# Which elliptic curves over totally real fields are modular these days?

As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is extractable from some papers of, say, Kisin, but I am not an expert in this material and hesitate to try that myself.

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All of them are potentially modular (this is because there exists a rational prime l such that E has good ordinary reduction at every prime in F above l). To prove that a specific example is actually modular might be hard, as all the works I am familiar with assume either residual modularity or large image (but I am definitely no expert either). – Olivier Sep 16 '10 at 5:24
But is one of those things enough? For example, if the curve has an irreducible mod-3 representation, then you can get started via Langlands-Tunnell - but are other things required? – David Hansen Sep 26 '10 at 18:04
It is known (potential modularity + Solomon's induction theorem) that the $L$-function of any elliptic curve over any totally real field is meromorphic on ${\mathbb C}$ and satisfies the expected functional equation. For many applications this is as good as modularity, but I don't whether whether this is of help to you. – Tim Dokchitser Nov 18 '10 at 23:11
I've recently asked a similar question in case you are still interested. mathoverflow.net/questions/96289/… – Eugene May 12 '12 at 1:10

Elliptic curves over real quadratic fields were proven to be modular very recently by Freitas, Le Hung and Siksek:

Some progress was alredy present in the thesis of Richard Taylor's student Le Hung.

Potentially modular is known unconditionally over arbitrary totally real fields, that is:

Theorem. Let $E/F$ be an elliptic curve over a totally real field. Then there is some totally real extension $F'/F$ such that $E/F'$ is modular.

As mentioned in the comments, for practical purposes this is almost as good as modularity.

The attribution of this theorem is complicated. There's some information about this in section 7 of our own Kevin Buzzard's very interesting survey on modularity:

You can also consult:

For arbitrary totally real fields, modularity is still not known.

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