# Extensions of the modularity theorem

In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, Diamond, and Taylor.

I'm asking this as a matter of interest. Are there any other fields over which elliptic curves are known to be modular? Are there any known fields for which this is not true for?

Also, is much research being conducted on this matter?

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...and Taylor, for the full modularity theorem. –  David Roberts May 8 '12 at 5:49
In the opposite direction, attaching an elliptic curve to a Hilbert modular form (working over totally real fields) is complete modulo the Absolute Hodge Conjecture (so, not very complete!) and is known unconditionally under some assumptions. See math.ucla.edu/~blasius/papers/echmf.pdf –  B R May 8 '12 at 14:05
Right. I was initially trying to avoid redundancy. –  Eugene May 9 '12 at 5:17

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 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.)

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Regarding extensions to non-totally real fields: mathoverflow.net/questions/12416/… –  B R May 8 '12 at 13:53

In the article in the Notices of the AMS which came out when the BCDT proof was announced, it says

Generalizations to other number fields. A number of ingredients in Wiles’s method have been significantly simplified, by Diamond and Fujiwara among others. Fujiwara, Skinner, and Wiles have been able to extend Wiles’s results to the case where the field $\mathbb{Q}$ is replaced by a totally real number field $K$. In particular, this yields analogues of the Shimura-Taniyama-Weil conjecture for a large class of elliptic curves defined over such a field.

Unfortunately it doesn't say what sorts of elliptic curves are covered by these results.

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This is over ten years ago, though. :) –  David Roberts May 9 '12 at 0:31