If p is a prime number and E is an elliptic curve over Q, 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 TaniymaShimura theorem states:
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 TaniyamaShimura 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 nonmodular elliptic curve. Kenneth Ribet later proved this result. In 1995, Andrew Wiles and Richard Taylor[?] proved a special case of the TaniyamaShimura theorem (the case of semistable elliptic curves[?]) which was strong enough to yield a proof of Fermat's Last Theorem.
The full TaniyamaShimura 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 TaniyamaShimura theorem. For example: no cube can be written as a sum of two relatively prime nth 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.
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