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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.

flag	asked Sep 16 2010 at 4:39 David Hansen 8,469●2●19●48

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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 2010 at 5:24

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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 2010 at 18:04

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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 2010 at 23:11

Eugene · Answer 1 · 2012-05-12 01:10:06Z UTC

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I've recently asked a similar question in case you are still interested.

http://mathoverflow.net/questions/96289/extensions-of-the-modularity-theorem

answered May 12 2012 at 1:10

Eugene
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Linking related questions is good practice (+1). – Ralph May 12 2012 at 1:35

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

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