Pierre Colmez has pointed out a few passages published in the Serre-Tate Correspondence where Serre and Tate express their opinions about the correct way to identify that conjecture about which so much ink has been spilled. The date is October 21, 1995, the papers of Wiles and Taylor-Wiles have been published, and Tate is confiding that
I am tired of the Sh-Ta-We question. But it doesn’t go away.
Joe Silverman had just written to him about the correct nomenclature for the Japanese translation of his book (presumably one of his books about elliptic curves):
Springer-Tokyo wondered if we still wanted to call it the “Taniyama-Weil Conjecture,” since they say that everyone in Japan now calls it the “Shimura-Taniyama Conjecture.” I certainly agree that Shimura’s name should be added to conjecture (Serge’s file is quite convincing), but I don’t feel strongly about whether Weil’s name should be omitted. I hope you won’t mind that I told Ina that for the Japanese edition it would be all right to call it « Shimura-Taniyama », although I suggested that they add a phrase « now proven in large part by Andrew Wiles ».
Tate is finished for the moment with Sh-Ta-We but wonders whether the new theorem should be named after Wiles or Taylor-Wiles, at which point he wrote the censored passage above.
Before we move on to the substance of the question, let’s speculate as to the reason for Tate’s ellipsis. We know that Tate was born in Minneapolis, and we heard a lot about Minnesota nice earlier in this election cycle; could it just be the way of Minnesotans to talk in ellipses? Or maybe the passage was censored by the Société Mathématique de France, publisher of the Serre-Tate Correspondence.
The significance of Tate’s expletival indifference is easier to ascertain. If anyone occupies the pinnacle of charisma in contemporary number theory, it’s John Tate; but here he is choosing not to exercise his charisma to manipulate public opinion in favor of one historical label or another. There’s no doubt in my mind that his survey article on elliptic curves was largely responsible for popularizing the conjecture and associating it with Weil’s name (“Weil  has the following precise conjecture…”) — but there’s no reason to think this was his intention; he had called it Weil’s-Shimura’s conjecture in a letter to Serre dated August 4, 1965 (on p. 262 of the first volume of the Serre-Tate Correspondence — it’s August and Tate writes “Bonnes Vacances” at the end of his letter).
Colmez adds a footnote to the 1995 letter. I quote verbatim:
Il me semble qu’une bonne part de la dispute vient de ce que l’on s’obstine à considérer deux conjectures bien distinctes comme une et une seule conjecture. Si E est une courbe elliptique définie sur Q, considérons les deux énoncés suivants :
La théorie d’Eichler-Shimura [Ei 54, Sh 58], complétée par des travaux d’Igusa [Ig 59] et de Carayol [Cara 86], permet de prouver que le second énoncé implique le premier. Réciproquement, le second est une conséquence de la conjonction du premier, de la théorie d’Eichler-Shimura, et de la conjecture de Tate pour les courbes elliptiques sur Q. Comme la véracité de la conjecture de Tate n’est pas vraiment une trivialité, cela donne une indication de la différence entre les deux énoncés. Une différence encore plus nette apparaît quand on essaie de généraliser les deux énoncés à un motif. Le premier se généralise sans problème en : la fonction L d’un motif est une fonction L automorphe – une incarnation de la correspondance de Langlands globale. Généraliser le second énoncé est plus problématique, et il n’est pas clair que ce soit vraiment possible.
Colmez’s mathematical gloss is impeccable, but he provides no evidence that anyone actually confused the two statements. Shimura certainly did not; when I was in Princeton the year after Faltings proved the Tate conjecture for abelian varieties, he (unwisely) asked me whether I thought the proof was correct, instead of consulting any of the numerous experts on hand.
Only a professional historian of mathematics can determine the accuracy of Colmez’s explanation for the disagreement. Serre, in any case, was never convinced by Lang’s file. In his message to Tate on October 22, 1995, he wrote
La contribution de Weil (rôle des constantes d’équations fonctionnelles + conducteur) me parait bien supérieure à celle de Shimura (qui se réduit à des conversations privées plus ou moins discutables).
Serre clarified his position in a letter to David Goss, dated March 30, 2000, and reprinted two years later in the Gazette of the Société Mathématique de France. I alluded to this letter in the text quoted in the earlier post. To my mind, the most interesting part of this letter is his explanation of what he sees as Weil’s contribution to the problem.
b) He suggests that, not only every elliptic curve over Q should be modular, but its “level” (in the modular sense) should coincide with its “conductor” (defined in terms of the local Néron models, say).
Part b) was a beautiful new idea ; it was not in Taniyama, nor in Shimura (as Shimura himself wrote to me after Weil’s paper had appeared). Its importance comes from the fact that it made the conjecture checkable numerically (while Taniyama’s statement was not). I remember vividly when Weil explained it to me, in the summer of 1966, in some Quartier Latin coffee house. Now things really began to make sense. Why no elliptic curve with conductor 1 (i.e. good reduction everywhere)? Because the modular curve X0(1) of level 1 has genus 0, that’s why!
One can agree or disagree with Serre’s criterion — who apart from Serre has proposed numerical verifiability (or falsifiability!) as the boundary between meaningful and questionable (“discutable”) conjectures? — but at least it has the merit of being stated clearly enough to serve as the starting point for a philosophical consideration of authorship of a conjecture. It’s not merely rhetorical.