Some History of the Shimura-Taniyama Conjecture

Lang S.

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.