Modular elliptic curve

A modular elliptic curve is an elliptic curve E that admits a parametrisation X₀(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on ideas posed by Yutaka Taniyama. In the West it became well known through a 1967 paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Taniyama–Shimura–Weil conjecture. It states that every rational elliptic curve is modular.

On a separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve. The curve consists of all points in the plane whose coordinates (x, y) satisfy the relation

Such an elliptic curve would enjoy very special properties, which are due to the appearance of high powers of integers in its equation and the fact that aⁿ + bⁿ = cⁿ is an nth power as well.

In the summer of 1986, Ken Ribet demonstrated that, just as Frey had anticipated, a special case of the Taniyama–Shimura conjecture (still not proved at the time), together with the now proved epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama–Shimura conjecture is true for semistable elliptic curves, then Fermat's Last Theorem would be true. However this theoretical approach was widely considered unattainable, since the Taniyama–Shimura conjecture was itself widely seen as completely inaccessible to proof with current knowledge. For example, Wiles' ex-supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".

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