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Andrew Wiles' proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves released by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the Modularity Theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, seen as virtually impossible to prove using current knowledge. Wiles first announced his proof on Wednesday 23 June 1993 at a lecture in Cambridge entitled "Elliptic Curves and Galois Representations." However, the proof was found to contain an error in September 1993. One year later, on Monday 19 September 1994, in what he would call "the most important moment of () working life," Wiles stumbled upon a revelation, "so indescribably beautiful... so simple and so elegant," that allowed him to correct the proof to the satisfaction of the mathematical community. The correct proof was published in May 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat. The proof itself is over 150 pages long and consumed seven years of Wiles' research time.〔 John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the century.〔http://www.pbs.org/wgbh/nova/transcripts/2414proof.html〕 For solving Fermat's Last Theorem, he was knighted, and received other honours. ==Progress of the previous decades== Fermat's Last Theorem states that no three positive integers ''a'', ''b'', and ''c'' can satisfy the equation : if ''n'' is an integer greater than two. In the 1950s and 1960s the modularity theorem, a connection between elliptic curves and modular forms, was conjectured by the Japanese mathematician Goro Shimura based on ideas posed by Yutaka Taniyama. In the West it became well known through a 1967 paper by André Weil, who gave conceptual evidence for it; thus, it is sometimes called the Taniyama–Shimura–Weil conjecture or the Taniyama-Shimura conjecture. It states that every rational elliptic curve is modular. On a separate branch of development, in the late 1960s, 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. The curve consists of all points in the plane whose coordinates (''x'', ''y'') satisfy the relation : Such an elliptic curve would enjoy very special properties, which are due to the appearance of high powers of integers in its equation and the fact that ''a''''n'' + ''b''''n'' = ''c''''n'' is an ''n''th power as well. In 1982–1985, 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. Again, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates ''x'' and ''y'' of the points on it. Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero ''a'', ''b'', ''c'' and ''n'' greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. As such, a proof or disproof of either of Fermat's Last Theorem or the modularity theorem would simultaneously prove or disprove the other.〔Singh, pp. 194–198; Aczel, pp. 109–114.〕 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 modularity theorem would imply Fermat's Last Theorem. Serre did not provide a complete proof of his proposal; the missing part became known as the epsilon conjecture or ε-conjecture (now known as Ribet's theorem). Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations, which would imply the modularity theorem. Although in the preceding twenty or thirty years much evidence had been accumulated to form conjectures about elliptic curves, the main reason to believe that these various conjectures were true lay not in the numerical confirmations, but in a remarkably coherent and attractive mathematical picture that they presented. Equally it could happen that one or more of these conjectures were actually untrue. Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to show that Frey's intuition was correct: that the above elliptic curve (now known as a Frey curve), if it exists, is always non-modular. Frey did not quite succeed in proving this rigorously; the missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was noticed by Jean-Pierre Serre〔 and proved in 1986 by Ken Ribet. Second, it was necessary to prove the modularity theorem—or at least to prove it for the sub-class of cases (known as semistable elliptic curves) which included Frey's equation. : * Ribet's theorem—if proved—would show that any solution to Fermat's equation could be used to generate a semistable elliptic curve that was not modular; : * The modularity theorem—if proved for Frey's equation—would show that all such elliptic curves must be modular. : * The contradiction implies that no solutions can exist to Fermat's equation, thus proving Fermat's Last Theorem. In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture. (His article was published in 1990.) He demonstrated that, just as Frey had anticipated, a special case of the modularity theorem (still not proved at the time), together with the now proved epsilon conjecture, implies Fermat's Last Theorem. Thus, if the modularity theorem is true for semistable elliptic curves, then Fermat's Last Theorem would be true. However this theoretical approach was widely considered unattainable, since the modularity theorem was itself widely seen as completely inaccessible to proof with current knowledge.〔 For example, Wiles' ex-supervisor John Coates states that it seemed "impossible to actually prove",〔 and Ken Ribet considered himself "one of the vast majority of people who believed () was completely inaccessible".〔 Hearing of the 1986 proof of the epsilon conjecture, Wiles decided to begin working exclusively towards a proof of the modularity theorem. Ribet later commented that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove ()." 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wiles' proof of Fermat's Last Theorem」の詳細全文を読む スポンサード リンク
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