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?

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    $\begingroup$ I think we should use the term "statement" or "question" rather than "conjecture" here, because Taniyama didn't formulate a precise conjecture. Regarding elliptic curves over number fields, it seems that Taniyama was maybe too optimistic when he made his statement. Formulating what it means for an elliptic curve over a number field to be modular is much more involved than in the case of elliptic curves over the rationals. Taniyama's original statement is explained in Shimura's book "The map of my life" (Appendix A1). $\endgroup$ – François Brunault Apr 16 '11 at 21:44
  • $\begingroup$ Thank you for your answer. Unfortunately I don't own this book and it is quite difficult for me to get an access to it now. So would you please be kind enough to give further details about this "statement" that Taniyama made? $\endgroup$ – Sylvain JULIEN Apr 16 '11 at 23:26
  • $\begingroup$ @Sylvain Julien, perhaps the article I mention can serve as a substitue for the book. It is my understanding that the points of view of Lang and Shimura on this subject were/are rather similar. $\endgroup$ – user9072 Apr 16 '11 at 23:40

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.)

  • $\begingroup$ @unknown : Thank you for the transcription of Taniyama's problems. I wasn't aware of Problem 13. A slight precision about the view expressed in my comment : I was saying it's fair to acknowledge that Taniyama's statement cannot be considered as a precise conjecture (in the modern sense). But of course, we should also acknowledge Taniyama's brilliant intuition. Clearly he had asked here « the good question », although its formulation was not entirely correct from a mathematical point of view. $\endgroup$ – François Brunault Apr 17 '11 at 7:20
  • $\begingroup$ According to Shimura, Taniyama also said : « Modular functions alone will not be enough. I think other special types of automorphic functions are necessary. » (see Shimura Collected Papers, Vol. 4, p. 10). According to Shimura again, here Taniyama meant Hecke's nonmodular triangle functions. But as it turns out, these functions are not relevant for the modularity of elliptic curves over the rationals. $\endgroup$ – François Brunault Apr 17 '11 at 7:47
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    $\begingroup$ @François Brunault: thank you for the clarification. This is basically how I understood your comment. The main point I wanted to make is just that to learn about this circle of ideas nowadays, the original problem does not seem like the best starting point. $\endgroup$ – user9072 Apr 17 '11 at 11:45

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