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This may be a very naive question. We always hear about elliptic curves over the rational numbers, or over other arithmetically significant fields or rings. But, are there open problems or recent fertile theories related to elliptic curves over the complex numbers or is everything considered "classical" and well known?

For example, what about moduar parametrizations: are they only interesting as far as they involve curves defined over "arithmetic bases"? If yes, why?

What about elliptic cohomology, $\mathrm{tmf}$ and the like: if I'm not mistaken, the moduli spaces considered in that theory are defined over $\mathbb{Z}$; anything relevant/interesting happens over $\mathbb{C}$?

(I asked this question after having read this question -which, I must say, I'm not able to understand due to my ignorance of arithmetic geometry- in which one is lead to consider even points valued in -I think- de Rham differential forms...)

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Your question is very vague. I don't know of any open problem about elliptic curves over the complex numbers per se, although one could come up with some unproven identity among elliptic functions or modular functions and say it's "about" elliptic curves. Then again, I am a number theorist.

As for your more specific question about modular parametrizations. If an elliptic curve admits a non-constant map from a modular curve (in the usual sense) to it, then the elliptic curve is defined over a number field, since there are only finitely many such elliptic curves for each modular curve (which is an easy consequence of Poincaré's complete reducibility or de Franchis's theorem or whatever).

Finally, about the other MO question you link to, the ring $B_{dR}$ has very little to do with de Rham differentials directly. It's one of Fontaine's rings of $p$-adic periods.

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Before every elliptic curve over ${\bf Q}$ was known to be modular, Mazur, in his Monthly article, sketched a proof of the fact that "if an elliptic curve defined over ${\bf Q}$ admits a nonconstant mapping from $X(N)$ defined over ${\bf C}$, for some $N$, then it admits a nonconstant mapping from $X_0(N)$ defined over ${\bf Q}$ as well (but possibly for a different value of $N$". – Chandan Singh Dalawat Oct 11 '11 at 2:57

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