Modularity theorem in the context of Semistable elliptic curve

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a + b = c for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.

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 previous knowledge by almost all mathematicians at the time.

Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". 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 in the journal Annals of Mathematics in the form of two articles, one authored by Wiles and the other co-authored by Wiles and Richard Taylor. Together, the two papers are 129 pages long and consumed more than seven years of Wiles's research time.

Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Knight Commander of the Order of the British Empire in 2000. In 2018, Wiles was appointed the first Regius Professor of Mathematics at Oxford. Wiles is also a 1997 MacArthur Fellow.

Wiles was born in Cambridge to theologian Maurice Frank Wiles and Patricia Wiles. While spending much of his childhood in Nigeria, Wiles developed an interest in mathematics and in Fermat's Last Theorem in particular. After moving to Oxford and graduating from there in 1974, he worked on unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur's generalizations of Iwasawa theory. In the early 1980s, Wiles spent a few years at the University of Cambridge before moving to Princeton University, where he worked on expanding out and applying Hilbert modular forms. In 1986, upon reading Ken Ribet's seminal work on Fermat's Last Theorem, Wiles set out to prove the modularity theorem for semistable elliptic curves, which implied Fermat's Last Theorem. By 1993, he had been able to convince a knowledgeable colleague that he had a proof of Fermat's Last Theorem, though a flaw was subsequently discovered. After an insight on 19 September 1994, Wiles and his student Richard Taylor were able to circumvent the flaw, and published the results in 1995, to widespread acclaim.

Brian Conrad (born November 20, 1970) is an American mathematician and number theorist, working at Stanford University. Previously, he taught at the University of Michigan and at Columbia University.

Conrad and others proved the modularity theorem, also known as the Taniyama-Shimura Conjecture. He proved this in 1999 with Christophe Breuil, Fred Diamond and Richard Taylor, while holding a joint postdoctoral position at Harvard University and the Institute for Advanced Study in Princeton, New Jersey.

Fred Irvin Diamond (born November 19, 1964) is a mathematician, known for his role in proving the modularity theorem for elliptic curves. His research interest is in modular forms and Galois representations.