# Elliptic curves over the complex numbers: everything “well known”?

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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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.
Before every elliptic curve over ${\bf Q}$ was known to be modular, Mazur, in his Monthly article mathdl.maa.org/images/upload_library/22/Chauvenet/Mazur.pdf, 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