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I shall deal specifically with the history of the conjecture which asserts that every elliptic curve over Q (the field of rational numbers) is modular. In other words, it is a rational image of a modular curve X0 (N), or equivalently of its Jacobian variety J0 (N). This conjecture is one of the most important of the century. The connection of this conjecture with the Fermat problem is explained in the introduction to Wiles's paper (Ann. of Math. May 1995), and I shall not return here to this connection. However, over the last thirty years, there have been false attributions and misrepresentations of the history of this conjecture, which has received incomplete or incorrect accounts on several important occasions. For ten years, I have systematically gathered documentation which I have distributed as the "Taniyama-Shimura File". Ribet refers to this file and its availability in [Ri 95]. It is therefore appropriate to publish a summary of some relevant items from this file, as wellas some more recent items, to document a more accurate history. I call the conjecture the Shimura-Taniyama conjecture for specific reasons which will be made explicit.
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forum.qxp 4/27/98 2:05 PM Page 1301
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Some History of the ShimuraTaniyama Conjecture
risque de confusion avec d’autres conjectures de Weil. On est passé de là à “conjecture de Taniyama-Weil”; c’est la terminologie utilisée ici. Plus récemment, on trouve “conjecture de Shimura-Taniyama-Weil”, ou même “conjecture de Shimura-Taniyama”, le nom de Shimura étant ajouté en hommage à son étude des quotients de J 0 (N) . Le lecteur choisira. L’essential est qu’il sache qu’il s’agit du même énoncé.
Serge Lang I shall deal specifically with the history of the conjecture which asserts that every elliptic curve over Q (the field of rational numbers) is modular. In other words, it is a rational