Modularity Theorem - History

History

Taniyama (1956) stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikko. Goro Shimura and Taniyama worked on improving its rigor until 1957. Weil (1967) rediscovered the conjecture, and showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form.

The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. He did this by attempting to show that any counterexample to Fermat's Last Theorem would give rise to a non-modular elliptic curve. However, his argument was not complete. The extra condition which was needed to link Taniyama-Shimura-Weil to Fermat's Last Theorem was identified by Serre (1987) and became known as the epsilon conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem. Wiles (1995), with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad, Diamond & Taylor (1999), and Breuil et al. (2001) who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 3. (The case n = 3 was already known by Euler.)