History
In his thesis, 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.. If ℓ is an odd prime and a, b, and c are positive integers such that
then a corresponding Frey curve is an algebraic curve given by the equation
This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q. In 1982 Gerhard Frey called attention to the unusual properties of the same curve as Hellegouarch, now called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. However, his argument was not complete. In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-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.
Read more about this topic: Ribet's Theorem
Famous quotes containing the word history:
“It’s a very delicate surgical operation—to cut out the heart without killing the patient. The history of our country, however, is a very tough old patient, and we’ll do the best we can.”
—Dudley Nichols, U.S. screenwriter. Jean Renoir. Sorel (Philip Merivale)
“Gossip is charming! History is merely gossip. But scandal is gossip made tedious by morality.”
—Oscar Wilde (1854–1900)
“There has never been in history another such culture as the Western civilization M a culture which has practiced the belief that the physical and social environment of man is subject to rational manipulation and that history is subject to the will and action of man; whereas central to the traditional cultures of the rivals of Western civilization, those of Africa and Asia, is a belief that it is environment that dominates man.”
—Ishmael Reed (b. 1938)