[Math] Taniyama’s original conjecture

Question

I've just read on Wikipedia that the original Taniyama conjecture about L-functions of elliptic curves over an arbitrary number field was still unproven.

This made me want to know more about this conjecture, but after a quick glance at the first results displayed by Google, I couldn't find out anything else than the modularity theorem.

So what did Yutaka Taniyama have on his mind?

Answer 1

There is an article by S. Lang on this subject that appeared in the Notices of the AMS (11/1995); it contains the exact statements of Taniyama's problem(s) on this subject. I reproduce two below (this is taken from a longer list of problems, not all are relevant to the question, and not all are in the article). However, this is mainly a historical article, and part of a controversy regarding proper attribution/naming of the problem over the rationals; thus, while having no personal opinion on this, I should perhaps point out that the views expressed in the article I link to might not be universally accepted. As said by François Brunault (at least this is how I interprete his statement) what precisely was asked in 1955, today seems mainly of historical (not so much mathematical) relevance.

12. Let C be an elliptic curve defined over an algebraic number field $k$, and $L_C(s)$ denote the $L$-function of $C$ over $k$. Namely, $$\zeta_C(s) = \frac{\zeta_k(s) \zeta_k(s − 1)}{L_C(s)}$$ is the zeta function of $C$ over $k$. If a conjecture of Hasse is true for $\zeta_C(s)$, then the Fourier series obtained from $L_C(s)$ by the inverse Mellin transformation must be an automorphic form of dimension $−2$, of some special type (cf. Hecke). If so, it is very plausible that this form is an elliptic differential of the field of that automorphic functions. The problem is to ask if it is possible to prove Hasse’s conjecture for $C$; by going back this considerations, and by finding a suitable automorphic form from which $L_C(s)$ may be obtained.

13. Concerning the above problem, our new problem is to characterize the field of elliptic modular functions of "Stufe" $N$, and especially, to decompose the Jacobian variety $J$ of this function field into simple factors, in the sense of isogeneity. It is well known, that, in case N = q is a prime number, satisfying $q = 3 \pmod{4}$, $J$ contains elliptic curves with complex multiplication. Is this true for general $N$?

After reproducing these two problems the article continues "As Shimura has pointed out, there were some questionable aspects to the Taniyama formulation in problem 12. First, the simple Mellin transform procedure would make sense only for elliptic curves defined over the rationals; the situation over number fields is much more complicated and is not properly understood today, even conjecturally."

(ADDED: It occured to me that the combination of my answer and my comment, could lead to a misinterpretation. The content I reproduced, in particular the final assertion, are not those aspects on which different opinions exist.)