The claim that every elliptic curve is also a modular form in disguise.

Also called the Shimura-Taniyama-Wiles Conjecture, STW is an ambitious attempt to connect two apparently unrelated areas of mathematics: the theory of numbers and the theory of shapes.

I find amusing one commentator's attempt to clarify this mindbendingly esoteric topic:

"Only front-line mathematicians will really understand the STW conjecture. But you could say 'there exists a modular form of weight two and level N which is an Eigenform under the Hecke series and has a Fourier series.'"

STW was used by Andrew Wiles as part of his 1993 proof of Fermat's Last Theorem.

The Taniyama-Shimura conjecture was originally made by the Japanese mathematician Yukata Taniyama in 1955. Taniyama worked with fellow Japanese mathematician Goro Shimura on the conjecture until the former's suicide in 1958. It says something about the breadth and generality of the TSC that it includes Fermat's Last Theorem, one of the longest-standing curiosities of mathematics, as a special sub-case! The conjecture implies a very deep connection between number theory and topology, two disciplines of mathematics which (while both rather abstract) are usually thought of as being altogether divorced.

I will here state the conjecture formally. You may wish to skip this section unless you want an idea of the actual mathematics involved.

The TSC suggests that, for the elliptic curve y² = Ax3 + Bx² + Cx + D over the rational numbers, there are variable modular functions f(x) and g(x) of the same level such that:

f²(x) = Ag²(x) + Cg(x) + D.

Another way to phrase this, although I'd be lying if I said I understand it myself, is that for every elliptic curve, there is a matching modular form with the same Dirichlet L-series.

In his proof of Fermat's Last Theorem, Andrew Wiles proved the conjecture true only for certain cases, a special subset of the original conjecture. Specifically, Wiles proved that every semistable elliptic curve was a "modular form in disguise" as Spuunbenda phrased it. A semistable elliptic curve is a curve with only squarefree conductors (if that helps). These are the only cases applicable to the truth or falsity of Fermat's Last Theorem itself. In his original refuted proof of 1993, Wiles had not proved the STW for a sufficiently broad variety of cases. A former student of Wiles, Richard Taylor, came to Princeton in 1994 to help resolve this issue and by the end of the year they had fixed the flaw. Wiles republished in 1994.

His proof was then extended by others, and by 1998 a proof had been constructed that applied to all conductors except those divisible by 27. This may seem bizarre but, for topological reasons that are beyond my understanding, remained the case for some time. Finally, on June 21st, 1999, Kenneth Ribet announced at a mathematician's conference that the full Taniyama-Shimura conjecture had been proved at last. Expanding on the foundations laid by Wiles and Taylor, a team of four mathematicians had proved the general case. The conjecture had been proved in full, and as such should no longer be called a conjecture - just as Fermat's Last Theorem should not formally have been called a theorem until very recently.

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