharmayum have shown, for instance, that every semistable elliptic curve over $\mathbb{Q}(\sqrt{2})$ is modular.)

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Yes, this is a very active area -- one of the major themes of current research in number theory.

Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than modularity. An elliptic curve $E$ over a number field $K$ is said to be potentially modular if there is a finite extension $L / K$ such that $E$ becomes modular over $L$. This notion of potential modularity has been much studied by Richard Taylor and his coauthors, and turns out to be almost as good for most purposes as knowing modularity over $K$.

It's now known, for instance, that any elliptic curve over a totally real number field $K$ becomes modular over some totally real extension $L / K$; a bit of googling turns up www2.math.kyushu-u.ac.jp/~virdol/basechange2.pdf (which shows that one can choose $L$ in a rather specific way, using work of Taylor and Skinner-Wiles to do the heavy lifting).

I'm not an expert in the area, but my impression from talking to genuine experts is that current methods are very much limited to the case where the elliptic curve is defined over a field which is either totally real or CM -- outside these situations modularity is much less well understood.