Modularity theorem

The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. Despite the name, which was a carry over from the Taniyama-Shimura conjecture, the theorem is the work of Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.

If p is a prime number and E is an elliptic curve over Q (the field of rational numbers), we can reduce the equation defining E modulo p; for all but finitely many values of p we will get an elliptic curve over the finite field F_p, with n_p elements, say. One then considers the sequence

a_p = n_p − p,

which is an important invariant of the elliptic curve E. Every modular form also gives rise to a sequence of numbers, by Fourier transform. An elliptic curve whose sequence agrees with that from a modular form is called modular. The Taniyama-Shimura theorem states:

"All elliptic curves over Q are modular."

This theorem was first conjectured by Yutaka Taniyama in September 1955. With Goro Shimura he improved its rigor until 1957. Taniyama died in 1958. In the 1960s it became associated with the Langlands program of unifying conjectures in mathematics, and was a key component thereof. The conjecture was picked up and promoted by André Weil in the 1970s, and Weil's name was associated with it in some quarters. Despite the interest, some considered it beyond proving.

It attracted considerable interest in the 1980s when Gerhard Frey suggested that the Taniyama-Shimura conjecture (as it was then called) 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. Ken Ribet later proved this result. In 1995, Andrew Wiles and Richard Taylor proved a special case of the Taniyama-Shimura theorem (the case of semistable elliptic curves) which was strong enough to yield a proof of Fermat's Last Theorem.

The full Taniyama-Shimura theorem was finally proved in 1999 by Breuil, Conrad, Diamond, and Taylor who, building on Wiles' work, incrementally chipped away at the remaining cases until the full result was proved.

Several theorems in number theory similar to Fermat's last theorem follow from the Taniyama-Shimura 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.)

In March 1996 Wiles shared the Wolf Prize with Robert Langlands. Although neither of them had originated nor finished the proof of the full theorem that had enabled their achievements, they were recognized as having had the decisive influences that led to its finally being proven.

References

Henri Darmon: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, Notices of the American Mathematical Society, Vol. 46 (1999), No. 11. Contains a gentle introduction to the theorem and an outline of the proof.
Brian Conrad, Fred Diamond, Richard Taylor: Modularity of certain potentially Barsotti-Tate Galois representations, Journal of the American Mathematical Society 12 (1999), pp. 521–567. Contains the proof.