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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asked Sep 16, 2010 at 4:39
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1$\begingroup$ 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). $\endgroup$– OlivierCommented Sep 16, 2010 at 5:24
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1$\begingroup$ 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? $\endgroup$– David HansenCommented Sep 26, 2010 at 18:04
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5$\begingroup$ 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. $\endgroup$– Tim DokchitserCommented Nov 18, 2010 at 23:11
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$\begingroup$ I've recently asked a similar question in case you are still interested. mathoverflow.net/questions/96289/… $\endgroup$– EugeneCommented May 12, 2012 at 1:10
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1 Answer
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Elliptic curves over real quadratic fields were proven to be modular very recently by Freitas, Le Hung and Siksek:
- Nuno Freitas, Bao Le Hung, Samir Siksek, Elliptic Curves over Real Quadratic Fields are Modular (2015)
Some progress was alredy present in the thesis of Richard Taylor's student Le Hung.
- Bao Le Hung, Modularity of some elliptic curves over totally real fields (2013)
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:
- Kevin Buzzard, Potential modularity—a survey (2011)
You can also consult:
- Jean-Pierre Wintenberger, Appendix: Potential modularity of elliptic curves over totally real fields (2010)
For arbitrary totally real fields, modularity is still not known.