Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using current knowledge by almost all current mathematicians at the time.[1]: 203–205, 223, 226
Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations".[2] However, in September 1993 the proof was found to contain an error. One year later on 19 September 1994, in what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995.[3]
Wiles's 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, significant number theoretic ideas from Iwasawa theory, and other 20th-century techniques which were not available to Fermat. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.
Together, the two papers which contain the proof are 129 pages long,[4][5] and consumed over seven years of Wiles's 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 [20th] century."[6] Wiles's path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems. For proving Fermat's Last Theorem, he was knighted, and received other honours such as the 2016 Abel Prize. When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a "stunning proof".[3]
Andrew Wiles[edit]
Hearing of Ribet's 1986 proof of the epsilon conjecture, English mathematician Andrew Wiles, who had studied elliptic curves and had a childhood fascination with Fermat, decided to begin working in secret towards a proof of the Taniyama–Shimura–Weil conjecture, since it was now professionally justifiable,[11] as well as because of the enticing goal of proving such a long-standing problem.
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 [it]."[1]: 223
Mathematical detail of Wiles's proof[edit]
Overview[edit]
Wiles opted to attempt to match elliptic curves to a countable set of modular forms. He found that this direct approach was not working, so he transformed the problem by instead matching the Galois representations of the elliptic curves to modular forms. Wiles denotes this matching (or mapping) that, more specifically, is a ring homomorphism: